Topological Optimization and Optimal Transport: In the Applied Sciences 9783110430417, 9783110439267

Citation preview

Maïtine Bergounioux, Édouard Oudet, Martin Rumpf, Guillaume Carlier, Thierry Champion, Filippo Santambrogio (Eds.) Topological Optimization and Optimal Transport

Radon Series on Computational and Applied Mathematics

| Managing Editor Ulrich Langer, Linz, Austria Editorial Board Hansjörg Albrecher, Lausanne, Switzerland Heinz W. Engl, Linz/Vienna, Austria Ronald H. W. Hoppe, Houston, TX, USA Karl Kunisch, Linz/Graz, Austria Harald Niederreiter, Linz, Austria Christian Schmeiser, Vienna, Austria

Volume 17

Topological Optimization and Optimal Transport | In the Applied Sciences Edited by Maïtine Bergounioux Édouard Oudet Martin Rumpf Guillaume Carlier Thierry Champion Filippo Santambrogio

Mathematics Subject Classification 2010 49J20, 49J45, 49M15, 49M29, 49Q20, 49Q15, 49Q10, 74P05, 74P15, 74P20, 74B20

ISBN 978-3-11-043926-7 e-ISBN (PDF) 978-3-11-043041-7 e-ISBN (EPUB) 978-3-11-043050-9 Set-ISBN 978-3-11-043042-4 ISSN 1865-3707 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2017 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

Contents Part I Graziano Crasta and Ilaria Fragalà Geometric issues in PDE problems related to the infinity Laplace operator | 5 1.1 Introduction | 5 1.2 On the Dirichlet problem | 6 1.3 On the overdetermined problem: the simple (web) case | 8 1.4 On stadium-like domains | 11 1.5 On the overdetermined problem: the general (non-web) case | 13 1.6 Open problems | 18 H. Harbrecht and M. D. Peters Solution of free boundary problems in the presence of geometric uncertainties | 20 2.1 Introduction | 20 2.2 Modelling uncertain domains | 22 2.2.1 Notation | 22 2.2.2 Random interior boundary | 22 2.2.3 Random exterior boundary | 23 2.2.4 Expectation and variance of the domain | 24 2.2.5 Stochastic quadrature method | 26 2.2.6 Analytical example | 27 2.3 Computing free boundaries | 28 2.3.1 Trial method | 28 2.3.2 Discretizing the free boundary | 29 2.3.3 Boundary integral equations | 30 2.3.4 Expectation and variance of the potential | 31 2.4 Numerical results | 32 2.4.1 First example | 32 2.4.2 Second example | 34 2.4.3 Third example | 34 2.4.4 Fourth example | 36 2.5 Conclusion | 38 M. Hintermüller and D. Wegner Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies | 40 3.1 Introduction | 40

VI | Contents

3.2 3.3 3.3.1 3.3.2 3.3.3 3.4 3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.6

Optimal control problem for the time discretization | 43 Yosida approximation and gradient method | 47 Sequential Yosida approximation | 47 Steepest descent method with expansive line search | 48 Newton’s method for the primal system | 49 Finite element approximation | 49 Numerical results | 51 Disk to a ring segment | 53 Ring to disks | 54 Grid pattern of disks | 57 Grid pattern of finger-like regions | 59 Conclusions | 60

Victor A. Kovtunenko High-order topological expansions for Helmholtz problems in 2D | 64 4.1 Introduction | 64 4.2 Background Helmholtz problem | 66 4.2.1 Inner asymptotic expansion by Fourier series in near field | 68 4.3 Helmholtz problems for geometric objects under Neumann (sound hard) boundary condition | 74 4.3.1 Outer asymptotic expansion by Fourier series in far field | 77 4.3.2 Uniform asymptotic expansion of solution of the Neumann problem | 83 4.3.3 Inverse Helmholtz problem under Neumann boundary condition | 85 4.4 Helmholtz problems for geometric objects under Dirichlet (sound soft) boundary condition | 92 4.4.1 Outer and inner asymptotic expansions by Fourier series | 93 4.4.2 High-order uniform asymptotic expansion of the Dirichlet problem | 100 4.4.3 Inverse Helmholtz problem under Dirichlet boundary condition | 102 4.5 Helmholtz problems for geometric objects under Robin (impedance) boundary condition | 105 4.5.1 Outer asymptotic expansion by Fourier series in far field | 107 4.5.2 Combined uniform asymptotic expansion of the Robin problem | 111 4.5.3 Inverse Helmholtz problem under Robin boundary condition | 113 4.5.4 Necessary optimality condition for the topology optimization | 118 Elie Bretin and Simon Masnou On a new phase field model for the approximation of interfacial energies of multiphase systems | 123 5.1 Introduction | 123 5.2 Derivation of the phase-field model | 125

Contents |

5.2.1 5.2.2 5.2.3 5.2.4 5.3 5.4 5.4.1 5.4.2 5.5 5.5.1 5.5.2

VII

The classical constant case σ i,j = 1 | 125 Additive surface tensions | 127 ℓ1 -embeddable surface tensions | 128 Derivation of the approximation perimeter for ℓ1 -embeddable surface tensions | 130 Convergence of the approximating multiphase perimeter | 133 L2 -gradient flow and some extensions | 134 Additional volume constraints | 134 Application to the wetting of multiphase droplets on solid surfaces | 135 Numerical experiments | 136 Evolution of partitions | 137 Wetting of multiphase droplets on solid surfaces | 138

Anca-Maria Toader and Cristian Barbarosie Optimization of eigenvalues and eigenmodes by using the adjoint method | 142 6.1 Introduction | 142 6.2 Setting of the problem and the objective functionals | 144 6.3 The derivatives of the eigenvalues and eigenmodes of vibration | 145 6.4 The derivative of the objective functional by the adjoint method | 148 6.5 Multiple eigenvalues | 149 6.6 The adjoint method in the framework of Bloch waves | 152 Blanche Buet, Gian Paolo Leonardi, and Simon Masnou Discrete varifolds and surface approximation | 159 7.1 Introduction | 159 7.2 Varifolds | 160 7.3 Discrete varifolds | 161 7.4 Approximation of rectifiable varifolds by discrete varifolds | 162 7.5 Curvature of a varifold: a new convolution approach | 164 7.5.1 Regularization of the first variation and conditions of bounded first variation | 164 7.5.2 ϵ-Approximation of the mean curvature vector | 165 7.6 Mean curvature of point-cloud varifolds | 166

VIII | Contents

Part II Jean-David Benamou and Brittany D. Froese Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem | 175 8.1 Introduction | 175 8.2 Duality of Aleksandrov and Pogorelov solutions | 178 8.2.1 Properties of the Legendre–Fenchel dual | 178 8.2.2 Aleksandrov solutions | 180 8.2.3 Geometric characterisation | 182 8.3 Mixed Aleksandrov–viscosity formulation | 183 8.3.1 Viscosity solutions | 183 8.3.2 Mixed Aleksandrov–viscosity solutions | 184 8.3.3 Characterisation of subgradient measure | 186 8.4 Approximation scheme | 188 8.4.1 Monge–Ampère operator and boundary conditions | 188 8.4.2 Discretization of subgradient measure | 189 8.4.3 Properties of the approximation scheme | 190 8.5 Numerical results | 191 8.5.1 Comparison to viscosity solver | 191 8.5.2 Comparison to exact solver | 193 8.5.3 Multiple Diracs | 194 8.6 Conclusions | 197 A Convexity | 197 B Discretization of Monge–Ampère | 199 C Extension to non-constant densities | 200 Simone Di Marino, Augusto Gerolin, and Luca Nenna Optimal transportation theory with repulsive costs | 204 9.1 Why multi-marginal transport theory for repulsive costs? | 207 9.1.1 Brief introduction to quantum mechanics of N-body systems | 207 9.1.2 Probabilistic Interpretation and Marginals | 211 9.1.3 Density functional theory (DFT) | 212 9.1.4 ‘Semi-classical limit’ and optimal transport problem | 214 9.2 DFT meets optimal transportation theory | 217 9.2.1 Couplings and multi-marginal optimal transportation problem | 217 9.2.2 Multi-marginal optimal transportation problem | 218 9.2.3 Dual formulation | 221 9.2.4 Geometry of the Optimal Transport sets | 223 9.2.5 Symmetric Case | 224 9.3 Multi-marginal OT with Coulomb cost | 226

Contents | IX

9.3.1 9.3.2 9.4 9.5 9.6 9.6.1 9.6.2 9.6.3 9.6.4 9.7

General theory: duality, equivalent formulations, and many particles limit | 226 The Monge problem: deterministic examples and counterexamples | 228 Multi-marginal OT with repulsive harmonic cost | 231 Multi-marginal OT for the determinant | 238 Numerics | 242 The regularized problem and the iterative proportional fitting procedure | 243 Numerical experiments: Coulomb cost | 246 Numerical experiments: repulsive Harmonic cost | 248 Numerical experiments: Determinant cost | 250 Conclusion | 252

Roméo Hatchi Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations | 257 10.1 Introduction | 257 10.1.1 Presentation of the general discrete model | 257 10.1.2 Assumptions and preliminary results | 259 10.2 Equivalence with Beckmann problem | 264 10.3 Characterization of minimizers via anisotropic elliptic PDEs | 267 10.4 Regularity when the 𝑣k ’s and c k ’s are constant | 269 10.5 Numerical simulations | 273 10.5.1 Description of the algorithm | 273 10.5.2 Numerical schemes and convergence study | 275 Paul Pegon On the Lagrangian branched transport model and the equivalence with its Eulerian formulation | 281 11.1 The Lagrangian model: irrigation plans | 282 11.1.1 Notation and general framework | 282 11.1.2 The Lagrangian irrigation problem | 284 11.1.3 Existence of minimizers | 285 11.2 The energy formula | 290 11.2.1 Rectifiable irrigation plans | 290 11.2.2 Proof of the energy formula | 294 11.2.3 Optimal irrigation plans are simple | 296 11.3 The Eulerian model: irrigation flows | 297 11.3.1 The discrete model | 298 11.3.2 The continuous model | 298 11.4 Equivalence between models | 300

X | Contents

11.4.1 11.4.2 11.4.3

From Lagrangian to Eulerian | 300 From Eulerian to Lagrangian | 301 The equivalence theorem | 302

Maxime Laborde On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows | 304 12.1 Introduction | 304 12.2 Wasserstein space and main result | 305 12.2.1 The Wasserstein distance | 306 12.2.2 Main result | 307 12.3 Semi-implicit JKO scheme | 308 12.4 κ-flows and gradient estimate | 313 12.4.1 κ-flows | 313 12.4.2 Gradient estimate | 315 12.5 Passage to the limit | 317 12.5.1 Weak and strong convergences | 318 12.5.2 Limit of the discrete system | 321 12.6 The case of a bounded domain Ω | 324 12.7 Uniqueness of solutions | 328 B. Maury and A. Preux Pressureless Euler equations with maximal density constraint: a time-splitting scheme | 333 13.1 Introduction | 333 13.2 Time-stepping scheme | 339 13.2.1 Time discretization strategy | 339 13.2.2 The scheme | 341 13.3 Numerical illustrations | 347 13.3.1 Space discretization scheme | 347 13.3.2 Numerical tests | 349 13.4 Conclusive remarks | 350 Horst Osberger and Daniel Matthes Convergence of a fully discrete variational scheme for a thin-film equation | 356 14.1 Introduction | 356 14.1.1 The equation and its properties | 356 14.1.2 Definition of the discretization | 358 14.1.3 Main results | 360 14.1.4 Relation to the literature | 361 14.1.5 Key estimates | 362 14.1.6 Structure of the paper | 363

Contents | XI

14.2 14.2.1 14.2.2 14.2.3 14.2.4 14.3 14.3.1 14.3.2 14.3.3 14.4 14.5 14.5.1 14.5.2 14.5.3 A

Definition of the fully discrete scheme | 364 Ansatz space and discrete entropy/information functionals | 364 Discretization in time | 366 Spatial interpolations | 368 A discrete Sobolev-type estimate | 369 A priori estimates and compactness | 370 Energy and entropy dissipation | 370 Compactness | 372 Convergence of time interpolants | 374 Weak formulation of the limit equation | 377 Numerical results | 392 Nonuniform meshes | 392 Implementation | 393 Numerical experiments | 393 Appendix | 397

F. Al Reda and B. Maury Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance | 400 15.1 Introduction | 400 15.2 Preliminaries | 403 15.3 Discretization of the Fokker–Planck equation | 411 15.4 Conclusive remarks, perspectives | 414 Index | 417

| Part I

Part I

| 3

Shape Optimization and topology optimization is a rapidly evolving field connecting various branches of mathematics ranging from geometric analysis and multiscale methods to numerical analysis and scientific computing. This book examplarily presents a collection of recent trends. Harbrecht and Peters study Bernoulli’s exterior free boundary value problem with geometric uncertainties. The paper by Hintermüller and Wegner studies an optimal control problem in the context of flow with phase separation. Bretin and Masnou study multiphase systems and interface configuration in these systems. Kovtunenko’s article discusses the problem of identifying shapes from physical data measurements at a distance boundary. The article by Crasta and Fragalà asks for a characterization of domains for the infinity Laplace operator. Toader and Barbarosie investigate in their paper shape optimization with cost functions depending on the eigenvalues of an elliptic operator. Finally, Buet, Leonardi and Masnou discuss the approximation of surface via discrete varifolds and how to define a proper notion of the first variation of area.

Graziano Crasta and Ilaria Fragalà

1 Geometric issues in PDE problems related to the infinity Laplace operator Abstract: We review some recent results related to the homogeneous Dirichlet problem for the infinity Laplace equation with a constant source in a bounded domain. We characterize the geometry of domains for which an overdetermined problem admits a viscosity solutions. An essential tool is a regularity result for viscosity solutions in convex domains, obtained by the convex envelope method introduced by Alvarez, Lasry, and Lions. Keywords: Overdetermined problems, infinity Laplacian AMS Classification: Primary 49K20, Secondary 49K30, 35J70, 35N25.

1.1 Introduction Our primary interest in partial differential equation (PDE) problems for the infinity Laplacian operator raised from the following overdetermined problem: −∆∞ u = 1 { { { u=0 { { { {|∇u| = c

in Ω on ∂Ω

(1.1)

on ∂Ω ,

whose study was firstly proposed in [6]. Let us recall that the infinity Laplacian is the strongly nonlinear and highly degenerated differential operator defined for smooth functions u by ∆∞ u := ∇2 u∇u ⋅ ∇u . It was firstly discovered by Aronsson in the sixties in connection with the so-called absolutely minimizing Lipschitz extensions and later in the nineties a fundamental advance concerning the existence and uniqueness of solutions came by Jensen. In the last decade, also due to their connection with tug-of-war games, boundary value problems involving the infinity Laplace operator have received a great impulse thanks to the contribution of several authors; without any attempt of completeness, let us quote the papers [2–4, 15, 16, 23–25, 27], where the reader may find further related references. Graziano Crasta, Dipartimento di Matematica “G. Castelnuovo,” Univ. di Roma I – P.le A. Moro 2 – 00185 Roma, Italy, [email protected] Ilaria Fragalà, Dipartimento di Matematica, Politecnico – Piazza Leonardo da Vinci, 32 –20133 Milano, Italy, [email protected]

6 | 1 Inhomogeneous infinity Laplacian

On the other hand, starting from the fundamental paper by Serrin [26], overdetermined problems of the type (1.1) have been studied for many operators (the basic examples being the Laplace and p-Laplace operator; see for instance [5, 14, 18, 19, 26]), not including the infinite Laplacian operator. In all these cases it is known that, if the overdetermined problem (1.1) admits a solution, then Ω is a ball. An intriguing discovery is that this is not the case for the infinity Laplacian, unless more regularity (and topological) assumptions are required on the domain Ω. Motivated by the aim of characterizing the shape of domains where problem (1.1) admits a solution, we were led to study a number of geometrical and regularity matters, going from the concavity properties of the unique solution to the Dirichlet problem given by the first two equations in (1.1), to the study of sets with positive reach and empty interior in ℝn . In this chapter, we review our achievements on these topics to this day. Our choice is in favor of an intuitive presentation: though the results are rigorously stated, they are introduced in an informal way, enlightening the main ideas and avoiding all technicalities. In this spirit, we invoke more than once heuristic arguments, and we limit ourselves to sketch the proofs, referring for all details to the original papers. The outline of the chapter is as follows. In Section 1.2, we recall some basic facts concerning existence, uniqueness, and regularity for the homogeneous Dirichlet problem with a constant source term. In Section 1.3, we deal with a simplified version of problem (1.1) where solutions are searched in the family of functions having prescribed level lines, and precisely the same level lines as the distance function from ∂Ω. Studying the problem in this setting leads to introduce a class of domains, that we call “stadium-like,” for which the cut locus agrees with the set of maximal distance from the boundary. In Section 1.4, we present the geometric results we obtained for stadium-like domains, which rely on a new classification of closed sets with positive reach and empty interior. These results are essentially two-dimensional. In Section 1.5, we deal with problem (1.1) in its general and quite challenging formulation. To pursue our attempt of showing that the field is extremely rich, and many relevant questions remain unsolved, we conclude the chapter with a short section of open problems.

1.2 On the Dirichlet problem In this section, we briefly discuss the Dirichlet problem for the infinity Laplace equation with a constant source term: {−∆∞ u = 1 in Ω , { u=0 on ∂Ω . {

(2.1)

1.2 On the Dirichlet problem |

7

y g

R

t

Fig. 1: Radial solution of the Dirichlet problem (2.1).

We begin with a basic example in order to get a feeling with the problem and motivate the use of viscosity solutions. Example 2.1. Let Ω = B R (0) be the ball of radius R centered at the origin. Let us look for a radial solution to problem (2.1) of the form u(x) = g(R − |x|), where g : [0, R] → ℝ is a continuous function, of class C2 in the interval (0, R). The Dirichlet boundary condition gives g(0) = 0. On the other hand, if we want u to be differentiable at x = 0 (which is a posteriori justified by Theorem 2.2 stated hereafter) we have to require that g󸀠 (R) = 0. Hence, we have to solve the following one-dimensional boundary value problem for the function g: −∆∞ u(x) = −g󸀠󸀠 (R − |x|) [g󸀠 (R − |x|)]2 = 1,

g(0) = 0,

g󸀠 (R) = 0 .

We easily obtain g(t) = c0 [R4/3 − (R − t)4/3 ],

t ∈ [0, R]

(c0 = 34/3 /4)

(see Figure 1). The function u(x) = g(R − |x|) is of class C1,1/3 (B R ) ∩ C2 (B R \ {0}). This shows that there are no radial solutions of class C2 (B R ). We shall turn back to the lackness of classical (i.e., C2 ) solutions for the Dirichlet problem (2.1) in arbitrary domains in Section 1.5. By the moment, we limit ourselves to consider the above example as a heuristic explanation why solutions to problem (2.1) cannot be expected to be classical. Moreover, we also observe that the notion of weak solutions is ruled out, because the equation is fully nonlinear and cannot be written in the divergence form. In fact, the right notion of solution to problem (2.1) is one of the viscosity solution. We shortly recall it below, for the benefit of the reader, referring to [8] for more details. A viscosity subsolution to the equation −∆∞ u − 1 = 0 is a function u ∈ C(Ω) which, for every x0 ∈ Ω, satisfies − ∆∞ φ(x0 ) − 1 ≤ 0

whenever φ ∈ C2 (Ω) and u − φ has a local maximum at x0 , (2.2)

or equivalently − ⟨Xp, p⟩ − 1 ≤ 0

2,+

∀(p, X) ∈ J Ω u(x0 ) .

(2.3)

8 | 1 Inhomogeneous infinity Laplacian Here the second-order superjet J 2,+ Ω u(x 0 ) of a function u ∈ C(Ω) at a point x 0 ∈ Ω n×n such that denotes the set of pairs (p, A) ∈ ℝn × ℝsym u(y) ≤ u(x0 ) + ⟨p, y − x0 ⟩ +

1 ⟨A(y − x0 ), y − x0 ⟩ + o (|y − x0 |2 ) 2

as y → x0 , y ∈ Ω. Similarly, a viscosity supersolution to the equation −∆∞ u − 1 = 0 is a function u ∈ C(Ω) which, for every x0 ∈ Ω, satisfies −∆∞ φ(x0 )−1 ≥ 0

whenever φ ∈ C2 (Ω) and u − φ has a local minimum at x0 , (2.4)

or equivalently − ⟨Xp, p⟩ − 1 ≥ 0

∀(p, X) ∈ J 2,− Ω u(x 0 )

(2.5)

J 2,− Ω u(x 0 )

(the second-order subjet is defined analogously to the superjet with the inequality reversed). Finally, a viscosity solution to problem (2.1) is a function u ∈ C(Ω) such that u = 0 on ∂Ω and u is a viscosity solution to −∆∞ u = 1 in Ω, meaning it is both a viscosity subsolution and a viscosity supersolution on Ω, according to the above definition. We are now in a position to recall the basic known facts concerning existence, uniqueness, and regularity for viscosity solutions to problem (2.1). Theorem 2.2 (Basic properties of viscosity solutions to (2.1)). The Dirichlet problem (2.1) admits a unique viscosity solution u. Moreover, u is differentiable at every point of Ω. Both existence and uniqueness of viscosity solution have been obtained by Lu and Wang in [24], by adapting the nowadays standard approach for viscosity solutions of nondegenerate second-order fully nonlinear equations. In particular, existence is obtained by Perron’s method, while uniqueness is a consequence of the following comparison principle. Theorem 2.3 (Comparison principle). Let u, 𝑣 ∈ C(Ω) be, respectively, viscosity suband supersolutions of −∆∞ u = 1 in Ω. If u ≤ 𝑣 on ∂Ω, then u ≤ 𝑣 in Ω. The fact that the unique solution u to (2.1) is differentiable everywhere has been recently proved by Lindgren [23], by adapting the method of Evans and Smart [16] for infinity harmonic functions.

1.3 On the overdetermined problem: the simple (web) case In this section, we consider a simplified version of the overdetermined problem (1.1) and introduce a class of domains where such a simplified version turns out to admit a solution.

1.3 On the overdetermined problem: the simple (web) case

| 9

To follow an intuitive approach, let us present a heuristic argument. Assume for a moment that u is a smooth solution to (1.1), and consider the gradient flow associated with u, i.e., the flow generated by the ordinary differential equation ̇ = ∇u(x(t)) . x(t) Solutions of this differential equation will be called characteristics. If x(t), t ∈ [0, T), is a characteristic, and φ(t) := u(x(t)) denotes the restriction of u along this solution, we have ̇ = |∇u(x(t))|2 , φ(t) ̈ = 2 ⟨D2 u(x)∇u(x), ∇u(x)⟩ = 2∆∞ u(x) = −2 φ(t) ̇ i.e., φ(t) = φ(0) + φ(0) t − t2 . Moreover, if x(0) = y ∈ ∂Ω, from the conditions u(y) = 0 and |∇u(y)| = c we can determine explicitly φ as φ(t) = √c t − t2 .

(3.1)

On the other hand, from this information we cannot reconstruct the expression of the solution u, because in general we do not know the geometry of characteristics, which clearly depends on the solution itself! However, there is a special case when this geometry is explicitly known, namely when the function u belongs to the following class: Definition 3.1 (Web functions). We say that u is a web function if it only depends on the distance d from the boundary of ∂Ω, that is it can be written for some function w as u(x) = w(d(x)). As we are going to realize immediately, when dealing with problem (1.1) within the class of web functions, there are two subsets of Ω related with the geometry of d which turn out to play a crucial role. We introduce them below: Definition 3.2 (Cut locus and high ridge). The cut locus Σ(Ω) of Ω is the closure in Ω of the set Σ(Ω) of points of non differentiability of d. The high ridge M(Ω) of Ω is the set where d achieves its maximum over Ω (called the inradius ρ Ω of the set Ω). Figure 2 shows the cut locus and the high ridge when Ω is a rectangle.

Fig. 2: Cut locus (solid), characteristics (dotted), high ridge (dashed).

10 | 1 Inhomogeneous infinity Laplacian Observe now that, for a generic domain Ω, if u is a web function, ∇u is parallel to ∇d, and hence the characteristics of u are line segments normal to the boundary. More precisely, a characteristic is a line segment which starts at a point of the boundary, is normal to the boundary itself, and reaches a point of the cut locus (for instance, some characteristics of a web function on a rectangle are the dotted line segments in Figure 2). Moreover, if u is written as w(d), we have |∇u(y)| = w󸀠 (0) for every y ∈ ∂Ω, so that the condition |∇u| = c on ∂Ω is automatically satisfied, with c = w󸀠 (0). Thus, asking that the unique viscosity solution to problem (2.1) is a web function we immediately obtain a solution to the overdetermined problem (1.1). By arguing as in Example 2.1, namely solving a one-dimensional boundary value problem for the function w, we obtain w(t) = c0 (R4/3 − (R − t)4/3 )

c0 :=

34/3 , 4

R :=

c3 . 3

If we now impose that u is differentiable, then we find that all characteristics must have the same length R, and that this length R must coincide with the inradius ρ Ω . In other words, the requirement that all characteristics must have the same length is equivalent to ask a precise geometric condition on Ω, which is the coincidence between cut locus and high ridge. Accordingly, we set the following: Definition 3.3 (Stadium-like domains). A set Ω ⊂ ℝn is said to be a stadium-like domain if M(Ω) = Σ(Ω). Clearly, the rectangle is not a stadium-like domain. Some examples of stadium-like domains are represented in Figure 3. The heuristic arguments presented above can be made rigorous and yield the following result. It has been proved in [6] in the regular case (for C1 solutions and C2 domains) and in [10] in the general case (with no regularity assumption on u and Ω). Theorem 3.4 (Web-viscosity solutions). The unique viscosity solution to problem (2.1) is a web function if and only if Ω is a stadium-like domain. In this case, the web-viscosity solution is given by 4/3

u(x) = ψ Ω (x) := g(d(x)) = c0 [ρ Ω − (ρ Ω − d(x))4/3 ] .

Σ =M

Fig. 3: Stadium-like domains.

(3.2)

1.4 On stadium-like domains

|

11

1.4 On stadium-like domains In view of Theorem 3.4, a natural question is whether and how is it possible to characterize the geometry of stadium-like domains. A complete classification of them has been given in [9] in dimension n = 2; a similar statement in higher dimensions has been proved until now only under the convexity assumption. To prepare our results, we have to recall the fundamental notion of set of positive reach introduced by Federer in [17]. Definition 4.1 (Set of positive reach). Let S ⊂ ℝn be a nonempty closed set, and let d S denote the distance function from S. We say that S is a set of positive reach if there exists r S > 0 (called radius of proximal smoothness) such that every point of the tubular neighborhood {x ∈ ℝn : 0 < d S (x) < r S } (4.1) has a unique projection on S. Federer himself proved that S has positive reach if and only if S is proximally C1 , which means that the distance function d S is of class C1 in a tubular neighborhood of the form (4.1). (If this is the case, it can be proved that d S is of class C1,1 in such tubular neighborhood.) In [9, Theorem 2], we have obtained the following complete characterization of planar sets with positive reach and empty interior: Theorem 4.2 (Characterization of planar proximally C1 sets with empty interior). Let S ⊂ ℝ2 be closed, proximally C 1 , with empty interior, and connected. Then S is either a singleton, or a one-dimensional manifold of class C1,1 . Sketch of the proof. The proof is of marked geometric stamp, and here we limit ourselves to give a rough idea of it. It consists basically in performing a careful analysis of the so-called contact set. Namely, we fix a point p ∈ S and a positive r smaller than the radius of proximal smoothness, and study the contact set of p into S r , which is defined as the set C r (p) where the circumference of radius r centered at p meets the boundary of the tubular neighborhood {d S (x) < r}. The main issue in the proof amounts to show that C r (p) consists either of two antipodal points, or of a semicircumference. Once one has this geometric characterization of the contact set, it is rather easy to deduce that S is locally the graph of a Lipschitz function g. Finally, the fact that it is of class C1,1 comes from the fact that g is both semiconcave and semiconvex. We explicitly note that a one-dimensional connected manifold can be with boundary (two points) or without boundary (a closed curve), see Figure 4. It is interesting to observe that, as soon as we require d S to be of class C2 in a tubular neighborhood of S, then the second case in Figure 4 (manifold with boundary) cannot happen. More precisely, let us set the following:

12 | 1 Inhomogeneous infinity Laplacian

Fig. 4: Planar proximally C 1 sets with empty interior.

Definition 4.3 (Proximally C k sets). We say that a nonempty closed subset S of ℝn is proximally C k if there exists r S > 0 such that d S is of class C k in a tubular neighborhood of S of the form (4.1). Then, we have (see [9, Theorem 3]): Theorem 4.4 (Characterization of proximally C2 sets with empty interior). Let S ⊂ ℝ2 be closed, proximally C2 , with empty interior, and connected. Then S is either a singleton, or a one-dimensional manifold of class C2 without a boundary. The above statement can be generalized to the case when, with the analogous meaning as in Definition 4.3, the set S is proximally C k,α , for some k ≥ 2 and α ∈ [0, 1], or proximally C∞ , or proximally C ω . Accordingly, S turns out to be a manifold, respectively, of class C k,α , C∞ , or C ω (cf. [9, Remark 4 (iii) and Remark 23]). A direct consequence of Theorems 4.2 and 4.4 is the following characterization of stadium-like domains. To understand it, one has to think of S as playing the role of the set M(Ω) = Σ(Ω), which is a nonempty closed set with empty interior (notice in fact that the high ridge M(Ω) cannot have interior points, since otherwise there would be points where ∇d = 0). Accordingly, the set Ω has to be thought as a tubular neighborhood of S. Theorem 4.5 (Characterization of planar domains with M = Σ). Let Ω ⊂ ℝ2 be an open bounded connected set with M(Ω) = Σ(Ω). Then Ω is either a disk or a parallel neighborhood of a one-dimensional C1,1 manifold. If in addition Ω is C2 , then Ω is either a disk or a parallel neighborhood of a onedimensional C2 manifold with no boundary. If Ω is also simply connected, then Ω is a disk.

Fig. 5: Stadium-like domains.

1.5 On the overdetermined problem: the general (non-web) case

|

13

The three possibilities are shown in Figure 5. In [9, Theorem 12], we also proved a partial extension for convex sets in higher dimension. Theorem 4.6 (Extension to higher dimensions). Let Ω ⊂ ℝn be an open bounded convex set. If M(Ω) = Σ(Ω) and Ω is C2 , then Ω is a ball. Now our Theorem 3.4 can be rephrased in the following much more “visual” way: Theorem 4.7 (Web-viscosity solutions). The unique viscosity solution to problem (2.1) is a web function if and only the shape of Ω can be characterized as in Theorem 4.5 (in dimension n = 2) and 4.6 (in any dimension provided Ω is assumed to be convex).

1.5 On the overdetermined problem: the general (non-web) case Up to now, we have characterized the geometry of sets for which the overdetermined problem (1.1) admits a solution in the class of web functions. (We stress once more that, in this class of functions, the overdetermined problem (1.1) is equivalent to the Dirichlet problem (2.1), since the condition |∇u| constant on ∂Ω is automatically satisfied.) In this section, we are going to consider what happens in the general case, i.e., without the restriction to web functions. Recalling the heuristic argument given at the beginning of Section 1.3, we see that we have to face with a number of additional difficulties. In particular, the following two main problems emerge. – Since u is unknown and, a priori, its level lines do not have any specific form, the geometry of the trajectories of the gradient flow is unknown. – Even worse, we do not know if the gradient flow is well posed. Namely, in general, we only know that ∇u is locally bounded, and it is never locally Lipschitz, as we shall see in Theorem 5.4 that u never belongs to C1,1 (Ω). This means that we cannot use the standard Cauchy–Lipschitz theory for ordinary differential equations for the gradient flow ẋ = ∇u(x). Moreover, even if we were able to prove an intermediate regularity result between local boundedness and local Lipschitzianity for ∇u (e.g., that it is locally in BV or in some Sobolev space), we could not even apply the Ambrosio–Di Perna–Lions theory of regular Lagrange flows, because we do not have a lower bound for the measure div ∇u. Our approach is motivated by the above remarks, and in particular it stems from the will of recovering the well-posedness of the gradient flow. In this respect it is well known that, in order to have at least forward well posedness, it is enough u to be locally semiconcave. By definition, this means that there exists a constant C ≥ 0 such that u(x + h) + u(x − h) − 2u(x) ≤ C|h|2

∀[x − h, x + h] ⊂ Ω ,

14 | 1 Inhomogeneous infinity Laplacian where [x − h, x + h] denotes the segment in ℝn joining the two points x − h and x + h. In fact, the forward uniqueness of solutions follows from the property ⟨∇u(y) − ∇u(x), y − x⟩ ≤ C|y − x|2 , which is the analogous, for differentiable semiconcave functions, of the monotonicity of the gradient of a (differentiable) concave function. Now, if x(t) and y(t) are two solutions of the gradient flow defined in a common interval [0, τ), setting w(t) := |y(t) − x(t)|2 /2 we obtain ̇ w(t) = ⟨∇u(y(t)) − ∇u(x(t)), y(t) − x(t)⟩ ≤ 2 C w(t) . Hence, if w(t0 ) = 0 for some t0 ∈ [0, τ), i.e., if x(t0 ) = y(t0 ), then by Gronwall’s inequality we obtain that w(t) = 0 for every t ∈ [t0 , τ), i.e., x(t) = y(t) for every t ∈ [t0 , τ). For a review on semiconcave functions, we refer to [7]. In this perspective, our first step will be to set up a regularity result for u, proving that u is locally semiconcave. Unfortunately, we are not able to obtain such a result in full generality, but we have to restrict to convex domains without corners. More precisely, we are going to assume that Ω is convex and satisfies an interior sphere condition.

(HΩ)

Theorem 5.1 (Power concavity and semiconcavity of solutions). Assume (HΩ) and let u be the viscosity solution to the Dirichlet problem (2.1). Then, u 3/4 is concave in Ω. In particular, u is locally semiconcave in Ω. Sketch of the proof. Let us outline the strategy we adopt in order to prove that the function w := −u 3/4 is convex in Ω. For the detailed proof, we refer to [11, Theorem 1]. We first observe that w is well defined (since u > 0 in Ω), and it is the unique viscosity solution of the Dirichlet problem 3

{−∆∞ w − w1 [ 13 |∇w|4 + ( 34 ) ] = 0 in Ω , (5.1) { w=0 on ∂Ω . { At first sight, the equation satisfied by w looks more complicate than the original one for u. On the other hand, thanks to the structure of such equation (we refer in particular to the factor 1/w in front of the brackets), we are enabled to adapt the convex envelop method developed by Alvarez et al. (see [1]). It consists essentially in the following steps. (i) Prove that the convex envelope w∗∗ of w is a viscosity supersolution to (5.1). This is the most challenging task, where the structure of the equation intervenes. The detailed proof can be found in [11]. The main ingredients are: – the representation of the convex envelope as k

k

k

w∗∗ (x) = inf { ∑ λ i w(x i ) : x = ∑ λ i x i , x i ∈ Ω, λ i > 0, ∑ λ i = 1} ; k≤n+1

i=1

i=1

i=1

1.5 On the overdetermined problem: the general (non-web) case

|

15

–

the fact that, since the normal derivative of w with respect to the external normal is +∞ at every boundary point of Ω, in our case the points x i in the formula above cannot lie on the boundary of Ω; – Proposition 1 in [1]; – the concavity of the map Q 󳨃→ 1/tr((p ⊗ p)Q−1 ). (ii) By Step (i) and the comparison principle (that for Equation (5.1) has been proved in [24, Theorem 3]), it follows that w∗∗ ≥ w in Ω. (iii) By definition of convex envelope, it is immediate that w∗∗ ≤ w in Ω. By combining Steps (ii) and (iii), we conclude that w coincides with its convex envelope, so that w = −u 3/4 is a convex function. From this power-concavity property of u, it is straightforward to conclude that u is locally semiconcave in Ω. Since u is locally semiconcave and differentiable everywhere, we obtain at once the following regularity property (see [7, Prop. 3.3.4]). Corollary 5.2 (C1 -regularity of solutions). Assume (HΩ) and let u be the viscosity solution to the Dirichlet problem (2.1). Then, u is continuously differentiable in Ω. Let us now turn back to the overdetermined boundary value problem (1.1), in the light of the regularity results obtained so far for the solution u to problem (2.1) in Ω. In order not to face with boundary regularity matters for u at the boundary of Ω (for which however some results are available in the literature, see [20, 21, 28]), in the following we will assume that u is C1 up to the boundary, namely that ∃ δ > 0 : u is of class C1 on {x ∈ Ω : d(x) < δ} .

(Hu)

As a consequence of Corollary 5.2 and assumption (Hu), for every initial point x0 ∈ Ω the Cauchy problem { ẋ = ∇u(x) , { x(0) = x0 { turns out to admit a unique forward solution X(⋅, x0 ), defined on some maximal interval [0, T(x0 )). Moreover, we can prove that t 󳨃→ X(t, x0 ) reaches in finite time a maximum point of u and then stops there. Characteristics are now back at our disposal! So, let us resume the heuristic approach started in Section 1.3, consisting in studying the solution along such curves. Assume for a moment that the solution u of the Dirichlet problem (2.1) is smooth enough (let’s say C2 ), and consider the P-function P(x) :=

1 4

|∇u(x)|4 + u(x) .

If x(⋅) = X(⋅, y) is a characteristic, then d P(x(t)) = |∇u(x)|2 ⟨D2 u(x)∇u(x), ∇u(x)⟩ + |∇u(x)|2 = 0 , dt so that the P-function is constant along the gradient flow.

16 | 1 Inhomogeneous infinity Laplacian If, in addition, we require the overdetermined condition |∇u| = c on ∂Ω to hold, we have that P(y) = c4 /4 at every point y ∈ ∂Ω. From this information, it follows that the P-function is constant along the set spanned by the gradient flow, i.e., on the whole Ω. In turn, the constancy of P over Ω allows us to characterize the expression of u and the shape of Ω exactly in the same way as was done in Section 1.3 in the web setting. Indeed, the following result holds. Theorem 5.3 (P-function). Under the assumptions (HΩ)–(Hu), let u be the unique so4/3 lution to problem (2.1). If P(x) = λ ≤ c0 ρ Ω for a.e. x ∈ Ω, then u is the web function defined in (3.2) and Ω is a stadium-like domain (for which the conclusions of Theorem 4.7 hold). Sketch of the proof. The function ψ Ω in (3.2) is the unique viscosity solution of the Hamilton–Jacobi equation H(u, ∇u) := 14 |∇u|4 + u − λ = 0 . On the other hand, u ∈ C 1 (Ω) is a classical solution of the same equation (since P is continuous and so P = λ in Ω). Therefore, u = ψ Ω . In particular, since u is a web function, the conclusions of Theorem 4.7 hold. Unfortunately, in general, u is not regular enough to prove that P is constant a.e. in Ω. Actually, the heuristic argument leading to the constancy of P can be made rigorous only provided u is at least of class C1,1 , and this kind of regularity never occurs. More precisely, the optimal expected regularity is C1,1/3 according to the result below, which is obtained essentially by dealing with ODE’s along the gradient flow of u, and, in particular, exploiting the expression of u along characteristics given by equation (3.1). Theorem 5.4 (Regularity threshold). If the unique solution u to problem (2.1) is of class C1,1 (A \ K), where K := argmaxΩ (u) and A is a neighborhood of K, then for any α > 1/3 it cannot occur that u is of class C 1,α (A). Nevertheless, not everything is lost. Still by exploiting characteristics, we can argue to obtain, in place of the constancy of the P-function, some useful upper and lower bounds for it. Theorem 5.5 (P-function inequalities). Under the assumptions (HΩ)–(Hu), let u be the unique solution to problem (2.1). Then, min ∂Ω

|∇u|4 ≤ P(x) ≤ max u 4 Ω

∀x ∈ Ω .

Sketch of the proof. Observe that, if y ∈ ∂Ω then P(X(0, y)) = P(y) = |∇u(y)|4 /4; on the other hand, for t large enough, X(t, y) is a maximum point of u, so that P(X(t, y)) = max u. Then to prove the statement, it is enough to show that P is nondecreasing along the gradient flow. To this end, in order to obtain a bit more of regularity, we consider

1.5 On the overdetermined problem: the general (non-web) case |

p

17

p

O

q

q Fig. 6: Domains considered in Theorem 5.6.

the supremal convolutions u ε (x) = sup {u(y) − y

|x − y|2 }. 2ε

By the local semiconcavity of u, these convolutions are of class C1,1 . Moreover, thanks to the so-called magic properties of their superjets, they turn out to be subsolutions of the PDE. Hence the corresponding approximated P-functions P ε :=

|∇u ε |4 + uε 4

are nondecreasing along the gradient flow of u ε . Finally by passing to the limit as ε → 0+ we get the desired monotonicity property for P. The bounds for the P-function obtained in Theorem 5.5 do not give us enough information to deduce a complete characterization of domains where the overdetermined problem (1.1) admits a solution. However, they are quite helpful to get at least a partial target. Namely we can prove the following result, showing that the same conclusions of Theorem 4.7 continue to hold without asking the solution to be a web function, provided some a priori geometric restrictions on Ω are imposed. Theorem 5.6 (Serrin-type theorem for ∆∞ ). Assume (HΩ)–(Hu). Further assume that there exists an inner ball B of radius ρ Ω which meets ∂Ω at two diametral points (see Figure 6 left). If there exists a solution u to the overdetermined problem (1.1), then u is the web function defined in (3.2), and Ω is a stadium-like domain (for which the conclusions of Theorem 4.7 hold). Sketch of the proof. Let p, q ∈ ∂Ω be the two diametral points belonging to ∂B ∩ ∂Ω, and let D be a stadium-like domain D that contains Ω and is tangent to Ω at p and q (see Figure 6, right). Let u B and u D denote, respectively, the solutions of the Dirichlet problem (2.1) in B and D. By comparison, we have uB ≤ u ≤ uD

in B .

In particular, this implies that u = u B = u D on the segment [p, q] and that ∇u = ∇u B = ∇u D at p and q, so that |∇u B | = |∇u D | = c at these two points. In turn, this gives

18 | 1 Inhomogeneous infinity Laplacian max u D = c4 /4 and hence, by Theorem 5.5, we obtain c4 |∇u|4 c4 = min ≤ P(x) ≤ max u ≤ . 4 4 4 ∂Ω Ω Now the conclusion follows from Theorem 5.3.

1.6 Open problems We list below some open questions related to the results reviewed above, which are in our opinion interesting challenges for further research. – Problem 1. Provide a complete characterization of stadium-like domains in higher dimensions (i.e., remove the convexity assumption in Theorem 4.6). – Problem 2. Provide a general version of Serrin theorem for ∆∞ (i.e., remove the geometric restrictions on Ω in Theorem 5.6). – Problem 3. Prove that the solution to the Dirichlet problem (2.1) is actually of class C1,1/3 (Ω) (i.e., show that the regularity threshold of Theorem 5.4 is achieved). – Problem 4. To some extent surprisingly, the geometric condition Σ(Ω) = M(Ω) appears independently in the paper [29], where it is shown that on stadium-like domains the infinity Laplacian admits a unique ground state. (An infinity ground state is, roughly speaking, the limit as p → +∞ of a sequence of solutions to the Euler–Lagrange equation for the nonlinear Rayleigh quotient associated with the p-Laplacian). As recently shown in [22], the uniqueness of an infinity ground state is false, in general, and the geometric characterization of domains where it is true is a completely open problem. It would be interesting to understand whether ∞ground states are unique in all convex domains or just in stadium-like ones. Note added in proof. Recently, some of the results presented in this chapter have been generalized to the case of the normalized infinity Laplace operator, see [12]. Moreover, we address to the forthcoming paper [13] for some developments on Problem 2.

Bibliography [1] [2] [3] [4] [5]

O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. 76(9):265–288, 1997. G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.) 41:439–505, 2004. E. N. Barron, R. R. Jensen and C. Y. Wang, The Euler equation and absolute minimizers of L ∞ functionals, Arch. Ration. Mech. Anal. 157:255–283, 2001. T. Bhattacharya and A. Mohammed, Inhomogeneous Dirichlet problems involving the infinityLaplacian, Adv. Differential Equations 17:225–266, 2012. F. Brock and A. Henrot, A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative, Rend. Circ. Mat. Palermo 51(2):375–390, 2002.

Bibliography

[6] [7]

[8] [9] [10] [11]

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

| 19

G. Buttazzo and B. Kawohl, Overdetermined boundary value problems for the ∞-Laplacian, Int. Math. Res. Not. IMRN, 237–247, 2011. P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations and optimal control, Progress in Nonlinear Differential Equations and their Applications 58, Birkhäuser, Boston, 2004. M. G. Crandall, H. Ishii and P. L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27:1–67, 1992. G. Crasta and I. Fragalà, On the characterization of some classes of proximally smooth sets, ESAIM Control Optim. Calc. Var. 22(3):710–727, 2016. G. Crasta and I. Fragalà, A symmetry problem for the infinity Laplacian, Int. Math. Res. Not. IMRN 18:8411–8436, 2015. G. Crasta and I. Fragalà, On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: regularity and geometric results, Arch. Ration. Mech. Anal. 218(3):1577–1607, 2015. G. Crasta and I. Fragalà, A C 1 regularity result for the inhomogeneous normalized infinity Laplacian, Proc. Amer. Math. Soc. 144(6):2547–2558, 2016. G. Crasta and I. Fragalà, Characterization of stadium-like domains via boundary value problems for the infinity Laplacian, Nonlinear Anal. 133:228–249, 2016. L. Damascelli and F. Pacella, Monotonicity and symmetry results for p-Laplace equations and applications, Adv. Differential Equations 5:1179–1200, 2000. L. C. Evans and O. Savin, C 1,α regularity for infinity harmonic functions in two dimensions, Calc. Var. Partial Differential Equations 32:325–347, 2008. L. C. Evans and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations 42:289–299, 2011. H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. I. Fragalà, F. Gazzola and B. Kawohl, Overdetermined problems with possibly degenerate ellipticity, a geometric approach, Math. Z. 254:117–132, 2006. N. Garofalo and J. L. Lewis, A symmetry result related to some overdetermined boundary value problems, Amer. J. Math. 111:9–33, 1989. G. Hong, Boundary differentiability for inhomogeneous infinity Laplace equations, Electron. J. Differential Equations 72:6, 2014. G. Hong, Counterexample to C 1 boundary regularity of infinity harmonic functions, Nonlinear Anal. 104:120–123, 2014. R. Hynd, C. Smart and Y. Yu, Nonuniqueness of infinity ground states, Calc. Var. Partial Differential Equations 48:545–554, 2013. E. Lindgren, On the regularity of solutions of the inhomogeneous infinity Laplace equation, Proc. Amer. Math. Soc. 142:277–288, 2014. G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Adv. Math. 217:1838–1868, 2008. Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22:167–210, 2009. J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43:304–318, 1971. J. Siljander, C. Wang and Y. Zhou, Everywhere differentiability of viscosity solutions to a class of Aronsson’s equations, preprint arXiv:1409.6804, 2014. C. Wang and Y. Yu, C 1 -boundary regularity of planar infinity harmonic functions, Math. Res. Lett. 19:823–835, 2012. Y. Yu, Some properties of the ground states of the infinity Laplacian, Indiana Univ. Math. J. 56:947–964, 2007.

H. Harbrecht and M. D. Peters

2 Solution of free boundary problems in the presence of geometric uncertainties Abstract: This chapter is concerned with solving Bernoulli’s exterior free boundary problem in the case of an interior boundary which is random. We model this random free boundary problem such that the expectation and the variance of the sought domain can be defined. In order to numerically approximate the expectation and the variance, we propose a sampling method like the (quasi-) Monte Carlo quadrature. The free boundary is determined for each sample by the trial method which is a fixed-pointlike iteration. Extensive numerical results are given in order to illustrate the model. Keywords: Bernoulli’s exterior free boundary problem, random boundary

2.1 Introduction Let T ⊂ ℝn denote a bounded domain with boundary ∂T = Γ. Inside the domain T, we assume the existence of a simply connected subdomain S ⊂ T with boundary ∂S = Σ. The resulting annular domain T \ S is denoted by D. The topological situation is visualized in Figure 1.

Σ

Ω

Γ

Fig. 1: The domain D and its boundaries Γ and Σ.

We consider the following overdetermined boundary value problem in the annular domain D: ∆u = 0 in D , ‖∇u‖ = f on Γ , (1.1) u=0 on Γ , u=1 on Σ , H. Harbrecht, Helmut Harbrecht, Universität Basel, Departement Mathematik und Informatik, Spiegelgasse 1, 4051 Basel, Schweiz, [email protected] M. D. Peters, Michael D. Peters, Universität Basel, Departement Mathematik und Informatik, Spiegelgasse 1, 4051 Basel, Schweiz, [email protected]

2.1 Introduction

|

21

where f > 0 is a given constant. We like to stress that the non-negativity of the Dirichlet data implies that u is positive in D. Hence, there holds the identity ‖∇u‖ ≡ −

∂u ∂n

on Γ

(1.2)

since u admits homogeneous Dirichlet data on Γ. We arrive at Bernoulli’s exterior free boundary problem if the boundary Γ is unknown. In other words, we seek a domain D with a fixed boundary Σ and unknown boundary Γ such that the overdetermined boundary value problem (1.1) is solvable. This problem has many applications in engineering sciences such as fluid mechanics, see [10], or electromagnetics, see [6, 7] and references therein. In the present form, it models, for example, the growth of anodes in electrochemical processes. For the existence and uniqueness of solutions, we refer the reader to, e.g., [3, 4, 17]; see also [9] for the related interior free boundary problem. Results concerning the geometric form of the solutions can be found in [1] and references therein. In this chapter, we try to model and solve the free boundary problem (1.1) in the case that the interior boundary is uncertain, i.e., if Σ = Σ(ω) with an additional parameter ω ∈ Ω. This model is of practical interest in order to treat, for example, tolerances in fabrication processes or if the interior boundary is only known by measurements which typically contain errors. We are thus looking for a tuple (D(ω), u(ω)) such that it holds ∆u(ω) = 0 in D(ω) , ‖∇u(ω)‖ = f

on Γ(ω) ,

u(ω) = 0

on Γ(ω) ,

u(ω) = 1

on Σ(ω) .

(1.3)

The questions to be answered in the following are: – How to model the random domain D(ω)? What is the associated expectation and the variance? – Do the expectation and the variance exist and are they finite? – What is the expectation and the variance of the potential u(ω) if the domain D(ω) is uncertain? – How to compute the solution to the random free boundary problem numerically? For the sake of simplicity, we restrict our consideration to the two-dimensional situation. Nevertheless, the extension to higher dimensions is straightforward and is left to the reader. The rest of this chapter is organized as follows. Section 2.2 is dedicated to answering the first two questions. We start by defining appropriate function spaces to define the stochastic model. Afterward, we define the random inner boundary and the resulting random outer boundary. Especially, we provide a theorem which guarantees the well posedness of the random free boundary problem under consideration. Moreover,

22 | 2 Solution of free boundary problems with geometric uncertainties

we introduce here the expectation and the variance of the domain’s boundaries. Finally, we give an analytic example which shows that the solution of the free boundary problem depends nonlinearly on the stochastic parameter. In Section 2.3, we answer the latter two questions. We propose the use of boundary integral equations for the solution of the underlying boundary value problem. This significantly decreases the effort for the numerical solution. In particular, we can describe the related potential of the free boundary problem in terms of Green’s representation formula. This also allows us to define its expectation and its variance. For the numerical approximation of the free boundary, we apply a trial method in combination with a Nyström discretization of the boundary integral equations. Section 2.4 is then devoted to the numerical examples. We will present here four different examples in order to illustrate different aspects of the proposed approach. We especially show that there is a clear difference between the expected free boundary and the free boundary which belongs to the expected interior boundary. As an important result, it follows thus that one cannot ignore random influences in numerical simulations. Finally, in Section 2.5, we state some concluding remarks.

2.2 Modelling uncertain domains 2.2.1 Notation In the sequel, let (Ω, F, ℙ) denote a complete and separable probability space with σ-algebra F and probability measure ℙ. Here, complete means that F contains all p ℙ-null sets. In the sequel, for a given Banach space X, the Bochner space Lℙ (Ω; X), 1 ≤ p ≤ ∞, consists of all equivalence classes of strongly measurable functions 𝑣 : Ω → X whose norm

‖𝑣‖Lℙp (Ω;X)

{ { p { { {(∫ ‖𝑣(⋅, ω)‖X dℙ(ω)) := { Ω { { { {ess sup ‖𝑣(⋅, ω)‖X , { ω∈Ω

1/p

,

p 0 has to be in the Bochner space L2 (Ω; C2per (I)), where C2per (I) denotes the Banach space of periodic, twice continuously differentiable functions, i.e., C2per (I) := {f ∈ C(I) : f (i) (0) = f (i) (2π), i = 0, 1, 2} , equipped with the norm 2

󵄨 󵄨 ‖f‖C2per (I) := ∑ max 󵄨󵄨󵄨f (i) (x)󵄨󵄨󵄨 . i=0

x∈I

For our purposes, we assume that q(ϕ, ω) is described by its expectation 𝔼[q](ϕ) = ∫ q(ϕ, ω) dℙ(ω) Ω

and its covariance Cov[q](ϕ, ϕ󸀠 ) = 𝔼[q(ϕ, ω)q(ϕ󸀠 , ω)] = ∫ q(ϕ, ω)q(ϕ󸀠 , ω) dℙ(ω) . Ω

Then, q(ϕ, ω) can be represented by the so-called Karhunen–Loève expansion, cf. [16], N

q(ϕ, ω) = 𝔼[q](ϕ) + ∑ q k (ϕ)Y k (ω) . k=1

Herein, the functions {q k (ϕ)} k are scaled versions of the eigenfunctions of the Hilbert– Schmidt operator associated with Cov[q](ϕ, ϕ󸀠 ). Common approaches to numerically recover the Karhunen–Loève expansion from these quantities are, e.g., given in [13] and the references therein. By construction, the random variables {Y k (ω)}k in the Karhunen–Loève expansion are uncorrelated. For our modelling, we shall also require that they are independent, which is a common assumption. Moreover, we suppose that they are identically distributed with img Y k (ω) = [−1, 1]. Note that it holds 2

N

2

𝕍[q](ϕ) = ∫ {q(ϕ, ω) − 𝔼[q](ϕ)} dℙ(ω) = ∑ (q k (ϕ)) . Ω

k=1

2.2.3 Random exterior boundary If the interior boundary Σ(ω) is starlike, then also the exterior boundary Γ(ω) is starlike. In particular, it also follows that the free boundary Γ(ω) is C∞ -smooth, see [2]

24 | 2 Solution of free boundary problems with geometric uncertainties

for details. Hence, the exterior boundary can likewise be represented via its parameterization: Γ(ω) = {x = γ(ϕ, ω) ∈ ℝ2 : γ(ϕ, ω) = r(ϕ, ω)er (ϕ), ϕ ∈ I} .

(2.1)

The following theorem guarantees us the well posedness of the problem under consid2 eration, cf. [4, 17]. It shows that it holds r(ϕ, ω) ∈ L∞ ℙ (Ω, C per (I)) if q(ϕ, ω) is almost surely bounded and thus that γ(ϕ, ω) is well defined. Theorem 2.1. Assume that q(ϕ, ω) is uniformly bounded almost surely, i.e., q(ϕ, ω) ≤ R

for all ϕ ∈ I and ℙ-almost every ω ∈ Ω .

(2.2)

Then, there exists a unique solution (D(ω), u(ω)) to (1.3) for almost every ω ∈ Ω. Especially, with some constant R > R, the radial function r(ϕ, ω) of the associated free boundary (2.1) satisfies q(ϕ, ω) < r(ϕ, ω) ≤ R

for all ϕ ∈ I and ℙ-almost every ω ∈ Ω .

Proof. In view of (2.2), it follows that Σ(ω) ⊂ B R (0) := {x ∈ ℝ2 : ‖x‖ < R} for almost every ω ∈ Ω. Hence, for fixed ω ∈ Ω, [17, Theorem 1] guarantees the unique solvability of (1.3). In particular, there exists a constant R > R such that Γ(ω) ⊂ B R (0) whenever Σ(ω) ⊂ B R (0). Therefore, the claim follows since q(ϕ, ω) is supposed to be uniformly bounded in ω ∈ Ω.

2.2.4 Expectation and variance of the domain Having the parameterizations σ(ω) and γ(ω) at hand, we can obtain the expectation and the variance of the domain D(ω). Theorem 2.2. The expectation of the domain D(ω) is given via the expectations of its boundaries’ parameterizations in accordance with 𝔼[∂D(ω)] = 𝔼[Σ(ω)] ∪ 𝔼[Γ(ω)] , where 𝔼[Σ(ω)] = {x ∈ ℝ2 : x = 𝔼[q(ϕ, ω)]er (ϕ), ϕ ∈ I} , 𝔼[Γ(ω)] = {x ∈ ℝ2 : x = 𝔼[r(ϕ, ω)]er (ϕ), ϕ ∈ I} . Proof. For the proof, we introduce the global parameterization δ : [0, 4π) → ∂D(ω) given by { σ(ϕ, ω), ϕ ∈ [0, 2π) , δ(ϕ, ω) = { (2.3) γ(ϕ − 2π, ω), ϕ ∈ [2π, 4π) . {

2.2 Modelling uncertain domains

|

25

Then, it holds per definition that 𝔼[∂D(ω)] = {x ∈ ℝ2 : x = 𝔼[δ(ϕ, ω)], ϕ ∈ [0, 4π)} . Therefore, the expected boundary 𝔼[∂D(ω)] consists of all points x ∈ ℝ2 with {𝔼[σ(ϕ, ω)], x={ 𝔼[γ(ϕ − 2π, ω)], {

ϕ ∈ [0, 2π) , ϕ ∈ [2π, 4π) .

This is equivalent to {𝔼[q(ϕ, ω)]er (ϕ), x={ 𝔼[r(ϕ − 2π, ω)]er (ϕ − 2π), {

ϕ ∈ [0, 2π) , ϕ ∈ [2π, 4π) ,

which immediately implies the assertion. The variance of the domain D(ω) is obtained in a similar way as the expectation. In particular, it suffices to take only the radial part of the variance into account due to the star shapedness. Theorem 2.3. The variance of the domain D(ω) in the radial direction is given via the variances of its boundaries parameterizations in accordance with 𝕍[∂D(ω)] = 𝕍[Σ(ω)] ∪ 𝕍[Γ(ω)] where 𝕍[Σ(ω)] = {x ∈ ℝ2 : x = 𝕍[q(ϕ, ω)]er (ϕ), ϕ ∈ I} , 𝕍[Γ(ω)] = {x ∈ ℝ2 : x = 𝕍[r(ϕ, ω)]er (ϕ), ϕ ∈ I} . Proof. We shall again employ the global parameterization δ(ϕ, ω) from (2.3). For the sake of notational convenience, we denote its centered version by δ 0 (ϕ, ω) := δ(ϕ, ω) − 𝔼[δ(ϕ, ω)] , and likewise for σ(ϕ, ω) and γ(ϕ, ω). The variance of D(ω) can be determined as the trace of the covariance Cov[∂D(ω)] = {X ∈ ℝ2×2 : X = 𝔼[δ 0 (ϕ, ω)δ 0 (ϕ󸀠 , ω)⊺ ], ϕ ∈ [0, 4π)} . From this representation, one concludes that Cov[∂D(ω)] consists of all (2 × 2) matrices X with {𝔼[σ0 (ϕ, ω)σ 0 (ϕ󸀠 , ω)⊺ ], { { { { {𝔼[σ0 (ϕ, ω)γ 0 (ϕ󸀠 − 2π, ω)⊺ ], X={ {𝔼[γ (ϕ − 2π, ω)σ 0 (ϕ󸀠 , ω)⊺ ], { 0 { { { 󸀠 ⊺ 𝔼[γ 0 (ϕ − 2π, ω)γ 0 (ϕ − 2π, ω) ], {

ϕ, ϕ󸀠 ∈ [0, 2π) , ϕ ∈ [0, 2π), ϕ󸀠 ∈ [2π, 4π) , ϕ ∈ [2π, 4π), ϕ󸀠 ∈ [0, 2π) , ϕ, ϕ󸀠 ∈ [2π, 4π) .

26 | 2 Solution of free boundary problems with geometric uncertainties The situation ϕ = ϕ󸀠 can only appear in the first or last case. These can be reformulated with ϕ, ϕ󸀠 ∈ [0, 2π) as Cov[σ, σ](ϕ, ϕ󸀠 ) = 𝔼[σ 0 (ϕ, ω)σ 0 (ϕ󸀠 , ω)⊺ ] = 𝔼[(q(ϕ, ω) − 𝔼[q](ϕ))(q(ϕ󸀠 , ω) − 𝔼[q](ϕ))]er (ϕ)er (ϕ󸀠 )⊺ and likewise as Cov[γ, γ](ϕ, ϕ󸀠 ) = 𝔼[γ 0 (ϕ, ω)γ 0 (ϕ󸀠 , ω)⊺ ] = 𝔼[(r(ϕ, ω) − 𝔼[r](ϕ))(r(ϕ󸀠 , ω) − 𝔼[r](ϕ))]er (ϕ)er (ϕ󸀠 )⊺ . By setting ϕ = ϕ󸀠 , we arrive at Cov[σ, σ](ϕ, ϕ) = 𝕍[q](ϕ)er (ϕ)er (ϕ)⊺ and Cov[γ, γ](ϕ, ϕ) = 𝕍[q](ϕ)er (ϕ)er (ϕ)⊺ . To get the radial part of the variances, we multiply the last expression by the radial direction er which yields the desired assertion. Consequently, in view of having 𝔼[q(ϕ, ω)] and 𝕍[q(ϕ, ω)] at hand, we need just to compute the expectation 𝔼[r(ϕ, ω)] and the variance 𝕍[r(ϕ, ω)] to obtain the expectation and the variance of the random domain D(ω).

2.2.5 Stochastic quadrature method For numerical simulation, we aim at approximating 𝔼[r(ϕ, ω)] and 𝕍[r(ϕ, ω)] with the aid of a (quasi-) Monte Carlo quadrature. To that end, we first parameterize the stochastic influences in q(ϕ, ω) by considering the parameter domain ◻ := [−1, 1]N and setting N

for y = [y1 , . . . , y N ]⊺ ∈ ◻ .

q(ϕ, y) = 𝔼[q](ϕ) + ∑ q k (ϕ)y k k=1

Especially, we have q(ϕ, y) ∈ L∞ (◻; C2per (I)) if q(ϕ, ω) ∈ L∞ (Ω; C2per (I)). Here, the space L∞ (◻; C2per (I)) is equipped with the pushforward measure ℙY , where Y = [Y 1 , . . . , Y N ]⊺ . This measure is of product structure due to the independence of the random variables. If the measure ℙY is absolutely continuous with respect to the Lebesgue measure, then there exists a density ρ(y), which is also of product structure, such that there holds 𝔼[q](ϕ) = ∫ q(ϕ, ω) dℙ(ω) = ∫ q(ϕ, y)ρ(y) dy . ◻

Ω

In complete analogy, we have for the variance 2

2

2

2

𝕍[q](ϕ) = ∫ (q(ϕ, ω)) dℙ(ω) − (𝔼[q](ϕ)) = ∫ (q(ϕ, y)) ρ(y) dy − (𝔼[q](ϕ)) . Ω

◻

27

2.2 Modelling uncertain domains |

Now, if F : L∞ (Ω; C2per (I)) → L∞ (Ω; C2per (I)),

q(ϕ, ω) 󳨃→ r(ϕ, ω)

(2.4)

denotes the solution map, the expectation and the variance of r(ϕ, ω) are given according to 𝔼[r](ϕ) = 𝔼[F(q)](ϕ) and 𝕍[r](ϕ) = 𝕍[F(q)](ϕ) . In view of this representation, we can apply a (quasi-) Monte Carlo quadrature in order to approximate the desired quantities. The Monte Carlo quadrature and the quasi-Monte Carlo quadrature approximate the integral of a sufficiently smooth function f over ◻ by a weighted sum according to ∫ f(y) dy ≈ ◻

1 M ∑ f(yi ) . M i=1

Herein, the sample points {y1 , . . . , yM } are either chosen randomly with respect to the uniform distribution, which results in the Monte Carlo quadrature, or according to a deterministic low-discrepancy sequence, which results in the quasi-Monte Carlo quadrature. The Monte Carlo quadrature can be shown to converge, in the mean square sense, with a dimension-independent rate of M −1/2 . The quasi-Monte Carlo quadrature based, for example, on Halton points, cf. [11], converges instead with the rate M δ−1 for arbitrary δ > 0. Although, for the quasi-Monte Carlo quadrature, the integrand has to provide bounded first-order mixed derivatives. For more details on this topic, see [5] and the references therein. In our particular problem under consideration, the expectation 𝔼[r](ϕ) and the variance 𝕍[r](ϕ) are finally computed in accordance with 𝔼[r](ϕ) = 𝔼[F(q)](ϕ) ≈

1 M ∑ F(q(ϕ, yi ))ρ(yi ) M i=1

and 2

𝕍[r](ϕ) = 𝕍[F(q)](ϕ) ≈

2 1 M 1 M ∑ (F(q(ϕ, yi ))) ρ(yi ) − ( ∑ F(q(ϕ, yi ))ρ(yi )) . M i=1 M i=1

2.2.6 Analytical example The calculations can be performed analytically if the interior boundary Σ(ω) is a circle around the origin with radius q(ω). Then, due to symmetry, also the free boundary Γ(ω) will be a circle around the origin with unknown radius r(ω). We shall thus focus on this particular situation in order to verify that the radius r(ω) depends nonlinearly on the stochastic input q(ω). Hence, on the associated expected domain 𝔼[D(ω)], the overdetermined boundary value problem (1.1) has, in general, no solution.

28 | 2 Solution of free boundary problems with geometric uncertainties Using polar coordinates and making the ansatz |u(ρ, ϕ)| = y(ρ), the solution with respect to the prescribed Dirichlet boundary condition of (1.1) has to satisfy y󸀠󸀠 +

y󸀠 = 0, ρ

y(q(ω)) = 1,

y(r(ω)) = 0 .

The solution to this boundary value problem is given by y(ρ) =

ρ log ( r(ω) )

log ( q(ω) r(ω) )

.

The desired Neumann boundary condition at the free boundary r(ω) yields the equation 1 ! −y󸀠 (r(ω)) = =f, r(ω) r(ω) log ( q(ω) ) which can be solved by means of Lambert’s W-function: r(ω) =

1 . 1 fW( fq(ω) )

(2.5)

Thus, the free boundary r(ω) depends nonlinearly on q(ω) since it generally holds 𝔼[r(ω)] = 𝔼[

1 1 fW( fq(ω) )

] ≠

1 1 fW( f 𝔼[q] )

.

(2.6)

Notice that the right-hand side would be the (unique) solution of the free boundary problem in the case of the interior circle of radius 𝔼[q(ω)]. Thus, indeed the overdetermined boundary value problem (1.1) will, in general, not be fulfilled on the expected domain 𝔼[D(ω)].

2.3 Computing free boundaries 2.3.1 Trial method For computing the expected domain 𝔼[D(ω)] and its variance 𝕍[D(ω)], we have to be able to determine the free boundary Γ(ω) for each specific realization of the fixed boundary Σ(ω). This will be done by the so-called trial method, which is a fixed point type iterative scheme. For the sake of simplicity in representation, we omit the stochastic variable ω in this section, i.e., we assume that ω ∈ Ω is fixed. The trial method for the solution of the free boundary problem (1.1) requires an update rule. Suppose that the current boundary in the k-th iteration is Γ k and let the current state u k satisfy in D k , ∆u k = 0 uk = 1 ∂u k =f − ∂n

on Σ , on Γ k .

(3.1)

2.3 Computing free boundaries | 29

The new boundary Γ k+1 is now determined by moving the old boundary into the radial direction, which is expressed by the update rule γ k+1 = γ k + δr k er . The update function δr k ∈ C2per ([0, 2π]) is chosen such that the desired homogeneous Dirichlet boundary condition is approximately satisfied at the new boundary Γ k+1, i.e., !

0 = u k ∘ γ k+1 ≈ u k ∘ γ k + (

∂u k ∘ γ k ) δr k ∂er

on [0, 2π] ,

(3.2)

where u k is assumed to be smoothly extended into the exterior of the domain D k . We decompose the derivative of u k in the direction er into its normal and tangential components ∂u k ∂u k ∂u k ⟨er , n⟩ + ⟨er , t⟩ on Γ k = (3.3) ∂er ∂n ∂t to arrive finally at the following iterative scheme (cf. [9, 12, 18]): (1) Choose an initial guess Γ0 of the free boundary. (2a) Solve the boundary value problem with the Neumann boundary condition on the free boundary Γ k . (2b) Update the free boundary Γ k such that the Dirichlet boundary condition is approximately satisfied at the new boundary Γ k+1 : uk δr k = − ∂u = − k

∂er

uk f⟨n, er ⟩ +

∂u k ∂t ⟨t, e r ⟩

.

(3.4)

(3) Repeat step 2 until the process becomes stationary up to a specified accuracy. Notice that the update equation (3.4) is always solvable at least in a neighborhood of the optimum free boundary Γ since there it holds −∂u/∂er = f⟨er , n⟩ > 0 due to ∂u k /∂t = 0, f > 0 and ⟨er , n⟩ > 0 for starlike domains.

2.3.2 Discretizing the free boundary For the numerical computations, we discretize the radial function r k associated with the boundary Γ k by a trigonometric polynomial according to r k (ϕ) =

a0 n−1 an + ∑ {a i cos(iϕ) + b i sin(iϕ)} + cos(nϕ) . 2 2 i=1

(3.5)

This obviously ensures that r k is always an element of C2per (I). To determine the update function δr k , represented likewise by a trigonometric polynomial, we insert the m ≥ 2n equidistantly distributed points ϕ i = 2πi/m into the update equation (3.4): δr k = −

uk f⟨n, er ⟩ +

∂u k ∂t ⟨t, e r ⟩

in all the points ϕ1 , . . . , ϕ m .

30 | 2 Solution of free boundary problems with geometric uncertainties

This is a discrete least-squares problem which can simply be solved by the normal equations. In view of the orthogonality of the Fourier basis, this means just a truncation of the trigonometric polynomial.

2.3.3 Boundary integral equations Our approach to determine the solution u k of the state equation (3.1) relies on the reformulation as a boundary integral equation by using Green’s fundamental solution G(x, y) = −

1 log ‖x − y‖2 . 2π

Namely, the solution u k (x) of (3.1) is given in each point x ∈ D by Green’s representation formula u k (x) = ∫ {G(x, y) Γ k ∪Σ

∂G(x, y) ∂u k (y) − u k (y)} dσ y . ∂n ∂ny

(3.6)

Using the jump properties of the layer potentials, we obtain the direct boundary integral formulation of the problem ∂G(x, y) 1 ∂u k u k (y) dσ y , u k (x) = ∫ G(x, y) (y) dσ y − ∫ 2 ∂n ∂ny Γ k ∪Σ

(3.7)

Γ k ∪Σ

where x ∈ Γ k ∪ Σ. If we label the boundaries by A, B ∈ {Γ k , Σ}, then (3.7) includes the single-layer operator V : C(A) → C(B),

(V AB ρ)(x) = −

1 ∫ log ‖x − y‖2 ρ(y) dσ y 2π

(3.8)

A

and the double-layer operator K : C(A) → C(B),

(KAB ρ)(x) =

⟨x − y, ny ⟩ 1 ρ(y) dσ y ∫ 2π ‖x − y‖22

(3.9)

A

with the densities ρ ∈ C(A) being the Cauchy data of u on A. Equation (3.7) in combination with (3.8) and (3.9) indicates the Neumann-to-Dirichlet map, which for problem (3.1) induces the following system of integral equations: 1

[2

I + KΓΓ KΓΣ

u k |Γ −VΣΓ VΓΓ ] [ ∂u k 󵄨󵄨 ] = [ −VΣΣ V 󵄨 ΓΣ ∂n 󵄨 Σ

−KΣΓ −f ][ ] . −( 12 I + KΣΣ ) 1

(3.10)

The boundary integral operator on the left-hand side of this coupled system of the boundary integral equation is continuous and satisfies a Gårding inequality with respect to the product Sobolev space L2 (Γ) × H −1/2 (Σ) provided that diam(Ω) < 1.

2.3 Computing free boundaries | 31

Since its injectivity follows from potential theory, this system of integral equations is uniquely solvable according to the Riesz–Schauder theory. The next step to the solution of the boundary value problem is the numerical approximation of the integral operators included in (3.10) which first requires the parameterization of the integral equations. To that end, we insert the parameterizations σ and γ k of the boundaries Σ and Γ k , respectively. For the approximation of the unknown Cauchy data, we use the collocation method based on trigonometric polynomials. Applying the trapezoidal rule for the numerical quadrature and the regularization technique along the lines of [15] to deal with the singular integrals, we arrive at an exponentially convergent Nyström method provided that the data and the boundaries and thus the solution are arbitrarily smooth.

2.3.4 Expectation and variance of the potential We shall comment on the expectation and the variance of the potential. To that end, we consider a specific sample ω ∈ Ω and assume that the associated free boundary Γ(ω) is known. Then, with the aid of the parameterizations σ(ω) : [0, 2π] → Σ(ω)

and

γ(ω) : [0, 2π] → Γ(ω) ,

we arrive, in view of (3.6), for x ∈ D(ω) at the potential representation 2π

u(x, ω) =

∑

V K K ∫ {k V A (x, ϕ, ω)ρ A (ϕ, ω) − k A (x, ϕ, ω)ρ A (ϕ, ω)} dϕ ,

(3.11)

A∈{Σ(ω),Γ(ω)} 0

where 󸀠 kV Σ(ω) (x, ϕ, ω) = G(x, σ(ϕ, ω))‖σ (ϕ, ω)‖2 , 󸀠 kV Γ(ω) (x, ϕ, ω) = G(x, γ(ϕ, ω))‖γ (ϕ, ω)‖2 ,

and kK Σ(ω) (x, ϕ, ω) =

∂G(x, σ(ϕ, ω)) 󸀠 ‖σ (ϕ, ω)‖2 , ∂ny

kK Γ(ω) (x, ϕ, ω) =

∂G(x, γ(ϕ, ω)) 󸀠 ‖γ (ϕ, ω)‖2 . ∂ny

Moreover, the related densities are given according to ∂u (σ(ϕ, ω)), ∂n ρK Σ(ω) (ϕ, ω) = u(σ(ϕ, ω)), ρV Σ(ω) (ϕ, ω) =

∂u (γ(ϕ, ω)) , ∂n ρK Γ(ω) (ϕ, ω) = u(γ(ϕ, ω)) . ρV Γ(ω) (ϕ, ω) =

These densities coincide with the Cauchy data of the potential u(ω) on the boundary ∂D(ω).

32 | 2 Solution of free boundary problems with geometric uncertainties In view of the representation (3.11), we observe that the expectation 𝔼[u](x) and the variance 𝕍[u](x) of the potential depend nonlinearly on the random parameter ω ∈ Ω. This is due to the fact that, in contrast to, e.g., [8], not only the density depends on ω but also the kernel function because of the parameterization. Nevertheless, if desired, these quantities can easily be approximated by sampling the expression (3.11) and its squared form for different realizations of the random parameter ω ∈ Ω by a (quasi-) Monte Carlo method as already discussed in Section 2.2.4 for r(ϕ, ω).

2.4 Numerical results In this section, we provide several numerical examples in order to illustrate our approach. For the numerical solution of the free boundary problem for each instance of the random parameter ω ∈ Ω, we apply the trial method proposed in the preceding section. The iteration is stopped if the ℓ∞ -norm of the update becomes smaller than 10−7 . For the discretization of the free boundary, we employ a trigonometric polynomial of order 32, i.e., n = 16 in (3.5). For the collocation method, we use m = 200 collocation points.

2.4.1 First example Our first example refers to the analytical example presented in Section 2.2.6. In particular, we want to illustrate that the expectation of the free boundary Γ(ω) differs from the free boundary obtained for given 𝔼[Σ]. To that end, we consider the following situation: In (1.3), we set u(ω) = 1 on Σ(ω) and ‖∇u(ω)‖2 = 10 on Γ(ω). Moreover, we set q(ϕ, ω) = 0.2 + 0.195X(ω), where X is distributed with respect to the counting measure μ(x) = 0.5 ⋅ δ −1 (x) + 0.5 ⋅ δ1 (x). Therefore, we can exactly determine the expectation and the variance of the free boundary by just two realizations of q(ϕ, ω). On the left-hand side of Figure 2, a visualization of the random domain’s statistics is found. The green line belongs to the expectation of the inner boundary 𝔼[Σ]. The expectation of the free boundary 𝔼[Γ] is indicated by the blue line. The gray shaded area refers to the standard deviation of Γ with respect to the expectation, i.e., the area which is bounded by 𝔼[Γ] ± √𝕍[Γ]. Moreover, we have depicted the solution to the free boundary problem for the fixed inner boundary 𝔼[Σ], i.e., the boundary related to the radius F(𝔼[q]), see (2.4), by the black dashed line. It can clearly be seen that this solution differs from the expectation 𝔼[Γ] due to the nonlinearity of the problem. This is also indicated by the plot of the related radial functions on the right-hand side of Figure 2. Here, we show the expectation 𝔼[r](ϕ) (blue line), the radius F(𝔼[q]) of the solution for the fixed inner boundary 𝔼[Σ] (black dashed line), the radius of the inner boundary 𝔼[q](ϕ) (green) and the radius of the standard deviation √𝕍[Γ](ϕ) (red).

2.4 Numerical results | 33

0.4 0.35 0.3

Radius

0.25 0.2 F ()[q ])

0.15

)[r ] 0.1

)[q ]

0.05

s td[r ]

0 0

1

2

3

4

5

6

Polar angle

Fig. 2: Expectation of the solution to the free boundary problem (left) and related expectations of the radii (right) for the first example.

In order to make the nonlinearity in the problem better visible, Figure 3, shows the radius r(ϕ, ω) (blue) computed by (2.5) with respect to q(ϕ, ω) ∈ [0.005, 0.395]. Moreover, we have depicted the sensitivity of r(ϕ, ω) with respect to q(ϕ, ω), i.e., the derivative with respect to q(ϕ, ω), in red. As it turns out, we have a strong nonlinearity only for very small values of q(ϕ, ω). For larger values of q(ϕ, ω), the problem exhibits a rather linear behavior. Finally, the black dot in the picture refers to the radius r that is obtained for 𝔼[q] = 0.2, i.e., F(0.2), and the blue dot on the secant connecting the extremal values of r(ϕ, ω) (green) refers to 𝔼[r] = 0.5(F(0.005) + F(0.395)), cf. (2.4). As they obviously do not coincide, this also confirms the statement (2.6).

0

r

10

r = F(q) 10

∂r/∂q

1

secant 

0.05

0.1

0.15

0.2

q

0.25

0.3

0.35

0.4

Fig. 3: Dependency and sensitivity of r(ϕ, ω) on q(ϕ, ω).

34 | 2 Solution of free boundary problems with geometric uncertainties

2.4.2 Second example For the second example, the same boundary conditions are chosen as in the previous example. Moreover, the radial function of Σ(ω) is defined according to √2

5

q(ϕ, ω) = 0.25 + 0.05 ∑ k=1

k

{ sin(kϕ)X2k−1 (ω) + cos(kϕ)X2k (ω)} ,

where the random variables {X k }k are independent and distributed with respect to the counting measure μ as before. In the spirit of the previous example, here we have to determine the 1024 realizations of the free boundary related to the 1024 possible realizations of q(ϕ, ω) in order to exactly determine the expectation and the variance of the free boundary. Thus, this example may be considered a more complex version of the previous one. 0.4 0.35 0.3

Radius

0.25

)[r ]

0.2

F ()[q ]) )[q ]

0.15

s td[r ]

0.1 0.05 0 0

1

2

3

4

5

6

Polar angle

Fig. 4: Expectation of the solution to the free boundary problem (left) and related expectations of the radii (right) for the second example.

Figure 4 visualizes the expectation and the standard deviation of the free boundary and the related radii. On the left hand side, one finds the random domain’s statistics and on the right-hand side the associated radial functions. Again, we see that there is a mismatch between the domain’s expectation (blue line) and the free boundary which belongs to the expected interior boundary (black dashed line).

2.4.3 Third example In our third example, we consider the approximation of the expectation and the variance of the solution to (1.3) in the case of a perturbed potato shaped inner domain. For

2.4 Numerical results | 35

the data, we prescribe the boundary conditions u(ω) = 1 on Σ(ω) and ‖∇u(ω)‖2 = 6 on Γ(ω). The radial function for Σ(ω) is given by √2 { sin(kϕ)X2k−1 (ω) + cos(kϕ)X2k (ω)} , k k=1 10

q(ϕ, ω) = 0.2 + 0.01f(ϕ) + ∑

where f(ϕ) is a trigonometric polynomial with coefficients, cf. (3.5), [a5 , . . . , a0 , b 1 , . . . , b 4 ] = [0.33, 0.26, 0.51, 0.70, 0.89, 0.48, 0.55, 0.14, 0.15, 0.26] . The random variables {X k }k are chosen to be independent and uniformly distributed in [−1, 1]. The approximation of the expectation 𝔼[r](ϕ) and the variance 𝕍[r](ϕ) is performed by the application of a quasi-Monte Carlo quadrature based on 10 000 Halton points, cf. [11]. As already pointed out in Section 2.2.4, the application of the quasiMonte Carlo quadrature requires mixed smoothness of the integrand. Although this is not proven here, we have strong evidence that the function r(ϕ, ω) exhibits this smoothness. This is also validated by this example. 0.4 0.35 0.3

Radius

0.25 0.2 )[r ]

0.15

F ()[q §Û ])

0.1

)[q ] s td[r ]

0.05 0 0

1

2

3

4

5

6

Polar angle

Fig. 5: Expectation of the solution to the free boundary problem (left) and related expectations of the radii (right) for the third example.

On the left-hand side of Figure 5, a visualization of the random domain’s statistics is found. The left-hand side of Figure 5 shows the expectation 𝔼[Σ] (green) and the expectation 𝔼[Γ] (blue). The gray shaded area refers to the standard deviation of Γ. Moreover, the free boundary that corresponds to 𝔼[Σ] is indicated by the black dashed line. It differs again clearly from the expectation 𝔼[Γ]. The right-hand side of Figure 5 shows the related radius functions. Here, the standard deviation is indicated in red. Finally, in order to justify the application of the quasi-Monte Carlo quadrature based on Halton points, we have also considered the convergence of the Monte Carlo

36 | 2 Solution of free boundary problems with geometric uncertainties

10

1

2

l



error

10

10

3

error in )[Σ] error in )[Γ] rate M –1/2 1

10

2

10

3

10

4

10

Samples

5

10

6

10

Fig. 6: Convergence of the Monte Carlo quadrature to the approximation based on the quasi-Monte Carlo quadrature.

quadrature toward the approximation of the expectation obtained by the quasi-Monte Carlo quadrature. The related plot is found in Figure 6. The green line refers to the approximation of the expectation of the inner boundary 𝔼[Σ], which is a linear problem. The blue line indicates the convergence of the expectation of the outer boundary 𝔼[Γ], which is a nonlinear problem. We have measured here the relative error in the ℓ∞ -norm of the boundaries evaluated in the collocation points. The theoretical rate of convergence, given by M −1/2 , where M denotes the number of Monte Carlo samples, is visualized by the black dashed line. As it turns out, we obtain in both cases convergence of the Monte Carlo quadrature toward the solution obtained by the quasi-Monte Carlo quadrature. This validates the approximation obtained by the quasi-Monte Carlo quadrature, which will also be used as the stochastic quadrature method in the following example.

2.4.4 Fourth example Finally, we consider an example where the inner boundary Σ(ω) is given by four circles of radius 0.05 with randomly varying midpoints [0.1(−1)i + 0.04X2(2i+j) (ω), 0.1(−1)j + 0.04X2(2i+j)+1 (ω)]

⊺

for i, j = 0, 1 .

Here, the random variables X1 , . . . , X8 are independent and uniformly distributed on [−1, 1]. The radii and midpoints of the circles are chosen such that they cannot overlap. In order to illustrate the situation under consideration, we have depicted six different realizations of Σ(ω) and of the related free boundaries Γ(ω) in Figure 7. For this example, the boundary conditions are chosen as u(ω) = 1 on Σ(ω) and ‖∇u(ω)‖2 = 8 on Γ(ω). The visualization of the computed expectation and the standard deviation of the free boundary as well as the related radii are presented in Figure 8. Even though the interior boundaries vary a lot, the difference between the free boundary related to 𝔼[Σ] and 𝔼[Γ] is relatively small.

2.4 Numerical results | 37

0.2

0.2

0.1

0.1

0

0

0.1

0.1

0.2

0.2

0.2

0.1

0

0.1

0.2

0.2

0.2

0.1

0.1

0

0

0.1

0.1

0.2

0.2

0.2

0.1

0

0.1

0.2

0.2

0.2

0.1

0.1

0

0

0.1

0.1

0.2

0.2

0.2

0.1

0

0.1

0.2

0.2

0.1

0

0.1

0.2

0.2

0.1

0

0.1

0.2

0.2

0.1

0

0.1

0.2

Fig. 7: Different realizations of the random boundary Σ(ω) and corresponding free boundary Γ(ω) for the fourth example.

38 | 2 Solution of free boundary problems with geometric uncertainties

0.25

Radius

0.2 0.15

F ()[q§Û ]) )[r ]

0.1

s td[r ] 0.05 0 0

1

2

3



4

5

6

Fig. 8: Expectation of the solution to the free boundary problem (left) and related expectations of the radii (right) for the fourth example.

2.5 Conclusion In the present chapter, Bernoulli’s exterior free boundary problem has been considered in the case of an interior boundary which is random. Such uncertainties may arise from tolerances in fabrication processes or from measurement errors. We modeled this problem mathematically and showed its well posedness. Expectation and variance of the resulting random domain have been introduced and numerically computed. Establishing regularity results with respect to the random parameter will be subject of future work in order to rigorously prove the convergence of the present quasi-Monte Carlo method and even more sophisticated quadrature methods.

Bibliography [1] [2]

[3] [4] [5] [6] [7]

A. Acker, On the geometric form of Bernoulli configurations, Mathematical Methods in the Applied Sciences, 10(1):1–14, 1988. A. Acker and R. Meyer, A free boundary problem for the p-laplacian: uniqueness, convexity, and successive approximation of solutions, Electronic Journal of Differential Equations, 8:1–20, 1995. H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, Journal für die reine und angewandte Mathematik, 325:105–144, 1981. A. Beurling, On free boundary problems for the Laplace equation, Seminars on Analytic functions, Institute for Advanced Study, Princeton, NJ, 1:248–263, 1957. R. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numerica, 7:1–49, 1998. M. Crouzeix, Variational approach of a magnetic shaping problem, European Journal of Mechanics B-Fluids, 10:527–536, 1991. J. Descloux, Stability of the solutions of the bidimensional magnetic shaping problem in absence of surface tension, European Journal of Mechanics B-Fluids, 10:513–526, 1991.

Bibliography |

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

39

J. Dölz, H. Harbrecht and M. Peters, H-matrix accelerated second moment analysis for potentials with rough correlation, Journal of Scientific Computing, 65:387–410, 2015. M. Flucher and M. Rumpf, Bernoulli’s free-boundary problem, qualitative theory and numerical approximation, Journal für die Reine und angewandte Mathematik, 486:165–204, 1997. A. Friedman, Free boundary problem in fluid dynamics, Astérisque, 118:55–67, 1984. J. H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals, Numerische Mathematik, 2(1):84–90, 1960. H. Harbrecht and G. Mitrou, Improved trial methods for a class of generalized Bernoulli problems, Journal of Mathematical Analysis and Applications, 420(1):177–194, 2014. H. Harbrecht, M. Peters and M. Siebenmorgen, Efficient approximation of random fields for numerical applications, Numerical Linear Algebra with Applications, 22:596–617, 2015. E. Hille and R. S. Phillips, Functional analysis and semi-groups, volume 31, American Mathematical Society, Providence, 1957. R. Kress, Linear integral equations, Vol. 82 of Applied Mathematical Sciences. Springer, New York, 2nd edition, 1999. M. Loève, Probability theory. I+II, Number 45 in Graduate Texts in Mathematics. Springer, New York, 4th edition, 1977. D. E. Tepper, On a free boundary problem, the starlike case, SIAM Journal on Mathematical Analysis, 6(3):503–505, 1975. T. Tiihonen and J. Järvinen, On fixed point (trial) methods for free boundary problems, In S. N. Antontsev, A. M. Khludnev and K.-H. Hoffmann, editors, Free Boundary Problems in Continuum Mechanics, volume 106 of International Series of Numerical Mathematics, pp. 339–350. Birkäuser, Basel, 1992.

M. Hintermüller and D. Wegner

3 Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies Abstract: This chapter is concerned with optimal control problems for the coupled Cahn–Hilliard (CH)/Navier–Stokes (NS) system related to ‘model H’ of Hohenberg and Halperin [20]. It proposes a time discretization allowing suitable energy estimates, that in particular force the total energy to decrease without control action, and it considers distributed as well as boundary control of the fluid. For nonsmooth potentials, including the double-obstacle potential contained in the associated Ginzburg– Landau energy, a regularization procedure based on a mollified Moreau–Yosida approximation is applied. The resulting regularized problems are discretized by finite elements and solved via a gradient descent method. Several numerical examples document the behavior of the algorithm as well as the controlled CH–NS system for boundary and for distributed control. Keywords: Cahn–Hilliard/Navier–Stokes system, double-obstacle potential, mathematical programming with equilibrium constraints, optimal boundary control, optimal distributed control, Yosida regularization

3.1 Introduction The renowned Cahn–Hilliard (CH) equation [9] is widely used for modeling phase separation and coarsening processes with a diffusive interface in multiphase systems. Whenever hydrodynamic effects are present, CH has to be coupled with an equation that describes the motion of the fluid. The resulting coupled system is used to model polymer blends, proteins crystallization, cf. [23] and references therein, or the solidification of liquid metal alloys [12]. It is utilized in the simulation of bubble dynamics (as in nitrobenzene), in Taylor flows [1, 4] or for pinch-offs of liquid–liquid jets [22]. Moreover, it is applied to describe the effects of surfactants such as colloid particles at fluid–fluid interfaces in gels and emulsions used in food, pharmaceutical, cosmetic, or petroleum industries [2, 5, 26]. Even the simulation of cooling systems in nuclear power plants or applications in computer graphics are conducted by using these models [3].

M. Hintermüller, Humboldt-Universität zu Berlin, Department of Mathematics, Germany D. Wegner, Humboldt-Universität zu Berlin, Department of Mathematics, Germany

3.1 Introduction

| 41

In this chapter, we consider the following model related to ‘model H’ of Hohenberg and Halperin [20]: 1 ∆w + ∇c ⋅ 𝑣 = 0 , Pe −w − ε∆c + Φ󸀠 (c) ∋ 0 ,

∂t c −

1 ∆𝑣 + (𝑣 ⋅ ∇)𝑣 − Kw∇c + ∇π = 0 , ∂t𝑣 − Re div 𝑣 = 0 ,

(1.1) (1.2) (1.3) (1.4)

which models a two-phase flow subject to phase separation. The system (1.1)–(1.2) is the CH system, and (1.3)–(1.4) is the Navier–Stokes (NS) system for modeling fluid flow. Here, these equations are assumed to hold in a suitable spatial domain Ω ⊂ ℝN with N ∈ {1, 2, 3} and for all times in a given time interval (0, T) with T > 0. The quantity c is the concentration of the fluid phase with c = 1 or −1 describing the pure phases, respectively, and c ∈ (−1, 1) an associated mixed state. Hence, c attains values within the interval [−1, 1] only. The related chemical potential is given by w, the velocity of the fluid is denoted by 𝑣 and the corresponding pressure by π. Moreover, the constant 1 Pe is the mobility coefficient of the system with Pe being the Péclet number, K measures the strength of the capillary forces, ε relates to the thickness of the interfacial region and Re is the Reynolds number. The terms ∇c⋅𝑣 in (1.1) and Kw∇c in (1.3) couple both systems. The function Φ in (1.2) is the homogeneous free energy density in the Ginzburg– Landau energy model associated with the CH system. Whenever it is nonconvex, a homogeneous mixture of the fluid is, in general, not a minimizer of the pertinent free energy functional. As a consequence, phase separation can occur. Popular choices of Φ assume the decomposition of Φ into the smooth nonconvex part − 12 c2 and a convex, but possibly nonsmooth part ψ(c), i.e., Φ(c)(x) = ψ(c(x)) − 12 c2 (x). Within this family of potentials, the double-well potential corresponding to ψ(c) = c4 is smooth and admits the concentration c to attain values outside of the physically meaningful range [−1, 1]. While the differentiability properties of the double-well potential yield mathematical convenience, the possibly unphysical range of c limits its ability to predict long-time behavior of the phase separation process. On the other hand, the logarithmic potential ψ(c) = (1 + c) ln(1 + c) + (1 − c) ln(1 − c) forces the concentration to take values inside (−1, 1) only, rendering the pure phase c = −1 and c = +1 singular. This potential was already considered in Cahn’s and Hilliard’s original work [9] and it is used in the Flory–Huggins theory for modeling the phase separation in polymer solutions. Finally, the double-obstacle potential with ψ(c) := 0 if |c| ≤ 1 and ψ(c) := +∞ if |c| > 1 ensures the concentration to attain physically meaningful values only and it admits the pure phases c = ±1. This potential is also used in the context of polymer solutions and it appears appropriate in cases of deep quenches and rapid wall hardening, cf. [25]. In contrast to the double well and the logarithmic potential that possess differentiability properties and where ψ󸀠 (c) (and thus ∂ψ(c) = {ψ󸀠 (c)}) is

42 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system

uniquely given, the derivative of the convex part ψ of the double-obstacle potential has to be understood in the sense of the (possibly set-valued) subdifferential ∂ψ for convex functions. Hence, in a pure phase, i.e., either c = 1 or −1, ∂ψ is both multi-valued and unbounded, which clearly complicates the analytic as well as the numerical treatment of the system (1.1)–(1.4). In many applications, one is interested in influencing the phase separation process. For instance, in binary alloys one typically tries to avoid or reduce separation effects since they drastically reduce the durability and the lifetime of the alloy. In the formation of polymeric membranes by an immersion precipitation process, where a polymer solution is immersed into a coagulation bath containing a nonsolvent, the decomposition of the polymeric solution has to be controlled in such a way that the final polymer has a given porosity pattern. The resulting morphological structure significantly influences the properties of the polymer membrane [33]. In this chapter, we therefore consider an optimal control problem for the coupled CH–NS system where the control u acts either as a distributed force entering (1.3) as a right-hand side or as a boundary control for the fluid velocity. While boundary control of the fluid can be realized by various devices, e.g., blowing and suction, a typical distributed control would rely on electro-magnetic fluids. For reasons of well-posedness, we close the system by appropriate boundary conditions on c, w, and 𝑣, and initial conditions on c and 𝑣. Thus, we study the problem min J(c, 𝑣, u) subject to the semidiscrete CH–NS system with control u ,

(P)

where J represents a suitably chosen optimization objective (cost function). Note that in contrast to other forms of controlling this system like receding horizon or instantaneous control techniques [10, 19, 21], where optimization of the corresponding control actions is only considered within one time step or over a small time horizon, we consider an optimal control problem over the whole time interval. Hence, the entire semi-discrete version of CH–NS is taken as a constraint system in the minimization context leading to an optimal balance with respect to the cost function between control costs and, e.g., closeness to a desired profile. Upon analyzing stationarity conditions for (P), the discretization process is completed by applying the finite element method in space. We note that the derivation of stationarity conditions for (P) is a delicate matter. This is in particular true when Φ involves the nonsmooth double-obstacle potential. In fact, in the latter case, (1.2) is equivalent to a variational inequality which is known to represent a degenerate constraint [7, 17, 18] in (P). As a consequence, one cannot apply the Karush–Kuhn–Tucker theory [34] for deriving stationarity conditions. In order to overcome this difficulty, in [18] a technique utilizing a mollified Moreau–Yosida approximation of Ψ is applied. Upon passage to the limit with the involved smoothing parameters a so-called C-stationarity system is obtained. On the numerical level, this proof technique requires the solution of a sequence of nonlinear programs in Banach space where the limit process with respect to the smoothing parameters is based on

3.2 Optimal control problem for the time discretization

| 43

a path-following scheme. While [18] merely contains theoretical results for boundary control problems only, the present work extends the scope of [18] to distributed control problems and, in particular, it offers a discretization and gradient-based solution scheme along with a report on various test cases involving the double-obstacle potential. Let us further mention that the literature on the optimal control of the coupled semidiscrete CH–NS system is, to the best of our knowledge, essentially void apart from the analytic result given in [18]. Instantaneous control of the semidiscrete CH– NS system has been considered in [19] and optimal control problems in one dimension for the viscous as well as for the convective Cahn–Hilliard equation has been studied in [31] and [32]. For the sole CH or the Allen–Cahn equation optimal control problems were studied from an analytic perspective in [14, 15, 17, 29, 30]. The rest of this chapter is organized as follows. In Section 2, we present a semidiscrete version of (1.1)–(1.4) and related optimal control problems with boundary or distributed control, respectively. Moreover, the corresponding first-order optimality conditions are given for smooth potentials ψ. Section 3 presents a mollified Yosida approximation that we apply to the subdifferential of ψ in order to find a stationarity condition for the original optimal control problem with the double-obstacle potential upon passage to the limit with the smoothing parameters in the mollified Yosida approximation. The resulting, smooth auxiliary problems can be solved by a gradient descent method and a path-following scheme which yields C-stationary points for the optimal control problem in the limit. An outline of the algorithm and its numerical realization is presented. The utilized finite element spaces are introduced in Section 4. Various examples showing the behavior of the algorithm for boundary as well as for distributed control are contained in Section 5. Section 6 summarizes the present work and the appendix provides our expansion of the Armijo line search algorithm.

3.2 Optimal control problem for the time discretization For its numerical realization, we discretize the above system (1.1)–(1.4) in time by using a semi-implicit Euler scheme. For this purpose, we fix a final time T > 0, a number of time steps M ∈ ℕ, and the corresponding time step size τ := T/M. The value of the concentration at time t i = iτ, i ∈ {0, . . ., M}, is denoted by c i and c = (c0 , . . ., c M ) is the vector of concentrations at the discrete times t i , i = 0, . . ., M. We proceed analogously with other time-dependent quantities. The time-discrete CH–NS system is given by 1 1 (c i+1 − c i ) − ∆w i+1 + ∇c i ⋅ 𝑣i+1 = C1i τ Pe −w i+1 − ε∆c i+1 − c i + ∂ψ(c i+1 ) ∋ C2i 1 1 (1) (𝑣i+1 − 𝑣i ) − ∆𝑣i+1 + (𝑣i ⋅ ∇)𝑣i+1 − Kw i+1 ∇c i + ∇π i = η(1) u i τ Re

(in Ω) ,

(2.1)

(in Ω) ,

(2.2)

(in Ω) ,

(2.3)

44 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system

∫ c i = 0,

∫ w i = 0,

Ω

Ω

∇c i ⋅ n⃗ = 0,

div 𝑣i = 0

∇w i ⋅ n⃗ = 0, c0 = c a ,

(2)

𝑣i = η(2) u i

𝑣0 = 𝑣a

(in Ω) ,

(2.4)

(on ∂Ω) ,

(2.5)

(in Ω) .

(2.6)

Here, the convex part of the homogeneous free energy density is given by a proper, convex and lower semicontinuous functional ψ : ℝ → ℝ, and C i = (C0i , . . ., C iM ) ∈ ℝM+1 for i ∈ {1, 2} denotes the tuple of constant functions on Ω with possibly different values on each time slice. The reason for using these constants is the following. By condition (2.4), we enforce c i and w i to have vanishing mean values, respectively. For w, this provides no restriction and is used in order to obtain a unique w (note here that a shift w i 󳨃→ w i + C does not change the dynamics of the system in the sense that K(w i+1 + C)∇c i + ∇π i = Kw i+1 ∇c i + ∇(π i + Cc i ) would only replace the pressure π i by π i + Cc i ). For c, on the other hand, ∇c i ⋅ 𝑣i+1 can only be expected to have mean value 0 if c i 𝑣i+1 has no normal contribution on the boundary. Indeed, one has ∫ K∇c i ⋅ 𝑣i+1 = K ∫ c i 𝑣i+1 ⋅ n⃗ − K ∫ c i div 𝑣i+1 = K ∫ c i 𝑣i+1 ⋅ n⃗ Ω

∂Ω

Ω

∂Ω

for smooth c i and 𝑣i+1 . While the mean-value property is problematic in the boundary control case, it is satisfied in the case of distributed control. For boundary control problems of the fluid, we simplify our model by projecting the concentration onto the space of functions with zero mean value. Hence, we impose the conditions (2.4) and introduce in equation (2.1) the correction term C1i which corresponds to the mean value of ∇c i ⋅𝑣i+1 . As a consequence, we obtain ∫Ω c i+1 = 0. In order to keep these corrections small, for boundary control we only use examples with one pure phase in a boundary layer as initial data. In this case, ∇c i can be expected to be small, if not zero. In this chapter, we consider either the case of Dirichlet boundary control (which corresponds to η(1) = 0) or the case of distributed control acting as an additional force on the fluid entering the system as a right-hand side in the balance of momentum (in this case set η(2) = 0). Of course, also a combined control action consisting of both types of controls could be studied using the techniques discussed in this chapter. For the concentration c and the chemical potential w, Neumann boundary conditions are imposed by (2.5). Note that in combination with the boundary control of 𝑣, this implies that only material which has the same concentration as the material on the boundary can be injected. Although perhaps demanding with respect to its technical realization, this allows us to use the standard weak formulation of the CH system corresponding to these boundary conditions. The conditions (2.4) forces both quantities to preserve a vanishing mean value and the velocity to be divergence free. Note that in our examples below, we normally assume initial values for the concentration which do not meet this mean-value condition. Hence, we shift the whole problem via 1 ĉ := c − c with c := |Ω| ∫Ω c and use ĉ instead of c. The shifted concentration ĉ satisfies

3.2 Optimal control problem for the time discretization

45

|

the mean-value condition in (2.4). As a consequence, the interval [−1, 1] is shifted to [−1 − c, 1 − c] and the potential ψ has to be adapted accordingly. Apart from the additional right-hand side in (2.3) corresponding to the distributed control, this setting was used in [18] in order to prove the existence of optimal boundary controls and to derive stationarity conditions. We note that the above discretization scheme allows to derive energy estimates for the total energy of the system in each time step. In particular, in the case that no control action is applied, the total energy decreases monotonically. This was a key property in the proof of existence of optimal controls; see [18] for details. Also note that in discretization schemes, as for instance the one used in [16] where the Cahn– Hilliard and the NS part decouple from each other, such energy estimates cannot be expected to hold true. Here, we exemplarily study the following cost functional on H 1 (Ω) × U (1) × U (2) given by M

(1)

(2)

J(c, u (1) , u (2) ) := ∑ λ i ‖c i − c∗i ‖2L2 + η(1) ‖u i ‖2L2 + η(2) ‖u i ‖2H 1/2

(2.7)

i=0

for closed subspaces U (1) of H 1σ (Ω) := {𝑣 ∈ H 1 (Ω; ℝN ) | div 𝑣 = 0} and U (2) of U := {Tr 𝑣 | 𝑣 ∈ H 1 (Ω; ℝN ), div 𝑣 = 0} which itself is a closed subspace of H 1/2 (∂Ω; ℝN ). Here and below Tr denotes the trace operator. We note that, of course, other objectives are conceivable. We equip the space H 1/2 (∂Ω; ℝN ) with the norm ‖u‖H 1/2 := {inf ‖𝑣‖H 1 (Ω;ℝN ) | 𝑣 ∈ H 1 (Ω; ℝN ), Tr 𝑣 = u} and H 1 (Ω; ℝN ) with the standard norm ‖𝑣‖H 1 := (∑ Nn=1 ‖𝑣n ‖2H 1 )1/2 for 𝑣 = (𝑣1 , . . ., 𝑣N ) ∈ H 1 (Ω; ℝN ), ‖𝑣‖H 1 := (‖𝑣‖2L2 + ‖ |∇𝑣| ‖2L2 )1/2 for 𝑣 ∈ H 1 (Ω) with the Euclidian norm | ⋅ | : ℝN → ℝ. The corresponding optimal control problem (P ψ ) depending on the choice of the potential ψ is then given by min {J(c, u (1) , u (2) ) | (c, w, 𝑣, u (1) , u (2) ) ∈ (H 1 (Ω)×H 1 (Ω)×H 1 (Ω; ℝN )×U (1) ×U (2) )

M+1

satisfies (2.1)–(2.5) in the weak sense}. In particular, we are interested in the case ψ = φ(0) of the double-obstacle potential, i.e., {0 if|r| ≤ 1 , (2.8) φ(0) : ℝ → ℝ, φ(0) (r) := { ∞ otherwise. { For (Pψ ), the existence of a minimizer together with first-order optimality conditions and stationarity conditions was established in [18] in the case of boundary control. Using the arguments of [18] and proceeding in a similar way for the problem setting in̂ (1) , u (2) ) := cluding distributed control, the gradient of the reduced cost functional J(u (1) (2) (1) (2) J(c(u , u ), u , u ) for a solution (c, w, 𝑣, u) of (2.1)–(2.5) and for a smooth potential ψ can be written, with the help of J, the duality mapping of [(U (1) × U (2) )]M+1 , as J −1 J󸀠̂ (u (1) , u (2) ) = (η(1) u (1) − z(1) , η(2) u (2) − z(2) )

46 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system (i)

with (z(1) , z(2) ) ∈ (U (1) × U (2) )M+1 given by z0 = 0 for i = 1, 2 and 1 1 1 − (p i+1 −p i )+(−ε∆+DA(c i+1 ))∗ (− ∆p i −K∇c i ⋅q i )−(− ∆p i+1 −K∇c i+1 ⋅q i+1 ) τ Pe Pe (2.9) − div(p i+1 𝑣i+2 ) + K div(w i+2 q i+1 ) = C3i + λ i J W0 (c i − c∗i ), 1 1 ∗ − (q i+1 − q i ) − ∆ q i + b 1 (𝑣i+2 , q i+1 ) τ Re +b 2 (𝑣i , q i ) + p i ∇c i = −∇π2i , ∫ p i = 0,

div q i = 0 ,

(2.10) (2.11)

Ω (1)

−J −1 I∗ q = z i+1 , U (1) U (1) →V −1 i

(2.12)

1 1 ∆q i + b 1 (𝑣i+2 , q i+1 ) H [− (q i+1 − q i ) − τ Re (2)

+ b 2 (𝑣i , q i ) + p i ∇c i ] = z i+1 .

(2.13)

Here, A := ∂ψ, I U (1) →V−1 is the canonical injection of U (1) into V−1 and H : (H 1 (Ω; ℝN ))∗ → U (2) denotes the composition H = P U (2) ∘ Tr ∘P Z ∘ J −1 conH1 (2) sisting of the orthogonal projection P U (2) of U onto U , the orthogonal projection P Z of H 1 (Ω; ℝN ) onto Z := {z ∈ H 1 (Ω; ℝN ) | div z = 0, (z | 𝑣)H 1 = 0 ∀𝑣 ∈ H01 (Ω; ℝN ), div 𝑣 = 0} and the inverse of J H 1 . Here, for a Hilbert space X, the operator J X denotes the canonical isomorphism from X onto X ∗ given by the Rieszrepresentation theorem [27]. By (⋅ | ⋅)H 1 , we denote the inner product of H 1 (Ω; ℝN ), and b 1 and b 2 are related to the convective part of the NS equation by ⟨b 1 (y2 , y3 ), y1 ⟩ := ⟨b 2 (y1 , y3 ), y2 ⟩ := ∫ y3 (x)(y1 (x) ⋅ ∇)y2 (x) dx Ω

with ⟨., .⟩ denoting the duality pairing between H 1σ (Ω)∗ and H 1σ (Ω). For an operator S, S∗ denotes its adjoint. Moreover, since (p, q) take values ((p0 , . . ., p M−1 ), (q0 , . . ., q M−1 )), we use the convention (p M , q M ) = 0 which was already used in the system (2.9)–(2.13). As a consequence of this convention, we do not have to denote the final time conditions on (p, q) explicitly. We further note that the notation of the reduced cost functional alluded to above corresponds to the situation where c (and w) is no longer considered an independent variable, but rather it is viewed as a function of u (1) and u (2) , i.e., c = c(u (1) , u (2) ) where c and u (1) , u (2) are linked by solving (2.1)–(2.6).

3.3 Yosida approximation and gradient method | 47

It is possible to rewrite (2.9) in a fashion more similar to (2.1)–(2.2). In fact, we introduce another variable r which satisfies ∫Ω r = 0 as well as the system 1 1 − (p i+1 − p i ) + (−ε∆ + DA(c i+1 ))∗ r i − I ∗ (− ∆p i+1 − K∇c i+1 ⋅ q i+1 ) τ Pe − div(p i+1 𝑣i+2 ) + K div(w i+2 q i+1 ) = C3i + λ i J W0 (c i − c∗i ) , −r i + (−

1 ∆p i − K∇c i ⋅ q i ) = 0 . Pe

3.3 Yosida approximation and gradient method 3.3.1 Sequential Yosida approximation In order to derive first-order optimality conditions for characterizing solutions of (Pψ ) in the case where ψ = φ(0) , in [18] we replaced the potential φ(0) by a mollified Moreau–Yosida approximation. For this purpose, let ρ ∈ C1 (ℝ) denote a fixed mollifier with supp ρ ⊂ [−1, 1], ∫ℝ ρ = 1 and 0 ≤ ρ ≤ 1 almost everywhere (a.e.) on ℝ

and let ϵ : ℝ+ → ℝ+ be a function with ϵ(α) > 0 for α > 0 and ϵ(α) α → 0 as α → 0. We consider the Yosida approximation β α (with parameter α > 0) of β := ∂φ(0) (for the general definition of the Yosida approximation, we refer to [6]) and define 1 s ρ(ϵ)(s) := ρ( ), ϵ ϵ

s

β̃ α := β α ∗ ρ ϵ(α) ,

γ α (s) := ∫ β α , 0

φ α (c) := ∫ γ α ∘ c , Ω

where ‘∗’ denotes the usual convolution operator and ‘∘’ represents composition. Here, β̃ α is the mollification of the Yosida approximation β α by a dilation of the convolution kernel ρ. Further, φ α is a regularization of the potential φ(0) which is very similar to the Moreau–Yosida approximation but it enjoys higher regularity. For α n > 0 we write φ(n) := φ α n and A(n) := ∂φ(n) for the ease of notation. Let (α n )n∈ℕ , α n > 0 for all n ∈ ℕ, denote a subsequence of reals with α n → 0. In order to solve the optimization problem (Pψ ), we make use of the fact that a sequence of optimal controls for the problems (Pφ(n) ) converges weakly in (H 1 (W1 ) × U (1) × U (2) )M+1 to a solution of (Pψ ) with even strong convergence in the first component; for a proof in the boundary control case, we refer to [18, Theorem 3.6]. For the numerical solution of (Pψ ), we therefore fix such a sequence (α n )n∈ℕ and solve the sequence of problems (Pφ(n) )n∈ℕ . In this chapter, each (Pφ(n) ) is solved by a steepest descent method, which is initialized by an (approximate) stationary point (n−1) of (Pφ ) for n ≥ 1. The entire process is initialized by picking some u (0) ∈ (U (1) × U (2) )M+1 .

48 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system

3.3.2 Steepest descent method with expansive line search ̂ Algorithmically, the steepest descent direction for the reduced objective J(u)| u=u(n,k) , where u (n,k) , k ∈ ℕ, denotes an actual approximation of u (n) , a stationary point of (Pφ(n) ) is computed as follows: Given u (n,k) ∈ (U (1) ×U (2) )M+1 one solves the primal system (2.1)–(2.6), which yields (c(n,k+1) , w(n,k+1) , 𝑣(n,k+1) ). With this information at hand, the adjoint system (2.9)–(2.13) is solved yielding (p(n,k+1) , q(n,k+1) ). Upon applying the Riesz-representation theorem [27], the gradient-based descent direction is given by d(n,k) = −J −1 J󸀠̂ (u (n,k) ) = (η(1) u (1) − z(1) , η(2) u (2) − z(2) ) with z(1) and z(2) as given in (2.12) and (2.13) and J denoting the duality mapping of [(U (1) × U (2) )]M+1 . In order to avoid the implementation of the H 1/2 (Ω; ℝN ) norm, we represent functions in U as their norm-minimal continuation into the space H 1σ (Ω), which we equip with the norm induced by H 1 (Ω; ℝN ). Then equations (2.12) and (2.13) (Ω; ℝN ) for every t i , i = for z(1) and z(2) , respectively, involve the applications of J −1 H1 1, . . ., M. Since the norm of H 1 (Ω; ℝN ) involves gradient terms, the computation of J −1 (Ω; ℝN ) requires to solve a Dirichlet problem. In fact, for u (2) ∈ H 1/2 (Ω; ℝN ), one H1 needs to solve the following problem û (2) = argmin {‖u 󸀠 ‖H 1 (Ω;ℝN ) : u 󸀠 ∈ U (1) , Tr u 󸀠 = u (2) } . Then, û (2) represents the norm-minimal extension of u (2) to H 1σ (Ω). The steepest descent direction d(n,k) is then used in an Armijo-type line search with expansion (see Algorithm 1 in the appendix) in order to find a suitable step length s(n,k) > 0 along d(n,k). The next control iterate is then given by u (n,k+1) := u (n,k) + s(n,k) d(n,k) and this process is repeated. We expand the standard Armijo line search (see, e.g., [8]) in order to reduce the number of evaluations of the reduced objective since each evaluation corresponds to solving the entire primal system. For this purpose, we initialize the line search parameter s by the value s(n,k−1) computed in the previous application of the line search algorithm. Then, the line search algorithm tests and, if necessary, replaces s by w c s for a constant factor w c < 1 until the Armijo rule is fulfilled or s becomes smaller than the lower limit smin > 0. In the latter case, the line search algorithm stops with a failure. In order to also allow increases of the step size, the algorithm checks whether the Armijo rule is satisfied for w e s, with some fixed w e > 1, in case the last z z applications of the line search procedure terminated successfully with the respective initial step size. This expansion is applied for at most z m consecutive steps. We continue the gradient descent method as long as the decrease of the values of the objective is sufficiently large or there are still sufficient large changes in the primal variables. More precisely, if in a number z s of consecutive gradient steps the new value J of the objective does not drop below (1 − θ n )Jold with Jold denoting the value of

3.4 Finite element approximation

|

49

the objective after the application of the previous descent step, and the changes ‖c − cold ‖L2 and ‖𝑣 − 𝑣old ‖L2 are smaller than ϑ n ‖cold ‖ and ϑ n ‖𝑣old ‖, respectively, then we stop the gradient descent algorithm and proceed to the next Yosida parameter α n+1 . Here, ϑ n , θ n > 0 denote parameters which depend on n.

3.3.3 Newton’s method for the primal system The primal system is solved iteratively forward in time by using Newton’s method. The latter converges at a local superlinear convergence rate. The adjoint system is linear and can be solved directly and iteratively backward in time. We note that the arguments given in [18] provide sufficient conditions for the existence of solutions to the respective system. Next, we exemplarily setup the primal system as it is solved by Newton’s method. ̃ , 𝑣̃) = (c i+1 , w i+1 , 𝑣i+1 ) Given (c i , w i , 𝑣i ) ∈ H 1 (Ω)×H 1 (Ω)×H 1σ (Ω) at t = t i , the tuple (̃c , w is computed such that ̃ , 𝑣̃) = 0 . F(̃c , w (3.1) For all (z c , z w , z𝑣 ) ∈ H 1 (Ω) × H 1 (Ω) × H 1σ (Ω) with ∫Ω z c = ∫Ω z w = 0, z𝑣 |∂Ω = 0, the variational form of (3.1) reads 1 ̃ | ∇z c ) (̃c − c i + τ∇c i ⋅ 𝑣̃ | z c ) + τ Pe (∇w 󸀠 ̃ , 𝑣̃), (z c , z w , z𝑣 )⟩ = ( ̃ − c i + ψ (̃c ) | z w ) + ε(∇̃c | ∇z w ) ⟨F(̃c , w ) (−w 1 ̃ ∇c i ) | z𝑣 ) + Re (̃ 𝑣 − 𝑣i + (𝑣i ⋅ ∇)̃ 𝑣 − Kw (∇̃ 𝑣 | ∇z𝑣 )

̃(0) , 𝑣̃(0) ) ∈ H 1 (Ω) × H 1 (Ω) × H 1σ (Ω) and with (f | g) := ∫Ω fg. Then, given some (̃c(0) , w setting l := 0, the Newton iterations take the form ̃(l+1) , 𝑣̃(l+1) ) = (̃c(l) , w ̃ (l) , 𝑣̃(l) ) + d(l) , (̃c(l+1) , w ̃ (l) , 𝑣̃(l) )d, z⟩ = −⟨F(̃c(l) , w ̃ (l) , 𝑣̃(l) ), z⟩ for all z = (z c , z w , z𝑣 ) ∈ for d satisfying ⟨DF(̃c(l) , w 1 1 1 H (Ω) × H (Ω) × H σ (Ω) with ∫Ω z c = ∫Ω z w = 0, z𝑣 |∂Ω = 0.

3.4 Finite element approximation For the numerical realization of the aforementioned gradient descent procedure, the systems (2.1)–(2.6) and (2.9)–(2.13) are discretized in space. For this purpose, let T denote a shape regular simplicial triangulation of Ω with Ω = ⋃ T∈T T. The corresponding set of faces of the elements in T is denoted by E. We consider the following finite element spaces: W T := {w ∈ C0 (Ω) | w|T ∈ P1 (T) T

∞

V := {𝑣 ∈ L (Ω) | 𝑣|T ∈ P1 (T),

∀T ∈ T} , 𝑣|T1 (E m ) = 𝑣|T2 (E m )

∀T, T1 , T2 ∈ T, E = T1 ∩ T2 ∈ E} ,

CE.1 Please define LBB.

50 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system where E m denotes the midpoint of a face E ∈ E and P1 (T) is the space of affine functions on T. We use the space W T of P1 -finite elements for the discretization of c, w, and p and the LBB CE.1 -stable Crouzeix–Raviart element space V T (see [11]) for 𝑣 and q. The spatial discretization of (2.1)–(2.6) leads to finding (c, w, 𝑣) ∈ (W T × W T × (2) T M+1 such that ∫Ω c = ∫Ω w = 0, div 𝑣 = 0, 𝑣i+1 |∂Ω = η(2) u i+1 , c0 = c a , 𝑣0 = 𝑣a V ) and 1 1 (∇w i+1 | ∇z c ) = 0 , ( (c i+1 − c i ) + ∇c i ⋅ 𝑣i+1 | z c ) + τ Pe (−w i+1 − c i + πT ∂ψ(c i+1 ) | z w ) + ε(∇c i+1 | ∇z w ) = 0 , 1 1 (∇𝑣i+1 | ∇z𝑣 ) = η(1) (u (1) | z𝑣 ) ( (𝑣i+1 − 𝑣i ) + (𝑣i ⋅ ∇)𝑣i+1 − Kw i+1 ∇c i | z𝑣 ) + τ Re for all (z c , z w , z𝑣 ) ∈ (W T × W T × V T )M with div z𝑣 = 0 and ∫Ω z c = ∫Ω z w = 0.

Here, πT : C(Ω) → W T denotes the Lagrange interpolation operator [13]. Differential operators on vector fields in V T are understood in the element-wise sense, i.e., (div 𝑣)|T = div(𝑣|T ) a.e. and for all T ∈ T, for instance. For the adjoint system, we proceed similarly and use the spatially discretized version 1 (p i+1 − p i ) + K∇c i+1 ⋅ q i+1 − div(p i+1 𝑣i+2 ) τ 1 ∇p i+1 | ∇z p ) +K div(w i+2 q i+1 ) | z c ) + (ε∇r i − Pe +(r i | DA(c i+1 )z p ) = (λ i (c i − c e i ) | z p ) , (−

(−r i − K∇c i ⋅ q i | z r ) +

1 (∇p i | ∇z r ) = 0 , Pe

1 (− (q i+1 − q i ) + b 1 (𝑣i+2 , q i+1 ) + b 2 (𝑣i , q i ) + p i ∇c i | z q ) τ 1 + (∇q i | ∇z q ) = 0 , Re ∫ p = 0,

∫ r = 0,

Ω

Ω

div q = 0 ,

I∗ q = z1 , −J −1 U (1) U (1) →V −1 1 1 ∆q i + b 1 (𝑣i+2 , q i+1 ) + b 2 (𝑣i , q i ) + p i ∇c i ] = z2 . H [− (q i+1 − q i ) − τ Re We emphasize that the above discretization yields the exact gradient of the optimal control problem for the spatially discretized semidiscrete CH–NS system. As mentioned above, we avoid the implementation of the H 1/2 (Ω; ℝN ) norm by representing functions in U as their norm-minimal continuation into the space H 1σ (Ω), where the latter space is equipped with the induced norm of H 1 (Ω; ℝN ). This extension operator will be denoted by F : U → H 1σ (Ω). Moreover, we reformulate (2.3) by

3.5 Numerical results |

51

replacing 𝑣 by 𝑣̃ + Fu and obtain 1 1 𝑣i + Fu i ⋅ ∇)̃ 𝑣i+1 − Kw i+1 ∇c i + ∇π (̃ 𝑣i+1 − 𝑣̃i ) − ∆̃ 𝑣i+1 + (̃ τ Re 1 1 𝑣i + Fu i ⋅ ∇)Fu i+1 ] , ∆u i+1 + (̃ = η(1) u (1) − [F (u i+1 − u i ) − τ Re

(4.1)

where 𝑣̃ satisfies homogeneous Dirichlet boundary conditions. In order to solve a single time step of either the adjoint system or the linearized primal system, where the latter corresponds to computing one Newton step, the solution to a linear problem of the form zT M 1 x = zT f

∀z : M2 z = 0,

M2 x = 0

(4.2)

has to be calculated. Here, x = (c i , w i , 𝑣̃i ) with 𝑣̃i = 𝑣i − Fu i or x = (p i , r i , q i ), respectively. M1 denotes the system matrix and f the right-hand side incorporating values of the state or the adjoint at former times. The matrix M2 corresponds to the discretization of the mapping (z c , z w , z𝑣 ) 󳨃→ (∫Ω z c , ∫Ω z w , div z𝑣 , z𝑣 |∂Ω ), which assembles the mean-value conditions on c, w and p, r, respectively, the incompressibility of 𝑣, respectively q, the boundary conditions on 𝑣̃, respectively q, as well as on the corresponding test functions. In order to solve (4.2), note that this problem can be rewritten as a saddle point system, which we solve in exchange (

M1 M2

M2T z f )( ) = ( ) . 0 λ 0

(4.3)

Our implementation uses MATLAB and a direct solver to compute the solution to (4.3). This equation corresponds to solving one time step of the dual system (2.9)–(2.13) or to solving the linearization of the primal system (2.1)–(2.3) in Newton’s method. Of course, a suitable preconditioned iterative solver using the Krylov-subspace method could be applied as well to solve (4.3). In order to reduce the computational effort for the direct solver, we use the null space method (see, e.g., [24]). For this purpose, note that in two dimensions a spanning system for all divergence-free vector fields 𝑣 ∈ V T is given by the curls of the nodal basis functions joined by the set 𝑣E ∈ V T for E ∈ E, where 𝑣E (E m ) is tangential to E and of unit length, and it satisfies 𝑣E (E󸀠m ) = 0 for E ≠ E󸀠 ∈ E (cf. [28]).

3.5 Numerical results This section reports on the numerical results which we obtained from our algorithm for solving several test problems including boundary or distributed controls. Here, as well as in many applications, one is interested in tracking a desired concentration profile c∗M at final time. Our objective in (2.7), however, includes a desired trajectory, where each L2 -distance to c∗i is weighted by λ i ≥ 0. Clearly, for properly chosen (c∗i )M i=0

52 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system and λ i > 0 this may help the control action to reach c M close to c∗M . In (2.7) the time instances, where a desired concentration is prescribed, coincide with (t i )M i=0 . Of course, it is conceivable to select desired concentration profiles at different times. In our numerical tests, we select c∗i = c∗M for all i = 0, . . ., M − 1 as well as a se(n) quence ((λ i )M−1 i=0 )n∈ℕ which decreases in each component as the Yosida parameter α n tends to 0. We first present two examples of boundary control and then test cases for distributed control. The ability to steer the phase concentration in the semidiscrete CH– NS system to a desired profile at final time is more challenging (if possible at all) for boundary control when compared to distributed control actions. Topological changes resulting from boundary control, like the splitting of regions of pure phases, seem possible only if the given phase configuration is energetically highly unfavorable. On the other hand, for the distributed control, a wide range of target profiles including many relevant ones for applications, are reachable. Examples for such phase patterns can be observed in Sections 3.5.3 and 3.5.4. Unless otherwise specified, the following parameters are used: We use T = 0.2 for the time horizon, and the time interval [0, T] is divided into M = 30 subintervals. We choose Ω = (0, 1)2 as a domain and present the first results of the optimal control of the CH–NS system using a uniform triangulation of mesh width h = 1/128. An adaptive finite-element implementation based on a residual type a posteriori error estimator is currently under investigation. The gradient descent method is performed for a fixed number nmax of different Yosida parameters. For the examples of Sections 3.5.1, 3.5.2, and 3.5.4, we use nmax = 5, and for that of Section 3.5.3, we use nmax = 4. As an initial Yosida parameter α1 , we choose α 1 = 10−3 for the examples of Sections 3.5.1, 3.5.2, and 3.5.3 and α1 = 10−2 in n−1 −6 Section 3.5.4. The other values of α n are given by α 1 ( 10α1 ) nmax −1 for n = 2, . . ., nmax . Then, α nmax = 10−6 and the values of α form a geometric sequence. For the constants entering the semidiscrete CH–NS system we set Pe = 10, K = 2, ε = 6 ⋅ 10−4 , and Re = 200. Furthermore, for the extended Armijo line search algorithm (Algorithm 1 in the appendix), we use the parameters w c = 1/3, w e = 2, z z = 3, z m = 4 and ν = 10−2 for boundary control and ν = 10−3 for distributed control, respectively. As a criterion for stopping with failure we set smin = 10−7 . Finally, we stop our Newton iteration as soon as the residual drops below 10−9 in the respective norm. For stopping the n−1 gradient descent method, we choose the parameters z s = 7, θ n = 10−4 × 0.1 nmax −1 and n−1 ϑ n = 10−3 × 0.1 nmax −1 for n = 1, . . ., nmax . Then, θ nmax = 10−5 , ϑ nmax = 10−4 and θ and ϑ form geometric sequences.

3.5 Numerical results |

53

3.5.1 Disk to a ring segment Our first example of boundary control is the deformation of a disk into a sector of a ring (i.e., we start with a configuration where we have the pure phase c = 1 in the given disk and c = −1 outside). Figure 1 shows the evolution of the concentration under the applied control, where the initial data containing pure phases only is included in the upper left corner. The desired profile is depicted in the bottom right corner of Figure 1. The action of the control on the boundary is indicated by the vector field on the boundary of the respective box. The length of an arrow indicates the intensity of the control action, and the direction of the control field is given by the orientation of the arrow. Moreover, interfacial regions are displayed in green color. The aim of this example was the deformation of a stable disk-shaped region into some region which is neither circular nor convex anymore. The figure shows that the algorithm was capable of finding a numerical locally optimal control inducing a phase concentration which is rather close to the desired profile. This is apparently achieved by first shifting the disk to the target location and then stretching it by pulling the ends apart (outflow at the lower corners) as well as pushing the middle part upward due to an inflow from the middle portion of the bottom boundary. Note that the final configuration is unstable, in the sense that the ring segment would again deform into a circular region if the system could evolve naturally, i.e., without a control action. Table 1 provides the iteration numbers for each run with different parameter settings (Yosida parameter and coefficients in the cost functional). It includes the Yosida parameter, the number of gradient steps in the corresponding setting, the average number of line search steps per gradient iterations, and the average number of Newton steps used in order to solve the primal system in a single time step. Table 1: Iteration numbers for the example “disk to a ring segment,” cf. Figure 1. Yosida parameter

Grad. steps

Line search steps

Newton steps

⌀ Newton/t

1.00000e−03 1.77828e−04 3.16228e−05 5.62341e−06 1.00000e−06

119 27 23 14 42

185 56 38 22 85

32 615 4964 3209 2663 17 320

6.08 3.06 2.91 4.17 7.03

The results show that the number of line search steps per gradient step is around 2 and that the largest average number of Newton steps per time step occurs in the first and the last time step and lies between 6 and 7. The related values of the cost functional are depicted in Figure 2. It can be observed that the graph is piecewise continuous with each piece belonging to one choice of the Yosida parameter. One further observes that each branch is monotonically decreasing over the iterations (horizontal axis). More-

54 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system

t=0

t=4

t=8

t = 12

t = 16

t = 20

t = 24

t = 28

c∗M

Fig. 1: Evolution of the concentration with boundary control to deform a disk into a ring segment (row-wise and from left to right). First and last image are the initial and the desired concentrations, respectively.

over, the (final) values of the cost functional increase from one branch to the next. This indicates that approximating the original problem and the involved variational inequality makes it more difficult for the control action to steer the system toward the desired state. The rather significant increase of the objective value at the beginning of the branch starting at around iteration 190 can be attributed to the fact that the numerical solution for the previous α-value does not provide a sufficiently good initial value for the Yosida parameter associated with the branch starting at approximately iteration 190.

3.5.2 Ring to disks The second example consists of deforming a ring region into four separate disks by using optimal boundary control. The evolution of the phase field and the corresponding control obtained by our algorithm is shown in Figure 3. The control appears to be al-

3.5 Numerical results |

55

Fig. 2: Values of the cost functional after each gradient step (e.g., Section 3.5.1). Each branch corresponds to a specific Yosida parameter α n .

most constant in time and with an inflow perpendicular to the boundary at the middle of the faces of Ω and an outflow at the corners. Moreover, the control action drives the concentration very close to the desired profile, which is depicted in the bottom right plot of Figure 3. The resulting disk-like regions deviate only slightly from the desired shapes and show the expected diffuse interface indicated in green. Note that the starting configuration possesses a large surface area compared to the area of the region itself and is therefore energetically unfavorable. This instability is the main reason for the success of the control action in this example. Once initiated by the control, the phase separation process helps in splitting up the ring and drives the evolution forward to the desired disks. The qualitative behavior of the values of the cost functional is similar to the one in the first test example. The iteration numbers of the algorithm given in Table 2 show that the number of gradient steps decreases with decreasing Yosida parameter. Moreover, the average numbers of Newton iterations for solving the primal systems in the individual time steps, first decrease with decreasing Yosida parameter. But for our smallest choice of the Yosida parameter, we notice a significant increase in Newton steps. This behavior can be attributed to the fact that the limit problem is nonsmooth, which causes an increase in the curvature of the mollified Yosida approximation, thus reducing the radius for fast local convergence of Newton’s method.

56 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system

t=0

t=4

t=8

t = 12

t = 16

t = 20

t = 24

c∗M

t = 28

Fig. 3: Evolution of the concentration with boundary control in order to transform a ring (first image) into four disks.

Table 2: Iteration numbers for the example “Ring to disks,” cf. Figure 3. Yosida parameter

Grad. steps

Line search steps

Newton steps

ø Newton/t

1.00000e−03 1.77828e−04 3.16228e−05 5.62341e−06 1.00000e−06

18 16 11 9 7

35 34 27 15 13

5626 3810 2838 2130 3788

5.54 3.86 3.62 4.90 10.05

3.5 Numerical results |

57

3.5.3 Grid pattern of disks Our first example involving distributed control concerns the deformation of a diskshaped region into a grid pattern of smaller disks. The evolution of the concentration (phase field) and the applied control are given in Figure 4. In order to depict the force field given by u in the domain Ω, we use a representation where different colors indicate different directions. The correspondence between a color and a direction is shown in Figure 5. Moreover, the modulus (length) of the vector field is encoded in the intensity of the colors where intense colors belong to large moduli and pale colors to small moduli, respectively. In Figure 4, it can be seen that the control enjoys a rich spatial structure that changes over time and which drives the phase field quite close to the desired state. But since the circular regions of the target profile are very close to each other, the control found by the algorithm is not capable of completely separating the different in-

t=0

t = 20

t=6

t = 24

t = 12

t = 28

t = 16

c∗M

Fig. 4: Evolution of the concentration (first and third row) and the corresponding distributed control (second and forth row) to reach a grid of smaller disks from a large disk. The direction of the vector field of the control is represented in a color scheme explained in Figure 5.

58 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system

Fig. 5: Correspondence between colors and directions (left) and the valued of the cost functional after each gradient step of example 3.5.3 (right).

terfaces from each other and a blending of regions occurs. Nevertheless, the results of this rather challenging example show the potential of distributed control once the technical realization of the control action can be guaranteed. Table 3 provides the iteration numbers for this example. Here, the average number of line search steps per gradient step is below 3 and the average number of Newton iterations used in order to solve one time step of the primal system does not exceed 4.4. Table 3: Iteration numbers for the example "Grid pattern of disks", cf. Figure 4. Yosida-parameter

grad. steps

line search steps

Newton steps

ø Newton/t

1.00000e−03 1.00000e−04 1.00000e−05 1.00000e−06

341 39 42 24

608 87 104 63

77 428 6807 8670 6402

4.39 2.70 2.87 3.50

The right plot in Figure 5 presents the evolution of the objective values along the iterations. In the branch belonging to the initial Yosida parameter a seesaw between iterations with steep descent and iterations with only small descent can be observed. For all of the other parameters, the evolution of the cost functional is almost flat indicating that the numerically optimal control found for large Yosida parameters is numerically almost optimal for smaller parameters as well.

3.5 Numerical results |

59

3.5.4 Grid pattern of finger-like regions A second example of optimal distributed control of the semidiscrete CH–NS system is shown in Figure 6. Here, the aim is to deform the initial profile of the concentration – again a disk-shaped region – into a pattern of horizontally aligned finger-like regions. The figure shows that the force field given by the control pushes the inner fluid (red) outside along the desired finger pattern, whereas the outer fluid (blue) is pushed vertically to the middle along the desired vacancies between the fingers. This behavior is exactly as one would expect from the optimal control in order to approach the desired profile: distribute both fluids spatially with minimal effort and then extend the regions occupied by both fluids in those directions that match the desired profile. In this example, apart from the inevitable formation of an interfacial region the final concentration almost perfectly fits the desired profile. Since the finger-like regions of the given target are sufficiently separated from each other blending of interfacial regions, as it occurred in the previous example, does not occur.

t=0

t = 20

t=6

t = 24

t = 12

t = 28

t = 16

c∗M

Fig. 6: Evolution of the concentration (first and third row) and the corresponding distributed control (second and forth row) to obtain finger-like regions out a disk.

60 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system

Table 4 provides the information on the iterations for this example. Table 4: Iteration numbers for the example “Grid pattern of finger-like regions,” cf. Figure 6. Yosida parameter

Grad. steps

Line searches

Newton steps

ø Newton/t

1.00000e−02 1.00000e−03 1.00000e−04 1.00000e−05 1.00000e−06

701 19 53 20 11

961 27 110 36 16

158 925 3235 9829 3565 2841

5.70 4.13 3.08 3.41 6.12

3.6 Conclusions This chapter presented optimal control problems for a time-discretized two-phase flow model. The governing state equations consisted of a coupled CH–NS system and we considered boundary as well as distributed control of the NS part. First-order optimality conditions were given. An algorithmic scheme was presented which is based on a sequence of Yosida-type approximations of the original problem linked via a path-following method. Upon spatial discretization, the individual Yosida regularized problems were solved using a gradient descent method and an extended Armijo line search. We concluded our discussion with reports on the behavior of our algorithm for several numerical test cases. Acknowledgment: The authors gratefully acknowledge support by DFG Research Center MATHEON under project C28 “Optimal control of phase separation phenomena” and by DFG SPP 1506 “Transport Processes at Fluidic Interfaces.” M. H. further acknowledges support through FWF under START-program Y305-N18 “Interfaces and Free Boundaries.”

Bibliography |

61

Algorithm 1: Armijo line search with expansion. Input: d(n,k)(= −J −1 J󸀠̂ (u (n,k) )), s(n,k−1) , z ∈ ℕ. Parameters: ν ∈ (0, 1), w c ∈ (0, 1), w e > 1, z z , z m ∈ ℕ, smin > 0. step 0. step 1.

step 2.

step 3.

step 4. step 5.

step 6.

Set s(n,k,l) := s(n,k−1) , z(l) := z, z e := 0. If ̂ (n,k) +s(n,k,l) d(n,k) ) ≤ J(u ̂ (n,k) )+νs(n,k,l)⟨J󸀠̂ (u (n,k) ), d(n,k) ⟩, then J(u if z e = 0 then continue with step 2. else continue with step 5. end else if z e = 0 then continue with step 3. else continue with step 6. end end Set z(l) := z(l) + 1. if z(l) ≥ z z then continue with step 4. else stop with s(n,k) := s(n,k,l) , z := z(l) end Set s(n,k,l+1) := w c s(n,k,l) , l := l + 1, z(l) := 0. if s(n,k,l) < s min then stop with failure, else return to step 1. end Set s(n,k,l+1) := w e s(n,k,l), l := l + 1, z(l) := 0, z e := z e + 1 and return to step 1. If z e < z m then return to step 4 else stop with s(n,k) := s(n,k,l) , z := z(l) end (n,k,l) , z := z(l) . Stop with s(n,k) := w−1 a s

Bibliography [1]

[2] [3]

S. Aland, S. Boden, A. Hahn, F. Klingbeil, M. Weismann and S. Weller, Quantitative comparison of Taylor flow simulations based on sharp-interface and diffuse-interface models, International Journal for Numerical Methods in Fluids, 73(4):344–361, 2013. S. Aland, J. Lowengrub and A. Voigt, Particles at fluid-fluid interfaces: A new Navier–Stokes– Cahn–Hilliard surface-phase-field-crystal model, Phys. Rev. E, 86:046321, 2012. S. Aland, S. Schwarz, J. Fröhlich and A. Voigt, Modeling and numerical approximations for bubbles in liquid metal, The European Physical Journal Special Topics, 220(1):185–194, 2013.

62 | 3 Distributed and boundary control problems for the Cahn–Hilliard/Navier–Stokes system

[4]

[5]

[6] [7] [8] [9] [10] [11] [12]

[13] [14]

[15]

[16]

[17]

[18] [19]

[20] [21] [22]

S. Aland and A. Voigt, Benchmark computations of diffuse interface models for twodimensional bubble dynamics, International Journal for Numerical Methods in Fluids, 69(3):747–761, 2012. S. Aland and A. Voigt, Simulation of common features and differences of surfactant-based and solid-stabilized emulsions, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 413(0):298–302, 2012. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest, 1976. V. Barbu, Optimal Control of Variational Inequalities, volume 100 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1984. J. F. Bonnans, J. C. Gilbert, C. Lemaréchal and C. A. Sagastizábal, Numerical Optimization. Theoretical and Practical Aspects. Transl. from the French, Universitext. Berlin: Springer, 2003. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J Chem. Phys., (2):258–266, 1958. H. Choi, M. Hinze and K. Kunisch, Instantaneous control of backward-facing step flows, Appl. Numer. Math., 31(2):133–158, 1999. M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I., 1974. S. Eckert, P. Nikrityuk, B. Willers, D. Räbiger, N. Shevchenko, H. Neumann-Heyme, V. Travnikov, S. Odenbach, A. Voigt and K. Eckert, Electromagnetic melt flow control during solidification of metallic alloys, The European Physical Journal Special Topics, 220(1):123–137, 2013. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences 159. New York, Springer, 2004. H. Farshbaf-Shaker, A penalty approach to optimal control of Allen–Cahn variational inequalities: MPEC-view, Preprint 06, Department of Mathematics, University of Regensburg, Germany, 2011. R. Glowinski and A. M. Ramos, A numerical approach to the Neumann control of the Cahn– Hilliard equation, In Computational Methods for Control and Applications. Gakuto International Series: Mathematical Sciences and Applications, Vol. 16, pp. 111–155, Tokyo, 2002. Gakkotosho Co. M. Hintermüller, M. Hinze and C. Kahle, An adaptive finite element Moreau–Yosida-based solver for a coupled Cahn–Hilliard/Navier–Stokes system, Journal of Computational Physics, 235(0):810–827, 2013. M. Hintermüller and D. Wegner, Distributed optimal control of the Cahn–Hilliard system including the case of a double-obstacle homogeneous free energy density, SIAM J. Control Optim., 50:388–418, 2013. M. Hintermüller and D. Wegner, Optimal control of a semidiscrete Cahn–Hilliard–Navier–Stokes system, SIAM J. Control Optim., 52(1):747–772, 2014. M. Hinze and C. Kahle, A nonlinear model predictive concept for control of two-phase flows governed by the Cahn–Hilliard Navier–Stokes system, D. Hömberg (ed.) et al., System modeling and optimization. 25th IFIP TC 7 conference on system modeling and optimization, Berlin, Germany, September 12–16, 2011. IFIP Advances in Information and Communication Technology 391, 348–357 (2013), 2013. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49:435–479, 1977. K. Ito and K. Kunisch, Receding horizon optimal control for infinite dimensional systems, ESAIM, Control Optim. Calc. Var., 8:741–760, 2002. J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for Cahn–Hilliard fluids, Journal of Computational Physics, 193(2):511–543, 2004.

Bibliography

| 63

[23] J. Kim and J. Lowengrub, Interfaces and multicomponent fluids, In J-P. Françoise, G. L. Naber and T. S. Tsun, editors, Encyclopedia of Mathematical Physics, pp. 135–144. Academic Press, Oxford, 2006. [24] J. Nocedal and S. J. Wright, Numerical optimization. 2nd edn., Springer Series in Operations Research and Financial Engineering. New York, NY: Springer, 2006. [25] Y. Oono and S. Puri, Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling, Phys. Rev. A, 38(1):434–453, 1988. [26] S. Praetorius and A. Voigt, A phase field crystal model for colloidal suspensions with hydrodynamic interactions, arXiv:1310.5495, 2013. [27] W. Rudin, Functional Analysis. 2nd edn., McGraw–Hill, New York, 1991. [28] I. A. Segal, Finite element methods for the incomressible Navier–Stokes equations, Delft University of Technology, 2012. [29] Q.-F. Wang and S. Nakagiri, Weak solutions of Cahn–Hilliard equations having forcing terms and optimal control problems, Research Institute for Mathematical Sciences, Kokyuroku, 1128:172–180, 2000. [30] J. Yong and S. Zheng, Feedback stabilization and optimal control for the Cahn–Hilliard equation, Nonlin. Analysis. Theory, Meth. & Appls., 17(5):431–444, 1991. [31] X. Zhao and C. Liu, Optimal control problem for viscous Cahn–Hilliard equation, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 74(17):6348–6357, 2011. [32] X. Zhao and C. Liu, Optimal control of the convective Cahn–Hilliard equation, Appl. Anal., 92(5):1028–1045, 2013. [33] B. Zhou, Massachusetts Institute of Technology. Dept. of Materials Science, and Engineering, Simulations of Polymeric Membrane Formation in 2D and 3D, Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, 2006. [34] J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optimization, 5:49–62, 1979.

Victor A. Kovtunenko

4 High-order topological expansions for Helmholtz problems in 2D Abstract: Methods of topological analysis are inherently related to singular perturbations. For topology variation, a trial geometric object put in a test domain is examined by reducing the object size from a finite to an infinitesimal one. Based on the singular perturbation of the forward Helmholtz problem, a topology optimization approach, which is a direct one, is described for the inverse problem of object identification from boundary measurements. Relying on the 2d setting in a bounded domain, the high-order asymptotic result is proved rigorously for the Neumann, Dirichlet, and Robin-type conditions stated at the object boundary. In particular, this implies the first-order asymptotic term called a topological derivative. For identifying arbitrary test objects, a variable parameter of the surface impedance is successful. The necessary optimality condition of minimum of the objective function with respect to trial geometric variables is discussed and realized for finding the center of the test object. Keywords: Inverse problem, object identification, shape and topology optimization, topological derivative, derivative-free optimality condition, singular perturbation, asymptotic analysis, variational method, Helmholtz problem AMS Classification: 35R30, 35Q93, 49Q10, 65K10

4.1 Introduction We consider both the forward and inverse Helmholtz problems with respect to a variable geometric object put in a bounded test domain. It has numerous applications for testing methodologies by scattering with acoustic, elastic, and electromagnetic waves. The literature on this subject is numerous, so we give selected references only. The classic methods of analysis available for the Helmholtz problem, see [16, 20, 22, 39, 58], are based mostly on the potential operator theory which is well established in the case of unbounded domains. Possible approaches to inverse problems can be distinguished to iterative, see [23, 35], as well as noniterative ones called direct. Within direct approaches to the inverse scattering, there are well known sampling, probe, factorization, singular source, MUSIC type, and other relevant techniques, e.g., [2, 4, 27, 32, 50]. Victor A. Kovtunenko, Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstraße 36, 8010 Graz, Austria; and, Lavrent’ev Institute of Hydrodynamics, Sibirian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russia, [email protected]

4.1 Introduction

| 65

The inverse Helmholtz problem of identification of an unknown geometric object belongs to the field of shape and topology optimization [1, 24, 25, 34, 42, 44, 55], as well as parameter estimation; see [10, 15, 40, 48] for the common methods here. Recently, the concept of topological derivatives was adapted to this field, e.g., in [6, 11, 56]. The reason is that a trial object of finite size in comparison to infinitesimal one implies variation of topology of the test domain. For the analysis of topological changes we refer to the methods of singular perturbations in [28, 33, 49]. The high-order topological expansions were considered, e.g., in [5, 13, 17, 18, 30] for the Poisson equation, and in [47] for a screened Poisson equation. They were the subject of discussion in [12, 54]. Regarding the inverse problems, in [17, 18] the Kohn– Vogelius cost functional over a domain was suggested as alternative to the boundary misfit functional. Numerically, the high-order terms usually have a high computational cost because they require to solve a partial differential equation (PDE) at every point where the topological derivative is computed. To derive even formally asymptotic representations under singular perturbations is itself a hard task requiring a huge number of fine calculations. Its rigorous mathematical justification is the challenging problem. In this chapter, we obtain the highorder asymptotic formula for solutions of the singularly perturbed state problems and the corresponding optimal value functions. The first-order term implying a topological derivative is the particular case here. All subsequent asymptotic expansions are proved by rigorous estimates of the residual error in appropriate function spaces. Our consideration addresses the topology optimization problem for the identification and reconstruction of arbitrary geometries. By “arbitrary geometry,” we mean the implicitly defined set of geometric variables which are parameterized by admissible triples of shape–center–size. Such general geometric assumptions can be treated within a proper variational formulation. We provide the underlying singularly perturbed, forward, and inverse, Helmholtz problem with suitable primal and dual variational principles according to the Fenchel– Legendre duality. The asymptotic terms are expressed by auxiliary Helmholtz problems given in bounded domains as well as boundary layers described by the Laplace problems in exterior domains. They all are stated in the weak form in proper Sobolev spaces. In the exterior domains, weighted Sobolev spaces are useful. A crucial issue of the topological analysis concerns the fact that boundary conditions of the test object should be prescribed a-priori. In the present chapter, relying on a 2d spatial setting, in a unified way, we consider the Helmholtz problem under Dirichlet (sound soft), Neumann (sound hard), and Robin (impedance) conditions stated at the object boundary, respectively, in Sections 4.3–4.5. The background problem in the reference domain is described in Section 4.2. From the perspective of topology optimization, the geometric variables enter the objective in a fully implicit way through the geometry-dependent state problem. This fact does not allow us to find optimality conditions based on directional derivatives

66 | 4 High-order topological expansions for Helmholtz problems in 2D

of the objective. Nevertheless, in Section 4.5.4 we show that an unknown parameter of the boundary impedance is well suitable for the purpose of variation of trial geometries. In fact, passing its imaginary part to the limit in the asymptotic expansion of the optimal value function, this gets a necessary optimality condition determining the optimal center, and this condition is derivative free. We start with the description of the reference configuration.

4.2 Background Helmholtz problem Let Ω ⊂ ℝ2 be a reference domain with the Lipschitz boundary ∂Ω. With ν = (ν1 , ν2 )⊤ , the unit normal vector at the boundary ∂Ω and outward to Ω is denoted. Let ∂Ω = Γ N ∪ Γ D consist of two nonempty, mutually disjoint, parts Γ N and Γ D associated to the Neumann and Dirichlet boundary conditions, respectively. For the fixed Neumann and Dirichlet data g ∈ L2 (Γ N ; ℂ) and h ∈ H 1/2 (Γ D ; ℂ), and for a wave number k ∈ ℝ+ , the reference (called background) Helmholtz problem for the wave potential u 0 (x), x = (x1 , x2 )⊤ ∈ Ω, is given by −[∆ + k 2 ]u 0 = 0 in Ω , 0

(2.1a)

=g

on Γ N ,

(2.1b)

u =h

on Γ D .

(2.1c)

∂u ∂ν 0

∂ In (2.1) and in what follows, ∆ is the Laplace operator, the usual notation ∂ν := ν ⋅ ∇ = ⊤ ν ∇ stands for the normal derivative, “⋅” means for the inner product of vectors, ∇ is the gradient, and the upper ⊤ denotes transposition swapping columns and rows. In the weak form, (2.1) is described by the following variational problem. Find u 0 ∈ H 1 (Ω; ℂ) such that

u0 = h ∫(∇u 0 ⋅ ∇u − k 2 u 0 u) dx = ∫ gu dS x Ω

on Γ D ,

(2.2a)

for all u ∈ H 1 (Ω; ℂ) : u = 0 on Γ D .

(2.2b)

ΓN

Here u = Re(u) − ıIm(u) is the complex conjugate of u = Re(u) + ıIm(u), and ı is the imaginary unit. Proposition 2.1. For the strong solution to (2.1), the boundary value problem (2.1) and the variational equation (2.2) are equivalent. There exists the unique weak solution u 0 to (2.2). Moreover, it implies the first-order necessary optimality condition for the Cherkaev–Gibiansky variational principle: P(u 0 ) = min max P(𝑣) Re(𝑣) Im(𝑣)

over 𝑣 ∈ H 1 (Ω; ℂ) : 𝑣 = h on Γ D

(2.3)

4.2 Background Helmholtz problem | 67

with the Lagrangian P : H 1 (Ω; ℂ) 󳨃→ ℝ of the form } { } { (2.4) P(𝑣) = Re { 12 ∫(∇𝑣 ⋅ ∇𝑣 − k 2 𝑣2 ) dx − ∫ g𝑣 dS x } . } { ΓN } { Ω Proof. The weak formulation can be verified by standard variational arguments. Indeed, (2.2) is derived by multiplying equation (2.1a) with a test function u ∈ H 1 (Ω; ℂ) and subsequent integration by parts over Ω due to boundary conditions (2.1b) and u = 0 on Γ D . In return, the following Green’s formula holds for every function u ∈ H 1 (Ω; ℂ) such that ∆u ∈ L2 (Ω; ℂ): ∫(∇u ⋅ ∇u + u ∆u) dx = ⟨ ∂u ∂ν , u⟩ Γ N

for all u ∈ H 1 (Ω; ℂ) : u = 0 on Γ D .

(2.5)

Ω 1/2

Here ⟨ ∂u ∂ν , u⟩ Γ N stands for the duality pairing between u ∈ H00 (Γ N ; ℂ) and 1/2 H00 (Γ N ; ℂ)⋆

∂u ∂ν

∈

in the Lions–Magenes dual space; see, e.g., [36, Section 1.4] for detail. With the help of (2.5), we obtain from (2.2b) − ∫ ([∆ + k 2 ]u 0 ) u dx = ⟨g −

∂u0 ∂ν , u⟩ Γ N ,

⟨g, u⟩Γ N = ∫ gu dS x ΓN

Ω

and derive (2.1a) and (2.1b) by the fundamental lemma of the calculus of variations when varying the test function u such that first u = 0 on Γ N and then u ≠ 0 on Γ N . The unique solvability of the variational equation (2.2) can be argued as usually with a Garding inequality and injectivity by using the Fredholm alternative; see, e.g., [46, Theorem 5.2, Chapter 3]. Now rewriting (2.4) componentwise for 𝑣 = Re(𝑣) + ıIm(𝑣) as P(𝑣) =

1 2

∫{|∇(Re(𝑣))|2 − |∇(Im(𝑣))|2 Ω

− k 2 (Re(𝑣)2 − Im(𝑣)2 )} dx − ∫ (Re(g)Re(𝑣) − Im(g)Im(𝑣)) dS x ΓN

and differentiating P(𝑣) with respect to Re(𝑣) and Im(𝑣), the necessary optimality condition for (2.3) implies two variational inequalities ∂ P (u 0 ) , Re (𝑣 − u 0 )⟩ ≥ 0, ⟨ ∂Re(𝑣)

∂ ⟨ ∂Im(𝑣) P (u 0 ) , Im (𝑣 − u 0 )⟩ ≤ 0

holding for all 𝑣 ∈ H 1 (Ω; ℂ) such that 𝑣 = h on Γ D . Inserting here 𝑣 = u 0 ± u with u ∈ H 1 (Ω; ℂ) such that u = 0 on Γ D , we obtain the following two variational equations: ∫ (∇Re (u 0 ) ⋅ ∇Re(u) − k 2 Re (u 0 ) Re(u)) dx = ∫ Re(g)Re(u) dS x , Ω

ΓN

∫ (∇Im (u 0 ) ⋅ ∇Im(u) − k 2 Im (u 0 ) Im(u)) dx = ∫ Im(g)Im(u) dS x . Ω

ΓN

68 | 4 High-order topological expansions for Helmholtz problems in 2D The summation of these equations for u = u and for u = ıu constitutes, respectively, the real and the imaginary parts of (2.2b), see (5.5a). This completes the proof. In the following section, we give a local representation of the solution to (2.2) in the near field of a trial point, which is called “inner” asymptotic expansion.

4.2.1 Inner asymptotic expansion by Fourier series in near field For a fixed center x0 ∈ Ω, a local polar coordinate system associated to x0 can be introduced with the help of the polar radius ρ ∈ ℝ+ and the polar angle θ ∈ (−π, π] such that x − x0 = ρ ̂x, that is ̂x :=

ρ := |x − x0 |,

x−x 0 |x−x 0 |

= (cos θ, sin θ)⊤ ,

̂x󸀠 = (− sin θ, cos θ)⊤ .

(2.6)

We set R > 0 such that B R (x0 ) ⊂ Ω, where B R (x0 ) denotes the ball of radius R and center x0 . In what follows, Bessel functions of the first kind J n (kρ) and the second kind (called the Neumann functions) Y n (kρ), n ∈ ℕ0 , will be employed for the argument kρ. These functions are two linearly independent solutions of the Bessel equation: (u n )󸀠󸀠ρ + 1ρ (u n )󸀠ρ + (k 2 −

n2 ) un ρ2

= 0 for kρ 󳨃→ u n : ℝ+ 󳨃→ ℝ .

(2.7)

In particular, J0 , J 1 , and Y0 yield the expansions for kρ ↘ +0: ∞

J 0 (kρ) = ∑ m=0

(−1)m (m!)2

(

kρ 2m 2 )

= 1 + a0 (kρ),

J 1 (kρ) = −J 0󸀠 (kρ) = 12 (kρ + a1 (kρ)), Y0 (kρ) =

2 π

(ln

kρ 2

4 a0 (kρ) = − (kρ) 4 + O((kρ) ) , 2

a1 (kρ) = −

+ γ) J0 (kρ) + a2 (kρ),

(kρ)3 8

(2.8a)

+ O((kρ)5 ) ,

(2.8b)

2

(2.8c)

a2 (kρ) = O((kρ) )

with the Euler constant γ > 0. Using (2.6)–(2.8), we prove the following truncated Fourier series. Lemma 2.2. The solution u 0 of (2.2) admits the local asymptotic representation u 0 (x) = u 0 (x0 )J 0 (kρ) + U00 (x)

for x ∈ B R (x0 ) ⊂ Ω

(2.9)

holding in the near field, with the residual U00 ∈ H 1 (B R (x0 ); ℂ) such that π

∫ U00 dθ = 0 ,

(2.10a)

−π

U00 (x0 + ρ ̂x) = O(ρ)

for ρ ∈ [0, R), θ ∈ (−π, π] .

(2.10b)

Proof. In the ball B δ (x0 ) with δ ∈ [0, R), we set π

u 00 (ρ)

:=

1 2π

∫ u 0 dθ −π

and

U00 := u 0 − u 00 ,

(2.11)

4.2 Background Helmholtz problem

| 69

thus decomposing u 0 into the radial u 00 and residual U00 functions: u 0 (x) = u 00 (ρ) + U00 (x)

for x ∈ B δ (x0 ) .

(2.12)

According to (2.11), the residual U00 ∈ H 1 (B R (x0 ); ℂ) and it fulfills (2.10a). Using (2.12) and (2.10a), the substitution of a smooth cut-off function η(ρ) supported in B δ (x0 ) as the test function u = η into (2.2b) and integration by parts gives 0 = ∫ (∇u 0 ⋅ ∇η − k 2 u 0 η) dx B δ (x 0 ) δ

π

π

{∂ } = ∫ { ∂ρ ( ∫ (u 00 + U00 ) dθ) η󸀠 − k 2 ∫ (u 00 + U00 ) dθ η} ρ dρ −π −π 0 { } δ

δ

󸀠

= 2π ∫ ((u 00 )󸀠ρ η󸀠 − k 2 u 00 η) ρ dρ = −2π ∫ ((ρ(u 00 )󸀠ρ )ρ + k 2 ρu 00 ) η dρ 0

0

for all η. This results in the Bessel equation (2.7) for n = 0: (u 00 )󸀠󸀠ρ + 1ρ (u 00 )󸀠ρ + k 2 u 00 = 0 for ρ ∈ (0, δ) .

(2.13)

Its general solution has the form u 00 (ρ) = K00 J 0 (kρ) + S00 Y0 (kρ),

K00 , S00 ∈ ℂ .

But the Neumann function Y0 (kρ) = O(| ln ρ|) in (2.8c) disagrees the inclusion u 00 ∈ H 1 ((0, δ); ℂ) in (2.11), hence, S00 = 0 and u 00 (ρ) = K00 J 0 (kρ),

K00 ∈ ℂ .

(2.14)

To justify (2.10b), we apply the Saint–Venant estimate to the residual U00 in (2.12). Equation (2.13) implies −[∆ + k 2 ]u 00 = 0 which together with (2.1a) yields the Helmholtz equation for the residual − [∆ + k 2 ]U00 = 0

in B δ (x0 ) .

(2.15)

With the help of (2.15) after integration by parts, we have 󵄨 󵄨2 󵄨 󵄨2 I(δ) := ∫ 󵄨󵄨󵄨󵄨∇U00 󵄨󵄨󵄨󵄨 dx = ∫ k 2 󵄨󵄨󵄨󵄨U00 󵄨󵄨󵄨󵄨 dx + B δ (x 0 )

B δ (x 0 )

∫

∂U 00 0 ∂ρ U 0 dS x

.

(2.16)

∂B δ (x 0 )

Thanks to (2.10a), the Poincare and the Wirtinger (cf. (2.29)) inequalities hold: 󵄨 󵄨2 ∫ 󵄨󵄨󵄨󵄨U00 󵄨󵄨󵄨󵄨 dx ≤

(2δ)2 π2

B δ (x 0 ) π

󵄨 󵄨2 ∫ 󵄨󵄨󵄨󵄨∇U00 󵄨󵄨󵄨󵄨 dx ,

π

󵄨󵄨 ∂U 0 󵄨󵄨2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨󵄨U00 󵄨󵄨󵄨󵄨 dθ ≤ ∫ 󵄨󵄨󵄨󵄨 ∂θ0 󵄨󵄨󵄨󵄨 dθ . 󵄨 󵄨

−π

(2.17a)

B δ (x 0 )

−π

(2.17b)

70 | 4 High-order topological expansions for Helmholtz problems in 2D

From (2.17b), we can estimate the boundary integral in (2.16) as π

∫

∂U 00 0 ∂ρ U 0

π

dS x = ∫

∂U 00 0 ∂ρ U 0

−π

∂B δ (x 0 )

󵄨󵄨 ∂U 0 󵄨󵄨2 δ dθ ≤ ∫ ( 2δ 󵄨󵄨󵄨󵄨 ∂ρ0 󵄨󵄨󵄨󵄨 + 󵄨 󵄨

1 2δ

−π

π

󵄨󵄨 ∂U 0 󵄨󵄨2 ≤ ∫ ( 2δ 󵄨󵄨󵄨󵄨 ∂ρ0 󵄨󵄨󵄨󵄨 + 󵄨 󵄨 −π

󵄨 0 󵄨2 1 󵄨󵄨󵄨 ∂U 0 󵄨󵄨󵄨 2δ 󵄨󵄨 ∂θ 󵄨󵄨 ) 󵄨

󵄨

δ dθ =

󵄨󵄨 0 󵄨󵄨2 󵄨󵄨U0 󵄨󵄨 ) δ dθ 󵄨 󵄨 ∫

δ 2

∂B δ (x 0 )

󵄨󵄨 0 󵄨󵄨2 󵄨󵄨∇U0 󵄨󵄨 dS x . 󵄨 󵄨

Therefore, together with (2.17a) and the coarea formula d dδ

󵄨 󵄨2 ∫ 󵄨󵄨󵄨󵄨∇U00 󵄨󵄨󵄨󵄨 dx =

B δ (x 0 )

∫ ∂B δ (x 0 )

󵄨󵄨 0 󵄨󵄨2 󵄨󵄨∇U0 󵄨󵄨 dS x , 󵄨 󵄨

(2.18)

from (2.16), we obtain the differential inequality for I(δ): 2

(1 − k 2 4δ ) I(δ) ≤ π2

δ d 2 dδ I(δ)

.

(2.19)

Integrating (2.19) with respect to δ ∈ (r, R) as R

ln ( I(R) I(r) )

=∫ r

R dI I

≥ ∫ ( 2δ −

8k2 δ) dδ π2

2

= ln ( Rr ) −

4k2 (R2 π2

− r2 )

r

we derive the resulting estimate 4k2 󵄨 󵄨2 2 (R 2 −r 2 ) = O(r2 ) . ∫ 󵄨󵄨󵄨󵄨∇U00 󵄨󵄨󵄨󵄨 dx = I(r) ≤ ( Rr ) I(R)e π2

(2.20)

B r (x 0 )

Due to the fundamental theorem of calculus and using homogeneity argument, the function oscillation in ℝ2 can be estimated from above (see, e.g., [38]) as sup

x,y∈B r (x 0 )

󵄨 󵄨󵄨 0 󵄨2 󵄨2 󵄨󵄨U0 (x) − U00 (y)󵄨󵄨󵄨 ≤ C ∫ (󵄨󵄨󵄨∇U00 󵄨󵄨󵄨 + r2 |∆U00 |2 ) dx 󵄨 󵄨 󵄨 󵄨 B r (x 0 )

󵄨2 󵄨 󵄨2 󵄨 = C ∫ (󵄨󵄨󵄨󵄨∇U00 󵄨󵄨󵄨󵄨 + r2 k 2 󵄨󵄨󵄨󵄨U00 󵄨󵄨󵄨󵄨 ) dx,

(C > 0)

(2.21)

B r (x 0 )

where we have used (2.15). Combining estimates (2.17a) for δ = ρ with (2.20) and (2.21) for r = ρ, where ρ ∈ [0, R), we derive that U00 (x0 + ρ ̂x) = U00 (x0 ) + O(ρ)

in B R (x0 ) .

But U00 (x0 ) = 0 due to (2.10a), thus following (2.10b). Now passing ρ ↘ +0 in (2.12), in view of (2.8a) and (2.10b) from (2.14), we find the factor K00 = u 0 (x0 ), hence (2.9) holds. This completes the proof. We generalize Lemma 2.2 to the high-order inner asymptotic expansion; see the related result in [49].

4.2 Background Helmholtz problem | 71

Proposition 2.3. For every N ∈ ℕ, the solution u 0 of (2.2) admits the following local asymptotic representation in the near-field B R (x0 ) ⊂ Ω: N

u 0 (x) = K00 J 0 (kρ) + ∑ J n (kρ)K 0n ⋅ ̂x n + U N0 (x) ,

(2.22)

n=1

where the notation ̂x n := (cos(nθ), sin(nθ))⊤ with the convention ̂x1 = x̂ for n = 1, and K 0n = ((K 0n )1 , (K 0n )2 ) ∈ ℂ2 . The residual U N0 ∈ H 1 (B R (x0 ); ℂ) is such that π

π

∫ U N0 ̂x n dθ = 0

∫ U N0 dθ = 0, −π

U N0 (x0 + ρ ̂x) = O(ρ

−π N+1

)

for n = 1, . . . , N ,

(2.23a)

for ρ ∈ [0, R), θ ∈ (−π, π] .

(2.23b)

Proof. Starting with u00 and U00 given in (2.9) and (2.10), for every n = 1, . . . , N, we set recursively the radial and residual functions: π

u 0n (ρ)

:=

∫ u 0 ̂x n dθ

1 π

and

0 U n0 := U n−1 − u 0n ⋅ x̂ n ,

(2.24)

−π

which constitute the local asymptotic representation with δ ∈ [0, R): N

u 0 (x) = u 00 (ρ) + ∑ u 0n (ρ) ⋅ x̂ n + U N0 (x)

for x ∈ B δ (x0 ) .

(2.25)

n=1

According to (2.24), all the residuals U n0 , n = 1, . . . , N, fulfill π

π

∫ U n0 dθ = 0,

∫ U n0 ̂x m dθ = 0 for m = 1, . . . , n ,

−π

−π

(2.26)

and (2.26) for n = N implies (2.23a). We show that every radial function u 0n satisfies the respective Bessel equation (2.7). Indeed, for every n = 1, . . . , N, plugging into (2.2b) the test vector function u = n ̂x η(ρ) supported in B δ (x0 ) and the representation u 0 = u 00 + ∑nm=1 u 0m ⋅ x̂ m + U n0 according to (2.25), recalling trigonometric calculus for x̂ n = (̂x1n , ̂x2n )⊤ : π ∂̂x 1n ∂θ

=

−n ̂x2n ,

∂̂x 2n ∂θ

=

n ̂x1n ,

∫ x̂ n dθ = 0,

n, m ∈ ℕ ,

−π π

{π for m = n and i = j ̂n ∫ x̂ m i x j dθ = { 0 otherwise −π {

,

i, j = 1, 2

(2.27)

72 | 4 High-order topological expansions for Helmholtz problems in 2D

and using orthogonality conditions in (2.23a), it succeeds in 0 = ∫ (∇u 0 ⋅ ∇(̂x n η) − k 2 u 0 ̂x n η) dx B δ (x 0 ) δ

=

π

+

n

( ∫ (u 00 + ∑ u 0m ⋅ ̂x m + U n0 ) ̂x n dθ) η󸀠

∂ ∫ { ∂ρ 0

m=1

−π π

η ρ2

n

∫ ( ∑ u 0m ⋅

∂̂x m ∂θ

+

∂U n0 ∂̂x n ∂θ ) ∂θ

dθ

m=1

−π π

n

− k η ∫ (u 00 + ∑ u 0m ⋅ ̂x m + U n0 ) dθ} ρ dρ 2

m=1

−π δ

󸀠

= π ∫ ((u 0n )ρ η󸀠 +

n2 0 u η ρ2 n

− k 2 u 0n η) ρ dρ .

0

After integration by parts, we obtain the Bessel equation (2.7) possessing the general solution u 0n (ρ) = K 0n J n (kρ), K 0n ∈ ℂ2 (2.28) since the Neumann function Y n (kρ) is singular for ρ ↘ +0 and disagrees the H 1 regularity of the solution u 0 . It remains to prove (2.23b). For this task, since (2.23a) holds, we refine the Wirtinger inequality (compare with (2.17b)): π

󵄨 󵄨2 ∫ 󵄨󵄨󵄨󵄨U N0 󵄨󵄨󵄨󵄨 dθ ≤

π

1 (N+1)2

−π

󵄨󵄨 ∂U 0 󵄨󵄨2 ∫ 󵄨󵄨󵄨󵄨 ∂θN 󵄨󵄨󵄨󵄨 dθ . 󵄨 󵄨

(2.29)

−π

Indeed, employing Fourier series U N0

=

c0N

∞

+∑

π

c Nn

⋅ ̂x n ,

c0N

:=

1 2π

n=1

∫

π

U N0

dθ,

c Nn

:=

1 π

−π

∫ U N0 ̂x n dθ −π

with scalar c0N ∈ ℂ and vectors c Nn ∈ ℂ2 , together with the derivative ∂U N0 ∂θ

∞

= ∑ nc Nn ⋅ (̂x n )󸀠 ,

where (̂x n )󸀠 = (− sin(nθ), cos(nθ))⊤ ,

n=1

this leads to the following Parseval identities: π

1 2π

󵄨 󵄨2 ∫ 󵄨󵄨󵄨󵄨U N0 󵄨󵄨󵄨󵄨 dθ = |c0N |2 +

−π

1 2

∞

∑ n=1

π

|c Nn |2 ,

1 2π

󵄨󵄨 ∂U 0 󵄨󵄨2 ∫ 󵄨󵄨󵄨󵄨 ∂θN 󵄨󵄨󵄨󵄨 dθ = 󵄨 󵄨

−π

1 2

∞

∑ n2 |c Nn |2 . n=1

4.2 Background Helmholtz problem | 73

Conditions (2.23a) imply c0N = ⋅ ⋅ ⋅ = c NN = 0 that allows us to estimate π

∞ 󵄨 󵄨2 ∫ 󵄨󵄨󵄨󵄨U N0 󵄨󵄨󵄨󵄨 dθ = π ∑ |c Nn |2 ≤ n=N+1

−π

π (N+1)2

∞

π

∑ n

2

|c Nn |2

=

n=N+1

1 (N+1)2

󵄨󵄨 ∂U 0 󵄨󵄨2 ∫ 󵄨󵄨󵄨󵄨 ∂θN 󵄨󵄨󵄨󵄨 dθ , 󵄨 󵄨

−π

thus concluding with (2.29). Due to (2.29), the boundary integral in I N (δ) introduced below can be estimated as π

∂U N0 0 ∂ρ U N

∫

󵄨󵄨 ∂U 0 󵄨󵄨2 δ 󵄨󵄨 N 󵄨󵄨 + dS x ≤ ∫ ( 2(N+1) 󵄨󵄨 ∂ρ 󵄨󵄨 󵄨 󵄨

N+1 2δ

−π

∂B δ (x 0 )

π

󵄨󵄨 ∂U 0 󵄨󵄨2 δ 󵄨󵄨 N 󵄨󵄨 + ≤ ∫( 2(N+1) 󵄨󵄨 ∂ρ 󵄨󵄨 󵄨 󵄨 −π

=

δ 2(N+1)

󵄨󵄨 0 󵄨󵄨2 󵄨󵄨U N 󵄨󵄨 ) δ dθ 󵄨 󵄨

󵄨󵄨 ∂U 0 󵄨󵄨2 1 󵄨󵄨 N 󵄨󵄨 2δ(N+1) 󵄨󵄨 ∂θ 󵄨󵄨 ) δ dθ 󵄨

󵄨

󵄨 󵄨2 ∫ 󵄨󵄨󵄨󵄨∇U N0 󵄨󵄨󵄨󵄨 dS x .

∂B δ (x 0 )

Therefore, considering similar to (2.16) the residual in the energy norm 󵄨 󵄨2 󵄨 󵄨2 I N (δ) := ∫ 󵄨󵄨󵄨󵄨∇U N0 󵄨󵄨󵄨󵄨 dx = ∫ k 2 󵄨󵄨󵄨󵄨U N0 󵄨󵄨󵄨󵄨 dx + B δ (x 0 )

B δ (x 0 )

∂U N0 0 ∂ρ U N

∫

dS x ,

∂B δ (x 0 )

applying here the Poincare inequality from (2.17a) and the co-area formula from (2.18), we derive the corresponding differential inequality 2

(1 − k 2 4δ ) I N (δ) ≤ π2

δ d 2(N+1) dδ I N (δ)

.

(2.30)

Integration of (2.30) over δ ∈ (r, R) leads to the estimate 0 ≤ I N (r) ≤ ( Rr )

2(N+1)

I(R)e

4(N+1)k2 π2

(R 2 −r 2 )

= O (r2(N+1) ) .

(2.31)

Applying to U N0 (x0 + ρ ̂x ) the point-wise estimate in the manner of (2.21), hence 󵄨2 󵄨󵄨 0 󵄨󵄨U N (x0 + ρ ̂x)󵄨󵄨󵄨 ≤ CI N (ρ) 󵄨 󵄨

with C > 0 ,

(2.32)

from (2.31) and (2.32) it follows (2.23b). The proof is completed. As a consequence, we specify Proposition 2.3 for N = 1, where K00 = u 0 (x0 ) and K10 = 2 0 k ∇u (x 0 ) are calculated due to (2.8a) and (2.8b). Corollary 2.4. The solution u 0 of (2.2) admits the first-order local asymptotic representation in the near-field B R (x0 ) ⊂ Ω as u 0 (x) = u 0 (x0 )J 0 (kρ) + 2k J 1 (kρ)∇u 0 (x0 ) ⋅ ̂x + U10 (x)

(2.33)

74 | 4 High-order topological expansions for Helmholtz problems in 2D with the residual U10 ∈ H 1 (B R (x0 ); ℂ) such that π

∫

π

U10

−π

dθ = ∫ U10 ̂x dθ = 0 ,

(2.34a)

−π

U10 (x0 + ρ ̂x ) = O(ρ 2 ) for ρ ∈ [0, R), θ ∈ (−π, π] .

(2.34b)

Moreover, differentiating (2.33) according to (2.6) and using the rule ∂ ∂x 1

∂ = cos θ ∂ρ −

sin θ ∂ ρ ∂θ ,

∂ = sin θ ∂ρ +

∂ ∂x 2

cos θ ∂ ρ ∂θ

yield the following formula for the gradient in B R (x0 ): ∇u 0 (x) = ∇u 0 (x0 ) + b 0u (x) + ∇U10 (x), with the term

b 0u (x)

:= (u +

0

(x0 )ka󸀠0 (kρ)

+

a 1 (kρ) 0 kρ (∇u (x 0 )

∇U10 = O(ρ)

a󸀠1 (kρ)∇u 0 (x0 ) ⋅ x̂󸀠 )̂x󸀠 ,

(2.35a)

⋅ ̂x) ̂x

b 0u = O(ρ) .

(2.35b)

In the following sections, we proceed with the “outer” asymptotic expansion which will be given in the far field with respect to a test geometric object put in the reference domain. We will see that it depends crucially on conditions imposed at the boundary of the test object. The Neumann, Dirichlet, and Robin boundary conditions will be considered separately in Sections 4.3, 4.4, and 4.5, respectively.

4.3 Helmholtz problems for geometric objects under Neumann (sound hard) boundary condition We start with the geometric description of a test object (inclusion, obstacle, scatterer). Let ω stand for a generic geometric shape implying the compact set in ℝ2 with the piecewise Lipschitz boundary ∂ω and the normal vector ν = (ν1 , ν2 )⊤ outward to ω. We require that 0 ∈ ω and the unit ball B1 (0) separating the near and far fields is the minimum enclosing ball centered at origin 0 such that ω ⊂ B1 (0). Consequently, the shapes are invariant to translations and isotropic scaling. We call by Gω the set of such shapes ω. Definition 3.1. Rescaling a shape ω ∈ Gω by a size parameter ε > 0, it produces a family of admissible geometric objects ω ε (x0 ) = {x ∈ ℝ2 :

x−x 0 ε

∈ ω} ⊂ B ε (x0 )

(3.1)

posed at a center x0 ∈ ℝ2 . Such admissible geometries in (3.1) admit parametrization by the triple of geometric variables (ω, ε, x0 ) ∈ Gω × ℝ+ × ℝ2 . In particular, the shape ω itself is equal to the parametrized object ω1 (0).

4.3 Helmholtz problems for geometric objects under Neumann boundary condition

| 75

Definition 3.2. We call by admissible geometries G = Gω × Gε × Gx in the reference domain Ω those triples: the shape ω ∈ Gω , the size ε ∈ Gε ⊂ ℝ+ , and the center x0 ∈ Gx ⊂ Ω of objects ω ε (x0 ) in (3.1) which satisfy the consistency condition: ω ε (x0 ) ⊂ B ε (x0 ) ⊂ Ω .

(3.2)

The set G of admissible geometries in (3.2) will be used further for the sake of shape variation of test objects in the reference domain Ω. Moreover, passing ε ↘ +0 diminishes the object and represents the topology change. For forward problems, we fix the object ω ε (x0 ) in Ω, or, equivalently, (ω, ε, x0 ) ∈ G, and we mark the dependence of functions on the size ε for the subsequent asymptotic analysis as ε ↘ +0. Given the boundary data g ∈ L2 (Γ N ; ℂ) and h ∈ H 1/2 (Γ D ; ℂ), the Neumann (sound hard) problem for the Helmholtz equation (compare with the background problem (2.1)) is considered for the wave potential u ε (x), x ∈ Ω \ ω ε (x0 ), satisfying −[∆ + k 2 ]u ε = 0 in Ω \ ω ε (x0 ) , ∂u ε ∂ν ∂u ε ∂ν ε

(3.3a)

= 0 on ∂ω ε (x0 ) ,

(3.3b)

=g

on Γ N ,

(3.3c)

u =h

on Γ D .

(3.3d)

The weak solution to (3.3) is described by the variational problem. Find u ε ∈ H 1 (Ω \ ω ε (x0 ); ℂ) such that uε = h ∫

on Γ D ,

(3.4a)

(∇u ε ⋅ ∇u − k 2 u ε u) dx = ∫ gu dS x

Ω\ω ε (x 0 )

ΓN 1

for all u ∈ H (Ω \ ω ε (x0 ); ℂ) : u = 0

on Γ D .

(3.4b)

For every wave number k ∈ ℝ+ , which can be also large, the well-posedness of problem (3.4) can be argued by using the Fredholm alternative, similar to Proposition 2.1. Moreover, (3.4) follows necessarily from the corresponding variational principle (compare with (2.3) and (2.4)): Pε (u ε ) = min max Pε (𝑣) Re(𝑣) Im(𝑣)

over 𝑣 ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : 𝑣 = h on Γ D

(3.5)

with the Lagrangian Pε : H 1 (Ω \ ω ε (x0 ); ℂ) 󳨃→ ℝ given by { } { } Pε (𝑣) = Re { 12 ∫ (∇𝑣 ⋅ ∇𝑣 − k 2 𝑣2 ) dx − ∫ g𝑣 dS x } . { } ΓN { Ω\ω ε (x0 ) }

(3.6) ε

From the variational problem (3.4), we can determine the boundary traction ∂u ∂ν in (3.3b) and (3.3c) in the weak sense using the following Green’s formula. By recalling

76 | 4 High-order topological expansions for Helmholtz problems in 2D that ν denotes both the outer normal on ∂Ω and ∂ω ε (x0 ), we have for every function u ∈ H 1 (Ω \ ω ε (x0 ); ℂ) such that ∆u ∈ L2 (Ω \ ω ε (x0 ); ℂ) (cf. (2.5)) ∫

∂u (∇u ⋅ ∇u + u ∆u) dx = ⟨ ∂u ∂ν , u⟩ Γ N − ⟨ ∂ν , u⟩ ∂ω ε (x 0 )

(3.7)

Ω\ω ε (x 0 ) 1

for all u ∈ H (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D . 1/2 (∂ω (x ); ℂ) and Here, ⟨ ∂u ε 0 ∂ν , u⟩ ∂ω ε (x 0 ) denotes the duality pairing between u ∈ H ∂u −1/2 (∂ω (x ); ℂ). ∈ H ε 0 ∂ν In order to derive residual error estimates, we require the uniform inf-sup condition (see, e.g., [41]): there exists β 0 > 0 such that 󵄨󵄨 󵄨󵄨 2 󵄨󵄨∫ 󵄨 󵄨󵄨 Ω\ω ε (x0 ) (∇u ⋅ ∇𝑣 − k u𝑣) dx󵄨󵄨󵄨 (3.8) 0 < β 0 ≤ inf sup u ‖u‖H1 (Ω\ω ε (x );ℂ) ‖𝑣‖H1 (Ω\ω ε (x );ℂ) 𝑣 0

0

for all u, 𝑣 ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : u = 𝑣 = 0 on Γ D , admissible geometries (ω, ε, x0 ) ∈ G, and moderate wave numbers k ∈ [0, k 0 ], k 0 > 0. By (3.8), the well-posedness of (3.4) follows directly from the Babuska–Lions–Necas–Lax–Milgram theorem; see [31, Section 4.4]. Proposition 3.3. The solutions u 0 of (2.2) and u ε of (3.4) satisfy the residual estimate 󵄩󵄩 ε 󵄩 󵄩󵄩u − u 0 󵄩󵄩󵄩 (3.9) 󵄩 󵄩H1 (Ω\ω ε (x0 );ℂ) = O(ε) . Proof. With the help of strong formulation (2.1), we derive from Green’s formula (3.7) the variational equation for u 0 over the perturbed domain Ω \ ω ε (x0 ) in the form ∫ (∇u 0 ⋅ ∇u − k 2 u 0 u) dx = ∫ gu dS x − Ω\ω ε (x 0 )

∫

∂u0 ∂ν u dS x

(3.10)

∂ω ε (x 0 )

ΓN

for all u ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D . 0

2 0 Here we have used the fact that ∂u ∂ν ∈ L (∂ω ε (x 0 ); ℂ) since the solution u is locally 2 H -smooth; see, e.g., [46, Theorem 10.1, Chapter 3]. Subtracting (3.10) from (3.4) and inserting the test function u = u ε − u 0 we have

∫ Ω\ω ε (x 0 )

󵄨 󵄨 󵄨2 󵄨2 (󵄨󵄨󵄨󵄨∇(u ε − u 0 )󵄨󵄨󵄨󵄨 − k 2 󵄨󵄨󵄨󵄨u ε − u 0 󵄨󵄨󵄨󵄨 ) dx =

∫

∂u0 ε ∂ν (u

− u 0 ) dS x .

∂ω ε (x 0 )

Applying here the inf-sup condition (3.8) and the Cauchy–Schwarz inequality, it results in the following estimate: 󵄩󵄩 ε 󵄩2 󵄩󵄩 u − u 0 󵄩󵄩󵄩 󵄩 󵄩H1 (Ω\ω ε (x0 );ℂ) ≤

1 β0

󵄩󵄩 ∂u0 󵄩󵄩 󵄩󵄩 ε 0󵄩 󵄩󵄩 󵄩󵄩 ∂ν 󵄩󵄩 󵄩 󵄩 󵄩L2 (∂ω ε (x0 );ℂ) 󵄩󵄩 u − u 󵄩󵄩L2 (∂ω ε (x0 );ℂ) .

(3.11)

For u ∈ H 1 (Ω \ ω ε (x0 ); ℂ), the boundary trace theorem provides with 0 < c < c: c‖u‖H1 (Ω\ω ε (x

0 );ℂ)

≤ ‖u‖H1/2 (∂Ω;ℂ) + ‖u‖H1/2 (∂ω ε (x

0 );ℂ)

≤ c‖u‖H1 (Ω\ω ε (x

0 );ℂ)

,

(3.12)

4.3 Helmholtz problems for geometric objects under Neumann boundary condition

| 77

where, by homogeneity argument, the H 1/2 -norm at ∂ω ε (x0 ) implies ‖u‖2H1/2 (∂ω

ε (x0 );ℂ)

= 1ε ‖u‖2L2 (∂ω

ε (x0 );ℂ)

|u(x)−u(y)|2 |x−y|2 ∂ω ε (x 0 )

+∫ ∫

dS x dS y .

(3.13)

Therefore, applying (3.12) and (3.13) with u = u ε − u 0 to (3.11), we obtain ‖u ε − u 0 ‖H1 (Ω\ω ε (x

0 );ℂ)

From (2.35) in Corollary 2.4, it follows

c √ ∂u0 β 0 ε‖ ∂ν ‖L2 (∂ω ε (x0 );ℂ)

≤

∂u0 ∂ν

.

= O(1) for ρ = O(ε) at ∂ω ε (x0 ), hence 1/2

󵄩󵄩 ∂u0 󵄩󵄩 󵄩󵄩 ∂ν 󵄩󵄩 󵄩 󵄩L2 (∂ω ε (x0 );ℂ) = ( ∫

∂ω ε (x 0 )

󵄨󵄨 ∂u0 󵄨󵄨2 󵄨󵄨 ∂ν 󵄨󵄨 dS x ) 󵄨 󵄨

= O(√ ε)

(3.14)

and we conclude with (3.9). The proof is completed. We observe that the last term on the right-hand side of (3.10) expresses the residual error near ∂ω ε (x0 ). This boundary integral constitutes the leading order O(ε) in the residual error estimate (3.9). To refine this estimate in Proposition 3.3 to the order o(ε), 0 0 we construct a corrector for ∂u ∂ν = ∇u (x 0 ) ⋅ ν + O(ε) (see (3.44)) in the form of the boundary layer; see, e.g., [59]. In will be expressed via outer asymptotic expansion in the far field with respect to the reference geometric object ω.

4.3.1 Outer asymptotic expansion by Fourier series in far field In this section, we state an auxiliary problem in the exterior domain for y ∈ ℝ2 \ ω 0 with respect to the stretched variable y = x−x ε according to Definition 3.1. For this reason, we introduce the weighted Sobolev spaces (see [7, 51, 52]): 1,p

W μ (ℝ2 \ ω; ℂ) = {𝑣 :

𝑣 μ , ∇𝑣

∈ L p (ℝ2 \ ω; ℂ)},

in ℝ2 \ B2 (0),

μ(y) = O(|y| ln |y|)

μ(y) = O(1)

p ∈ (1, ∞) , in B2 (0) \ ω ,

with the weight μ suggested by the weighted Poincare inequality in exterior domains ∫

2

𝑣 ( |y| ln |y| ) dy ≤ 4

ℝ2 \B2 (0)

|∇𝑣|2 dy

∫ ℝ2 \B2 (0)

∫ 𝑣 dS x = 0 .

if

(3.15)

∂B2 (0)

We note that the constant function is allowed for p ≥ 2 and logarithm for p > 2 in 1,p Wμ . For p = 2, we consider the real-valued exterior Neumann problem. Find vector 2 1,2 function w ν (y) = ((w ν )1 , (w ν )2 )⊤ ∈ (W μ (ℝ2 \ ω; ℝ) \ ℙ0 ) such that ∫ Dw ν ∇𝑣 dy = ∫ ν𝑣 dS y ℝ2 \ω

∂ω

1,2

for all 𝑣 ∈ W μ (ℝ2 \ ω; ℝ) ,

(3.16)

78 | 4 High-order topological expansions for Helmholtz problems in 2D where the derivative matrix (Dw ν )ij = (w ν )i,j for i, j = 1, 2, the normal vector ν = (ν1 , ν2 )⊤ , and ℙ0 stands for polynomials of degree zero, i.e., constant. Excluding constant solutions, this implies the boundary value problem: −∆w ν = 0

in ℝ2 \ ω ,

(3.17a)

Dw ν ν = −ν

on ∂ω ,

(3.17b)

as |y| ↗ ∞ .

(3.17c)

wν =

1 O ( |y| )

The existence of a solution to (3.16) follows from the result of [7]. After rescaling y = x−x 0 ε we reduce the problem to the bounded domain Ω \ ω ε (x 0 ) as follows. ε 0 Lemma 3.4. The rescaled solution w εν (x) := w ν ( x−x ε ) to (3.16) implies the function w ν ∈ 1 2 H (Ω \ ω ε (x0 ); ℝ) which fulfills the following relation:

Dw εν ∇u dx = ∫ (Dw εν ν)u dS x +

∫ Ω\ω ε (x 0 )

1 ε

∫

νu dS x

∂ω ε (x 0 )

ΓN

(3.18)

1

for all u ∈ H (Ω \ ω ε (x0 ); ℝ) : u = 0 on Γ D . It admits the far-field representation in the Fourier series w εν (x) =

ε 1 ρ 2π M ω

̂x + W νε (x)

for x ∈ ℝ2 \ B ε (x0 ) ,

(3.19)

with the 2 × 2 real matrix M ω and the residual function W νε = ((W νε )1 , (W νε )2 )⊤ ∈ H 1 (Ω \ B ε (x0 ); ℝ)2 such that π

∫

π

W νε

−π

dθ = ∫ W νε ̂x dθ = 0 ,

(3.20a)

−π 2

W νε (x0 + ρ ̂x) = O (( ρε ) )

for ρ > ε, θ ∈ (−π, π] .

(3.20b)

Moreover, the uniform estimates hold ‖Dw εν ‖L2 (Ω\ω ε (x

2×2 0 );ℝ)

‖w εν ‖L2 (∂ω ε (x0 );ℝ)2

= O(1), = O(√ ε),

‖w εν ‖L2 (Ω\ω ε (x

2 0 );ℝ)

‖Dw εν ν‖L2 (∂ω ε (x0 );ℝ)2

= O(ε√| ln ε|) ,

(3.21a)

1 = O ( √ε ) .

(3.21b)

Proof. The local coordinate system (2.6) after stretching implies the polar radius |y| ∈ ℝ+ and the polar angle θ ∈ (−π, π] such that y=

x−x 0 ε

= |y|̂x ,

ρ

|y| = ε ,

̂x = (cos θ, sin θ)⊤ .

(3.22)

By using the radial vector functions x̂ n from Proposition 2.3, the harmonic vectorvalued function in (3.17) admits the Fourier series in the far field as ∞

w ν (y) = ∑ n=1

ν ̂n 1 |y|n C n x ,

C νn ∈ ℝ2×2 ,

for y ∈ ℝ2 \ B1 (0)

(3.23)

4.3 Helmholtz problems for geometric objects under Neumann boundary condition

| 79

with unknown coefficient matrices C νn , n ∈ ℕ. Formula (3.23) implies 1 1 ̂ |y| 2π M ω x

w ν (y) =

+ W ν (y),

1 2π M ω

:= C1ν ,

y ∈ ℝ2 \ B1 (0)

(3.24)

with the square matrix M ω ∈ ℝ2×2 and the vector-valued residual function W ν = ((W ν )1 , (W ν )2 )⊤ ∈ W μ1,2 (ℝ2 \ ω; ℝ)2 such that π

π

∫ W ν dθ = ∫ W ν ̂x dθ = 0 , −π

(3.25a)

−π

W ν (y) = O(|y|2 )

for |y| > 1, θ ∈ (−π, π] ,

(3.25b)

where (3.25a) is obtained with the help of trigonometric calculus in (2.27). Applying 0 the coordinate transformation y = x−x ε to (3.24) and (3.25), due to (3.22) it follows x−x 0 ε ε 0 straightforwardly (3.19) and (3.20) for w ν (x) := w ν ( x−x ε ) and W ν (x) := W ν ( ε ). x−x 0 Next, we apply the coordinate transformation y = ε to the problem (3.17) and employ the differential calculus according to (3.22) ∂ ∂y

∂ = ε ∂x ,

dy =

1 ε2

dx,

dS y =

1 ε

dS x

(3.26)

to derive the following relations: −∆w εν = 0 Dw εν ν 1,2 W μ (ℝ2

=

in ℝ2 \ ω ε (x0 ) ,

− 1ε ν

(3.27a)

on ∂ω ε (x0 ) .

); ℝ)2

w εν

H 1 (Ω

(3.27b)

Since ∈ \ ω ε (x0 follows ∈ \ ω ε (x0 and in view of (3.27a), the Green formula (3.7) can be applied to the vector function u = w εν : w εν

); ℝ)2

(Dw εν ∇u + u ∆w εν ) dx = ⟨Dw εν ν, u⟩Γ N − ⟨Dw εν ν, u⟩∂ω ε (x0 )

∫ Ω\ω ε (x 0 )

for all u ∈ H 1 (Ω \ ω ε (x0 ); ℝ) : u = 0 on Γ D . As a result, using (3.27) and the fact that the solution w εν is locally smooth near Γ N , hence Dw εν ν ∈ L2 (Γ N ; ℝ)2 , we arrive at formulation (3.18) in the bounded domain. It remains to justify estimates (3.21). The first inequality in (3.21a) follows from |Dw εν |2 dx ≤

∫ Ω\ω ε (x 0 )

|Dw εν |2 dx = ∫ |Dw ν |2 dy = O(1)

∫ ℝ2 \ω ε (x 0 )

ℝ2 \ω

which is calculated according to (3.26). To obtain the second inequality in (3.21a), we inscribe Ω in a ball B R (x0 ) of radius R > 0 sufficiently large and decompose it in B R (x0 ) \ B ε (x0 ) and B ε (x0 ) \ ω ε (x0 ) such that ∫ |w εν |2 dx ≤ Ω\ω ε (x 0 )

∫

|w εν |2 dx +

B R (x 0 )\B ε (x 0 )

∫

|w εν |2 dx

B ε (x 0 )\ω ε (x 0 )

π R 1 M ω x̂|2 + |W νε |2 ) ρ dρ dθ + ε2 ∫ |w ν |2 dy = O(ε2 | ln ε|) = ∫ ∫ (( ρε )2 | 2π −π ε

due to (3.19), (3.20), and (3.26).

B1 (0)\ω

80 | 4 High-order topological expansions for Helmholtz problems in 2D

Finally, by homogeneity and the trace theorem from (3.12) and (3.13) it follows 1 ‖w εν ‖L2 (∂ω ε (x0 );ℝ)2 √ε

≤ c‖w εν ‖H1 (Ω\ω ε (x

2 0 );ℝ)

= O(1)

and the former inequality in (3.21b), while the latter one is confirmed from (3.27b) by ‖Dw εν ν‖2L2 (∂ω

ε (x0 );ℝ)2

=

󵄨 󵄨2 ∫ 󵄨󵄨󵄨 νε 󵄨󵄨󵄨 dS x =

1 meas2 (∂ω ε (x0 )) ε2

= 1ε meas2 (∂ω)

∂ω ε (x 0 )

where meas2 (∂ω ε (x0 )) and meas2 (∂ω) mean the Hausdorff measure of the sets in ℝ2 . This completes the proof. We remark that the matrix M ω in Lemma 3.4 is called added or virtual mass tensor in [53, Note G]. Its properties are given in Lemma 3.5 following [6, 14, 26, 45]. Lemma 3.5. The entries of M ω have the implicit expression: (M ω )ij = δ ij meas2 (ω) + ∫ (w ν )i ν j dS y ,

i, j = 1, 2 .

(3.28)

∂ω

The matrix M ω ∈ Spsd(ℝ2×2 ), i.e., symmetric positive semidefinite, and positive definite if meas2 (ω) > 0. For ellipsoidal shapes ω it has the explicit expression M ω = Θ(α)M ω󸀠 Θ(α)⊤ , M ω󸀠 = π(a + b) (

b 0

Θ(α) := (

cos α sin α

− sin α ) , cos α

(3.29a)

0 ) a

(3.29b)

with the ellipse major a = 1 and minor b ∈ (0, 1] semiaxes, where the major axis has an angle of α ∈ (− π2 , 2π ) with the y1 -axis counted in the anticlockwise direction. Proof. We split the exterior domain in the far-field ℝ2 \ B1 (0) and the near-field B1 (0)\ ω. In the far field, the truncated Fourier series (3.24) holds. In the near field, for i, j = 1, 2 we have the second Green formula 0=

∫ {∆(w ν )i y j − (w ν )i ∆y j } dy B1 (0)\ω

=

∂(w ν )i j ν )i ∫ { ∂(w ∂|y| y j − (w ν )i ∂|y| } dS y − ∫ { ∂ν y j − (w ν )i ∂y

∂B1 (0)

∂y j ∂ν

∂y j ∂|y|

= x̂ j according to (3.22), y j = x̂ j since

= ν j it follows

ν )i ̂ − ∫ { ∂(w ∂|y| − (w ν )i } x j dS y = ∫ {ν i y j + (w ν )i ν j } dS y .

∂B1(0)

.

∂ω

Using here the Neumann condition (3.17b), |y| = 1 at ∂B1 (0), and

∂y j ∂ν } dS y

∂ω

(3.30)

4.3 Helmholtz problems for geometric objects under Neumann boundary condition

|

81

We employ (3.24), (3.25a), and the trigonometric calculus (2.27) to calculate the integral over ∂B1 (0) on the left-hand side of (3.30) as ν )i ̂ ∫ {− ∂(w ∂|y| + (w ν )i } x j dS y

∂B1 (0) π

2

1 = ∫ { 2π ∑ (M ω )il x̂ l −

2

∂(W ν )i ∂|y|

+

1 2π

l=1

−π

∑ (M ω )il x̂ l + (W ν )i } ̂x j dθ l=1

π 2

=

1 π

∫ ∑ (M ω )il ̂x l ̂x j dθ = (M ω )ij . −π l=1

Applying to the right-hand side of (3.30) the divergence theorem ∫ ν i y j dS y = ∫ y j,i dy = δ ij meas2 (ω), ω

∂ω

{1 , δ ij = { 0, {

if i = j , if i ≠ j ,

(3.31)

this together results in expression (3.28). To prove the symmetry of M ω , we insert 𝑣 = (w ν )j as the test function in (3.16): ∫ ∇(w ν )i ⋅ ∇(w ν )j dy = ∫ ν i (w ν )j dS y = ∫ ν j (w ν )i dS y , ℝ2 \ω

∂ω

∂ω

written component-wisely for i, j = 1, 2, which follows (M ω )ij = (M ω )ji in (3.28). For arbitrary ξ ∈ ℝ2 , from (3.16) we have the nonnegative linear combinations 2

0 ≤ ∫ |∇ (ξ1 (w ν )1 + ξ2 (w ν )2 ) |2 dy = ∑ ∫ (w ν )i ξ i ν j ξ j dS y . i,j=1 ∂ω

ℝ2 \ω

Therefore, multiplying (3.28) with ξ i ξ j and summing the result over i, j = 1, 2, the positive semidefiniteness, which is strict if meas2 (ω) > 0, follows: 2

2

∑ (M ω )ij ξ i ξ j = |ξ|2 meas2 (ω) + ∑ ∫(w ν )i ξ i n j ξ j dS y ≥ |ξ|2 meas2 (ω) . i,j=1

i,j=1 ω

Finally, we derive the explicit representation of M ω for ellipsoidal shapes ω. Let the canonical ellipse ω󸀠 enclosed in the ball B1 (0) have the major a = 1 and the minor b ∈ (0, 1] semiaxes with respect to y󸀠 -coordinates, i.e., 2

y󸀠

ω󸀠 = {y󸀠 ∈ ℝ2 : ( a1 ) + (

y󸀠2 2 b)

< 1} ,

a=1.

Let the major axis of the reference ellipse ω written in y-coordinates have an angle of α ∈ (− 2π , 2π ) with the y1 -axis counted in the anticlockwise direction, i.e., 2

2

2 sin α 2 cos α ) + ( −y1 sin α+y ) < 1} , ω = {y ∈ ℝ2 : ( y1 cos α+y a b

a=1.

82 | 4 High-order topological expansions for Helmholtz problems in 2D The y-coordinates are after rotation of y󸀠 -coordinates with the angle of −α, and y󸀠 ∈ ω󸀠 when Θ(−α)y = Θ⊤ (α)y ∈ ω with the orthogonal matrix Θ(α) given in (3.29a). Therefore, we will prove formula (3.29b) for ω󸀠 and then transform y󸀠 = Θ⊤ y. We introduce the elliptic coordinates r ∈ ℝ+ and ψ ∈ (−π, π] such that y󸀠1 = c cosh(r) cos ψ,

c = √ a2 − b 2 ,

y󸀠2 = c sinh(r) sin ψ,

a=1,

(3.32)

where c is called the linear eccentricity. By setting the distance r0 ∈ ℝ+ such that a = c cosh(r0 )

b = c sinh(r0 ) ,

(3.33a)

the geometry ω󸀠 can be restated as ∂ω󸀠 = {r = r0 , ψ ∈ (−π, π]},

ℝ2 \ ω󸀠 = {r > r0 , ψ ∈ (−π, π]} .

(3.33b)

From (3.32), the differential calculus in elliptic coordinates follows: ∂ { ∂y󸀠 = 1 { ∂ = ∂y { 󸀠2

1 𝜘2 (r,ψ) 1 𝜘2 (r,ψ)

∂ ∂ − c cosh(r) sin ψ ∂ψ (c cosh(r) cos ψ ∂r ) , ∂ ∂ (c cosh(r) sin ψ ∂r + c sinh(r) cos ψ ∂ψ ) ,

dy󸀠 = 𝜘2 (r, ψ) dr dψ,

(3.34)

𝜘(r, ψ) = c√sinh2 (r) + sin2 ψ

with the scale factor 𝜘(r, ψ). In particular, at the ellipse boundary as r = r0 in (3.33b) with the normal vector ν󸀠 , using (3.32) for constant ψ and (3.33a), we have ν󸀠 =

1 (b cos ψ, a sin ψ)⊤ , 𝜘(r 0 ,ψ)

∂ ∂ν 󸀠

=

∂ 1 𝜘(r 0 ,ψ) ∂r

, (3.35)

𝜘(r0 , ψ) = √a2 cos2 ψ + b 2 sin2 ψ .

dS y󸀠 = 𝜘(r0 , ψ) dψ,

Applying the coordinate transformation (3.32) and calculus in (3.34) and (3.35), the exterior problem (3.17), formulated for w ν󸀠 in ℝ2 \ω󸀠 in y󸀠 -coordinates, can be rewritten in elliptic coordinates. In fact, similar to Proposition 2.3, the harmonic vector function w ν󸀠 admits the following Fourier series in ℝ2 \ ω󸀠 (cf. (3.23)): ∞

󸀠

w ν󸀠 = ∑ e−nr C νn (cos(nψ), sin(nψ))⊤

for r > r0

(3.36)

n=1 󸀠

according to (3.17c) for r ↗ ∞. Coefficient matrices C νn ∈ ℝ2×2 in (3.36) can be found from the boundary condition (3.17b) written due to (3.35) and (3.36) as ∂w ν󸀠 󵄨󵄨󵄨 1 𝜘(r 0 ,ψ) ∂r 󵄨󵄨󵄨 r=r 0

∞

=−∑ n=1

⊤ ne −nr0 ν 󸀠 𝜘(r 0 ,ψ) C n (cos(nψ), sin(nψ))

󸀠

=−

(b cos ψ,a sin ψ)⊤ 𝜘(r 0 ,ψ)

.

󸀠

Henceforth, e−r0 C1ν (cos ψ, sin ψ)⊤ = (b cos ψ, a sin ψ)⊤ , C νn = 0 for all n ≥ 2, and we obtain the following analytic expression for the solution: w ν󸀠 = e r0 −r (b cos ψ, a sin ψ)⊤

for r ≥ r0 .

(3.37)

| 83

4.3 Helmholtz problems for geometric objects under Neumann boundary condition

Now the matrix M ω󸀠 can be calculated analytically. After substitution of (3.37) in the representation formula (3.28), it implies the following two vectors for j = 1, 2: (M ω󸀠 )( ⋅ ,j) = δ( ⋅ ,j) meas2 (ω󸀠 ) + ∫ (b cos(ψ)ν󸀠j , a sin(ψ)ν󸀠j )⊤ dS y󸀠 .

(3.38)

∂ω󸀠

We extend (b cos ψ, a sin ψ)⊤ from the boundary inside ω󸀠 with the smooth function ( ba y󸀠1 , ab y󸀠2 )⊤ and use the divergence theorem to calculate the elliptic integral in (3.38) {( b , 0)⊤ , ∫ ( ba y󸀠1 ν󸀠j , ab y󸀠2 ν󸀠j )⊤ dS y󸀠 = ∫ ( ba y󸀠1,j ba y󸀠2,j )⊤ dy󸀠 = meas2 (ω󸀠 ) { a (0, ab )⊤ , { ∂ω󸀠 ω󸀠

j = 1, j = 2.

Together with meas2 (ω󸀠 ) = πab from (3.38) we arrive at (3.29b). The transformation formula in (3.29a) can be justified by rotation y󸀠 = Θ⊤ y applied to the variational equation in the manner of (3.16): ∫ Dw ν (Θy󸀠 )∇𝑣 dy󸀠 = ∫ ν𝑣 dS y󸀠

1,2

for all 𝑣 ∈ W μ (ℝ2 \ ω󸀠 ; ℝ) ,

∂ω󸀠

ℝ2 \ω󸀠

which, after the left multiplication with Θ⊤ , results in ∫ D(Θ⊤ w ν (Θy󸀠 ))∇𝑣 dy󸀠 = ∫ Θ⊤ ν𝑣 dS y󸀠 .

(3.39)

∂ω󸀠

ℝ2 \ω󸀠

Since Θ⊤ ν = ν󸀠 in (3.39) and using the representation (3.24) this proves the identity w ν󸀠 (y󸀠 ) = Θ⊤ w ν (Θy󸀠 ) = =

Θy󸀠 1 1 ⊤ Θ M ω |Θy 󸀠| |Θy󸀠 | 2π

󸀠 1 1 ⊤ Θ M ω Θ |yy󸀠 | |y󸀠 | 2π

+ Θ⊤ W ν (Θy󸀠 )

+ Θ⊤ W ν (Θy󸀠 ) =

󸀠 1 1 M 󸀠 y |y󸀠 | 2π ω |y󸀠 |

+ W ν󸀠 (y󸀠 ) ,

which implies Θ⊤ M ω Θ = M ω󸀠 , thus (3.29a), and completes the proof. We remark that, in the limit case when b ↘ +0, formulas in Lemma 3.5 describe the singular matrix of virtual mass for the straight crack; see [9, 45].

4.3.2 Uniform asymptotic expansion of solution of the Neumann problem With the help of the boundary layer w εν described in Lemma 3.4, we can improve the residual error estimate (3.9) according to (3.43) in Theorem 3.6: ‖u ε − u 0 − ε∇u 0 (x0 ) ⋅ w εν ‖H1 (Ω\ω ε (x

0 );ℂ)

= O(ε2 √| ln ε|) .

(3.40)

The leading order here is contributed by εw εν over the domain Ω \ ω ε (x0 ) due to the second inequality in (3.21a). Further, we construct a refined asymptotic term of order

84 | 4 High-order topological expansions for Helmholtz problems in 2D ε2 using the auxiliary Helmholtz problem. Find u 1 ∈ H 1 (Ω; ℂ)2 such that u 1 = 0 on Γ D , ∫(Du 1 ∇u − k 2 u 1 u) dx =

k2 ε √| ln ε|

Ω

(3.41a) w εν u dx

∫ Ω\ω ε (x 0 )

for all u ∈ H (Ω; ℂ) : u = 0 on Γ D . 1

(3.41b)

We stress that the solution u 1 to problem (3.41) is estimated uniformly with respect to ε since the right-hand side of (3.41b) is bounded by (3.21a) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 ε 󵄨󵄨 ∫ w u dS x 󵄨󵄨󵄨 ≤ ν 󵄨󵄨 ε√| ln ε| 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 Ω\ω ε (x 0 )

‖w εν ‖

L2 (Ω\ω ε (x0 );ℝ)2

ε √| ln ε|

‖u‖L2 (Ω\ω ε (x

0 );ℂ)

≤ C‖u‖L2 (Ω\ω ε (x

0 );ℂ)

with C > 0. Using Green’s formula (3.7) and the local H 2 -regularity of the solution u 1 in B ε (x0 ) ⊃ ω ε (x0 ), we restate (3.41b) over the perturbed domain ∫

(Du 1 ∇u − k 2 u 1 u) dx =

Ω\ω ε (x 0 )

k2 ε √| ln ε|

w εν u dx −

∫ Ω\ω ε (x 0 )

∫ (Du 1 ν)u dS x (3.42)

∂ω ε (x 0 )

for all u ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D . Theorem 3.6. The solutions u 0 of (2.2), u ε of (3.4), w ν of (3.16), and u 1 of (3.41) satisfy the following residual error estimate: ‖q1ε ‖H1 (Ω\ω ε (x

0 );ℂ)

= O(ε2 ) ,

(3.43a)

q1ε : = u ε − u 0 − ε∇u 0 (x0 ) ⋅ (w εν + ε√| ln ε|u 1 ) .

(3.43b)

Proof. Subtracting from (3.4b) equations (3.10), (3.18) multiplied with ε∇u 0 (x0 ), and (3.42) multiplied with ε2 √| ln ε|∇u 0 (x0 ), using ∂u0 ∂ν

− ∇u 0 (x0 ) ⋅ ν = b 0u ⋅ ν +

∂U 10 ∂ν

= O(ε)

on ∂ω ε (x0 )

(3.44)

according to (2.35), the differential identity ∇ (∇u 0 (x0 ) ⋅ (w εν + ε√| ln ε|u 1 )) = ∇u 0 (x0 ) ⋅ (Dw εν + ε√| ln ε|Du 1 ) , and the notation of q1ε introduced in (3.43b), we obtain the variational equation ∫

(∇q1ε ⋅ ∇u − k 2 q1ε u) dx = −ε ∫ (∇u 0 (x0 ) ⋅ Dw εν ν)u dS x

Ω\ω ε (x 0 )

+

∫

ΓN

(b 0u ⋅ ν +

∂U 10 ∂ν

+ε2 √| ln ε|∇u 0 (x0 ) ⋅ Du 1 ν) u dS x

∂ω ε (x 0 )

for all u ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D .

(3.45)

4.3 Helmholtz problems for geometric objects under Neumann boundary condition

| 85

One difficulty is that q1ε is inhomogeneous, namely q1ε = −ε∇u 0 (x0 ) ⋅ w εν = O(ε2 ) at Γ D due to (3.19). For its lifting, we take a smooth cut-off function η Γ D supported in a neighborhood of Γ D such that η Γ D = 1 at Γ D . Henceforth, for Q1ε := q1ε + R1ε ,

R1ε := ε(∇u 0 (x0 ) ⋅ w εν )η Γ D = O(ε2 ),

Q1ε = 0 on Γ D ,

applying to (3.45) with q1ε = Q1ε − R1ε the Cauchy–Schwarz inequality, the asymptotic estimates in (3.19), (3.44), and the trace theorems (3.12) and (3.13), we obtain 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∫ (∇Q ε ⋅ ∇u − k 2 Q ε u) dx󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨 ∫ (∇R ε ⋅ ∇u − k 2 R ε u) dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 1 1 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨Ω\ω ε (x0 ) 󵄨󵄨 󵄨󵄨Ω\ω ε (x0 ) 󵄨󵄨 + C1 ε2 ‖u‖L2 (Γ

N ;ℂ)

+ C2 ε

∫

|u| dS x ≤ Cε2 ‖u‖H1 (Ω\ω ε (x

0 );ℂ)

,

C, C1 , C2 > 0 .

∂ω ε (x 0 )

This upper bound together with the inf-sup condition (3.8) proves ‖Q1ε ‖H1 (Ω\ω ε (x );ℂ) = 0 O(ε2 ), hence (3.43a) and the assertion of the theorem. As a consequence, from (3.45), we infer the boundary value problem for q1ε : −[∆ + k 2 ]q1ε = 0 ∂q1ε ∂ν ∂q1ε ∂ν q1ε

in Ω \ ω ε (x0 ) ,

(3.46a)

= −ν ⋅ (b 0u + ∇U10 + ε2 √| ln ε|(Du 1 )∇u 0 (x0 ))

on ∂ω ε (x0 ) ,

(3.46b)

= −ε∇u 0 (x0 ) ⋅ Dw εν ν

on Γ N ,

(3.46c)

on Γ D .

(3.46d)

0

= −ε∇u (x0 ) ⋅

w εν

Theorem 3.6 is useful for the asymptotic expansion with respect to ε ↘ +0 of the state-constrained objective function as suggested in the following section.

4.3.3 Inverse Helmholtz problem under Neumann boundary condition In the inverse setting of the problem, the shape ω⋆ ∈ Gω , the size ε⋆ ∈ Gε , and the center x⋆ ∈ Gx for an unknown geometric object ω⋆ε⋆ (x⋆ ) being tested are to be identified and reconstructed from the known boundary measurement u ⋆ ∈ L2 (Γ N ; ℂ). The admissible set G = Gω × Gε × Gx is introduced in Definitions 3.1 and 3.2. For this purpose, a trial geometric object ω ε (x0 ) with admissible (ω, ε, x0 ) ∈ G is put in Ω. For such trial variables, we find a family of solutions u ε to problem (3.4) and determine the square function of the misfit at the boundary J : G 󳨃→ ℝ+ ,

J(ω, ε, x0 ) :=

1 2

∫ |u ε − u ⋆ |2 dS x .

(3.47)

ΓN

The objective function (3.47) serves for the state-constrained, topology optimization problem. Find (ω⋆ , ε⋆ , x⋆ ) ∈ G which is the argument of the trivial minimum 0 = J(ω⋆ , ε⋆ , x⋆ ) =

min

(ω,ε,x 0)∈G

J(ω, ε, x0 )

subject to (3.4) .

(3.48)

86 | 4 High-order topological expansions for Helmholtz problems in 2D Since the test geometry (ω⋆ , ε⋆ , x⋆ ) ∈ G is feasible, the trivial minimum in (3.48) is ⋆ ⋆ attained at the solution u ε of (3.4) for the test object ω⋆ε⋆ (x⋆ ) when u ε = u ⋆ at Γ N in (3.47). Uniqueness of the minimum is open. To bring (3.48) in a form suitable for analysis, we give primal-dual arguments. While u ε in (3.47) implies the primal state variable, a dual state variable 𝑣ε can be obtained by a Fenchel–Legendre duality corresponding to the variational principle: Lε (u ε , 𝑣ε ) =

min

max

Re(u),Re(𝑣) Im(u),Im(𝑣)

L ε (u, 𝑣) (3.49)

over u, 𝑣 ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : u = h, 𝑣 = 0 on Γ D , where the Lagrangian Lε has the form (compare with (3.5) and (3.6)): { { L ε (u, 𝑣) := Re { ∫ (∇u ⋅ ∇𝑣 − k 2 u𝑣) dx − ∫ g𝑣 dS x + { ΓN {Ω\ω ε (x0 )

1 2

} } ∫ (u − u ⋆ )2 dS x } . (3.50) } ΓN }

Lemma 3.7. The first-order necessary optimality conditions for (3.49) imply the primal problem (3.4) together with the dual variational problem. Find 𝑣ε ∈ H 1 (Ω \ ω ε (x0 ); ℂ) such that 𝑣ε = 0 on Γ D , ∫

(3.51a)

(∇𝑣ε ⋅ ∇u − k 2 𝑣ε u) dx = − ∫ (u ε − u ⋆ )u dS x

Ω\ω ε (x 0 )

ΓN

for all u ∈ H (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D . 1

(3.51b)

Proof. Indeed, using variational calculus from Proposition 2.1, the first-order optimality condition for (3.49) necessitates four variational inequalities: ∂ ⟨ ∂Re(𝑣) L ε (u ε , 𝑣ε ), Re(𝑣 − 𝑣ε )⟩ ≥ 0, ∂ ⟨ ∂Re(u) L ε (u ε , 𝑣ε ), Re(u − u ε )⟩ ≥ 0,

∂ ⟨ ∂Im(𝑣) L ε (u ε , 𝑣ε ), Im(𝑣 − 𝑣ε )⟩ ≤ 0 , ∂ ⟨ ∂Im(u) L ε (u ε , 𝑣ε ), Im(u − u ε )⟩ ≤ 0

holding for all u, 𝑣 ∈ H 1 (Ω \ ω ε (x0 ); ℂ) such that u = h, 𝑣 = 0 on Γ D . Inserting here 𝑣 = 𝑣ε ± v and u = u ε ± u with v, u ∈ H 1 (Ω \ ω ε (x0 ); ℂ) such that v = u = 0 on Γ D we obtain the following four variational equations: (∇Re(u ε ) ⋅ ∇Re(v) − k 2 Re(u ε )Re(v)) dx = ∫ Re(g)Re(v) dS x ,

∫ Ω\ω ε (x 0 )

∫

ΓN

(∇Im(u ε ) ⋅ ∇Im(v) − k 2 Im(u ε )Im(v)) dx = ∫ Im(g)Im(v) dS x ,

Ω\ω ε (x 0 )

ΓN

(∇Re(𝑣ε ) ⋅ ∇Re(u) − k 2 Re(𝑣ε )Re(u)) dx = ∫ Re(u ⋆ − u ε )Re(u) dS x ,

∫ Ω\ω ε (x 0 )

∫ Ω\ω ε (x 0 )

ΓN

(∇Im(𝑣ε ) ⋅ ∇Im(u) − k 2 Im(𝑣ε )Im(u)) dx = ∫ Im(u ⋆ − u ε )Im(u) dS x . ΓN

4.3 Helmholtz problems for geometric objects under Neumann boundary condition

|

87

The summation of the first and the second equations for v = u and v = ıu constitutes the real and imaginary parts of (3.4b), while the third and the fourth equations for u = u and u = ıu contribute to (3.51b), respectively. This completes the proof. We emphasize that the Helmholtz problem (3.51) is analogous to (3.4) and differs from it by the boundary data at ∂Ω. Therefore, the well-posedness result stated for u ε remains true also for 𝑣ε . It implies the weak solution to (cf. (3.3)) −[∆ + k 2 ]𝑣ε = 0 ∂𝑣ε ∂ν ∂𝑣ε ∂ν ε

=0

in Ω \ ω ε (x0 ) ,

(3.52a)

on ∂ω ε (x0 ) ,

(3.52b)

⋆

ε

= −(u − u ) on Γ N ,

𝑣 =0

on Γ D .

(3.52c) (3.52d)

Plugging the representation (3.52c) in (3.47), we establish the following equivalence. Proposition 3.8. The topology optimization problem (3.48) can be equivalently stated with respect to the primal and dual state variables. Find (ω⋆ , ε⋆ , x⋆ ) ∈ G such that 0=

{ } { } ε Re {− 12 ∫ (u ε − u ⋆ ) ∂𝑣 dS x } ∂ν { } (ω,ε,x 0)∈G { ΓN } min

subject to (3.4) and (3.51) .

(3.53)

Further we provide asymptotic analysis of the objective as ε ↘ +0. As ε = 0, problem (3.51) turns into the dual background problem stated in the reference domain Ω. Find 𝑣0 ∈ H 1 (Ω; ℂ) such that 𝑣0 = 0 on Γ D ,

(3.54a)

∫(∇𝑣0 ⋅ ∇u − k 2 𝑣0 u) dx = − ∫ (u 0 − u ⋆ )u dS x ΓN

Ω 1

for all u ∈ H (Ω; ℂ) : u = 0 on Γ D ,

(3.54b)

which implies the weak solution to (cf. (2.1)) −[∆ + k 2 ]𝑣0 = 0 0

∂𝑣 ∂ν 0

in Ω ,

= −(u 0 − u ⋆ ) on Γ N ,

𝑣 =0

on Γ D .

(3.55a) (3.55b) (3.55c)

Problem (3.54) is similar to the primal background problem (2.2), henceforth, all the results of Section 4.2 hold true for (3.54), too. In particular, the inner asymptotic expansion holds in the near-field B R (x0 ) ⊂ Ω in the form 𝑣0 (x) = 𝑣0 (x0 )J 0 (kρ) + V00 (x),

V00 (x) = 2k J 1 (kρ)∇𝑣0 (x0 ) ⋅ ̂x + V10 (x)

(3.56)

88 | 4 High-order topological expansions for Helmholtz problems in 2D with the residuals V00 , V10 ∈ H 1 (B R (x0 ); ℂ) such that π

∫

π

V00

dθ = ∫

−π

π

dθ = ∫ V10 ̂x dθ = 0 ,

V10

−π

(3.57a)

−π

V00 (x0 + ρ ̂x ) = O(ρ),

V10 (x0 + ρ ̂x) = O(ρ 2 )

for θ ∈ (−π, π] ,

(3.57b)

and it implies similar to (2.35) representation of the gradient ∇𝑣0 (x) = ∇𝑣0 (x0 ) + b 0𝑣 (x) + ∇V10 (x), b 0𝑣 (x)

0

:= (𝑣 +

(x0 )ka󸀠0 (kρ)

+

a 1 (kρ) 0 kρ (∇𝑣 (x 0 )

∇V10 = O(ρ) ,

a󸀠1 (kρ)∇𝑣0 (x0 ) ̂󸀠

̂󸀠

⋅ x )x ,

b 0𝑣

(3.58a)

⋅ x̂) ̂x

= O(ρ) .

(3.58b)

We note that the expansion of 𝑣ε − 𝑣0 for ε ↘ +0 in the manner of Theorem 3.6 would be a hard task since the right-hand side of problem (3.51) itself depends on u ε . In this respect, Proposition 3.8 will not be helpful. Instead, decomposing u ε − u ⋆ = u ε − u 0 + u 0 − u ⋆ and using (3.55b), we express the objective in (3.47) equivalently { } { } 0 J(ω, ε, x0 ) = J 0 − Re { ∫ (u ε − u 0 ) ∂𝑣 dS x } + ∂ν { } {Γ N } J0 := 12 ∫ |u 0 − u ⋆ |2 dS x = O(1), ΓN

1 2

1 2

∫ |u ε − u 0 |2 dS x , ΓN

(3.59)

∫ |u ε − u 0 |2 dS x = O(ε4 | ln ε|) ΓN

for ε ↘ +0 due to Theorem 3.6. From (3.59), we infer the asymptotic result below. Theorem 3.9. The objective in (3.47) admits the high-order asymptotic expansion J(ω, ε, x0 ) = J 0 + Re {ε2 J 1N (ω, x0 ) + J 2ε + J 3ε + J 4ε } + O(ε4 | ln ε|) ,

(3.60)

where the asymptotic terms are expressed by formulas: J1N (ω, x0 ) := −∇u 0 (x0 )⊤ M ω ∇𝑣0 (x0 ) + k 2 meas2 (ω)u 0 (x0 )𝑣0 (x0 ) ,

(3.61a)

π 1 0 0 ̂ J2ε := ε3 √| ln ε| ∫ ∇u 0 (x0 ) ⋅ ( ∂u ∂ρ 𝑣 (x 0 ) − u ∇𝑣 (x 0 ) ⋅ x ) dθ 1

−π

= O(ε

3√

| ln ε|) ,

(3.61b)

π

J 3ε := ε2 ∫ ∇u 0 (x0 ) ⋅ (

∂W νε 0 ∂ρ V1

− W νε

∂V 10 ∂ρ ) dθ

−π

−

∫ ν ⋅ (q1ε ∇𝑣0 (x0 ) + (b 0u + ∇U10 )V00 ) dS x = O(ε3 ) , ∂ω ε (x 0 )

(3.61c)

4.3 Helmholtz problems for geometric objects under Neumann boundary condition

| 89

π 1 󸀠 0 0 ̂ J 4ε := ε2 √| ln ε|(ε ∫ ∇u 0 (x0 ) ⋅ { ∂u ∂ρ ε∇𝑣 (x 0 ) ⋅ x − u (𝑣 (x 0 )ka0 + 1

∂V 10 ∂ρ )} dθ

−π

−

ν ⋅ (Du 1 )∇u 0 (x0 )V00 dS x ) = O(ε4 √| ln ε|) .

∫

(3.61d)

∂ω ε (x 0 )

Proof. To verify (3.60), it needs to expand up to O(ε4 | ln ε|)-order asymptotic terms in the boundary integral in (3.59) 0

I(u ε − u 0 , 𝑣0 ) := − ∫ (u ε − u 0 ) ∂𝑣 ∂ν dS x . ΓN

For this task, we employ the second Green formula in Ω \ B ε (x0 ) and rewrite I as I(u ε − u 0 , 𝑣0 ) =

(∆(u ε − u 0 )𝑣0 − (u ε − u 0 )∆𝑣0 ) dx

∫ Ω\B ε (x 0 )

+

∫

( ∂(u∂ρ−u ) 𝑣0 − (u ε − u 0 ) ∂𝑣 ∂ρ ) dS x , ε

0

0

(3.62)

∂B ε (x 0 )

where the domain integral disappears according to the Helmholtz equations (2.1a), (3.3a), and (3.55a). On the one hand, we note that I is an invariant integral which can be written over arbitrary Lipschitz-smooth boundary ∂O of a domain O such that ω ε (x0 ) ⊆ O ⊂ Ω. On the other hand, the integral over the circle ∂B ε (x0 ) is advantageous while it can be calculated by substitution of the uniform expansion (3.43) for u ε − u 0 and the inner expansion (3.56) and (3.58) for 𝑣0 . By doing so, we decompose I(u ε − u 0 , 𝑣0 ) = I(q1ε , 𝑣0 ) + I(u ε − u 0 − q1ε , 𝑣0 ) with the residual q1ε from Theorem 3.6 and calculate these two integrals separately. First, applying to I(q1ε , 𝑣0 ) the second Green formula in B ε (x0 ) \ ω ε (x0 ) we obtain I(q1ε , 𝑣0 ) :=

∫

(

∂B ε (x 0 )

=−

∂q1ε 0 ∂ρ 𝑣

0

− q1ε ∂𝑣 ∂ρ ) dS x =

∫

(

∂q1ε 0 ∂ν 𝑣

0

− q1ε ∂𝑣 ∂ν ) dS x

∂ω ε (x 0 )

∫

ν ⋅ {q1ε (∇𝑣0 (x0 ) + b 0𝑣 + ∇V10 )

∂ω ε (x 0 )

+ (b 0u + ∇U10 + ε2 √| ln ε|(Du 1 )∇u 0 (x0 )) (𝑣0 (x0 )(1 + a0 ) + V00 )} dS x after substitution of (3.46b), (3.56), and (3.58a). The divergence theorem provides −

∫

ν ⋅ (b 0u + ∇U10 + ε2 √| ln ε|(Du 1 )∇u 0 (x0 )) 𝑣0 (x0 ) dS x

∂ω ε (x 0 )

= − ∫ div (b 0u + ∇U10 + ε2 √| ln ε|(Du 1 )∇u 0 (x0 )) 𝑣0 (x0 ) dx . ω ε (x 0 )

(3.63)

90 | 4 High-order topological expansions for Helmholtz problems in 2D 2

Here − div(∇U10 ) = −∆U10 = k 2 U10 due to (2.1a) and (2.33), b 0u = −u 0 (x0 ) k2ρ ̂x + O(ρ 2 ) according to (2.8) and (2.35b), and since ρ ̂x = x − x0 then q

I1 :=

∫ div (u 0 (x0 ) k

2

(x−x 0 ) 0 𝑣 (x0 )) dx 2

= k 2 u 0 (x0 )𝑣0 (x0 ) ∫ dx .

ω ε (x 0 )

(3.64)

ω ε (x 0 )

With the help of (3.63) and (3.64), we collect the asymptotic terms of the same order q

q

q

(the term I2 of order O(ε3 √| ln ε|) is zero) ,

q

q

I(q1ε , 𝑣0 ) = I1 + I3 + I4 + I5 q

I3 := −

ν ⋅ (q1ε ∇𝑣0 (x0 ) + (b 0u + ∇U10 )V00 ) dS x = O(ε3 ) ,

∫ ∂ω ε (x 0 )

q I4

:= −ε2 √| ln ε|

ν ⋅ (Du 1 )∇u 0 (x0 )V00 dS x = O(ε4 √| ln ε|) ,

∫ ∂ω ε (x 0 )

q I5

:= − ∫ div (b 0u + u 0 (x0 ) k

2

(x−x 0 ) 2

+ ε2 √| ln ε|(Du 1 )∇u 0 (x0 )) 𝑣0 (x0 ) dx

ω ε (x 0 )

−

∫ {ν ⋅ (b 0u + ∇U10 + ε2 √| ln ε|(Du 1 )∇u 0 (x0 )) 𝑣0 (x0 )a0 ∂ω ε (x 0 )

+ q1ε (b 0𝑣 + ∇V10 )} dS x + ∫ k 2 U10 𝑣0 (x0 ) dx = O(ε4 ) ω ε (x 0 )

in view of the following asymptotic relations: ∂q1ε ∂ρ

q1ε = U10 = V10 = O(ε2 ), V00 = b 0u = b 0𝑣 = O(ε),

=

∂U 10 ∂ρ

=

∂V 10 ∂ρ

u1 =

= O(ε),

∫ dx = ε2 meas2 (ω), ω ε (x 0 )

∂u1 ∂ρ

= O(1) ,

∫ dS x = εmeas1 (∂ω) ,

(3.65)

∂ω ε (x 0 )

which hold in B ε (x0 ) due to (2.34b), (2.35b), (3.43), and (3.57b). Second, inserting in I the representations (3.19), (3.43), (3.56), and (3.58a) we have I(u ε − u 0 − q1ε , 𝑣0 ) :=

∫

(

∂(u ε −u0 −q1ε ) 0 𝑣 ∂ρ

0

− (u ε − u 0 − q1ε ) ∂𝑣 ∂ρ ) dS x

∂B ε (x 0 ) π 1 = ε ∫ ∇u 0 (x0 ) ⋅ {(− 2πε M ω ̂x +

∂W νε ∂ρ

0 + ε√| ln ε| ∂u ∂ρ ) (𝑣 (x 0 )(1 + a0 ) 1

−π

+ (ε +

a1 0 k )∇𝑣 (x 0 )

1 ⋅ ̂x + V10 ) − ( 2π M ω ̂x + W νε + ε√| ln ε|u 1 )

× (∇𝑣0 (x0 ) ⋅ x̂ + b 0𝑣 ⋅ ̂x +

∂V 10 ∂ρ )} ε dθ

= I1w + I2w + I3w + I4w + I5w . (3.66)

4.3 Helmholtz problems for geometric objects under Neumann boundary condition

| 91

We calculate the asymptotic terms I1w , I2w , I3w , I4w , I5w in (3.66) by using the orthogonality (3.20a) and (3.57a), calculus (2.27) providing π

∫ ∇u 0 (x0 ) ⋅

1 2π (M ω

̂x)(∇𝑣0 (x0 ) ⋅ ̂x) dθ = 12 ∇u 0 (x0 )⊤ M ω ∇𝑣0 (x0 ) ,

(3.67)

−π

and the following relations hold at ∂B ε (x0 ) due to (2.8), (3.20b), and (3.58b): b 0𝑣 ⋅ ̂x = 𝑣0 (x0 )ka󸀠0 + a󸀠1 ∇𝑣0 (x0 ) ⋅ ̂x , a󸀠0 = O(ε),

a0 = O(ε2 ),

W νε = O(1),

a1 = O(ε3 ),

∂W νε ∂ρ

= O( 1ε ) ,

a󸀠1 = O(ε2 ) .

(3.68)

With the help of (3.65), (3.67), and (3.68) the calculation results in I1w = −ε2 ∇u 0 (x0 )⊤ M ω ∇𝑣0 (x0 ) , π

I2w

:= ε

3√

1 3√ 0 0 ̂ | ln ε| ∫ ∇u 0 (x0 ) ⋅ ( ∂u | ln ε|) , ∂ρ 𝑣 (x 0 ) − u ∇𝑣 (x 0 ) ⋅ x ) dθ = O(ε 1

−π π

I3w := ε2∫∇u 0 (x0 ) ⋅ (

∂W νε 0 ∂ρ V1

− W νε

∂V 10 ∂ρ ) dθ

= O(ε3 ) ,

−π π 1 󸀠 0 0 ̂ I4w := ε3 √| ln ε| ∫ ∇u 0 (x0 ) ⋅ { ∂u ∂ρ ε∇𝑣 (x 0 ) ⋅ x − u (𝑣 (x 0 )ka0 + 1

∂V 10 ∂ρ )} dθ

−π

= O(ε4 √| ln ε|) , π 0 I5w := ε3 √| ln ε| ∫ ∇u 0 (x0 ) ⋅ { ∂u ∂ρ (𝑣 (x 0 )a0 + 1

a1 0 kε ∇𝑣 (x 0 )

⋅ ̂x + V10 )

−π

−u

1

(a󸀠1 ∇𝑣0 (x0 )

⋅ ̂x)} dθ − 2ε ( ak1 + εa󸀠1 )∇u 0 (x0 )⊤ M ω ∇𝑣0 (x0 ) = O(ε4 ) .

q

q

q

Finally, the summation I1 + I1w = ε2 J 1N , I2w = J 2ε , I3 + I3w = J 3ε , and I4 + I4w = J 4ε gathers the asymptotic terms in (3.61) and finishes the proof. We make a few remarks on corollaries following from Theorem 3.9. The first-order asymptotic term Re(J1N ) given in formula (3.61a) is called the topological derivative of the objective J following the terminology of [57] since Re(J 1N (ω, x0 )) = lim

1 2 ε↘+0 ε

(J(ω, ε, x0 ) − J 0 ) .

(3.69)

It was found, e.g., in [3, 8]. After approximation by (3.60), the optimization problem (3.48) forces the shapetopological control problem. For fixed ε ∈ Gε , find (ω⋆ , x⋆ ) ∈ Gω × Gx such that Re(J1N (ω⋆ , x⋆ )) =

min

(ω,x 0 )∈Gω ×Gx

Re(J 1N (ω, x0 )) .

(3.70)

92 | 4 High-order topological expansions for Helmholtz problems in 2D

The approximated objective in (3.70) does not depend on the perturbed state. It is expressed by the reference solutions u 0 and 𝑣0 of the primal (2.2) and the dual (3.54) background Helmholtz problems, as well as the solution w ν of the exterior Neumann problem for the Laplace operator (3.16). The boundary layer w ν enters formula (3.61a) via the virtual mass tensor M ω . Therefore, employing the explicit description of M ω given for ellipsoidal shapes in Lemma 3.5, the control problem (3.70) can be relaxed by reducing the set of admissible shapes Gω to a family of ellipses depending on rotation and compression. The problem of identification of the center x⋆ of the test object will be discussed further in Section 4.5.4. In Section 4.4, we modify our methods to treat forward and inverse problems for the Helmholtz equation under Dirichlet boundary conditions at ∂ω ε (x0 ).

4.4 Helmholtz problems for geometric objects under Dirichlet (sound soft) boundary condition Given g ∈ L2 (Γ N ; ℂ) and h ∈ H 1/2 (Γ D ; ℂ), the (forward) Dirichlet problem for the Helmholtz equation consists in finding the wave potential u ε (x) fulfilling −[∆ + k 2 ]u ε = 0 in Ω \ ω ε (x0 ) , ε

u = 0 on ∂ω ε (x0 ) , ∂u ε ∂ν ε

(4.1a) (4.1b)

=g

on Γ N ,

(4.1c)

u =h

on Γ D ,

(4.1d)

which is described by the variational problem. Find u ε ∈ H 1 (Ω \ ω ε (x0 ); ℂ) such that uε = h ∫

on Γ D ,

u ε = 0 on ∂ω ε (x0 ) ,

(4.2a)

(∇u ε ⋅ ∇u − k 2 u ε u) dx = ∫ gu dS x

Ω\ω ε (x 0 )

ΓN 1

for all u ∈ H (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D ∪ ∂ω ε (x0 ) .

(4.2b)

The well-posedness of the Dirichlet problem (4.2) is argued similar to the Neumann problem (3.4). It implies necessary condition for the variational principle with Pε from (3.6): Pε (u ε ) = min max Pε (𝑣) Re(𝑣) Im(𝑣) (4.3) 1 over 𝑣 ∈ H (Ω \ ω ε (x0 ); ℂ) : 𝑣 = h on Γ D , 𝑣 = 0 on ∂ω ε (x0 ) . To evaluate the difference of u ε from the background solution u 0 , the outer asymptotic expansion in the far field is needed; see [33, Section 3.3].

4.4 Helmholtz problems for geometric objects under Dirichlet boundary condition

|

93

4.4.1 Outer and inner asymptotic expansions by Fourier series We construct three auxiliary problems: two boundary layers in the exterior domain ℝ2 \ ω and a regularized Helmholtz problem in Ω for the logarithm. First, we define the kernel of the Laplace operator in ℝ2 by means of the logarithmic capacity; see, e.g., [21]. We consider the following homogeneous Dirichlet problem: −∆w00 = 0 w00 = 0

in ℝ2 \ ω ,

(4.4a)

on ∂ω ,

(4.4b)

w00 = O(ln |y|) as |y| ↗ ∞ .

(4.4c)

The weak variational formulation to (4.4) can be given in the weighted Sobolev spaces introduced in Section 4.3.1. For p > 2 and p󸀠 < 2 such that 1p + p1󸀠 = 1, find w00 ∈ 1,p

W μ (ℝ2 \ ω; ℝ) such that w00 = 0

on ∂ω ,

(4.5a) 1,p

∫ ∇w00 ⋅ ∇𝑣 dy = 0

for all 𝑣 ∈ W μ

󸀠

(ℝ2 \ ω; ℝ) : 𝑣 = 0 on ∂ω .

(4.5b)

ℝ2 \ω

The existence of a nontrivial solution to (4.5) is argued as follows. Following [7], ̃ ∈ there exists a unique solution of the inhomogeneous Dirichlet problem. Find w 1,2 W μ (ℝ2 \ ω; ℝ) such that ̃ = ln |y| on ∂ω , w ̃ ⋅ ∇𝑣 dy = 0 ∫ ∇w

for all 𝑣 ∈ W μ1,2 (ℝ2 \ ω; ℝ) : 𝑣 = 0 on ∂ω .

ℝ2 \ω 1,p ̃ solves (4.5). It admits the Since ln |y| ∈ W μ (ℝ2 \ ω; ℝ) for p > 2, then w00 = ln |y| − w 2 Fourier series (cf. (3.23)) in ℝ \ B1 (0):

w00 (y) = − ln(cap(ω)) + ln |y| + W00 ,

∞

W00 = ∑ n=1

00 ̂ n 1 |y|n C n x

,

(4.6)

2×2 , n ∈ ℕ, and cap(ω) ∈ ℝ called logarithmic capacity of the set ω. with C00 + n ∈ℝ 0 After rescaling y = x−x ε , the exterior problem (4.5) can be reduced to the bounded domain Ω \ ω ε (x0 ) with the help of the following Green formula holding for every function u ∈ H 1 (Ω \ ω ε (x0 ); ℂ) such that ∆u ∈ L2 (Ω \ ω ε (x0 ); ℂ) (cf. (3.7)):

∫

(∇u ⋅ ∇u + u ∆u) dx = ⟨ ∂u ∂ν , u⟩ Γ N (4.7)

Ω\ω ε (x 0 ) 1

for all u ∈ H (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D ∪ ∂ω ε (x0 ) . Then relations (4.5)–(4.7) prove the assertion of Lemma 4.1.

94 | 4 High-order topological expansions for Helmholtz problems in 2D ε 0 Lemma 4.1. The rescaled solution w00 (x) := w00 ( x−x ε ) to (4.5) fulfills ε w00 = 0 on ∂ω ε (x0 ) , ε ∇w00 ⋅ ∇u dx = ∫

∫ Ω\ω ε (x 0 )

(4.8a)

ε ∂w 00 ∂ν u dS x

ΓN

1

for all u ∈ H (Ω \ ω ε (x0 ); ℝ) : u = 0 on Γ D ∪ ∂ω ε (x0 ) .

(4.8b)

It admits the far-field representation in the Fourier series ε ε w00 (x) = − ln(εcap(ω)) + ln ρ + W00 (x)

for x ∈ ℝ2 \ B ε (x0 ) ,

(4.9)

ε with the residual function W00 ∈ H 1 (Ω \ ω ε (x0 ); ℝ) such that π ε ∫ W00 dθ = 0,

ε W00 = O( ρε )

for ρ > ε, θ ∈ (−π, π] .

(4.10)

−π

Second, to compensate the logarithm in (4.9) which is unbounded as ρ ↘ +0, we construct the regularized Helmholtz problem in Ω. Find u ln ∈ H 1 (Ω; ℝ) such that u ln = ln ρ ∫(∇u ln ⋅ ∇u − k 2 u ln u) dx = ∫ Ω

on Γ D ,

∂(ln ρ) ∂ν u dS x

(4.11a) − ∫ k 2 u ln ρ dx

ΓN

Ω

1

for all u ∈ H (Ω; ℝ) : u = 0 on Γ D .

(4.11b)

Since ∆[ln ρ] = 0, the solution of (4.11) describes the boundary value problem: −[∆ + k 2 ]u ln = −[∆ + k 2 ] ln ρ ln

∂u ∂ν ln

u

=

∂(ln ρ) ∂ν

= ln ρ

in Ω ,

(4.12a)

on Γ N ,

(4.12b)

on Γ D .

(4.12c)

Using Green’s formula (4.7), we restate (4.11b) over the perturbed domain as ∫ (∇u ln ⋅ ∇u − k 2 u ln u) dx = ∫ Ω\ω ε (x 0 )

∂(ln ρ) ∂ν u dS x

ΓN

−

∫

k 2 u ln ρ dx

Ω\ω ε (x 0 )

(4.13)

1

for all u ∈ H (Ω \ ω ε (x0 ); ℝ) : u = 0 on Γ D ∪ ∂ω ε (x0 ) . Similarly to Lemma 2.2, we establish in the following the inner asymptotic expansion for u ln . Lemma 4.2. The solution u ln of (4.11) admits the representation in the near field u ln (x) = u ln (x0 ) + (u ln (x0 ) − ln ρ)a0 − 2π a2 + U0ln (x)

in B R (x0 ) ⊂ Ω ,

(4.14)

with a0 and a2 given in (2.8) and the residual U0ln ∈ H 1 (B R (x0 ); ℝ) such that π

∫ U0ln dθ = 0, −π

U0ln = O(ρ)

for ρ ∈ [0, R), θ ∈ (−π, π] .

(4.15)

4.4 Helmholtz problems for geometric objects under Dirichlet boundary condition

|

95

Proof. For B R (x0 ) ⊂ Ω, we decompose u ln into the radial and residual functions: ln u ln (x) = u ln 0 (ρ) + U 0 (x)

in B δ (x0 ), δ ∈ [0, R),

where

π

u ln 0 (ρ)

:=

1 2π

π ln

∫ u dθ,

U0ln

ln

:= u −

u ln 0 ,

hence ∫ U0ln dθ = 0 .

−π

(4.16)

−π

Using (4.16), we substitute a smooth cut-off function η(ρ) supported in B δ (x0 ) as the test function u = η into (4.11b) and integrate it by parts to derive that 0 = ∫ (∇u ln ⋅ ∇η − k 2 (u ln − ln ρ) η) dx B δ (x 0 ) δ

󸀠

󸀠 2 ln = 2π ∫ ((u ln 0 ) ρ η − k (u 0 − ln ρ) η) ρ dρ 0 δ

󸀠

󸀠 2 ln = −2π ∫ ((ρ(u ln 0 )ρ ) ρ + k ρ (u 0 − ln ρ)) η dρ 0

which implies the inhomogeneous Bessel equation 󸀠󸀠 2 ln 2 1 ln 󸀠 (u ln 0 )ρ + ρ (u 0 )ρ + k u 0 = k ln ρ

for ρ ∈ (0, δ) .

(4.17)

Together with the particular integral ln ρ, its general solution has the form ln ln 2 u ln 0 (ρ) = K 0 (1 + a0 ) + S 0 ( π (ln

k 2

+ ln ρ + γ)(1 + a0 ) + a2 ) + ln ρ

(4.18)

with the Bessel and Neumann functions written according to (2.8a) and (2.8c), and two ln π unknown coefficients K0ln , Sln 0 ∈ ℝ. The factor S 0 = − 2 avoids the leading logarithmic ln 1 term, thus providing the function regularity u 0 ∈ H ((0, δ); ℝ) in (4.16). The inhomogeneous Helmholtz equations (4.12a) and (4.17) imply −[∆+k 2 ]U0ln = 0 π which argues a Fourier series representation of U0ln for ρ ↘ +0. From ∫−π U0ln dθ = 0 in (4.16) we obtain the Wirtinger inequality (2.17b) for the residual U0ln . This follows the asymptotic order U0ln = O(ρ) in (4.15); see for detail the proof of Lemma 2.2. Now passing ρ ↘ +0 in (4.16) and (4.18) we find K0ln = u ln (x0 )+ (ln 2k + γ) and, consequently, we arrive at formula (4.14). This completes the proof. With the help of Lemmas 4.1 and Lemma 4.2, we construct the first-order correction ε function wln to u ε − u 0 which will be used further in Theorem 4.5. ε to (4.5) and the solution uln to (4.11) Lemma 4.3. Combining the rescaled solution w00 1 the first-order correction function in H (Ω \ ω ε (x0 ); ℝ) is formed ε ε := w00 + ln(εcap(ω)) − u ln wln

(4.19)

96 | 4 High-order topological expansions for Helmholtz problems in 2D

which fulfills the following relations: ε ε = W00 wln ε wln

on Γ D ,

(4.20a) ln

ln

= ln(εcap(ω)) − u (x0 ) + (ln ρ − u (x0 ))a0 + ε ∂W 00

ε ε ⋅ ∇u − k 2 wln u) dx = ∫ ∫ (∇wln Ω\ω ε (x 0 )

∂ν

π 2 a2

−

U0ln

on ∂ω ε (x0 ) , (4.20b)

ε ∫ k 2 W00 u dx

u dS x −

Ω\ω ε (x 0 )

ΓN 1

for all u ∈ H (Ω \ ω ε (x0 ); ℝ) : u = 0

on Γ D ∪ ∂ω ε (x0 )

(4.20c)

and admits the representation in the Fourier series ε ε = (ln ρ − u ln (x0 )) (1 + a0 ) + W00 + 2π a2 − U0ln wln

in B R (x0 ) \ B ε (x0 ) .

(4.21)

Proof. Indeed, formulas (4.20) and (4.21) are obtained by substitution of the represenε as well as the representations (4.11a), (4.13), and tations (4.8) and (4.9) holding for w00 ε ln (4.14) for u in the combination of functions w00 and u ln defined in (4.19). Third, we construct a boundary layer which realizes the second-order correction to u ε − u 0 as it will be proved further in Theorem 4.5. For this task, we consider the vector-valued exterior Dirichlet problem: −∆w y = 0 w y = −y wy =

1 O ( |y| )

in ℝ2 \ ω ,

(4.22a)

on ∂ω ,

(4.22b)

as |y| ↗ ∞ .

(4.22c)

The boundary value problem (4.22) admits the following weak formulation. Find w y = 1,2 ((w y )1 , (w y )2 )⊤ ∈ W μ (ℝ2 \ ω; ℝ)2 such that w y = −y ∫ Dw y ∇𝑣 dy = 0

on ∂ω ,

(4.23a) 1,2

for all 𝑣 ∈ W μ (ℝ2 \ ω; ℝ) : 𝑣 = 0 on ∂ω .

(4.23b)

ℝ2 \ω

The unique solution to (4.23) is guaranteed by existence theorems in [7]. Moreover, Dw y ν ∈ H −1/2 (∂ω; ℝ)2 is well defined at the boundary ∂ω by Green’s 1,2 formula holding for harmonic functions u ∈ W μ (ℝ2 \ ω; ℂ) with ∆u = 0 (see [7]): ∫ ∇u ⋅ ∇u dy = −⟨ ∂u ∂ν , u⟩ ∂ω

1,2

for all u ∈ W μ (ℝ2 \ ω; ℂ)

(4.24)

ℝ2 \ω 1/2 (∂ω; ℂ) and ∂u ∈ H −1/2 (∂ω; ℂ). with the paring ⟨ ∂u ∂ν , u⟩ ∂ω between u ∈ H ∂ν x−x 0 After rescaling y = ε , we reduce the exterior Dirichlet problem to the bounded domain Ω \ ω ε (x0 ) which is described in the following lemma.

4.4 Helmholtz problems for geometric objects under Dirichlet boundary condition

| 97

0 Lemma 4.4. The rescaled solution w εy (x) := w y ( x−x ε ) to (4.23) implies the vector funcε 1 2 tion w y ∈ H (Ω \ ω ε (x0 ); ℝ) which fulfills the following relations: 0 w εy = − x−x ε

on ∂ω ε (x0 ) ,

∫

Dw εy ∇u dx = ∫ (Dw εy ν)u dS x

Ω\ω ε (x 0 )

ΓN

1

for all u ∈ H (Ω \ ω ε (x0 ); ℝ) : u = 0 on Γ D ∪ ∂ω ε (x0 ) .

(4.25a)

(4.25b)

It admits the far-field representation in the Fourier series w εy (x) =

ε 1 ρ 2π P ω

̂x + W yε (x)

for x ∈ ℝ2 \ B ε (x0 ) ,

(4.26)

with P ω called polarization matrix in [53, Note G] and the residual function W yε = ((W yε )1 , (W yε )2 )⊤ ∈ H 1 (Ω \ ω ε (x0 ); ℝ)2 such that for ρ > ε, θ ∈ (−π, π] it holds π

∫

π

W yε

dθ = ∫ W yε ̂x dθ = 0,

−π

W yε = O (( ρε )2 ) .

(4.27)

−π

The entries of the 2 × 2 matrix P ω have the implicit expression (cf. (3.28)): (P ω )ij = −δ ij meas2 (ω) − ⟨

∂(w y )i ∂ν , y j ⟩ ∂ω ,

i, j = 1, 2 .

(4.28)

−P ω is symmetric positive semidefinite (Spsd), and symmetric positive definite (Spd) if meas2 (ω) > 0. For ellipsoidal shapes ω, it has the explicit expression (cf. (3.29)) P ω = Θ(α)P ω󸀠 Θ(α)⊤ ,

P ω󸀠 = −π(a + b) (

a 0

0 ) b

(4.29)

with the ellipse major a = 1 and minor b ∈ (0, 1] semiaxes, where the major axis has an angle of α ∈ (− 2π , 2π ) with the y1 -axis counted in the anticlockwise direction. Proof. Following the proof of Lemma 3.4 and employing the radial functions ̂x n from Proposition 2.3, the harmonic function w y in (4.22) obeys the Fourier series x ∞

w y (y) = ∑ n=1

y n 1 ̂ |y|n C n x

for y ∈ ℝ2 \ B1 (0)

y

with unknown coefficient matrices C n ∈ ℝ2×2 , n ∈ ℕ. This implies w y (y) =

1 1 ̂ |y| 2π P ω x

+ W y (y),

1 2π P ω

y

:= C1 ,

y ∈ ℝ2 \ B1 (0)

(4.30)

with the residual W y = ((W y )1 , (W y )2 )⊤ ∈ W μ (ℝ2 \ ω; ℝ)2 such that 1,2

π

π

∫ W y dθ = ∫ W y ̂x dθ = 0, W y = O(|y|2 ) for |y| > 1, θ ∈ (−π, π] . −π

−π

(4.31)

98 | 4 High-order topological expansions for Helmholtz problems in 2D 0 After the transformation y = x−x ε using the calculus (3.22), from (4.30) and (4.31) we x−x 0 ε ε 0 obtain (4.26) and (4.27) for w y (x) := w y ( x−x ε ) and W y (x) := W y ( ε ). x−x 0 The coordinate transformation y = ε applied to the boundary value problem (4.22) and supported by the differential calculus in (3.26) leads to the relations

−∆w εy = 0 w εy

=

0 − x−x ε

in ℝ2 \ ω ε (x0 ) ,

(4.32a)

on ∂ω ε (x0 ) .

(4.32b)

The solution to (4.32a) is locally H 2 -smooth, hence Dw εy ν ∈ L2 (Γ N ; ℝ)2 , and the inclusion w εy ∈ W μ1,2 (ℝ2 \ ω ε (x0 ); ℝ)2 implies w εy ∈ H 1 (Ω \ ω ε (x0 ); ℝ)2 . Therefore, applying Green’s formula (4.7) to w εy , from (4.32) we derive the weak formulation (4.25) of the transformed problem in the bounded domain. To obtain the expressions of matrix P ω , we follow the lines in the proof of Lemma 3.5. In the near-field B1 (0) \ ω, due to (4.22a) from the second Green formula, we have ∫ {

∂(w y )i ∂|y| y j

∂y

− (w y )i ∂|y|j } dS y = ⟨

∂(w y )i ∂ν

, y j ⟩∂ω − ∫(w y )i

∂B1 (0)

∂y j ∂ν

dS y ,

i, j = 1, 2 ,

∂ω

with the duality pairing defined in (4.24), and the Dirichlet condition (4.22b) follows − ∫ {

∂(w y )i ∂|y|

− (w y )i } ̂x j dS y = −⟨

∂(w y )i ∂ν , y j ⟩ ∂ω

− ∫ y i ν j dS y .

∂B1 (0)

(4.33)

∂ω

Using (4.30) and (4.31), we calculate the integral on the left-hand side of (4.33) as ∂(w ) {− ∂|y|y i

∫

+ (w y )i } ̂x j dS y =

π 2

1 π

∫ ∑ (P ω )il ̂x l x̂ j dθ = (P ω )ij −π l=1

∂B1 (0)

due to (2.27). Applying to the right-hand side of (4.33) the divergence theorem (3.31) this results in expression (4.28). With the help of the Green formula (4.24) and relations (4.22), we derive that ∫ ∇(w y )i ⋅ ∇(w y )j dy = ⟨

∂(w y )i ∂ν , y j ⟩ ∂ω

=⟨

∂(w y )j ∂ν , y i ⟩ ∂ω

,

(4.34)

ℝ2 \ω

which proves the symmetry (P ω )ij = (P ω )ji in (4.28) as well as the nonnegativeness 2

0 ≤ ∫ |∇ (ξ1 (w y )1 + ξ2 (w y )2 ) |2 dy = ∑ ⟨

∂(w y )i ∂ν ξ i , y j ξ j ⟩ ∂ω

i,j=1

ℝ2 \ω

for arbitrary ξ = (ξ1 , ξ2 )⊤ ∈ ℝ2 . Therefore, multiplying (4.28) with −ξ i ξ j , we obtain 2

2

− ∑ (P ω )ij ξ i ξ j = |ξ|2 meas2 (ω) + ∑ ⟨ i,j=1

∂(w y )i ∂ν ξ i , y j ξ j ⟩ ∂ω

≥ |ξ|2 meas2 (ω) .

i,j=1

This implies that −P ω ∈ Spsd(ℝ2×2 ), and −P ω ∈ Spd(ℝ2×2 ) if meas2 (ω) > 0.

4.4 Helmholtz problems for geometric objects under Dirichlet boundary condition

| 99

Finally, let ω󸀠 be the canonical ellipsoidal shape with the major a = 1 and the minor b ∈ (0, 1] semiaxes in respect to Cartesian coordinates (y󸀠1 , y󸀠2 )⊤ . The ellipse can be written in the elliptic coordinates (3.32) in the form (3.33). Composing the Fourier series for the solution w y󸀠 of (4.22) in ℝ2 \ ω󸀠 (cf. (3.36)): ∞

y󸀠

w y󸀠 = ∑ e−nr C n (cos(nψ), sin(nψ))⊤

for r > r0 ,

(4.35)

n=1 y󸀠

the unknown coefficient matrices C n ∈ ℝ2×2 in (4.35) are determined from the boundary condition (4.22b) ∞

y󸀠 󵄨 w y󸀠 󵄨󵄨󵄨r=r0 = ∑ e−nr0 C n (cos(nψ), sin(nψ))⊤ = −(a cos ψ, b sin ψ)⊤

on ∂ω󸀠

n=1

y󸀠

y󸀠

as e−r0 C1 (cos ψ, sin ψ)⊤ = −(a cos ψ, b sin ψ)⊤ and C n = 0 for all n ≥ 2. Henceforth, we obtain the solution analytically w y󸀠 = −e r0 −r (a cos ψ, b sin ψ)⊤

for r ≥ r0 .

(4.36)

Now, we calculate the matrix P ω󸀠 explicitly by substituting (4.36) in the representation formula (4.28). Indeed, using (3.35) the normal derivative of w y󸀠 at ∂ω󸀠 is found ∂w y󸀠 ∂ν 󸀠

=

∂w y󸀠 󵄨 1 󵄨 𝜘(r 0 ,ψ) ∂r 󵄨󵄨r=r 0

=

⊤ 1 𝜘(r 0 ,ψ) (a cos ψ, b sin ψ)

= ( ab ν󸀠1 , ba ν󸀠2 )⊤ ,

and it leads to the following two vectors in (4.28) for j = 1, 2: (P ω󸀠 )( ⋅ ,j) = −δ( ⋅ ,j) meas2 (ω󸀠 ) − I( ⋅ ,j) ,

⊤

I( ⋅ ,j) := ∫ ( ab ν󸀠1 y󸀠j , ba ν󸀠2 y󸀠j ) dS y .

(4.37)

∂ω󸀠

Using the divergence theorem, we calculate two elliptic integrals I( ⋅ ,j) in (4.37) {( a , 0)⊤ I( ⋅ ,j) = ∫ ( ab y󸀠j,1 ba y󸀠j,2 )⊤ dy󸀠 = meas2 (ω󸀠 ) { b (0, ba )⊤ { ω󸀠

for j = 1, for j = 2,

and arrive at the second formula in (4.29). The first formula in (4.29) is justified by the rotation of y󸀠 ∈ ω󸀠 to Θ⊤ (α)y ∈ ω using the orthogonal matrix Θ(α) given in (3.29a) with the angle of α ∈ (− π2 , 2π ). Indeed, the coordinate transformation y = Θy󸀠 applied to the Dirichlet problem (4.23) reads w y (Θy󸀠 ) = −Θy󸀠 ∫ Dw y (Θy󸀠 )∇𝑣 dy󸀠 = 0

on ∂ω󸀠 , for all 𝑣 ∈ W μ (ℝ2 \ ω󸀠 ; ℝ) : 𝑣 = 0 on ∂ω󸀠 . 1,2

ℝ2 \ω󸀠

Henceforth, w y󸀠 (y󸀠 ) = Θ⊤ w y (Θy󸀠 ) and the representation (4.30) provides Θ⊤ P ω Θ = P ω󸀠 , for detail see the proof of Lemma 3.5. Our proof is finished. We remark that, in the limit case when b ↘ +0, explicit formulas (4.29) in Lemma 4.4 describe the singular polarization matrix P ω for the straight crack; see [45].

100 | 4 High-order topological expansions for Helmholtz problems in 2D

4.4.2 High-order uniform asymptotic expansion of the Dirichlet problem ε and w εy given in LemWith the help of the first- and second-order correction terms wln ε 0 mas 4.3 and 4.4 we decompose the residual u − u for the Dirichlet problem (4.2).

Theorem 4.5. The solutions u 0 of (2.2), u ε of (4.2), w00 of (4.5) and u ln of (4.11) comε posed together in the function wln given in (4.19), and the solution w y of (4.23) satisfies the following residual error estimate: ‖q2ε ‖H1 (Ω\ω ε (x

0 );ℂ)

= O(√

ε ) | ln ε|

q2ε := u ε − u 0 −

,

(4.38a)

ε u0 (x 0 )w ln uln (x 0 )−ln(εcap(ω))

− ε∇u 0 (x0 ) ⋅ w εy .

(4.38b)

Proof. For q2ε introduced in (4.38b), we use the boundary conditions (2.2a), (4.2a), and (4.20a) at Γ D to obtain (4.41a). We apply the local representations (4.9), (4.21), and u 0 (x) = u 0 (x0 )(1 + a0 ) + (ρ +

a1 0 k )∇u (x 0 )

⋅ ̂x + U10

in B ε (x0 )

(4.39)

holding due to (2.8) and (2.33), the boundary conditions (4.32b) and (4.20b) implying ε w ln uln (x 0 )−ln(εcap(ω))

= −1 +

π (ln ρ−uln (x 0 ))a 0 + 2 a 2 −U 0ln uln (x 0 )−ln(εcap(ω))

on ∂ω ε (x0 ) ,

(4.40)

and ρ ̂x = x − x0 to calculate the residual at ∂ω ε (x0 ): q2ε = −u 0 (x0 )a0 −

a1 0 k ∇u (x 0 )

⋅ ̂x − U10 − u 0 (x0 )

π (ln ρ−uln (x 0 ))a 0 + 2 a 2 −U 0ln uln (x 0 )−ln(εcap(ω))

,

hence (4.41b). We subtract from (4.2b) equations (3.10), (4.20c) multiplied with (u 0 (x0 ))/(u ln (x0 ) − ln(εcap(ω))), and (4.25b) multiplied with ε∇u 0 (x0 ), thus obtaining the relations: ε u0 (x 0 )W 00 0 )−ln(εcap(ω))

q2ε = − uln (x q2ε =

− ε∇u 0 (x0 ) ⋅ w εy

π (ln ρ−ln(εcap(ω)))a 0 + 2 a 2 −U 0ln −u 0 (x0 ) uln (x 0 )−ln(εcap(ω))

(∇q2ε ⋅ ∇u − k 2 q2ε u) dx =

∫ Ω\ω ε (x 0 )

on Γ D , −

(4.41a)

a1 0 k ∇u (x 0 )

⋅ ̂x − U10

ε u0 (x 0 )W 00 0 )−ln(εcap(ω))

k 2 ( uln (x

∫

on ∂ω ε (x0 ) ,

(4.41b)

+ ε∇u 0 (x0 ) ⋅ w εy ) u dx

Ω\ω ε (x 0 ) u

∂ − ∫u ∂ν ( uln (x

0

ε (x 0 )W 00

0 )−ln(εcap(ω))

+ ε∇u 0 (x0 ) ⋅ w εy ) dS x

ΓN

for all u ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D ∪ ∂ω ε (x0 ) .

(4.41c)

For lifting in the boundary conditions (4.41a) and (4.41b), respectively, the cut-off functions η Γ D supported in a neighborhood of Γ D such that η Γ D = 1 at Γ D and η εx0 with a local support in B2ε (x0 ) such that η εx0 = 1 in B ε (x0 ) are taken. We define R2ε := (u 0 (x0 )

π (ln ρ−ln(εcap(ω)))a 0 + 2 a 2 −U 0ln ln u (x 0 )−ln(εcap(ω))

ε u0 (x 0 )W 00 0 )−ln(εcap(ω))

+ ( uln (x

+

a1 0 k ∇u (x 0 )

⋅ ̂x + U10 ) η εx0

+ ε∇u 0 (x0 ) ⋅ w εy ) η Γ D = O ( | lnε ε| ) ,

Q2ε := q2ε + R2ε . (4.42)

4.4 Helmholtz problems for geometric objects under Dirichlet boundary condition

| 101

Since (1)/(u ln (x0 ) − ln(εcap(ω))) = O(1/(| ln ε|)), the asymptotic order in (4.42) is provided by a0 = a2 = U10 = O(ε2 ), a1 = O(ε3 ), U0ln = O(ε) holding due to the representaε tions (2.8), (2.34b), and (4.15) in B2ε (x0 ). At Γ D it is argued that w εy = W00 = O(ε) due to (4.10), (4.26), and (4.27). Applying the Cauchy–Schwarz inequality to (4.41c) and using (4.42), where ‖η εx0 ‖H1 (Ω\ω ε (x );ℝ) = O(1), we estimate with C, C3 , C4 > 0: 0

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 ∫ (∇Q2ε ⋅ ∇u − k 2 Q2ε u) dx󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨 ∫ (∇R2ε ⋅ ∇u − k 2 R2ε u) dx󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨Ω\ω (x ) 󵄨󵄨 󵄨󵄨Ω\ω (x ) 󵄨󵄨 󵄨 ε 0 󵄨 󵄨 ε 0 󵄨 + ≤

C3 ε ‖u‖L2 (Ω\ω ε (x );ℂ) √| ln ε| 0

Cε ‖u‖H1 (Ω\ω ε (x );ℂ) √| ln ε| 0

+

C4 ε | ln ε| ‖u‖L2 (Γ N ;ℂ)

.

(4.43)

ε = O(ε) at Γ N due to (4.10), (4.26), and (4.27). Here we have utilized w εy = W00 ε The asymptotic order √ in Ω \ ω ε (x0 ) was calculated as follows. | ln ε|

Inscribing Ω in a ball B R (x0 ) of radius R > 0 sufficiently large, we decompose it in the far-field B R (x0 ) \ B ε (x0 ) and the near-field B ε (x0 ) \ ω ε (x0 ). In the far field, R

∫

ε 2 |W00 | dx

≤

Ω\B ε (x 0 )

∫

ε 2 |W00 |

dx ≤ C ∫( ρε )2 ρ dρ = O(ε2 | ln ε|)

B R (x 0 )\B ε (x 0 )

due to (4.10), hence ‖ uln (x

ε W 00

0 )−ln(εcap(ω))

(4.44)

ε

‖L2 (B

R (x0 )\ω ε (x0 );ℂ)

= O ( √| ε

ln ε|

), and analog

R

∫ |w εy |2 dx ≤ Ω\B ε (x 0 )

∫

|w εy |2 dx ≤ C ∫( ρε )2 ρ dρ = O(ε2 | ln ε|)

B R (x 0 )\B ε (x 0 )

(4.45)

ε

due to (4.26) and (4.27), with C > 0, hence ‖εw εy ‖L2 (B (x )\ω ε (x );ℂ) = O(ε2 √| ln ε|). In the 0 R 0 0 near field, the transformation y = x−x with the calculus (3.26) provides ε ∫

ε 2 |W00 | dx = ε2

B ε (x 0 )\ω ε (x 0 )

∫

|W00 |2 dy = O(ε2 )

(4.46)

B1 (0)\ω

since W00 in (4.6) does not depend on ε. Analogously for w y from (4.23), we obtain ∫ B ε (x 0 )\ω ε (x 0 )

|w εy |2 dx = ε2

∫

|w y |2 dy = O(ε2 ) .

(4.47)

B1 (0)\ω

The estimates (4.42) and (4.43) together with the inf-sup condition (3.8) holding for all u, 𝑣 ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : u = 𝑣 = 0 on Γ D ∪ ∂ω ε (x0 ) prove the asymptotic order in (4.38a) and the assertion of the theorem. As a corollary of Theorem 4.5, we state two asymptotic expansions of the low order in two propositions as follows.

102 | 4 High-order topological expansions for Helmholtz problems in 2D Proposition 4.6. The solutions u 0 of (2.2), u ε of (4.2), w00 of (4.5) and u ln of (4.11) ε given in (4.19), satisfy the error estimate composed together in the function wln ‖q1ε ‖H1 (Ω\ω ε (x

0 );ℂ)

q1ε := u ε − u 0 −

= O(ε),

Indeed, using the coordinate transformation y = ∫

|Dw εy |2 dx ≤

Ω\ω ε (x 0 )

hence

x−x 0 ε

ε u0 (x 0 )w ln 0 )−ln(εcap(ω))

.

(4.48)

and calculus (3.26), we have

|Dw εy |2 dx = ∫ |Dw y |2 dy = O(1) ,

∫ ℝ2 \ω ε (x 0 )

‖w εy ‖H1 (Ω\ω ε (x );ℂ)2 0

uln (x

ℝ2 \ω

= O(1) due to (4.45) and (4.47), and from (4.38) it follows

‖q1ε ‖H1 (Ω\ω ε (x

0 );ℂ)

= ‖q2ε + ε∇u 0 (x0 ) ⋅ w εy ‖H1 (Ω\ω ε (x

0 );ℂ)

= O(ε) .

Proposition 4.7. The solutions u 0 of (2.2) and u ε of (4.2) satisfy the error estimate ‖u ε − u 0 ‖H1 (Ω\ω ε (x

0 );ℂ)

= O ( | ln1 ε| ) .

(4.49)

Moreover, at Γ N the estimate (4.49) can be improved as ‖u ε − u 0 ‖L2 (Γ

N ;ℂ)

= O(√

ε ) | ln ε|

.

(4.50)

Proof. The homogeneity argument and Lemma 4.3 supported by estimates (4.44) and ε (4.46) similarly provide ‖wln ‖H1 (Ω\ω ε (x );ℂ) = O(1) and leads to (4.49) for u ε − u 0 = q1ε + 0

ε (u 0 (x0 )wln )/(u ln (x0 ) − ln(εcap(ω))) in view of Proposition 4.6. ε The estimate (4.50) is argued by the representation (4.38) and w εy = W00 = O(ε) at Γ N holding due to (4.10), (4.26), and (4.27). The proof is completed.

We note that the residual estimate (4.50) will be needed for (4.53) in the next section.

4.4.3 Inverse Helmholtz problem under Dirichlet boundary condition For the objective function J defined in (3.47) and characterizing the misfit of the solution u ε to (4.2) from the known measurement u ⋆ at the boundary Γ N , we consider a state-constrained topology optimization problem (cf. (3.48)). Find the feasible test geometry (ω⋆ , ε⋆ , x⋆ ) ∈ G such that 0 = J(ω⋆ , ε⋆ , x⋆ ) =

min

(ω,ε,x 0)∈G

J(ω, ε, x0 ) subject to (4.2) .

(4.51)

For the primal state variable u ε entering the objective J, the Fenchel–Legendre duality argues a dual state variable 𝑣ε resulting necessarily from the variational principle: L ε (u ε , 𝑣ε ) =

min

max

Re(u),Re(𝑣) Im(u),Im(𝑣)

L ε (u, 𝑣) over u, 𝑣 ∈ H 1 (Ω \ ω ε (x0 ); ℂ)

such that u = h, 𝑣 = 0 on Γ D and u = 𝑣 = 0 on ∂ω ε (x0 ) , where the Lagrangian Lε has the form (3.50).

(4.52)

4.4 Helmholtz problems for geometric objects under Dirichlet boundary condition

|

103

Using the primal background solution u 0 to (2.2) and the dual background solution 𝑣0 to (3.54), similar to (3.59), we decompose the objective { } { } 0 dS x } + J(ω, ε, x0 ) = J 0 − Re { ∫ (u ε − u 0 ) ∂𝑣 ∂ν { } {Γ N } J0 := 12 ∫ |u 0 − u ⋆ |2 dS x = O(1),

∫ |u ε − u 0 |2 dS x , ΓN

(4.53) 2

∫ |u ε − u 0 |2 dS x = O ( | lnε ε| ) ,

1 2

ΓN

1 2

ΓN

where the asymptotic order in (4.53) is justified by (4.50) in Proposition 4.7. By analogy with Theorem 3.9, we prove the high-order asymptotic expansion. Theorem 4.8. The objective in (4.53) admits the high-order asymptotic expansion J(ω, ε, x0 ) = J 0 + Re { uln (x

J 1D (x 0 ) )−ln(εcap(ω)) 0

2

+ J6ε + J7ε + J 8ε + J9ε } + O( | lnε ε| )

(4.54)

for ε ↘ +0, where the asymptotic terms are expressed by formulas: J 1D (x0 ) := 2πu 0 (x0 )𝑣0 (x0 ) ,

(4.55a)

π

J6ε := ε ∫

∂q2ε 0 ∂ρ 𝑣 (x 0 ) dθ

−π π

J 7ε := ε ∫ −π

= O(√

u0 (x 0 ) uln (x 0 )−ln(εcap(ω))

(

ε ) | ln ε|

ε ∂W 00 0 ∂ρ V0

,

(4.55b)

ε − W00

∂V 00 ∂ρ ) dθ

= O( | lnε ε| ) ,

(4.55c)

π

J 8ε

2

:= ε ∫ ∇u 0 (x0 ) ⋅ (

∂W yε 0 ∂ρ V0

− W yε

∂V 00 ∂ρ ) dθ

−π

+ ε2 uln (x

ln ε

0 )−ln(εcap(ω))

k2 D 2 J 1 (x 0 )

= O (ε2 ) ,

(4.55d)

π

J 9ε := ε ∫ (

∂q2ε 0 ∂ρ V0

− q2ε

∂V 00 0 ∂ρ ) 𝑣 (x 0 ) dθ

−π

= O(√ ε

2

| ln ε|

) .

(4.55e)

Proof. The proof is based on the asymptotic expansions obtained in Section 4.4.2. We express the invariant integral in (4.53) according to (3.62) equivalently 0

I(u ε − u 0 , 𝑣0 ) := − ∫ (u ε − u 0 ) ∂𝑣 ∂ν dS x =

∫

ΓN

∂B ε (x 0 )

( ∂(u∂ρ−u ) 𝑣0 − (u ε − u 0 ) ∂𝑣 ∂ρ ) dS x . (4.56) ε

0

0

Substituting here the representations (3.56), with J 0 from (2.8a), and (4.38b), we have I(u ε − u 0 , 𝑣0 ) π

= ∫ ( ∂(u∂ρ−u ) 𝑣0 − (u ε − u 0 ) ∂𝑣 ∂ρ ) ε dθ ε

−π

0

0

104 | 4 High-order topological expansions for Helmholtz problems in 2D π

= ∫ {( uln (x −π

u0 (x 0 )

0 )−ln(εcap(ω))

− ( uln (x y

y

ε ∂w ln ∂ρ

u0 (x 0 ) ε wln 0 )−ln(εcap(ω)) y

y

y

∂w εy ∂ρ

+ ε∇u 0 (x0 ) ⋅

+

∂q2ε 0 ∂ρ ) (𝑣 (x 0 )(1

+ a0 ) + V00 ) ∂V 00 ∂ρ )} ε dθ

+ ε∇u 0 (x0 ) ⋅ w εy + q2ε ) (𝑣0 (x0 )ka󸀠0 +

y

= I1 + I2 + I3 + I4 + I5 + I6 y

y

y

y

y

y

with the integrals I1 , I2 , I3 , I4 , I5 , I6 which are defined and calculated as follows. ε Using the formula (4.21) of wln and the orthogonality in (4.10), (4.16), we find π y I1

:= ∫ −π

u0 (x 0 )𝑣0 (x 0 ) uln (x 0 )−ln(εcap(ω))

(

ε ∂w ln ∂ρ (1

ln 󸀠 0 × {( 1+a ε − u (x 0 )ka0 +

=

J 1D (x 0 ) (1 uln (x 0 )−ln(εcap(ω))

+

k2 2 2 ε

ε + a0 ) − wln ka󸀠0 ) ε dθ = 2πε uln (xu πka 󸀠2 2 ) (1

0

(x 0 )𝑣0 (x 0 )

0 )−ln(εcap(ω))

+ a0 ) − ((ln ε − u ln (x0 ))(1 + a0 ) +

πa 2 󸀠 2 ) ka0 }

3 (since a󸀠0 (kε) = − kε 2 + O(ε ))

2

ln ε) + O ( | lnε ε| )

with the notation introduced in (4.55a), while the orthogonality in (3.57a) provides π y

I2 := ∫ −π π

= ∫ −π

u0 (x 0 ) uln (x 0 )−ln(εcap(ω))

(

ε ∂w ln 0 ∂ρ V0

u0 (x 0 ) uln (x 0 )−ln(εcap(ω))

(

ε ∂(W 00 −U 0ln ) 0 V0 ∂ρ

ε − wln

∂V 00 ∂ρ )

ε dθ

ε − (W00 − U0ln )

∂V 00 ∂ρ )

ε dθ

π u0 (x 0 ) uln (x 0 )−ln(εcap(ω))

=ε ∫ −π

∂U 0ln ∂ρ

due to U0ln = V00 = O(ε), ε ∂W 00 ∂ρ

=

(

∂V 00 ∂ρ

ε ∂W 00 0 ∂ρ V0

ε − W00

∂V 00 ∂ρ ) dθ

2

+ O ( | lnε ε| )

ε = O(1) in (4.15) and (3.57b), and W00 = O(1),

= O( 1ε ) in (4.10) for ρ = ε. The expressions (4.26) and (4.27) for w εy imply that π y

I3 := ∫ ε∇u 0 (x0 ) ⋅ (

∂w εy ∂ρ (1

+ a0 ) − w εy ka󸀠0 ) 𝑣0 (x0 ) ε dθ = 0 ,

−π

and with the asymptotic order W yε = O(1),

∂W yε ∂ρ

= O( 1ε ) for ρ = ε it yields

π y I4

:= ∫ ε∇u 0 (x0 ) ⋅ (

∂w εy 0 ∂ρ V0

− w εy

∂V 00 ∂ρ ) ε dθ

−π π

= ε ∫ ∇u 0 (x0 ) ⋅ ( 2

−π

∂W yε 0 V0 ∂ρ

− W yε

∂V 00 dθ ∂ρ )

= O(ε2 ) .

4.5 Helmholtz problems for geometric objects under Robin (impedance) boundary condition |

105

Finally, the asymptotic order given in (4.38a) in Theorem 4.5 allows us to estimate π y I5

:= ∫

π ∂q ε ( ∂ρ2 (1

+ a0 ) −

q2ε ka󸀠0 ) 𝑣0 (x0 ) ε dθ

−π π y

I6 := ∫ (

∂q2ε 0 ∂ρ 𝑣 (x 0 ) dθ

=ε∫ −π

∂q2ε 0 V ∂ρ 0

− q2ε

∂V 00 𝑣0 (x0 ) ε dθ ∂ρ )

−π

since q2ε = O( √

∂q ε ε ) and ∂ρ2 | ln ε|

= O( √

1 ) at | ln ε|

= O(√ ε

2

| ln ε|

+ O ( √|ε

3

ln ε|

) ,

)

∂B ε (x0 ) by the trace theorem. y

y

y

y

y

y

Collecting the terms of the same order in I1 , I2 , I3 , I4 , I5 , I6 , we arrive at (4.55), thus proving the assertion of the theorem. We remark the asymptotic decomposition of the ε-dependent factor in (4.54) 1 uln (x 0 )−ln(εcap(ω))

=

1 − ln ε

+

uln (x 0 ) ln ε(uln (x 0 )−ln(εcap(ω)))

=

1 − ln ε

+ O ( | ln1ε|2 ) .

(4.57)

With the help of (4.57), from Theorem 4.8, we infer the corollary (cf. (3.69)): Re(J 1D (x0 )) = lim

1 −1 ε↘+0 | ln ε|

(J(ω, ε, x0 ) − J 0 ) .

(4.58)

Here the first-order asymptotic term Re(J 1D ) is given by formula (4.55a). It is called the topological derivative of the objective J for the Dirichlet problem (4.2), and in the form (4.58) was found, e.g., in [56]. We note that in comparison to the first asymptotic term Re(J 1N (ω, x0 )) for the Neumann problem, (4.58) does not depend on the shape ω. For the generalization of the concept of topological derivatives, we refer to [29, 37]. In the following section, we bridge the gap between the Neumann and Dirichlet problems by employing Robin condition with unknown parameter of the boundary impedance.

4.5 Helmholtz problems for geometric objects under Robin (impedance) boundary condition For the test object ω ε (x0 ), we introduce a parameter α ∈ ℂ of the boundary impedance which enters the Robin boundary condition for the forward Helmholtz problem: −[∆ + k 2 ]u (ε,α) = 0 (ε,α) − ∂u∂ν

+ αu

(ε,α) (ε,α)

∂u ∂ν (ε,α)

u

in Ω \ ω ε (x0 ) ,

(5.1a)

=0

on ∂ω ε (x0 ) ,

(5.1b)

=g

on Γ N ,

(5.1c)

=h

on Γ D .

(5.1d)

We mark the dependence of (5.1) on α which, on the one hand, turns into the Neumann problem (3.3) in the case of small |α| ↘ 0, and, on the other hand, into the Dirichlet

106 | 4 High-order topological expansions for Helmholtz problems in 2D problem (4.1) in the case of large |α| ↗ ∞. In this way, problem (5.1) accounts for arbitrary boundary conditions of the test object depending on the parameter α. The weak solution to (5.1) is described by the following variational problem. Find u (ε,α) ∈ H 1 (Ω \ ω ε (x0 ); ℂ) such that u (ε,α) = h (∇u (ε,α) ⋅ ∇u − k 2 u (ε,α) u) dx +

∫ Ω\ω ε (x 0 )

on Γ D ,

(5.2a)

αu (ε,α) u dS x = ∫ gu dS x

∫ ∂ω ε (x 0 )

ΓN

for all u ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D .

(5.2b)

Following Proposition 2.1 the corresponding variational principle is established below. Proposition 5.1. The variational equation (5.2) results necessarily from the variational principle: P(ε,α) (u (ε,α) ) = min max P(ε,α) (𝑣) Re(𝑣) Im(𝑣) (5.3) 1 over 𝑣 ∈ H (Ω \ ω ε (x0 ); ℂ) : 𝑣 = h on Γ D , where the Lagrangian P(ε,α) : H 1 (Ω \ ω ε (x0 ); ℂ) 󳨃→ ℝ is determined (cf. (3.6)) by { { P(ε,α) (𝑣) := Re { 12 ∫ (∇𝑣 ⋅ ∇𝑣 − k 2 𝑣2 ) dx + { { Ω\ω ε (x0 )

1 2

} } α𝑣2 dS x − ∫ g𝑣 dS x } . } ΓN ∂ω ε (x 0 ) } ∫

(5.4)

Proof. Component-wisely for 𝑣 = Re(𝑣) + ıIm(𝑣) the functional in (5.4) reads P(ε,α) (𝑣) =

{|∇(Re(𝑣))|2 − |∇(Im(𝑣))|2 − k 2 (Re(𝑣)2 − Im(𝑣)2 )} dx

∫

1 2

Ω\ω ε (x 0 )

+

{ 12 Re(α) (Re(𝑣)2 − Im(𝑣)2 ) − Im(α)Re(𝑣)Im(𝑣)} dS x

∫ ∂ω ε (x 0 )

− ∫ (Re(g)Re(𝑣) − Im(g)Im(𝑣)) dS x . ΓN

Therefore, for 𝑣 = u (ε,α) ± u in (5.3) with u ∈ H 1 (Ω \ ω ε (x0 ); ℂ) such that u = 0 on Γ D , we obtain the first-order necessary optimality conditions: ∫

(∇Re(u (ε,α) ) ⋅ ∇Re(u) − k 2 Re(u (ε,α))Re(u)) dx − ∫ Re(g)Re(u) dS x

Ω\ω ε (x 0 )

ΓN

+

∫ ∂ω ε (x 0 )

(Re(α)Re(u (ε,α)) − Im(α)Im(u (ε,α) )) Re(u) dS x = 0 ,

4.5 Helmholtz problems for geometric objects under Robin boundary condition

∫

107

|

(∇Im(u (ε,α) ) ⋅ ∇Im(u) − k 2 Im(u (ε,α) )Im(u)) dx − ∫ Im(g)Im(u) dS x

Ω\ω ε (x 0 )

ΓN

+

(Re(α)Im(u (ε,α) ) + Im(α)Re(u (ε,α) )) Im(u) dS x = 0 .

∫ ∂ω ε (x 0 )

Summing these two equations, first for u = u, and then for u = ıu, and accounting for the identities: gu = Re(g)Re(u) + Im(g)Im(u) + ı (Im(g)Re(u) − Re(g)Im(u)) ,

(5.5a)

similar ones for u (ε,α) u and the inner product ∇u (ε,α) ⋅ ∇u, and CE.2 αu (ε,α) u = Re(α) (Re(u (ε,α))Re(u) + Im(u (ε,α))Im(u)) + Im(α) (Im(u (ε,α))Re(u) − Re(u (ε,α) )Im(u)) + ı{Re(α)(Im(u (ε,α))Re(u) − Re(u (ε,α) )Im(u)) + Im(α) (Re(u (ε,α) )Re(u) + Im(u (ε,α) )Im(u))} (5.5b) it constitutes the real and imaginary parts of (5.2b), thus finishing the proof. For fixed k, under reasonable assumptions on α, there exists a unique solution of the variational equation (5.2); see, e.g., [19]. Summarizing these assumptions, α should be either not too large or it should have a definite sign such that Re(α) ≥ 0 and either Im(α) ≥ 0 or Im(α) ≤ 0. The latter case is realized for the Dirichlet boundary condition when |α| ↗ ∞. We refer to the admissible set Gα ⊂ ℂ of such α which allows the solvability of (5.2). Similar to (3.8), the uniform inf-sup condition assumes β 1 > 0 such that 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∫ (∇u ⋅ ∇𝑣 − k 2 u𝑣) dx + ∫ αu𝑣 dS x 󵄨󵄨󵄨󵄨 󵄨󵄨 ∂ω (x ) ε 0 󵄨󵄨 󵄨󵄨 󵄨󵄨Ω\ω ε (x0 ) 󵄨󵄨 (5.6) 0 < β 1 ≤ inf sup u 𝑣 ‖u‖H1 (Ω\ω ε (x );ℂ) ‖𝑣‖H1 (Ω\ω ε (x );ℂ) 0

0

H 1 (Ω\ ω

for all u, 𝑣 ∈ ε (x 0 ); ℂ) : u = 𝑣 = 0 on Γ D , admissible geometries (ω, ε, x 0 ) ∈ G, impedances α ∈ Gα , and wave numbers k ∈ [0, k 1 ], k 1 > 0.

4.5.1 Outer asymptotic expansion by Fourier series in far field To cancel the leading asymptotic term in (5.20) it needs the exterior Neumann problem: −∆w0 = 0 ∂w 0 ∂ν

=1

w0 = O(ln |y|)

in ℝ2 \ ω ,

(5.7a)

on ∂ω ,

(5.7b)

as |y| ↗ ∞ .

(5.7c)

CE.2 The sense of the sentence “The summation first for. . . ” seems to be unclear. Please check.

108 | 4 High-order topological expansions for Helmholtz problems in 2D

The unique weak solution to (5.7) can be defined in the weighted Sobolev spaces. For 1,p p > 2 and p󸀠 < 2 such that 1p + p1󸀠 = 1, find w0 ∈ W μ (ℝ2 \ ω; ℝ) \ ℙ0 such that ∫ ∇w0 ⋅ ∇𝑣 dy = − ∫ 𝑣 dS y ℝ2 \ω

1,p 󸀠

for all 𝑣 ∈ W μ

(ℝ2 \ ω; ℝ) ,

(5.8)

∂ω

since constant test functions 𝑣 are excluded in (5.8) for p󸀠 < 2 and no solvability condition for the Neumann data arises. If p = p󸀠 = 2, then (5.8) is not solvable. Indeed, to argue the solvability of (5.8), we employ the function w00 from the nontrivial kernel of the Laplace operator given in (4.4). Applying to w00 the Green formula in B1 (0) \ ω in the manner of (4.24), for the constant test function, we calculate π

0=

∂w 00 ∂|y|

∫

00 dS y − ⟨ ∂w ∂ν , 1⟩ ∂ω = ∫

−π

∂B1 (0)

∂(ln |y|) 󵄨󵄨 ∂|y| 󵄨󵄨|y|=1

00 dθ − ⟨ ∂w ∂ν , 1⟩ ∂ω

due to the representation (4.6) and, consequently, it fulfills the solvability condition 2π 00 0 = 2π − ⟨ ∂w ∂ν , 1⟩ ∂ω = ⟨ |∂ω| −

∂w 00 ∂ν , 1⟩ ∂ω

=

2π |∂ω| ⟨1

−

|∂ω| ∂w 00 2π ∂ν , 1⟩ ∂ω

,

where |∂ω| := meas1 (∂ω). Therefore, there exists a solution of the corresponding Neũ 0 ∈ W μ1,2 (ℝ2 \ ω; ℝ) such that mann problem defined up to a constant. Find w ∂w 00 ̃ 0 ⋅ ∇𝑣 dy = ⟨ |∂ω| ∫ ∇w 2π ∂ν − 1, 𝑣⟩ ∂ω

for all 𝑣 ∈ W μ1,2 (ℝ2 \ ω; ℝ) .

(5.9)

ℝ2 \ω

̃0 − |∂ω| From (5.9) and Green’s formula (4.24) it follows that w0 = w 2π w00 solves (5.8). The constant-free solution w0 admits logarithm in the Fourier series in ℝ2 \ B1 (0): w0 (y) = c00 ln |y| + W0 , ∞

W1 = ∑ n=2

1 0 |y|n c n

⋅ x̂ n ,

W0 =

1 0 |y| c 1

c0n ∈ ℝ2 ,

⋅ ̂x + W1 , c01 =:

1 2π m ω

(5.10)

.

To determine unknown c00 ∈ ℝ in (5.10), we apply the divergence theorem in B1 (0) \ ω: π

∫ ∆w0 dy = B1 (0)\ω

∫ ∂B1 (0)

∂w 0 ∂|y|

dS y − ∫ ∂ω

∂w 0 ∂ν

|y|) 󵄨󵄨 dS y = ∫ c00 ∂(ln ∂|y| 󵄨󵄨|y|=1 dθ − ∫dS y −π

∂ω

which implies 0 = 2πc00 − |∂ω|, hence c00 = |∂ω| 2π . 0 After rescaling y = x−x with the help of calculus (3.26) and using the Green forε mula (4.7), we reduce the exterior problem to the bounded domain Ω \ ω ε (x0 ), and from relations (5.7)–(5.10), we derive the following result:

4.5 Helmholtz problems for geometric objects under Robin boundary condition

| 109

0 Lemma 5.2. The rescaled solution w0ε (x) := w0 ( x−x ε ) to (5.8) fulfills

∇w0ε ⋅ ∇u dx = − 1ε

∫ Ω\ω ε (x 0 )

∫

u dS x + ∫

∂ω ε (x 0 )

∂w 0ε ∂ν u dS x

(5.11)

ΓN

for all u ∈ H (Ω \ ω ε (x0 ); ℝ) : u = 0 on Γ D . 1

It admits the far-field representation in the Fourier series for x ∈ ℝ2 \ B ε (x0 ) w0ε (x) =

|∂ω| 2π (ln ρ

− ln ε) + W0ε (x),

W0ε (x) =

ε 1 ρ 2π (m ω

⋅ ̂x) + W1ε (x)

(5.12)

x−x 0 ε 0 with the residual functions W0ε (x) := W0 ( x−x ε ) and W1 := W1 ( ε ) such that π

π

π

∫ W0ε dθ = ∫ W1ε dθ = ∫ W1ε ̂x dθ = 0 , −π

−π

W0ε

=

(5.13a)

−π

W1ε

O( ρε ),

= O (( ρε ))

2

for ρ > ε, θ ∈ (−π, π] .

(5.13b)

Entries of the coefficient vector m ω ∈ ℝ2 have the implicit expression: (m ω )i = ∫ (w0 ν i − y i ) dS y ,

i = 1, 2 ,

(5.14)

∂ω

and m ω = 0 is zero for ellipsoidal shapes ω. Proof. It suffices to prove (5.14). Indeed, the second Green formula (cf. (3.30)) gives ∂w 0 0 ̂ − ∫ ( ∂w ∂|y| − w0 ) x dS y = − ∫ ( ∂ν y − w0 ν) dS y = ∫ (w0 ν − y) dS y . ∂B1 (0)

∂ω

∂ω

Using (5.10), we calculate the integral on the left-hand side here as π

− ∫

0 ( ∂w ∂|y|

− w0 ) ̂x dS y = ∫ (− |∂ω| 2π +

1 2π m ω

⋅ ̂x +

∂W 1 ∂|y|

+

1 2π m ω

⋅ x̂ + W1 )̂x dθ

−π

∂B1 (0)

π

= ∫ 1π (m ω ⋅ x̂)̂x dθ = m ω , −π

which succeeds in (5.14). Resetting (5.8) in the Cartesian coordinates y󸀠 = Θ⊤ y, for the ellipse ω󸀠 written in the elliptic coordinates (3.32) in the form (3.33), we have the Fourier series ∞

w󸀠0 = c󸀠00 + c󸀠0 r + ∑ e−nr c󸀠n ⋅ (cos(nψ), sin(nψ))

for r > r0

n=1

for the solution, and due to (3.35) the Neumann boundary condition (5.7b) implies ∂w 󸀠0 󵄨󵄨 1 𝜘(r 0 ,ψ) ∂r 󵄨󵄨 r=r 0

=

1 𝜘(r 0 ,ψ)

∞

(c󸀠0 − ∑ ne−nr0 c󸀠n ⋅ (cos(nψ), sin(nψ))) = 1 . n=1

110 | 4 High-order topological expansions for Helmholtz problems in 2D

Therefore, thanks to the orthogonality of the Fourier basis functions, we calculate explicitly ∂w 󸀠0 󸀠 ∂ν 󸀠 y

󸀠

∫ y dS y󸀠 = ∫ ∂ω󸀠

π

dS y󸀠 = ∫ 𝜘(r10 ,ψ) −π

∂ω󸀠 π

∂w 󸀠0 󵄨󵄨 ⊤ ∂r 󵄨󵄨 r=r 0 (a cos ψ, b sin ψ) 𝜘(r0 , ψ) dψ

= − ∫ e−r0 ((c󸀠1 )1 cos ψ + (c󸀠1 )2 sin ψ) (a cos ψ, b sin ψ)⊤ dψ −π

⊤

= −πe−r0 ((c󸀠1 )1 a, (c󸀠1 )2 b) . On the other hand, due to the symmetry of ω󸀠 when changing ψ = ϕ − π it follows 0

π

∫ y󸀠 dS y󸀠 = ∫(a cos ψ, b sin ψ)⊤𝜘(r0 , ψ) dψ + ∫(a cos ψ, b sin ψ)⊤𝜘(r0 , ψ) dψ ∂ω󸀠

−π

0

π

π ⊤

= ∫(a cos(ϕ − π), b sin(ϕ − π)) 𝜘(r0 , ϕ) dϕ + ∫(a cos ψ, b sin ψ)⊤𝜘(r0 , ψ) dψ 0

0

=0 and leads to c󸀠1 = 0. Inserting these expressions in (5.14), we conclude π

∞

m ω󸀠 = ∫ (w󸀠0 ν󸀠 − y󸀠 ) dS y󸀠 = ∫ (c󸀠00 + c󸀠0 r0 + ∑ e−nr0 c󸀠n ⋅ (cos(nψ), sin(nψ))) n=2

−π

∂ω󸀠

×

⊤ 1 𝜘(r 0 ,ψ) (b cos ψ, a sin ψ) 𝜘(r0 , ψ) dψ

=0.

After the rotation of the Cartesian coordinates y = Θy󸀠 in (5.8), we obtain w󸀠0 (y󸀠 ) = w0 (Θy󸀠 ), hence m ω = Θm ω󸀠 = 0 is zero as well. This proves the assertion of the lemma. To compensate the logarithm in (5.12), we employ the result of Section 4.4.1 for the regularized problem (4.11), and using Lemma 4.2, we state the corrector w1ε as follows. Lemma 5.3. The combination of w0ε from Lemma 5.2 and the solution u ln to (4.11) w1ε := w0ε +

|∂ω| 2π

(ln ε − u ln )

(5.15)

fulfills the following relations: ln −∆w1ε = k 2 |∂ω| 2π (ln ρ − u ) ∂w 1ε ∂ν

where

= 1ε +

|∂ω| 2π

(b ln ⋅ ν −

∂U 0ln ∂ν

)

b ln := ((ln ρ − u ln (x0 ))ka󸀠0 +

a0 ρ

+

πka 󸀠2 ̂ 2 )x

in Ω \ ω ε (x0 ) ,

(5.16a)

on ∂ω ε (x0 ) ,

(5.16b)

= O(ρ| ln ρ|) ,

(5.16c)

4.5 Helmholtz problems for geometric objects under Robin boundary condition

| 111

it satisfies the variational equation w1ε = W0ε

on Γ D ,

(5.17a)

(∇w1ε ⋅ ∇u − k 2 w1ε u) dx

∫ Ω\ω ε (x 0 )

∂W 0ε ∂ν u dS x

=∫

−

∂w 1ε ∂ν u dS x

∫

k 2 (w0ε +

∫

|∂ω| 2π (ln ε

− ln ρ)) u dx

Ω\ω ε (x 0 )

∂ω ε (x 0 )

ΓN

−

1

for all u ∈ H (Ω \ ω ε (x0 ); ℝ) : u = 0 on Γ D

(5.17b)

and admits the far-field representation for x ∈ B R (x0 ) \ B ε (x0 ): w1ε (x) =

|∂ω| 2π

((ln ρ − u ln (x0 ))(1 + a0 ) +

πa 2 2

− U0ln ) + W0ε .

(5.18)

Proof. The assertion is derived directly by substitution of the asymptotic representations (4.14) in (5.15) for u ln and (5.12) for w0ε . With the notation in (5.16c), this leads to (5.18) and (5.16b), obtained after differentiation of the radial function in (5.18) by the ∂ ̂ rule ∇ = ∂ρ x. Relations (5.17) follow from (4.11a), (4.13), and (5.11). Combining the boundary layers w εν and w1ε constructed in Lemmas 3.4 and Lemma 5.3, respectively, in the following section we expand the solution of the Robin problem (5.2).

4.5.2 Combined uniform asymptotic expansion of the Robin problem Analogous to Sections 4.3.2 and 4.4.2, the following uniform asymptotic expansion holds. Theorem 5.4. The solutions u 0 of (2.2), w ν of (3.16), u (ε,α) of (5.2), and w1ε of (5.17) satisfy the residual error estimate (ε,α)

‖q1

‖H1 (Ω\ω ε (x

0 );ℂ)

(ε,α)

q1

= O ({(1 + |α|)√| ln ε| + |α|2 } ε2 ) , := u (ε,α) − u 0 − ε (αu 0 (x0 )w1ε + ∇u 0 (x0 ) ⋅ w εν ) .

(ε,α)

(∇q0

:= u (ε,α) − u 0 of (5.2) and (3.10) using (3.44)

(ε,α)

u) dx +

⋅ ∇u − k 2 q0

Ω\ω ε (x 0 )

=

∫

(ε,α)

αq0

u dS x

∂ω ε (x 0 ) 0

0 ∫ ( ∂u ∂ν − αu ) u dS x = ∂ω ε (x 0 )

(5.19b)

(ε,α)

Proof. If we write the residual q0 ∫

(5.19a)

∫ (∇u 0 (x0 ) ⋅ ν − αu 0 (x0 ))u dS x + o(ε) ,

(5.20)

∂ω ε (x 0 )

then we observe that the linear combination of the boundary layers introduced in (5.19b) will cancel the leading asymptotic term due to (3.27b) and (5.16b).

112 | 4 High-order topological expansions for Helmholtz problems in 2D In detail, subtracting from (5.2) equations (3.10), (5.17) multiplied with εαu 0 (x0 ), (ε,α) and (3.18) multiplied with ε∇u 0 (x0 ), for the residual q1 in (5.19b) we derive (ε,α)

q1

= −εαu 0 (x0 )W0ε − ε∇u 0 (x0 ) ⋅ w εν = O ((1 + |α|)ε2 ) (ε,α) (∇q1

∫

on Γ D ,

(5.21a)

for all u ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D ,

(5.21b)

⋅ ∇u −

(ε,α) k 2 q1 u) dx

+

Ω\ω ε (x 0 )

(ε,α) αq1 u dS x

∫ ∂ω ε (x 0 )

= I1 + I2 + I3

where the integrals I1 , I2 , I3 are defined and estimated as follows. According to (3.19) and (5.13), we have W0ε = w εν = O(ε) at Γ D , hence the asymptotic order in (5.21a), and

∂W 0ε ∂ν

= Dw εν ν = O(ε) at Γ N providing

I1 := −ε ∫ (αu 0 (x0 )

∂W 0ε ∂ν

+ ∇u 0 (x0 ) ⋅ (Dw εν ν)) u dS x = O ((1 + |α|)ε2 ) .

ΓN

Based on the expressions (2.9) and (2.35) for u 0 and (5.16b) for I2 :=

0

0 0 ( ∂u ∂ν − αu + εαu (x 0 ) (

∫

∂w 1ε ∂ν

∂w 1ε ∂ν

it follows

− αw1ε ) − ε∇u 0 (x0 ) ⋅ ( νε + αw εν )) u dS x

∂ω ε (x 0 )

=

{b 0u ⋅ ν +

∫

∂U 10 ∂ν

− αu 0 (x0 )a0 − αU00 + εαu 0 (x0 ) |∂ω| 2π (b ln ⋅ ν −

∂U 0ln ∂ν )

∂ω ε (x 0 )

− εα2 u 0 (x0 )w1ε − εα∇u 0 (x0 ) ⋅ w εν } u dS x = O ((1 + |α| + |α|2 )ε2 ) due to (2.8a), (3.19), (4.16), and (5.16c), where the estimate (3.21b) implying w εν = O(1) and similarly w1ε = O(1) at ∂ω ε (x0 ) were used. The representation over domain ∫

I3 := ε

k 2 {αu 0 (x0 ) (w0ε +

|∂ω| 2π (ln ε

− ln ρ)) + ∇u 0 (x0 ) ⋅ w εν } u dx

Ω\ω ε (x 0 )

= O ((1 + |α|)ε2 √| ln ε|) is argued by (3.21a) and by the similar estimate resulting by homogeneity from Lemma 5.2: √ ‖w0ε + |∂ω| 2π (ln ε − ln ρ)‖L2 (Ω\ω ε (x );ℝ)2 = O(ε | ln ε|) . 0

Next lifting with the help of a cut-off function η Γ D such that η Γ D = 1 at Γ D , we set (ε,α)

R1

:= ε (αu 0 (x0 )W0ε + ∇u 0 (x0 ) ⋅ w εν ) η Γ D = O ((1 + |α|)ε2 ) .

(5.22)

Applying to (5.21b) the Cauchy–Schwarz inequality, the estimates of I1 , I2 , I3 succeed 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 (ε,α) (ε,α) (ε,α) 2 󵄨󵄨󵄨 ∫ (∇Q1 ⋅ ∇u − k Q1 u) dx + ∫ αQ1 u dS x 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ∂ω ε (x 0 ) 󵄨󵄨 󵄨󵄨Ω\ω ε (x0 ) ≤ C ({(1 + |α|)√| ln ε| + |α|2 } ε2 ) ‖u‖H1 (Ω\ω ε (x

0 );ℂ)

,

(ε,α)

Q1

(ε,α)

:= q1

(ε,α)

+ R1

with C > 0, which together with the inf-sup condition (5.6) proves the theorem.

4.5 Helmholtz problems for geometric objects under Robin boundary condition

| 113

Based on Theorem 5.4, in the following section, we treat asymptotically for ε ↘ +0 the optimal value function J in (3.47) depending on the boundary impedance α.

4.5.3 Inverse Helmholtz problem under Robin boundary condition In contrast to Sections 4.3.3 and 4.4.3, here the test parameter of surface impedance α ⋆ ∈ Gα ⊂ ℂ (see (5.6)) is unknown a-priori and has to be identified together with the test object ω⋆ε⋆ (x⋆ ). For a feasible trial parameter α ∈ Gα and the solution u (ε,α) of (5.2), we reset the objective J : G × Gα 󳨃→ ℝ+ ,

J(ω, ε, x0 , α) :=

1 2

∫ |u (ε,α) − u ⋆ |2 dS x

(5.23)

ΓN

and the topology optimization problem. Find (ω⋆ , ε⋆ , x⋆ , α ⋆ ) ∈ G × Gα such that 0 = J(ω⋆ , ε⋆ , x⋆ , α ⋆ ) =

min

(ω,ε,x 0 ,α)∈G×Gα

J(ω, ε, x0 , α) subject to (5.2) .

(5.24)

Analogously to Lemma 3.7 and Proposition 5.1, a Fenchel–Legendre duality provides the primal–dual variational principle: L(ε,α)(u (ε,α) , 𝑣(ε,α)) =

min

max

Re(u),Re(𝑣) Im(u),Im(𝑣)

L(ε,α)(u, 𝑣)

(5.25)

over u, 𝑣 ∈ H 1 (Ω \ ω ε (x0 ); ℂ) : u = h, 𝑣 = 0 on Γ D , where the Lagrangian L(ε,α) has the form (compare with (3.49) and (3.50)): L(ε,α)(u, 𝑣) := Re{

∫

(∇u ⋅ ∇𝑣 − k 2 u𝑣) dx − ∫ g𝑣 dS x

Ω\ω ε (x 0 )

+

∫

ΓN

αu𝑣 dS x +

∂ω ε (x 0 )

1 2

∫ (u − u ⋆ )2 dS x } ,

(5.26)

ΓN

thus arguing necessarily the dual variational problem. Find 𝑣(ε,α) ∈ H 1 (Ω \ ω ε (x0 ); ℂ) such that 𝑣(ε,α) = 0 on Γ D , ∫

(∇𝑣(ε,α) ⋅ ∇u − k 2 𝑣(ε,α) u) dx +

Ω\ω ε (x 0 )

(5.27a) ∫ ∂ω ε (x 0 )

1

for all u ∈ H (Ω \ ω ε (x0 ); ℂ) : u = 0 on Γ D .

α𝑣(ε,α) u dS x = ∫ (u ⋆ − u (ε,α))u dS x ΓN

(5.27b)

Again utilizing the primal u 0 and the dual 𝑣0 background solutions of (2.2) and (3.54), which are independent of the test object as well as its surface impedance, sim-

114 | 4 High-order topological expansions for Helmholtz problems in 2D

ilar to (3.59) and (4.53), we express the objective in (5.23) equivalently as J(ω, ε, x0 , α) = J 0 + Re {I(u (ε,α) − u 0 , 𝑣0 )} +

∫ |u (ε,α) − u 0 |2 dS x ,

1 2

ΓN 0

I(u (ε,α) − u 0 , 𝑣0 ) := ∫ (u 0 − u (ε,α) ) ∂𝑣 ∂ν dS x ,

J 0 :=

ΓN 1 2

∫ |u 0 − u ⋆ |2 dS x = O(1),

1 2

ΓN 2

∫ |u (ε,α) − u 0 |2 dS x = O ({(1 + |α|)√| ln ε| + |α|2 } ε4 )

(5.28)

ΓN

with the asymptotic order due to Theorem 5.4. The refinement of (5.28) follows. Theorem 5.5. The objective in (5.28) admits the high-order asymptotic expansion (ε,α)

D 2 N J(ω, ε, x0 , α) = J 0 + Re {ε |∂ω| 2π αJ 1 (x 0 ) + (ε J 1 (ω, x 0 ) + αJ 2 (ε,α)

+J5

(ε,α)

+ J6

(ε,α)

) + J3

(ε,α)

+ J4

2

} + O ({(1 + |α|)√| ln ε| + |α|2 } ε4 ) ,

(5.29)

where J 1N and J1D are expressed in (3.61a) and (4.55a), and the asymptotic terms are (ε,α)

J2

:=

(εαu 0 (x0 )w1ε + ∇u 0 (x0 ) ⋅ (x + εw εν )) 𝑣0 (x0 ) dS x

∫ ∂ω ε (x 0 )

− ε2 u 0 (x0 )(m ω ⋅ ∇𝑣0 (x0 )) = O((1 + |α|)ε2 ) , (ε,α)

J3

:= − εα

(5.30a)

3 0 u 0 (x0 ) |∂ω| 2π (b ln ⋅ ν)𝑣 (x 0 ) dS x = O(|α|ε | ln ε|) ,

∫

(5.30b)

∂ω ε (x 0 ) (ε,α) J4

:=

(ε,α)

∫ αq1

(𝑣0 (x0 ) − ∇𝑣0 (x0 ) ⋅ ν) dS x +

∂ω ε (x 0 )

∫

(ε,α)

∂q1 ∂ρ

V00 dS x

∂B ε (x 0 )

= O ((1 + |α|) {(1 + |α|)√| ln ε| + |α|2 } ε3 ) , (ε,α)

J5

:=

(αu 0 (x0 )a0 + αU10 − (b 0u + u 0 (x0 )

∫

(5.30c)

k2 ρ ̂ 2 x)

⋅ ν) 𝑣0 (x0 ) dS x

∂ω ε (x 0 ) π

+ ε2 ∫ {αu 0 (x0 ) [

∂W 1ε 0 ∂ρ V1

− W1ε

∂V 10 ∂ρ

−

|∂ω| 2π

(

∂U 0ln 0 ∂ρ V0

− U0ln

∂V 00 ∂ρ )]

−π

+ ∇u 0 (x0 ) ⋅ (

∂W νε 0 ∂ρ V1

− W νε

∂V 10 ∂ρ )} dθ

+ ε (2a0 + ε

πka 󸀠2 |∂ω| D 2 ) 2π αJ 1 (x 0 )

= O((1 + |α|)ε3 ) , (ε,α)

J6

:=

∫

(

(ε,α) ∂q1

∂ρ

(5.30d) (ε,α)

𝑣0 (x0 )a0 − q1

(b 0𝑣 + ∇V10 (x0 )) ⋅ ̂x) dS x

∂B ε (x 0 )

= O ({(1 + |α|)√| ln ε| + |α|2 } ε4 ) .

(5.30e)

4.5 Helmholtz problems for geometric objects under Robin boundary condition

|

115

Proof. Employing the equivalent expression (4.56) over the circle ∂B ε (x0 ) for the invariant integral I(u (ε,α) − u 0 , 𝑣0 ) in (5.28) and substituting here the asymptotic representation (5.19b) for u (ε,α) − u 0 , we have I(u ε − u 0 , 𝑣0 ) =

{(εαu 0 (x0 )

∫

∂w 1ε ∂ρ

+ ε∇u 0 (x0 ) ⋅

(ε,α)

∂w εν ∂ρ

+

∂q1 ∂ρ

(ε,α)

)

∂𝑣0 ∂ρ } dS x

) 𝑣0

∂B ε (x 0 )

− (εαu 0 (x0 )w1ε + ε∇u 0 (x0 ) ⋅ w εν + q1

= I1α + I2α + I3α .

The integrals I1α , I2α , I3α are set as follows. On the one hand, the first-order representation 𝑣0 = 𝑣0 (x0 )(1+ a0 )+ V00 from (3.56) and (2.8a) provides the expression π

I1α :=

εαu 0 (x0 ) (

∫

∂w 1ε 0 ∂ρ 𝑣

0

0 0 − w1ε ∂𝑣 ∂ρ ) dS x = ∫ εαu (x 0 )𝑣 (x 0 )(

∂w 1ε ∂ρ (1

+ a0 )

−π

∂B ε (x 0 ) π

−

w1ε ka󸀠0 ) ε dθ

+ ∫ εαu 0 (x0 ) (

∂w 1ε 0 ∂ρ V0

− w1ε

∂V 00 ∂ρ )

α α ε dθ =: I11 + I12 .

−π

Inserting in I1α the representations (5.12) for w0ε and (5.18) for w1ε from Lemmas 5.2 and 5.3 and using the orthogonality conditions for U0ln , W0ε , and W1ε , we calculate π 1+a 0 α = ∫ εαu 0 (x0 )𝑣0 (x0 ) |∂ω| I11 2π {( ε +

πka 󸀠2 2 ) (1

+ a0 ) −

πa 2 󸀠 2 ka0 }

ε dθ

−π

= ε|∂ω|αu 0 (x0 )𝑣0 (x0 ) {1 + (2a0 + ε

πka 󸀠2 2 )

󸀠 󸀠 + (a20 + ε πk 2 (a2 a0 − a2 a0 ))} ,

π α I12

= − ε α ∫ u 0 (x0 ) |∂ω| 2π ( 2

∂U 0ln 0 ∂ρ V0

− U0ln

∂V 00 ∂ρ ) dθ

α + I13 ,

−π π α I13

2

π

:= ε α ∫ u

0

∂W ε (x0 ) ( ∂ρ0 V00

−

0

∂V W0ε ∂ρ0 ) dθ

1 = ε α ∫ u 0 (x0 ){(− 1ε 2π (m ω ⋅ ̂x) + 2

−π

−π

((ε +

a1 0 k ) (∇𝑣 (x 0 )

⋅ ̂x) +

1 V10 ) − ( 2π (m ω

⋅ ̂x) + W1ε ) ((1 + a󸀠1 )(∇𝑣0 (x0 ) ⋅ ̂x )

π

+

∂V 10 ∂ρ )} dθ

= ε2 α ∫ u 0 (x0 ) (

∂W 1ε 0 ∂ρ V1

− W1ε

∂V 10 ∂ρ ) dθ

α + I14 ,

−π π α 1 I14 := − ε2 α ∫ u 0 (x0 ) 2π (m ω ⋅ ̂x) (1 +

a1 εk

+ 1 + a󸀠1 ) (∇𝑣0 (x0 ) ⋅ ̂x) dθ

−π

= − ε αu 0 (x0 )(m ω ⋅ ∇𝑣0 (x0 )) (1 + 12 ( aεk1 + a󸀠1 )) . 2

∂W 1ε ∂ρ )

116 | 4 High-order topological expansions for Helmholtz problems in 2D On the other hand, the second-order expansion of 𝑣0 = 𝑣0 (x0 )(1 + a0 ) + (ρ + x) + V10 in the manner of (4.39) and its derivative from (3.58) yields 0) ⋅ ̂

(∇𝑣0 (x

I2α :=

ε∇u 0 (x0 ) ⋅ (

∫

∂w εν 0 ∂ρ 𝑣

a1 k )

0

− w εν ∂𝑣 ∂ρ ) dS x

∂B ε (x 0 ) π ∂w εν ∂ρ (ε

= ∫ ε∇u 0 (x0 ) ⋅ (

a1 0 k )(∇𝑣 (x 0 )

+

⋅ x̂) − w εν (∇𝑣0 (x0 ) + b 0𝑣 ) ⋅ ̂x ) ε dθ

−π π

+ ∫ ε∇u 0 (x0 ) ⋅ (

∂w εν 0 ∂ρ V1

− w εν

∂V 10 ∂ρ )

α α ε dθ =: I21 + I22 ,

−π π

where we have used ∫−π w εν dθ = 0 due to (3.19) and (3.20). Inserting in I2α the representation (3.19) of w εν it recalls the calculation in (3.66)–(3.68) getting α I21 = −ε2 ∇u 0 (x0 )⊤ M ω ∇𝑣0 (x0 ) (1 +

1 2

( aεk1 + a󸀠1 )) ,

π α = ε2 ∫ ∇u 0 (x0 ) ⋅ ( I22

∂W νε 0 ∂ρ V1

− W νε

∂V 10 ∂ρ ) dθ

.

−π (ε,α)

It remains to estimate I3α which includes the residual q1 I3α :=

∫

(ε,α)

(

∂q1 ∂ρ

(

∂q1 ∂ρ

given in Theorem 5.4:

(ε,α) ∂𝑣0 ∂ρ ) dS x

𝑣0 − q1

∂B ε (x 0 )

=

∫

(ε,α)

(ε,α)

𝑣0 (x0 ) − q1

∇𝑣0 (x0 ) ⋅ ̂x ) dS x +

∂B ε (x 0 )

+

∫

(ε,α)

∂q1 ∂ρ

V00 dS x

∂B ε (x 0 )

∫

(ε,α)

∂q1 ( ∂ρ

(ε,α)

α α α (b 0𝑣 + ∇V10 (x0 )) ⋅ ̂x) dS x =: I31 + I32 + I33 .

𝑣0 (x0 )a0 − q1

∂B ε (x 0 ) α the second Green formula in B ε (x0 ) \ ω ε (x0 ) it can be rewritten as By applying to I31 α =⟨ I31

(ε,α)

∂q1 ∂ν

, 𝑣0 (x0 )⟩

∂ω ε (x 0 ) (ε,α)

We extract the normal derivative (ε,α)

∂q1 ∂ν

(ε,α)

= αq1

+ αu 0 −

∂u0 ∂ν

∂q1 ∂ν

−

∂ν

+ εαu 0 (x0 ) (αw1ε −

+ εαu 0 (x0 ) (αw1ε −

(∇𝑣0 (x0 ) ⋅ ν) dS x .

(5.31)

at ∂ω ε (x0 ) from I2 in (5.21b):

(ε,α)

−

q1

∂ω ε (x 0 )

+ ε∇u 0 (x0 ) ⋅ (αw εν − Dw εν ν) = αq1 ∂U 10

(ε,α)

∫

|∂ω| 2π

∂w 1ε ∂ν )

+ αu 0 (x0 )a0 + αU00 − b 0u ⋅ ν

(b ln ⋅ ν −

∂U 0ln ∂ν ))

+ εα∇u 0 (x0 ) ⋅ w εν

(5.32)

4.5 Helmholtz problems for geometric objects under Robin boundary condition

| 117

and substitute it in (5.31). Analog to (3.64), the divergence theorem in ω ε (x0 ) provides ν ⋅ (u 0 (x0 )

∫

k2 ρ ̂ 2 x

ln 0 − ∇U10 + εαu 0 (x0 ) |∂ω| 2π ∇U 0 ) 𝑣 (x 0 ) dS x

∂ω ε (x 0 )

=

∫ div (u 0 (x0 )

k2 ρ ̂ 2 x

ln 0 − ∇U10 + εαu 0 (x0 ) |∂ω| 2π ∇U 0 ) 𝑣 (x 0 ) dx

ω ε (x 0 ) ln 0 = ε2 meas2 (ω)k 2 u 0 (x0 )𝑣0 (x0 ) + ∫k 2 (U10 − εαu 0 (x0 ) |∂ω| 2π U 0 ) 𝑣 (x 0 ) dx

(5.33)

ω ε (x 0 )

since div(ρ ̂x ) = 2, ∆U10 = −k 2 U10 , and ∆U0ln = −k 2 U0ln . With the help of (5.32), (5.33), and due to U00 = ∇u 0 (x0 ) ⋅ (x + ak1 ̂x) + U10 , from (5.31) we infer α I31 =α

(εαu 0 (x0 )w1ε + ∇u 0 (x0 ) ⋅ (x + εw εν )) 𝑣0 (x0 ) dS x

∫ ∂ω ε (x 0 )

+ ε2 meas2 (ω)k 2 u 0 (x0 )𝑣0 (x0 ) +

(ε,α)

∫

{(αq1

+ αu 0 (x0 )a0

∂ω ε (x 0 )

+

α( ak1 ∇u 0 (x0 ) (ε,α)

−q1

⋅ x̂ +

U10 )

−

(b 0u

0 + u 0 (x0 ) k2ρ ̂x) ⋅ ν − εαu 0 (x0 ) |∂ω| 2π b ln ⋅ ν)𝑣 (x 0 ) 2

ln 0 (∇𝑣0 (x0 ) ⋅ ν)} dS x + ∫ k 2 (U10 − εαu 0 (x0 ) |∂ω| 2π U 0 ) 𝑣 (x 0 ) dx . ω ε (x 0 )

After collection of the asymptotic terms of the same order in view of the asymptotic relations (3.65) and (3.68) in the proof of Theorem 3.9, (4.15) in Lemma 4.2, (5.13) in Lemma 5.2, (5.16c) in Lemma 5.3, and (5.19) in Theorem 5.4, it follows the expansion α α α α α α α D + I12 + I21 + I22 + I31 + I32 + I33 = ε |∂ω| I(u ε − u 0 , 𝑣0 ) = I11 2π αJ 1 (x 0 ) (ε,α)

+ (ε2 J 1N (ω, x0 ) + αJ2

(ε,α)

) + J3

(ε,α)

+ J4

(ε,α)

+ J5

(ε,α)

with J 1N and J 1D given in (3.61a) and (4.55a), the terms from J2 (5.30), and the residual term collected by (ε,α)

J7

󸀠 󸀠 := ε (a20 + ε πk 2 (a2 a0 − a2 a0 ))

|∂ω| D 2π αJ 1 (x 0 )

− ∇u 0 (x0 )⊤ M ω ∇𝑣0 (x0 )} ( aεk1 + a󸀠1 ) +

−

∫

(ε,α)

+ J6

(ε,α)

to J 6

0 ε2 2 {αu (x 0 )(m ω

(ε,α)

+ J7

described in

⋅ ∇𝑣0 (x0 ))

α ak1 (∇u 0 (x0 ) ⋅ ̂x )𝑣0 (x0 ) dS x

∂ω ε (x 0 ) ln 4 0 + ∫ k 2 (U10 − εαu 0 (x0 ) |∂ω| 2π U 0 ) 𝑣 (x 0 ) dx = O((1 + |α|)ε ) . ω ε (x 0 )

The proof of the assertion of the theorem is completed. We finish with the important generalizations and consequences of Theorem 5.5. For a variable parameter of the surface impedance α ∈ L∞ (∂ω; ℂ), following [43], |∂ω| 0 the expansion (5.29) can be generalized by replacing α with α( x−x ε ), 2π α with the 1 ∞ average 2π ∫∂ω α(y) dS y , and the modulus |α| with the L -norm.

118 | 4 High-order topological expansions for Helmholtz problems in 2D Sections 4.3 and 4.4 correspond, respectively, to the case |α| ↘ +0 and a special case of |α| ↗ ∞ which will be explained further. Moreover, they provide our construction with the auxiliary functions w εν , w00 , and u ln as well as the expressions of J1N and J 1D used here. Therefore, these sections are not redundant. For fixed α ∈ ℂ, the concept of the topological derivatives from (3.69) and (4.58) can be generalized to the Robin problem (5.2): |∂ω| 2π

Re(αJ 1D (x0 )) = lim

1 ε↘+0 ε

(J(ω, ε, x0 , α) − J 0 )

(5.34)

utilizing the leading asymptotic term in the α-dependent asymptotic expansion (5.29). However, the leading term in (5.29) changes when varying |α| ↘ +0 or |α| ↗ ∞. In fact, on the one hand, passing |α| ↘ +0 in (5.29), we obtain the representation lim J(ω, ε, x0 , α) = J 0 + ε2 Re(J 1N (ω, x0 )) + O (√| ln ε|ε3 )

|α|↘+0

(5.35)

with the residual estimate according to (5.30). Asymptotically, the limit in (5.35) coincides with formulas (3.60) and (3.61) for the Neumann problem in Theorem 3.9. On the one hand, taking the limit as |α| ↗ ∞, which corresponds to the Dirichlet problem, in the next section we derive a derivative-free necessary optimality condition following from the topology optimization problem (5.24).

4.5.4 Necessary optimality condition for the topology optimization Recalling the topology optimization problem (5.24) for an unknown parameter of the test impedance α ⋆ ∈ ℂ, it allows, in particular, the both limit cases |α ⋆ | ↘ +0 and |α ⋆ | ↗ ∞. In the latter case, we derive the necessary optimality condition which does not appear in the limit optimization problems (3.48) and (4.51), when either the Neumann or Dirichlet boundary conditions of the test object are assumed a-priori. For ε ↘ +0, the first-order approximation of the objective J from (5.29) implies D D J(ω, ε, x0 , α) = J 0 + ε |∂ω| 2π {Re(α)Re(J 1 (x 0 )) − Im(α)Im(J 1 (x 0 ))}

+ O((1 + |α| + |α|2 )ε2 ),

where |α| = √|Re(α)|2 + |Im(α)|2 .

(5.36)

The general α-dependent representation (5.36) turns into the particular one |∂ω| D D 1 − ln ε Re(J 1 (x 0 )) − ε 2π Im(α)Im(J 1 (x 0 )) + O ( | ln1ε|2 + (|Im(α)| + |Im(α)|2 )ε2 ) ,

J(ω, ε, x0 , α) = J 0 + when

Re(α) =

2π 1 ε(− ln ε) |∂ω|

+ O( ε| ln1 ε|2 )

.

(5.37a) (5.37b)

Due to (4.57), formula (5.37a) coincides with the low-order asymptotic terms of expansion (4.54) from Theorem 4.8 for the Dirichlet problem, when |Im(α)| = O( ε| ln1 ε|2 ). Nevertheless, for fixed ε and variable α it differs.

4.5 Helmholtz problems for geometric objects under Robin boundary condition

|

119

Indeed, for the test parameters (ω, ε, x0 ) = (ω⋆ , ε⋆ , x⋆ ) and α = α ⋆ , the optimal objective value J(ω⋆ , ε⋆ , x⋆ , α ⋆ ) = 0 in (5.24). In particular, when |α ⋆ | ↗ ∞, too. The finite (zero) limit in (5.37a) when |Im(α⋆ )| ↗ ∞ (hence |α ⋆ | ↗ ∞) and arbitrary ε⋆ can be preserved only if its complement Im(J1D (x⋆ )) is zero. We note that this argument holds true also for finite optima 0 ≠ J(ω⋆ , ε⋆ , x⋆ , α ⋆ ) < ∞ when the test parameters are infeasible. Thus, recalling the form (4.55a) of J1D , we have proved the following main result. Theorem 5.6. For arbitrary geometric parameters (ω⋆ , ε⋆ , x⋆ ) ∈ G of the test object under the Dirichlet boundary condition, the necessary optimality condition of (4.51) Im (u 0 (x⋆ )𝑣0 (x⋆ )) = 0

(5.38)

is expressed by the primal and dual background solutions u 0 and 𝑣0 to (2.2) and (3.54). Based on Theorem 5.6, in [43] the imaging function is introduced f : {u ⋆ ∈ Gu } 󳨃→ C(Ω; ℝ),

f u⋆ (x) := Im (u 0 (x)𝑣0 (x))

(5.39)

over a set of feasible boundary measurements u ⋆ ∈ Gu ⊂ L2 (Γ N ; ℂ). Thanks to (5.38), its zero-level set contains the test center x⋆ : x⋆ ∈ L=0 (f u⋆ ) := {x ∈ Ω :

f u⋆ (x) = 0} .

(5.40)

The formalism (5.39) and (5.40) guarantee a high-precision numerical solution of the inverse problem of identification of the center of an unknown test object from boundary measurements. In [43], the following numerical findings are reported in detail: – Implementation of the imaging function f u⋆ in (5.39) has low computational costs since it needs to solve the background Helmholtz problems in the fixed domain. – The center x⋆ from (5.40) can be detected as the intersection point of zero-level sets x⋆ = ⋂di=1 L=0 (f u⋆i ) (5.41) – – –

from two and three different measurements u ⋆i in 2d and 3d, respectively. For low wave numbers k, formula (5.41) holds in arbitrary spatial dimensions d ∈ {1, 2, 3} and for arbitrary feasible test objects ω⋆ε⋆ (x⋆ ) ⊂ Ω. The numerical identification result by (5.41) is exact when x⋆ coincides with a computational mesh node, it is highly stable to discretization and noise errors. Although Theorem 5.6 is stated for the Dirichlet problem, (5.39) and (5.41) are applicable numerically as well to finite α ⋆ ∈ ℝ for the Robin problem, in particular, to α ⋆ = 0 for the Neumann problem.

Funding: The results were obtained with the support of the Austrian Science Fund (FWF) in the framework of the project P26147-N26: “Object identification problems: numerical analysis” (PION) and NAWI Graz.

120 | 4 High-order topological expansions for Helmholtz problems in 2D

Acknowledgment: The author thanks the organizers, especially K. Kunisch and M. Rumpf, for the support during Workshop 1 of the Special Semester on “New Trends in Calculus of Variations” held at the RICAM (Linz, Austria), October 13–17, 2014.

Bibliography [1] [2]

[3] [4] [5]

[6] [7]

[8] [9] [10] [11] [12] [13] [14] [15]

[16] [17] [18]

G. Allaire, F. de Gournay, F. Jouve and A.-M. Toader, Structural optimization using topological and shape sensitivity via a level set method, Control Cybernet. 34:59–80, 2005. H. Ammari, Differential electromagnetic imaging, in I. Graham, U. Langer, J. Melenk and M. Sini, eds., Direct and Inverse Problems in Wave Propagation and Applications, Radon Ser. Comput. Appl. Math. 14, 1–50, de Gruyter, Berlin, 2013. H. Ammari, J. Garnier, V. Jugnon and H. Kang, Stability and resolution analysis for a topological derivative based imaging functional, SIAM J. Control Optim. 50:48–76, 2012. H. Ammari, R. Griesmaier and M. Hanke, Identification of small inhomogeneities: Asymptotic factorization, Math. Comp. 76:1425–1448, 2007. H. Ammari and H. Kang, High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of inhomogeneities of small diameter, SIAM J. Math. Anal. 34:1152–1166, 2003. H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities From Boundary Measurements, Springer-Verlag, Berlin, 2004. C. Amrouche, V. Girault and J. Giroire, Dirichlet and Neumann exterior problems for the ndimensional Laplace operator: an approach in weighted Sobolev spaces, J. Math. Pures Appl. 76:55–81, 1997. S. Amstutz and N. Dominquez, Topological sensitivity analysis in the context of ultrasonic nondestructive testing, Eng. Anal. Boundary Elem. 32:936–947, 2008. I. I. Argatov and V. A. Kovtunenko, A kinking crack: generalization of the concept of the topological derivative, 2009 Ann. Bull. Australian Inst. High Energetic Materials 1:124–130, 2010. H. T. Banks, K. Holm and F. Kappel, Comparison of optimal design methods in inverse problems, Inverse Probl. 27:075002, 2011. C. Bellis, M. Bonnet and F. Cakoni, Acoustic inverse scattering using topological derivative of far-field measurements-based L 2 cost functionals, Inverse Probl. 29:075012, 2013. M. Bonnet, Discussion of “Second order topological sensitivity analysis” by J. Rocha de Faria et al., Int. J. Solids Structures 45:705–707, 2008. M. Bonnet, Higher-order topological sensitivity for 2-D potential problems. Application to fast identification of inclusions, Int. J. Solids Structures 46:2275–2292, 2009. M. Brühl, M. Hanke and M. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities, Numer. Math. 93:631–654, 2003. N. D. Cahill, B. Jadamba, A. A. Khan, M. Sama and B. C. Winkler, A first-order adjoint and a second-order hybrid method for an energy output least-squares elastography inverse problem of identifying tumor location, Bound. Value Probl. Springer 2013:263, 2013. F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, AMS 188, Springer, 2014. A. Canelas, A. Laurain and A. A. Novotny, A new reconstruction method for the inverse potential problem, J. Comput. Phys. 268:417–431, 2014. A. Canelas, A. Laurain and A. A. Novotny, A new reconstruction method for the inverse potential problem, Inverse Probl. 31:075009, 2015.

Bibliography |

121

[19] S. N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal. 37:598–618, 2005. [20] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, SpringerVerlag, Berlin, 1998. [21] V. Coscia and R. Russo, Some remarks on the Dirichlet problem in plane exterior domains, Ricerche di Matematica 56:31–41, 2007. [22] R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20:439–467, 2013. [23] H. W. Engl, M. Hanke and A Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996. [24] H. A. Eschenauer, V. V. Kobelev and A. Schumacher, Bubble method for topology and shape optimization of structures, Struct. Multidiscip. Optim. 8:42–51, 1994. [25] K. Fellner and V. A. Kovtunenko, A singularly perturbed nonlinear Poisson–Boltzmann equation: uniform and super-asymptotic expansions, Math. Meth. Appl. Sci. 38:3575–3586, 2015. [26] A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Arch. Rational Mech. Anal. 105:299–326, 1989. [27] H. Harbrecht and T. Hohage, Fast methods for three-dimensional inverse obstacle scattering, J. Integral Equations Appl. 19:237–260, 2007. [28] A. Herwig, Elliptische Randwertprobleme zweiter Ordnung in Gebieten mit einer Fehlstelle, Z. Anal. Anwend. 8:153–161, 1989. [29] M. Hintermüller and V. A. Kovtunenko, From shape variation to topology changes in constrained minimization: a velocity method-based concept, Optimization Meth. Software 26:513–532, 2011. [30] M. Hintermüller, A. Laurain and A. A. Novotny, Second-order topological expansion for electrical impedance tomography, Adv. Comput. Math. 36:235–265, 2012. [31] F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Springer-Verlag, New York, 1998. [32] M. Ikehata and H. Itou, Extracting the support function of a cavity in an isotropic elastic body from a single set of boundary data, Inverse Probl. 25:10500, 2009. [33] A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, AMS, Providence, RI 1992. [34] L. Jackowska-Strumillo, J. Sokolowski, A. Zochowski and A. Henrot, On numerical solution of shape inverse problems, Comput. Optim. Appl. 23:231–255, 2002. [35] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Problems, de Gruyter, Berlin, New York, 2008. [36] A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, WIT-Press, Southampton, Boston, 2000. [37] A. M. Khludnev, V. A. Kovtunenko and A. Tani, On the topological derivative due to kink of a crack with nonpenetration. Anti-plane model, J. Math. Pures Appl., 94:571–596, 2010. [38] A. M. Khludnev and V. A. Kozlov, Asymptotics of solutions near crack tips for Poisson equation with inequality type boundary conditions, Z. Angew. Math. Phys. 59:264–280, 2008. [39] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, Dordrecht, Heidelberg, London, 2011. [40] M. V. Klibanov, A. V. Kuzhuget, S. I. Kabanikhin and D. V. Nechaev, The quasi-reversibility method for the thermoacoustic tomography and a coefficient inverse problem, Appl. Anal. 87:1227–1254, 2008. [41] R. V. Kohn, D. Onofrei, M. S. Vogelius and M. I. Weinstein, Cloaking via change of variables for the Helmholtz equation, Commun. Pure Appl. Math. 63:973–1016, 2010.

122 | 4 High-order topological expansions for Helmholtz problems in 2D

[42] V. A. Kovtunenko, Primal-dual methods of shape sensitivity analysis for curvilinear cracks with non-penetration, IMA J. Appl. Math. 71:635–657, 2006. [43] V. A. Kovtunenko and K. Kunisch, High precision identification of an object: optimality conditions based concept of imaging, SIAM J. Control Optim. 52:773–796, 2014. [44] V. A. Kovtunenko, K. Kunisch and W. Ring, Propagation and bifurcation of cracks based on implicit surfaces and discontinuous velocities, Comput. Visual Sci. 12:397–408, 2009. [45] V. A. Kovtunenko and G. Leugering, A shape-topological control problem for nonlinear crack – defect interaction: the anti-plane variational model, SIAM J. Control Optim. 54:1329–1351, 2016. [46] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Equations of Elliptic Type, Academic Press, New York, 1968. [47] A. Laurain, M. Hintermüller, M. Freiberger and H. Scharfetter, Topological sensitivity analysis in fluorescence optical tomography, Inverse Probl. 29:025003, 2013. [48] M. M. Lavrentiev, A. V. Avdeev, M. M. Lavrentiev jr. and V. I. Priimenko, Inverse Problems of Mathematical Physics, VSP Publ., Utrecht, Boston, 2003. [49] V. G. Maz’ya, S. A. Nazarov and B. A. Plamenevski, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Birkhäuser, Basel, 2000. [50] G. Nakamura, R. Potthast and M. Sini, Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test, Commun. Part. Diff. Eq. 31:1505–1528, 2006. [51] P. Neittaanmäki and G. F. Roach, Weighted Sobolev spaces and exterior problems for the Helmholtz equation, Proc. Roy. Soc. London Ser. A 410:373–383, 1987. [52] D. Pauly and S. Repin, Functional a posteriori error estimates for elliptic problems in exterior domains, J. Math. Sci. (N.Y.) 162:393–406, 2009. [53] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton, 1951. [54] J. Rocha de Faria, A. A. Novotny, R. A. Feijoo, E. Taroco and C. Padra, Second order topological sensitivity analysis. Int. J. Solids Structures 44:4958–4977, 2007. [55] M. Rumpf and B. Wirth, Variational methods in shape analysis, In O. Scherzer (ed.), Handbook of Mathematical Methods in Imaging, 1363–1401, Springer, 2011. [56] B. Samet, S. Amstutz and M. Masmoudi, The topological asymptotic for the Helmholtz equation, SIAM J. Control Optim. 42:1523–1544, 2003. [57] J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization, SIAM J. Control Optim. 37:1251–1272, 1999. [58] I. N. Vekua, On Metaharmonic Functions, Lecture Notes of TICMI 14, Tbilisi University Press, 2013. [59] M. I. Vishik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspehi Mat. Nauk (N.S.) 12:3–122, 1957. (in Russian)

Elie Bretin and Simon Masnou

5 On a new phase field model for the approximation of interfacial energies of multiphase systems Abstract: This chapter is devoted to a new multiphase field approximation model for a category of interfacial energies of multiphase systems which appear in material sciences or image processing. Our model has several advantages when the surface tensions satisfy a suitable embedding property (namely the ℓ1 -embeddability): (i) it can be explicitly derived from the surface tensions; (ii) the Γ-convergence to the multiphase perimeter can be proven; (iii) it is convenient for robust numerical approximation of the associated gradient flow. Several applications are presented, in particular to droplets dynamics. Keywords: phase field, multiphase perimeter, Γ-convergence, droplets, material sciences, image processing.

5.1 Introduction This chapter is a short account of results more completely described in [8], which are related to the approximation with a phase field model of an N-phase perimeter of the form 1 N ∑ σ i,j H d−1 (∂∗ Ω i ∩ ∂∗ Ω j ) . (1.1) P(Ω1 , . . . , Ω N ) = 2 i,j=1 Here, Ω1 , . . . , Ω N denote the sets of the finite perimeter covering Ω, i.e., Ω = ⋃Ni=1 Ω i up to a Lebesgue negligible set, |Ω i ∩ Ω j | = 0 for all i ≠ j (denoting as | ⋅ | the Lebesgue measure), H d−1 is the (d−1)-dimensional Hausdorff measure, and ∂∗ Ω i is the reduced boundary of Ω i in Ω (i.e., the sets of boundary points of Ω i in Ω where an approximate normal exists); see [2, 20] for details on functions of bounded variation (BV) and sets of finite perimeter. In the sequel, we will often denote as P the BV perimeter, i.e., if A ⊂ Ω has finite perimeter in Ω, we define P(A) = H d−1 (∂∗ A). In (1.1), σ i,j is the surface tension associated with Γ i,j = ∂∗ Ω i ∩ ∂∗ Ω j for i, j = 1, . . . , N. It is physically sound to assume that the surface tensions belong to the following space: S N = {σ = (σ i,j ) ∈ ℝN×N , σ i,j = σ j,i > 0 if i ≠ j and σ i,i = 0} In order to guarantee the lower semicontinuity of the N-phase perimeter, it is necessary and sufficient to assume that the surface tensions satisfy the triangle inequalElie Bretin, Université de Lyon, CNRS UMR 5208, INSA de Lyon, Institut Camille Jordan, 20 Avenue Albert Einstein, F-69621 Villeurbanne Cedex, France, [email protected] Simon Masnou, Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, [email protected]

124 | 5 On a new multiphase field model for interfacial energies ity [9, 20, 23], i.e., σ i,j + σ j,k ≥ σ i,k for any i, j, k. Although mathematically sound, this property is however not fulfilled by all physical systems, so that the approximation issue needs also to be addressed in the nontriangle case. There are many applications in material science where the multiphase perimeter plays a role (with either uniform or nonuniform, isotropic or anisotropic surface tensions): it determines the arrangement of grains in polycrystalline materials, the structure of soap foams, honeycombs, or nanowires, etc. [15]. Other applications can be found in image processing, where multiphase perimeters are very useful in the context of image segmentation, image restoration, optical flow estimation or stereo reconstruction [10, 24, 26]. In this chapter, we focus on multiphase perimeters which involve possibly nonuniform but isotropic surface tensions. The most simple instance of a multiphase system is the binary system with constant surface tension whose perimeter’s gradient flow is the celebrated mean curvature flow. There is a vast literature devoted to its numerical approximation, and the methods can be roughly classified into five categories [13]: parametric methods, level set methods, convolution/thresholding-type algorithms, convexification methods, and phase-field approaches. The literature on the approximation of multiphase perimeters is more reduced, but there have been contributions in the same five categories of methods. The method that we propose in this chapter belongs to the category of phase-field approaches. Both theoretical and numerical contributions are related to this topic, see for instance [3, 5, 16–19, 25, 28] and the numerous references therein. Before entering into more details, let us sketch the main properties of our model: (P1) It is a phase-field model with a potential term that can be derived consistently and explicitly from a given matrix σ ∈ S N of surface tensions, as soon as σ can be associated with ℓ1 distances between vectors in ℝM for a suitable dimension M (such σ is called ℓ1 -embeddable; see more details below). As will be discussed later, the consistent derivation of the potential is a major difference with the derivations that can be found in the literature; (P2) In the strict ℓ1 -embeddability case (see below), the Γ-convergence of our approximating model to the multiphase perimeter can be proven. In view of the previous item, it is to the best of our knowledge the first contribution where one can derive the model from the surface tensions and recover them back from the Γ-convergence; (P3) Our approximating perimeter falls in a large category of phase-field models studied with great accuracy by Garcke et al. [17] where it is shown using the matched asymptotic expansion method that the correct evolution laws for simple and triple points are recovered asymptotically; (P4) The L2 -gradient flow yields an Allen–Cahn system with a linear diffusive part, which allows simple and robust numerical schemes with a very good spatial accuracy;

5.2 Derivation of the phase-field model | 125

(P5) Various interesting constraints can be easily added to the model, e.g., volume constraints or stationarity constraint on a phase, which can be very useful for simulating wetting phenomena.

5.2 Derivation of the phase-field model 5.2.1 The classical constant case σ i,j = 1 In the standard and simple case where all surface tensions are constant, i.e., σ i,j = 1, one has P(Ω1 , . . ., Ω N ) = 12 ∑Ni,j=1 H d−1 (Γ i,j ) = 12 ∑Ni=1 H d−1 (∂∗ Ω i ), and the latter expression depends on full interface boundaries ∂∗ Ω i (full with respect to Ω), and not partial interface boundaries Γ i,j = ∂∗ Ω i ∩ ∂∗ Ω j . Denoting u = (u 1 , u 2 , . . . , u N ) ∈ ℝN and Σ = {u ∈ ℝN ; ∑Ni=1 u i = 1}, it is well known that P can be easily approximated by { 1 ∑N ∫ ( ϵ |∇u i |2 + 1ϵ F(u i )) dx, P ϵ (u) = { 2 i=1 Ω 2 +∞ {

if u ∈ Σ otherwise

where ϵ is a small parameter that characterizes the width of the diffuse interface, and F(s) = (s2 (1 − s)2 )/2 is a double-well potential. This follows from Modica–Mortola’s theorem [21] that states that the family of functionals J ϵ defined by ϵ 1 J ϵ (u) = ∫ ( |∇u|2 + F(u)) dx 2 ϵ Ω 1

approximates (in the sense of Γ-convergence) c F P with c F = ∫0 √2F(s) ds and {|Du|(Ω) P(u) = { +∞ {

when u ∈ BV(Ω, {0, 1} otherwise,

where |Du|(Ω) denotes the total variation of u ∈ BV(Ω). Using this approximation for every phase u i yields P ϵ , of which the Γ-convergence to c W P can be obtained as in [3, 25]. Moreover, the L2 -gradient flow of P ϵ reads ∂t ui =

1 1 (∆u i − 2 F 󸀠 (u i )) + λ(t), 2 ϵ

for all i = 1, . . . , N ,

where λ(t) is a Lagrange multiplier associated with the constraint u ∈ Σ that can be explicitly computed as λ(t) = Nϵ1 2 ∑Ni=1 F 󸀠 (u i ). The flow is an Allen–Cahn system that can be easily approximated numerically, for instance, using a splitting method with an implicit resolution of the diffusion term in Fourier space coupled with an explicit treatment of the reaction term [11].

126 | 5 On a new multiphase field model for interfacial energies The case of a general surface tension σ ∈ S N is more involved. In [17, 18], Garcke et al. studied phase-field approximations of the general form {∫ ϵf(u, ∇u) + 1ϵ W(u) dx P ϵ (u) = { +∞ {

if u ∈ Σ otherwise.

The category of functions f and W for which the results obtained in the chapter apply is quite general, but the authors have a preference for N

α i,j 󵄨 󵄨󵄨u i ∇u j − u j ∇u i 󵄨󵄨󵄨2 , 󵄨 󵄨 2 i,j=1

f(u, ∇u) = ∑

with (α i,j )i,j ∈ S N ,

(2.1)

and W a positive multiwell potential defined on Σ and vanishing only at each standard unit vector of the canonical basis (e1 , . . . , e N ) of ℝN . Each vector ei corresponds to a phase, and an N-phase system is given by u = ∑Ni=1 u i ei with ∑ u i = 1. The authors of [18] propose a multiwell potential of the form N

1 α i,j u 2i u 2j + ∑ α i,j,k u 2i u 2j u 2k 2 i,j=1 i 3 follows from the computation above. Indeed, for all u ∈ Σ N

N 1 N 1 N W σ (u) = ∑ σ i F(u i ) = ∑ σ i u 2i (1 − u i )2 = ∑ σ i u 2i ( ∑ u j ) 2 i=1 2 i=1 i=1 j=1,j=i̸

=

1 N 1 ∑ σ i,j u 2i u 2j + 4 i,j=1 2

2

N

∑ i 0 for all i = 1, . . . , N. Then the phase-field perimeter

where

{∫ (− ϵ σ∇u ⋅ ∇u + 1ϵ W σ (u)) dx if u ∈ Σ P ϵ (u) = { Ω 4 +∞ otherwise, { N 1 N ∑ (σ i + σ j )u 2i u 2j + ∑ σ k u i u j u 2k , W σ (u) = 4 i,j=1 i 0, ∀i ∈ {1, . . . , N}. Remark 2.5. It is easily seen that a necessary condition for strict ℓ1 -embeddability is the strict triangle inequality, i.e., σ i,k < σ i,j + σ j,k whenever i ≠ j ≠ k. When N ≤ 3, the strict triangle inequality is also a sufficient condition for strict ℓ1 -embeddability due to the fact that surface tensions which satisfy the strict triangle inequality are additive with positive coefficients, and the Cut Cone Property follows with coefficients σ {i} = σ i > 0. Whenever N = 4, the same conclusion follows from Remark 2.6. Lastly, the same remark and the last paragraph before Section 5.3 yield that the ℓ1 -embeddability together with the strict triangle inequality imply the strict ℓ1 -embeddability when N > 4.

130 | 5 On a new multiphase field model for interfacial energies

5.2.4 Derivation of the approximation perimeter for ℓ1 -embeddable surface tensions In the case of additive surface tensions, as we saw above, the multiphase perimeter can be directly written as a nonnegative combination of integrals on boundaries of sets (and not subsets of boundaries), which allows a multiphase approximation. As it follows from Lemma 2.3, a similar decomposition holds for ℓ1 -embeddable surface tensions. Thus, a multiphase approximation is again possible, and a natural candidate to approximate P is given by 󵄨 󵄨2 {∫ [∑ σ S ( 2ϵ 󵄨󵄨󵄨∇(∑i∈S u i )󵄨󵄨󵄨 + 1ϵ F(∑ i∈S u i ))] dx P ϵ (u) = { Ω S⊂{1,2,...,N} +∞ {

if u ∈ Σ otherwise ,

where the coefficients σ S are given by the Cut Cone Property (2.4). Note that for N ≥ 4, the decomposition is not unique. This expression has a drawback: the σ S ’s are unknown. We will now derive another expression which can be explicitly computed from the surface tension matrix σ = (σ i,j ) as soon as σ i,j > 0 whenever i ≠ j. 5.2.4.1 A condensed form for the approximating multiphase perimeter Let α i = ∑S⊂{1,2,...,N} σ S δ i∈S and α i,j = ∑S⊂{1,2,...,N} σ S δ i∈S δ j∈S with δ i∈S = 1 if i ∈ S, 0 otherwise. Since σ is assumed to be ℓ1 -embeddable, it follows from the Cut Cone Property (2.4) that σ i,j =

∑

σ S (δ i∈S δ j∈S̸ + δ i∈S̸ δ j∈S ) = α i + α j − 2α i,j .

S⊂{1,2,...,N}

Then, we calculate that, for all u ∈ Σ (see [8]) 󵄨󵄨2 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 σ S 󵄨󵄨󵄨∇ (∑ u i )󵄨󵄨󵄨 = − σ∇u ⋅ ∇u , 󵄨󵄨 󵄨󵄨 2 S⊂{1,2,...,N} i∈S 󵄨 󵄨 ∑

therefore, the approximating perimeter introduced above can be rewritten as {∫ − ϵ σ∇u ⋅ ∇u + 1ϵ W σ (u) dx P ϵ (u) = { Ω 4 +∞ {

if u ∈ Σ , otherwise,

where the multiwell potential W σ (u) reads W σ (u) =

∑

σ S F (∑ u i ) .

S⊂{1,2,...,N}

i∈S

(2.5)

5.2 Derivation of the phase-field model

| 131

5.2.4.2 Rewriting the potential when N = 4 When N = 4, any surface tension matrix satisfying the triangle inequality is ℓ1 -embeddable [14], thus the perimeter is decomposable as P(Ω1 , Ω2 , . . . , Ω4 ) =

1 4 ∑ σ i,j H d−1 (Γ i,j ) = ∑ σ S H d−1 (∂∗ (∪i∈S Ω i )) , 2 i,j=1 S⊂{1,2,...,4}

(2.6)

but no explicit formula is known for the coefficients σ S . Considering the whole collection of sets ∪i∈S Ω i , S ⊂ {1, . . . , N}, we define Q1 = Ω1 ,

Q2 = Ω2 ,

Q3 = Ω3 ,

Q4 = Ω4

and Q5 = Ω1 ∪ Ω2 = (Ω3 ∪ Ω4 )c , Q6 = Ω1 ∪ Ω3 = (Ω2 ∪ Ω4 )c , Q7 = Ω2 ∪ Ω3 = (Ω1 ∪ Ω4 )c . Defining σ̃ 1 { { { { { {σ̃ 2 { { σ̃ 3 { { { { {σ̃ 4

= (σ 12 + σ 13 + σ 14 ) /2 = (σ 12 + σ 23 + σ 24 ) /2

and

= (σ 13 + σ 23 + σ 34 ) /2 = (σ 14 + σ 24 + σ 34 ) /2

σ̃ 5 = − (σ 12 + σ 34 ) /2 { { { σ̃ 6 = − (σ 13 + σ 24 ) /2 { { { {σ̃ 7 = − (σ 14 + σ 23 ) /2 ,

we calculate from (2.6) that P(Ω1 , Ω2 , Ω3 , Ω4 ) = ∑7i=1 σ̃ i H d−1 (∂∗ Q i ). Denoting σ min = max {−σ̃ i } , i=5,6,7

σ max =

and

min { σ̃ i } ,

i=1,2,3,4

we have that σ min ≤ σ max . Let us now choose arbitrarily σ ∗ ∈ [σ min , σ max ] and define { σ i = σ̃ i − σ ∗ { σ = σ̃ i + σ ∗ { i

for i = 1, . . . , 4 , otherwise .

Obviously P(Ω1 , Ω2 , Ω3 , Ω4 ) = ∑7i=1 σ i H d−1 (∂∗ Q i ), and, from our observations above, σ i ≥ 0 for all i ∈ {1, . . . , 7}. Remark 2.6. In the particular case where (σ i,j ) satisfies the strict triangle inequality, then σ min < σ max thus one can choose σ ∗ = (σ min + σ max )/2 so that σ i > 0 for every i ∈ {1, . . . , 7}, i.e., σ is strictly ℓ1 -embeddable. The previous argument also shows that for all decompositions of P of the form 7

P(Ω1 , Ω2 , Ω3 , Ω4 ) = ∑ α i H d−1 (∂∗ Q i ) , i=1

with α i ≥ 0, we can associate a coefficient {α i = σ̃ i − σ ∗ { α = σ̃ i + σ ∗ { i

σ∗

∈ [σ min , σ max ] such that

for i = 1, . . . , 4 , for i = 5, . . . , 7 .

132 | 5 On a new multiphase field model for interfacial energies

The decomposition that we have obtained leads to a natural potential for the phasefield approximation, i.e., Wσ can be chosen as W σ (u) =

1 2

4

4

4

i=2

i=1

σ i,j (F(u i ) + F(u j ) − F(u i + u j )) + σ ∗ ( ∑ F(u 1 + u i ) − ∑ F(u i ))

∑ i,j=1,i 0 1 such that 2ε > 2N(N−1) + 2N‖u‖∞ and ε ≤ d/8; in this way it is true that if we define d w x0 (x1 ) =

N { } 1 − ∑ u(x i )} , ∑ { |x − x | |x i −x 0 |≥ε,i=2,...,N i j i=2 {1≤i d/2 − 2ε ≥ d/4 for every i ≠ j, and so we have w ≤ 2N(N−1) + N‖u‖∞ but then it is clear that for d any (x2 , . . . , x N ) such that |x i − x0 | ≤ ε we will have ∑ 1≤i w(x1 ) |x i − x j | i=2 2ε

∀x1 ∈ B(x0 , ε) ;

this proves that in fact w x0 = w on the set B(x0 , ε) and so, in particular, also in B(x0 , ε/2). But in this set w x0 is Lipschitz and semi concave since it is an infimum of uniformly C∞ functions on B(x0 , ε/2). Moreover the bounds on the first and second derivatives do not depend on x0 but only on ε, that is fixed a priori, and so by a covering argument we obtain the thesis. Another interesting reformulation of the Coulomb-like problem, or more generally when we have only interaction between two particles, can be found in [38] where, seeking a dimensional reduction of the problem, the authors use the fact that if γ ∈ Π N (μ) is a symmetric plan then ∫

∑

ℝNd

1≤i 0 ,

{−1 − 2x S(x) = { 1 − 2x {

if x ≤ 0 if x > 0 .

We have T♯ μ = μ and S♯ μ = μ, and moreover x + T(x) + S(x) = 0; in particular (Id, S, T)♯ μ is a flat optimal plan and so the thesis. Example 4.14 (Optimal transport diffuse plan). Let us consider the same problem as in Example 4.13. Now we consider a general symmetric plan γ = 12 H2 |H f(max{|x|, |y|, |z|}), where H is defined as H = {x + y + z = 0} ∩ {|x| ≤ 1, |y| ≤ 1, |z| ≤ 1} and f is a function to be chosen later. This is a symmetric flat optimal plan; now we compute the marginals. Since it is clear that γ is invariant under x 󳨃→ −x, it is sufficient to consider the marginal on the set x > 0. But then we can make the computation ∫ ϕ(x) dγ(x, y, z) =

∬

√3ϕ(x)f(max{|x|, |y|, |x + y|}) dx dy

|x|,|y|,|x+y|≤1 , x≥0

x≥0

1 1−x

1 0

= ∫ ∫ √3ϕ(x)f(x + y) dy dx + √3 ∫ ∫ ϕ(x)f(x) dy dx 0 −x

0 0 1 −x

+ √3 ∫ ∫ ϕ(x)f(|y|) dy dx 0 −1 1

1

= ∫ √3ϕ(x) (xf(x) + 2 ∫ f(t) dt) dx . 0

x

In particular the choice f(x) = √63 x gives the marginals equal to 12 L|[−1,1] . We also notice that any even density ρ = h(|x|) for some decreasing function h : [0, 1] → [0, ∞) can be represented in this way: in fact it is sufficient to choose f(x) = √13 ( h(x) x − 1 h(t) t3

2x ∫x

dt).

Remark 4.15 (A Counterexample on the uniqueness N > 3). As mentioned in Theorem 2.16 and in [38], it was already understood by Pass that in these high-dimensional cases, the solution of repulsive costs may also be non unique, as opposite to the 2-marginals case. On a higher dimensional surface there can be enough wiggle room

238 | 9 Optimal transport for repulsive costs: old, new and numerics

to construct more than one measure with common marginals, as shown in Examples 4.9–4.14. In most of the cases, the non-uniqueness seems to be given by the symmetries of the problem, but in Example 4.14 and Corollary 4.8 this is not the case, as we exploit the fact that the dimension of the set c(x + y + z) − ϕ(x) − ϕ(y) − ϕ(z) = 0, is greater than the minimal one. Finally, the last proposition of this section states that when d = 1 and odd N we have no hope in general to find piecewise regular cyclic optimal transport maps. Proposition 4.16. Let μ = L|[0,1] and N ≥ 3 be an odd number. Then, the infimum inf { ∫ c(x, T(x), T (2) (x), . . . , T (N−1) ) dμ :

T♯ μ = μ , }. TN = I

is not attained by a map T which is differentiable almost everywhere. Proof. First, we notice that if T is differentiable almost everywhere then also T (2) has the same property, thanks to the fact that T♯ μ = μ. In particular, since the Lusin property holds true for T (i) for every i = 1, . . . , N and T♯ μ = μ, the change of variable formula holds and in particular we have (T (i) )󸀠 (x) = ±1 for almost every x (notice also that T is bijective almost everywhere since T (N) (x) = x). Since μ is N-flat, we have that the condition on T in order to be an optimal cyclical map is x + T(x) + T (2) (x) + ⋅ ⋅ ⋅ + T (N−1) (x) = N/2; now we can differentiate this identity and so we will get 1 + T 󸀠 (x) + (T (2) )󸀠 (x) + ⋅ ⋅ ⋅ + (T (N−1) )󸀠 (x) = 0

for a.e. x .

But this is absurd since on the left-hand side we have an odd number of ±1 and their sum will be an odd number. In conclusion, also in the case of the repulsive harmonic cost, the picture is far from being clear: an interesting structure appears when {μ i }Ni=1 is a flat N-tuple of measures but we still cannot characterize this property. Moreover, in the flat case in which μ i = μ, for example, when μ = 12 L1 |[−1,1] , we have both diffuse optimal plan and a cyclical optimal map. An interesting open problem is whether for any N-flat measure μ, say absolutely continuous with respect to the Lebesgue measure, we have a cyclical optimal map and a diffuse plan.

9.5 Multi-marginal OT for the determinant We are going to give a short overview of the main results in [17], where Carlier and Nazaret consider the following optimal transport problems for the determinant: (MKDet )

sup γ∈Π((ℝd )d ,μ 1 ,...,μ d )

∫ det(x1 , . . . , x d ) dγ(x1 , . . . , x d ) (ℝd )d

(5.1)

9.5 Multi-marginal OT for the determinant |

239

and (MK| Det | )

sup γ∈Π((ℝd )d ,μ 1 ,...,μ d )

∫ | det(x1 , . . . , x d )| dγ(x1 , . . . , x d ) ,

(5.2)

(ℝd )d

where μ1 , . . . , μ d are absolutely continuous probability measures in ℝd . In addition, in order to guarantee existence of a solution, we assume that there exist p1 , . . . , p d ∈ [1, ∞[ such that d

1 = 1, p i=1 i ∑

d

and

∑∫ i=1

ℝd

|x i |p i dμ(x i ) < +∞ . pi

Notice that for this particular cost, the problem (2.1) makes sense only when N = d. In the following, we will focus on problem (5.1) and exhibit explicit minimizers γ in the radial case. Clearly, the difference between (5.1) and (5.2) is that the second one admits positively and negatively oriented basis of vectors, while the first one ‘chooses’ only the positive ones. Moreover, if we assume that among marginals μ 1 , . . . , μ d there exist two symmetric probability measures μ i , μ j , i ≠ j, i.e. μ i = (− Id)♯ μ i and μ j = (− Id)♯ μ j , then any solution γ of (5.1) satisfies det(x1 , . . . , x d ) ≥ 0 γ-almost everywhere and so also solves (5.2) (Proposition 6, [17]). Similarly to the Gangbo–Świe¸ch cost [41], the Monge–Kantorovich problem for the determinant (5.1) can be seen as a natural extension of classical optimal transport problem with 2-marginals and so it is equivalent to the 2-marginals repulsive harmonic cost (4.1). Indeed, we can write in the 2-marginals case, det(x1 , x2 ) = ⟨x1 , Rx2 ⟩, where R : ℝ2 → ℝ2 is the rotation of angle −π/2. Hence, since μ1 and μ2 have finite second moments, up to a change of variable x̃ 2 = Rx2 , the problem (MKDet ) in (5.1) is equivalent to the classical Brenier’s optimal transportation problem: argmax ∫ det(x1 , x2 ) dγ(x1 , x2 ) = argmax ∫ ⟨x1 , x̃ 2 ⟩ dγ(x1 , x2 )

γ∈Π(μ 1 ,μ 2 )

γ∈Π(μ 1 ,μ 2 )

ℝ2

ℝ2

= argmax ∫ ⟨x1 , x2 ⟩ dγ̃ ̃ γ∈Π(μ 1 , μ̃ 2 )

ℝ2

= argmin ∫ γ∈Π(μ 1 , μ̃ 2 )

ℝ2

|x1 − x2 |2 dγ(x1 , x2 ) − C 2

where C = 1/2(∫ |x1 |2 dμ1 + ∫ |x2 |2 dμ2 ) and μ̃ 2 = R♯ μ 2 . In the sequel, we are going to construct maximizers for (5.1), thanks to some properties of the Kantorovich potentials of the dual problem associated to (5.1) (see Theorem 5.1), d

(KDet N )

N

inf { ∫ ∑ u i (x i ) dμ i (x i ) : det(x1 , . . . , x d ) ≤ ∑ u i (x i )} . ℝd

i=1

i=1

(5.3)

240 | 9 Optimal transport for repulsive costs: old, new and numerics

In [17], the authors provide a useful characterization of optimal transport plans through the potentials u i , given by Theorem 5.1. In addition, by means of a standard convexification trick we obtain regularity results on the Kantorovich potentials. Theorem 5.1. A coupling γ ∈ Π((ℝd )d , μ1 , . . . , μ d ) is optimal in (5.1) if and only if there exists lower semi-continuous convex functions u i : ℝd → ℝ ∪ {∞} such that for all i ∈ {1, . . . , d}, d

∑ u j (x j ) ≥ u ∗i ((−1)i+1 ⋀ x j ) ,

j=1,j=i̸

on

d

(ℝd ) ;

j=i̸

d

∑ u j (x j ) = u ∗i ((−1)i+1 ⋀ x j ) ,

γ-almost everywhere ;

(−1)i+1 ⋀ x j ∈ ∂u i (x i ),

γ-almost everywhere .

j=1,j=i̸

j=i̸

j=i̸

where ⋀ di=1 x j denotes the wedge product and, for every i, u ∗i is the convex dual of the Kantorovich potential u i . Now, the main idea is to use the geometrical constraints on the Kantorovich potentials u i (5.3), given by Theorem 5.1, in order to construct an explicit solution. We illustrate Theorem 5.1 and explain how to construct a particular optimal γ for (5.1) by an example in the 3-marginals case. Let μ i = ρ i L3 , i = 1, 2, 3 radially symmetric probability measures on the 3-dimensional ball B. In this particular situation, the optimizers of (5.1) and (5.2) have a natural geometric interpretation: what is the best way to place three random vectors x, y, z, distributed by probability measures μ1 , μ2 , μ3 on the sphere, such that the simplex generated by those three vectors (x, y, z) has maximum average volume? Suppose γ ∈ Π(B, μ 1 , μ2 , μ3 ) optimal in (5.1) when d = 3. From optimality of γ, we have u 1 (x) + u 2 (y) + u 3 (z) = det(x, y, z), γ-almost everywhere . Applying Theorem 5.1, we obtain u 2 (y) + u 3 (z) = u ∗1 (y ∧ z) { { { u (x) + u 3 (z) = u ∗2 (−x ∧ z), { { 1 { ∗ { u 1 (x) + u 2 (y) = u 3 (x ∧ y)

γ-almost everywhere ,

and, ∇u 1 (x) = y ∧ z { { { ∇u 2 (y) = −x ∧ z, { { { {∇u 3 (z) = x ∧ y

γ-almost everywhere .

(5.4)

9.5 Multi-marginal OT for the determinant | 241

It follows from (5.4), given a vector x in the ball, the conditional probability of y given x is supported in a meridian M(x) M(x) = {y ∈ S2 : ⟨∇u 1 (x), y⟩ = 0} , where S2 is the 2-sphere. Finally, assuming that ⟨x, ∇u 1 (x)⟩ ≠ 0, the conditional probability of z given the pair (x, y) is simply given by a delta function on z z=

∇u 1 (x) ∧ ∇u 2 (y) . ⟨x, ∇u 1 (x)⟩

In particular, we have ⟨x, ∇u 1 (x)⟩ + ⟨y, ∇u 2 (y)⟩ + ⟨z, ∇u 3 (z)⟩ = det(x, y, z) = det(∇u 1 (x), ∇u 2 (y), ∇u 3 (z)). Example 5.2 (An explicit solution in the ball B ⊂ ℝ3 , see also [17]). Suppose μ i = L3B , i = 1, 2, 3, the 3Th-dimensional Lebesgue measure in the ball B ⊂ ℝ3 . The following coupling γ∗ ∫ f dγ∗ = B3

1 L3 (B)

∫ ( ∫ f(x, |x|y, x ∧ y) B

dH1 (y) ) dx, ∀f ∈ C(B3 , ℝ). 2π

(5.5)

M(x)

is an optimizer for (5.1) with d = 3. Indeed, from this we can show explicit potentials u∗1 (x) = u ∗2 (x) = u ∗3 (x) = |x|3 /3; clearly we have det(x, y, z) ≤ |x||y||z| ≤ |x|3 /3 + |y|3 /3 + |z|3 /3,

∀ (x, y, z) ∈ B ,

with equality when |x| = |y| = |z| and x, y, z are orthogonal. Since γ∗ is concentrated on this kind of triples of vector we have the optimality. Finally, by a suitable change of variable, it is easy to see that γ∗ ∈ Π3 (L3B ). Some comments on the radially symmetric d-marginals case: In [17], for d radially symmetric probability measures, the authors exhibit explicit optimal couplings γ∗. In their proof, two aspects were crucial: the first one is remark that if {μ i }di=1 are radially symmetric measures in ℝd , then the optimal Kantorovich potentials u i (x i ) = u i (|x i |) are also radially symmetric; in particular the system (5.4) for general d implies that the support of γ∗ is contained in the set of an orthogonal basis. The second observation is to notice that in the support of γ∗ we have G i (|x1 |) = |x i |, where G i is the unique x monotone increasing map such that (G̃ i )♯ μ 1 = μ i , where G̃ i (x) = |x| G i (|x|). This is done analyzing the corresponding radial problem (with cost c(r1 , . . . , r d ) = r1 ⋅ ⋅ ⋅ r d ), using the optimality condition ϕ󸀠i (r i ) = ∂ i c and the fact that in this case r i ϕ󸀠i (r i ) = r1 ϕ󸀠1 (r1 ) = c ≥ 0; then, exploiting the convexity of ϕ i we get r i = G i (r1 ) for some increasing function G i , which is uniquely determined. Existence of Monge-type solutions: In the 3-marginals case in the unit ball, by construction of the coupling γ∗ in (5.5) or, more generally, the optimal coupling in the

242 | 9 Optimal transport for repulsive costs: old, new and numerics

d-marginal case (see Theorem 4 in [17]), we can see that their support are not concentrated in the graph of cyclic maps T, T 2 , . . . , T d−1 or simply on the graph of maps T1 , . . . , T d−1 as we could expect from Corollary 2.15. In other words, γ∗ in (5.5) is not Monge-type solution. The existence of Monge-type solutions for the determinant cost is still an open problem for odd number of marginals. From the geometric conditions we discussed above, in the case in which μ i = μ a radial measure, if Monge solutions exist then, for every x ∈ ℝd , (x, T1 (x), . . . , T d−1 (x)) should be an orthogonal basis, and |T i (x)| = |x|, i = 1, . . . , d − 1. For the interesting even-dimensional case, we can observe a similar phenomena remarked in the repulsive harmonic costs, concerning the existence of trivial evendimensional solutions for the Monge problem in (5.1). We expect the existence of nonregular optimal transport map also to this case. Example 5.3 (Carlier and Nazaret, [17]). The even-dimensional phenomenon: as in the repulsive harmonic cost, it is easy to construct Monge minimizers for the determinant cost for even number of marginals ≥ 4. For instance, suppose c(x1 , x2 , x3 , x4 ) = det(x1 , x2 , x3 , x4 ), alle the marginals equal to the uniform measure μ on the ball, define transport maps T1 , T2 , T3 : B → ℝ4 by, for x = (x1 , x2 , x3 , x4 ) ∈ B −x2 x1 T1 (x) = ( ), −x4 x3

−x3 x4 T2 (x) = ( ), x1 −x2

−x4 −x3 T3 (x) = ( ) . x2 x1

We can see that γ T = (Id, T1 , T2 , T3 )♯ μ is a Monge-type optimal transport plan for (5.1) and (5.2).

9.6 Numerics Numerics for the multi-marginal problems have so far not been extensively developed. Discretizing the multi-marginal problem leads to a linear program where the number of constraints grows exponentially in the number of marginals. Carlier et al. [18] studied the matching of teams problem and they were able to reformulate the problem as a linear program whose number of constraints grows only linearly in the number of marginals. More recently a numerical method based on an entropic regularization has been developed and it has been applied to various optimal transport problems in [9, 10, 29]. Let us mention that the Coulomb cost has been treated numerically in recent paper as in [45, 63], where the solutions are based on the analytical form of transport maps given in [45], in [64], where parameterized functional forms of the Kantorovich potential are used, in [19, 20], where a linear programming approach has been developed to solve the one-dimensional problem, and, as already mentioned,

9.6 Numerics

| 243

in [10]. In this section, we focus on the regularized method proposed in [9, 10, 29] and we, finally, present some numerical experiments for the costs studied above.

9.6.1 The regularized problem and the iterative proportional fitting procedure Let us consider the problem (MK)

∫ c(x1 , . . . , x N )γ(x1 , . . . , x N ) dx1 , . . . , dx N ,

inf

γ∈Π(ℝdN ,μ 1 ,...,μ N )

(6.1)

ℝdN

where N is the number of marginals μ i , which are probability distributions over ℝd , c(x1 , . . ., x N ) is the cost function, γ the coupling, is the probability distribution over ℝdN Remark 6.1. From now on the marginals μ i and the coupling γ are densities and when the optimal coupling γ is induced by maps T i , we write it as γ = μ1 (x1 )δ(x2 − T2 (x1 )) ⋅ ⋅ ⋅ δ(x N − T N (x1 )). In order to discretize (6.1), we use a discretization with M d points of the support of the kth marginal as {x j k }j k =1,...,M d . If the densities μ k are approximated by μ k ≈ ∑j k μ j k δ x jk , we get min ∑ c j1 ,...,j N γ j1 ,...,j N , (6.2) γ∈Π k

j1 ,...,j N

where Π k is the discretization of Π, c j1 ,...,j N = c(x j1 , . . . , x j N ) and the coupling support for each coordinate is restricted to the points {x j k }j k =1,...,M d thus becoming a (M d )N matrix again denoted by γ with elements γ j1 ,...,j N . The marginal constraints Ci (such that Π k = ⋂Nk=1 Ck ) becomes } γ j1 ,...,j N = μ j k , ∀j k = 1, . . . , M d } . j1 ,...,j k−1 ,j k+1 ,...,j N }

{ (M )N Ck := { γ ∈ ℝ+ d : {

∑

(6.3)

As in the continuous framework, the problem (6.1) admits a dual formulation N

Md

max ∑ ∑ u j k μ j k u jk

k=1 j k =1 N

s.t. ∑ u j k ≤ c j1 ⋅⋅⋅j N

∀ j k = 1, . . . , M d ,

(6.4)

k=1

where u j k = u k (x j k ) is the kth Kantorovich potential. One can notice that the primal (6.1) has (M d )N unknowns and M d × N linear constraints and the dual problem (6.4) has M d × N unknowns, but (M d )N constraints. This actually makes the problems computationally unsolvable with standard linear programming methods even for small cases.

244 | 9 Optimal transport for repulsive costs: old, new and numerics

Remark 6.2. We underline that in many applications we have presented (as in DFT) the marginals μ j k are equal (μ j k = μ j , ∀ k ∈ {1, . . . , N}). Thus the dual problem can be re-written in a more convenient way (but still computationally unfeasible for many marginals) M

max ∑ Nu j μ j uj

j=1 N

s.t. ∑ u j k ≤ c j1 ⋅⋅⋅j N

∀j k = 1, . . . , M d ,

(6.5)

k=1

where u j = u j k = u(x j k ). Now the dual problem has M d unknown, but (M d )N linear constraints. A different approach consists in computing the problem (6.1) regularized by the entropy of the joint coupling. This regularization dates to Schrödinger [81] and, as mentioned above, it has been recently introduced in many applications involving optimal transport [9, 10, 29, 40]. Thus, we consider the following discrete regularized problem min ∑ c j1 ,...,j N γ j1 ,...,j N + ϵE(γ) , γ∈C

(6.6)

j1 ,...j N

where E(γ) is defined as follows: {∑ γ j ,...,j N log(γ j1 ,...,j N ) ifγ ≥ 0 E(γ) = { j1 ,...j N 1 +∞ otherwise , {

(6.7)

and C is the intersection of the set associated to the marginal constraint (we remark that the entropy is a penalization of the non-negative constraint on γ). After elementary computations, we can re-write the problem as min KL(γ|γ)̄ , γ∈C

(6.8)

where KL(γ|γ)̄ = ∑i1 ,...,i N γ i1 ,...,i N log((γ i1 ,...,i N )/(γ̄ i1 ,...,iN )) is the Kullback–Leibler distance and c j1 ,...,j N − ϵ ̄γ i1 ,...,i N = e . (6.9) As explained in Section 9.2, when the transport plan γ is concentrated on the graph of a transport map which solves the Monge problem, after discretization of the densities, this property is lost along but we still expect the matrix γ to be sparse. The entropic regularization spreads the support and this helps to stabilize the computation as it defines a strongly convex program with a unique solution γ ϵ . Moreover the solution γ ϵ can be obtained through elementary operations. The regularized solutions γ ϵ then converge to γ⋆ (see Figure 2), the solution of (6.1) with minimal entropy, as ϵ → 0 (see [26] for a detailed asymptotic analysis and the proof of exponential convergence).

245

9.6 Numerics |

ϵ = 0.2

ϵ = 0.1

ϵ = 0.05

ϵ = 0.025

ϵ = 0.0125

ϵ = 0.006

Fig. 2: Support of the coupling γ ϵ for the Coulomb cost and μ1 = μ2 = (1 + cos(πx/2))/2. The simulation has been performed on a discretization of [−2, 2] with M d = 1000.

In order to introduce the Iterative Proportional Fitting Procedure (IPFP), we consider the 2-marginals problem min ∑ c ij γ ij . (6.10) γ∈C

i,j

The aim of the IPFP is to find the KL projection of γ̄ on the set C = {γ ij ∈ ℝM d × ℝM d : ∑j γ ij = μ i } ∩ {γ ij ∈ ℝM d × ℝM d : ∑i γ ij = ν j }. By writing down the Lagrangian associated to (6.10) and computing the optimality condition, we find that γ ij can be written as γ ij = a i b j γ̄ ij with a i = e u i /ϵ , b j = e𝑣j /ϵ , (6.11) where u i and 𝑣j are the regularized Kantorovich potential. Then, a i and b j can be uniquely determined by the marginal constraint ai =

μi , ∑j b j γ̄ ij

bj =

νj . ∑i a i γ̄ ij

(6.12)

Thus, we can now define the following iterative method: b n+1 = j

νj , ∑i a ni γ̄ ij

a n+1 = i

μi γ̄ ij ∑j b n+1 j

.

(6.13)

246 | 9 Optimal transport for repulsive costs: old, new and numerics

Remark 6.3. In [77] Rüschendorf proves that the iterative method (6.13) converges to the KL-projection of γ̄ on C. Remark 6.4. Rüschendorf and Thomsen (see [78]) proved, in the continous measure framework, that a unique KL-projection exists and takes the form γ(x, y) = a(x) ⊗ ̄ y) (where a(x) and b(y) are non-negative functions). b(y)γ(x, The extension to the multi-marginal framework is straightforward but cumbersome to write. It leads to a problem set on N M d -dimensional vectors a j,i (⋅) , j = 1, . . . , N, i(⋅) = 1, . . . , M d . Each update takes the form a n+1 j,i j =

ρ ij n+1 ∑i1 ,i2 ,...i j−1 ,i j+1 ,...,i N γ̄ i1 ,...,i N a1,i 1

n+1 n n a2,i . . .a n+1 j−1,i j−1 a j+1,i j+1 . . .a N,i N 2

.

Example 6.5 (IPFP and 3-marginals). In order to clarify the extension of the IPFP to the multi-marginal case, we consider 3-marginals and we write down the updates of the algorithm ρ i1 n+1 a1,i = , n n 1 a3,i ∑i2 ,i3 γ̄ i1 ,i2 ,i3 a2,i 2 3 ρ i2 n+1 = , a2,i n+1 n 2 ̄ a3,i3 ∑i1 ,i3 γ i1 ,i2 ,i3 a1,i 1 ρ i3 n+1 = . a3,i n+1 n+1 3 ̄ ∑i1 ,i2 γ i1 ,i2 ,i3 a1,i a2,i 1

2

In the following sections, we present some numerical results obtained by using the IPFP.

9.6.2 Numerical experiments: Coulomb cost We present now some results for the multi-marginal problem with Coulomb cost in the real line. We consider the case where the N marginals are equal to a density μ (we work in a DFT framework so the marginals rapresent the electrons which are indistinguishable). We recall that if we split μ into N μ̃ i with equal mass (∫ μ̃ i (x) dx = N1 ∫ μ(x) dx), then we expect an optimal plan induced by a cyclical map such that T♯ μ̃ i = μ̃ i+1 i = 1, . . . , N − 1 and T♯ μ̃ N = μ̃ 1 . The simulations in Figure 3 are all performed on a discretization of [−5, 5] with N M d = 200, with marginals μ i = μ(x) = 10 (1 + cos( π5 x)) i = 1, . . . , N and ϵ = 0.02. If we focus on the support of the coupling γ̃ 12 (x, y) = (e1 , e2 )♯ γ(x, y, z) we can notice that the numerical solution correctly reproduces the prescribed behavior: the transport plan is induced by a cyclical optimal map (see Section 9.3). We refer the reader to [10] and [32] for examples in higher dimension. Remark 6.6. Theorem 3.7 actually works also for other costs functions as c(x1 , . . . , x N ) = ∑Ni 0, we take a sequence of finite oriented networks Ω ε = (N ε , E ε ). The set of nodes in Ω ε is N ε and E ε is the set of pairs (x, e) with x ∈ N ε and e ∈ ℝd such that |e| is of the order ε, the segment [x, x + e] is included in Ω and x + e still belongs to N ε . We will simply identify arcs to pairs (x, e). We assume |E ε | := max{|e|, there exists x such that (x, e) ∈ E ε } = ε.

Roméo Hatchi, CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France, [email protected]

258 | 10 Wardrop equilibria

10.1.1.1 Masses and congestion Let us denote the traffic flow on the arc (x, e) by m ε (x, e). There is a function g ε : E ε × ℝ+ → ℝ+ such that for each (x, e) ∈ E ε and m ≥ 0, g ε (x, e, m) represents the traveling time of arc (x, e) when the mass on (x, e) is m. The function g ε is positive and increasing in its last variable. This describes the congestion effect. We will denote the collection of all arc-masses m ε (x, e) by mε . 10.1.1.2 Marginals There is a distribution of sources f ε− = ∑x∈N ε f ε− (x)δ x and sinks f ε+ = ∑x∈N ε f ε+ (x)δ x which are discrete measures with same total mass on the set of nodes N ε (that we can assume to be 1 as a normalization) ∑ f ε− (x) = ∑ f ε+ (y) = 1 . x∈N ε

y∈N ε

The numbers f ε− (x) and f ε+ (x) are nonnegative for every x ∈ N ε . 10.1.1.3 Paths and equilibria A path is a finite set of successive arcs (x, e) ∈ E ε on the network. C ε is the finite set of loop-free paths on Ω ε and may be partitioned as Cε =

⋃ (x,y)∈N ε ×N ε

ε C εx,y = ⋃ C εx,⋅ = ⋃ C⋅,y , x∈N ε

y∈N ε

ε ) is the set of loop-free paths starting at the origin x (rewhere C εx,⋅ (respectively C⋅,y ε . spectively stopping at the terminal point y) and C εx,y is the intersection of C εx,⋅ and C⋅,y ε Then the travel time of a path γ ∈ C is given by ε τm ∑ g ε (x, e, m ε (x, e)) . ε (γ) := (x,e)⊂γ

The mass commuting on the path γ ∈ C ε will be denoted by w ε (γ). The collection of all path-masses w ε (γ) will be denoted by wε . We may define an equilibrium that satisfies optimality requirements compatible with the distribution of sources and sinks and such that all paths used minimize the traveling time between their extremities, taking into account the congestion effects. In other words, we have to impose mass conservation conditions that relate arc-masses, path-masses and the data f ε− and f ε+ : f ε− (x) = ∑ w ε (γ), f ε+ (y) = ∑ w ε (γ) , γ∈C εx,⋅

∀(x, y) ∈ N ε × N ε

(1.1)

ε γ∈C ⋅,y

and m ε (x, e) =

∑ γ∈C ε :

(x,e)⊂γ

w ε (γ) ,

∀(x, e) ∈ E ε .

(1.2)

10.1 Introduction |

259

ε We define Tm ε to be the minimal length functional, that is, ε Tm ε (x, y) := min ε

γ∈C x,y

∑ g ε (x, e, m ε (x, e) . (x,e)⊂γ

Let Π(f ε− , f ε+ ) be the set of discrete transport plans between f ε− and f ε+ , that is, the set of collection of nonnegative elements (φ ε (x, y))(x,y)∈N ε 2 such that ∑ φ ε (x, y) = f ε− (x)

and

y∈N ε

∑ φ ε (x, y) = f ε+ (y) ,

for every (x, y) ∈ N ε × N ε .

x∈N ε

This results in the concept of Wardrop equilibrium that is defined precisely as follows: Definition 1.1. A Wardrop equilibrium is a configuration of nonnegative arc-masses mε : (x, e) → (m ε (x, e)) and of nonnegative path-masses wε : γ → w ε (γ), that satisfy the mass conservation conditions (1.1) and (1.2) and such that: (1) For every (x, y) ∈ N ε × N ε and every γ ∈ C εx,y , if w ε (γ) > 0 then ε ε 󸀠 τm ε (γ) = min τ mε (γ ) , ε

(1.3)

γ 󸀠 ∈C x,y

(2) If we define Π ε (x, y) = ∑γ∈C εx,y w ε (γ) then Π ε is a minimizer of inf

ε φ ε (x, y)Tm ε (x, y) .

∑

φ ε ∈Π(f ε− ,f ε+ ) (x,y)∈N ε ×N ε

(1.4)

Condition (1.3) means that users behave rationally and always use shortest paths, taking in consideration congestion, that is, travel times increase with the flow. In [1, 16], the main discrete model studied is short term, that is, the transport plan is prescribed. Here we work with a long-term variant as in [6, 8]. It means that we have fixed only the marginals (that are f ε− and f ε+ ). So the transport plan now is an unknown and must be determined by some additional optimality condition that is (1.4). Condition (1.4) requires that there is an optimal transport plan between the fixed marginals for the transport cost induced by the congested metric. So we also have an optimal transportation problem.

10.1.2 Assumptions and preliminary results A few years after the work of Wardrop, Beckmann et al. [2] observed that Wardrop equilibria coincide with the minimizers of a convex optimization problem: Theorem 1.2. A flow configuration (wε , mε ) is a Wardrop equilibrium if and only if it minimizes m

∑ (x,e)∈E ε

G ε (x, e, m ε (x, e))

where

G ε (x, e, m) := ∫ g ε (x, e, α) dα

(1.5)

0

subject to nonnegativity constraints and the mass conservation conditions (1.1) and (1.2).

260 | 10 Wardrop equilibria

The problem (1.5) is interesting since it easily implies existence results and numerical schemes. However, it requires knowing the whole path flow configuration wε so that it may quickly be untractable for dense networks. However, a similar issue was recently studied in [16]. Under structural assumptions, it is shown that we may pass to a continuous limit which will simplify the structure. Here, we will not see all these hypotheses, only the main ones. So we refer to [16] for more details. The only noticeable difference is that we take here β = 1 in Remark 1 of [16] for physical reasons. Assumption 1.3. The discrete measures (f ε− )ε>0 and (f ε− )ε>0 weakly star converge to some probability measures f − and f + on Ω: lim ∑ (φ(x)f ε− (x) + ψ(x)f ε+ (x)) = ∫ φdf − + ∫ ψdf + ,

ε→0+

x∈N ε

Ω

∀(φ, ψ) ∈ C(Ω)2 .

Ω

Assumption 1.4. There exists N ∈ ℕ, {𝑣k }k=1,...,N ∈ C1 (Ω, 𝕊d−1 )N and {c k }k=1,...,N ∈ C1 (Ω, ℝ∗+ )N such that E ε weakly converges in the sense that lim

ε→0+

∑ (x,e)∈E ε

|e|d φ (x,

e )= |e|

∫

φ(x, 𝑣) θ(dx, d𝑣) ,

∀φ ∈ C(Ω × 𝕊d−1 ) ,

Ω×𝕊d−1

where θ ∈ M+ (Ω × 𝕊d−1 ) and θ is of the form N

θ(dx, d𝑣) = ∑ c k (x)δ𝑣k (x) dx . k=1

Moreover, there exists a constant C > 0 such that for every (x, z, ξ) ∈ Ω × 𝕊d−1 × ℝ+N , there exists Z̄ ∈ ℝ+N such that |Z|̄ ≤ C and N

Z̄ ⋅ ξ = min {Z ⋅ ξ; Z = (z1 , . . . , z N ) ∈ ℝ+N and ∑ z k 𝑣k (x) = z} .

(1.6)

k=1

The c k ’s are the volume coefficients and the 𝑣k ’s are the directions in the network. The measure θ depends on the discretization of Ω, i.e., the sequence {Ω ε }ε . The last subassumption (1.6) allows us to keep some control on an optimal conical decomposition of all z ∈ ℝd in the family of directions {𝑣k (x)} k for every x ∈ Ω. There always exists a conical decomposition of z in {𝑣k (x)}, not too large with respect to z. The next assumption focuses on the congestion functions g ε . Assumption 1.5. g ε is of the form g ε (x, e, m) = |e|g (x,

e m ) , , |e| |e|d−1

∀ε > 0, (x, e) ∈ E ε , m ≥ 0 ,

(1.7)

where g : Ω × 𝕊d−1 × ℝ+ 󳨃→ ℝ is a given continuous, nonnegative function that is increasing in its last variable.

10.1 Introduction

|

261

In ℝ2 , it is very natural: the traveling time on an arc of length |e| is of the order |e| and depends on the flow per unit of length m/|e|. In ℝd , it is a bit less natural. The traveling time is always of the order |e| but now depends on m/|e|d−1 . We have also removed the ε-dependence on the g ε . We then have G ε (x, e, m) = |e|d G (x,

e m ) , |e| |e|d−1

m

where

G(x, 𝑣, m) := ∫ g(x, 𝑣, α) dα . 0

We also add assumptions on G: Assumption 1.6. There exists a closed neighborhood U of Ω such that for k = 1, . . . , N, 𝑣k may be extended on U in a function C1 (still denoted 𝑣k ). Moreover, each function (x, m) ∈ U ×ℝ+ 󳨃→ G(x, 𝑣k (x), m) is Carathéodory, convex nondecreasing in its second argument with G(x, 𝑣k (x), 0) = 0 a.e. x ∈ U and there exists 1 < q < d/(d − 1) and two constants 0 < λ ≤ Λ such that for every (x, m) ∈ U × ℝ+ one has λ(m q − 1) ≤ G(x, 𝑣, m) ≤ Λ(m q + 1) .

(1.8)

Lq

The q-growth is natural since we want to work in in the continuous limit. The condition on q has a technical reason. It means that the conjugate exponent p of q is > d, which allows us to use Morrey’s inequality in the proof of the convergence ([16]). The extension on U will serve to use regularization by convolution and Moser’s flow argument. Examples of models that satisfy these assumptions are regular decompositions. In two-dimensional networks, there exists three different regular decompositions: cartesian, triangular, and hexagonal. In these models, the length of an arc in E ε is ε. The c k ’s and 𝑣k ’s are constant. In the cartesian case, N = 4, (𝑣1 , 𝑣2 , 𝑣3 , 𝑣4 ) := ((1, 0), (0, 1), (−1, 0), (0, −1)), and c k = 1 for k = 1, . . . , 4. For more details, see [16]. Now, before presenting the continuous limit problem, let us set some notations. Let us write the set of generalized curves L = {(γ, ρ) : γ ∈ W 1,∞ ([0, 1], Ω) , ρ ∈ Pγ ∩ L∞ ([0, 1])N } , where N

̇ = ∑ 𝑣k (γ(t)) ρ k (t) a.e.} . Pγ = {ρ : t ∈ [0, 1] → ρ(t) ∈ ℝ+N and γ(t) k=1

We can notice that Pγ is never empty thanks to Assumption 1.4. Let us denote Q ∈ Q(f − , f + ) the set of Borel probability measures Q on L such that the mass-conservation constraints are satisfied Q (f − , f + ) := {Q ∈ M1+ (L) : e0# Q = f − , e1# Q = f + } , where e t (γ, ρ) = γ(t), t ∈ [0, 1], (γ, ρ) ∈ L. For k = 1, . . . , N let us then define the nonnegative measures on Ω × 𝕊d−1 , m Qk by 1

∫ Ω×𝕊d−1

φ(x, 𝑣)dm Qk (x, 𝑣)

= ∫ (∫ φ(γ(t), 𝑣k (γ(t)))ρ k (t) dt) dQ(γ, ρ) , L

0

(1.9)

262 | 10 Wardrop equilibria for every φ ∈ C(Ω × 𝕊d−1 , ℝ). Then write simply m Q = ∑Nk=1 m Qk , nonnegative measure on Ω × 𝕊d−1 . Finally assume that Qq (f − , f + ) := {Q ∈ Q(f − , f + ) : m Q ∈ L q (θ)} ≠ 0 . It is true when for instance, f + and f − are in L q (Ω) and Ω is convex. Indeed, first for Q ∈ M1+ (W 1,∞ ([0, 1], Ω)), let us define i Q ∈ M+ (Ω) as follows: 1

∫ φ di Q = Ω

̇ (∫ φ(γ(t))|γ(t)| dt) dQ(γ) for φ ∈ C(Ω, ℝ) .

∫

0

W 1,∞ ([0,1],Ω)

It follows from the regularity results of [11, 22] that there exists Q ∈ M1+ (W 1,∞ ([0, 1], Ω)) such that e0# Q = f − , e1# Q = f + and i Q ∈ L q . For each curve γ, let ρ γ ∈ Pγ such γ ̇ that ∑k ρ k (t) ≤ C|γ(t)| (we have the existence due to Assumption 1.4). Then we set ⋅ ̃ Q = (id, ρ )# Q. We have Q̃ ∈ Qq (f − , f + ) so that we have proved the existence of such kind of measures. A necessary and sufficient condition to ensure Qq (f − , f + ) ≠ 0 is that f + − f − ∈ W −1,q (Ω) (see [7]). Then Wardrop equilibria at scale ε converge as ε → 0+ to solutions of the following problem: ∫

inf

Q∈Qq (f − ,f + )

G(x, 𝑣, m Q (x, 𝑣))θ(dx, d𝑣)

(1.10)

Ω×𝕊d−1

(see [16]). Nevertheless this problem (1.10) is posed over probability measures on generalized curves and it is not obvious at all that it is simpler to solve than the discrete problem (1.5). So in this chapter, we want to show that problem (1.10) is equivalent to another problem that will roughly amount to solve an elliptic PDE. This problem is } { } { infσ { ∫ G (x, 𝑣, ϱ(x, 𝑣)) θ(dx, d𝑣); −div σ = f } , } σ∈L q (Ω,ℝd ) ϱ∈P { } {Ω×𝕊d−1 inf

(1.11)

where N

Pσ = {ϱ : Ω × 𝕊d−1 → ℝ+ ; ∀x ∈ Ω, σ(x) = ∑ 𝑣k (x)ϱ(x, 𝑣k (x))} , k=1

f = f + − f − and the equation − div(σ) = f is defined by duality ∫ ∇u ⋅ σ = ∫ u df, for all u ∈ C1 (Ω) , Ω

Ω

so the homogeneous Neumann boundary condition σ ⋅ ν Ω = 0 is satisfied on ∂Ω in the weak sense. For the sake of clarity, let us define N

G(x, σ) := infσ ∑ c k (x)G(x, 𝑣k (x), ϱ k ) := infσ G(x, ϱ) ϱ∈Px

k=1

ϱ∈Px

10.1 Introduction

| 263

where N

N

Pσx = {ϱ ∈ ℝ+N ; σ = ∑ 𝑣k (x)ϱ k }

and

G(x, ϱ) := ∑ c k (x)G(x, 𝑣k (x), ϱ k ) ,

k=1

k=1

for x ∈ Ω, σ ∈ ℝd . We recall that the c k ’s are the volume coefficients in θ. G is convex in the second variable (since G is convex in its last variable). The minimization problem (1.11) can then be rewritten as inf

σ∈L q (Ω,ℝd )

{ } {∫ G(x, σ(x)) dx; −div σ = f } . {Ω }

(1.12)

This problem (1.12) looks like the ones introduced by Beckmann [3] for the design of an efficient commodity transport program. The dual problem of (1.12) takes the form sup u∈W 1,p (Ω)

{ } ∗ {∫ u df − ∫ G (x, ∇u(x)) dx} , Ω {Ω }

(1.13)

where p is the conjugate exponent of q and G∗ is the Legendre transform of G(x, ⋅). In order to solve (1.12), we can first solve the Euler–Lagrange equation of its dual formulation and then use the primal–dual optimality conditions. Nevertheless, in our typical congestion models, the functions G(x, 𝑣, ⋅) have a positive derivative at zero (i.e., g(x, 𝑣, 0)). Indeed, going at infinite speed – or teleportation – is not possible even when there is no congestion. So we have a singularity in the integrand in (1.12). Then G∗ and the Euler–Lagrange equation of (1.13) are extremely degenerate. Moreover, the prototypical equation of [8] is the following: p−1

− div ((|∇u| − 1)+

∇u )=f . |∇u|

Here, for well chosen g, we obtain anisotropic equation of the form d

N

p−1

− ∑ ∂ l [ ∑ (∇u ⋅ 𝑣k (x) − δ k c k (x))+ 𝑣kl (x)] = f . l=1

k=1

where 𝑣k (x) = (𝑣1k (x), . . . , 𝑣kd (x)) for k = 1, . . . , N and x ∈ Ω. In the cartesian case (in ℝ2 ), we can separate the variables in the sum (since here G(x, σ 1 , σ 2 ) = G1 (x, σ 1 ) + G2 (x, σ 2 )). But in the hexagonal one (d = 2), it is impossible. The previous equation degenerates in an unbounded set of values of the gradient and its study is delicate, even if all the δ k ’s are zero. It is more complicated than the one in [6]. Indeed, the studied model in [6] is the cartesian one and the prototypical equation is 2

p−1

− ∑ ∂ k ((|∂ k u| − δ k )+ k=1

∂k u )=f . |∂ k u|

264 | 10 Wardrop equilibria

The plan of the paper is as follows. In Section 2, we formulate some relationship between (1.10) and (1.12). Section 3 is devoted to optimality conditions for (1.12) in terms of solutions of (1.13). We also present the kind of PDEs that represent realistic anisotropic models of congestion. In Section 4, we give some regularity results in the particular case where the c k ’s and the 𝑣k ’s are constant. Finally, in Section 5, we describe numerical schemes that allow us to approximate the solutions of the PDEs.

10.2 Equivalence with Beckmann problem Let us study the relationship between problems (1.10) and (1.11). We still assume that all specified hypotheses in Section 10.1 are satisfied. Let us notice that, thanks to Assumption 1.4, for every σ ∈ L q (Ω, ℝd ), there exists ϱ̂ ∈ Pσ such that ϱ̂ ∈ L q (θ) and ϱ̂ minimizes the following problem: } { } { infσ { ∫ G(x, 𝑣, ϱ(x, 𝑣)) θ(dx, d𝑣)} . } ϱ∈P { } {Ω×𝕊d−1 For ϱ ∈ Pσ , define ϱ̄ : Ω → ℝ+N where ϱ̄ k (x) = ϱ(x, 𝑣k (x)), for every x ∈ Ω, k = 1, . . . , N. Now, we only consider ϱ̄ that we simply write ϱ (by abuse of notations). Theorem 2.1. Under all previous assumptions, we have inf (1.10) = inf (1.12) . Proof. We adapt the proof in [6]. We will show the two inequalities. Step 1. inf (1.10) ≥ inf (1.12). Let Q ∈ Qq (f − , f + ). We build σ Q ∈ L q (Ω, ℝd ) that will allow us to obtain the desired inequality, we define it as follows: 1

̇ dt dQ(γ, ρ) , ∫ φ dσ = ∫ ∫ φ(γ(t)) ⋅ γ(t) Q

∀φ ∈ C(Ω, ℝd ) .

L 0

Ω

In particular, we have −div σ Q = f since Q ∈ Q(f − , f + ). We now justify that N

σ Q (x) = ∫ 𝑣 m Q (x, 𝑣) d𝑣 = ∑ 𝑣k (x)m Q (x, 𝑣k (x)) a.e. x ∈ Ω . k=1

𝕊d−1

Recall that for every ξ ∈ C(Ω × 𝕊d−1 , ℝ), 1

∫ Ω×𝕊d−1

N

ξdm Q = ∫ ∫ ( ∑ ξ(γ(t), 𝑣k (γ(t)))ρ k (t)) dt dQ(γ, ρ) . L 0

k=1

(2.1)

10.2 Equivalence with Beckmann problem |

265

By taking ξ of the form ξ(x, 𝑣) = φ(x) ⋅ 𝑣 with φ ∈ C(Ω, ℝd ), we get 1

∫

N

Q

φ(x) ⋅ 𝑣 dm (x, 𝑣) = ∫ ∫ ( ∑ ρ k (t)φ(γ(t)) ⋅ 𝑣k (γ(t))) dt dQ(γ, ρ) L 0

Ω×𝕊d−1

k=1

= ∫ φ dσ Q . Ω Q

Moreover, since ≥ 0, we obtain that m Q ∈ Pσ (and so that σ Q ∈ L q ) and the desired inequality follows. mQ

Step 2. inf (1.10) ≤ inf (1.12). Now prove the other inequality. We will use Moser’s flow method (see [8, 10, 20]) and a classical regularization argument. Fix δ > 0. Let σ ∈ L q (Ω, ℝd ) and ϱ ∈ Pσ ∩ L q (Ω, ℝN ) such that ∫ G(x, 𝑣, ϱ(x, 𝑣)) θ(dx, d𝑣) ≤ inf (1.12) + δ , Ω×𝕊d−1 d with −div σ = f . We extend them outside Ω by 0. Let then η ∈ C∞ c (ℝ ) be a positive function, supported in the unit ball B1 and such that ∫ℝd η = 1. For ε ≪ 1 so that Ω ε := Ω + εB1 ⋐ U, we define η ε (x) := ε−d η(ε−1 x), σ ε := η ε ⋆ σ, and ϱ εk (x) := η ε ⋆ ϱ k (x) for k = 1, . . . , N. By construction, we thus have that σ ε ∈ C∞ (Ω ε ) and

− div (σ ε ) = f ε+ − f ε−

in

Ωε

and

σ ε = 0 on

∂Ω ε , ε

where f±ε = η ε ⋆ (f± 1Ω ) + ε. But the problem is that we do not have ϱ ε ∈ Pσ . We shall ε build a sequence (P ε ) in Pσ that converges to ρ in L q (U, ℝN ). Notice that N

σ ε (x) = ∑ ∫ η ε (y)ϱ k (x − y)𝑣k (x − y) dy k=1 N

N

= ∑ ϱ εk (x)𝑣k (x) + ∑ ∫ η ε (y)ϱ k (x − y)(𝑣k (x − y) − 𝑣k (x)) dy . k=1

k=1

There exists p εk ∈ L q (Ω ε ) such that for every k = 1, . . . , N, p εk ≥ 0, p εk → 0 and for x ∈ Ω ε , we have N

N

I ε (x) = ∑ ∫ η ε (y)ϱ k (x − y)(𝑣k (x − y) − 𝑣k (x)) dy = ∑ p εk (x)𝑣k (x) . k=1

k=1

Such a family exists since ∈ and → 0 (by using the fact that the 𝑣k ’s are in C1 (U)) and we can estimate p εk with I ε due to Assumption 1.4. Then if we set P ε = ε ϱ ε + p ε , we have P ε ∈ Pσ and P ε → ϱ in L q . Define g ε (t, x) := (1 − t)f ε− (x) + tf ε+ (x) ∀t ∈ [0, 1], x ∈ Ω ε , let then X ε be the flow of the vector field 𝑣ε := σ ε /g ε , that is, Iε

Lq

Iε

{ Ẋ tε (x) = 𝑣ε (t, X tε (x)) { ε X (x) = x, (t, x) ∈ [0, 1] × Ω ε . { 0

266 | 10 Wardrop equilibria We have ∂ t g ε + div (g ε 𝑣ε ) = 0. Since 𝑣ε is smooth and the initial data is g ε (0, ⋅) = f ε− , we have X tε # f ε− = g ε (t, ⋅). Let us define the set of generalized curves L ε = {(γ, ρ) : γ ∈ W 1,∞ ([0, 1], Ω ε ), ρ ∈ Pγ ∩ L∞ ([0, 1])N } . Let us consider the following measure Q ε on L ε Q ε = ∫ δ(X⋅ε (x),P ε (X⋅ε (x))/g ε (⋅,X⋅ε (x))) df ε− (x) . Ωε ε

We then have e t# Q ε = X tε # f ε− = g ε (t, ⋅) for t ∈ [0, 1]. We define σ Q and m Qk as in (2.1) ε and (1.9), respectively, by using test functions defined on Ω ε . We then have σ Q = σ ε . Indeed, for φ ∈ C(Ω ε , ℝd ), we have ε

1

∫ φ dσ Ωε

Qε

= ∫ ∫ φ(X tε (x)) ⋅ 𝑣ε (t, X tε (x))f ε− (x) dt dx Ωε 0 1

= ∫ ∫ φ(x) ⋅ 𝑣ε (t, x)g ε (t, x) dx dt 0 Ωε

= ∫ φ dσ ε Ωε

which gives the equality. We used the definition of Q ε , the fact that X tε # f ε− = g ε (t, ⋅) and ε ε that 𝑣ε g ε = σ ε and Fubini’s theorem. In the same way, we have m Q ∈ Pσ . To prove it, we take the same arguments as in the end of Step 1 and in the previous calculation. For φ ∈ C(Ω ε , ℝd ), we have 1

∫ Ω ε ×𝕊d−1

N

Qε

φ(x) ⋅ 𝑣 m (dx, d𝑣) = ∫ (∫ ∑ φ(X tε (x)) ⋅ 𝑣k (X tε (x)) 0

Ω ε k=1

1

= ∫ (∫ φ(X tε (x)) ⋅ 0

Ωε

P εk (X tε (x)) − f (x) dx) dt g ε (t, X tε (x)) ε

σ ε (X tε (x)) − f (x) dx) dt g ε (t, X tε (x)) ε

1

= ∫ (∫ φ(x) ⋅ σ ε (x) dx) dt 0

Ωε

= ∫ φ dσ ε . Ωε ε

Moreover, more precisely, we have m Qk (dx, d𝑣) = δ𝑣k (x) P εk (x) dx. Then we conclude as in [6]. First for any Lipschitz curve φ, let us denote by φ̃ its constant speed reparame-

10.3 Characterization of minimizers via anisotropic elliptic PDEs |

267

̃ = φ(s−1 (t)), where terization, that is, for t ∈ [0, 1], φ(t) t

1

0

0

1 ̇ ̇ s(t) = ∫ |φ(u)| du with l(φ) = ∫ |φ(u)| du . l(φ) For (φ, ρ) ∈ L, let ρ̃ be the reparameterization of ρ, i.e., ρ̃ k (t) :=

l(σ) ρ k (s−1 (t)) , ̇ |σ(s−1 (t))|

∀t ∈ [0, 1], k = 1, . . . , N .

̃ We have Let us denote by Q̃ the push forward of Q through the map (φ, ρ) 󳨃→ (φ,̃ ρ). ̃ ε ̃ m Qk = m Qk and σ Q = σ Q . Then arguing as in [16], the L q bound on m Q yields the tightness of the family of Borel measures Q̃ ε on C([0, 1], ℝd ) × L∞ ([0, 1])N . So Q ε ⋆weakly converges to some measure Q (up to a subsequence). Let us remark that Q̃ ε has its total mass equal to that of f ε+ , that is, 1 + ε|Ω ε |. Thus one can show that Q(L) = 1) (due to the fact that Q(L) = limε→0+ Q(L ε ) = 1). Moreover, we have Q ∈ Q(f − , f + ) ε thanks to the ⋆-weak convergence of Q̃ ε to Q. Recalling the fact that P εk = m Q (⋅, 𝑣k (⋅)) strongly converges in L q to ϱ k (ϱ ∈ Pσ ) and due to the same semicontinuity argument as in [9, 16], we have m Q (⋅, 𝑣k (⋅)) ≤ ϱ k in the sense of measures. Then m Q (⋅, 𝑣k (⋅)) ∈ L q so that Q ∈ Qq (f − , f + ). It follows from the monotonicity of G(x, 𝑣, ⋅) that ∫

G(x, 𝑣, m Q (x, 𝑣)) θ(dx, d𝑣) ≤

Ω×𝕊d−1

∫

G(x, 𝑣, ϱ(x, 𝑣)) θ(dx, d𝑣)

Ω×𝕊d−1

≤ inf (1.12) + δ . Letting δ → 0+ , we have the desired result. In fact, we showed in the previous proof a stronger result. We proved the following equivalence: Q solves (1.10) ⇐⇒ σ Q solves (1.11) and moreover, (m Q (⋅, 𝑣k (⋅))) k=1,...,N ∈ Pσ

Q

is optimal for (1.10). We also built a minimizing sequence for (1.10) from a regularization of a solution σ of (1.11) by using Moser’s flow argument.

10.3 Characterization of minimizers via anisotropic elliptic PDEs Here, we study the primal problem (1.12) and its dual problem (1.13). Recalling that f = f + − f − has zero mean, we can reduce the problem (1.13) only to zero-mean W 1,p (Ω) functions. Besides for (x, 𝑣) ∈ Ω × 𝕊d−1 and k = 1, . . . , N, the functions g k (x, 𝑣, ⋅) are continuous positive and increasing on ℝ+ since it is the time per unit of length to leave from the point x in the direction 𝑣 when the intensity of traffic in this direction is m.

268 | 10 Wardrop equilibria Since G(x, 𝑣, ⋅) has a positive derivative (i.e., g k (x, 𝑣, ⋅)), G is strictly convex in its last variable then so is G(x, ⋅) for x ∈ Ω. Thus G∗ is C1 . However, G is not differentiable so that G∗ (x, ⋅) is degenerate. By standard convex duality (Fenchel–Rockafellar’s theorem, see [13] for instance), we have min (1.12) = max (1.13) and we can characterize the optimal solution σ of (1.12) (unique by strict convexity) as follows: σ(x) = ∇G∗ (x, ∇u(x)) , where u is a solution of (1.13). In other terms, u is a weak solution of the Euler– Lagrange equation − div (∇G∗ (x, ∇u(x))) = f in Ω , { ∇G∗ (x, ∇u(x)) ⋅ ν Ω = 0 on ∂Ω , in the sense that ∫ ∇G∗ (x, ∇u(x)) ⋅ ∇φ(x) dx = ∫ φ(x) df(x) , Ω

∀φ ∈ W 1,p (Ω) .

Ω

Let us remark that if u is not unique, σ is. A typical example is g(x, 𝑣k (x), m) = g k (x, m) = a k (x)m q−1 + δ k with δ k > 0 and the weights a k are regular and positive. We can explicitly compute G∗ (x, z). Let us notice that for every x ∈ Ω, z ∈ ℝd , we have G∗ (x, z) = sup (z ⋅ σ − G(x, σ)) = sup (z ⋅ σ − infσ G(x, ϱ)) σ∈ℝd

ϱ∈Px

σ∈ℝd

N

= sup (z ⋅ σ − G(x, ϱ)) = sup { ∑ (z ⋅ 𝑣k (x))ϱ k − G(x, ϱ)} . σ,ϱ

ϱ∈ℝ+N

k=1

A direct calculus then gives N

G∗ (x, z) = ∑ k=1 1 − q−1

where b k = (a k c k )

N

b k (x) p (z ⋅ 𝑣k (x) − δ k c k (x))+ , p

. The PDE then becomes d

p−1

− ∑ ∑ ∂ l [b k (x)𝑣kl (x)(∇u ⋅ 𝑣k (x) − δ k c k (x))+ ] = f ,

(3.1)

k=1 l=1

where 𝑣k (x) = (𝑣1k (x), . . . , 𝑣kd (x)). p For k = 1, . . . , N, G∗k (x, z) = b kp(x) (z⋅𝑣k (x)−δ k )+ vanishes if z⋅𝑣k (x) ∈]−∞, δ k c k (x)] so that any u whose gradient satisfies ∇u(x) ⋅ 𝑣k (x) ∈] − ∞, δ k c k (x)], ∀x ∈ Ω, k = 1, . . . , N is a solution of the previous PDE with f = 0. In consequence, we cannot hope to obtain estimates on the second derivatives of u or even oscillation estimates on ∇u from (3.1). Nevertheless, we will see that we have some regularity results on the

10.4 Regularity when the 𝑣k ’s and c k ’s are constant | 269

vector field σ = (σ 1 , . . . , σ d ) that solves (1.12) in the case where the directions and the volume coefficients are constant, that is, N

p−1

σ(x) = ∑ [b k (x)(∇u(x) ⋅ 𝑣k − δ k c k )+ ] 𝑣k , k=1

for every x ∈ Ω.

10.4 Regularity when the 𝑣k ’s and c k ’s are constant Our aim here is to get some regularity results in the case where the 𝑣k ’s and the c k ’s are constant. We will strongly base on [6] to prove this regularity result. Let us consider the model equation N

p−1

− ∑ div ((∇u(x) ⋅ 𝑣k − δ k c k )+ 𝑣k ) = f ,

(4.1)

k=1

where 𝑣k ∈ 𝕊d−1 , c k > 0 and b k ≡ 1 for k = 1, . . . , N. Define for z ∈ ℝd N

p−1

F(z) = ∑ F k (z), with F k (z) = (z ⋅ 𝑣k − δ k c k )+ 𝑣k

(4.2)

k=1

and N

p

H(z) = ∑ H k (z), with H k (z) = (z ⋅ 𝑣k − δ k c k )+2 𝑣k .

(4.3)

k=1

Here we assume only p ≥ 2. We have the following lemma that establishes some connections between F and H. Lemma 4.1. Let F and H be defined as above with p ≥ 2, then for every (z, w) ∈ ℝd ×ℝd , the following inequalities are true for k = 1, . . . , N: |F k (z)| ≤ |z|p−1 , |F k (z) − F k (w)| ≤ (p − 1) (|H k (z)|

(4.4) p−2 p

and (F k (z) − F k (w)) ⋅ (z − w) ≥

+ |H k (z)|

p−2 p

) |H k (z) − H k (w)| ,

4 |H k (z) − H k (w)|2 . p2

(4.5)

(4.6)

Proof. The first one is trivial. For the second one, from [18] one has the general result: for all (a, b) ∈ ℝd × ℝd , the following inequality holds: 󵄨 p−2 p−2 p−2 p−2 󵄨 󵄨󵄨 p−2 󵄨 󵄨󵄨|a| a − |b|p−2 b 󵄨󵄨󵄨 ≤ (p − 1) (|a| 2 + |b| 2 ) 󵄨󵄨󵄨󵄨|a| 2 a − |b| 2 b 󵄨󵄨󵄨󵄨 . 󵄨 󵄨 󵄨 󵄨

(4.7)

Choosing a = (z ⋅ 𝑣k − δ k c k )+ 𝑣k and b = (w ⋅ 𝑣k − δ k c k )+ 𝑣k in (4.7), we then obtain (4.5).

270 | 10 Wardrop equilibria Let us now prove the third inequality. It is trivial if both z ⋅ 𝑣k and w ⋅ 𝑣k are less than δ k c k . If z ⋅ 𝑣k > δ k c k and w ⋅ 𝑣k ≤ δ k c k , we have p−1

p

(F k (z) − F k (w)) ⋅ (z − w) = (z ⋅ 𝑣k − δ k c k )+ (z ⋅ 𝑣k − w ⋅ 𝑣k ) ≥ (z ⋅ 𝑣k − δ k c k )+ = |H k (z)|2 . For the case z ⋅ 𝑣k > δ k c k and w ⋅ 𝑣k > δ k c k , we use the following inequality (again [18]) (|a|p−2 a − |b|p−2 b) ⋅ (a − b) ≥

2 p−2 p−2 4 (|a| 2 a − |b| 2 b) . 2 p

Again taking a = (z ⋅ 𝑣k − δ k c k )+ 𝑣k and b = (w ⋅ 𝑣k − δ k c k )+ 𝑣k , we have 4 |H k (z) − H k (w)|2 ≤ (|F k (z)| − |F k (w)|)𝑣k ⋅ ((z ⋅ 𝑣k − δ k c k )+ − (w ⋅ 𝑣k − δ k c k )+ )𝑣k p2 = (|F k (z)| − |F k (w)|)(z − w) ⋅ 𝑣k , which gives (4.6). 1,q

Let us fix f ∈ Wloc (Ω), where q is the conjugate exponent of p and let us consider the equation − div F(∇u) = f . (4.8) Thanks to Nirenberg’s method of incremental ratios, we then have the following result that is strongly inspired of Theorem 4.1 in [6]: 1,p

Theorem 4.2. Let u ∈ Wloc (Ω) be a local weak solution of (4.8). Then H := H(∇u) ∈ 1,2 1,2 Wloc (Ω). More precisely, for every k = 1, . . . , N, Hk := H k (∇u) ∈ Wloc (Ω). Proof. For the sake of clarity, write F := F(∇u) and similarly, Fk , Hk (note that Fk ∈ q Lloc (Ω) and Hk ∈ L2loc (Ω) due to (4.4) and (4.5). Let us define the translate of the function φ by the vector h by τ h φ := φ(⋅ + h). Let φ ∈ W 1,q (Ω) be compactly supported in Ω and h ∈ ℝd \{0} be such that |h| < dist(supp(φ), ℝd \{0}), we then have ∫ Ω

τhF − F τh f − f ⋅ ∇φ dx = ∫ ⋅ φ dx . |h| |h|

(4.9)

Ω

C∞ c (Ω)

Let ω ⋐ ω0 ⋐ Ω and ξ ∈ such that supp(ξ ) ⊂ ω0 , 0 ≤ ξ ≤ 1 and ξ = 1 on ω d and h ∈ ℝ \{0} such that |h| ≤ r0 < 12 dist(ω0 , ℝd \Ω). In what follows, we denote by C a nonnegative constant that does not depend on h but may change from one line to another. We then introduce the test function φ = ξ 2 |h|−1 (τ h u − u) , 1,p

1,q

in (4.9). Let us fix ω󸀠 := ω0 + B(0, r0 ). It follows from u ∈ Wloc (Ω), f ∈ Wloc (Ω) and the Hölder inequality that |h|−2 ∫(τ h F − F) ⋅ (ξ 2 (τ h ∇u − ∇u) + 2ξ∇ξ(τ h u − u)) ≤ ‖∇f‖L q (ω󸀠 ) ‖∇u‖L p (ω󸀠 ) . Ω

10.4 Regularity when the 𝑣k ’s and c k ’s are constant | 271

The left-hand side of the previous inequality is the sum of 2N terms I11 + I12 + ⋅ ⋅ ⋅ + I N1 + I N2 where for every k = 1, . . . , N, I k1 := |h|−2 ∫ ξ 2 (F k (τ h ∇u) − F k (∇u) ⋅ (τ h ∇u − ∇u) , Ω

and I k2 := |h|−2 ∫ ξ 2 (F k (τ h ∇u) − F k (∇u) ⋅ ∇ξξ(τ h u − u) . Ω

Let k = 1, . . . , N fixed. We will find estimations on I k1 and I k2 . Due to (4.5), I k1 satisfies I k1 ≥

4 ‖ξ|h|−1 (τ h Hk − Hk )‖2L2 . p2

For I k2 , if p > 2, it follows from (4.6) and the Hölder inequality with exponents 2, p and 2p/(p − 2) that |I k2 | ≤ |h|−2 ∫ |ξ∇ξ||τ h u − u||τ h Hk − Hk | (|τ h Hk |

p−2 p

+ |Hk |

p−2 p

)

Ω p−2 2p

−1

−1

≤ C‖|h| (τ h u − u)‖L p (ω0 ) ‖ξ|h| (τ h Hk − Hk )‖L2 ( ∫ |Hk | + |τ h Hk | ) 2

2

ω0 −1

≤ C‖ξ|h| (τ h Hk − Hk )‖L2 , and if p = 2, we simply use Cauchy–Schwarz inequality and obtain |I k2 | ≤ C‖ξ|h|−1 (τ h Hk − Hk )‖L2 . Bringing together all estimates, we then obtain N 󵄩 N 󵄩 󵄩󵄩 󵄩󵄩 τ h Hk − Hk 󵄩󵄩󵄩2 󵄩 󵄩󵄩 ≤ C (1 + ∑ 󵄩󵄩󵄩ξ τ h Hk − Hk 󵄩󵄩󵄩 ) . ∑ 󵄩󵄩󵄩ξ 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 2 󵄩 h h 󵄩L 󵄩L k=1 󵄩 k=1 󵄩

and we finally get

N 󵄩 󵄩󵄩 τ h Hk − Hk 󵄩󵄩󵄩2 󵄩󵄩 ∑ 󵄩󵄩󵄩 󵄩󵄩 2 ≤ C , 󵄩 h 󵄩 L (ω) k=1 󵄩

for some constant C that depends on p, ‖f‖W 1,q , ‖u‖W 1,p and the distance between ω 1,2 and ∂Ω, but not on h. We have the desired result, that is, Hk ∈ Wloc (Ω), for k = 1, . . . , N, and so also H. If we consider the variational problem of Beckmann type N { } 1 q ∫ infσ ∑ c k ( ϱ k + δ k ϱ k ) : − div σ = f } , inf { q σ∈L (Ω) q ϱ∈Px k=1 {Ω }

(4.10)

we then have the following Sobolev regularity result for the unique minimizer that generalizes Corollary 4.3 in [6].

272 | 10 Wardrop equilibria 1,r Corollary 4.3. The solution σ of (4.10) is in the Sobolev space Wloc (Ω), where

{2 { { r = {any value < 2, { { dp { dp−(d+p)+2 ,

if p = 2 , if p > 2 and d = 2 , if p > 2 and d > 2 .

Proof. By duality, we know the relation between σ and any solution of the dual problem u N

p−1

σ = ∑ (∇u ⋅ 𝑣k − δ k c k )+ 𝑣k . k=1

W 1,q (Ω) is a weak solution of the Euler–Lagrange equation (4.1), using The-

Since u ∈ orem 4.2 and Lemma 4.1, we have the vector fields p

Hk (x) = (∇u(x) ⋅ 𝑣k − δ k c k )+2 𝑣k , k = 1, . . . , N , 1,2 (Ω). We then notice that σ = ∑Nk=1 σ k with are in Wloc

σ k = |Hk |

p−2 p

Hk , k = 1, . . . , N .

1,2 (Ω). For the other cases, we The first case is trivial; we simply have σ k = Hk ∈ Wloc use the Sobolev theorem. If p > 2 and d > 2, then Hk ∈ L2∗ loc (Ω) with

1 1 1 = − . 2∗ 2 N Applying (4.5) with z = τ h ∇u and w = ∇u, we have 󵄨󵄨 τ H − H 󵄨󵄨 󵄨󵄨 τ σ − σ 󵄨󵄨 p−2 p−2 k 󵄨󵄨 k 󵄨󵄨 󵄨 h k 󵄨󵄨 h k 󵄨󵄨 󵄨󵄨 ≤ (p − 1) (|τ h Hk | p + |Hk | p ) 󵄨󵄨󵄨 󵄨󵄨 . 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 |h| |h| 󵄨 Since |Hk |

p−2 p

2∗ p p−2 r ∈ Lloc (Ω), we have the right-hand side term is in Lloc (Ω) with r given by

1 p−2 1 = ∗ + . r 2 p 2 We can then control this integral 󵄨󵄨 τ σ − σ 󵄨󵄨r k 󵄨󵄨 󵄨 h k ∫ 󵄨󵄨󵄨 󵄨󵄨 dx . 󵄨󵄨 󵄨󵄨 |h| s (Ω) for For the case p > 2 and d = 2, it follows from the same theorem that Hk ∈ Lloc every s < +∞ and the same reasoning allows us to conclude.

This Sobolev regularity result can be extended to equations with weights such as N

p−1

− ∑ div (b k (x)(∇u(x) ⋅ 𝑣k − δ k c k )+ 𝑣k ) = f .

(4.11)

k=1

An open problem is to investigate if one can generalize this Sobolev regularity result to the case where the 𝑣k ’s and c k ’s are in C1 (Ω).

10.5 Numerical simulations

|

273

10.5 Numerical simulations 10.5.1 Description of the algorithm We numerically approximate by finite elements solutions of the following minimization problem: inf J(u) := G∗ (∇u) − ⟨f, u⟩ (5.1) u∈W 1,p (Ω)

with G∗ (Φ)

G∗ (x, Φ(x)) dx

= ∫Ω for Φ ∈ L p (Ω)d and ⟨f, w⟩ = ∫Ω u df for w ∈ L p (Ω). Let us recall that Ω is a bounded domain of ℝd with Lipschitz boundary and f = f + − f − is in the dual of W 1,p (Ω) with zero mean ∫Ω f = 0. We will use the augmented Lagrangian method described in [5] (that we will recall later). ALG2 is a particular case of the Douglas–Rachford splitting method for the sum of two nonlinear operators (see [19] or more recently [21]). ALG2 was used for transport problems for the first time in [4]. Let a regular triangulation of Ω with typical mesh size h, let E h ⊂ W 1,p (Ω) be the corresponding finite-dimensional space of P2 finite elements of the order 2, whose generic elements are denoted by u h . Moreover, if necessary, we approximate the terms f by f h ∈ E h (again with ⟨f h , 1⟩ = 0) and G by a convex function Gh ∈ E h . Let us consider the approximating problem inf J h (u h ) := G∗h (∇u h ) − ⟨f h , u h ⟩ .

(5.2)

sup {−Gh (σ h ) : − divh (σ h ) = f h } ,

(5.3)

u h ∈E h

and its dual σ h ∈F hd

where F h is the space of P1 finite elements of the order 1 and − divh (σ h ) may be given as ⟨σ h , ∇u h ⟩F d = −⟨divh (σ h ), u h ⟩E h . h

Theorem 5.1. If u h solves (5.2) then up to a subsequence, u h converges as h → 0 to a u weakly in W 1,p (Ω) such that u solves (5.1). It is a direct application of a general theorem (see [5] and [15] for similar results and more details). Using the discretization by finite elements, (5.1) becomes inf J(u) := F(u) + G∗ (Λu) ,

u∈ℝn

(5.4)

where F : ℝn → ℝ ∪ {+∞}, G : ℝm → ℝ ∪ {+∞} are two convex l.s.c. and proper functions and Λ is an m × n matrix with real entries. Λ is the discrete analogue of ∇. The dual of (5.4) then reads as sup −F∗ (−Λ T σ) − G(σ) .

(5.5)

σ∈ℝm

We say that a pair (u,̄ σ)̄ ∈ ℝn × ℝm satisfies the primal–dual extremality relations if: ̄ σ̄ ∈ ∂G∗ (Λ u)̄ . − Λ T σ̄ ∈ ∂F(u),

(5.6)

274 | 10 Wardrop equilibria It means that ū solves (5.4) and that σ̄ solves (5.5) and moreover, (5.4) and (5.5) have the same value (no duality gap). It is equivalent to find a saddle point of the augmented Lagrangian function for r > 0 (see, e.g., [14, 15]) r L r (u, q, σ) := F(u)+G∗ (q)+σ⋅(Λu−q)+ |Λu−q|2 , 2

∀(u, q, σ) ∈ ℝn ×ℝm ×ℝm . (5.7)

It is the discrete formulation of the corresponding augmented Lagrangian function L r (u, q, σ) := ∫ G∗ (x, q(x)) dx − ⟨u, f⟩ + ⟨σ, ∇u − q⟩ +

r |∇u − q|2 2

(5.8)

Ω

and the variational problem of (5.4) is { } inf {∫ G∗ (x, q(x)) dx − ∫ u(x)f(x) dx} , u,q Ω {Ω }

(5.9)

subject to the constraint that ∇u = q. The augmented Lagrangian algorithm ALG2 involves building a sequence (u k , q k , σ k ) ∈ ℝn × ℝm × ℝm from initial data (u 0 , q0 , σ 0 ) as follows: (1) Minimization problem with respect to u: u k+1 := argminu∈ℝn {F(u) + σ k ⋅ Λu +

r |Λu − q k |2 )} . 2

That is equivalent to solve the variational formulation of Laplace equation −r(∆u k+1 − div(q k )) = f + div(σ k ) in Ω with the Neumann boundary condition r

∂u k+1 = rq k ⋅ ν − σ k ⋅ ν on ∂Ω . ∂ν

This is where we use the Galerkin discretization by finite elements. (2) Minimization problem with respect to q: q k+1 := argminq∈ℝm {G∗ (q) − σ k ⋅ q +

r |Λu k+1 − q|2 )} . 2

(3) Using the gradient ascent formula for σ σ k+1 = σ k + r(Λu k+1 − q k+1 ) . Theorem 5.2. Given r > 0. If there exists a solution to the primal–dual extremality relations (5.6) and Λ has full column-rank, then there exists an (u,̄ σ)̄ ∈ ℝn × ℝm satisfying (5.6) such that the sequence (u k , q k , σ k ) generated by the ALG2-scheme above satisfies u k → ū ,

q k → Λ ū ,

σ k → σ̄

as

k → +∞ .

(5.10)

We directly apply a general theorem whose proof can be found in [12] (Theorem 8), following contributions of [14, 15, 19] to the analysis of splitting methods.

10.5 Numerical simulations

|

275

10.5.2 Numerical schemes and convergence study We use the software FreeFem++ (see [17]) to implement the numerical scheme. We take the Lagrangian finite elements and notations used in Section 10.5.1, P2 FE for u h and P1 FE for (q h , σ h ). Λu h is the projection on P1 of the operator Λ, that is, ∇u h . The first step and the third one are always the same and only the second one varies with our different test cases. We indicate the numerical convergence of ALG2 iterations by the ⋅k superscript and the convergence of finite elements discretization by the ⋅h subscript. For our numerical simulations, we work with the space dimension d = 2 and we choose for Ω a 2D square (x = (x1 , x2 ) ∈ [0, 1]2 ). We make tests with different f : f1− := e−40∗((x1 −0.75) f2−

:= e

2

+(x 2 −0.25)2 )

−40∗((x 1 −0.5)2 +(x 2 −0.15)2 )

and

f1+ := e−40∗((x1 −0.25)

and

f2+

:= e

2

+(x 2 −0.65)2 )

−40∗((x 1 −0.5)2 +(x 2 −0.75)2 )

,

,

In the third case, we take f3− a constant density and f3+ is the sum of three concentrated Gaussians f3+ (x1 , x2 ) = e−400∗((x−0.25)

2

+(y−0.75)2 )

+ e−400∗((x−0.85)

2

+ e−400∗((x−0.35)

+(y−0.7)2 )

2

+(y−0.15)2 )

.

We also make tests with nonconstant c k : h(x1 , x2 ) = 3 − 2 ∗ e−10∗((x1 −0.5)

2

+(y2 −0.5)2 )

.

As specified above, we use a triangulation of the unit square with n = 1/h element on each side. We use the following convergence criteria: (1) DIV.Error = (∫Ω (div σ kh + f)2 )1/2 is the L2 error on the divergence constraint. h

(2) BND.Error = (∫∂Ω (σ kh ⋅ ν)2 )1/2 is the L2 (∂Ω h ) error on the Neumann boundary h condition. (3) DUAL.Error = maxx j |G(x j , σ kh (x j )) + G∗ (x j , ∇u kh (x j )) − ∇u kh (x j ) ⋅ σ kh (x j )| where the maximum is with respect to the vertices x j . The first two criteria represent the optimality conditions for the minimization of the Lagrangian with respect to u and the third one is for maximization with respect to σ. We make tests for two models. In the first one, the directions are the same as in the cartesian model and the volume coefficients are not necessarily constant. In the second one, the directions are the same than in the hexagonal one and the volume coefficients are equal to 1 (it is simpler to compute G(x, σ)). That is, 𝑣k = exp(ikπ/3) and δ k c k = 1 for k = 1, . . . , 6. We call these models still the cartesian one, the hexagonal one respectively. The cartesian one is much easier since we can separate variables. p G = G1 + G2 with Gi (x, q) = bpi (|q i | − δ i c i (x))+ so that the second step of ALG2 is equivalent to solve the pointwise problem inf q

1 r p (|q| − c(x))+ + |q − q̃ k |2 , p 2

276 | 10 Wardrop equilibria

Table 1: Convergence of the finite element discretization for all test cases. Test case

DIV.Error

BND.Error

DUAL.Error

Time execution (seconds)

1 2 3 4 5 6 7

8.4745e-05 2.2536e–05 5.2141e–05 1.1823e–05 1.1629e–05 3.5553e–04 4.1373e–04

0 8.8705e–04 1.4736e–04 7.6776e–04 0 1.2406 1.1710

3.6126e–06 3.0663e–05 1.1556e–02 8.7412e–06 9.7498e–04 2.1083e–06 4.8113e–04

436 4764 792 170 285 431 4657

where q̃ k = Λu k+1 +

σk r . This amounts to set p−1

(λ|q̃ k | − c(x))+

q k+1 = λ q̃ k and to solve this equation in λ

+ rλ|q̃ k | = r|q̃ k | = 0

with λ ≥ 0. We can use the dichotomy algorithm. For the hexagonal one, we use Newton’s method. Since the function of which we seek the minimizer has its Hessian matrix that is definite positive, we can use the inverse of this Hessian matrix. We show the results of numerical simulations after 200 iterations for both models. Figures represent σ. We notice that length of arrows are proportional to transport density. Level curves correspond to the density term of the source/sink data to be transported. In Figure 3, the case p = 1.01 means that there is much congestion. The case p = 2 is reasonable congestion and in the last one p = 100, there is little congestion. When there are obstacles, the criteria BND.Error is not very good. Indeed, the flow comes right on the obstacle and it turns fast. On the other side of the obstacle, the flow is tangent to the border. Many other cases may of course be examined (other boundary conditions, obstacles, coefficients depending on x, different exponents p for the different components of the flow. . . ). Acknowledgment: The author would like to thank Guillaume Carlier for his extensive help and advice as well as Jean-David Benamou and Ahmed-Amine Homman for their explanations on FreeFem ++.

10.5 Numerical simulations

Fig. 1: Test case 1: cartesian case (d = 2) with f = f3 , c k constant and p = 10.

Fig. 2: Test case 2: hexagonal case (d = 2) with f = f3 , c k constant and p = 3.

|

277

278 | 10 Wardrop equilibria

Fig. 3: Test cases 3, 4, and 5: cartesian case (d = 2) with f = f2 , c k constant and p = 1.01, 2, 100.

10.5 Numerical simulations

|

279

Fig. 4: Test case 6: cartesian case (d = 2) with f = f1 , c 1 = h and c 2 = 1, p = 3 and two obstacles.

Fig. 5: Test case 7: hexagonal case (d = 2) with f = f1 , c k constant, p = 3 and an obstacle.

280 | 10 Wardrop equilibria

Bibliography [1] [2] [3] [4] [5] [6] [7] [8]

[9] [10]

[11] [12]

[13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

[23]

J.-B. Baillon and G. Carlier, From discrete to continuous Wardrop equilibria, Networks and Heterogenous Media, 7(2), 2012. M. Beckmann, C. McGuire and C. Winsten, Studies in the Economics of Transportation, Technical report, 1956. M. Beckmann, A continuous model of transportation, Econometrica: Journal of the Econometric Society, pp. 643–660, 1952. J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge– Kantorovich mass transfer problem, Numerische Mathematik, 84(3):375–393, 2000. J.-D. Benamou and G. Carlier, Augmented Lagrangian methods for transport optimization, Mean-Field Games and degenerate PDEs, 2014. L. Brasco and G. Carlier, Congested traffic equilibria and degenerate anisotropic PDEs, Dynamic Games and Applications, 3(4):508–522, 2013. L. Brasco and M. Petrache, A continuous model of transportation revisited, Journal of Mathematical Sciences, 196(2):119–137, 2014. L. Brasco, G. Carlier and Filippo Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations, Journal de mathématiques pures et appliquées, 93(6):652– 671, 2010. G. Carlier, C. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and Wardrop equilibria, SIAM Journal on Control and Optimization, 47(3):1330–1350, 2008. B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobian determinant, In Annales de l’Institut Henri Poincaré. Analyse non linéaire, volume 7, pp. 1–26. Elsevier, 1990. L. De Pascale, L. C. Evans and A. Pratelli, Integral estimates for transport densities, Bulletin of the London Mathematical Society, 36(03):383–395, 2004. J. Eckstein and D. P. Bertsekas, On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55(1–3):293– 318, 1992. I. Ekeland and R. Temam, Convex analysis and variational problems, 1976. M. Fortin and R. Glowinski, Augmented Lagrangian methods, volume 15 of studies in mathematics and its applications, 1983. D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers & Mathematics with Applications, 2(1):17–40, 1976. R. Hatchi, Wardrop equilibria: rigorous derivation of continuous limits from general networks models, 2015. F. Hecht, New development in FreeFem++, J. Numer. Math., 20(3–4):251–265, 2012. P. Lindqvist, Notes on the p-Laplace equation, Univ., 2006. P.-L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis, 16(6):964–979, 1979. J. Moser, On the volume elements on a manifold, Transactions of the American Mathematical Society, pp. 286–294, 1965. N. Papadakis, G. Peyré and E. Oudet, Optimal transport with proximal splitting, SIAM Journal on Imaging Sciences, 7(1):212–238, 2014. F. Santambrogio, Absolute continuity and summability of transport densities: simpler proofs and new estimates, Calculus of Variations and Partial Differential Equations, 36(3):343–354, 2009. J. G. Wardrop, Road paper. some theoretical aspects of road traffic research, In ICE Proceedings: Engineering Divisions, volume 1, pp. 325–362. Thomas Telford, 1952.

Paul Pegon

11 On the Lagrangian branched transport model and the equivalence with its Eulerian formulation Abstract: First, we present two classical models of branched transport: the Lagrangian model introduced by Bernot et al. and Maddalena et al. [3, 7], and the Eulerian model introduced by Xia [13]. An emphasis is put on the Lagrangian model, for which we give a complete proof of existence of minimizers in a – hopefully – simplified manner. We also treat in detail some σ-finiteness and rectifiability issues to yield rigorously the energy formula connecting the irrigation cost Iα to the Gilbert energy Eα . Our main purpose is to use this energy formula and exploit a Smirnov decomposition of vector flows, which was proved via the Dacorogna–Moser approach in [9], to establish the equivalence between the Lagrangian and Eulerian models, as stated in Theorem 4.7. Keywords: optimal transport, branched transport, Gilbert energy, rectifiability, Smirnov decomposition Mathematics Subject Classification 2010: 49J45, 49Q10, 28A75, 90B20

Introduction Branched transport may be seen as an extension of the classical Monge–Kantorovich mass transportation problem (see [10] for a general reference). In this problem, we are given two probability measures μ and ν representing the source and the target mass distributions and want to find a map T which sends μ to ν in the most economical way. In the original problem from Monge, the cost of moving some mass m along a distance l is proportional to m × l and each particle moves independently to its destination along a straight line. For example, if one wants to transport a uniform mass on the segment [−1, 1] to a mass 2 located at the point y = (0, 1), the mass will travel along straight lines departing from each point in [−1, 1] to y; hence there are infinitely many transport rays (in fact uncountably many). This is obviously not the most economical way to transport mass if we think, for example, of ground transportation networks; in this case you do not want to build infinitely many roads but you prefer to build a unique larger road, which ramifies near the source and the destination to collect and dispatch the goods. Hence in accurate models, one should expect some branching structure to arise, which we actually observe in the optimal structures for the irrigation costs we deal with here. This branching behavior may be seen in many supply–demand distribution systems such as road, pipeline or communication networks, but also natural systems like Paul Pegon, Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay Cedex, France, [email protected]

282 | 11 Equivalence between branched transport models

blood vessels, roots, or river basins. It is usually due to “economy of scale” principles, which say, roughly speaking, that building something bigger will cost more, but proportionally less: once you have built the infrastructures, it does not cost much more to increase the traffic along the network. Thus it is in many cases more economically relevant to consider concave costs w.r.t. the mass, for instance costs of the form m α × l with α ∈ [0, 1[, which are strictly subadditive in m and will force the mass to travel jointly as much as possible. Notice that to model such behavior either in Lagrangian or Eulerian frameworks, one needs to look at the paths actually followed by each particle, and this could not be done via transport maps T or transports plans π ∈ Π(μ, ν), which only describe how much mass goes from a location x to another location y. We shall present here the two main models of branched transport: the Eulerian model developed by Maddalena et al. [7], later studied by Bernot et al. [3], and the Lagrangian model introduced by Xia [13] as an extension to a discrete model model proposed by Gilbert [6]. In Section 11.1, we put an emphasis on the Lagrangian model and give a detailed and simple proof of existence of optimal irrigation plans without resorting to parameterizations as done in [3], thus avoiding unnecessary technicalities and measurability issues. Also notice that, in this way, the Lagrangian branched transport model fits the general framework of dynamical transport problems with measures on curves, as in [4] for incompressible Euler and in [2, 5] for traffic congestion. Section 11.2 is devoted to a rigorous proof of the energy formula, which connects the irrigation cost Iα to the Gilbert energy Eα , tackling some issues of σ-finiteness and rectifiability. In Section 11.3, we give a brief description of the discrete model by Gilbert and the continuous extension proposed by Xia. The purpose of this chapter lies in Section 11.4, which establishes the equivalence between the two models, using the energy formula and a Smirnov decomposition (see [11]) obtained by Santambrogio [9] via a Dacorogna–Moser approach, as stated in Theorem 4.7.

11.1 The Lagrangian model: irrigation plans 11.1.1 Notation and general framework Let K be a compact subset of ℝd . We denote by Γ(K) (or simply Γ) the space of 1-Lipschitz curves in K parameterized on ℝ+ , embedded with the topology of uniform convergence on compact sets. Recall that it is a compact metrizable space¹.

1 One may use the distance d(γ, γ 󸀠 ) := supn∈ℕ⋆

1 n |γ

− γ 󸀠 |L∞ ([0,n]) .

11.1 The Lagrangian model: irrigation plans | 283

Length and stopping time If γ ∈ Γ, we define its stopping time and its length, respectively, by T(γ) = inf{t ≥ 0 : γ is constant on [t, +∞[} , ∞

̇ dt , L(γ) = ∫ |γ(t)| 0

which are valued in [0, +∞]. Since curves are 1-Lipschitz, L(γ) ≤ T(γ). Moreover, one may prove that T and L are both lower semicontinuous functions and, as such, are Borel. We denote by Γ1 (K) the set of curves γ with finite length, i.e. those satisfying L(γ) < ∞. Irrigation plans We call irrigation plan any probability measure η ∈ Prob(Γ) satisfying L(η) := ∫ L(γ) η(dγ) < +∞ .

(1.1)

Γ

Notice that any irrigation plan is concentrated on Γ1 (K). If μ and ν are two probability measures on K, one says that η ∈ IP(K) irrigates ν from μ if one recovers the measures μ and ν by sending the mass of each curve, respectively, to its starting point and to its terminating point, which means that (π 0 )# η = μ ,

(π ∞ )# η = ν ,

where π0 (γ) = γ(0), π ∞ (γ) = γ(∞) := limt→+∞ γ(t) and f# η denotes the push-forward of η by f whenever f is a Borel map². We denote by IP(μ, ν) the set of irrigation plans irrigating ν from μ: IP(μ, ν) = {η ∈ IP(K) : (π 0 )# η = μ, (π ∞ )# η = ν} . Speed normalization We say that a curve γ ∈ Γ is parameterized by arc length or normalized if it has unit ̇ speed up until it stops, i.e. |γ(t)| = 1 for a.e. t ∈ [0, T(γ)[. If an irrigation plan η ∈ IP(K) is such that η-a.e. curve γ is normalized, we say that η is itself parameterized by arc length or normalized. Set sn : γ 󳨃→ γ̃ the map, which associates to each curve γ ∈ Γ(K) its speed normalization³. If η ∈ IP(K) is a general irrigation plan, one may define its speed normalization as η̃ := sn# η and check that (π0 )# η = (π0 )# η̃ and (π∞ )# η = (π ∞ )# η.̃ 2 Notice that limt→∞ γ(t) exists if γ ∈ Γ 1 (K), and this is all we need since any irrigation plan is concentrated on Γ 1 (K). 3 One may check that it is Borel.

284 | 11 Equivalence between branched transport models

Multiplicity Given an irrigation plan η ∈ IP(K), let us define the multiplicity θ η : K → [0, ∞] as θ η (x) = η(γ ∈ Γ(K) : x ∈ γ) , which is also denoted by |x|η . It represents the amount of mass that passes at x through curves of η. We call domain of η the set D η of points with positive multiplicity (points that are really visited by η): D η := {x ∈ K : θ η (x) > 0} . Simplicity If γ ∈ Γ, we denote by m(x, γ) = #{t ∈ [0, T(γ)] ∩ ℝ+ : γ(t) = x} ∈ ℕ ∪ {∞} the multiplicity of x on γ, which is the number of times the curve γ visits x. We call simple points of γ ∈ Γ those which are visited only once, i.e. such that m(x, γ) = 1 and denote by S γ the set of such points. We say that γ is simple if γ \ S γ = 0 and essentially simple if H1 (γ \ S γ ) = 0. As usual, we extend these definitions to irrigation plans, saying that η is simple (resp. essentially simple) if η-a.e. curve is simple (resp. essentially simple). Finally, we set m η (x) := ∫ m(x, γ)η(dγ) Γ

which represents the mean number of times curves visit x. Notice that θ η (x) = ∫ 𝟙x∈γ η(dγ) ≤ ∫ m(x, γ)η(dγ) ≐ m η (x) Γ

Γ

so that m η (x) is in a way the “full” multiplicity at x.

11.1.2 The Lagrangian irrigation problem Notation If ϕ : K → ℝ+ and ψ : K → ℝd are Borel functions on K, we set ∞

̇ dt ∫ ϕ(x) |dx| := ∫ ϕ(γ(t)) | γ(t)| γ

and

0 ∞

̇ dt ∫ ψ(x) ⋅ dx := ∫ ψ(γ(t)) ⋅ γ(t) γ

̇ provided t 󳨃→ |ψ(γ(t))||γ(t)| is integrable.

0

11.1 The Lagrangian model: irrigation plans | 285

Irrigation costs For α ∈ [0, 1], we consider the irrigation cost Iα : IP(K) → [0, ∞] defined by Iα (η) := ∫ ∫ |x|α−1 |dx| η(dγ) , η Γ γ

with the conventions 0α−1 = ∞ if α < 1, 0α−1 = 1 otherwise, and ∞ × 0 = 0. If μ, ν are two probability measures on K, we want to minimize the cost Iα on the set of irrigation plans, which send μ to ν, which reads min

η∈IP(μ,ν)

∫ ∫ |x|α−1 |dx| η(dγ) . η

(LI α )

Γ γ

Notice that Iα is invariant under speed normalization; thus we will often assume without loss of generality that irrigation plans are normalized. The following result gives a sufficient condition for irrigability with finite cost, and may be found in [3]. Proposition 1.1 (Irrigability). If α > 1 − 1d , then for every pair of measures (μ, ν) ∈ Prob(K), there is an irrigation plan η ∈ IP(μ, ν) of finite α-cost, i.e. such that Iα (η) < ∞. Remark 1.2. One can actually show that the uniform measure on a unit cube can be irrigated from a unit Dirac mass if and only if α > 1 − 1d ; hence this condition is necessary for an arbitrary pair (μ, ν) to be irrigable with finite α-cost, provided, for example, that K has nonempty interior.

11.1.3 Existence of minimizers In this section, we prove the necessary lower semicontinuity and compactness results leading to the proof of existence of minimizers by the direct method of calculus of variations. We recall here that, unless stated otherwise, continuity properties on Γ relate to the topology of uniform convergence on compact sets and on IP(K) to the weak-⋆ topology in the duality with C(K). A tightness result For C > 0, we define IPC (K) as the set of irrigation plans η on K such that T(η) := ∫ T(γ)η(dγ) ≤ C Γ

and IPC (μ, ν) := IPC (K) ∩ IP(μ, ν). Notice that for a normalized irrigation plan η one has T(η) ≐ ∫ T(γ)η(dγ) = ∫ L(γ)η(dγ) ≐ L(η) . Γ

Γ

286 | 11 Equivalence between branched transport models Remark 1.3. Since T is lower semicontinuous on Prob(Γ) and L ≤ T, IPC (K) is a closed – thus compact – subset of Prob(Γ). The following lemma results directly from Markov’s inequality. Lemma 1.4 (Tightness). Given C > 0, for all η ∈ IP C (K) one has η(γ : T(γ) > M) ≤

C . M

This can be considered as a tightness result because it means that all irrigation plans η ∈ IPC (K) are almost concentrated (uniformly) on the sets Γ M := {γ ∈ Γ : T(γ) ≤ M}, which are compact for the uniform topology. Continuity results When A is a closed subset of K, we set [A] = {γ ∈ Γ1 : γ ∩ A ≠ 0} ,

|A|η = η([A]) ,

so that |x|η = |{x}|η . One may show that [A] is Borel. Our first continuity result is that of | ⋅ |η along decreasing sequences of closed sets. Lemma 1.5. If (A n )n∈ℕ is a decreasing sequence of closed sets and A = ⋂↓n A n then |A|η = lim |A n |η . n→∞

Proof. Let us prove that [A] = ⋂↓n [A n ]. The inclusion [A] ⊆ ⋂n [A n ] is clear since [A] ⊆ [A n ] for all n. Let us take γ ∈ ⋂n [A n ]. Since belonging to some [B] only depends on the trajectory of γ, we may assume that it is parameterized by arc length. Because γ has finite length L, there is a sequence (A n )n and a sequence (t n )n ∈ [0, L] such that γ(t n ) ∈ A n for all n. One may extract a converging subsequence, still denoted by (t n )n n→∞ such that t n 󳨀󳨀󳨀󳨀→ t ∈ [0, L]. Since the (A n )’s are decreasing closed sets, γ(t) belongs to their intersection A, hence γ ∈ [A]. By the monotone convergence theorem lim |A n |η ≐ lim η([A n ]) = η([A]) = |A| η . n

n

Proposition 1.6. For C > 0, the map θ:

K × IPC (K) (x, η)

󳨀→ 󳨃󳨀→

[0, 1] |x| η

is upper semicontinuous. ̄ ϵ), hence Proof. Given x n → x and η n ⇀ η, take ϵ > 0. If n is large enough, x n ∈ B(x, ̄ lim supn |x n |η n ≤ lim supn |B(x, ϵ)|η n . Besides, using Lemma 1.4 one gets 󵄨󵄨 ̄ 󵄨 ̄ 󵄨󵄨B(x, ϵ)󵄨󵄨󵄨 ≤ η n ({T > M}) + η n ({T ≤ M} ∩ [B(x, ϵ)]) 󵄨 󵄨ηn ≤ C/M + η n (A) ,

11.1 The Lagrangian model: irrigation plans |

287

̄ where we set A := {T ≤ M} ∩ [B(x, ϵ)]. It is easy to check A is closed since T is lsc and the ball is closed. Hence passing to the lim sup in n yields 󵄨 ̄ 󵄨 ̄ 󵄨 󵄨 lim sup 󵄨󵄨󵄨󵄨B(x, ϵ)󵄨󵄨󵄨󵄨 η ≤ C/M + η(A) ≤ C/M + 󵄨󵄨󵄨󵄨B(x, ϵ)󵄨󵄨󵄨󵄨 η . n

(1.2)

n

Taking M → ∞, one gets 󵄨 ̄ 󵄨 ̄ 󵄨 󵄨 lim sup |x n |η n ≤ lim sup 󵄨󵄨󵄨󵄨B(x, ϵ)󵄨󵄨󵄨󵄨 η ≤ 󵄨󵄨󵄨󵄨B(x, ϵ)󵄨󵄨󵄨󵄨 η , n n n then we pass to the limit ϵ → 0 using Lemma 1.5: 󵄨 ̄ 󵄨 lim sup |x n |η n ≤ lim 󵄨󵄨󵄨󵄨B(x, ϵ)󵄨󵄨󵄨󵄨 η = |x|η . ϵ→0 n For any η ∈ IP(K), we define the α-cost of a curve γ ∈ Γ w.r.t. η by Z α,η (γ) = ∫ θ α−1 η (x) |dx| γ

and we set Zα :

Γ × IP(K) (γ, η)

󳨀→ 󳨃󳨀→

ℝ . Z α,η (γ)

Proposition 1.7. For any C > 0, the function Z α is lower semicontinuous on Γ × IPC . Proof. The case α = 1 is clear since Z1 (γ, η) = L(γ); hence we assume α < 1. We know that the map f : (x, η) 󳨃→ |x|α−1 is lsc on K × IPC since θ : (x, η) 󳨃→ |x|η is usc and η α − 1 < 0. Now take γ n → γ and η n ⇀ η, then since T is lsc, for ϵ > 0 and n large enough, we have T(γ) ≤ T(γ n ) + ϵ, which implies that T(γ n )

T(γ)−ϵ

∫ f(γ n (t), η n ) |γ̇ n (t)| dt ≥

∫ f(γ n (t), η n ) | γ̇ n (t)| dt .

0

0

(1.3)

Suppose for a moment that f is continuous on K ×IPC which is a compact metric space, hence it is uniformly continuous. Since γ n converges uniformly to γ on [0, T(γ)−ϵ)], the function g n : t 󳨃→ f(γ n (t), η n ) converges uniformly to g : t 󳨃→ f(γ(t), η) on [0, T(γ) − ϵ]. Now we have to take care of the |γ̇ n (t)| factor. Since the sequence (γ̇ n )n is bounded L∞

in L∞ ([0, T(γ) − ϵ]) one may extract a subsequence (γ̇ n k )k such that γ̇ n k 󳨀󳨀⇀ γ̇ and L∞

̇ |γ̇ n k | 󳨀󳨀⇀ u. It is a classical result that |γ(t)| ≤ u(t) almost everywhere on [0, T(γ) − ϵ]. Denoting by ⟨⋅, ⋅⟩ the duality bracket L1 − L∞ on [0, T(γ) − ϵ], we have T(γ)−ϵ

󵄨 󵄨 󵄨 󵄨 k→∞ ∫ f(γ n k (t), η n k ) 󵄨󵄨󵄨 γ̇ n k (t)󵄨󵄨󵄨 dt ≐ ⟨g n k , 󵄨󵄨󵄨γ̇ n k 󵄨󵄨󵄨⟩ 󳨀󳨀󳨀󳨀→ ⟨g, u⟩ ≥ 0

T(γ)−ϵ

̇ dt ∫ f(γ(t), η) | γ(t)| 0

288 | 11 Equivalence between branched transport models since g n k → g uniformly hence strongly in L1 . Prior to extracting the subsequence (γ̇ n k )k , we could have taken first a subsequence of γ̇ n such that the left-hand side conT(γ)−ϵ verged to lim inf n ∫0 f(γ n (t), η n )|γ̇ n (t)| dt. Thus, we actually obtain T(γ)−ϵ

T(γ)−ϵ

lim inf ∫ f(γ n (t), η n ) | γ̇ n (t)| dt ≥ n

0

̇ ∫ f(γ(t), η) | γ(t)| dt . 0

Finally, we use this inequality together with (1.3) and pass to the limit ϵ → 0 thanks to the monotone convergence theorem, which yields T(γ n )

T(γ)

̇ lim inf ∫ f(γ n (t), η n ) | γ̇ n (t)| dt ≥ ∫ f(γ(t), η) | γ(t)| dt . n

0

(1.4)

0

In general f is not continuous but only lsc, but (1.4) still holds for our function f by considering an increasing sequence of continuous functions f k ↑ f , writing the inequality with f k and using the monotone convergence theorem as k → ∞. We have therefore proven that lim inf Z α (γ n , η n ) ≥ Z α (γ, η) , n

hence Z α is lsc on Γ × IPC (K). Notice that our cost Iα may be written as Iα (η) = ∫ Z α (γ, η)η(dγ) , Γ

hence its lower semicontinuity on IP C will be obtained as a corollary to the following lemma. Lemma 1.8. Let X be a closed subset of Prob(Γ). (1) If f : Γ × X → ℝ is continuous, then the functional F : η 󳨃→ ∫Γ f(γ, η) η(dγ) is continuous on X. (2) If f : Γ × X → [0, +∞] is lsc, then F : η 󳨃→ ∫Γ f(γ, η) η(dγ) is lsc on X. Proof. Let us prove the first item. Take η n ⇀ η. Since X is a compact metrizable space, so is Γ × X and by Heine’s theorem, f is uniformly continuous on Γ × X. Setting g n (γ) = f(γ, η n ) and g(γ) = f(γ, η), this implies that g n → g strongly in C(Γ). Since η n ⇀ η weakly ⋆, we have ∫Γ f(γ, η n ) η n (dγ) = ⟨η n , g n ⟩ → ⟨η, g⟩ = ∫Γ f(γ, η) η(dγ) where ⟨⋅, ⋅⟩ denotes the C(Γ) − M(Γ) duality bracket. The second item is a straightforward consequence of the fact that f can be written as the increasing limit of continuous functions f k , and of the monotone convergence theorem. Corollary 1.9. For α ∈ [0, 1], the functional Iα is lower semincontinus on IPC (K).

11.1 The Lagrangian model: irrigation plans | 289

Proof. We know by Remark 1.3 that IPC (K) is a closed subset of Prob(Γ), which gives the result by the previous lemma applied to the function f = Z α defined on IPC (K) × Γ. Finally, we are left to investigate the continuity of the maps π 0 : η 󳨃→ (π 0 )# η and π ∞ : η 󳨃→ (π ∞ )# η on IP(K). Proposition 1.10. The maps π 0 , π ∞ : IPC (K) → Prob(K) are continuous⁴. In particular, IP C (μ, ν) is closed. Proof. Clearly the map π0 : γ 󳨃→ γ(0) is continuous on Γ, hence π 0 is continuous on IP(K). However, π ∞ : γ 󳨃→ γ(∞) defined on Γ1 is not necessarily continuous thus π ∞ needs not be continuous on IP(K). Nevertheless, π∞ is continuous on all the sets Γ M for M > 0 and the tightness result of Lemma 1.4 allows us to conclude. Indeed take C η n ⇀ η in IPC (K). Take ϵ > 0 and M large enough so that M ≤ ϵ. For any ϕ ∈ C(K), one has ∫ ϕ(γ(∞))η n (dγ) = ∫ ϕ(γ(M))η n (dγ) + ∫ ϕ(γ(∞))η n (dγ) Γ

(Γ M )c

ΓM

and ∫ ϕ(γ(∞))η(dγ) = ∫ ϕ(γ(M))η(dγ) + ∫ ϕ(γ(∞))η(dγ) Γ

(Γ M )c

ΓM

thus 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫ ϕ(γ(∞))η n (dγ) − ∫ ϕ(γ(∞))η(dγ)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Γ 󵄨󵄨 Γ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ≤ 󵄨󵄨󵄨 ∫ ϕ(γ(M))η n (dγ) − ∫ ϕ(γ(M))η(dγ)󵄨󵄨󵄨󵄨 + 2ϵ|ϕ|∞ . 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨Γ M ΓM We pass to the lim supn using the continuity of π M : γ 󳨃→ γ(M) on Γ, then pass to the limit ϵ → 0 to yield ∫ ϕ(x)(π ∞ )# η n (dx) → ∫ ϕ(x)(π ∞ )# η(dx) K

K

which means that π ∞ is continuous on IPC (K). The existence theorem We are now able to prove the existence theorem for the minimization problem (LI α ).

4 Recall that we always endow spaces of measures with their weak-⋆ topology.

290 | 11 Equivalence between branched transport models

Theorem 1.11. If μ, ν are probability measures on K, there exists a minimizer η of the problem min

η∈IP(μ,ν)

∫ ∫ |x|α−1 |dx| η(dγ) . η

(LIα )

Γ γ

Proof. We assume Iα ≢ +∞, otherwise there is nothing to prove. Let us take a minimizing sequence η n , which we may assume to be normalized. In particular Iα (η n ) ≤ C for some C > 0. Consequently T(η n ) = L(η n ) ≐ ∫ ∫ |dx| η n (dγ) ≤ ∫ ∫ |x|α−1 |dx| η n (dγ) ≐ Iα (η n ) ≤ C , η Γ γ

Γ γ

which implies that η n ∈ IPC (μ, ν). Thanks to Proposition 1.10, IP C (μ, ν) is a closed subset of Prob(Γ) which is compact (and metrizable) by Banach–Alaoglu’s theorem, hence it is itself compact and we can extract a converging sequence η n ⇀ η ∈ IPC (μ, ν) up to some renaming. By Corollary 1.9, Iα is lsc on IPC (K), thus Iα (η) ≤ lim inf Iα (η n ) = inf (LIα ) , n

which shows that η is a minimizer to the problem (LI α ).

11.2 The energy formula The goal of this section is to establish the following formula: ∫ ∫ |x|α−1 |dx| η(dγ) = ∫ |x|αη H1 (dx) , η Γ γ

(EF)

K

provided η satisfies some hypotheses (namely essential simplicity and rectifiability). The term on the right-hand side is the so-called Gilbert energy denoted by Eα (η). The proof relies solely on the correct use of Fubini–Tonelli’s theorem, which requires σ-finiteness of measures. The next subsection is therefore devoted to the rectifiability of irrigation plans.

11.2.1 Rectifiable irrigation plans Intensity and flow We define the intensity i η ∈ M+ (K) and flow 𝑣η ∈ Md (K) of an irrigation plan η ∈ IP(μ, ν) by the formulas ⟨i η , ϕ⟩ = ∫ ∫ ϕ(x) |dx| η(dγ) , Γ γ

⟨𝑣η , ψ⟩ = ∫ ∫ ψ(x) ⋅ dx η(dγ) , Γ γ

11.2 The energy formula

| 291

for all ϕ ∈ C(K), ψ ∈ C(K)d . The quantity i η (dx) represents the mean circulation at x and 𝑣η (dx) the mean flow at x. Concentration and rectifiability Let A be a Borel subset of K. By definition of i η , the following assertions are equivalent: (1) i η is concentrated on A, i.e. i η (A c ) = 0, (2) for η-a.e. γ ∈ Γ, γ ⊆ A up to an H1 -null set, i.e. H1 (γ \ A) = 0. In that case, we say (with a slight abuse) that η is concentrated on A. An irrigation plan η ∈ IP(K) is termed σ-finite if it is concentrated on a σ-finite set w.r.t. H1 and rectifiable if it is concentrated on a 1-rectifiable set⁵. The intensity i η has a simple expression when η is σ-finite, as shown below. Proposition 2.1. If η is an irrigation plan concentrated on a σ-finite set A, then i η = m η H1 A . Proof. For any Borel set B, one has i η (B) = i η (A ∩ B) = ∫ ∫ 𝟙 A∩B |dx| η(dγ) Γ γ

= ∫ ∫ m(x, γ) H 1 A (dx)η(dγ) Γ B

= ∫ ∫ m(x, γ)η(dγ) H 1 A (dx) B Γ

= ∫ m η (x) H1 A (dx) . B

The equality on the second line follows from the coarea formula, the next from Fubini– Tonelli’s theorem which holds because measures are σ-finite, and the last one from the definition of m η . In order to prove that the domain D η ≐ {x ∈ K : θ η (x) > 0} of an irrigation plan is 1-rectifiable, we will need a few notions and non-trivial lemmas of geometric measure theory. Density of a set If A is a subset of ℝd , we define the upper and lower 1-density of A at x as Θ(x, A) = lim sup r↓0

H1 (A ∩ B(x, r)) , 2r

Θ(x, A) = lim inf r↓0

H1 (A ∩ B(x, r)) . 2r

5 A 1-rectifiable set is the union of an H1 -null set with a countable union of Lipschitz curves.

292 | 11 Equivalence between branched transport models

When these quantities are equal, we call their common value Θ(x, A) the 1-density of A at x. The first lemma we will need is proved in [8, Chapter 8]. Lemma 2.2. If B ⊆ ℝd , H1 (B) = sup {H1 (K) : K ⊆ B compact such that H1 (K) < ∞} . The second is due to Besicovitch and may be obtained as a particular case of [8, Theorem 17.6]. Lemma 2.3. Let E be an H1 -measurable set such that H1 (E) < ∞. If its 1-density Θ(x, E) exists and is equal to 1 for H1 -a.e. x in E then it is 1-rectifiable. Finally, we will need the following result which is included in [8, Theorem 6.2]. Lemma 2.4. If E is a set such that H1 (E) < ∞, then the upper 1-density Θ(x, E) is less than 1 for H1 -a.e. x in E. Proposition 2.5 (Rectifiability of the domain). If η ∈ IP(K) is an irrigation plan, its domain D η is 1-rectifiable. Proof. First of all, since the domain does not change under normalization, we may assume that η is parameterized by arc length. We have D η = ⋃ n>0 D n where D n := {x : θ η (x) >

1 } . n

Let us show that H1 (D n ) < ∞. By contradiction assume that for some n, H1 (D n ) = ∞, hence thanks to Lemma 2.2 one can find for M > 0 as large as we want a compact subset K 󸀠 ⊆ D n such that M ≤ H1 (K 󸀠 ) < ∞. Since H1 (K 󸀠 ) < ∞ one can use Fubini– Tonelli’s theorem to get L(η) ≐ ∫ L(γ)η(dγ) ≥ ∫ H1 (γ ∩ K 󸀠 )η(dγ) = ∫ ∫ 𝟙x∈γ H1 (dx)η(dγ) Γ

Γ

Γ K󸀠

= ∫ ∫ 𝟙x∈γ η(dγ) H1 (dx) K󸀠 Γ

= ∫ θ η (x) H1 (dx) > K󸀠

M . n

The inequality L(η) > must be true for all M > 0, i.e. L(η) = ∞, which contradicts the definition of an irrigation plan, hence H1 (D n ) < ∞. Now we shall prove that Θ(x, D n ) = 1 a.e. on D n . Since D n has finite H1 -measure, we already know Θ(x, D n ) ≤ 1 for H1 -a.e. x ∈ D n by Lemma 2.4, thus it remains only to prove Θ(x, D n ) ≥ 1. If A is a Borel subset of ℝ, we denote Leb(A) the set of Lebesgue points of A, which are points t such that M n

lim r↓0

|A ∩ [t − r, t + r]| =1, 2r

11.2 The energy formula |

293

where |X| denotes the Lebesgue measure of X ⊆ ℝ. Recall that by Lebesgue’s theorem, we have |A \ Leb(A)| = 0. For any γ ∈ Γ1 , we set 1 1 ) for all n s.t. |γ(t)| η > } , n n ̇ B γ = {t ∈]0, T(γ)[ : γ(t) exists} ,

A γ = {t : t ∈ Leb (s : |γ(s)| η > D γ = γ(A γ ∩ B γ ) .

Notice that |[0, T(γ)] \ (A γ ∪ B γ )| = 0 hence H1 (γ \ D γ ) = 0 since γ is Lipschitz. Finally, we set D󸀠 = ⋃ D γ . γ∈Γ1

Let us check that H1 (D η \ D󸀠 ) = 0. We obtain ∫ θ η (x) H 1 (dx) = ∫ ∫ 𝟙 x∈γ η(dγ) H1 (dx) D η \D󸀠

D η \D󸀠 Γ

=∫ ∫

𝟙x∈γ H1 (dx)η(dγ)

Γ D η \D󸀠

= ∫ H1 (D η ∩ γ \ D󸀠 )η(dγ) Γ1

=0. The use of Fubini–Tonelli’s theorem is justified since D η = ⋃ n D n is σ-finite and the last equality follows from H 1 (D η ∩γ\D󸀠 ) ≤ H1 (γ\D γ ) = 0. This implies H1 (D η \D󸀠 ) = 0 since θ η > 0 on D η . Now take any x ∈ D n ∩ D󸀠 . By construction of D󸀠 there is a curve γ ∈ Γ1 and a t ∈ A γ ∩ B γ such that x = γ(t), which implies that |s ∈ [t − r, t + r] : γ(s) ∈ D n | r↓0 󳨀󳨀→ 1 2r

(2.1)

and H1 (γ([t − r, t + r]) \ D n ) r↓0 󳨀󳨀→ 0 . (2.2) 2r ̄ It follows from (2.2) and the fact that γ([t − r, t + r]) ⊆ B(x, r) because γ is 1-Lipschitz that Θ(x, D n ) ≐ lim inf r↓0

H1 (B(x, r) ∩ D n ) H1 (γ([t − r, t + r])) ≥ lim inf . 2r r↓0 2r

But γ has a derivative e at t which has unit norm. Moreover, the H1 -measure of γ([t − r, t + r]), which is a compact connected set, is larger than the distance between γ(t − r) and γ(t + r), and since γ(t ± r) = x ± re + o(r) one has H1 (γ([t − r, t + r])) ≥ |γ(t + r) − γ(t − r)| = 2r + o(r) , which yields Θ(x, D n ) ≥ 1. This proves that Θ(x, D n ) exists and is equal to 1 for H 1 -a.e. x ∈ D n hence D n is 1-rectifiable by Lemma 2.3 and D η = ⋃n D n as well.

294 | 11 Equivalence between branched transport models

At this stage, we have shown that the domain of any irrigation plan is rectifiable, yet this does not mean that any irrigation plan is rectifiable (this is obviously not the case) since η needs not be concentrated on D η . However, it is essentially the only candidate rectifiable set (or even candidate σ-finite set) on which η could be concentrated, as stated below. Corollary 2.6. Given η ∈ IP(K) an irrigation plan, the following assertions are equivalent: (i) η is concentrated on D η , (ii) η is rectifiable, (iii) η is σ-finite. Proof. It is enough to prove (iii) ⇒ (i) by the previous proposition. If η is concentrated on a σ-finite set A, we know by Proposition 2.1 that i η = m η H1 A . Therefore i η is also concentrated on {x : m η (x) > 0} = D η . Remark 2.7. From this and Proposition 2.1, we get i η = m η H1 if η is rectifiable. The most important consequence of Proposition 2.5 is the following rectifiability result. Theorem 2.8 (Rectifiability). If η has finite α-cost with α ∈ [0, 1[, it is rectifiable. Proof. Because of the previous statement, we need only show that η is concentrated on D η . We have Iα (η) ≐ ∫Γ ∫γ |x|α−1 η | dx|η(dγ) < ∞ hence for η-almost every curve γ, for H1 -almost every x in γ, |x|α−1 < ∞, which implies that |x|η > 0, i.e. x ∈ D η . By η definition, it means that η is concentrated on D η .

11.2.2 Proof of the energy formula We define the Gilbert energy Eα : IP(K) → [0, ∞] as 1 α { { {∫ θ η (x) H (dx) Eα (η) = {K { { {+∞

if η is rectifiable , otherwise ,

and a variant Ē α : IP(K) → [0, ∞] (kind of a “full” energy) 1 α−1 { { {∫ θ η (x)m η (x) H (dx) Ē α (η) = {K { { {+∞

if η is rectifiable , otherwise .

Assuming α ∈ [0, 1[, we would like to establish the energy formula Iα (η) = Eα (η) .

(EF)

11.2 The energy formula |

295

This does not hold in general. Actually, we are going to show that Iα (η) = Ē α (η) for all irrigation plan η ∈ IP(K) and that Ē α (η) = Eα (η) provided η is essentially simple. Theorem 2.9 (Energy formula). Assuming α ∈ [0, 1[, the following formula holds: Iα (η) = Ē α (η) .

(EF’)

Moreover, if η is essentially simple this rewrites Iα (η) = Eα (η) .

(EF)

Proof. By Theorem 2.8, if η is not rectifiable then Iα (η) = Eα (η) = Ē α (η) = ∞ and the result is clear. Now, we assume that η is rectifiable, which means that it is concentrated on the rectifiable set D η , according to Theorem 2.8 and Corollary 2.6. Notice that by the coarea formula, we have m(x, γ) H1 (dx)η(dγ) , ∫ ∫ |x|α−1 |dx| η(dγ) = ∫ ∫ |x| α−1 η η Γ γ

Γ K

thus the goal is to reverse the order of integration. Here Fubini–Tonelli’s theorem applies because η is concentrated on its domain, which is rectifiable, which yields Iα (η) = ∫ ∫ |x|α−1 m(x, γ) H 1 (dx)η(dγ) η Γ Dη

m η (x) H1 (dx) = ∫ |x| α−1 η Dη 1 ̄ = ∫ θ α−1 η (x)m η (x) H (dx) = E α (η) . K

and (EF’) holds. Now if η is essentially simple then in all the previous calculations m η (x) = θ η (x) so that 1 Iα (η) = ∫ ∫ θ α−1 η (x)θ γ (x)η(dγ) H (dx) Dη Γ

= ∫ θ αη (x) H 1 (dx) = Eα (η) , K

thus getting (EF). Remark 2.10. Actually, the proof shows that the equality Iα (η) = Ē α (η) (and Iα (η) = Eα (η) if η is essentially simple) also holds for α = 1 provided η is rectifiable. However, one may find η non-rectifiable such that I1 (η) ∈]0, ∞[ while H1 (D η ) = 0. In that case, one has 0 = ∫K m η (x) H 1 (dx) < I1 (η) < Ē 1 (η) = ∞. Also notice that 1 0 = ∫K θ α−1 η (x)m η (x) H (dx) < I α (η) = ∞ for α ∈ [0, 1[, which explains why we ̄ imposed Eα (η) = Eα (η) = ∞ if η is not rectifiable.

296 | 11 Equivalence between branched transport models

11.2.3 Optimal irrigation plans are simple In this section, we shall prove that optimal irrigation plans are necessary simple using the energy formula. “Reduced” intensity We associate to any irrigation plan η ∈ IP(K) a “reduced” intensity j η by ⟨j η , ϕ⟩ = ∫ ∫ ϕ(x) H 1 (dx)η(dγ) , Γ γ

for all ϕ ∈ C(K). It is a positive finite measure, since the total mass is |j η | = ∫ H1 (γ)η(dγ) ≤ ∫ L(γ)η(dγ) = L(η) < ∞ . Γ

Γ

Remark 2.11. Notice that if A is a Borel set, j η (A) = 0 ⇔ i η (A) = 0 hence by definition η is rectifiable if and only if j η is concentrated on a rectifiable set, in which case it is concentrated on the rectifiable domain D η and one has j η = θ η H1 using Fubini– Tonelli’s theorem. Lemma 2.12 (Simple replacement). Let η ∈ IP(μ, ν) be an irrigation plan. Consider the minimization problem min L(ζ) : j ζ ≤ j η and ζ ∈ IP(μ, ν) .

(LENη )

Then (1) this problem admits minimizers which are all simple, (2) if η is rectifiable, all minimizers ζ are also rectifiable and j ζ ≤ j η rewrites θζ ≤ θη

H1 -almost everywhere .

(2.3)

Any minimizer of (LENη ) is called a simple replacement of η. Proof. Let us call m the infimum of (LENη ) and show that it admits a minimizer. Take a minimizing sequence (ζ n )n such that every ζ n is normalized, in particular ζ n ∈ IPC (K) for some C > 0. Up to extraction, we have convergence ζ n ⇀ ζ , and since IPC (μ, ν) is closed by Proposition 1.10, ζ ∈ IP C (μ, ν). Moreover L(ζ) = m by lower semicontinuity of L ≐ I1 on IPC (K), which we proved in Corollary 1.9. Now in order to show that ζ is a solution of (LENη ), we only have to check the last constraint j ζ ≤ j η . Take any open set O. One has j ζ n (O) = ∫ H1 (γ ∩ O)ζ n (dγ) . Γ

By a generalization of Golab’s Theorem (see [1]), the following holds: H1 (γ ∩ O) ≤ lim inf H1 (γ n ∩ O) , n

11.3 The Eulerian model: irrigation flows | 297

if γ n → γ uniformly on compact sets, which means that γ 󳨃→ H1 (γ ∩ O) is lower semicontinuous on Γ. Consequently ζ 󳨃→ ∫Γ H1 (γ ∩ O)ζ(dγ) is lower semicontinuous and one gets j ζ (O) ≐ ∫ H1 (γ ∩ O)ζ(dγ) ≤ lim inf ∫ H1 (γ ∩ O)ζ n (dγ) ≤ j η (O) n

Γ

Γ

for all open set O. This implies that j ζ ≤ j η by regularity of finite measures; hence ζ is a minimizer of (LENη ). Let us check that any minimizer ζ is simple. By contradiction, if it was not simple there would be a set Γ󸀠 ⊆ Γ such that ζ(Γ󸀠 ) > 0 and every γ ∈ Γ󸀠 has a loop. One may define a Borel map r : γ 󳨃→ r(γ) which removes from γ ∈ Γ󸀠 the loop with maximal length (the first one in case there are several), and is identical on Γ \ Γ󸀠 . Then set ζ ̄ := r(ζ). Obviously, one has L(ζ ̄ ) < L(ζ), ζ ̄ ∈ IP(μ, ν) and j ζ ̄ ≤ j ζ , which contradicts the optimality of ζ in (LENη ). Finally, suppose η is rectifiable and take ζ a minimizer of our problem. According to Remark 2.11, the inequality j ζ ≤ j η implies that ζ is rectifiable and j η = θ η H1 , j ζ = θ ζ H1 , which yields (ii). Proposition 2.13. Given α ∈ [0, 1], if η ∈ IP(μ, ν) is optimal with finite α-cost, then it is simple. Proof. The case α = 1 is straightforward from Lemma 2.12 since L = I1 . Now we assume that α < 1 and take η optimal, in which case the finiteness of the α-cost implies the rectifiability of η by Theorem 2.8. We only need to show that η is a minimizer of (LENη ). Take η̃ as a simple replacement of η. Then since η, η̃ are rectifiable and θ η̃ ≤ θ η H1 a.e., one has 1 α 1 α 1 ̃ . Iα (η) = ∫ θ α−1 η m η d H ≥ ∫ θ η d H ≥ ∫ θ η̃ d H = I α (η) K

K

K

Since η is optimal, we have equality everywhere, which means that m η = θ η = θ η̃ = m η̃ H1 -a.e. Consequently L(η) = ∫ m η (x) H1 (dx) = ∫ m η̃ (x) H1 (dx) = L(η)̃ K

K

hence η minimizes (LEN η ) and is as such simple by Lemma 2.12.

11.3 The Eulerian model: irrigation flows In this section, we present the Eulerian model of branched transport, which was introduced by Xia [13] as a continuous extension to a discrete model proposed by Gilbert [6].

298 | 11 Equivalence between branched transport models

11.3.1 The discrete model Oriented Graph An oriented graph in K is a pair G = (E, w) where E = E(G) is a set of oriented segments (e1 , . . . , e n ) called edges and w : E(G) →]0, ∞[ is a function, which gives a weight to any edge. An oriented segment e simply consists of an ordered pair of points (e− , e+ ) in K which we call starting and ending point of e. We denote by |e| := |e + − e− | its + − d−1 its orientation provided e + ≠ e − , and set G(K) to be the length, by ê := |ee+ −e −e − | ∈ 𝕊 set of oriented graphs on K. Irrigation graphs Given two atomic probability measures μ = ∑i α i δ x i and ν = ∑i β i δ y i , we say that G irrigates ν from μ if it satisfies the so-called Kirchhoff condition, well-known for electric circuits: incoming mass at 𝑣 = outcoming mass at 𝑣 , for all vertex 𝑣 of the graph. By “incoming mass”, we mean the total weight of edges with terminating point 𝑣, increased by a i if 𝑣 = x i , and by “outcoming mass”, we mean the total weight of edges with starting point 𝑣, increased by b j if 𝑣 = y j . Indeed μ and ν are seen as mass being, respectively, pushed in and out of the graph. The set of graphs irrigating ν from μ is denoted by G(μ, ν). Discrete irrigation problem With a slight abuse, we define the α-cost of a graph as Eα (G) = ∑ w(e)α |e| , e∈E(G)

which means that the cost of moving a mass m along a segment of length l is m α ⋅ l. Given μ, ν two atomic probability measures, we want to minimize the cost of irrigation among all graphs sending μ to ν, which reads min

G∈G(μ,ν)

Eα (G) .

(DIα )

11.3.2 The continuous model From now on, we assume that α ∈ [0, 1[. Irrigation flow We call irrigation flow on K any vector measure 𝑣 ∈ Md (K) such that ∇ ⋅ 𝑣 ∈ M(K), where ∇ ⋅ 𝑣 is the divergence of 𝑣 in the sense of distribution. We denote by IF(K) the set of irrigation flows.

11.3 The Eulerian model: irrigation flows |

299

Remark 3.1. These objects have several names. They are called one-dimensional normal currents in the terminology of geometric measure theory, and are called traffic paths by Xia [13]. If E ⊆ K is an H1 -measurable set, τ : E → 𝕊d−1 is H1 -measurable and θ : E → ℝ+ is H1 -integrable, we define the vector measure [E, τ, θ] ∈ Md (K) by ⟨[E, τ, θ], ψ⟩ = ∫ θ(x)ψ(x) ⋅ τ(x) H 1 (dx) , E

for all ψ ∈ C(K, ℝd ). In other terms [E, τ, θ] ≐ θτ H1 E . Rectifiable irrigation flow Recall that if E is a 1-rectifiable set, at H1 -a.e x ∈ E there is an approximate tangent line (see [8, Chapter 17]) denoted by Tan(x, E). An irrigation flow of the form 𝑣 = [E, τ, θ] where E is 1-rectifiable and τ(x) ∈ Tan(x, E) for H1 -a.e. x ∈ E is termed rectifiable. From discrete to continuous Consider a graph G ∈ G(K). One can define the vector measure 𝑣G by 𝑣G := ∑ [e, e,̂ w(e)] . e∈E(G)

One can check that ∇ ⋅ 𝑣G = ∑e∈E(G) w(e)(δ e− − δ e+ ) ∈ M(K) and that it is a rectifiable irrigation flow on K. Also, both the cost Eα and the constraint G ∈ G(μ, ν) can be expressed solely in terms of 𝑣G . Indeed, if we identify 𝑣G with its H1 -density, one has Eα (G) ≐ ∑ w(e)α |e| = ∫ |𝑣G (x)| α H1 (dx) . e∈E(G)

K

And the Kirchhoff condition is expressed in terms of the divergence ∇ ⋅ 𝑣G : Proposition 3.2. If G is a graph, G ∈ G(μ, ν) if and only if ∇ ⋅ 𝑣G = μ − ν. This leads to defining the following cost on IF(K): { {∫ |𝑣(x)| α H1 (dx) M α (𝑣) = { { {+∞

if 𝑣 is rectifiable , otherwise ,

which is called the α-mass of 𝑣. Actually, Xia gave a different definition of Mα in [13], as a relaxation of the Eα functional: M α (𝑣) = inf lim inf Eα (G n ) , n N 𝑣G n 󳨀 ⇀𝑣 N

where 𝑣n 󳨀 ⇀ 𝑣 means that 𝑣n ⇀ 𝑣 and ∇ ⋅ 𝑣n ⇀ ∇ ⋅ 𝑣 weakly ⋆ as measures on K. These definitions coincide on IF(K) as shown in [14], and M α (𝑣) = ∫ |𝑣(x)|α H1 (dx) as soon

300 | 11 Equivalence between branched transport models as 𝑣 has an H1 -density. Finally, we say that 𝑣 sends μ to ν if ∇ ⋅ 𝑣 = μ − ν and denote by IF(μ, ν) the set of such irrigation flows. Eulerian irrigation problem We are now able to formulate an Eulerian irrigation problem in a continuous setting. Given two probability measures μ, ν ∈ Prob(K), we want to find an irrigation flow 𝑣 sending μ to ν which has minimal α-mass. This reads min

𝑣∈IF(μ,ν)

M α (𝑣) .

(EIα )

Xia proved the following theorem in [13]. Theorem 3.3 (Existence of minimizers). For any μ, ν ∈ Prob(K), there is a minimizer 𝑣 to the problem (EI α ). Moreover if 1 − 1d < α < 1 the minimum is always finite.

11.4 Equivalence between models In this section, we show that the Lagrangian and Eulerian irrigation problems are equivalent, in the sense that they have same minimal value, and one can build minimizers of one problem from minimizers of the other. In this section, we assume 1 − 1d < α < 1.

11.4.1 From Lagrangian to Eulerian Recall that we have associated to any irrigation plan η ∈ IP(K) an intensity i η ∈ M+ (K) and a flow 𝑣η ∈ Md (K). We will show that 𝑣η is an irrigation flow sending μ to ν and satisfying M α (𝑣η ) ≤ Iα (η) under some hypotheses. Proposition 4.1. If η ∈ IP(μ, ν) then 𝑣η ∈ IF(μ, ν). Proof. Let us calculate the distributional divergence of 𝑣η . For ϕ ∈ C(K), we have ∞

̇ dt η(dγ) ⟨∇ ⋅ 𝑣η , ϕ⟩ = −⟨𝑣η , ∇ϕ⟩ = − ∫ ∫ ∇ϕ(γ(t) ⋅ γ(t) Γ 0

= ∫ (ϕ(γ(0)) − ϕ(γ(∞))) η(dγ) Γ1

= ∫ ϕ(x)μ(dx) − ∫ ϕ(x)ν(dx) , K

K

thus ∇ ⋅ 𝑣η = μ − ν ∈ M(K), which implies that 𝑣η ∈ IF(μ, ν).

11.4 Equivalence between models |

301

Proposition 4.2. If η is an essentially simple and rectifiable irrigation plan, in particular, if η is optimal, we have M α (𝑣η ) ≤ Iα (η) . Proof. By Remark 2.7, we know that i η = m η H1 = θ η H1 . Since |𝑣η | ≤ i η = θ η H1 , 𝑣η has an H1 -density which is less than θ η . Using the energy formula, we have 󵄨 󵄨α Iα (η) = Eα (η) = ∫ θ αη (x) H1 (dx) ≥ ∫ 󵄨󵄨󵄨𝑣η (x)󵄨󵄨󵄨 H1 (dx) = M α (𝑣η ) . K

K

We have therefore proven that, we have inf M α ≤ inf Iα ,

IF(μ,ν)

IP(μ,ν)

and that if η is optimal, 𝑣η is a good optimal candidate for the Eulerian problem (EIα ).

11.4.2 From Eulerian to Lagrangian Given an irrigation flow 𝑣 ∈ IF(μ, ν) of finite cost M α , we would like to build an irrigation plan η ∈ IP(μ, ν) such that 𝑣 = 𝑣η (and whose cost is less than 𝑣). This is not true in general but a Smirnov decomposition gives the result if 𝑣 is optimal for (EIα ). Cycle If 𝑣 ∈ IF(K), we say that w ∈ IF(K) is a cycle of 𝑣 if |𝑣| = |w| + |𝑣 − w| and ∇ ⋅ w = 0. It is easy to check that if 𝑣 is rectifiable then w and 𝑣 − w are also rectifiable. The following Smirnov decomposition is proved by Santambrogio via a Dacorogna–Moser approach in [9]. Theorem 4.3 (Irrigation flow decomposition). Given an irrigation flow 𝑣 ∈ IF(μ, ν), there is an irrigation plan η ∈ IP(μ, ν) and a cycle w ∈ IF(K) satisfying (1) 𝑣 = 𝑣η + w, (2) i η ≤ |𝑣|. Corollary 4.4. If 𝑣 is an optimal irrigation flow in IF(μ, ν), there is an irrigation plan η ∈ IP(μ, ν) such that (1) 𝑣 = 𝑣η , (2) |𝑣η | = i η . Proof. Let us take 𝑣η , w as in the previous theorem. Since M α (𝑣) < ∞, 𝑣 and 𝑣η are rectifiable, and by optimality of 𝑣, one has α 󵄨 󵄨α 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑣η (x)󵄨󵄨󵄨 H1 (dx) = M α (𝑣η ) ≥ M α (𝑣) = ∫ (󵄨󵄨󵄨𝑣η (x)󵄨󵄨󵄨 + |w(x)|) H1 (dx) , K

K

302 | 11 Equivalence between branched transport models thus we must have |w(x)| = 0 H1 -a.e., which means w = 0, thus 𝑣 = 𝑣η and (i) holds. This implies |𝑣η | ≤ i η ≤ |𝑣| = |𝑣η | and thus we have the equality |𝑣η | = i η wanted in (ii). Proposition 4.5. If 𝑣 is an optimal irrigation flow in IF(μ, ν), one can find an irrigation plan η ∈ IP(μ, ν) such that Iα (η) ≤ M α (𝑣) . Proof. Take η as in the previous corollary. Since M α (𝑣) < ∞, 𝑣 is rectifiable and i η = |𝑣| is concentrated on a rectifiable set, which means by definition that η is rectifiable. As a consequence |𝑣| = i η = m η H1 and we have 󵄨 󵄨α M α (𝑣) = ∫ 󵄨󵄨󵄨𝑣η (x)󵄨󵄨󵄨 H1 (dx) = ∫ m αη (x) H1 (dx), K

K

while 1 Iα (η) = Ē α (η) = ∫ θ α−1 η (x)m η (x) H (dx) . K

We would like Iα (η) ≤ M α (𝑣), which is a priori not necessarily the case for the η we have constructed, since it is not necessarily essentially simple. Instead, take a simple replacement η̃ ∈ IP(μ, ν) satisfying m η̃ = θ η̃ ≤ θ η ≤ m η . Then we get Iα (η)̃ = ∫ θ αη̃ H1 (dx) ≤ ∫ m αη (x) H 1 (dx) = M α (𝑣) , K

K

which yields the result. Remark 4.6. Since the minima in the Eulerian and Lagrangian problems are actually the same as we shall see in Theorem 4.7, the previous inequality is an equality, which implies that θ η = θ η̃ = m η H1 -a.e. thus η was actually optimal hence simple.

11.4.3 The equivalence theorem We are now able to formulate the equivalence between the Lagrangian and Eulerian models. Theorem 4.7 (Equivalence theorem). If 1 − 1d < α < 1 and μ, ν ∈ Prob(K), the Eulerian problem (EI α ) and the Lagrangian problem (LI α ) are equivalent in the following sense: (1) the minima are the same min Iα (η) = min M α (𝑣) ,

η∈IP(μ,ν)

𝑣∈IF(μ,ν)

(2) if 𝑣 is optimal in IF(μ, ν), it can be represented by an optimal irrigation plan, i.e. 𝑣 = 𝑣η for some optimal η ∈ IP(μ, ν), (3) if η is optimal in IP(μ, ν), then 𝑣η is optimal in IF(μ, ν) and i η = |𝑣η |.

Bibliography

| 303

Proof. It all follows from Propositions 4.2 and 4.5. The equality i η = |𝑣η | comes from 󵄨 󵄨α Iα (η) = ∫ i αη (x) H1 (dx) ≥ ∫ 󵄨󵄨󵄨𝑣η (x)󵄨󵄨󵄨 H1 (dx) = M α (𝑣η ) , K

K

since we have equality everywhere by optimality of 𝑣η and η. Remark 4.8. Notice, in particular, that the equality i η = |𝑣η | implies that curves of η have the same tangent vectors when they coincide. To be more precise, there is an H1 -a.e. defined function τ : D η 󳨃→ 𝕊d−1 such that for η-a.e. γ ∈ Γ, for H1 -a.e. x ∈ γ, ̇ = |γ(t)|τ(x) ̇ γ(t) whenever γ(t) = x.

Bibliography [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

A. Brancolini and G. Buttazzo, Optimal networks for mass transportation problems, ESAIM: Control, Optimisation and Calculus of Variations, 11(1):88–101, 3 2010. L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations, Journal de mathématiques pures et appliquées, 93(6):652–671, 2010. M. Bernot, V. Caselles and J.-M. Morel, Optimal transportation networks, Springer, 2009. Y. Brenier, A modified least action principle allowing mass concentrations for the early universe reconstruction problem, Confluentes Mathematici, 3(03):361–385, 2011. G. Carlier, C. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and wardrop equilibria, SIAM Journal on Control and Optimization, 47(3):1330–1350, 2008. E.-N. Gilbert, Minimum cost communication networks, Bell System Tech. J. (46), 2209–2227, 1967. F. Maddalena, J.-M. Morel and S. Solimini, A variational model of irrigation patterns, Interfaces and Free Boundaries, 5(4):391–416, 2003. P. Mattila, Geometry of sets and measures in Euclidean spaces: fractals and rectifiability, Number 44. Cambridge University Press, 1999. F. Santambrogio, A Dacorogna–Moser approach to flow decomposition and minimal flow problems, ESAIM: Proceedings and Surveys, 45:265–274, 2014. F. Santambrogio, Optimal Transport for the Applied Mathematician, Birkhäuser, 2015. S.-K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz, 5(4):206–238, 1993. C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, AMS, 2003. Q. Xia, Optimal paths related to transport problems, Communications in Contemporary Mathematics, 5(02):251–279, 2003. Q. Xia, Interior regularity of optimal transport paths, Calculus of Variations and Partial Differential Equations, 20(3):283–299, 2004.

Maxime Laborde

12 On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows Abstract: This chapter presents existence and uniqueness results for a class of parabolic systems with nonlinear diffusion and nonlocal interaction. These systems can be viewed as regular perturbations of Wasserstein gradient flows. Here, we extend results known in the periodic case [1] to the whole space and on a smooth bounded domain. Existence is obtained using a semi-implicit Jordan–Kinderlehrer–Otto scheme and uniqueness follows from a displacement convexity argument. Keywords: nonlinear diffusion, systems, interacting species, Wasserstein gradient flows, semi-implicit JKO scheme, nonlinear parabolic equations. Mathematics Subject Classification 2010: 35K15, 35K40, 49J40.

12.1 Introduction In this chapter, we study existence and uniqueness of solutions for systems of the form { ∂ t ρ i − div(ρ i ∇(V i [ρ])) − α i div(ρ i ∇F 󸀠i (ρ i )) = 0 on ℝ+ × Ω , { ρ (0, ⋅) = ρ i,0 on Ω , { i

(1.1)

where i ∈ [[1, l]] (l ∈ ℕ∗ ), Ω = ℝn or is a bounded set of ℝn and ρ := (ρ 1 , . . . , ρ l ) is a collection of densities. Our motivation for this system comes from its appearance in modeling interacting species. In the case of ∇(V i [ρ]) = 0 or V i [ρ] does not depend on ρ, this system can be seen as a gradient flow in the product Wasserstein space i.e ∇F 󸀠i (ρ i ) can be seen as the first variation of a functional Fi defined on measures. This theory started with the work of Jordan et al. [2], where they discovered that the Fokker–Planck equation can be seen as the gradient flow of ∫ℝn ρ log ρ + ∫ℝn Vρ. The method that they used to prove this result is often called JKO scheme. Now, it is well-known that the gradient flow method permits to prove the existence of solution under very weak assumptions on the initial condition for several evolution equations, such as the heat equation [2], the porous media equation [3], degenerate parabolic equations [4], Keller–Segel equation [5]. The general theory of gradient flow has been very much developed and is detailed in the book of Ambrosio et al. [6], which is the main reference in this domain. However, this method is very restrictive if we want to treat the case of systems with several interaction potentials. Indeed, Di Francesco and Fagioli show in the first part Maxime Laborde, CEREMADE, UMR CNRS 7534, Université Paris IX Dauphine, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16, FRANCE, [email protected]

12.2 Wasserstein space and main result |

305

of [7] that we have to take the same (or proportional) interaction potentials, of the form V[ρ] = W ∗ ρ for all densities. They prove an existence/uniqueness result of (1.1) using gradient flow theory in a product Wasserstein space without diffusion (α i = 0) and with l = 2, V1 [ρ 1 , ρ 2 ] := W1,1 ∗ ρ 1 + W1,2 ∗ ρ 2 and V2 [ρ 1 , ρ 2 ] := W2,2 ∗ ρ 2 + W2,1 ∗ ρ 1 where W1,2 and W2,1 are proportional. Nevertheless in the second part of [7], they introduce a new semi-implicit JKO scheme to treat the case where W1,2 and W2,1 are not proportional. In other words, they use the usual JKO scheme freezing the measure in V i [ρ]. The purpose of this chapter is to add a nonlinear diffusion in the system studied in [7]. Unfortunately, this term requires strong convergence to pass to the limit. This can be obtained using an extension of Aubin–Lions lemma proved by Rossi and Savaré [8] and recalled in Theorem 5.3. This theorem requires separately timecompactness and space-compactness to obtain a strong convergence in L m ((0, T)×Ω). The time-compactness follows from classical estimate on the Wasserstein distance in the JKO scheme. The difficulty is to prove the space-compactness. This problem has already been solved in [1] in the periodic case using the semi-implicit scheme of [7]. In this chapter, we extend this result on the whole space ℝn or on a smooth bounded domain. On the one hand in ℝn , we will use the same argument than in [1]. We use the powerful flow-interchange argument of Matthes et al. [9] and also used in the work of Di Francesco and Matthes [10]. The differences with the periodic case are that functionals are not, a priori, bounded from below and we cannot use Sobolev compactness-embedding theorem. On the other hand in a bounded domain, the flow-interchange argument is very restrictive because it forces us to work in a convex domain and to impose some boundary condition on V i [ρ]. To avoid these assumptions, we establish a BV estimate to obtain compactness in space and then to find the strong convergence needed. This chapter is comprised of seven sections. In Section 12.2, we start by recalling some facts on the Wasserstein space and we state our main result, Theorem 2.3. Sections 12.3, 12.4 and 12.5 are devoted to prove Theorem 2.3. In Section 12.3, we introduce a semi-implict JKO scheme, as in [7], and resulting standard estimates. Then, in Section 12.4, we recall the flow-interchange theory developed in [9] and we find a stronger estimate on the solution’s gradient, which can be done by differentiating the energy along the heat flow. In Section 12.5, we establish convergence results and we prove Theorem 2.3. Section 12.6 deals with the case of a bounded domain. In the final Section 12.7, we show uniqueness of (1.1) using a displacement convexity argument.

12.2 Wasserstein space and main result Before stating the main theorem, we recall some facts on the Wasserstein distance.

306 | 12 On some nonlinear evolution systems

12.2.1 The Wasserstein distance We introduce { } P2 (ℝn ) := { μ ∈ M(ℝn ; ℝ+ ) : ∫ dμ = 1 and M(μ) := ∫ |x|2 dμ(x) < +∞} , ℝn ℝn { } and we note P2ac (ℝn ) the subset of P2 (ℝn ) of probability measures on ℝn absolutely continuous with respect to the Lebesgue measure. The Wasserstein distance of the order 2, W2 (ρ, μ), between ρ and μ in P2 (ℝn ), is defined by 1/2

W2 (ρ, μ) :=

inf

γ∈Π(ρ,μ)

( ∫ |x − y| dγ(x, y)) 2

,

ℝn ×ℝn

where Π(ρ, μ) is the set of probability measures on ℝn × ℝn , whose first marginal is ρ and second marginal is μ. It is well known that P2 (ℝn ) equipped with W2 defines a metric space (see, e.g., [11–13]). Moreover if ρ ∈ P2ac (ℝn ) then W2 (ρ, μ) admits a unique optimal transport plan γ T and this plan is induced by a transport map, i.e γ T = (Id × T)# ρ, where T is the gradient of a convex function (see [14]). Now if ρ, μ ∈ P2 (ℝn )l , we define the product distance by l

W2 (ρ, μ) := ∑ W2 (ρ i , μ i ) , i=1

or every equivalent metric as W2 (ρ, μ) := (∑li=1 W22 (ρ i , μ i ))1/2 . We can also define the 1-Wasserstein distance by W1 (ρ, μ) :=

inf

γ∈Π(ρ,μ)

( ∫ |x − y| dγ(x, y)) , ℝn ×ℝn

and the Kantorovich duality formula (see [11–13]) gives { } W1 (ρ, μ) = sup { ∫ φ d(ρ − μ) : φ ∈ L1 (d|ρ − μ|) ∩ Lip1 (ℝn )} , {ℝn } where Lip1 is the set of 1-Lipschitz continuous functions. Then for all ρ, μ ∈ P2 (ℝn ) and φ ∈ Lip(ℝn ), we have ∫ φ d(ρ − μ) ≤ CW1 (ρ, μ) ≤ CW2 (ρ, μ) . ℝn

(2.1)

12.2 Wasserstein space and main result | 307

12.2.2 Main result Let l ∈ ℕ∗ and for all i ∈ [[1, l]], we define V i : P(ℝn )l → C2 (ℝn ) continuous such that: –

For all ρ = (ρ 1 , . . . , ρ l ) ∈ P(ℝn )l , V i [ρ] ≥ 0 ,

–

(2.2)

There exists C > 0 such that for all ρ ∈ P(ℝn )l , ‖∇(V i [ρ])‖L∞ (ℝn ) + ‖D2 (V i [ρ])‖L∞ (ℝn ) ≤ C ,

(2.3)

i.e V i [ρ] and ∇(V i [ρ]) are Lipschitz functions and the Lipschitz constants do not depend on the measure. –

There exists C > 0 such that for all ν, σ ∈ P(ℝn )l , ‖∇(V i [ν]) − ∇(V i [σ])‖L∞ (ℝn ) ≤ CW2 (ν, σ) .

(2.4)

Remark 2.1. The assumption (2.2) can be replaced by V i [ρ] is bounded by below uniformly in ρ. Let m ≥ 1, we define the class of functions Hm by Hm := {x 󳨃→ x log(x)}

if m = 1 ,

and, if m > 1, Hm is the class of strictly convex superlinear functions F : ℝ+ → ℝ, which satisfy F(0) = F 󸀠 (0) = 0 ,

F 󸀠󸀠 (x) ≥ Cx m−2

and

P(x) := xF 󸀠 (x) − F(x) ≤ C(x + x m ) . (2.5)

The first two assumptions imply that if m > 1 and F ∈ Hm , then F controls x m . Before giving a definition of solution of (1.1), we recall that the nonlinear diffusion term can be rewrite as div(ρ∇F 󸀠 (ρ)) = ∆P(ρ) , where P(x) := xF 󸀠 (x) − F(x) is the pressure associated to F. Definition 2.2. We say that (ρ 1 , . . . , ρ l ) : [0, +∞[→ P2ac (ℝn )l is a weak solution of (1.1) if for all i ∈ [[1, l]], ρ i ∈ C([0, T], P2ac (ℝn )), P i (ρ i ) ∈ L1 (]0, T[×ℝn ) for all T < ∞ n and for all φ1 , . . . , φ l ∈ C∞ c ([0, +∞[×ℝ ), +∞

∫ ∫ [(∂ t φ i − ∇φ i ⋅ ∇(V i [ρ])) ρ i + α i ∆φ i P i (ρ i )] = − ∫ φ i (0, x)ρ i,0 (x) . 0 ℝn

ℝn

308 | 12 On some nonlinear evolution systems

With this definition of solution, we have the following result. Theorem 2.3. For all i ∈ [[1, l]], let F i ∈ Hm i , with m i ≥ 1, and V i satisfy (2.2), (2.3) and (2.4). Let α 1 , . . . , α l be the positive constants. If ρ i,0 ∈ P2ac (ℝn ) satisfy Fi (ρ i,0 ) + Vi (ρ i,0 |ρ0 ) < +∞ ,

(2.6)

with {∫ n F i (ρ(x)) dx Fi (ρ) := { ℝ +∞ {

if ρ ≪ L n , otherwise,

and

V i (ρ|μ) := ∫ V i [μ]ρ dx . ℝ

then there exist (ρ 1 , . . . , ρ l ) : [0, +∞[→ P2ac (ℝn )l , continuous with respect to W2 , weak solution of (1.1). Remark 2.4. In the following, to simplify the proof, we take α i = 1.

12.3 Semi-implicit JKO scheme In this section, we introduce the semi-implicit JKO scheme, as [7], and we find the first estimates as in the usual JKO scheme. Let h > 0 be a time step, we construct l sequences with the following iterative discrete scheme: for all i ∈ [[1, l]], ρ 0i,h = ρ i,0 and for all k ≥ 1, ρ ki,h minimizes 2 k−1 k−1 Ei,h (ρ|ρ k−1 h ) := W2 (ρ, ρ i,h ) + 2h (Fi (ρ) + V i (ρ|ρ h )) , k−1 = (ρ 1,h , . . . , ρ k−1 on ρ ∈ P2ac (ℝn ), with ρ k−1 h l,h ).

In the next proposition, we show that all these sequences are well defined. We start to prove that these are well defined for one step and after Remark 3.2, we extend the result for all k. Proposition 3.1. Let ρ 0 = (ρ 1,0 , . . . , ρ l,0 ) ∈ P2ac (ℝn )l , there exists a unique ρ 1h = (ρ 11,h , . . . , ρ 1l,h ) ∈ P2ac (ℝn )l such that, for all i ∈ [[1, l]], ρ 1i,h = argmin Ei,h (ρ|ρ 0h ) .

(3.1)

Proof. First of all, we distinguish the case m i > 1 from m i = 1. If m i > 1, then Ei,h (ρ|μ) ≥ 0, for all ρ, μ 1 , . . . , μ l ∈ P2ac (ℝn ). Let ρ ν be a minimizing sequence. As Ei,h (ρ i,0 |ρ0 ) < +∞ (according to (2.6)), (Ei,h (ρ ν |ρ 0 ))ν is bounded above. So there exists C > 0 such that 0 ≤ Fi (ρ ν ) ≤ C

and

W2 (ρ ν , ρ i,0 ) ≤ C .

309

12.3 Semi-implicit JKO scheme |

From the second inequality, it follows that the second moment of ρ ν is bounded. Now if m i = 1, following [2], we obtain Ei,h (ρ|ρ 0h ) ≥

1 1 M(ρ) − C(1 + M(ρ))α − M(ρ 0i,h ) , 4 2

(3.2)

with some 0 < α < 1. And since x 󳨃→ 14 x − C(1 + x)α is bounded below, we see that Ei,h is bounded below. Let ρ ν be a minimizing sequence. Then, we have (Fi (ρ ν ))ν bounded above. Indeed, as Ei,h (ρ i,0 |ρ0 ) < +∞, (Ei,h (ρ ν |ρ 0 ))ν is bounded above and from (2.2) we get, ∫ V i [ρ 0 ](x)ρ ν (x) dx ≥ 0 , ℝ

so (Fi (ρ ν ))ν is bounded above. According to (3.2), (M(ρ ν ))ν is bounded. Consequently, (Fi (ρ ν ))ν is bounded because Fi (ρ) ≥ −C(1 + M(ρ))α . In both cases, using Dunford–Pettis’ theorem, we deduce that there exists ρ 1i,h ∈ P2ac (ℝn ) such that ρ ν ⇀ ρ 1i,h weakly in L1 (ℝn ) . It remains to prove that ρ 1i,h is a solution for the minimization problem. But since Fi and W22 (⋅, ρ i,0 ) are weakly lower semicontinuous in L1 (ℝn ), we have Ei,h (ρ 1i,h |ρ0 ) ≤ lim inf Ei,h (ρ ν |ρ0 ) . ν↗+∞

To conclude the proof, we show that the minimizer is unique. This follows from the convexity of Vi (⋅|ρ 0 ) and ρ ∈ P2ac (ℝn ) 󳨃→ W22 (ρ, ρ 0i,h ) and the strict convexity of Fi . Remark 3.2. By induction, Proposition 3.1 is still true for all k ≥ 1: the proof is similar when we take k − 1 instead of 0 and if we notice that for all i, Fi (ρ 1i,h ) + Vi (ρ 1i,h |ρ 1h ) ≤ Fi (ρ i,0 ) + Vi (ρ i,0|ρ 0 ) + CW2 (ρ 0 , ρ1h ) ≤ C . The last inequality is obtained from the minimization scheme and from the assumptions (2.2), (2.4), and (2.6). By induction it becomes, for all k ≥ 2, k−1

j−1

j

k−1 k−1 Fi (ρ k−1 i,h ) + V i (ρ i,h |ρ h ) ≤ Fi (ρ i,0) + V i (ρ i,0 |ρ0 ) + C ∑ W2 (ρ h , ρ h ) ≤ C . j=1

This inequality shows previous proof.

k−1 Ei,h (ρ k−1 i,h |ρ h )

< +∞ and so we can bound (Fi (ρ ν ))ν in the

Thus, we proved that sequences (ρ ki,h )k≥0 are well defined for all i ∈ [[1, l]]. Then, we define the interpolation ρ i,h : ℝ+ → P2ac (ℝn ) by, for all k ∈ ℕ, ρ i,h (t) = ρ ki,h if t ∈ ((k − 1)h, kh] .

(3.3)

The following proposition shows that ρ i,h are solutions of a discrete approximation of the system (1.1).

310 | 12 On some nonlinear evolution systems Proposition 3.3. Let h > 0, for all T > 0, let N such that N = ⌈ Th ⌉. Then for all n l (ϕ1 , . . . , ϕ l ) ∈ C∞ c ([0, T) × ℝ ) and for all i ∈ [[1, l]], T

N−1

∫ ∫ ρ i,h (t, x)∂ t ϕ i (t, x) dx dt = −h ∑ ∫ P i (ρ k+1 i,h (x))∆ϕ i (t k , x) dx k=0 ℝn

0 ℝn

N−1

+ h ∑ ∫ ∇(V i [ρ kh ]) ⋅ ∇ϕ i (t k , x)ρ k+1 i,h (x) dx k=0 ℝn N−1

+ ∑

∫ R[ϕ i (t k , ⋅)](x, y) dγ ki,h (x, y)

k=0 ℝn ×ℝn

− ∫ ρ i,0 (x)ϕ i (0, x) dx , ℝn

where t k = hk (t N := T) and γ ki,h is the optimal transport plan in W2 (ρ ki,h , ρ k+1 i,h ). Moren over, R is defined such that, for all ϕ ∈ C∞ c ([0, T) × ℝ ), |R[ϕ](x, y)| ≤

1 2 ‖D ϕ‖L∞ ([0,T)×ℝn ) |x − y|2 . 2

Proof. We split the proof in two steps. We first compute the first variation of Ei,h (⋅|ρ kh ) and then we integrate in time. In the following, i is fixed in [[1, l]]. First step: For all k ≥ 0, if γ ki,h is the optimal transport plan in W2 (ρ ki,h , ρ k+1 i,h ) then k k+1 ∫ φ i (x)(ρ k+1 i,h (x) − ρ i,h (x)) = h ∫ P i (ρ i,h (x))∆φ i (x) dx ℝn

ℝn

− h ∫ ∇(V i [ρ kh ])(x) ⋅ ∇φ i (x)ρ k+1 i,h (x) dx ℝn

−

∫ R[φ i ](x, y) dγ ki,h (x, y) , ℝn ×ℝn

n for all φ i ∈ C∞ c (ℝ ).

To obtain this equality, we compute the first variation of Ei,h (⋅|ρ kh ). Let ξ i ∈ τ > 0 and let Ψ τ defined by

n n C∞ c (ℝ , ℝ ) and

∂τ Ψτ = ξi ∘ Ψτ ,

Ψ0 = Id .

k+1 k+1 After we perturb ρ k+1 i,h by ρ τ = (Ψ τ )♯ ρ i,h . According to the definition of ρ i,h , we get

1 k (Ei,h (ρ τ |ρ kh ) − Ei,h (ρ k+1 i,h |ρ h )) ≥ 0 . τ

(3.4)

By standard computations (see, for instance, [2, 4]) we have lim sup τ↘0

1 1 2 1 k ( W (ρ τ , ρ ki,h ) − W22 (ρ k+1 i,h , ρ i,h )) ≤ τ 2 2 2

∫ (y − x)⋅ξ i (y) dγ ki,h (x, y), (3.5) ℝn ×ℝn

12.3 Semi-implicit JKO scheme |

311

where γ ki,h is the optimal transport plan in W2 (ρ ki,h , ρ k+1 i,h ), lim sup τ↘0

1 k+1 (Fi (ρ τ ) − Fi (ρ k+1 i,h )) ≤ − ∫ P i (ρ i,h (x)) div(ξ i (x)) dx , τ

(3.6)

ℝn

and lim sup τ↘0

1 k k k+1 (Vi (ρ τ |ρ kh ) − V i (ρ k+1 i,h |ρ h )) ≤ ∫ ∇ (V i [ρ h ]) (x) ⋅ ξ i (x)ρ i,h (x) dx . τ

(3.7)

ℝn

If we combine (3.4)–(3.7), we obtain ∫ (y − x) ⋅ ξ i (y) dγ ki,h (x, y) + h ∫ ∇ (V i [ρ kh ]) (x) ⋅ ξ i (x)ρ k+1 i,h (x) dx ℝn ×ℝn

ℝn

− h ∫ P i (ρ k+1 i,h (x)) div(ξ i (x)) dx ≥ 0 . ℝn

And if we replace ξ i by −ξ i , this inequality becomes an equality. To conclude the first part, we choose ξ i = ∇φ i and we notice, using Taylor’s expansion, that φ i (x) − φ i (y) = ∇φ i (y) ⋅ (x − y) + R[φ i ](x, y) , with R[φ i ] satisfying |R[φ i ](x, y)| ≤

1 2 ‖D φ i ‖L∞ ([0,T)×ℝn ) |x − y|2 . 2

n l Second step: For all (ϕ1 , . . . , ϕ l ) ∈ C∞ c ([0, T) × ℝ ) , extended, for all i ∈ [[1, l]], by ϕ i (0, ⋅) on [−h, 0), then T

tk

N

∫ ∫ ρ i,h (t, x)∂ t ϕ i (t, x) dx dt = ∑ ∫ ∫ ρ ki,h (x)∂ t ϕ i (t, x) dx dt 0 ℝn

k=0 t

k−1

ℝn

N

= ∑ ∫ ρ ki,h (x)(ϕ i (t k , x) − ϕ i (t k−1 , x)) dx k=0 ℝn N−1

= ∑ ∫ ϕ i (t k , x)(ρ ki,h (x) − ρ k+1 i,h (x)) dx k=0 ℝn

− ∫ ρ i,0 (x)ϕ i (0, x) dx . ℝn

Using the first part with φ i = ϕ i (t k , ⋅), we obtain the desired equality. The last proposition of this section gives usual estimates in gradient flow theory.

312 | 12 On some nonlinear evolution systems Proposition 3.4. For all T < +∞ and for all i ∈ [[1, l]], there exists a constant C < +∞ such that for all k ∈ ℕ and for all h with kh ≤ T and let N = ⌈ Th ⌉, we have M(ρ ki,h ) ≤ C ,

(3.8)

Fi (ρ ki,h )

(3.9)

≤C,

N−1

∑ W22 (ρ ki,h , ρ k+1 i,h ) ≤ Ch .

(3.10)

k=0

Proof. The proof combines some techniques used in [2] and [7]. In the following, i is k fixed in [[1, l]]. As ρ k+1 i,h is optimal and ρ i,h is admissible, we have k k k Ei,h (ρ k+1 i,h |ρ h ) ≤ E i,h (ρ i,h |ρ h ) .

In other words, 1 2 k k+1 k k k k W (ρ , ρ k+1 ) + h (Fi (ρ k+1 i,h ) + V i (ρ i,h |ρ h )) ≤ h (Fi (ρ i,h ) + V i (ρ i,h |ρ h )) . 2 2 i,h i,h From (2.3), we know that V i [ρ] is a C-Lipschitz function where C does not depend on the measure. Hence, because of (2.1), we have k k+1 k Vi (ρ ki,h |ρ kh ) − Vi (ρ k+1 i,h |ρ h ) ≤ CW2 (ρ i,h , ρ i,h ) .

Using Young’s inequality, we obtain k 2 Vi (ρ ki,h |ρ kh ) − Vi (ρ k+1 i,h |ρ h ) ≤ C h +

1 2 k+1 k W (ρ , ρ i,h ) . 4h 2 i,h

It yields 1 2 k 2 2 W (ρ , ρ k+1 ) ≤ h (Fi (ρ ki,h ) − Fi (ρ k+1 i,h )) + C h . 4 2 i,h i,h Summing over k, we can assert that N−1

∑ k=0

(3.11)

N−1 1 2 k 2 W2 (ρ i,h , ρ k+1 ) ≤ h ∑ (Fi (ρ ki,h ) − Fi (ρ k+1 ( i,h i,h )) + C T) 4 k=0

≤ h (Fi (ρ i,0 ) − Fi (ρ Ni,h ) + C2 T) . But by assumption, Fi (ρ i,0 ) < +∞ and −Fi (ρ) ≤ C(1 + M(ρ))α , with 0 < α < 1, then N

1 2 k N α 2 W2 (ρ i,h , ρ k+1 i,h ) ≤ h (Fi (ρ i,0 ) + C(1 + M(ρ i,h )) + C T) . 4 k=1 ∑

Thus, we are reduced to prove (3.8). But M(ρ ki,h ) ≤ 2W22 (ρ ki,h , ρ i,0) + 2M(ρ i,0 ) k−1 m+1 ≤ 2k ∑ W22 (ρ m i,h , ρ i,h ) + 2M(ρ i,0 ) m=0

≤ 8kh (Fi (ρ i,0 ) + C(1 + M(ρ ki,h ))α + C2 T) + 2M(ρ i,0 ) ≤ 8T (Fi (ρ i,0 ) + C(1 + M(ρ ki,h ))α + C2 T) + 2M(ρ i,0 ) .

(3.12)

12.4 κ-flows and gradient estimate |

313

As α < 1, we get (3.8). The second line is obtained with the triangle inequality and Cauchy–Schwarz inequality, while the third line is obtained because of (3.12). So we have proved (3.8) and (3.10). To have (3.9), we just have to use (3.11) and to sum. This implies Fi (ρ ki,h ) ≤ Fi (ρ i,0 ) + C2 T , which proves the proposition.

12.4 κ-flows and gradient estimate Estimates of Proposition 3.4 permit to obtain weak convergence in L1 (see Proposition 5.1). Unfortunately, it is not enough to pass to the limit in the nonlinear diffusion term P i (ρ i,h ). In this section, we follow the general strategy developed in [9] and used m /2 in [10] and [1] to get an estimate on the gradient of ρ i,hi . This estimate will be used in Proposition 5.2 to have a strong convergence of ρ i,h in L m i (]0, T[×ℝn ). In the following, we are only interested by the case where m i > 1 because if m i = 1, P i (ρ i,h ) = ρ i,h and the weak convergence is enough to pass to the limit in Proposition 3.3. In the first part of this section, we recall the definition of κ-flows (or contractive gradient flow) and some results on the dissipation of Fi + V i then, in the second part, we use these results with the heat flow to find an estimate on the gradient.

12.4.1 κ-flows Definition 4.1. A semigroup S Ψ : ℝ+ × P2ac (ℝn ) → P2ac (ℝn ) is a κ-flow for the functional Ψ : P2ac (ℝn ) → ℝ ∪ {+∞} with respect to W2 if, for all ρ ∈ P2ac (ℝn ), the curve s 󳨃→ S sΨ [ρ] is absolutely continuous on ℝ+ , S0Ψ = Id and satisfies the evolution variational inequality (EVI) κ 1 d+ |σ=s W22 (S sΨ [ρ], ρ)̃ + W22 (S sΨ [ρ], ρ)̃ ≤ Ψ(ρ)̃ − Ψ(S sΨ [ρ]) , 2 dσ 2

(4.1)

for all s > 0 and for all ρ̃ ∈ P2ac (ℝn ) such that Ψ(ρ)̃ < +∞, where f(t + s) − f(t) d+ f(t) := lim sup . + dt s s→0 In [6], the authors showed that the fact a functional admits a κ-flow is equivalent to λ-displacement convexity (see Section 12.7 for definition). The next two lemmas give results on the variations of ρ ki,h along specific κ-flows and are extracted from [10]. The goal is to use them with the heat flow.

314 | 12 On some nonlinear evolution systems Lemma 4.2. Let Ψ : P2ac (ℝn ) → ℝ∪{+∞} l.s.c on P2ac (ℝn ) which possesses a κ-flot SΨ . Define the dissipation Di,Ψ along SΨ by Di,Ψ (ρ|μ) := lim sup s↘0

1 (Fi (ρ) − Fi (S sΨ [ρ]) + V i (ρ|μ) − V i (S sΨ [ρ]|μ)) s

for all ρ ∈ P2ac (ℝn ) and μ ∈ P2ac (ℝn )l . k If ρ k−1 i,h and ρ i,h are two consecutive steps of the semi-implicit JKO scheme, then k k k−1 Ψ (ρ k−1 i,h ) − Ψ (ρ i,h ) ≥ hD i,Ψ (ρ i,h |ρ h ) +

κ 2 k W (ρ , ρ k−1 ) . 2 2 i,h i,h

(4.2)

k−1 Proof. Since the result is trivial if Ψ(ρ k−1 i,h ) = +∞, we assume Ψ(ρ i,h ) < +∞. Thus we can use the EVI inequality (4.1) with ρ := ρ ki,h and ρ̃ := ρ k−1 i,h . We obtain s k Ψ (ρ k−1 i,h ) − Ψ (S Ψ [ρ i,h ]) ≥

1 d+ κ 2 s k k−1 |σ=s W22 (S sΨ [ρ ki,h ], ρ k−1 i,h ) + 2 W2 (S Ψ [ρ i,h ], ρ i,h ) . 2 dσ

By lower semicontinuity of Ψ, we have k k−1 s k Ψ(ρ k−1 i,h ) − Ψ(ρ i,h ) ≥ Ψ(ρ i,h ) − lim inf Ψ(S Ψ [ρ i,h ]) s↘0

s k ≥ lim sup (Ψ(ρ k−1 i,h ) − Ψ(S Ψ (ρ i,h ))) s↘0

≥ lim sup ( s↘0

+

1 d+ |σ=s W22 (S sΨ [ρ ki,h ], ρ k−1 i,h )) 2 dσ

κ 2 k W (ρ , ρ k−1 ) . 2 2 i,h i,h

The last line is obtained thanks to the W2 -continuity of s 󳨃→ S sΨ [ρ ki,h ] in s = 0. Moreover, the absolute continuity of s 󳨃→ S sΨ [ρ ki,h ] implies lim sup ( s↘0

1 d+ |σ=s W22 (S sΨ [ρ ki,h ], ρ k−1 i,h )) 2 dσ 1 2 k k−1 ≥ lim sup (W22 (S sΨ [ρ ki,h ], ρ k−1 i,h ) − W2 (ρ i,h , ρ i,h )) . 2s s↘0

But since ρ ki,h minimizes Ei,h (⋅|ρ k−1 h ), we get, for all s ≥ 0, 2 k k−1 k s k W22 (S sΨ [ρ ki,h ], ρ k−1 i,h ) − W2 (ρ i,h , ρ i,h ) ≥ 2h (Fi (ρ i,h ) − Fi (S Ψ [ρ i,h ])) s k k−1 + 2h (Vi (ρ ki,h |ρ k−1 h ) − V i (S Ψ [ρ i,h ] |ρ h )) .

This concludes the proof. Corollary 4.3. Under the same hypotheses as in Lemma 4.2, let S Ψ a κ-flow such that, for all k ∈ ℕ, the curve s 󳨃→ SsΨ [ρ ki,h ] lies in L1 (ℝn ). Moreover, assume that s 󳨃→ F i (S sΨ [ρ ki,h ]) is differentiable for s > 0 and is continuous at s = 0 in L1 (ℝn ).

315

12.4 κ-flows and gradient estimate |

d (Fi (S σΨ [ρ ki,h ]) + Vi (S σΨ [ρ ki,h ]|ρ k−1 In addition, we assume that the family − dσ h )) is |σ=s bounded from below by an integrable function as s goes to 0 and let Ki,Ψ : P2ac (ℝn ) 󳨃→ ] − ∞, +∞] be a functional such that

lim inf (− s↘0

d k k−1 (Fi (S σΨ [ρ ki,h ]) + Vi (SσΨ [ρ ki,h ]|ρ k−1 h ))) ≥ K i,Ψ (ρ i,h |ρ h ) . dσ |σ=s

(4.3)

Then, for all k ∈ ℕ, k k k−1 Ψ (ρ k−1 i,h ) − Ψ (ρ i,h ) ≥ hK i,Ψ (ρ i,h |ρ h ) +

κ 2 k W (ρ , ρ k−1 ) . 2 2 i,h i,h

(4.4)

k−1 Proof. It is sufficient to show that Di,Ψ (⋅|ρ k−1 h ) is bounded below by K i,Ψ (⋅|ρ h ). The 1 proof is as in Corollary 4.3 of [10]. The hypotheses of L -regularity on F i (S sΨ [ρ ki,h ])

imply that s 󳨃→ Fi (S sΨ [ρ ki,h ]) is differentiable for s > 0 and continuous at s = 0. We have the same regularity for s 󳨃→ Vi (S sΨ [ρ ki,h ]|ρ k−1 h ). By the fundamental theorem of calculus, Di,Ψ (ρ ki,h |ρ k−1 h ) = lim sup s↘0

1 (Fi (ρ ki,h ) − Fi (S sψ [ρ ki,h ]) s

s k k−1 +Vi (ρ ki,h |ρ k−1 h ) − V i (S Ψ [ρ i,h ]|ρ h )) 1

= lim sup ∫ (− s↘0

0

1

≥ ∫ lim inf (− s↘0

0

d (Fi (S σΨ [ρ ki,h ]) + Vi (S σΨ [ρ ki,h ]|ρ k−1 h ))) dz dσ |σ=sz

d (Fi (S σΨ [ρ ki,h ]) + Vi (S σΨ [ρ ki,h ]|ρ k−1 h ))) dz dσ |σ=sz

≥ Ki,Ψ (ρ ki,h |ρ k−1 h ). The last line is obtained by Fatou’s lemma and assumption (4.3). To conclude, we apply Lemma 4.2.

12.4.2 Gradient estimate Proposition 4.4. For all i ∈ [[1, l]] such that m i > 1, there exists a constant C which depends only on ρ i,0 such that m /2

‖ρ i,hi ‖L2 ([0,T];H 1(ℝn )) ≤ C(1 + T) for all T > 0. Before starting the proof of Proposition 4.4, we recall the definition of the Entropy functional, E(ρ) = ∫ ρ log ρ, for all ρ ∈ Pac (ℝn ) . ℝn

316 | 12 On some nonlinear evolution systems We know that this functional possesses a κ-flow, with κ = 0 which is given by the heat semigroup (see, for instance [2, 15], or [12]). In other words, for a given η0 ∈ P2ac (ℝn ), the curve s 󳨃→ η(s) := S sE [η0 ] solves {∂ s η = ∆η { η(0) = η 0 {

on ℝ+ × ℝn , on ℝn ,

in the classical sense. η(s) is a positive density for all s > 0 and is continuously differentiable as a map from ℝ+ to C∞ ∩ L1 (ℝn ). Moreover, if η0 ∈ L m (ℝn ), then η(s) converges to η 0 in L m (ℝn ) when s ↘ 0. Proof. Based on the facts set out above, SE satisfies the hypotheses of Corollary 4.3. We just have to define a suitable lower bound Ki,E to use it. The spatial regularity of η(s) for all s > 0 allows the following calculations. Thus for all μ ∈ P2ac (ℝn )l , we have ∂ s (Fi (S sE [η0 ]) + Vi (S sE [η0 ]|μ)) = ∫ ∂ s F i (η) dx + ∫ V i [μ]∂ s η(s, x) dx ℝn

ℝn

= ∫ F 󸀠i (η(s, x))∆η(s, x) dx + ∫ V i [μ]∆η(s, x) dx ℝn

ℝn

2 = − ∫ F 󸀠󸀠 i (η(s, x))|∇η(s, x)| dx + ∫ ∆(V i [μ])η(s, x) dx . ℝn

ℝn

m i −2 thus According to (2.5), F 󸀠󸀠 i (x) ≥ Cx

∂ s (Fi (S sE [η0 ])+Vi (S sE [η0 ]|μ)) ≤ −C ∫ η(s, x)m i −2 |∇η(s, x)|2 dx + ∫ ∆(V i [μ])η(s, x) dx ℝn

ℝn

≤ −C ∫ |∇η(s, x)

m i /2 2

| dx + ∫ ∆(V i [μ])η(s, x) dx .

ℝn

ℝn

Since (2.3), we obtain a lower bound on the family −∂ s (Fi (S sE [η0 ])+V i (S sE [η0 ]|μ)). Indeed, −∂ s (Fi (S sE [η0 ]) + Vi (S sE [η0 ]|μ)) ≥ C ∫ |∇η(s, x)m i /2 |2 dx − ∫ ∆(V i [μ])η(s, x) dx , ℝn

ℝn

≥ −C because ‖S sE [η0 ]‖L1 (ℝn ) = 1. Then, we define Ki,E (ρ|μ) := C ∫ |∇(ρ(x)m i /2 )|2 dx − ∫ ∆(V[μ])ρ(x) dx . ℝn

ℝn

12.5 Passage to the limit

| 317

We shall now establish that Ki,E satisfies (4.3). First of all, we notice that d (Fi (S σE [ρ ki,h ]) + Vi (S σE [ρ ki,h ] |ρ k−1 h ))) dσ |σ=s d d Fi (S σE [ρ ki,h ])) + lim inf (− Vi (S σE [ρ ki,h ] |ρ k−1 ≥ lim inf (− h )) . s↘0 s↘0 dσ |σ=s dσ |σ=s

lim inf (− s↘0

(4.5)

Thanks to the proof of Lemma 4.4 and with Lemma A.1 of [10], we obtain lim inf (− s↘0

d (Fi (S σE [ρ ki,h ])) ≥ C ∫ |∇(ρ ki,h (x)m i /2 )|2 dx . dσ |σ=s

(4.6)

ℝn

Moreover, as S sE is continuous in L1 (ℝn ) at s = 0 and according to (2.3), lim inf (− s↘0

d k−1 k Vi (S σE [ρ ki,h ]|ρ k−1 h )) ≥ − ∫ ∆(V i [ρ h ])ρ i,h (x) dx . dσ |σ=s

(4.7)

ℝn

The combination of (4.5), (4.6), and (4.7) gives (4.3) for Ki,E . We apply Corollary 4.3 and get k k k−1 E(ρ k−1 (4.8) i,h ) − E(ρ i,h ) ≥ hK i,E (ρ i,h |ρ h ) . But since ∆(V i [ρ]) ∈ L∞ (ℝn ) uniformly on ρ (2.3), k Ch ∫ |∇(ρ ki,h (x)m i /2 )|2 dx ≤ E(ρ k−1 i,h ) − E(ρ i,h ) + Ch . ℝn

Now we sum on k from 1 to N = ⌈ Th ⌉ N

Ch ∑ ‖∇(ρ ki,h (x)m i /2 )‖2L2 (ℝn ) ≤ E(ρ i,0 ) − E(ρ Ni,h ) + CT .

(4.9)

k=1

According to [2] and [10], there exists a constant C > 0 and 0 < α < 1 such that for all ρ ∈ P2ac (ℝn ), −C(1 + M(ρ))α ≤ E(ρ) ≤ CFi (ρ) . Since for all k, h, M(ρ ki,h ) is bounded, according to (3.8) and the fact that Fi (ρ i,0 ) < +∞ by (2.6), we have N 󵄩 󵄩2 h ∑ 󵄩󵄩󵄩󵄩∇(ρ ki,h (x)m i /2 )󵄩󵄩󵄩󵄩 L2 (ℝn ) ≤ C(1 + T) . k=1

To conclude the proof, we use (2.5) and (3.9).

12.5 Passage to the limit In this section, we establish weak and strong convergences for sequences (ρ i,h ), in order to pass to the limit in the discrete system of Proposition 3.3.

318 | 12 On some nonlinear evolution systems

12.5.1 Weak and strong convergences The first convergence result is obtained using the estimates on the distance (3.10) and on the energy Fi (3.9). Proposition 5.1. Every sequences (h k )k∈ℕ of time steps which tends to 0 contains a subsequence, nonrelabeled, such that ρ i,h k converges, uniformly on compact time intervals, in W2 to a 12 -Hölder function ρ i : [0, +∞[→ P2ac (ℝn ). Proof. The estimation on the sum of distances (3.10) gives us for all t, s ≥ 0, W2 (ρ i,h (t, ⋅), ρ i,h (s, ⋅)) ≤ C(|t − s| + h)1/2 , with C independ of h. According to Proposition 3.3.1 of [6] and using a diagonal argument, at least for a subsequence, for all i, ρ i,h k converges uniformly on compact time intervals in W2 to a 12 -Hölder function ρ i : [0, +∞[→ P2 (ℝn ). To conclude, we show that for all t ≥ 0, ρ(t, ⋅) ∈ P2ac (ℝn ). But as F i is superlinear, Dunford–Pettis’ theorem completes the proof. With the previous proposition, we can pass to the limit in the case m i = 1 because P i (ρ i,h ) = ρ i,h and the term ∇(V i [ρ h ]) is controlled thanks to the hypothesis (2.4). Unfortunately, it is not enough to pass to the limit in P i (ρ i,h ) when m i > 1. In the next proposition, we use Proposition 4.4 to get a stronger convergence. Proposition 5.2. For all i ∈ [[1, l]] such that m i > 1, ρ i,h converges to ρ i in L m i (]0, T[×ℝn ) and P i (ρ i,h ) converges to P i (ρ i ) in L1 (]0, T[×ℝn ), for all T > 0. The proof of this proposition is obtained by using an extension of Aubin–Lions lemma given by Rossi and Savaré in [8] (Theorem 2) and recalled in [10] (Theorem 4.9). Theorem 5.3 (Theorem 2 in [8]). On a Banach space X, let be given – a normal coercive integrand G : X → ℝ+ , i.e, G is l.s.c and its sublevels are relatively compact in X, – a pseudodistance g : X × X → [0, +∞], i.e, g is l.s.c and [g(ρ, μ) = 0, ρ, μ ∈ X with G(ρ), G(μ) < ∞] ⇒ ρ = μ. Let U be a set of measurable functions u : ]0, T[→ X with a fixed T > 0. Under the hypotheses that T

sup ∫ G(u(t)) dt < +∞ u∈U

0

T−h

and

lim sup ∫ g(u(t + h), u(t)) dt = 0 , h↘0 u∈U

(5.1)

0

U contains a subsequence (u n )n∈ℕ which converges in measure with respect to t ∈]0, T[ to a limit u ⋆ : ]0, T[→ X.

12.5 Passage to the limit

| 319

To apply this theorem, we define on X := L m i (ℝn ), as in [10], g by {W2 (ρ, μ) g(ρ, μ) := { +∞ {

if ρ, μ ∈ P2 (ℝn ) , otherwise ,

and Gi by {‖ρ m i /2 ‖H 1 (ℝn ) + M(ρ) if ρ ∈ P2ac (ℝn ) and ρ m i /2 ∈ H 1 (ℝn ) , Gi (ρ) := { +∞ otherwise . { Now, we show that Gi satisfies Theorem 5.3 conditions. Lemma 5.4. For all i ∈ [[1, l]] such that m i > 1, Gi is l.s.c and its sublevels are relatively compact in L m i (ℝn ). Proof. The l.s.c of Gi on L m i (ℝn ) follows from Lemma A.1 in [10]. To complete the proof, we have to show that sublevels A c := {ρ ∈ L m i (ℝn ) | Gi (ρ) ≤ c} of Gi are relatively compact in L m i (ℝn ). To do this, we prove that B c := {η = ρ m i /2 | ρ ∈ A c } is relatively compact in L2 (ℝn ) and since the map j : L2 (ℝn ) → L m i (ℝn ), with j(η) = η 2/m i , is continuous, A c = j(B c ) will be relatively compact in L m i (ℝn ). We want to apply the Fréchet–Kolmogorov theorem to show that B c is relatively compact in L2 (ℝn ). B c is bounded in L2 (ℝn ): Since η 2 = ρ m i with Gi (ρ) ≤ c, it is straightforward to see ∫ η2 ≤ c . ℝn

B c is tight under translations: for every η ∈ B c and h ∈ ℝn , we have that 󵄨󵄨 1 󵄨󵄨2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ∫ |η(x + h) − η(x)| dx ≤ |h| ∫ 󵄨󵄨∫ |∇η(x + zh)| dz󵄨󵄨󵄨 dx 󵄨󵄨 󵄨󵄨 󵄨󵄨 ℝn ℝn 󵄨󵄨 0 2

2

≤ |h|2 ∫ |∇η(x)|2 dx ≤ c|h|2 , ℝn

thus the left-hand side converges to 0 uniformly on B c as |h| ↘ 0. Elements of B c are uniformly decaying at infinity: For all η ∈ B c and R > 0, we have ∫ η2 dx ≤

1 ∫ |x|1/n η1/nm i η2−1/nm i dx . R1/n n ℝ

|x|>R

If we use Hölder inequality with p = 2n and q =

2n 2n−1 ,

we get 2n−1 2n

1/2n 2

∫ η dx ≤ |x|>R

1 R1/n

2 2/m i

( ∫ |x| η ℝn

)

(∫ η ℝn

2(2m i −1/n)/m i (2−1/n)

)

.

320 | 12 On some nonlinear evolution systems As η2/m i = ρ with Gi (ρ) ≤ c, we have ∫ |x|2 η2/m i ≤ c . ℝn

To bound the other term, we use the Gagliardo–Nirenberg inequality: for 1 ≤ q, r ≤ +∞, we have ‖u‖L p ≤ C‖∇u‖αL r ‖u‖1−α Lq , for all 0 < α < 1 and for p given by 1 1 1 1 = α ( − ) + (1 − α) . p r n q i −1/n) We choose p = 2(2m m i (2−1/n) , q = r = 2, and α = 0 < α < 1) then we obtain

m i −1 2(2m i −1/n)

(1−α)p/2

αp/2

∫η

2(2m i −1/n)/m i (2−1/n)

ℝn

2

≤ ( ∫ |∇η| )

(since m i > 1, we have

2

(∫ η )

ℝn

,

ℝn

but since η = ρ m i /2 with Gi (ρ) ≤ c, the second term is bounded then ∫ η2 dx ≤ |x|>R

C →0, R1/n

as R goes to +∞. We conclude thanks to Fréchet–Kolmogorov theorem. Proof of Proposition 5.2. We want to apply Theorem 5.3 with X := L m i (ℝn ), G := Gi , g and U := {ρ i,h k | k ∈ ℕ}. According to Lemma 5.4, Gi satisfies the hypotheses of the theorem. It is obvious that it is the same for g. Thus, we only have to check conditions for usU. The first condition is satisfied because of (3.8) and Proposition 4.4 and the second is satisfied because of (3.10) (the proof is done in [10] Proposition 4.8, for example). According to Theorem 5.3 and using a diagonal argument, there exists a subsequence, not-relabeled, such that for all i with m i > 1, there exists ρ̃ i : ]0, T[→ L m i (ℝn ) such that ρ i,h k converges in measure with respect to t in L m i (ℝn ) to ρ̃ i . Moreover, as ρ i,h k (t) converges in W2 for all t ∈ [0, T] to ρ i (t) (Proposition 5.1) then ρ̃ i = ρ i . Now since convergence in measure implies a.e convergence up to a subsequence, we may also assume that ρ i,h k (t) converges strongly in L m i (ℝn ) to ρ i (t) t-a.e. Now, thanks to and (2.5) and (3.9), we have m

∫ ρ i,hi (t, x) dx ≤ CFi (ρ i,h (t, ⋅)) ≤ C , ℝn

then Lebesgue’s dominated convergence theorem implies that ρ i,h k converges strongly in L m i (]0, T[×ℝn ) to ρ i .

12.5 Passage to the limit

| 321

To conclude the proof, we have to show that P i (ρ i,h ) converges to P i (ρ i ) in L1 (]0, T[×ℝn ). First of all, up to a subsequence, we may assume that there exists g ∈ L m i (]0, T[×ℝn ) such that ρ i,h k → ρ i (t, x)-a.e.

and

ρ i,h k ≤ g (t, x)-a.e .

Thus according to (2.5) P i (ρ i,h k ) → P i (ρ i ) (t, x)-a.e.

and 0 ≤ P(ρ i,h k ) ≤ C(ρ i,h k + g m i ) (t, x)-a.e .

So when we pass to the limit we have (t, x)-a.e 0 ≤ P(ρ i ) ≤ C(ρ i + g m i ) ∈ L1 (]0, T[×ℝn ) . Then C(ρ i,h k + ρ i + 2g m i ) − |P i (ρ i,h k ) − P i (ρ i )| ≥ 0 and using the a.e convergence of ρ i,h k and P i (ρ i,h k ), 2CT + 2C

g(x)m i dx dt

∬ ]0,T[×ℝn

=

∬

lim inf (C(ρ i,h k + ρ i + 2g m i ) − |P i (ρ i,h k ) − P i (ρ i )|)

]0,T[×ℝn

≤ 2CT + 2C

∬

g m i (x) dx dt − lim sup

]0,T[×ℝn

∬

|P i (ρ i,h k ) − P i (ρ i )| .

]0,T[×ℝn

To do these computations, we used that ‖ρ i,h k ‖L1 (]0,T[×ℝn ) = ‖ρ i ‖L1 (]0,T[×ℝn ) = T and Fatou’s lemma. Since g ∈ L m i (]0, T[×ℝn ), we obtain lim sup

∬

|P i (ρ i,h k ) − P i (ρ i )| ≤ 0 ,

]0,T[×ℝn

which concludes the proof.

12.5.2 Limit of the discrete system In this section, we pass to the limit in the discrete system of Proposition 3.3. In the T n following, we consider ϕ i ∈ C∞ c ([0, T) × ℝ ) and N = ⌈ h ⌉. Proof of Theorem 2.3. We will pass to the limit in all terms in Proposition 3.3. Convergence of the remainder term: By definition of R, we have ∫ R[ϕ i (t k , ⋅)](x, y) dγ ki,h (x, y) ≤ ℝn ×ℝn

1 2 ‖∇ ϕ i ‖L∞ ([0,T]×ℝn ) W22 (ρ ki,h , ρ k+1 i,h ) . 2

322 | 12 On some nonlinear evolution systems

and according to the estimate (3.10), we get 󵄨󵄨N−1 󵄨󵄨 N−1 󵄨󵄨󵄨 󵄨󵄨 k 󵄨󵄨 ∑ ∫ R[ϕ i (t k , ⋅)](x, y) dγ (x, y)󵄨󵄨󵄨 ≤ C ∑ W 2 (ρ k , ρ k+1 ) ≤ Ch → 0 . 2 i,h 󵄨󵄨 󵄨󵄨 i,h i,h 󵄨󵄨 k=0 n n 󵄨󵄨 k=0 󵄨 ℝ ×ℝ 󵄨 Convergence of the linear term: 󵄨󵄨 󵄨󵄨 T T 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨∫ ∫ ρ i,h (t, x)∂ t ϕ i (t, x) dx dt − ∫ ∫ ρ i (t, x)∂ t ϕ i (t, x) dx dt󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 0 ℝ 0 ℝn ≤ CT sup W2 (ρ i,h (t, ⋅), ρ i (t, ⋅)) → 0 , t∈[0,T]

when h ↘ 0 because of (2.1) and Proposition 5.1. Convergence of the diffusion term: 󵄨󵄨 󵄨󵄨 T 󵄨󵄨 N−1 󵄨󵄨 󵄨󵄨 󵄨 k+1 󵄨󵄨h ∑ ∫ P i (ρ i,h (x)) ⋅ ∆ϕ i (t k , x) dx − ∫ ∫ P i (ρ i (t, x))∆ϕ i (t, x) dx dt󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k=0 ℝn 󵄨󵄨 0 ℝn 󵄨 󵄨 󵄨󵄨 T 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 3 ≤ C(1 + T)‖D ϕ i ‖L∞ h + 󵄨󵄨󵄨∫ ∫ (P i (ρ i,h (t, x)) − P i (ρ(t, x))) ∆ϕ i (t, x) dx dt󵄨󵄨󵄨󵄨 . 󵄨󵄨 n 󵄨󵄨 󵄨󵄨0 ℝ 󵄨󵄨 If m i = 1, the right-hand side converges to 0 because of Proposition 5.1 and otherwise it goes to 0 because of Proposition 5.2. Convergence of the interaction term: 󵄨󵄨 N−1 󵄨󵄨 󵄨󵄨 󵄨󵄨h ∑ ∫ ∇(V i [ρ kh ])(x) ⋅ ∇ϕ i (t k , x)ρ k+1 i,h (x) dx 󵄨󵄨 󵄨󵄨 k=0 ℝn

󵄨󵄨 󵄨󵄨 󵄨 − ∫ ∫ ∇(V i [ρ(t, ⋅)])(x) ⋅ ∇ϕ i (t, x)ρ i (t, x) dx dt󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 0 ℝn T

󵄨󵄨 N−1 󵄨󵄨 󵄨 ≤ 󵄨󵄨󵄨h ∑ ∫ ∇(V i [ρ kh ])(x) ⋅ ∇ϕ i (t k , x)ρ k+1 i,h (x) dx 󵄨󵄨 󵄨󵄨 k=0 ℝn N−1

t k+1

− ∑ ∫ ∫ ∇(V i [ρ(t, ⋅)])(x) ⋅ k=0 t ℝn k

󵄨󵄨 󵄨󵄨

󵄨󵄨 󵄨 ∇ϕ i (t k , x)ρ k+1 i,h (x) dx dt󵄨󵄨 󵄨󵄨 󵄨󵄨

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨N−1 t k+1 󵄨 󵄨󵄨 k+1 + 󵄨󵄨󵄨 ∑ ∫ ∫ ∇(V i [ρ(t, ⋅)])(x) ⋅ (∇ϕ i (t k , x) − ∇ϕ i (t, x))ρ i,h (x) dx dt󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 k=0 󵄨󵄨 󵄨󵄨 t k ℝn 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨N−1 t k+1 󵄨󵄨 󵄨 󵄨󵄨 (x) − ρ (t, x)) dx dt + 󵄨󵄨󵄨󵄨 ∑ ∫ ∫ ∇(V i [ρ(t, ⋅)])(x) ⋅ ∇ϕ i (t, x)(ρ k+1 i 󵄨󵄨 i,h 󵄨󵄨 󵄨󵄨 k=0 󵄨󵄨 󵄨󵄨 t k ℝn ≤ J1 + J2 + J3 .

12.5 Passage to the limit

| 323

As ρ i,h converges weakly L1 (]0, T[×ℝn ) to ρ i and (∇(V i [ρ]) ⋅ ∇ϕ i ) ∈ L∞ ([0, T] × ℝn ), then J 3 → 0 as h → 0 . For J 2 , we use the fact that ∇ϕ i is a Lipschitz function and that ∇(V i [ρ]) is bounded thanks to (2.3), and then, J 2 ≤ CT‖D2 ϕ i ‖L∞ ([0,T]×ℝn ) h → 0 . Using assumption (2.4), we have N−1

t k+1

J 1 ≤ C‖∇ϕ i ‖L∞ ([0,T]×ℝn ) ∑ ∫ W2 (ρ kh , ρ(t, ⋅)) dt k=0 t k

Then using triangle inequality and Cauchy–Schwarz inequality, we obtain t k+1

N−1

k+1 J 1 ≤ C‖∇ϕ i ‖L∞ ([0,T]×ℝn ) ∑ ∫ (W2 (ρ kh , ρ k+1 h ) + W2 (ρ h , ρ(t, ⋅))) k=0 t k N−1

≤ C‖∇ϕ i ‖L∞ ([0,T]×ℝn ) (h ∑

N−1

W2 (ρ kh , ρ k+1 h )

+ ∑ ∫ W2 (ρ k+1 h , ρ(t, ⋅)) dt)

k=0

k=0 t k T

N−1

≤ C‖∇ϕ i ‖L∞ ([0,T]×ℝn ) (T ∑

t k+1

W22 (ρ kh , ρ k+1 h )

+ ∫ W2 (ρ h (t, ⋅), ρ(t, ⋅)) dt)

k=0

0

According to (3.10), we obtain N−1

T ∑ W22 (ρ kh , ρ k+1 h ) ≤ CTh → 0 k=0

when h ↘ 0. Moreover, T

∫ W2 (ρ h (t, ⋅), ρ(t, ⋅)) dt ≤ T sup W2 (ρ h (t, ⋅), ρ(t, ⋅)) → 0 , t∈[0,T]

0

when h goes to 0, which proves that J 1 → 0 as h → 0 . If we combine all these convergences, Theorem 2.3 is proved.

324 | 12 On some nonlinear evolution systems

12.6 The case of a bounded domain Ω In this section, we work on a smooth bounded domain Ω of ℝn and only with one density but, as in the whole space, the result readily extends to systems. Our aim is to solve (1.1). We remark that Ω is not taken convex so we cannot use the flow-interchange argument anymore because this argument uses the displacement convexity of the Entropy. Moreover since Ω is bounded, the solution has to satisfy some boundary conditions contrary to the periodic case [1] or in ℝn . In our case, we study (1.1) with no flux boundary condition, which is the natural boundary condition for gradient flows, i.e, we want to solve ∂ t ρ − div(ρ∇(V[ρ])) − ∆P(ρ) = 0 on ℝ+ × Ω , { { { (6.1) (ρ∇(V[ρ]) + ∇P(ρ)) ⋅ ν = 0 on ℝ+ × ∂Ω , { { { n on ℝ , { ρ(0, ⋅) = ρ 0 where ν is the outward unit normal to ∂Ω. We say that ρ : [0, +∞[→ Pac (Ω) is a weak solution of (6.1), with F ∈ Hm , if ρ ∈ C([0, +∞[; Pac (Ω)) ∩ L m (]0, T[×Ω), P(ρ) ∈ L1 (]0, T[×Ω), ∇P(ρ) ∈ Mn ([0, T] × Ω) n for all T < ∞ and if for all φ ∈ C∞ c ([0, +∞[×ℝ ), we have ∞

∫ ∫ [(∂ t φ − ∇φ ⋅ ∇(V[ρ])) ρ − ∇P(ρ) ⋅ ∇φ] = − ∫ φ(0, x)ρ 0 (x) . 0 Ω

Ω n C∞ c ([0, +∞[×ℝ ),

we do not impose that they vanish on Since test functions are in the boundary of Ω, which give Neumann boundary condition. Theorem 6.1. Let F ∈ Hm for m ≥ 1 and let V satisfies (2.2)–(2.4). If we assume that ρ 0 ∈ Pac (Ω) satisfies F(ρ 0 ) + V(ρ 0 |ρ 0 ) < +∞ , (6.2) with {∫ F(ρ(x)) dx F(ρ) := { Ω +∞ {

n if ρ ≪ L|Ω ,

otherwise,

and

V(ρ|μ) := ∫ V[μ]ρ dx . Ω

then (6.1) admits at least one weak solution. The proof of this theorem is different from the one on ℝn because we will not use the flow-interchange argument of Matthes, McCann, and Savaré to find strong convergence since Ω is not assumed convex. First, we will find an a.e equality using the first variation of energies in order to have a discrete equation, as in Proposition 3.3. Then, we will derive a new estimate on the gradient of some power of ρ h from this a.e equality. To conclude, we will use again the refined version of Aubin–Lions lemma of Rossi and Savaré in [8].

12.6 The case of a bounded domain Ω

ρ kh

| 325

On Ω we can define, with the semi-implicit JKO scheme, the sequence (ρ kh )k , where minimizes ρ 󳨃→ Eh (ρ|ρ k−1 h ) :=

1 2 k−1 W (ρ, ρ k−1 h ) + F(ρ) + V (ρ|ρ h ) 2h 2

on P(Ω). The proof of existence and uniqueness of ρ kh is the same as in Proposition 3.1. It is even easier because on a bounded domain F is bounded from below for all m ≥ 1. We also find the same estimates than in Proposition 3.4 on the functional and the distance (see, e.g., [4],[1]). Now, we will establish a discrete equation satisfied by the piecewise interpolation of the sequence (ρ kh )k defined by, for all k ∈ ℕ, ρ h (t) = ρ kh

t ∈ ((k − 1)h, kh] .

if

Proposition 6.2. For every k ≥ 0, we have (y − T k (y))ρ k+1 + h∇(V[ρ kh ])ρ k+1 + h∇(P(ρ k+1 h h h )) = 0 a.e.

on

Ω,

(6.3)

where T k is the optimal transport map between ρ k+1 and ρ kh . Then ρ h satisfies h T

N−1

∫ ∫ ρ h (t, x)∂ t φ(t, x) dx dt = h ∑ ∫ ∇(V[ρ kh ])(x) ⋅ ∇φ(t k , x)ρ k+1 h (x) dx dt k=0 Ω

0 Ω

N−1

+ h ∑ ∫ ∇P(ρ k+1 h (x)) ⋅ ∇φ(t k , x) dx k=0 Ω N−1

+ ∑ ∫ R[φ(t k , ⋅)](x, y) dγ k (x, y) k=0 Ω×Ω

− ∫ ρ 0 (x)φ(0, x) dx , Ω n with N = ⌈ Th ⌉, for all ϕ ∈ C∞ c ([0, T) × ℝ ), γ k is the optimal transport plan in k+1 k W2 (ρ h , ρ h ) and 1 |R[ϕ](x, y)| ≤ ‖D2 ϕ‖L∞ (ℝ×ℝn ) |x − y|2 . 2

Proof. First, we prove the equality (6.3). As in Proposition 3.3, taking the first variation n in the semi-implicit JKO scheme, we find for all ξ ∈ C∞ c (Ω; ℝ ), k k+1 − h ∫ P(ρ k+1 ∫(y − T k (y)) ⋅ ξ(y)ρ k+1 h (y) dy + h ∫ ∇(V[ρ h ]) ⋅ ξρ h h ) div(ξ) = 0 , Ω

Ω

(6.4)

Ω

and ρ kh . Now we claim that where T k is the optimal transport map between ρ k+1 h k+1 1,1 m P(ρ h ) ∈ W (Ω). Indeed, since F controls x and P is controlled by x m then (3.9)

326 | 12 On some nonlinear evolution systems 1 gives P(ρ k+1 h ) ∈ L (Ω). Moreover, (6.4) gives

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 |y − T k (y)| k+1 󵄨 󵄨󵄨 k+1 ρ h + C] ‖ξ‖L∞ (Ω) 󵄨󵄨∫ P(ρ h ) div(ξ)󵄨󵄨󵄨 ≤ [∫ 󵄨󵄨 󵄨󵄨 h 󵄨󵄨 [Ω 󵄨󵄨Ω ] ≤[

W2 (ρ kh , ρ k+1 h ) + C] ‖ξ‖L∞ (Ω) . h

Id−T k k+1 k+1 k k+1 in Mn (Ω). And, This implies P(ρ k+1 h ) ∈ BV(Ω) and ∇P(ρ h ) = ∇V[ρ h ]ρ h + h ρ h Id−T k k+1 k+1 k k+1 1 1,1 since ∇V[ρ h ]ρ h + h ρ h ∈ L (Ω), we have P(ρ h ) ∈ W (Ω) and (6.3).

Now, we verify that ρ h satisfies (6.2). We start to take the scalar product between (6.3) n and ∇φ with φ ∈ C∞ c ([0, T) × ℝ ), and we find, for all t ∈ [0, T), k k+1 ∫ (y − T k (y)) ⋅ ∇φ(t, y)ρ k+1 h (y) dy + h ∫ ∇ (V[ρ h ]) (y) ⋅ ∇φ(t, y)ρ h (y) dy Ω

Ω

+

h ∫ ∇ (P (ρ k+1 h )) (y)

⋅ ∇φ(t, y) dy = 0 .

(6.5)

Ω

Moreover, if we extend φ by φ(0, ⋅) on [−h, 0), then T

tk

N

∫ ∫ ρ h (t, x)∂ t φ(t, x) dx dt = ∑ ∫ ∫ ρ kh (x)∂ t φ(t, x) dx dt k=0 t

0 Ω

k−1

Ω

N

= ∑ ∫ ρ kh (x)(φ(t k , x) − φ(t k−1 , x)) dx k=0 Ω N−1

= ∑ ∫ φ(t k , x)(ρ kh (x) − ρ k+1 h (x)) dx k=0 Ω

− ∫ ρ 0 (x)φ(0, x) dx . ℝn

And using the second-order Taylor–Lagrange formula, we find ∫ (φ(kh, x) − φ(kh, y)) dγ k (x, y) Ω×Ω

= ∫ ∇φ(kh, y) ⋅ (x − y) dγ k (x, y) + ∫ R[φ(t k , ⋅)](x, y) dγ k (x, y) . Ω×Ω

Ω×Ω

This concludes the proof if we sum on k and use (6.5). Remark 6.3. We remark that equality (6.3) is still true in ℝn . Indeed, the first part of the proof does not depend of the domain and we can use this argument on ℝn . This equality will be used in Section 12.7 to obtain uniqueness result.

12.6 The case of a bounded domain Ω

| 327

In the next proposition, we propose an alternative argument to the flow-interchange argument to get an estimate on the gradient of ρ h . Differences with the flow-interchange argument are that we do not need to assume the space convexity and boundary condition on ∇V[ρ]. Moreover, we do not obtain exactly the same estimate. Indeed, in m/2 Proposition 4.4, ∇ρ h is bounded in L2 ((0, T) × ℝn ) whereas in the following propo1 n sition, we establish a bound on ∇ρ m h in L ((0, T) × ℝ ) using (6.3). Proposition 6.4. There exists a constant C which does not depend on h such that 1 1,1 ‖ρ m h ‖L ([0,T];W (Ω)) ≤ CT

for all T > 0. Proof. According to (6.3), we have k k+1 h ∫ |∇(P(ρ k+1 h ))| dx ≤ W2 (ρ h , ρ h ) + hC . Ω

Then if we sum on k from 0 to N − 1, we get T

N−1

∫ ∫ |∇(P(ρ h ))| dx dt ≤ ∑ W2 (ρ kh , ρ k+1 h ) + TC k=0

0 Ω

N−1

≤ N ∑ W22 (ρ kh , ρ k+1 h ) + TC k=0

≤ CT , because of (3.10). If F(x) = x log(x) then P󸀠 (x) = 1 and if F satisfies (2.5), then F 󸀠󸀠 (x) ≥ Cx m−2 and 󸀠 P (x) = xF 󸀠󸀠 (x) ≥ Cx m−1 . In both cases, we have P󸀠 (x) ≥ Cx m−1 (with m = 1 for x log(x)). So T

T

∫ ∫ |∇(P(ρ h ))| dx dt = ∫ ∫ P󸀠 (ρ h )|∇ρ h | dx dt 0 Ω

0 Ω T

T

≥ C ∫ ∫ ρ m−1 |∇ρ h | dx dt = C ∫ ∫ |∇ρ m h | dx dt , h 0 Ω

0 Ω

which proves the proposition. Now, we introduce G : L m (Ω) → [0, +∞] defined by {‖ρ m ‖BV(Ω) G(ρ) := { +∞ {

if ρ ∈ Pac (Ω) and ρ m ∈ BV(Ω) , otherwise.

328 | 12 On some nonlinear evolution systems Proposition 6.5. G is lower semicontinuous on L m (Ω) and its sublevels are relatively compact in L m (Ω). Proof. First, we show that G is lower semicontinuous on L m (Ω). Let ρ n be a sequence which converges strongly to ρ in L m (Ω) with supn G(ρ n ) ≤ C < +∞. Without loss of generality, we assume that ρ n converges to ρ a.e. Since C < +∞, the functions ρ m n are uniformly bounded in BV(Ω). So we know that m ρ n converges weakly in BV(Ω) to μ. But since Ω is smooth and bounded, the injection m of BV(Ω) into L1 (Ω) is compact. We can deduce that μ = ρ m and ρ m n converges to ρ strongly in L1 (Ω). Then by lower semicontinuity of the BV-norm in L1 , we obtain G(ρ) ≤ lim inf G(ρ n ) . n↗+∞

Now, we have to prove that the sublevels, A c := {ρ ∈ L m (Ω) : G(ρ) ≤ c}, are relatively compact in L m (Ω). Since i : η ∈ L1 (Ω) 󳨃→ η1/m ∈ L m (Ω) is continuous, we just have to prove that B c := {η = ρ m : ρ ∈ A c } is relatively compact in L1 (Ω). So to conclude the proof, it is enough to notice that B c is a bounded subset of BV(Ω) and that the injection of BV(Ω) into L1 (Ω) is compact. Now, we can apply Rossi–Savaré theorem (Theorem 5.3) to have the strong convergence in L m (]0, T[×Ω) of ρ h to ρ and then we find the strong convergence in L1 (]0, T[×Ω) of P(ρ h ) to P(ρ), for all T > 0, using the fact that P is controlled by x m (2.5) and Krasnoselskii theorem (see [16], Chapter 2). Moreover, since T

∫ ∫ |∇(P(ρ h ))| dx dt ≤ CT , 0 Ω

we have ∇(P(ρ h ))dxdt ⇀ μ in Mn ([0, T] × Ω) ,

(6.6)

i.e T

T

∫ ∫ ξ ⋅ ∇(P(ρ h ))dxdt → ∫ ∫ ξ ⋅ dμ , 0 Ω

0 Ω

for all ξ ∈ Cb ([0, T] × Ω) (this means that we do not require ξ to vanish on ∂Ω). But since P(ρ h ) converges strongly to P(ρ) in L1 ([0, T] × Ω), μ = ∇(P(ρ)). To conclude, we pass to the limit in (6.2) and Theorem 6.1 follows.

12.7 Uniqueness of solutions Let F an energy defined on P2 (Ω). Let ρ, μ ∈ P2 (Ω), we recall that the geodesics for W2 are of the form ρ t := π t# γ, where γ is an optimal transport plan for W2 (ρ, μ) and

12.7 Uniqueness of solutions

|

329

π t (x, y) := (1 − t)x + ty. F is said displacement convex if t ∈ [0, 1] 󳨃→ F(ρ t ) is convex. Now, we state a general uniqueness argument based on geodesic convexity. Theorem 7.1. Assume that V i satisfy (2.3) and (2.4) and F i such that Fi is displacement convex. Let ρ 1 := (ρ 11 , . . . , ρ 1l ) and ρ 2 := (ρ 21 , . . . , ρ 2l ) two weak solutions of (1.1) or (6.1) with initial conditions ρ1i (0, ⋅) = ρ 1i,0 and ρ 2i (0, ⋅) = ρ 2i,0. If for all T < +∞, T l

T l

∫ ∑ ‖𝑣1i,t ‖L2 (ρ1i,t ) dt + ∫ ∑ ‖𝑣2i,t ‖L2 (ρ2i,t ) dt < +∞ , 0 i=1

(7.1)

0 i=1

with, for j ∈ {1, 2},

j

j

𝑣i,t := −

∇P i (ρ i,t ) j ρ i,t

j

− ∇V i [ρ t ] ,

then for every t ∈ [0, T], W22 (ρ 1t , ρ 2t ) ≤ e4Ct W22 (ρ 10 , ρ 20 ) . In particular, we have uniqueness for the Cauchy problems (1.1) and (6.1). The proof of this theorem can be found in Theorem 6.1 in [1]. The proof is a little perturbation of the one of Theorem 11.1.4 of [6] and is based on displacement convexity argument and Gronwall’s lemma. Remark 7.2. We say that F : [0, +∞) → ℝ satisfies McCann’s condition if x ∈ (0, +∞) 󳨃→ x n F(x−n ) is convex nonincreasing .

(7.2)

McCann showed in [17] that if F satisfies (7.2), then F is displacement convex. Then if for all i, F i satisfies McCann’s condition, we have uniqueness in (1.1). Moreover we re1 x m , m > 1 (porous medium mark that F(x) = x log(x) (linear diffusion) and F(x) = m−1 diffusion) satisfy this condition. In the following proposition, we will prove that assumption (7.1) holds if Ω is a smooth bounded convex subset of ℝn or if Ω = ℝn . Proposition 7.3. Let ρ := (ρ 1 , . . . , ρ l ) be a weak solution of (1.1) obtained with the previous semi-implicit JKO scheme. Then ρ i satisfies (7.1) for all i ∈ [[1, l]]. Proof. We do not separate the cases where Ω is a bounded set or is ℝn . We split the proof in two parts. First, we show that (7.1) is satisfied by ρ i,h defined in (3.3). Then by a l.s.c argument we will conclude the proof.

330 | 12 On some nonlinear evolution systems

In the first step, we show that ρ i,h satisfies T

∫ ∫ |∇F 󸀠i (ρ i,h ) + ∇V i [ρ h ]|2 ρ i,h dx dt ≤ C ,

(7.3)

0 Ω

where C does not depend of h. By Equality (6.3) and Remark 6.3, we have k ∇F 󸀠i (ρ k+1 i,h ) + ∇V i [ρ h ] =

T k (y) − y h

ρ k+1 i,h − a.e on Ω ,

k where T k is the optimal transport map between ρ k+1 i,h and ρ i,h . Then if we take the

square, multiply by ρ k+1 i,h and integrate on Ω, we find k 2 k+1 ∫ |∇F 󸀠i (ρ k+1 i,h ) + ∇V i [ρ h ]| ρ i,h dx =

1 2 k+1 k W (ρ , ρ i,h ) . h2 2 i,h

Ω

Now using (2.4), we get k+1 󸀠 k+1 k k k+1 |∇F 󸀠i (ρ k+1 i,h ) + ∇V i [ρ h ]| ≤ |∇F i (ρ i,h ) + ∇V i [ρ h ]| + |∇V i [ρ h ] − ∇V i [ρ h ]| k k+1 k ≤ |∇F 󸀠i (ρ k+1 i,h ) + ∇V i [ρ h ]| + CW2 (ρ h , ρ h )

So we have k+1 2 k+1 ∫ |∇F 󸀠i (ρ k+1 i,h ) + ∇V i [ρ h ]| ρ i,h dx ≤ C (

1 2 k+1 k k W (ρ , ρ i,h ) + W22 (ρ k+1 h , ρ h )) . h2 2 i,h

Ω

Then using (3.10), we finally get T

N−1

k+1 2 k+1 ∫ ∫ |∇F 󸀠i (ρ i,h ) + ∇V i [ρ h ]|2 ρ i,h dx dt = h ∑ ∫ |∇F 󸀠i (ρ k+1 i,h ) + ∇V i [ρ h ]| ρ i,h dx k=0 Ω

0 Ω

≤ C(

1 N−1 2 k+1 k ∑ W (ρ , ρ i,h ) + 1) h k=0 2 i,h

≤C. To conclude, we have to pass to the limit in (7.3). First, we claim that ∇P i (ρ i,h ) converges to ∇P i (ρ i ) in Mn ([0, T] × Ω). In a bounded set, this has been proved in (6.6). In ℝn thanks to the previous step, we have T

T

∫ ∫ |∇P i (ρ i,h )| dt = ∫ ∫ |∇F 󸀠i (ρ i,h )|ρ i,h dx dt 0 ℝn

0 ℝn T

≤ ∫ ∫ (|∇F 󸀠i (ρ i,h )|2 + 1)ρ i,h 0 ℝn

≤C,

12.7 Uniqueness of solutions

|

331

which gives the result because P i (ρ i,h ) strongly converges in L1 ([0, T] × ℝn ) to P i (ρ i ). Let ψ : ℝn+1 → ℝ ∪ {+∞} defined by |m|2

if (r, m) ∈]0, +∞[×ℝn ,

{ { { r ψ(r, m) := {0 { { {+∞

if (r, m) = (0, 0) , otherwise,

as in [18]. And define Ψ : M((0, T) × Ω) × M n ((0, T) × Ω) → ℝ ∪ {+∞}, as in [19], by T

{∫ ∫ ψ( dρ/ dL, dE/ dL) dx dt Ψ(ρ, E) := { 0 Ω +∞ {

if ρ ≥ 0 , otherwise,

where dσ/ dL is Radon–Nikodym derivative of σ with respect to L|[0,T]×Ω . We can remark that since ψ(0, m) = +∞ for any m ≠ 0, we have Ψ(ρ, E) < +∞ ⇒ E ≪ ρ . With this definition, we can rewrite (7.3) as T

Ψ(ρ i,h , ∇P i (ρ i,h ) + ∇V i [ρ h ]ρ i,h ) = ∫ ∫ |∇F 󸀠i (ρ i,h ) + ∇V i [ρ h ]|2 ρ i,h dx dt ≤ C , 0 Ω

which, in particular, implies that ∇P i (ρ i,h ) ≪ ρ i,h ≪ L|[0,T]×Ω . Moreover, according to [20], Ψ is lower semicontinuous on M([0, T] × Ω) × Mn ([0, T] × Ω). So, it holds Ψ(ρ i , ∇P i (ρ i ) + ∇V i [ρ]ρ i ) ≤ lim inf Ψ(ρ i,h , ∇P i (ρ i,h ) + ∇V i [ρ h ]ρ i,h ) ≤ C , h↘0

which imply ∇P i (ρ i ) ≪ ρ i ≪ L|[0,T]×Ω and conclude the proof because T

T 󵄨󵄨 ∇P (ρ ) 󵄨󵄨2 |∇P i (ρ i ) + ∇V i [ρ]ρ i |2 󵄨󵄨 i i 󵄨󵄨 ∫ ∫ 󵄨󵄨 + ∇V i [ρ]󵄨󵄨 ρ i dx dt = ∫ ∫ dx dt 󵄨󵄨 ρ i 󵄨󵄨 ρi 0 Ω

0 Ω

= Ψ(ρ i , ∇P i (ρ i ) + ∇V i [ρ]ρ i ) ≤C. Acknowledgment: The author wants to gratefully thank G. Carlier for his help and advices.

332 | 12 On some nonlinear evolution systems

Bibliography [1] [2] [3] [4] [5]

[6] [7] [8] [9] [10]

[11] [12] [13]

[14] [15] [16] [17] [18] [19] [20]

G. Carlier and M. Laborde, On systems of continuity equations with nonlinear diffusion and nonlocal drifts, 2015. R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal., 29(1):1–17, 1998. F. Otto, Double degenerate diffusion equations as steepest descent, 1996. M. Agueh, Existence of solutions to degenerate parabolic equations via the Monge– Kantorovich theory, Adv. Differential Equations, 10(3):309–360, 2005. A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak–Keller–Segel model, SIAM J. Numer. Anal., 46(2):691–721, 2008. L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005. M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, Nonlinearity, 26(10):2777–2808, 2013. R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2, 2003. D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34(10–12):1352–1397, 2009. M. Di Francesco and D. Matthes, Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations, Calc. Var. Partial Differential Equations, 50(1–2):199–230, 2014. F. Santambrogio, Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and Their Applications 87. Birkasauser Verlag, Basel, 2015. C. Villani, Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. C. Villani, Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009, Old and new. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44(4):375–417, 1991. S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40(3):1104–1122, 2008. D. G. De Figueiredo, Lectures on the Ekeland variational principle with applications and detours, Springer Berlin, 1989. R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128(1):153–179, 1997. J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge– Kantorovich mass transfer problem, Numer. Math., 84(3):375–393, 2000. G. Buttazzo, C. Jimenez and E. Oudet, An optimization problem for mass transportation with congested dynamics, SIAM J. Control Optim., 48(3):1961–1976, 2009. G. Bouchitté and G. Buttazzo, New lower semicontinuity results for nonconvex functionals defined on measures, Nonlinear Anal., 15(7):679–692, 1990.

B. Maury and A. Preux

13 Pressureless Euler equations with maximal density constraint: a time-splitting scheme Abstract: In this chapter, we consider the pressureless Euler equations with a congestion constraint. This system still raises many open questions and, aside from its onedimensional version, very little is known concerning its solutions. The strategy that we propose relies on previous works on crowd motion models with congestion in the framework of the Wasserstein space, and on a microscopic granular model with nonelastic collisions. We illustrate the approach by preliminary numerical simulations in the two-dimensional setting. Keywords: Pressureless Euler equations, congestion, optimal transportation

13.1 Introduction We are interested in the pressureless gas dynamics system that describes the free motion of inertial particles. The system simply expresses mass conservation and momentum conservation; it writes {

∂ t ρ + ∂ x (ρu) = 0 , ∂ t (ρu) + ∂ x (ρu 2 ) = 0 ,

(1.1)

in the one-dimensional setting. Here, ρ represents the density of particles and u is the velocity field. The fact that a single velocity can be locally defined is a strong implicit assumption here. The real free transport equation for non-interacting particles would be of the kinetic type (namely the collisionless Boltzmann equation), allowing for various velocities to coexist at the same place. In the previous system (1.1), even for smooth initial data, transport characteristics are likely to cross, leading to an incompatibility in terms of velocity. Preserving the monokinetic character of the representation calls for considering interactions between particles, although they do not explicitly appear in the equation. As described in [15] or [2], this model can create Dirac masses even if initial data are smooth. It is therefore necessary to define measure-valued solutions for this system. The existence of such solutions of (1.1) has been proven constructively in [3] and [4], by approximating the initial measure by Dirac masses (sticky particles). The motion of the corresponding collection of particles is then computed, following an B. Maury, Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France & DMA, École Normale Supérieure, 45 rue d’Ulm, Paris A. Preux, Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France

334 | 13 Pressureless Euler equations with congestion and optimal transportation

event-driven approach to handle binary collisions, and the corresponding sticky particle solutions are shown to converge as the number of particles tends to infinity. This approach has been introduced in [5] as a model describing the formation of galaxies in the early stage of the universe. This system has been studied in the framework of hyperbolic systems of conservation laws (see, e.g. [6] and [7]), although it lacks hyperbolicity. A proof of uniqueness has been given in [8] with appropriate conditions on entropy and energy. This system has also been formulated and analyzed as a firstorder differential inclusion, see [9, 10]. Some other approaches have been proposed, for instance, in [11] with an optimal transportation approach, with a viscosity regularization in [12] and [13], with finite size sticky particles in [14]. We are interested here in the situation where the density is subject to remain below a threshold value, say 1; it corresponds to the pressureless gas dynamics with congestion constraint: ∂ t ρ + ∂ x (ρu) = 0 { { { { { { ∂ t (ρu) + ∂ x (ρu 2 + π) = 0 { { { (1.2) ρ≤1 { { { { { (1 − ρ)π = 0 { { { { {π ≥ 0. A first approach and numerical algorithm have been given in [1], and the existence of solutions is proved in [16]. The approach is constructive, and dedicated to the onedimensional setting: it is based on a generalization of the sticky particles model. Particles have been replaced by macroscopic congested blocks, but the dynamic is similar, with a pressure π that is active in congested zone only (a typical collision between two identical blocks is represented in Figure 1 (top)). In [17] a similar system is obtained and analyzed as a limit of the Aw-Rascle model. Both systems (1.1) and (1.2) admit infinitely many solutions, some of them being highly non-physical. Consider, for example, the case without congestion (1.1), and ρ defined as follows: ρ is a Dirac mass at 0 during [0, 1], and splits at time 1 into two half Dirac masses, one going to the right at constant velocity u, and the other one at velocity −u, where u > 0 is arbitrary. This measure path, with its obvious associated velocity, verifies the system although it does not correspond to any physical reality. The fact that such solutions are compatible with (1.1), although no interaction force explicitly appears, can be explained: a Dirac mass can be considered as the sum of two half Dirac masses at the same point. Splitting is achieved by exerting an impulsion of one of the halves, and the opposite impulsion on the other. Since these impulsions sum up to zero, it can be done without any real action on the system, and the obtained path verifies the weak formulation of the equations. The splitting time is of course arbitrary, and each of the subparticles can be further split in lighter particles, at any time, as soon as local conservation of mass and momentum are respected. Similar bizarre solutions can be built for the congested case. An initially steady block

13.1 Introduction

| 335

can be suddenly splits in two halves, with a pressure field that is singular in time, but regular in space at the instant of the splitting (piecewise affine function vanishing at the ends of the block, maximal at the center). One may hope to recover uniqueness by demanding that the kinetic energy decreases. This is not enough, as the following example shows: consider two Dirac masses heading to 0 with opposite velocities u and −u. They may collide at 0 into a steady Dirac mass with weight 2, but they may also bounce against each other, and head back to −∞ and +∞ at velocities −eu and eu, respectively. For any e ∈ [0, 1] (restitution coefficient), the solution is energy decreasing (with conservation for the purely elastic case e = 1). To sum up, both systems allow for multiple solutions to exist, some of them being non-physical (energy increasing). But still infinitely many energy-decreasing solutions exist. A collision law is obviously lacking. Before writing the multidimensional version of (1.2), let us describe now the two main ingredients that we will use to write the model (with a collision law) and to elaborate a time-splitting strategy. First ingredient: microscopic granular model At the microscopic level, the motion of rigid particles can be modeled as follows. Consider N rigid spheres in ℝd , with common radius r > 0, unit mass, and centers q1 , . . . , q N . Denote by q ∈ ℝdN the vector of positions, and let D ij = |q j − q i | − 2r denote the inter-grain distance. Prescribing grain rigidity amounts to define a set of feasible configurations, that is K̃ = {q = (q1 , . . . , q N ) ∈ ℝdN , D ij ≥ 0

∀i ≠ j} .

The cone of feasible velocities (i.e. velocities that do not lead to overlapping between grains) is defined for any q ∈ K̃ as C K̃ (q) = {𝑣 ∈ ℝdN , D ij = 0 󳨐⇒ G ij ⋅ 𝑣 ≥ 0} , where G ij ∈ ℝdN is the gradient of D ij . The inertial motion of this system of grains submitted to a force field f (possibly including interactions, i.e. f may depend on q) with non-elastic collision can be written as (see, e.g. [18, 19]) dq/ dt = u , { { { { { du/ dt = f + ∑ π ij G ij , { { { { i 0 be a time step, ρ n ∈ P(Ω) ρ velocity at time nτ. The first step consists in projecting the current velocity field on the set of Mcρ n , to prevent particles from crossing. The density is then moved according to this projected velocity field, during τ. The obtained intermediate density is likely

342 | 13 Pressureless Euler equations with congestion and optimal transportation

to violate the constraint. The third step consists in projecting it (in the Wasserstein sense), on the feasible set K, to account for the saturation constraint. The final step consists in defining the new (post-collision) velocity, using the Lagrange map between the previous density and the new one. For (ρ n , u n ) given, the scheme reads ũ n = ℙMcρn (u n ) { { { { { n+1 { = (Id + τ ũ n )# ρ n { ρ̃ { W n+1 { = ℙK 2 (ρ̃ n+1 ) { {ρ { { { { { n+1 Id − (s n+1 )−1 ∘ r n+1 u = { τ

(2.3)

with ̄ ⟨𝑣(x) − 𝑣(y), x − y⟩ ≥ c − 1 |x − y|2 ρ n ⊗ ρ n -a.e.} , Mcρ n = {𝑣 ∈ L2ρ n (Ω), (2.4) τ and c ∈ (0, 1) a small parameter. K is the space of all those probability measures supported in Ω̄ that admit a density less than 1. The mappings involved in the last step (reconstruction of the velocity) are s n+1 = Id + τ ũ n , and r n+1 , that is the optimal transportation map between ρ n+1 and ρ̃ n+1 . Figure 2 illustrates how the time-stepping scheme treats a one-dimensional collision between two blocks. The figure reads from bottom to top. The first part of the scheme moves densities with the current velocity, while avoiding contact of pathlines (the velocity is projected on Mcρ n ). This intermediate density violates the congestion constraint, and the second step is meant to enforce it, by projection of the density on K. The three circles on the figure correspond to successive positions of a physical particle during the scheme: initially at x, it is moved to y, and then to z by projection. For this very particle, the reconstruction of the after-collision velocity (at the new position z) gives (z − x)/τ. As explained in the proof of Proposition 2.7, it may happen that a gap remains between the two blocks after this first partial treatment, in that case it may need two more time steps to fully handle the collision. Let us start by establishing some basic properties pertaining to the proposed scheme. The first lemma asserts that the first step (projection on M cρ n ) is well defined, and preserves the total momentum. In a second lemma, we show that the intermediate density ρ̃ n+1 is bounded. In the third one, we recall a known result concerning the projection on K. ̄ and u ∈ L2 (Ω), ̄ then there exists a unique ũ ∈ Mc such that: Lemma 2.1. Let ρ ∈ P(Ω) ρ ρ ̄ . |u − 𝑣|L2ρ (Ω)̄ ≥ |u − u|̃ L2ρ (Ω)̄ , ∀𝑣 ∈ L2ρ (Ω)

13.2 Time-stepping scheme |

343

z

projection on K

y

transport by un = PMc (un) n

x

Fig. 2: Time stepping scheme for a 1d collision.

Moreover, we have the following properties: (i) |u|̃ L2ρ (Ω)̄ ≤ |u|L2ρ (Ω)̄ ; ̃ dρ(x) = ∫Ω̄ u(x) dρ(x). (ii) ∫Ω̄ u(x)

Proof. Since M cρ is a closed and convex set, the projection ũ := ℙMcρ (u) is well defined. Inequality (i) comes from the fact Mcρ is also a cone pointed at 0. Let us now prove that the first step of the scheme does not change momentum (identity (ii)). If u ∈ Mcρ , then u + k is also in Mcρ for any k in ℝd . If ũ is the projection of u on Mcρ , we have |u − u|̃ 2L2 (Ω)̄ ≤ |u − ũ − k|2L2 (Ω)̄ ρ

ρ

and thus ⟨u − u,̃ Letting |k| tend to 0 yields ⟨u − u,̃

k |k| . ≤ ⟩ |k| L2ρ (Ω)̄ 2 k 2 ̄ |k| ⟩ L ρ ( Ω)

⟨u − u,̃

≤ 0, for all k ∈ ℝd and consequently

k =0. ⟩ |k| L2ρ (Ω)̄

344 | 13 Pressureless Euler equations with congestion and optimal transportation

Consider the total momenta before and after the projection of velocities: Etot = ∫ u(x) dρ(x)

and

̃ dρ(x) . Ẽ tot = ∫ u(x)

Ω̄

Ω̄

We have ⟨Etot , k⟩ = ⟨Ẽ tot , k⟩ for all k ∈ ℝd and finally Ẽ tot = Etot . ̄ and ũ ∈ Mc , we have Lemma 2.2. Let ρ̃ := s# ρ = (Id + τ u)̃ # ρ, with ρ ∈ L∞ (Ω), ρ 1 󵄨 󵄨 󵄨󵄨 ̃ 󵄨󵄨 󵄨󵄨ρ 󵄨󵄨L∞ (Ω)̄ ≤ d 󵄨󵄨󵄨ρ 󵄨󵄨󵄨L∞ (Ω)̄ . c Proof. We consider here that ũ is regular, and we refer to [29], Section 5.5, for the general case. Since ũ is in Mcρ , the eigenvalues of ∇s are larger than c and then det ∇s is ρ larger than c d . Finally, since ρ̃ = | det ∇s| ∘ s−1 we have 󵄩󵄩 󵄩󵄩 1 󵄨 1 󵄨 󵄨 ρ 󵄨 󵄩 󵄩󵄩 ̃ 󵄩󵄩 󵄩 ∘ s−1 󵄩󵄩󵄩 ≤ d 󵄨󵄨󵄨󵄨ρ ∘ s−1 󵄨󵄨󵄨󵄨L∞ (Ω)̄ ≤ d 󵄨󵄨󵄨ρ 󵄨󵄨󵄨L∞ (Ω)̄ , 󵄩󵄩ρ 󵄩󵄩L∞ (Ω)̄ = 󵄩󵄩󵄩 󵄩󵄩L∞ (Ω)̄ 󵄩󵄩 | det ∇s| c c which ends the proof. The following lemma concerns the projection on K. It is proven in [21]. ̄ There exists a unique ρ K ∈ K such that Lemma 2.3. Let ρ ∈ P(Ω). W2 (ρ, μ) ≥ W2 (ρ, ρ K ), ∀μ ∈ K , and there exists π ∈ {p ∈ H 1 | p ≥ 0, (1 − ρ)p = 0} such that ρ = (Id + ∇π)# ρ K . We may now prove that the scheme defines a unique density–velocity couple. ̄ The scheme (2.3) defines a unique (ρ n+1, u n+1 ) Proposition 2.4. Let (ρ n , u n ) ∈ K×L2ρ n (Ω). ̄ ∈ K × L2ρ n+1 (Ω). Proof. We can check that s n+1 is strictly monotone and consequently injective. We can define an inverse on s n+1 (spt(ρ n )) = spt(ρ̃ n+1 ). Thanks to Lemma 2.3, we have existence and uniqueness of r n+1 : spt(ρ n+1 ) → spt(ρ̃ n+1 ) and consequently the uniqueness of u n+1 and ρ n+1 . Moreover, we have 2

∫(u n+1 (x))2 ρ n+1 (x) dx = ∫ ( Ω̄

Ω̄

x − (s n+1 )−1 ∘ r n+1 (x) ) ρ n+1 (x) dx τ 2

= ∫( Ω̄

(r n+1 )−1 (x) − (s n+1 )−1 (x) ) ρ̃ n+1 (x) dx τ

13.2 Time-stepping scheme |

2

≤ 2 ∫ [( Ω̄

2

(r n+1 )−1 (x) − x x − (s n+1 )−1 (x) ) +( ) ] ρ̃ n+1 (x) dx τ τ 2

≤ 2∫( Ω̄

345

2

x − r n+1 (x) s n+1 (x) − x ) ρ n+1 (x) dx + 2 ∫ ( ) ρ n (x) dx τ τ Ω̄

≤ 2 ∫(∇π n+1 (x))2 ρ n+1 (x) dx + 2 ∫ ũ n (x)2 ρ n (x) dx Ω̄

Ω̄

≤ 4 ∫ u n (x)2 ρ n (x) dx Ω̄

since

2

∫ (∇π n+1 (x)) ρ n+1 (x) dx = W2 (ρ n+1 , ρ̃ n+1 ) ≤ W2 (ρ n , ρ̃ n+1 ) Ω̄

and W2 (ρ n , ρ̃ n+1 ) ≤ ∫ u n (x)2 ρ n (x) dx , Ω̄

which ends the proof. Proposition 2.5. Total mass and momentum are conserved by the scheme (2.3). Proof. Conservation of mass is inherent to the approach, which is a succession of transports and projections in the Wasserstein sense. As for the momentum, we have already seen that ∫Ω̄ E n (x) dx = ∫Ω̄ Ẽ n (x) dx, where Ẽ n = ρ n 𝑣ñ . ∫ Ẽ n (x)ϕ(x) dx = ∫ ρ n (x)ũ n (x)ϕ(x) dx Ω̄

Ω̄ −1

= ∫ ((s n+1 ) Ω̄

−1

= ∫ ((s n+1 ) Ω̄

= ∫ ρ n+1 (x) ( Ω̄

∘ r n+1 ) ρ n+1 (x)ũ n (x)ϕ(x) dx #

∘ r n+1 ) ρ n+1 (x) ( #

s n+1 (x) − x ) ϕ(x) dx τ

r n+1 − (s n+1 )−1 ∘ r n+1 ) ϕ((s n+1 )−1 ∘ r n+1 (x)) dx τ

= ∫ ρ n+1 (x) (∇π n+1 + u n+1 ) ϕ((s n+1 )−1 ∘ r n+1 (x)) dx . Ω̄

346 | 13 Pressureless Euler equations with congestion and optimal transportation So Ẽ nτ = ((s n+1 )−1 ∘ r n+1 )# (ρ n+1 ∇π n+1 + E n+1 ) and consequently ∫ Ẽ n (x) dx = ∫ ρ n+1 (x)(∇π n+1 (x) + u n+1 (x)) dx Ω̄

Ω̄

= ∫ ρ n+1 (x)∇π n+1 (x) + ∫ ρ n+1 (x)u n+1 (x) dx Ω̄

Ω̄

= ∫ E n+1 (x) dx . Ω̄

Remark 2.6. A direct consequence of the previous proposition is that the center of mass moves at a constant velocity U=

∫Ω̄ u n (x)ρ n (x) dx ∫Ω̄ ρ n (x) dx

.

The next proposition pertains to the one-dimensional setting. We consider the exact solution that corresponds to the collision of two saturated blocks. We establish that, except for the few times steps (no more than 3) that are needed to handle the collision at the discrete level, the discretization scheme recovers the exact solution. Proposition 2.7. Let ρ t be the density representing two blocks colliding at time t0 and η > 0. If c ≤ √2− 1 then there exists τ o > 0 such that for all time step τ ≤ τ o the solution given by the scheme is exact, for all step n ∈ {n ∈ ℕ, nτ ∉ [t0 − η, t0 + η]}. Proof. Let us start by two basic remarks. First, from Remark 2.6, the scheme computes in an exact way the motion of the mass center. Second, an isolated block moving at uniform velocity is preserved by the scheme (blocks stay blocks). As a consequence, we just have to check that the scheme makes the blocks stick together in a finite number of steps (no more than 3, as we shall see). Let ∆U n ≥ 0 be the velocity difference between the two blocks at time nτ and e n their distance. One can check that the updated velocity difference ∆U n+1 is equal to e n −e n+1 . The velocity field u n remains in Mcρ n if and only if the constraint is verified in τ the ends of blocks so that c−1 n 2 (e ) τ ⇐⇒ τ∆U n ≤ (1 − c)e n ∆U n (−e n ) ≥

Now, if τ∆U n > (1−c)e n and if the length of the smaller block L is sufficiently large (take τ such that L ≥ 2τ‖𝑣‖∞ ), the density ρ̃ n+1 exceeds the congestion constraint (see n n Figure 2) over two blocks of total size cτ∆U 1−c − ce . In this case, the mass M projected in the gap between the two blocks is larger than the half of over-congested mass, M≥

1 cτ∆U n 1 1 − ce n ) ≥ (τ∆U n + (c − 1)e n ) . ( − 1) ( 2 c 1−c 2

13.3 Numerical illustrations

|

347

Two cases have to be considered. If τ∆U n ≥ (1 + c)e n , the mass M is larger than the mass necessary to refill the gap, this implies e n+1 = 0 and τ∆U n+1 = e n . The following step allows blocks to have the same velocity, e n+2 = 0 and ∆U n+2 = 0. So, the scheme handles the collision with two steps. Now if τ∆U n ∈ ((1 − c)e n , (1 + c)e n ), the projection on the set of feasible velocities imposes that ẽ n+1 = ce n (the distance between the ends of blocks is exactly c times the previous one), then e n+1 ≤ ce n and consequently τ∆U n+1 = e n − e n+1 ≥ (1 − c)e n . Thus, (1 + c)e n+1 ≤ (1 + c)ce n ≤ (1 − c)e n ≤ τ∆U n+1 since c ≤ √2 − 1, conditions of the first case are obtained, the scheme handles the collision with three steps. L , Finally, taking τ o = min{ 2|𝑣|∞

2η 5 }

allows us to prove the property.

13.3 Numerical illustrations We briefly describe here a space discretization strategy that can be carried out to perfom actual simulations, and we present some numerical tests to illustrate the behavior of the numerical scheme in various situations.

13.3.1 Space discretization scheme We take Ω = (0, 1)2 and we discretize the domain in a collection of cells (C i )i∈I with a uniform size h × h and the center located at x i . The quantities used previously are approximated by constant functions in each cell f ≈ ∑ f(x i )𝟙 C i := ∑ f i 𝟙 C i , i∈I

i∈I

and we denote by the corresponding space (α = 1 for a scalar field, α = 2 for a vector field). From now on, we will assimilate f to the vector (f i ) ∈ (ℝd )I . We denote by I ρ the set of indices corresponding to non-zero values of the ρ i ’s: I ρ = {i ∈ I | ρ i > 0}. We may now describe the four steps of the numerical scheme. X hα

Suppose that we have (ρ n , 𝑣n ) ∈ X 1h × X 2h . Step 1. We replace Mcn by the space of fields 𝑣 in X 2h such that ∀i, j ∈ I ρ n , ⟨𝑣i − 𝑣j , x i − x j ⟩ ≥ (

c−1 ) |x i − x j |2 . τ

348 | 13 Pressureless Euler equations with congestion and optimal transportation

The projection can be formulated in a saddle-point form: Consider the function F ij := 2 𝑣 → ⟨𝑣i − 𝑣j , x i − x j ⟩ − ( c−1 τ )|x i − x j | , the projection onto this space is defined as { } 󵄨 󵄨 𝑣̃n = argmin { sup (󵄨󵄨󵄨𝑣̃ − 𝑣n 󵄨󵄨󵄨L2n − ∑ λ ij F ij (𝑣n ))} . ρ 2 I I ̃ 𝑣∈ℝ i,j∈(I ρn )2 { λ ij ∈ℝ+ } Step 2. We approximate the transport step by a projected Lagrangian scheme (see Remark 3.1): ρ̃ n+1 := (Id + τ 𝑣̃n )# ρ n ≈ ∑ (∫(Id + τ𝑣̃n )# ρ n (x) dx) 𝟙 C i , i∈I

Ci

Ẽ n+1 := (Id + τ 𝑣̃n )# E n ≈ ∑ (∫(Id + τ𝑣̃n )# E n (x) dx) 𝟙C i . i∈I

Ci

Step 3. In order to project on K, we define the discrete transport plan as the minimizer of a discrete counterpart of the transport cost, among all those plans ζ = (ζ ij )i,j∈I with ρ̃ n+1 as first marginal, and second marginal in K. The cost is defined as C(ζ) = ∑ ∑ β ij , i∈I ρ̃ n+1 j∈I

where the cell-to-cell costs β ij are defined (see Remark 3.2) by ζ2

ij { n+1 |x i − x j |2 β ij = { ρ̃ i {+∞

if C i is a neighbor of C j

(3.1)

otherwise ,

where cells are considered neighbors if they share an edge or a vertex. The infinite cost for non-neighboring cells prevents the transport from leaping over adjacent cells. This choice is consistent with the fact that very small time steps are used in the presented illustrations. Step 4. We reconstruct the velocity by E n+1 = ∑ ζ ij ( j i∈I ρ̃ n+1

Ẽ n+1 i ρ̃ n+1 i

+(

xj − xi )) , u n+1 = E n+1 /ρ n+1 . j j j τ

Remark 3.1. Note that the space X h is not stable by the transportation step. If ρ ∈ X 1h , then (Id + τ𝑣)# ρ is a piecewise constant function that is not, in general, in X 1h . A function in X 1h is obtained by L2 projection. Remark 3.2. The step 3 is undoubtedly the most problematic one. A natural counterpart of the continuous projection problem would consist in minimizing ∑ ζ ij |x i − x j |2 i,j

13.3 Numerical illustrations

|

349

over all those discrete transport plans (ζ ij ) with ρ̃ n+1 as first marginal, and such that the second marginal belongs to K. Yet, this straight discretization of the transport cost does not lead to a satisfying approximation of the projection. This feature is related to the numerical diffusion that is inherent to the space discretization of the finite volume type that we chose. This delicate issue will be addressed in a forthcoming paper. Let us simply say here that, if one considers the JKO scheme applied to the gradient flow that corresponds to transport at constant velocity, such a discretization leads to an untenable behavior. For small time steps, the discretized JKO scheme is static, i.e. the density is not transported. It comes from the fact that, because of numerical diffusion, the expression above highly overestimates the cost of the displacement from a cell to its neighbor so that immobility is cheaper than cell-to-cell motion. This locking phenomenon can be avoided by changing the cost for short displacements. More precisely, it can be checked in the transport case that changing the cost into ζ ij2 ρ̃ n+1 i

|x i − x j |2

makes it possible to recover the classical upwind scheme in the one-dimensional case. This explains the choice of the cost of transport between neighboring cells that appears in (3.1).

13.3.2 Numerical tests In this section, we illustrate our comments by comparing numerical solutions computed by the SCoPI software for the granular model (see [30–32] for details on the algorithm) with the solutions computed by our algorithm for the scheme described above. This comparison is not intended to rigorously validate the approach in any way, since the two models are different (see the next section), but it asserts a satisfactory behavior of the macroscopic model to mimic the motion of many-body suspensions. In the microscopic examples we take the number of particles (with a radius of r given below) that corresponds to the density of the macroscopic model.

First numerical example In this case, a small cloud of particles (on the left) is set at uniform (rightward) velocity toward another cloud of particles initially at rest. Macroscopic model: – Ω̄ = [−40, 40]2 , h = 0.3, τ = 0.0150; – ρ0 = 0.2𝟙x2 +y2 0 . Basically, we follow the ansatz from [29], but we shall deviate in the discretization of the potential of the flow. First, equation (1.1) is re-written in terms of Lagrangian coordinates: since each u(t, ⋅) is of fixed mass M, we can introduce time-dependent Lagrangian maps X(t, ⋅) : [0, M] → Ω X(t,ξ)

ξ = ∫ u(t, x) dx,

for each ξ ∈ [0, M] .

(1.9)

a

For the moment, we ignore the ambiguity in the definition of X(t, ξ) outside of the support of u(t). Expressed in terms of X, and after elementary manipulations, the Hele– Shaw equation (1.1) becomes: 1 1 ∂ t X = ∂ ξ ( Z 3 ∂ ξξ Z + Z 2 ∂ ξξ (Z 2 )) + V(X) , 2 4 1 = u(t, X(t, ξ)) . (1.10) where Z(t, ξ) := ∂ ξ X(t, ξ)

14.1 Introduction

| 359

It is easily seen that equation (1.10) is the L2 -gradient flow for M

M

0

0

1 1 2 1 E (X) := E (u ∘ X) = ∫ [ ] dξ + ∫ V(X) dξ , 2 Xξ ξ Xξ V

V

with respect to the usual L2 -norm on L2 ([0, M]; ℝ). This directly reflects the gradient flow structure of (1.1) with respect to the L2 -Wasserstein metric. Equation (1.10) is now discretized as follows. First, fix a spatiotemporal discretization parameter ∆ = (τ; δ), where τ > 0 is a time step size, and δ = M/K for some K ∈ ℕ defines an equidistant partition of [0, M] into K intervals [ξ k−1 , ξ k ] of length δ each, i.e., ξ k = kδ for k = 0, 1, . . . , K. Accordingly, introduce the central first- and secondorder finite difference operators D1δ and D2δ for discrete functions defined either on the ξ k ’s or on the interval midpoints ξ k+1/2 = (k + 1/2)δ in the canonical way, see Section 14.2.1 for details. At each time t = nτ, the Lagrangian map X(t, ⋅) is approximated by a monotone vector x⃗ n∆ = (x1n , . . . , x nK−1 ) ∈ ℝK−1 with a < x1n < ⋅ ⋅ ⋅ < x nK−1 < b in the sense that X(nτ, kδ) ≈ x nk . We will further use the convention that x0 = a and x K = b. For brevity, introduce the vectors z n∆ = z[x⃗ n∆ ] with entries ⃗ κ= (z[x])

δ 1 = 1 x κ+ 12 − x κ− 12 [Dδ x]⃗ κ

for

κ=

1 1 3 , ,...,K− , 2 2 2

(1.11)

and z− 1 = z 1 and z K+ 1 = z K− 1 by convention. These vectors approximate the function 2 2 2 2 Z in (1.10) such that Z(nτ, κδ) ≈ z nκ . The fully discrete evolution for the x⃗n∆ is now obtained from the following standard discretization of (1.10) with central finite differences: x nk − x n−1 1 1 k = D1δ [ (z⃗n )3 D2δ [z⃗ n ] + (z⃗n )2 D2δ [(z⃗n )2 ]] + V x (x k ) . τ 2 4 k

(1.12)

Note that there are infinitely many equivalent ways to re-write the right-hand side of equation (1.10), and accordingly infinitely many (nonequivalent!) central finitedifference discretizations. Another one, having different properties, is studied in [31]. Our convergence result only applies to the particular form (1.12), since only for that one, we obtain “the right” Lyapunov functionals that provide a priori estimates for the discrete-to-continuous limit. Finally, we define a time-dependent, spatially piecewise constant density function u n∆ : Ω → ℝ≥0 from the sequence x⃗ ∆ := (x⃗ n∆ )∞ n=0 via K

u n∆ = uδ [x⃗ n∆ ] := ∑ k=1

δ 𝕀(x ,x ] . x nk − x nk−1 k−1 k

(1.13)

360 | 14 Convergence of a scheme for a thin-film equation

By definition, the densities are nonnegative and of time-independent mass, K

xk

∫ u n∆ dx = ∑ ∫ k=1 x k−1

Ω

δ = Kδ = M . x nk − x nk−1

Finally, we introduce the piecewise constant interpolation {u ∆ }τ : ℝ≥0 × Ω → ℝ≥0 in time by (1.14) {u ∆ }τ (t) = u n∆ for (n − 1)τ < t ≤ nτ , and {u ∆ }τ (0) = u 0∆ .

14.1.3 Main results For the statement of our first result, fix a discretization parameter ∆ = (τ; δ). On mono⃗ introduce the functionals tone vectors x⃗ ∈ ℝK−1 with densities z⃗ = z[x], K

[1] Hδ (x)⃗ := δ ∑ log(z k− 12 ),

EVδ (x)⃗ :=

k=1

K δ K z k+ 12 + z k− 12 z k+ 12 − z k− 12 2 ∑ ( ) +δ ∑ V x (x k ), 2 k=1 2 δ k=0

which are discrete replacements for the entropy and the modified Dirichlet energy functionals, respectively. Theorem 1.1. From any monotone discrete initial datum x⃗ 0∆ , a sequence of monotone x⃗ n∆ satisfying (1.12) can be constructed by inductively defining x⃗ n∆ as a global minimizer of x⃗ 󳨃→

δ 2 V ⃗ ∑ (x k − x n−1 k ) + E δ (x) . 2τ k

(1.15)

This sequence of vectors x⃗ n∆ dissipates both the Boltzmann entropy and the discrete Dirichlet energy, [1] [1] Hδ (x⃗ n∆ ) ≤ Hδ (x⃗ n−1 ∆ )

and

EVδ (x⃗ n∆ ) ≤ EVδ (x⃗ n−1 ∆ ).

To state our main result about convergence, recall the definition (1.14) of the time interpolant. Further, ∆ symbolizes a whole sequence of mesh parameters from now on, and we write ∆ → 0 to indicate that τ → 0 and δ → 0 simultaneously. Theorem 1.2. Let a nonnegative initial condition u 0 ∈ H 1 (Ω) of finite second moment be given and fix a time horizon T > 0. Choose initial approximations x⃗ 0∆ such that u 0∆ = u[x⃗ 0∆ ] ⇀ u 0 weakly in H 1 (Ω) as ∆ → 0, and EV := sup EVδ (x⃗ 0∆ ) < ∞, ∆

[1]

H[1] := sup Hδ (x⃗ 0∆ ) < ∞.

(1.16)

∆

For each ∆, construct a discrete approximation x⃗ ∆ according to the procedure described in Theorem 1.1 above. Then, there are a subsequence with ∆ → 0 and a limit function u ∗ ∈ C([0, T] × Ω) such that:

14.1 Introduction

– – – –

| 361

{u ∆ }τ converges to u∗ locally uniformly on [0, T] × Ω, u ∗ ∈ L2 ([0, T]; H 1 (Ω)), u ∗ (0) = u 0 , u ∗ satisfies the following weak formulation of (1.1) with no-flux boundary conditions (1.2): T

T

∫ ∫ ∂ t ηρu ∗ dt dx + ∫ ηN(u ∗ , ρ) dt = 0 , 0 ℝ

(1.17)

0

for any test functions ρ ∈ C∞ (Ω) with ρ 󸀠 (a) = ρ 󸀠 (0) = 0, and η ∈ C∞ c ((0, T)), where the operator N is given by N(u, ρ) :=

1 ∫ ((u 2 )x ρ xxx + 3u 2x ρ xx ) dx + ∫ V x uρ x dx . 2 Ω

(1.18)

Ω

Remark 1.3. (1) Quality of convergence: Since {u ∆ }τ is piecewise constant in space and time, uniform convergence is obviously the best kind of convergence that can be achieved. (2) Rate of convergence: Numerical experiments with smooth initial data u 0 show that the rate of convergence if of the order τ + δ2 , see Section 14.5. (3) No uniqueness: Since our notion of solution is very weak, we cannot exclude that different subsequences of {u ∆ }τ converge to different limits. (4) Initial approximation: The assumptions in (1.16) are not independent: bounded[1] ness of EVδ (x⃗ 0∆ ) implies boundedness of Hδ (x⃗ 0∆ ) from above.

14.1.4 Relation to the literature The idea to derive numerical discretizations for solution of Wasserstein gradient flows from the Lagrangian representation is not new in the literature. A very general (but rather theoretical) treatise was given in [24]. Several practical schemes have been developed on grounds of the Lagrangian representation for this class of evolution problems, mainly for second-order diffusion equations [8, 9, 26, 32], but also for chemotaxis systems [7], for nonlocal aggregation equations [11, 13], and for variants of the Boltzmann equation [20]. Lagrangian schemes for fourth-order equations are relatively rare. Alternative Lagrangian discretizations for (1.1) or related thin-film-type equations have been proposed and analyzed in [14, 20], but no rigorous convergence analysis has been carried out. We also mention two schemes [17, 29] for the quantum drift diffusion equation, which is formally similar to (1.1). In [29], the idea to enforce dissipation of two Lyapunov functionals has been developed, and was used to rigorously study the discreteto-continuous limit. For the analysis here, we shall borrow various ideas from [29]. A comment is in place on related Lagrangian schemes in higher spatial dimensions. Here, and also in our related works [28, 29, 31], the most significant benefit

362 | 14 Convergence of a scheme for a thin-film equation

from working on a one-dimensional interval, is that the space of densities is flat with respect to the L2 -Wasserstein metric; it is of nonpositive curvature in higher dimensions, which makes the numerical approximation of the Wasserstein distance significantly more difficult. Just recently, a very promising approach for a truly structurepreserving discretization in higher space dimensions has been made [3]. There, a numerical solver for second-order drift diffusion equations with aggregation in multiple space dimensions is introduced that preserves – in addition to the Lagrangian and the gradient flow aspects – also “some geometry” of the optimal transport. These manifold structural properties enable the authors to rigorously perform a (partial) convergence analysis. It is currently unclear if that approach can be pushed further to deal with fourth-order equations as well. Among the numerous non-Lagrangian approaches to numerical discretization of thin-film equations, we are aware of two contributions [23, 34] in which the idea to enforce simultaneous dissipation of energy and entropy has been implemented. The discretization is performed using finite elements [23] and finite differences [34], respectively. The discrete-to-continuous limit has been rigorously analyzed for both schemes. In difference to the convergence result presented here, certain positivity hypotheses on the limit solution are either assumed a priori [34], or are incorporated in the weak form of the limit equation [23]. See, however, [22] for an improvement of the convergence result. The primary challenge in our convergence analysis is to carry out all estimates under no additional assumptions on the regularity of the limit solution u ∗ . In particular, we would like to deal with compactly supported solutions of a priori low regularity at the edge of the support. Also, we allow very general initial conditions u 0 . Without sufficient a priori smoothness, we cannot simply use Taylor approximations and the like to estimate the difference between {u ∆ }τ and u ∗ . Instead, we are forced to derive new a priori estimates directly from the scheme, using our two Lyapunov functionals. On the technical level, the main difficulty is that our scheme is fully discrete, which means that we are working with spatial difference quotients instead of derivatives. Lacking a discrete chain rule, the derivation of the relevant estimates turns out to be much harder than for the original problem (1.1). For instance, we are able to prove a compactness estimate for u ∆ , but not for its inverse distribution function, although both estimates would be equivalent in a smooth setting. This forces us to switch back and forth between the original (1.1) and the Lagrangian (1.10) formulation of the thinfilm equation.

14.1.5 Key estimates We give a very formal outline for the derivation of the two main a priori estimate on the fully discrete solutions.

14.1 Introduction

|

363

The first main estimate is related to the gradient flow structure of (1.1): it is the potential flow of the modified Dirichlet energy EV with respect to the Wasserstein metric W2 . The consequences, which are immediate from the abstract theory of gradient flows [1], are that t 󳨃→ EV (u(t)) is monotone, and that each solution “curve” t 󳨃→ u(t) is globally Hölder- 12 -continuous with respect to W2 . In order to inherit these properties to our discretization, the latter is constructed as a gradient flow of a flow potential EVδ (which approximates EV in a certain sense) with respect to a particular metric on the space of monotone vectors (which is related to W2 ). See Section 14.2.1 for details. The corresponding fully discrete energy estimates are collected in Proposition 3.1. We are not able to give a meaning to the full energy dissipation relation (1.6) on the discrete side, but this is irrelevant to our analysis. The second, equally important discrete estimate mimicks (1.7). Unfortunately, the L2 -norm of u xx is an inconvenient quantity to deal with, for two reasons. First, we need to perform most of the estimates in the Lagrangian picture, where M

∫ u 2xx dx = Ω

1 ∫ Z(Z 2 )2ξξ dξ 4 0

is algebraically more difficult to handle than the equivalent functional M

∫ u 2 (log u)2xx dx = ∫ Z 3 Z 2ξξ dξ , Ω

(1.19)

0

which we shall eventually work with, see Lemma 3.2. Our discretization (1.12) is tailormade in such a way that entropy dissipation yields a discrete version of (1.19). Second, the formulation of an H 2 -estimate would require a global C1,1 -interpolation of the piecewise constant densities u ∆ that respects positivity, which seems impractical. Instead, we settle for a control on the total variation of the first derivative ∂ ξ û ∆ of a simple C0,1 -interpolation û ∆ , see Proposition 3.3. This TV-control is a perfect replacement for the H 2 -estimate in (1.7), and is the source for compactness, see Proposition 3.7.

14.1.6 Structure of the paper Below, we start with a detailed description of our numerical scheme as a discrete Wasserstein-like gradient flow and discuss structural consistency of our approach. In Section 14.3, we derive various a priori estimates on the fully discrete solutions. This leads to the main convergence result in Proposition 3.7, showing the existence of a limit function u ∗ for ∆ → 0. This limit function satisfies the weak formulation of (1.1) stated in (1.17); this is shown in Section 14.4. Finally, we report on numerical experiments and discuss the observed rate of convergence in Section 14.5.

364 | 14 Convergence of a scheme for a thin-film equation

14.2 Definition of the fully discrete scheme The main aim of this section is to interpret the discrete equations (1.12) as time steps in the minimizing movement scheme for a suitable discretization EVδ of the functional EV with respect to a Wasserstein-like metric on a finite-dimensional submanifold Pδ (Ω) of P2 (Ω).

14.2.1 Ansatz space and discrete entropy/information functionals Fix K ∈ ℕ, let δ := 1/K, and define ξ k = Mk/K for k = 0, 1, . . . , K. For further reference, we introduce the sets of integer and half-integer indices 𝕀+K = {1, . . . , K − 1},

𝕀K = {0, 1, . . . , K},

1/2

and 𝕀K

1 3 1 ={ , ,...,K− } . 2 2 2

First- and second-order central difference operators D1δ and D2δ are defined in the usual way: if y⃗ = (yℓ )ℓ∈𝕀K is a discrete function defined for integer indices ℓ ∈ 𝕀K (i.e., on the 1/2 nodes ξℓ ), then D1δ y⃗ and D2δ y⃗ are defined on half-integer indices κ ∈ 𝕀K (i.e., on the intervals [ξ κ− 12 , ξ κ+ 12 ]), and on the “inner” integer indices k ∈ 𝕀+K , respectively, with [D1δ y]⃗ κ =

y k+ 1 − y k− 1 2

2

δ

,

[D2δ y]⃗ k =

y κ+1 − 2y κ + y κ−1 . δ2 1/2

If y⃗ = (y λ )λ∈𝕀1/2 is defined for half-integer indices λ ∈ 𝕀K instead, then these definiK

tions are modified in the obvious way to have D1δ y⃗ and D2δ y⃗ defined for integers k ∈ 𝕀K 1/2 and half-integers κ ∈ 𝕀K , respectively; y⃗ needs to be augmented with additional values for y− 1 and y K+ 1 in this case. 2 2 Next, we introduce the set of monotone vectors 󵄨 xδ := {(x0 , . . . , x K ) 󵄨󵄨󵄨 x0 < x1 < ⋅ ⋅ ⋅ < x K−1 < x K } ⊆ ℝK−1 . Each x⃗ ∈ xδ corresponds to a vector z⃗ = (z1/2 , z3/2 , . . . , z K−1/2 ) of density values z κ via (1.11). Our convention is that z− 1 = z 1 and z K+ 1 = z K− 1 . For a function f : xδ → ℝ, its 2 2 2 2 first and second differential, ∂x⃗ f : x → ℝK−1 and ∂2x⃗ f : x → ℝ(K−1)×(K−1) , respectively, ⃗ k = ∂ x k f(x)⃗ and by [∂ x⃗ f(x)] ⃗ kℓ = ∂ x k ∂ xℓ f(x). ⃗ Further, f ’s gradient are defined by [∂x⃗ f(x)] ⃗ For vectors v,⃗ w⃗ ∈ ℝK−1 , the scalar product ⟨⋅, ⋅⟩ δ ∇δ f is given by ∇δ f(x)⃗ = δ−1 ∂x⃗ f(x). is defined by K

⃗ δ = δ ∑ 𝑣k w k , ⟨v,⃗ w⟩ k=0

󵄩 󵄩 ⃗ δ. with induced norm 󵄩󵄩󵄩v⃗󵄩󵄩󵄩 δ = √⟨v,⃗ v⟩

Example 2.1. Each component z κ of z⃗ = z δ [x]⃗ is a function on xδ , and ∂x⃗ z κ = −z2κ

eκ+ 1 − eκ− 1 2

2

δ

,

(2.1)

where ek ∈ ℝK−1 is the kth canonical unit vector, with the convention e0 = eK = 0.

14.2 Definition of the fully discrete scheme |

365

The main object of interest is the finite-dimensional submanifold Pδ (Ω) of P2 (Ω) that ⃗ with uδ given consists of all locally constant density functions of the form u = uδ [x], in (1.13), where x⃗ ∈ xδ . To each density function u = uδ [x]⃗ ∈ Pδ (Ω) we associate its Lagrangian map as the monotonically increasing function X = X δ [x]⃗ : [0, M] → Ω that is piecewise linear with respect to (ξ0 , ξ1 , . . . , ξ K ) and satisfies X(ξ k ) = x k for k = 0, . . . , K. The density u and its Lagrangian map X are related as u∘X=

1 . Xξ

Remark 2.2. In one space dimension, the Wasserstein metric on P2 (Ω) is isometrically equivalent to the L2 -norm on the flat space of Lagrangian maps, see, e.g., [33]. Our 󵄩 󵄩 norm 󵄩󵄩󵄩x⃗ − y⃗ 󵄩󵄩󵄩δ is not identical but equivalent to the L2 -norm between the Lagrangian ⃗ Consequently, there exist K-independent constants c1 , c2 > 0, maps X δ [x]⃗ and X δ [y]. such that 󵄩 󵄩 ⃗ uδ [y]) ⃗ ≤ c2 󵄩󵄩󵄩󵄩x⃗ − y⃗ 󵄩󵄩󵄩󵄩δ , c1 󵄩󵄩󵄩x⃗ − y⃗ 󵄩󵄩󵄩δ ≤ W2 (uδ [x],

for all x,⃗ y⃗ ∈ xδ .

(2.2)

See [28, Lemma 7] for a proof. [1]

[2]

Next, consider two functionals Hδ , Hδ : xδ → ℝ given as follows: K

[1]

⃗ = ∫ uδ [x]⃗ log (uδ [x]) ⃗ dx = δ ∑ log(z k− 1 ) , Hδ (x)⃗ = H[1] (uδ [x]) 2 k=1

Ω [2]

⃗ = Hδ (x)⃗ = H[2] (uδ [x])

K

1 δ ⃗ 2 dx = ∑ z k− 1 . ∫ (uδ [x]) 2 4 4 k=1 Ω

[1]

[2]

Here Hδ is just the restriction of the the entropy H[1] to xδ , and Hδ is the restriction of the quadratic Renyi entropy M

H[2] (u) =

1 1 ∫ u 2 dx = ∫ Z dξ . 4 4 0

Ω

Using (2.1), we obtain an explicit representation of the gradients, eκ− 1 − eκ+ 1 eκ− 1 − eκ+ 1 δ [1] [2] 2 2 2 2 ∂x⃗ Hδ (x)⃗ = δ ∑ z κ , ∂x⃗ Hδ (x)⃗ = ∑ z2κ , δ 4 1/2 δ 1/2 κ∈𝕀K

(2.3)

κ∈𝕀K

and – for further reference – also of the Hessians, [1]

∂2x⃗ Hδ (x)⃗ = δ ∑ z2κ ( κ∈𝕀1/2 K [2]

∂2x⃗ Hδ (x)⃗ =

eκ− 12 − eκ+ 12 δ

)(

eκ− 12 − eκ+ 12 δ

T

) ,

eκ− 1 − eκ+ 1 eκ− 1 − eκ+ 1 T δ 2 2 2 2 ∑ z3κ ( )( ) . 2 1/2 δ δ

(2.4)

κ∈𝕀K

A key property of our simple discretization ansatz is the preservation of convexity.

366 | 14 Convergence of a scheme for a thin-film equation [1]

[2]

Lemma 2.3. The functionals Hδ and Hδ are convex on x. Proof. This follows by inspection of the Hessians (2.4). A conceptually different discretization is needed for the energy functional E from (1.5), which is identically +∞ on Pδ (Ω): [1]

[2]

⃗ ∇δ Hδ (x)⟩ ⃗ . Eδ (x)⃗ := ⟨∇δ Hδ (x),

(2.5)

δ

Substitution of the explicit representations (2.3) in the definition (2.5) yields [1]

[2]

⃗ ∇δ Hδ (x)⟩ ⃗ = Eδ (x)⃗ = ⟨∇δ Hδ (x), δ

z κ+ 12 + z κ− 12 z κ+ 12 − z κ− 12 2 δ ∑ ( ) . 2 1/2 2 δ k∈𝕀K

It remains to define a discrete counterpart for the potential V. A change of variables yields in the definition in (1.5) yields M

V(u) = ∫ V(x)u(x) dx = ∫ V(X) dξ , 0

Ω

Thus, a natural discretization Vδ of V is given by Vδ (x)⃗ = δ ∑ V(x k ) . k∈𝕀K

In summary, our discretization EVδ of EV is [1]

[2]

⃗ ∇δ Hδ (x)⟩ ⃗ + Vδ (x)⃗ . EVδ (x)⃗ = Eδ (x)⃗ + Vδ (x)⃗ = ⟨∇δ Hδ (x), δ

14.2.2 Discretization in time Next, the spatially discrete gradient flow equation ẋ⃗ = −∇δ EVδ (x)⃗

(2.6)

is also discretized in time, using minimizing movements. To this end, fix a time step with τ > 0; we combine the spatial and temporal mesh widths in a single discretization parameter ∆ = (τ; δ). For each y⃗ ∈ xδ , introduce the Yosida-regularized energy EV∆ (⋅; y)⃗ : xδ → ℝ by 1 󵄩󵄩 󵄩2 EV∆ (x;⃗ y)⃗ = 󵄩x⃗ − y⃗ 󵄩󵄩󵄩δ + EVδ (x)⃗ . 2τ 󵄩 A fully discrete approximation x⃗ ∆ = (x⃗ 0∆ , x⃗ 1∆ , . . . , x⃗ n∆ , . . . ) of (2.6) is now defined inductively from a given initial datum x⃗ 0∆ by choosing each x⃗ n∆ as a global minimizer of EV∆ (⋅; x⃗ n−1 ∆ ). Below, we prove that such a minimizer always exists, see Lemma 2.6.

14.2 Definition of the fully discrete scheme |

367

In practice, one wishes to define x⃗ n∆ as – preferably unique – solution of the Euler– Lagrange equations associated to EV∆ (⋅; x⃗ n−1 ∆ ), which leads to the implicit Euler time stepping: x⃗ − x⃗ n−1 1 [1] [2] [2] [1] ∆ ⃗ . = −∇δ EVδ (x)⃗ = − 2 (∂2x⃗ Hδ (x)⃗ ⋅ ∂x⃗ Hδ (x)⃗ + ∂2x⃗ Hδ (x)⃗ ⋅ ∂x⃗ Hδ (x)) τ δ

(2.7)

Using (2.3) and (2.4), a straightforward calculation shows that (2.7) is the precisely the numerical scheme (1.12) from the introduction. Equivalence of (2.7) and the minimization problem for EV∆ is guaranteed at least for sufficiently small τ > 0. Proposition 2.4. For each discretization ∆ and every initial condition x⃗ 0 ∈ xδ , the sequence of equations (2.7) can be solved inductively. Moreover, if τ > 0 is sufficiently small with respect to δ and EVδ (x⃗ 0 ), then each equation (2.7) possesses a unique solution with EVδ (x)⃗ ≤ Eδ (x⃗ 0 ), and that solution is the unique global minimizer of EV∆ (⋅; x⃗ n−1 ∆ ). Remark 2.5. In principle, the proof of Lemma 2.6 below provides a criterion on the smallness of τ > 0 that would guarantee the unique solvability of (2.7). We shall not make this criterion explicit, since in practice, we observe that the Newton method applied to (2.7) and initialized with x⃗ n−1 always converges to “the right” solution x⃗ n∆ , ∆ even for comparatively large steps τ and in rather degenerate situations; we refer the reader to our numerical results in Section 14.5. The proof of this proposition is a consequence of the following rather technical lemma. Lemma 2.6. Fix a spatial discretization parameter δ, and let C := EV (x⃗ 0 ). Then for every y⃗ ∈ xδ with EVδ (y)⃗ ≤ C, the following are true: – For each τ > 0, the function EV∆ (⋅; y)⃗ possesses at least one global minimizer x⃗ ∗ ∈ xδ , and that x⃗ ∗ satisfies the Euler–Lagrange equation x⃗ ∗ − y⃗ = −∇δ EVδ [x⃗∗ ] . τ –

There exists a τ C > 0 independent of y⃗ such that for each τ ∈ (0, τ C ), the global minimizer x⃗ ∗ ∈ xδ is strict and unique, and it is the only critical point of EV∆ (⋅; y)⃗ with EVδ (x)⃗ ≤ C.

Proof. Fix y⃗ ∈ xδ with EVδ (y)⃗ ≤ C, and define the nonempty (since it contains y)⃗ sub⃗ −1 ([0, C]) ⊂ xδ . Let z⃗ = z δ [x], ⃗ and observe that z κ ≥ δ/(b − a) level set A C := (EV∆ (⋅, y)) 1/2 for each z ∈ 𝕀K . From here, it follows further that zκ −

δ ≤ ∑ |z k+ 1 − z k− 1 | 2 2 b−a + k∈𝕀K

1 2

≤(∑ k∈𝕀+K

δ ) (δ ∑ (z k+ 12 + z k− 12 ) ( z k+ 12 + z k− 12 +

≤ (2(b − a))

k∈𝕀K

1/2 V Eδ (x)⃗ 1/2

≤ (4(b − a)C)1/2 .

z k+ 12 − z k− 12 δ

2

) )

1 2

368 | 14 Convergence of a scheme for a thin-film equation This implies that the differences x κ+ 1 − x κ− 1 = δ/z κ have a uniform positive lower 2 2 bound on A C . It follows that A C is a compact subset in the interior of xδ . Consequently, the continuous function EV∆ (⋅; y)⃗ attains a global minimum at x⃗ ∗ ∈ xδ . Since x⃗ ∗ ∈ A C lies in the interior of x,⃗ it satisfies ∂x⃗ EV∆ (x⃗ ∗ ; y)⃗ = 0, which is the Euler–Lagrange equation. This proves the first claim. Since EVδ : xδ → ℝ is smooth, its restriction to the compact set A C is λ C -convex with some λ C ∈ ℝ, i.e., ∂2x⃗ EVδ (x)⃗ ≥ λ C 1K−1 for all x⃗ ∈ A C . Independently of y,⃗ we have ∂2x⃗ EV∆ (x,⃗ y)⃗ = ∂2x⃗ EVδ (x)⃗ +

δ 1K−1 , τ

which means that x⃗ 󳨃→ EV∆ (x,⃗ y)⃗ is strictly convex on A C if 0 < σ < τ C :=

δ . (−λ C )

Consequently, each such EV∆ (⋅, y)⃗ has at most one critical point x⃗ ∗ in the interior of A C , and this x⃗ ∗ is necessarily a strict global minimizer.

14.2.3 Spatial interpolations Consider a fully discrete solution x⃗ ∆ = (x⃗ 0∆ , x⃗ 1∆ , . . . ). For notational simplification, we write the entries of the vectors x⃗ n∆ and z⃗ n∆ = z δ [x⃗ n∆ ] as x k and z κ , respectively, whenever there is no ambiguity in the choice of ∆ and the time step n. Recall that u n∆ = uδ [x⃗ n∆ ] ∈ Pδ (Ω) defines a sequence of densitites on Ω which are piecewise constant with respect to the (nonuniform) grid (a, x1 , . . . , x K−1 , b). To facilitate the study of convergence of weak derivatives, we also introduce piecewise ̂ n∆ : Ω → ℝ>0 . affine interpolations ̂z n∆ : [0, M] → ℝ>0 and u In addition to ξ k = kδ for k ∈ 𝕀K , introduce the intermediate points ξ κ = κδ for 1/2 κ ∈ 𝕀K . Accordingly, introduce the intermediate values for the vectors x⃗ n∆ and z⃗ n∆ : 1 1/2 (x 1 + x κ− 12 ) for κ ∈ 𝕀K , 2 κ+ 2 1 z k = (z k+ 12 + z k− 12 ) for k ∈ 𝕀+K . 2

xκ =

Now define – ̂z n∆ : [0, M] → ℝ as the piecewise affine interpolation of the values (z 12 , z 32 , . . . , z K− 1 ) with respect to the equidistant grid (ξ 1 , ξ 3 , . . . , ξ K− 1 ), and 2 2 2 2 – û n∆ : Ω → ℝ as the piecewise affine function with ̂ n∆ ∘ Xn∆ = ̂z n∆ . u

(2.8)

Our convention is that ̂z n∆ (ξ) = z 12 for 0 ≤ ξ ≤ δ/2 and ̂z n∆ (ξ) = z K− 12 for M−δ/2 ≤ ξ ≤ M, ̂ n∆ (x) = z 1 for x ∈ [a, x 1 ] and û n∆ (x) = z K− 1 for x ∈ [x K− 1 , b]. The and accordingly u 2 2 2 2

369

14.2 Definition of the fully discrete scheme |

definitions have been made such that ̂ (x k ) z k = ̂z (ξ k ) = u

x k = Xn∆ (ξ k ),

1/2

for all k ∈ 𝕀K ∪ 𝕀K .

(2.9)

̂ n∆ is piecewise affine with respect to the “double grid” (x0 , x 1 , x1 , . . . , Notice that u 2 x K− 12 , x K ), but in general not with respect to the subgrid (x0 , x1 , . . . , x K ). By direct calculation, we obtain for each k ∈ 𝕀+K that 󵄨 ∂ x û n∆ 󵄨󵄨󵄨(x

,x k )

=

󵄨 ∂ x û n∆ 󵄨󵄨󵄨(x k ,x 1 ) k+

=

k− 1 2

2

z k − z k− 1 2

x k − x k− 1 2 z k+ 1 − z k 2

x k+ 1 − x k

z k+ 1 − z k− 1

=

2

2

x k − x k−1 z k+ 1 − z k− 1

=

2

2

x k+1 − x k

2

= z k− 1

z k+ 1 − z k− 1 2

2

z k+ 1

δ − z k− 1

2

= z k+ 12

2

, (2.10)

2

.

δ

Trivially, we also have ∂ x û vanishes identically on the intervals (a, x 12 ) and (x K− 12 , b).

14.2.4 A discrete Sobolev-type estimate The following inequality plays a key role in our analysis. Recall the conventions that z− 12 = z 12 , z K+ 12 = z k− 12 , and that z k = 12 (z k+ 12 + z k− 12 ). Lemma 2.7. For any x⃗ ∈ xδ , δ ∑ zk (

z k+ 12 − z k− 12

k∈𝕀+K

δ

4

z κ+1 − 2z κ + z κ−1 2 9 δ ∑ z3κ ( ) . 4 δ2 1/2

) ≤

(2.11)

κ∈𝕀K

Proof. Define the left-hand side in (2.11) as (A). Then, 3

(A) = δ−3 ∑ z k (z k+ 12 − z k− 12 ) (z k+ 12 − z k− 12 ) k∈𝕀+K K

3

3

= δ−3 ∑ z k− 1 [z k−1 (z k− 1 − z k− 3 ) − z k (z k+ 1 − z k− 1 ) ] 2 2 2 2 2 k=1

= =

δ−3 2 δ−3 2

K

3

3

∑ z k− 12 [(z k− 12 + z k− 32 ) (z k− 12 − z k− 32 ) − (z k+ 12 + z k− 12 ) (z k+ 12 − z k− 12 ) ]

k=1 K

3

3

∑ z k− 1 [ (z k− 3 − z k− 1 ) (z k− 1 − z k− 3 ) + (z k− 1 − z k+ 1 ) (z k+ 1 − z k− 1 ) k=1

2

2

2

2

2

2

3

2

2

3

+ 2z k− 1 (z k− 1 − z k− 3 ) − 2z k− 1 (z k+ 1 − z k− 1 ) ] . 2

2

2

2

2

2

Rearranging terms yields K

3

3

(A) = −(A) − δ−3 ∑ z2k− 1 [(z k− 12 − z k− 32 ) − (z k+ 12 − z k− 12 ) ] k=1

2

2

370 | 14 Convergence of a scheme for a thin-film equation and further using the identity (a3 − b 3 ) = (a − b)(a2 + b 2 + ab), (A) = − =−

δ−3 K 2 3 ∑ z 1 [(z k− 1 − z k− 3 ) − (z k+ 1 − z k− 1 )3 ] 2 2 2 2 2 k=1 k− 2 δ−1 K 2 2 2 ∑ z 1 [D2 z] 1 [ (z k− 1 − z k− 3 ) + (z k+ 1 − z k− 1 ) 2 2 2 2 2 k=1 k− 2 δ k− 2

+ (z k− 1 − z k− 3 ) (z k+ 1 − z k− 1 ) ] . 2

2

2

2

Invoke Hölder’s inequality and the elementary estimate ab ≤ 12 (a2 + b 2 ) to conclude that 1 2

K 9 1 2 2 2 2 (A) ≤ (δ ∑ z3κ [D2δ z]κ ) (δ−3 ∑ z k− 1 [(z k− 1 − z k− 3 ) + (z k+ 1 − z k− 1 ) ] ) 2 2 2 2 2 2 4 1/2 k=1

1 2

κ∈𝕀K

1 2

1

2 K z 1 k− 2 3 2 4 4 ≤ (δ ∑ z3κ [D2δ z]κ ) (δ−3 ∑ [(z k− 12 − z k− 32 ) + (z k+ 12 − z k− 12 ) ]) 2 2 1/2 k=1

κ∈𝕀K

1 2

=

1 3 2 (δ ∑ z3κ [D2δ z]κ ) (A) 2 , 2 1/2

κ∈𝕀K

(2.12) where we have used an index shift and the conventions z− 1 = z 1 , z K+ 1 = z k− 1 in the 2 2 2 2 last step

14.3 A priori estimates and compactness 14.3.1 Energy and entropy dissipation Fix some discretization parameters ∆ = (τ; δ). Below, we derive a priori bounds on fully discrete solutions (x⃗ n∆ )∞ n=0 that are independent of ∆. Specifically, we shall prove two essential estimates: the first one is monotonicity of the energy EVδ , the second one [1]

is obtained from the dissipation of the auxiliary Lyapunov functional Hδ . We begin with the classical energy estimate. Proposition 3.1. One has that EVδ is monotone, i.e., EVδ (x⃗ n∆ ) ≤ EVδ (x⃗ n−1 ∆ ), and further: EVδ (x⃗ n∆ ) ≤ EVδ (x⃗ 0∆ ) n − x⃗ ∆ ‖2δ ∞ 󵄩 󵄩󵄩 x⃗ n − x⃗ n−1 󵄩󵄩󵄩2 󵄩󵄩 ∆ ∆ 󵄩 󵄩

‖x⃗ n∆

τ ∑ 󵄩󵄩 󵄩 n=1 󵄩 󵄩

τ

≤

2EVδ (x⃗ 0∆ ) (n

− n)τ

for all n ≥ 0 ,

(3.1)

for all n ≥ n ≥ 0 ,

(3.2)

∞ 󵄩 󵄩2 󵄩󵄩 = τ ∑ 󵄩󵄩󵄩󵄩∇δ EVδ (x⃗ n∆ )󵄩󵄩󵄩󵄩 δ ≤ 2EVδ (x⃗ 0∆ ) . 󵄩󵄩 n=1 󵄩δ

(3.3)

14.3 A priori estimates and compactness | 371

Proof. The monotonicity (3.5) follows (by induction on n) from the definition of x⃗ n∆ as minimizer of EV∆ (⋅; x⃗ n−1 ∆ ): 1 n 2 V V V V ⃗n ⃗ n ⃗ n−1 ⃗ n−1 ⃗ n−1 ⃗ n−1 ‖x⃗ − x⃗ n−1 ∆ ‖δ + E δ (x ∆ ) = E ∆ (x ∆ ; x ∆ ) ≤ E ∆ (x ∆ ; x ∆ ) = E δ (x ∆ ) . 2τ ∆ (3.4) Moreover, summation of these inequalities from n = n + 1 to n = n yields EVδ (x⃗ n∆ ) ≤

2

‖x⃗ n − x⃗ n−1 τ n n ∆ ‖δ ∑ [ ∆ ] ≤ EVδ (x⃗ ∆ ) − EVδ (x⃗ n∆ ) ≤ EVδ (x⃗ 0∆ ) . 2 n=n+1 τ For n = 0 and n → ∞, we obtain the first part of (3.3). The second part follows by (2.7). If instead we combine the estimate with Jensen’s inequality, we obtain 2 1/2 󵄩󵄩 ⃗ n 󵄩󵄩 n−1 n n n 󵄩󵄩x∆ − x⃗ n−1 n󵄩 󵄩󵄩 ⃗ n 󵄩δ ≤ (τ ∑ [ ‖x⃗ ∆ − x⃗ ∆ ‖δ ] ) (τ(n − n))1/2 , ∆ 󵄩 󵄩󵄩x∆ − x⃗ ∆ 󵄩󵄩󵄩 δ ≤ τ ∑ τ τ n=n+1 n=n+1

which leads to (3.2). The previous estimates were completely general. The following estimate is very particular for the problem at hand. [1]

[1]

[1]

Lemma 3.2. One has that Hδ is monotone, i.e., Hδ (x⃗ n∆ ) ≤ Hδ (x⃗ n−1 ∆ ). Moreover, it holds for any T > 0 that N

τ ∑ δ ∑ (z nκ )3 ( n=0

1/2

κ∈𝕀K

z κ+1 − 2z κ + z κ−1 2 ) ≤ 4(H[1] + ΛM(T + 1)) , δ2

(3.5)

for each N τ ∈ ℕ with N τ τ ∈ (T, T + 1). [1]

Proof. Convexity of Hδ implies that [1]

[1]

[1]

[1]

⃗n ⃗ n ⃗ n−1 − x⃗ n∆ ⟩ = τ ⟨∇δ Hδ (x⃗ n∆ ), ∇δ EVδ (x⃗ n∆ )⟩ , Hδ (x⃗ n−1 ∆ ) − H δ (x ∆ ) ≥ ⟨∇ δ H δ (x ∆ ), x ∆ δ

δ

for each n = 1, . . . , N τ . Summation of these inequalities over n yields Nτ

[1]

[1]

[1]

τ ∑ ⟨∇δ Hδ (x⃗ n∆ ), ∇δ EVδ (x⃗ n∆ )⟩ ≤ Hδ (x⃗ 0∆ ) − Hδ (x⃗ N∆ ) . δ

n=1

[1]

(3.6)

To estimate the right-hand side in (3.6), observe that Hδ (x⃗ 0∆ ) ≤ H[1] by hypothesis, [1] and that Hδ (x⃗ N∆ ) is bounded from below thanks to the convexity of s 󳨃→ s ln(s) and Jensen’s inequality, which yieds for any x⃗ ∈ xδ [1]

Hδ (x)⃗ = ∫ uδ [x]⃗ ln uδ [x]⃗ dx ≥ M ln ( Ω

M ) . b−a

372 | 14 Convergence of a scheme for a thin-film equation We turn to estimate the left-hand side in (3.6) from below. Recall that EVδ = Eδ +Vδ . For the component corresponding to Vδ , we find, using (1.3) and (2.3), [1]

⟨∇δ Hδ (x⃗ n∆ ), ∇δ Vδ (x⃗ n∆ )⟩ = δ ∑ z κ δ

1/2 κ∈𝕀K

V x (x κ− 1 ) − V x (x κ+ 1 ) 2

2

δ

≥ (inf V xx (x)) δ ∑ z κ x∈ℝ

x κ− 1 − x κ+ 1 2

2

δ

1/2 κ∈𝕀K

≥ −ΛM .

The component corresponding to Eδ is more difficult to estimate. Thanks to (2.3) and (2.4), we have [1]

⃗ ∇δ Hδ (x)⟩ ⃗ 4 ⟨∇δ Eδ (x), [1]

[2]

δ [1]

[2]

[1]

[1]

⃗ ∇2δ Hδ (x)∇ ⃗ δ Hδ (x)⟩ ⃗ ⃗ ∇2δ Hδ (x)∇ ⃗ δ Hδ (x)⟩ ⃗ = 4 ⟨∇δ Hδ (x), + 4 ⟨∇δ Hδ (x), δ

δ

z κ+1 − 2z κ + z κ−1 2 = 2δ ∑ z3κ ( ) δ2 1/2 κ∈𝕀K

+ δ ∑ z2κ ( 1/2

κ∈𝕀K

z2κ+1 − 2z2κ + z2κ−1 z κ+1 − 2z κ + z κ−1 ) ( ) . δ2 δ2

Further estimates are needed to control the second sum from below. Observing that z2κ+1 − 2z2κ + z2κ−1 z κ+1 − 2z κ + z κ−1 z κ+1 − z κ 2 z κ−1 − z κ 2 = 2z κ +( ) +( ) , 2 2 δ δ δ δ and that 2ab ≥ − 32 a2 − 23 b 2 for arbitrary real numbers a, b, we conclude that [1]

⃗ ∇δ Hδ (x)⟩ ⃗ 4 ⟨∇δ Eδ (x),

δ

z k+ 1 − z k− 1 4 3 z κ+1 − 2z κ + z κ−1 2 2δ 2 2 Th ≥ (4 − ) δ ∑ z3κ ( ) − z ( ∑ ) . k 2 3 κ∈𝕀 δ δ2 1/2 κ∈𝕀K

K

Now apply inequality (2.11).

14.3.2 Compactness The following lemma contains the key estimate to derive compactness of fully discrete solutions in the limit ∆ → 0. Below, we prove that from the entropy dissipation (3.5), ̂ n∆ . we obtain a control on the total variation of ∂ x u Several equivalent definitions of the total variation of f ∈ L1 (Ω) exist. In case of piecewise smooth functions with jump discontinuities, the most appropriate definition is } { J−1 TV [f ] = sup { ∑ |f(r j+1 ) − f(r j )| : J ∈ ℕ, a < r0 < r2 < ⋅ ⋅ ⋅ < r J < b } . } { j=0

(3.7)

373

14.3 A priori estimates and compactness |

Further recall the notation f x̄ = lim f(x) − lim f(x) . x↓ x̄

x↑ x̄

for the height of the jump in f(x)’s value at x = x.̄ Proposition 3.3. For any T > 0 and N τ ∈ ℕ with τN τ ∈ (T, T + 1), one has Nτ

̂ n∆ ]2 ≤ Č T . τ ∑ TV [∂ x u

(3.8)

n=1

Remark 3.4. The proof below yields for Č T the explicit value Č T = 50(H[1] + ΛM(T + 1)) . ̂ n∆ is locally constant on each interval (x k− 1 , x k ), and Proof. Fix n. The function ∂ x u 2 ̂ n∆ is given by the sum over equal to zero elsewhere. Therefore, the total variation of ∂ x u all jumps at the points of discontinuity, ̂ n∆ ] = ∑ 󵄨󵄨󵄨󵄨∂ x û n∆ x k 󵄨󵄨󵄨󵄨 + ∑ 󵄨󵄨󵄨󵄨∂ x û n∆ x κ 󵄨󵄨󵄨󵄨 . (3.9) TV [∂ x u k∈𝕀+K

1/2

κ∈𝕀K

The jumps can be evaluated by the direct calculation z k − z k− 1 z k+ 1 − z k− 1 z k+ 1 2 2 2 2 ̂ 󵄨󵄨󵄨󵄨(x 1 ,x ) = 1 ∂x u = = z k− k 2 k− x k − x k− 1 x k − x k−1 2 2 z k+ 1 − z k z k+ 1 − z k− 1 z k+ 1 2 2 2 2 ̂ 󵄨󵄨󵄨󵄨(x ,x 1 ) = = = z k+ 1 ∂x u k k+ 2 x 1 −x x −x k

k+ 2

2

k+1

k

− z k− 1 2

δ − z k− 1 2

δ

,

for k ∈ 𝕀K \{0} ,

,

for k ∈ 𝕀K \{K} . (3.10)

This implies that 󵄨󵄨 ̂ n 󵄨󵄨 󵄨󵄨∂ x u ∆ x k 󵄨󵄨 = δ (

z nk+ 1 − z nk− 1 2

2

δ

2

for k ∈ 𝕀+K ,

)

z n − 2z nκ + z nκ−1 1/2 󵄨󵄨 ̂ n 󵄨󵄨 ) for κ ∈ 𝕀K . 󵄨󵄨∂ x u ∆ x κ 󵄨󵄨 = δz nκ ( κ+1 δ2 We substitute this into (3.9), use Hölder’s inequality, and apply (2.11) to obtain as a consequence of elementary estimates that ̂ n∆ ] TV [∂ x u

≤δ ∑ (

z nk+ 1 − z nk− 1 2

2

δ

k∈𝕀+K

2

󵄨 󵄨 ) + δ ∑ z nκ 󵄨󵄨󵄨󵄨[D2δ z⃗ n∆ ]κ 󵄨󵄨󵄨󵄨 1/2

κ∈𝕀K

1

1

2 z nk+ 1 − z nk− 1 4 2 [ δ 2 2 [ ≤(∑ ) [(δ ∑ ( ) ) + (δ ∑ (z nκ )3 [D2δ z⃗ n∆ ]2κ ) δ + zk + k∈𝕀K k∈𝕀K κ∈𝕀1/2 K [ 1 2

≤

1 5 (2(b − a)) 2 (δ ∑ (z nκ )3 [D2δ z⃗n∆ ]2κ ) . 2 1/2

κ∈𝕀K

1 2

] ] ] ]

374 | 14 Convergence of a scheme for a thin-film equation We take both sides to the square, multiply by τ, and sum over n = 0, . . . , N τ . An application of the entropy dissipation inequality (3.5) yields the desired bound (3.8).

14.3.3 Convergence of time interpolants Lemma 3.5. There is a constant C > 0 just dependent on EV and (b − a), such that the following estimates hold uniformly as ∆ → 0: The functions {u ∆ }τ and {û ∆ }τ are uniformly bounded, and sup ‖∂ x {û ∆ } τ (t)‖L2 (Ω) ≤ C ,

(3.11)

̂ ∆ }τ (t) − {u ∆ }τ (t)‖L1 (Ω) ≤ Cδ , sup ‖ { u

(3.12)

̂ ∆ }τ (t)‖L∞ (Ω) ≤ C . sup ‖ { u

(3.13)

t∈ℝ≥0 t∈ℝ≥0 t∈ℝ≥0

Proof. For each n ∈ ℕ, z nk+ 1 − z nk 2 z nk − z nk− 1 2 󵄩󵄩 ̂ n 󵄩󵄩2 2 2 n n n n ) ] 󵄩󵄩 ∂ x u ∆ 󵄩󵄩L2 (Ω) = ∑ [(x k+ 1 − x k ) ( n n ) + (x k − x k− 1 ) ( n n 2 x 2 x − x 1 − xk + k k+ 2 k− 12 k∈𝕀K [ ] 2 2 n n n n n n n n − z − z z z x k − x k−1 x K−1 − x k k+ 12 k− 12 k+ 12 k− 12 [ = ∑ ) ] ( n ( n n ) + n 2 2 x − x x − x + K−1 k k k−1 k∈𝕀K [ ] 2 n n n n z k+ 1 + z k− 1 z k+ 1 − z k− 1 2 2 2 2 ( ) ≤ 2Eδ (x⃗ n∆ ) . =δ ∑ 2 δ + k∈𝕀K

This gives (3.11). For proving (3.12), we start with the elementary observation that |u n∆ (x) − û n∆ (x)| ≤ |z nk+ 1 − z nk− 1 | for all x ∈ [x k− 1 , x k+ 1 ] . 2

2

2

2

Therefore, 󵄨󵄨 z n 1 − z n 1 󵄨󵄨 k+ 2 k− 2 ‖u n∆ − û n∆ ‖L1 (Ω) ≤ δ ∑ δ 󵄨󵄨󵄨󵄨 δ 󵄨 k∈𝕀+K 󵄨󵄨

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨

1/2

δ ≤ δ( ∑ n) z k∈𝕀+ k K

(δ ∑ z nk ( k∈𝕀+K

z nk+ 1 − z nk− 1 2

2

δ

2

1/2

) )

≤ δ(2(b − a))1/2 Eδ (x⃗ n∆ )1/2 , which shows (3.12). Finally, (3.13) is a consequence of (3.11) and (3.12). First, note that 󵄩󵄩 ̂ 󵄩 ̂ ∆ } τ (t) − {u ∆ }τ (t)‖L1 (Ω) ≤ M + Cδ 󵄩󵄩{u ∆ }τ (t)󵄩󵄩󵄩 L1 (Ω) ≤ ‖ {u ∆ }τ (t)‖L1 (Ω) + ‖ { u

14.3 A priori estimates and compactness |

375

is uniformly bounded. Now apply the interpolation inequality 2/3

1/3

̂ ∆ } τ (t)‖L1 (Ω) ‖ { û ∆ } τ (t)‖L∞ (Ω) ≤ C‖∂ x {û ∆ }τ (t)‖L2 (Ω) ‖ { u to obtain the uniform bound in (3.13). Proposition 3.6. There exists a function u∗ : ℝ≥0 × Ω → ℝ≥0 that satisfies for any T > 0 u ∗ ∈ C1/2 ([0, T]; P2 (Ω)) ∩ L∞ ([0, T]; H 1 (Ω)) .

(3.14)

Furthermore, there exists a subsequence of ∆ (still denoted by ∆), such that the following are true: {u ∆ }τ (t) 󳨀→ u ∗ (t) {û ∆ } τ 󳨀→ u ∗ {X∆ }τ (t) 󳨀→ X∗ (t)

in P2 (Ω), uniformly with respect to time ,

(3.15)

uniformly on [0, T] × Ω ,

(3.16)

2

in L ([0, M]), uniformly with respect to t ∈ [0, T] ,

(3.17)

where X∗ ∈ C1/2 ([0, T]; L2 ([0, M])) is the Lagrangian map of u ∗ . Proof. From the discrete energy inequality (3.2) and the equivalence (2.2) of W2 with the usual L2 -Wasserstein metric W2 , it follows by elementary considerations that 2

W2 ( {u ∆ }τ (t), {u ∆ }τ (s)) ≤ C(|t − s|) ,

(3.18)

for all t, s ∈ [0, T]. Hence the generalized version of the Arzela–Ascoli theorem from [1, Proposition 3.3.1] is applicable and yields the convergence of a subsequence of ({u ∆ }τ ) to a limit u ∗ in P2 (Ω), locally uniformly with respect to t ∈ [0, ∞). The Hölder-type estimate (3.18) implies u ∈ C1/2 ([0, ∞); P2 (Ω)). The claim (3.17) is a consequence of the equivalence between the Wasserstein metric on P2 (Ω) and the L2 -metric on X, see Remark 2.2. In addition, the limit function u ∗ is bounded on [0, T]×Ω, thanks to (3.13). As an intermediate step toward proving uniform convergence of {û ∆ }τ , we show that ̂ ∆ (t) 󳨀→ u ∗ (t) in L2 (Ω), uniformly in t ∈ [0, T] . u (3.19) For t ∈ [0, T], we expand the L2 -norm as follows: ̂ ∆ }τ − u ∗ ) {u ∆ }τ ](t, x) dx ‖ { û ∆ } τ (t) − u ∗ (t)‖2L2 (Ω) = ∫ [( { u Ω

̂ ∆ } τ − u ∗ )( {û ∆ } τ − {u ∆ }τ )](t, x) dx + ∫ [( { u Ω

̂ ∆ } τ − u ∗ )u ∗ ](t, x) dx . − ∫ [( { u Ω

On the one hand, observe that sup ∫ [( {û ∆ } τ − u ∗ )( {û ∆ }τ − {u ∆ }τ )](t, x) dx

t∈[0,T]

Ω

̂ ∆ } τ (t)‖L∞ (Ω) + ‖u ∗ (t)‖L∞ (Ω) )‖ { u ̂ ∆ } τ (t) − {u ∆ }τ (t)‖L1 (Ω) ) , ≤ sup ((‖ { u t∈[0,T]

376 | 14 Convergence of a scheme for a thin-film equation which converges to zero as ∆ → 0, using both conclusions from Lemma 3.5. On the other hand, we can use a change of variables to write ∫ [( {û ∆ }τ − u ∗ ) {u ∆ }τ ](t, x) dx − ∫ [( {û ∆ } τ − u ∗ )u ∗ ](t, x) dx Ω

Ω M

M

̂ ∆ }τ − u ∗ ](t, {X ∆ }τ (t, x)) dξ − ∫ [ {û ∆ }τ − u ∗ ](t, X∗ (t, ξ)) dξ . = ∫ [ {u 0

0

We regroup terms under the integrals and use the triangle inequality. For the first term, we obtain 󵄨󵄨󵄨 M 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ̂ ∆ }τ (t, {X ∆ }τ (t, ξ)) − { u ̂ ∆ }τ (t, X∗ (t, ξ))) dξ 󵄨󵄨󵄨 sup 󵄨󵄨󵄨∫ ({ u 󵄨󵄨 t∈[0,T] 󵄨󵄨 󵄨󵄨 󵄨󵄨0 󵄨 M {X∆ }τ (t,ξ)

≤ sup ∫ t∈[0,T]

0

∫

󵄨󵄨 󵄨 󵄨󵄨∂ x {û ∆ }τ 󵄨󵄨󵄨 (t, y) dy dξ

X∗ (t,ξ)

M

≤ sup ∫ ‖ { û ∆ } τ ‖H 1 (Ω) |X∗ − {X ∆ }τ |(t, ξ)1/2 dξ t∈[0,T]

0 1/4

̂ ∆ }τ (t)‖H 1 (Ω) ‖X∗ (t) − {X ∆ }τ (t)‖L2 ([0,M]) ) . ≤ sup (‖ { u t∈[0,T]

̂ ∆ } τ . Together, A similar reasoning applies to the integral involving u ∗ in place of { u ∞ 1 this proves (3.19), and it further proves that u ∗ ∈ L ([0, T]; H (Ω)), since the uniform ̂ ∆ from (3.11) is inherited by the limit. bound on u Now the Gagliardo–Nirenberg inequality (A.1) provides the estimate 2/3

1/3

̂ ∆ }τ (t) − u ∗ (t)‖H 1 (Ω) ‖ { u ̂ ∆ }τ (t) − u ∗ (t)‖L2 (Ω) . (3.20) ‖ { û ∆ } τ (t) − u ∗ (t)‖C1/6 (Ω) ≤ C‖ { u Combining the convergence in L2 (Ω) by (3.19) with the boundedness in H 1 (Ω) from ̂ ∆ (t) → u ∗ (t) in C1/6 (Ω), uniformly in t ∈ [0, T]. This (3.11), it readily follows that u ̂ ∆ } τ → u ∗ uniformly on [0, T] × Ω. clearly implies that { u 1 Proposition 3.7. In the setting of Proposition 3.6, we have u∗ ∈ L∞ loc (ℝ≥0 ; H (Ω)), and

{û ∆ } τ → u ∗

strongly in L2 ([0, T]; H 1 (Ω))

(3.21)

for any T > 0 as ∆ → 0. ̂ n∆ is differentiable with local constant derivaProof. Fix [0, T] ⊆ ℝ≥0 . Remember that u 1/2 + ̂ n∆ (x) = 0 tives on any interval (x κ− 1 , x κ ] for κ ∈ 𝕀K ∪𝕀K ∪{K}, and it especially holds ∂ x u 2 for all x ∈ (a, a + δ/2) and all x ∈ (b − δ/2, b). Therefore, integration by parts and a

377

14.4 Weak formulation of the limit equation |

rearrangement of the terms yields 󵄩󵄩 ̂ n 󵄩󵄩2 󵄩󵄩∂ x u ∆ 󵄩󵄩L2 (Ω) =

xκ

∑

∫ ∂ x û n∆ ∂ x û n∆ dx =

κ∈𝕀+K ∪𝕀K ∪{K} x κ− 1

∑ κ∈𝕀+K ∪𝕀K ∪{K}

1/2

1/2

2

̂ n∆ (x)] [û n∆ (x)∂ x u

x=x κ −0 x=x κ− 1 +0 2

󵄩 ̂ n 󵄩󵄩 n ≤ 󵄩󵄩󵄩u ∆󵄩 󵄩L∞ (Ω) TV [∂ x û ∆ ] . Take further two arbitrary discretizations ∆1 , ∆2 and apply the above result on the dif̂ ∆2 } τ . Using that TV [f − g] ≤ TV [f ] + TV [g] we obtain by integration ference {û ∆1 } τ − { u w.r.t. time that T

󵄩 ̂ ∆1 } τ − ∂ x { u ̂ ∆2 } τ 󵄩󵄩󵄩󵄩2L2 (Ω) dt ∫ 󵄩󵄩󵄩 ∂ x { u 0 T

≤T

1/2

1/2

󵄩 ̂ ∆2 } τ 󵄩󵄩󵄩󵄩L∞ (Ω) (2 ∫ TV [∂ x { u ̂ ∆1 } τ ]2 + TV [∂ x { u ̂ ∆2 } τ ]2 dt) sup 󵄩󵄩󵄩{û ∆1 } τ − { u

t∈[0,T]

.

0

This shows that {û ∆ } τ is a Cauchy sequence in L2 ([0, T]; H 1 (Ω)) – remember (3.8) and especially the convergence result in (3.16) – and its limit has to coincide with u ∗ in the sense of distributions, due to the uniform convergence of {û ∆ }τ to u ∗ on [0, T]× Ω.

14.4 Weak formulation of the limit equation In the continuous theory, a suitable weak formulation for (1.1) is attained by applying purely variational methods, see for instance [19, 27]. More precisely, the weak formulation in (1.17) is obtained by studying the variation of the entropy E along a Wasserstein gradient flow generated by an arbitrary spatial test function ρ, which describes a transport along the velocity field ρ 󸀠 . The corresponding entropy functional is Φ(u) = ∫ℝ ρ(x)u(x) dx. It is therefore obvious to adapt this idea – similar as in [28, 29] – to show that {u ∆ }τ inherits a discrete analog to the weak formulation (1.17). Hence, we study the variations of the entropy Eδ along the vector field generated by the potential M

⃗ dξ Φ(x)⃗ = ∫ ρ(X δ [x]) 0

for any arbitrary smooth test function ρ ∈ C∞ (Ω) with ρ 󸀠 (a) = ρ 󸀠 (b) = 0. That is why we define M

⃗ v(⃗ x)⃗ = ∇δ Φ(x),

where

⃗ k = ∫ ρ 󸀠 (X(ξ))θ k (ξ) dξ, [∂x⃗ Φ(x)]

k = 1, . . . , K − 1 . (4.1)

0

Later on, we will use the compactness results from Section 14.3.3 to pass to the limit, which yields the weak formulation of our main result in Theorem 1.2. Therefore, the aim of this section is to show the following:

378 | 14 Convergence of a scheme for a thin-film equation Proposition 4.1. For every ρ ∈ C∞ (Ω) with ρ 󸀠 (a) = ρ 󸀠 (b) = 0, and for every η ∈ C∞ c (ℝ>0 ), the limit curve u ∗ satisfies ∞

∞

∫ ∫ ∂ t φu ∗ dt dx + ∫ N(u ∗ , φ) dt = 0 , 0 Ω

(4.2)

0

where the highly nonlinear term N from (1.18) is given by N(u, ρ) =

1 ∫(u 2 )x ρ 󸀠󸀠󸀠 + 3u 2x ρ 󸀠󸀠 dx + ∫ V x uρ 󸀠 dx . 2 Ω

(4.3)

Ω

The proof of this statement will be treated in two essential steps: (1) Show the validity of a discrete weak formulation for {u ∆ }τ , using a discrete flow interchange estimate. (2) Passing to the limit using Proposition 3.7. For definiteness, fix a spatial test function ρ ∈ C∞ (Ω) with ρ 󸀠 (a) = ρ 󸀠 (b) = 0, and a temporal test function η ∈ C∞ c (ℝ>0 ) with supp η ⊆ (0, T) for a suitable T > 0. Denote again by N τ ∈ ℕ an integer with τN τ ∈ (T, T + 1). Let B > 0 be chosen such that ‖ρ‖C4 (Ω) ≤ B

and ‖η‖C1 (ℝ≥0 ) ≤ B .

(4.4)

For convenience, we assume δ < 1 and τ < 1. In the estimates that follow, the nonexplicity constants possibly depend on Ω, T, B, and EV , but not on ∆. Lemma 4.2 (Discrete weak formulation). For any functions ρ ∈ C∞ (Ω) with ρ 󸀠 (a) = n n n ρ 󸀠 (b) = 0, and η ∈ C∞ c (ℝ>0 ), the solution x⃗ ∆ with u ∆ = u δ [x⃗ ∆ ] of the minimization problem (1.15) fulfills ∞

τ ∑ η((n − 1)τ) ( n=0

Φ(x⃗ n∆ ) − Φ(x⃗ n−1 ∆ ) − ⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 ⟩δ ) = O(τ) + O(δ1/4 ) , τ

(4.5)

where we use the short-hand notation ρ 󸀠 (x)⃗ := (ρ 󸀠 (x1 ), . . . , ρ 󸀠 (x K−1 )) for any x⃗ ∈ xδ . Proof. As a first step, we prove that both vectors ρ 󸀠 (x)⃗ and v(⃗ x)⃗ nearby coinside for any x⃗ ∈ xδ , i.e. it holds 󵄩󵄩 󵄩 󵄩󵄩v(⃗ x)⃗ − ρ 󸀠 (x)⃗ 󵄩󵄩󵄩 ≤ 2δ1/2 C , (4.6) 󵄩 󵄩δ for a constant C > 0 that only depends on B and Ω. Hence, denote by X = X δ [x]⃗ the corresponding Lagrangian map of x⃗ and choose any k = 0, . . . , K, then one easily gets M

⃗ k ) − ρ (x k )| = 󵄨󵄨󵄨󵄨[∇δ Φ(x)] ⃗ k − ρ 󸀠 (x k )󵄨󵄨󵄨󵄨 ≤ δ−1 ∫ 󵄨󵄨󵄨󵄨ρ 󸀠 (X(ξ))θ k (ξ) − ρ 󸀠 (x k )θ k (ξ)󵄨󵄨󵄨󵄨 dξ . |v(x 󸀠

0

First assume ξ ∈ [ξ k−1 , ξ k ], then a Taylor expansion for ρ 󸀠 (X(ξ)) yields ρ 󸀠 (X(ξ)) = ρ 󸀠 (x k ) − ρ 󸀠󸀠 (X(ξ̃k ))X ξ (ξ̃k )(ξ − ξ k ) = ρ 󸀠 (x k ) − ρ 󸀠󸀠 (X(ξ̃k ))(x k − x k−1 )θ k−1 (ξ)

14.4 Weak formulation of the limit equation |

ξ for a certain ξ̃k ∈ [ξ k−1 , ξ k ]. Consequently, the validity of ∫ξ k θ k θ k−1 dξ = k−1

3δ 2

379

yields

ξk

δ

−1

3 󵄨 󵄨 ∫ 󵄨󵄨󵄨ρ 󸀠 (X(ξ))θ k (ξ) − ρ 󸀠 (x k )θ k (ξ)󵄨󵄨󵄨 dξ ≤ B(x k − x k−1 ) . 2 ξ k−1

Similarly one proves the analog statement for ξ ∈ [ξ k , ξ K−1 ], hence ⃗ k ) − ρ 󸀠 (x k )| ≤ |v(x

3 3 ((x k − x k−1 ) + (x k+1 − x k )) = B(x k+1 − x k−1 ) . 2 2

Squaring the above term and summing-up over all k = 1, . . . , K−1 finally proves (4.6), due to (x k+1 − x k−1 ) ≤ 2(b − a) and K−1 9δ 2 󵄩󵄩 󵄩2 9δ 2 K−1 2 󵄩󵄩v(⃗ x)⃗ − ρ 󸀠 (x)⃗ 󵄩󵄩󵄩 ≤ B B ∑ (x − x ) ≤ (b − a) ∑ (x k+1 − x k−1 ) ≤ Cδ . k+1 k−1 󵄩 󵄩δ 4 2 k=1 k=1

Let us now invoke the proof of (4.5). a Taylor expansion of ρ for X, X󸀠 ∈ X yields ρ(X) − ρ(X󸀠 ) −

B (X − X󸀠 )2 ≤ ρ 󸀠 (X)(X − X󸀠 ) , 2

⃗n which implies for X󸀠 = X δ [x⃗ n−1 ∆ ] and X = X δ [x ∆ ] Φ(x⃗ n∆ )

−

Φ(x⃗ n−1 ∆ )

B − 2

M

󵄩󵄩2 K−1 n 󵄩󵄩 n 󸀠 󵄩 ≤ ∑ [x⃗ ∆ − x⃗ n−1 󵄩󵄩x⃗ ∆ − x⃗ n−1 ⃗n ∆ 󵄩 ∆ ]k ∫ ρ (X δ [x ∆ ])θ k (ξ) dξ 󵄩δ 󵄩 k=1

=

⟨x⃗ n∆

−

0 n−1 x⃗ ∆ , ∇δ Φ(x⃗ n∆ )⟩ δ

= τ ⟨∇δ EVδ (x⃗ n∆ ), ∇δ Φ(x⃗ n∆ )⟩ δ .

(4.7)

Thanks to (4.6), the last term can be estimated as follows: τ ⟨∇δ EVδ (x⃗ n∆ ), ∇δ Φ(x⃗ n∆ )⟩δ = τ ⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ n∆ )⟩δ + τ ⟨∇δ EVδ (x⃗ n∆ ), v(⃗ x⃗ n∆ ) − ρ 󸀠 (x⃗ n∆ )⟩δ 󵄩 󵄩 󵄩 󵄩 ≤ τ ⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ n∆ )⟩δ + τ 󵄩󵄩󵄩󵄩∇δ EVδ (x⃗ n∆ )󵄩󵄩󵄩󵄩 δ 󵄩󵄩󵄩󵄩v(⃗ x⃗ n∆ ) − ρ 󸀠 (x⃗ n∆ )󵄩󵄩󵄩󵄩 δ 󵄩 󵄩 ≤ τ ⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ n∆ )⟩δ + Cτδ1/2 󵄩󵄩󵄩󵄩∇δ EVδ (x⃗ n∆ )󵄩󵄩󵄩󵄩 δ . (4.8) So combining (4.7) and (4.7), add η((n − 1)τ) and summing-up over n = 1, . . . , N, one attains 󵄨󵄨 󵄨󵄨 N τ 󵄨󵄨 󵄨󵄨 Φ(x⃗ n∆ ) − Φ(x⃗ n−1 ∆ ) V ⃗n 󸀠 ⃗n 󵄨󵄨τ ∑ η((n − 1)τ) ( − ⟨∇δ Eδ (x∆ ), ρ (x∆ )⟩δ )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 τ 󵄨󵄨 n=1 󵄨 Nτ

B 2 n=1

≤ ‖η‖C0 ([0,T]) τ ∑

N

τ 󵄩󵄩2 󵄩 󵄩 󵄩󵄩 n 󵄩󵄩 + C‖η‖C0 ([0,T]) τ ∑ δ1/2 󵄩󵄩󵄩∇δ EVδ (x⃗ n∆ )󵄩󵄩󵄩 , 󵄩󵄩x⃗ ∆ − x⃗ n−1 ∆ 󵄩δ 󵄩 󵄩δ 󵄩

n=1

where the right-hand side is of the order O(τ) + O(δ1/2 ), due to (3.3). An analog calculation replacing ρ with −ρ finally leads to (4.5).

380 | 14 Convergence of a scheme for a thin-film equation

The identification of the weak formulation in (4.2) with the limit of (4.5) is splitted in two main steps: In the first one, we estimate the term that more or less describes the error that is caused by approximating the time derivative in (4.2) with the respective difference quotient in (4.5), 󵄨󵄨 T 󵄨󵄨󵄨 󵄨 e1,∆ := 󵄨󵄨󵄨󵄨 ∫ (η󸀠 (t) ∫ ρ(x) {u ∆ }τ (t, x) dx + η(t) {⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ n∆ )⟩ δ } (t)) dt󵄨󵄨󵄨󵄨 τ 󵄨󵄨 󵄨󵄨 0 Ω ≤ C(τ + δ1/2 ) .

(4.9)

The second much more challenging step is to prove the error estimate 󵄨󵄨 T 1 󵄨 e2,∆ := 󵄨󵄨󵄨󵄨 ∫ η(t)( ∫ ρ 󸀠󸀠󸀠 (x)∂ x ({u ∆ }2τ )(t, x) + 3ρ 󸀠󸀠 (x)∂ x {u ∆ }2τ (t, x) dx 2 󵄨󵄨 0 Ω 󵄨󵄨 󵄨 + ∫ V x (x) {u ∆ }τ ρ 󸀠 (x) dx − {⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ n∆ )⟩ δ } (t)) dt󵄨󵄨󵄨󵄨 ≤ Cδ1/4 , τ 󵄨󵄨 Ω

(4.10)

which, heuristically spoken, gives a rate of convergence of {⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ n∆ )⟩δ } toτ ward the nonlinear term N(u ∗ , ρ) from (4.3). The first estimate in (4.9) is a consequence of Lemma 4.2: Proof of (4.9). Using that η(nτ) = 0 for any n ≥ N τ , we obtain after “summation by parts”: T

− ∫ η󸀠 (t) (∫ ρ(x) {u ∆ }τ (t, x) dx) dt 0

Ω Nτ

nτ

n=1

(n−1)τ

= − ∑ ( ∫ η󸀠 (t) dt ∫ ρ(x)ū n∆ (x) dx) Ω M

Nτ

η(nτ) − η((n − 1)τ) = −τ ∑ ( ∫ ρ ∘ Xn∆ (ξ) dξ ) τ n=1 0

M

Nτ

= τ ∑ (η((n − 1)τ) ∫ n=1

ρ ∘ X n∆ (ξ) − ρ ∘ X n−1 ∆ (ξ) dξ ) . τ

(4.11)

0

Finally observe that 󵄨󵄨 󵄨󵄨 T 󵄨󵄨 󵄨󵄨 Nτ 󵄨 󵄨󵄨 󸀠 V V n 󸀠 n R := 󵄨󵄨󵄨∫ η(t) {⟨ρ (x⃗ ∆ ), ∇δ Eδ (x⃗ ∆ )⟩ δ } (t) dt − τ ∑ η((n − 1)τ) ⟨∇δ Eδ (x⃗ ∆ ), ρ (x⃗ ∆ )⟩ δ 󵄨󵄨󵄨󵄨 τ 󵄨󵄨 󵄨󵄨 n=1 󵄨󵄨 󵄨󵄨0 1/2 󵄨 󵄨󵄨 2 󵄨󵄨 nτ 1/2 N τ 󵄨󵄨 ∞ 󵄨󵄨 2 󵄨󵄨 1 2󵄩 V ⃗n 󵄩 󵄩 󵄨 󵄩 󵄩 󵄨 󵄩 󵄨 ∫ η(t) dt − η((n − 1)τ)󵄨󵄨 ) (τ ∑ B 󵄩󵄩∇δ Eδ (x∆ )󵄩󵄩 δ ) ≤ (τ ∑ 󵄨󵄨 τ 󵄨󵄨 n=1 󵄨󵄨󵄨 n=1 󵄨󵄨 󵄨 (n−1)τ ≤ ((T + 1)B2 τ2 )

1/2

(2B2 EVδ (x⃗ 0∆ ))1/2 = C󸀠 EVδ (x⃗ 0∆ )1/2 τ ,

14.4 Weak formulation of the limit equation |

381

using the energy estimate (3.3). We conclude that

e1,∆

󵄨M n−1 n 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨󵄨 ρ ∘ X ∆ (ξ) − ρ ∘ X ∆ (ξ) ≤ R + τ ∑ (󵄨󵄨η((n − 1)τ)󵄨󵄨 󵄨󵄨 ∫ dξ τ 󵄨󵄨 0 n=1 󵄨󵄨 󵄨 − ⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ n∆ )⟩ δ 󵄨󵄨󵄨󵄨) 󵄨󵄨 1/4 = O(τ) + O(δ ) , Nτ

(4.11)

M

where we have used (4.5), keeping in mind that Φ(x⃗ n∆ ) = ∫0 ρ(X n∆ ) dξ . The proof of (4.10) is treated essentially in two steps. In the first one, we rewrite the term ⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ n∆ )⟩δ (see Lemma 4.3), and use Taylor expansions to identify it with the corresponding integral terms of (4.3) up to some additional error terms, see Lemmata 4.6–4.10. Then we use the strong compactness result of Proposition 3.7 to pass to the limit as ∆ → 0 in the second step. Lemma 4.3. With the short-hand notation ρ 󸀠 (x)⃗ = (ρ 󸀠 (x1 ), . . . , ρ 󸀠 (x K−1 )) for any x⃗ ∈ xδ , one has − ⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ n∆ )⟩δ = A1n + A2n + A3n − A4n + A5n + A6n + A7n ,

(4.12)

where z nk+ 1 − z nk− 1

2

A1n

=δ ∑ (

A2n

z nk+ 1 − z nk− 1 2 ρ 󸀠 (x nk+1 ) − ρ 󸀠 (x nk ) δ 2 2 = ∑ ( ) (z nk+ 1 )2 ( ) , 4 + δ δ 2

2

2

δ

k∈𝕀+K

) ((z nk+ 1 )2 + (z nk− 1 )2 + z nk+ 1 z nk− 1 ) ( 2

2

2

2

k∈𝕀K

A3n =

z nk+ 1 − z nk− 1 2 ρ 󸀠 (x nk ) − ρ 󸀠 (x nk−1 ) δ 2 2 ∑ ( ) (z nk− 1 )2 ( ) , 4 + δ 2 δ k∈𝕀K

A4n

=δ ∑ (

z nk+ 1 − z nk− 1 2

δ

k∈𝕀+K

A5n = δ ∑ ( k∈𝕀+K

×(

2

2

) (

(z nk+ 1 )3 + (z nk− 1 )3 2

2

2z nk+ 1 z nk− 1 2

z nk+ 1 2

− δ

z nk− 1 2

)(

(z nk+ 1 )3 2

+

2

(z nk− 1 )3 2

2

ρ 󸀠 (x nk+1 ) − ρ 󸀠 (x nk ) − (x nk+1 − x nk )ρ 󸀠󸀠 (x nk ) δ2

) ρ 󸀠󸀠 (x nk ) ,

)

) ,

ρ 󸀠 (x nk+1 ) − ρ 󸀠 (x nk−1 ) ) , 2δ

382 | 14 Convergence of a scheme for a thin-film equation

A6n

=δ ∑ (

z nk+ 1 − z nk− 1 2

δ

k∈𝕀+K

×(

2

)(

(z nk+ 1 )3 + (z nk− 1 )3 2

2

2

ρ 󸀠 (x nk−1 ) − ρ 󸀠 (x nk ) − (x nk−1 − x nk )ρ 󸀠󸀠 (x nk ) δ2

)

) ,

A7n = δ ∑ V(x nk )ρ 󸀠 (x nk ) . k∈𝕀+K

Proof. Fix some time index n ∈ ℕ (omitted in the calculations below). Recall the representation of ∇δ EVδ as ∇δ EVδ (x)⃗ =

1 [1] [1] 2 [2] ⃗ ⃗ x⃗ H[2] ⃗ ⃗ (∂2 H (x)∂ δ (x) + ∂ x⃗ H δ (x)∂ x⃗ H δ (x)) δ2 x⃗ δ

with corresponding gradients and hessians in (2.3) and (2.4). Multiplication with ρ 󸀠 (x⃗ ∆ ) then yields − ⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ ∆ )⟩δ =

z κ+ 12 − 2z κ + z κ− 12 ρ 󸀠 (x κ+ 12 ) − ρ 󸀠 (x κ− 12 ) δ ∑ z3κ ( ) ( ) 2 1/2 δ δ2 κ∈𝕀K

z2κ+ 1 − 2z2κ + z2κ− 1 ρ 󸀠 (x κ+ 12 ) − ρ 󸀠 (x κ− 12 ) δ 2 2 2 ∑ zκ ( + ) ( ) 4 1/2 δ δ2 κ∈𝕀K

+ δ ∑ V(x k )ρ 󸀠 (x k ). k∈𝕀+K

Observing that z2κ+1 − 2z2κ + z2κ−1 z κ+1 − 2z κ + z κ−1 z κ+1 − z κ 2 z κ−1 − z κ 2 = 2z + ( + ( ) ) , κ δ δ δ2 δ2 we further obtain that − ⟨∇δ EVδ (x⃗ n∆ ), ρ 󸀠 (x⃗ ∆ )⟩ δ = δ ∑ z3κ (

z κ+ 1 − 2z κ + z κ− 1 2

1/2 κ∈𝕀K

2

δ2

)(

ρ 󸀠 (x κ+ 1 ) − ρ 󸀠 (x κ− 1 ) 2

2

δ

)

+ A2 + A3 + A7 . Hence, it remains to show that (A) = A1 − A4 + A5 + A6 , where (A) := δ ∑ z3κ (

z κ+ 12 − 2z κ + z κ− 12

1/2

κ∈𝕀K

δ2

)(

ρ 󸀠 (x κ+ 12 ) − ρ 󸀠 (x κ− 12 ) δ

) .

After “summation by parts” and an application of the elementary equality (for arbitrary numbers p± and q± ) p+ q+ − p− q− =

p+ + p− q+ + q− (q+ − q− ) + (p+ − p− ) , 2 2

14.4 Weak formulation of the limit equation |

383

one attains (A) =

z3k− 1 − z3k+ 1 z k+ 12 − z k− 12 ρ 󸀠 (x k+1 ) − ρ 󸀠 (x k−1 ) δ 2 2 ∑ ( )( )( ) 2 + δ δ δ k∈𝕀K

z3k− 1 + z3k+ 1 z k+ 1 − z k− 1 δ ρ 󸀠 (x k+1 ) − 2ρ 󸀠 (x k ) + ρ 󸀠 (x k−1 ) 2 2 2 2 + ∑ ( )( )( ) 2 + δ δ δ k∈𝕀K

= A1 +

z3k− 1 + z3k+ 1 z k+ 1 − z k− 1 δ ρ 󸀠 (x k+1 ) − 2ρ 󸀠 (x k ) + ρ 󸀠 (x k−1 ) 2 2 2 2 ∑ ( )( )( ) , 2 + δ δ δ k∈𝕀K

where we additionally used the identity (p3 − q3 ) = (p − q)(p2 + q2 + pq) in the last step. In order to see that the last sum in (14.4) equals to −A4 + A5 + A6 , simply observe that the identity z k+ 12 − z k− 12 1 1 x k+1 − x k x k−1 − x k − =− + = δ δ z k+ 1 z k− 1 z k+ 1 z k− 1 2

makes the coefficient of

ρ 󸀠󸀠 (x

2

2

2

k ) vanish.

For the analysis of the terms in (4.12), we need some sophisticated estimates presented in the following two lemmata. The first one gives a control on the oscillation of the zvalues at neighboring grid points: Lemma 4.4. For any p, q ∈ {1, 2} with p + q ≤ 3 one has Nτ

∑δ ∑ n=1

k∈𝕀+K

z nk

󵄨󵄨 z n − z n 󵄨󵄨 k+ 12 k− 12 󵄨󵄨 󵄨󵄨 δ 󵄨󵄨 󵄨

󵄨󵄨 󵄨󵄨p 󵄨󵄨 (z n )q 󵄨󵄨 󵄨󵄨 󵄨󵄨 k± 12 󵄨󵄨 ≤ Cδ1/4 . 󵄨󵄨 󵄨󵄨 − 1 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 (z n 1 )q 󵄨󵄨 󵄨󵄨 󵄨󵄨 k∓ 2

(4.13)

Proof. Instead of (4.13), we are going to prove that Nτ

τ∑δ ∑ n=1

k∈𝕀+K

z nk

󵄨󵄨 z n 1 − z n 1 󵄨󵄨 k+ 2 k− 2 󵄨󵄨 󵄨󵄨 δ 󵄨󵄨 󵄨

󵄨󵄨p 󵄨󵄨 z n 1 󵄨󵄨q 󵄨󵄨 󵄨󵄨 k± 2 󵄨󵄨 1/4 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 n − 1󵄨󵄨󵄨 ≤ Cδ 󵄨󵄨 󵄨󵄨 z k∓ 1 󵄨󵄨 󵄨 󵄨 󵄨 2

(4.14)

is satisfied for any p, q ∈ {1, 2} with p + q ≤ 3, which implies (4.13) because of the following considerations: The situation is clear for q = 1, thus assume q = 2 in (4.13). Then (4.14) is an upper bound on (4.13), due to (z nk± 1 )2 2

(z nk∓ 1 )2 2

for any n = 1, . . . , N τ .

−1= (

z nk± 1 2

z nk∓ 1 2

2

− 1) + 2 (

z nk± 1 2

z nk∓ 1 2

− 1)

384 | 14 Convergence of a scheme for a thin-film equation

To prove (4.14), we first apply Hölder’s inequality, Nτ

τ∑δ ∑ k∈𝕀+K

n=1

z nk

󵄨󵄨 z n 1 − z n 1 󵄨󵄨 k+ 2 k− 2 󵄨󵄨 󵄨󵄨 δ 󵄨󵄨 󵄨

Nτ

≤ (τ ∑ ∑ n=1 k∈𝕀+K

δz nk

(

󵄨󵄨q 󵄨󵄨 z n 1 − z n 1 󵄨󵄨p 󵄨󵄨 z n 1 Nτ 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 k± 2 k− 2 n 󵄨󵄨󵄨 k+ 2 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 n − 1󵄨󵄨 = τ ∑ δ ∑ z k 󵄨󵄨 δ 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 z k∓ 1 󵄨󵄨 n=1 k∈𝕀+K 󵄨 󵄨 󵄨 2

z nk+ 1 − z nk− 1 2

2

δ

p+q 4

4

) )

Nτ

(δτ ∑ ∑ n=1 k∈𝕀+K

z nk

(

q α

δ z nk± 1

q 󵄨󵄨p+q 󵄨󵄨 δ 󵄨󵄨 ( n ) 󵄨󵄨 z k∓ 1 󵄨󵄨 󵄨 2

α

) ) ,

2

(4.15) with α = 1 − p+q 4 . The first factor is uniformly bounded due to (2.11) and (3.5). For the second term, we use (3.13) and (A.2) to achieve Nτ

δτ ∑ ∑ n=1 k∈𝕀+K

z nk

(

q α

δ

q

z nk± 1

q

) ≤ (T + 1)δ(b − a) α ‖ { û } τ ‖L∞ ([0,T]×Ω) ≤ C(T + 1)δ(b − a) α ,

2

which shows (4.14), due to α ≥ 14 . Lemma 4.5. For any p ∈ {1, 2} one obtains that Nτ

τ∑δ ∑ k∈𝕀+K

n=1

z nk

󵄨󵄨 z n 1 − z n 1 󵄨󵄨 k+ 2 k− 2 󵄨󵄨 󵄨󵄨 δ 󵄨󵄨 󵄨

󵄨󵄨2 󵄨󵄨 󵄨󵄨 (x n − x n )p ≤ Cδ1/2 . 󵄨󵄨 k+1 k−1 󵄨󵄨 󵄨

(4.16)

Proof. Appling Hölder’s inequality, Nτ

τ∑δ ∑ n=1

k∈𝕀+K

z nk

󵄨󵄨 z n 1 − z n 1 󵄨󵄨 k+ 2 k− 2 󵄨󵄨 󵄨󵄨 δ 󵄨󵄨 󵄨

Nτ

≤ (τ ∑ δ ∑ n=1

k∈𝕀+K

z nk

󵄨󵄨2 󵄨󵄨 󵄨󵄨 (x n − x n )p 󵄨󵄨 k+1 k−1 󵄨󵄨 󵄨

󵄨󵄨 z n 1 − z n 1 󵄨󵄨 k+ 2 k− 2 󵄨󵄨 󵄨󵄨 δ 󵄨󵄨 󵄨

1/2 󵄨󵄨4 1/2 Nτ 󵄨󵄨 󵄨󵄨 ) (τ ∑ δ ∑ z n (x n − x n )2p ) . 󵄨󵄨 k k+1 k−1 󵄨󵄨 n=1 k∈𝕀+K 󵄨

The first sum is uniformly bounded thanks to (2.11) and (3.5), and the second one satisfies 1/2

Nτ

(τ ∑ δ ∑ z nk (x nk+1 − x nk−1 )2p ) n=1

k∈𝕀+K

1/2

̂ }τ ‖L∞ ([0,T]×Ω)(b − a)p , ≤ δ1/2 (T + 1)1/2 ‖ { u

where we used (3.13) and (A.2). Lemma 4.6. There is a constant C 1 > 0 expressible in Ω, T, B, and EV such that Nτ

M

n=1

0

󵄨󵄨 󵄨󵄨 R1 := τ ∑ 󵄨󵄨󵄨󵄨A1n − 3 ∫ ̂z n∆ (ξ)∂ ξ ̂z n∆ (ξ)2 ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ 󵄨󵄨󵄨󵄨 ≤ C1 δ1/4 . 󵄨 󵄨

385

14.4 Weak formulation of the limit equation |

Proof. Let us introduce the term

B1n := δ ∑ z nk (

2

z nk+ 1 − z nk− 1 2

δ

k∈𝕀+K

ξ k+ 1

2

3 2 ∫ ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ . ) δ ξ k− 1

2

First observe that by definition of

̂z n∆ ,

M

∫ ̂z n∆ (ξ)∂ ξ ̂z n∆ (ξ)2 ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ = ∑ ( k∈𝕀+K

0

z nk+ 1 − z nk− 1 2

2

δ

2 ξ k+ 12

)

∫ ̂z n∆ (ξ)ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ , ξ k− 1

2

hence we get for

B1n

󵄨󵄨 M 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 n n 2 󸀠󸀠 n n 󵄨󵄨󵄨3 ∫ ̂z ∆ (ξ)∂ ξ ̂z ∆ (ξ) ρ ∘ X∆ (ξ) dξ − B1 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 0 ≤ 3B ∑ (

z nk+ 1 − z nk− 1

k∈𝕀+K

2

2

δ

2 ξ k+ 12

󵄨󵄨 z n 1 − z n 1 󵄨 k− 2 󵄨󵄨̂n 󵄨󵄨 n 󵄨󵄨󵄨 k+ 2 ) ∫ 󵄨󵄨z ∆ (ξ) − z k 󵄨󵄨 dξ ≤ 3Bδ ∑ z k 󵄨󵄨 δ 󵄨 󵄨󵄨 k∈𝕀+K ξ k− 1 2

󵄨󵄨2 󵄨󵄨 z n 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 k+ 2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 n − 1󵄨󵄨󵄨 . 󵄨󵄨 󵄨󵄨 z k− 1 󵄨󵄨 󵄨 󵄨 󵄨 2

This especially implies, due to (4.13) that M

Nτ 󵄨 󵄨󵄨 󵄨 τ ∑ 󵄨󵄨󵄨󵄨B1n − 3 ∫ ̂z n∆ (ξ)∂ ξ ̂z n∆ (ξ)2 ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ 󵄨󵄨󵄨󵄨 ≤ Cδ1/4 . 󵄨 n=1 󵄨

(4.17)

0

For simplification of R1 , let us fix n (omitted in the following), and introduce x̃ +k ∈ [x k , x k+1 ] and x̃ −k ∈ [x k−1 , x k ] such that ρ 󸀠 (x k+1 ) − ρ 󸀠 (x k−1 ) ρ 󸀠 (x k+1 ) − ρ 󸀠 (x k ) ρ 󸀠 (x k ) − ρ 󸀠 (x k−1 ) = + 2δ 2δ 2δ + − x k+1 − x k x k+1 − x k 1 ρ 󸀠󸀠 (x̃ k ) ρ 󸀠󸀠 (x̃ k ) + ) . + ρ 󸀠󸀠 (x̃ +k ) = ( = ρ 󸀠󸀠 (x̃ +k ) 2δ 2δ 2 z k+ 1 z k− 1 2

2

Recalling that ξ k+1

∫ θ k (ξ) dξ = δ , ξ k−1

(4.18)

386 | 14 Convergence of a scheme for a thin-film equation one has for each k ∈ 𝕀+K , ξ k+ 1

ρ 󸀠󸀠 (x̃ +k ) ρ 󸀠󸀠 (x̃ −k ) 3 1 + )− ∫ z k ρ 󸀠󸀠 ∘ X∆ dξ (A) := (z2k+ 1 + z2k− 1 + z k+ 1 z k− 1 ) ( 2 2 2 z k+ 1 z k− 1 δ 2 2 2

2

2

ξ k− 1

2

=

z2k− 1

z k− 12 1 1 z k− 1 (2 + ) ρ 󸀠󸀠 (x̃ +k ) + z k+ 12 (2 + 2 2 ) ρ 󸀠󸀠 (x̃ +k ) 2 4 z k+ 12 4 z k+ 1 2

z k+ 12 1 1 + z k+ 1 (2 + ) ρ 󸀠󸀠 (x̃ −k ) + z k− 12 (2 + 2 1 4 z k− 2 4

z2k+ 1 2 z2k− 1 2

ξ k+ 1

2

3 ) ρ 󸀠󸀠 (x̃ −k ) − ∫ z k ρ 󸀠󸀠 ∘ X∆ dξ , δ ξ k− 1

2

and furthermore z2k− 1 z k− 12 1[ − 1) + z k+ 1 ( 2 2 − 1)] ρ 󸀠󸀠 (x̃ +k ) (A) = z k− 1 ( 2 2 4 z k+ 12 z k+ 1 [ ] 2 2 z k+ 1 z k+ 1 1 2 + [z k+ 12 ( − 1) + z k− 12 ( 2 2 − 1)] ρ 󸀠󸀠 (x̃ −k ) 4 z k− 1 z k− 1 2 ] [ 2 ξ k+ 1

ξ k+ 1

2

2

3 3 − ∫ z k [ρ 󸀠󸀠 ∘ X∆ − ρ 󸀠󸀠 (x̃ +k )] dξ − ∫ z k [ρ 󸀠󸀠 ∘ X∆ − ρ 󸀠󸀠 (x̃ −k )] dξ . 2δ 2δ ξ k− 1

ξ k− 1

2

2

Applying the trivial identity (for arbitrary numbers p and q) q(

p p2 − 1) = (p + q) ( − 1) , q q2

the above term finally reads as z k− 12 z k− 12 1 − 1) + 2z k ( − 1)] ρ 󸀠󸀠 (x̃ +k ) (A) = [z k− 1 ( 2 4 z k+ 12 z k+ 12 +

z k+ 1 z k+ 1 1 2 2 − 1) + 2z k ( − 1)] ρ 󸀠󸀠 (x̃ −k ) [z k+ 12 ( 4 z k− 1 z k− 1 2

ξ k+ 1

2

ξ k+ 1

2

2

3 3 − ∫ z k [ρ 󸀠󸀠 ∘ X∆ − ρ 󸀠󸀠 (x̃ +k )] dξ − ∫ z k [ρ 󸀠󸀠 ∘ X∆ − ρ 󸀠󸀠 (x̃ −k )] dξ . 2δ 2δ ξ k− 1

(4.19)

ξ k− 1

2

2

x̃ +k

Since lies between the values x k and x k+1 , and X∆ (ξ) ∈ [x k , x k+ 12 ] for each ξ ∈ [ξ k , ξ k+ 12 ], we conclude that |X∆ (ξ) − x̃ +k | ≤ x k+1 − x k , and therefore ξ k+ 1

2

3 3 󵄨 󵄨 ∫ z k 󵄨󵄨󵄨ρ 󸀠󸀠 ∘ X∆ (ξ) − ρ 󸀠󸀠 (x̃ +k )󵄨󵄨󵄨 dξ ≤ Bz k (x k+1 − x k ) . 2δ 2 ξk

(4.20)

14.4 Weak formulation of the limit equation |

387

A similar estimate is valid for the other integral over [ξ k− 1 , ξ k ] and for the integrals 2 with ρ 󸀠󸀠 (x̃ −k ). Thus, combining (4.19) and (4.20) with z nk± 1 ≤ 2z nk and the definition 2

of A1n , one attains that 󵄨󵄨 n 󵄨 󵄨󵄨A1 − B1n 󵄨󵄨󵄨 ≤ 2B ∑ z nk (

z nk+ 1 − z nk− 1 2

δ

k∈𝕀+K

+ 3B ∑ k∈𝕀+K

z nk

2

(

2

z nk− 1

) [( n − 1) + ( n 2 − 1)] z 1 z k+ 1 2 [ k− 2 ] 2

z nk+ 1 − z nk− 1 2

z nk+ 1

2

δ

2

) (x nk+1 − x nk−1 ) ,

and further, applying (4.13) and (4.16), Nτ

󵄨 󵄨 τ ∑ 󵄨󵄨󵄨A1n − B1n 󵄨󵄨󵄨 ≤ Cδ1/4 .

(4.21)

n=1

By triangle inequality, (4.17) and (4.21) provide the claim. Along the same lines, one proves the analogous estimate for A2n and A3n in place of A1n : Lemma 4.7. There are constants C 2 > 0 and C3 > 0 expressible in Ω, T, B, and EV such that M

Nτ 󵄨 󵄨󵄨 1 󵄨 R2 := τ ∑ 󵄨󵄨󵄨󵄨A2n − ∫ ̂z n∆ (ξ)∂ ξ ̂z n∆ (ξ)2 ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ 󵄨󵄨󵄨󵄨 ≤ C2 δ1/4 , 4 󵄨 n=1 󵄨 0

M

󵄨󵄨 󵄨󵄨 1 R3 := τ ∑ 󵄨󵄨󵄨󵄨A3n − ∫ ̂z n∆ (ξ)∂ ξ ̂z n∆ (ξ)2 ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ 󵄨󵄨󵄨󵄨 ≤ C3 δ1/4 . 4 󵄨 n=1 󵄨 Nτ

0

Lemma 4.8. There is a constant C4 > 0 expressible in Ω, T, B, and EV such that Nτ

M

n=1

0

󵄨󵄨 󵄨󵄨 R4 := τ ∑ 󵄨󵄨󵄨󵄨A4n − ∫ ̂z n∆ (ξ)∂ ξ ̂z n∆ (ξ)2 ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ 󵄨󵄨󵄨󵄨 ≤ C4 δ1/4 . 󵄨 󵄨 Proof. The proof is almost identical to the one for Lemma 4.6. As before, we introduce the term B4n

:= δ ∑ k∈𝕀+K

z nk

(

z nk+ 1 − z nk− 1 2

2

δ

2

ξ k+ 1

2

1 ∫ ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ ) δ ξ k− 1

2

and get due to (4.13), analogously to (4.17), that M

Nτ 󵄨 󵄨󵄨 󵄨 τ ∑ 󵄨󵄨󵄨󵄨B4n − ∫ ̂z n∆ (ξ)∂ ξ ̂z n∆ (ξ)2 ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ 󵄨󵄨󵄨󵄨 ≤ Cδ1/4 . 󵄨 n=1 󵄨 0

(4.22)

388 | 14 Convergence of a scheme for a thin-film equation

By writing 3

(z nk+ 1 ) + (z nk− 1 ) 2

3

= z nk (

2

2z nk+ 1 z nk− 1 2

2

z nk− 1

− 1) + z nk (

2

z nk+ 1 2

z nk+ 1 2

z nk− 1

− 1) + z nk ,

2

one obtains 3

3

(z nk+ 1 ) + (z nk− 1 ) 2

2

2

2z nk+ 1 z nk− 1 2

ξ k+ 1

1 ρ 󸀠󸀠 (x nk ) − ∫ z k ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ δ ξ k− 1

2

2

=

z nk

(

z nk− 1 2

z nk+ 1

− 1) +

z nk

(

2

z nk+ 1 2

z nk− 1

ξ k+ 1

2

1 − 1) − ∫ z nk [ρ 󸀠󸀠 ∘ Xn∆ (ξ) − ρ 󸀠󸀠 (x nk )] dξ . δ ξ k− 1

2

2

Observing – in analogy to (4.20) – that ξ k+ 1

2

1 󵄨 󵄨 ∫ z nk 󵄨󵄨󵄨ρ 󸀠󸀠 ∘ Xn∆ (ξ) − ρ 󸀠󸀠 (x nk )󵄨󵄨󵄨 dξ ≤ Bz nk (x nk+ 1 − x nk− 1 ) , δ 2 2 ξ k− 1

2

we obtain the same bound on |A4n − B4n | as before on |A1n − B1n |, i.e., z nk+ 1 − z nk− 1

󵄨 󵄨󵄨 n 󵄨󵄨A4 − B4n 󵄨󵄨󵄨 ≤ B ∑ z nk (

2

δ

k∈𝕀+K

+B ∑ k∈𝕀+K

2

z nk

(

2

z nk− 1 z nk+ 1 ) [( n 2 − 1) + ( n 2 − 1)] z 1 z k+ 1 2 [ k− 2 ]

z nk+ 1 − z nk− 1 2

2

δ

2

) (x nk+1 − x nk−1 ) .

Again, applying (4.13) and (4.16), we get Nτ

󵄨 󵄨 τ ∑ 󵄨󵄨󵄨A4n − B4n 󵄨󵄨󵄨 ≤ Cδ1/4 ,

(4.23)

n=1

and the estimates (4.22) and (4.23) imply the desired bound on R4 . Lemma 4.9. There is a constant C5 > 0 expressible in Ω, T, B, and EV such that 󵄨 󵄨󵄨 M 󵄨󵄨 N τ 󵄨󵄨󵄨 󵄨󵄨 n 1 󵄨 n n 󸀠󸀠󸀠 n R5 := τ ∑ 󵄨󵄨󵄨A5 − ∫ ̂z ∆ (ξ)∂ ξ ̂z ∆ (ξ)ρ ∘ X∆ (ξ) dξ 󵄨󵄨󵄨󵄨 ≤ C5 δ1/4 . 2 󵄨 󵄨󵄨 n=1 󵄨󵄨 󵄨󵄨 0 󵄨 Proof. The idea of the proof is the same as in the previous proofs. Let us define similar to B1n the term B5n

:= δ ∑ k∈𝕀+K

z nk+ 1 2

(

z nk+ 1 − z nk− 1 2

2

δ

ξ k+ 1

2

1 ) ∫ ρ 󸀠󸀠󸀠 ∘ Xn∆ (ξ) dξ . 2δ ξ k− 1

2

389

14.4 Weak formulation of the limit equation |

Note in particular that we weight the integral here with z nk+ 1 . Then 2

ξ k+ 1 󵄨󵄨 󵄨󵄨 M 2 󵄨󵄨 󵄨󵄨 z nk+ 1 − z nk− 1 1 󵄨 󵄨󵄨 1 󵄨 󵄨 2 2 n n 󸀠󸀠󸀠 n n 󵄨 󵄨󵄨 ∫ ̂z ∆ (ξ)∂ ξ ̂z ∆ (ξ)ρ ∘ X∆ (ξ) dξ − B5 󵄨󵄨 ≤ B ∑ ( ) ∫ 󵄨󵄨󵄨̂z n∆ (ξ) − z nk+ 1 󵄨󵄨󵄨 dξ 󵄨󵄨 2 󵄨󵄨 2 δ 2 + 󵄨󵄨 󵄨󵄨 0 k∈𝕀K ξ k− 1 󵄨 󵄨 2 󵄨󵄨 󵄨󵄨󵄨 z n 1 − z n 1 󵄨󵄨󵄨 󵄨󵄨󵄨 z n 1 󵄨󵄨 k− 2 󵄨󵄨 󵄨󵄨 k− 2 1 󵄨 k+ 2 󵄨󵄨 󵄨󵄨 n − 1󵄨󵄨󵄨 , ≤ Bδ ∑ z nk+ 1 󵄨󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 2 2 󵄨 δ 󵄨󵄨 󵄨󵄨 z k+ 1 󵄨󵄨 󵄨󵄨 k∈𝕀+K 󵄨󵄨 2

where we used that by definition of ̂z n∆ , M

∫ ̂z n∆ (ξ)∂ ξ ̂z n∆ (ξ)ρ 󸀠󸀠󸀠

∘ Xn∆ (ξ) dξ

0

= ∑ (

ξ k+ 1

z nk+ 1 − z nk− 1 2

2

) ∫ ̂z n∆ (ξ)ρ 󸀠󸀠󸀠 ∘ Xn∆ (ξ) dξ .

2

δ

k∈𝕀+K

ξ k− 1

2

This especially implies, due to (4.13) that M

󵄨󵄨 󵄨󵄨 1 τ ∑ 󵄨󵄨󵄨󵄨B5n − ∫ ̂z n∆ (ξ)∂ ξ ̂z n∆ (ξ)ρ 󸀠󸀠󸀠 ∘ Xn∆ (ξ) dξ 󵄨󵄨󵄨󵄨 ≤ Cδ1/4 . 2 󵄨 n=1 󵄨 Nτ

(4.24)

0

Furthermore, one can introduce intermediate values x̃ +k such that 1 n δ2 ρ 󸀠󸀠󸀠 (x̃ +k ) . (x k+1 − x nk )2 ρ 󸀠󸀠󸀠 (x̃ +k ) = 2 2(z nk+ 1 )2

ρ 󸀠 (x nk+1 ) − ρ 󸀠 (x nk ) − (x nk+1 − x nk )ρ 󸀠󸀠 (x nk ) =

2

Using the identity (

(z nk+ 1 )3 + (z nk− 1 )3 2

2

2

)

1 2(z nk+ 1 )2

=

z nk+ 1 2

2

+

z nk− 1 2

4

2

(

(z nk− 1 )2 2

(z nk+ 1 )2

− 1) +

z nk+ 1 2

4

(

z nk− 1 2

z nk+ 1

− 1) ,

2

2

we thus have – using again (4.18) – that

(

(z nk+ 1 )3 + (z nk− 1 )3 2

2

2

)(

ρ 󸀠 (x nk+1 ) − ρ 󸀠 (x nk ) − (x nk+1 − x nk )ρ 󸀠󸀠 (x nk ) δ2

ξ k+ 1

2

1 )− ∫ z nk ρ 󸀠󸀠󸀠 ∘ Xn∆ dξ 2δ ξ k− 1

2

=

z nk− 1 2

4

(

(z nk− 1 )2 2

(z nk+ 1 )2

− 1) +

z nk+ 1 2

4

(

z nk− 1 2

z nk+ 1 2

2

ξ k+ 1

2

− 1) −

1 ∫ z nk+ 1 [ρ 󸀠󸀠󸀠 ∘ Xn∆ − ρ 󸀠󸀠󸀠 (x̃ +k )] dξ . 2δ 2 ξ k− 1

2

Observing – in analogy to (4.20) – that ξ k+ 1

2

1 B 󵄨 󵄨 ∫ z nk 󵄨󵄨󵄨ρ 󸀠󸀠 ∘ Xn∆ (ξ) − ρ 󸀠󸀠 (x nk )󵄨󵄨󵄨 dξ ≤ z nk+ 1 (x nk+ 1 − x nk− 1 ) , 2δ 2 2 2 2 ξ k− 1

2

390 | 14 Convergence of a scheme for a thin-film equation and z nk+ 1 ≤ 2z nk , we obtain the following bound on |A5n − B5n |: 2

z nk+ 1 − z nk− 1 z nk− 1 (z nk− 1 )2 B 󵄨󵄨 n 2 2 2 2 n 󵄨󵄨 n [ A ≤ − B z ( − 1) + ( − 1)] ∑ ) ( 󵄨󵄨 5 5 󵄨󵄨 n n k 2 4 δ z (z ) 1 1 k+ k+ k∈𝕀+K 2 2 ] [ z nk+ 1 − z nk− 1 2 2 + B ∑ z nk ( ) (x nk+1 − x nk−1 ) . δ + k∈𝕀K

Again, applying (4.13) and (4.16), we get Nτ

󵄨 󵄨 τ ∑ 󵄨󵄨󵄨A5n − B5n 󵄨󵄨󵄨 ≤ Cδ1/4 ,

(4.25)

n=1

and the estimates (4.24) and (4.25) imply the desired bound on R5 . Arguing like in the previous proof, one shows the analogous estimate for A6n in place of A5n . It remains to analyze the potential term A7n , where we instantaneously identify the ξ -integral with the x-integral: Lemma 4.10. There is a constant C7 > 0 expressible in Ω, T, and B such that 󵄨󵄨 󵄨 N τ 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 n n R7 := τ ∑ 󵄨󵄨A7 − ∫ V x (x)u ∆ (x) dx󵄨󵄨󵄨 ≤ C7 δ . 󵄨󵄨 󵄨 n=1 󵄨󵄨󵄨 󵄨󵄨 Ω Proof. Since the product V x ρ x is a smooth function on the domain Ω, we can invoke the mean-value theorem and find intermediate values x̃ k , such that 󵄨󵄨 󵄨󵄨 M 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 n n n n 󵄨󵄨δ ∑ V x (x k )ρ x (x k ) − ∫ V x (X ∆ (ξ))ρ x (X ∆ (ξ)) dξ 󵄨󵄨󵄨 ≤ δ ∑ ∂ x (V x ρ x )(x̃ k )(x n 1 − x n 1 ) κ+ 2 κ− 2 󵄨󵄨 󵄨󵄨 + k∈𝕀+K 󵄨󵄨󵄨 󵄨󵄨󵄨 k∈𝕀K 0 󵄨 󵄨 ≤ δ(b − a) sup 󵄨󵄨󵄨V x (x)ρ x (x)󵄨󵄨󵄨 . x∈Ω

The claim then follows by a change of variables. It remains to identify the integral expressions inside R1 to R5 with those in the weak formulation (4.2). Lemma 4.11. One obtains M

∫ ̂z n∆ (ξ)∂ ξ ̂z n∆ (ξ)ρ 󸀠󸀠󸀠 ∘ Xn∆ (ξ) dξ = 0

1 2 ∫ ∂ x (û n∆ (x)) ρ 󸀠󸀠󸀠 (x) dx , 2 Ω

(4.26)

󵄨󵄨 M 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨 ̂ n∆ )2 (x)ρ 󸀠󸀠 (x) dx󵄨󵄨󵄨 ≤ C8 δ1/4 (4.27) R8 := τ ∑ 󵄨󵄨󵄨∫ ̂z ∆ (ξ)(∂ ξ ̂z n∆ )2 (ξ)ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ − ∫(∂ x u 󵄨󵄨 󵄨 󵄨󵄨 n=1 󵄨󵄨 0 Ω 󵄨 󵄨 Nτ

for a constant C8 > 0 expressible in Ω, T, B, and EV .

14.4 Weak formulation of the limit equation |

391

Proof. The starting point is the relation (2.8) between the locally affine interpolants û n∆ and ̂z n∆ that is ̂z n∆ (ξ) = û n∆ ∘ X n∆ (ξ)

(4.28)

for all ξ ∈ [0, M]. Both sides of this equation are differentiable at almost every ξ ∈ [0, M], with ∂ ξ ̂z n∆ (ξ) = ∂ x û n∆ ∘ X n∆ (ξ)∂ ξ X n∆ (ξ) . Substitute this expression for ∂ ξ ̂z n∆ (ξ) into the left-hand side of (4.26), and perform a change of variables x = X n∆ (ξ) to obtain the integral on the right. Next, observe that the x-integral in (4.27) can be written as M

∫(∂ x û n∆ )2 (x)ρ 󸀠󸀠 (x) dx = ∫(∂ ξ ̂z n∆ )2 (ξ) 0

Ω

1 ρ 󸀠󸀠 ∘ X n∆ (ξ) dξ , ∂ ξ X n∆

(4.29)

using (4.28). It hence remains to estimate the difference between the ξ -integral in (4.27) and (4.29), respectively. To this end, observe that for each ξ ∈ (ξ k , ξ k+ 12 ) with some k ∈ 𝕀+K , one has ∂ ξ X n∆ (ξ) = 1/z nk+ 1 and ̂z ∆ (ξ) ∈ [z k− 1 , z k+ 1 ]. Hence, for those ξ , 2

2

󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 z nk+ 1 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨1 − n 󵄨 ≤ 1− n 2 󵄨󵄨 ̂z ∆ (ξ)∂ ξ X n∆ (ξ) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 z k− 1 󵄨 󵄨 2

2

󵄨󵄨 󵄨󵄨 󵄨󵄨 . 󵄨󵄨 󵄨󵄨 󵄨

If instead ξ ∈ (ξ k− 12 , ξ k ), then this estimate is satisfied with the roles of z nk+ 1 and z nk− 1 2

2

interchanged. Consequently, 󵄨󵄨 M 󵄨󵄨 M 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󸀠󸀠 n 󵄨󵄨∫(∂ ξ ̂z n∆ )2 (ξ)̂z n∆ (ξ)ρ 󸀠󸀠 ∘ X n∆ (ξ) dξ − ∫(∂ ξ ̂z n∆ )2 (ξ) 󵄨󵄨 ρ ∘ X (ξ) dξ n ∆ 󵄨󵄨 󵄨󵄨 ∂ X (ξ) ξ ∆ 󵄨󵄨0 󵄨󵄨 0 󵄨 󵄨 M

≤ B ∫(∂ ξ ̂z n∆ )2 (ξ)̂z n∆ (ξ) (1 − 0

≤ Bδ ∑ k∈𝕀+K

z nk

(

z nk+ 1 − z nk− 1 2

2

δ

1 ) dξ ̂z n∆ (ξ)∂ ξ X n∆ (ξ)

󵄨󵄨 z 1 󵄨󵄨 󵄨󵄨 z 1 󵄨󵄨 󵄨󵄨 k+ 2 󵄨󵄨 󵄨󵄨 k− 2 󵄨󵄨 − 1󵄨󵄨󵄨 + 󵄨󵄨󵄨 − 1󵄨󵄨󵄨) , ) (󵄨󵄨󵄨 󵄨󵄨 z k− 1 󵄨󵄨 󵄨󵄨 z k+ 1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 2 2 2

1

which is again at least of the order O(δ 4 ), as we have seen before in (4.13). Proof of (4.10). Combining the discrete weak formulation (4.12), the change of variables formulae (4.26) and (4.27), and the definitions of R1 to R8 , it follows

392 | 14 Convergence of a scheme for a thin-film equation

that M 󵄨󵄨 M 3 󵄨󵄨 n n 󸀠󸀠󸀠 n 󵄨 ̂ ̂ ≤ BR8 + Bτ ∑ 󵄨󵄨 ∫ z ∆ (ξ)∂ ξ z ∆ (ξ)ρ ∘ X∆ (ξ) dξ + ∫ ̂z n∆ (ξ)(∂ ξ ̂z n∆ )2 (ξ)ρ 󸀠󸀠 ∘ Xn∆ (ξ) dξ 2 n=1 󵄨󵄨 0 0 7 󵄨󵄨󵄨 + ∫ V x (x) {u ∆ }τ (x)ρ 󸀠 (x) dx − ∑ A ni 󵄨󵄨󵄨󵄨 󵄨󵄨 i=1 Ω Nτ

e2,∆

7

7

≤ B ∑ R ni ≤ B ∑ C i δ1/4 . i=1

i=1

This implies the desired inequality (4.10). We are now going to finish the proof of this section’s main result, Proposition 4.1. Proof of Proposition 4.1. Owing to (4.9) and (4.10), we know that 󵄨󵄨 T 1 󵄨󵄨 󵄨󵄨 ∫ η󸀠 (t) ∫ ρ(x) {u ∆ }τ (t, x) dx + η(t) ∫ ρ 󸀠󸀠󸀠 (x)∂ x ({u ∆ }2τ )(t, x) 󵄨󵄨 2 󵄨0 Ω Ω 󵄨󵄨 󵄨 + 3ρ 󸀠󸀠 (x)(∂ x {u ∆ }τ )2 (t, x) dx + ∫ V x (x) {u ∆ }τ (t, x)ρ 󸀠 (x) dt󵄨󵄨󵄨󵄨 󵄨󵄨 Ω ≤ e1,∆ + e2,∆ ≤ C(τ + δ1/4 ) . To obtain (4.2) in the limit ∆ → 0, we still need to show the convergence of the integrals to their respective limits, but this is no challenging task anymore: Note that (3.21) implies ∂ x {u ∆ }τ 󳨀→ ∂ x u ∗ strongly in L2 ([0, T] × Ω) , (4.30) hence (∂ x {u ∆ }τ )2 converges to (∂ x u ∗ )2 in L1 ([0, T] × Ω). Furthermore, we have ∂ x ({u ∆ }2τ ) = 2 {u ∆ }τ ∂ x {u ∆ }τ 󳨀→ 2u ∗ ∂ x (u ∗ ) = ∂ x (u 2∗ )

(4.31)

in L2 ([0, T] × Ω). Here we used (4.30) and that {u ∆ }τ converges to u ∗ uniformly on [0, T]× Ω due to (3.16). Hence, (4.30) and (4.31) suffice to pass to the limit in the second integral. Finally remember the weak convergence result in (3.15), {u ∆ }τ → u ∗ in P2 (Ω) with respect to time, hence the convergence of the first and third integral is assured as well.

14.5 Numerical results 14.5.1 Nonuniform meshes An equidistant mass grid – as used in the analysis above – leads to a good spatial resolution of regions where the value of u 0 is large, but provides a very poor resolution in regions where u 0 is small. Since we are interested in regions of low density,

14.5 Numerical results |

393

and especially in the evolution of supports, it is natural to use a nonequidistant mass grid with an adapted spatial resolution, like the one defined as follows: The mass discretization of [0, M] is determined by a vector δ⃗ = (ξ0 , ξ1 , ξ2 , . . . , ξ K−1 , ξ K ), with 0 = ξ0 < ξ1 < ⋅ ⋅ ⋅ < ξ K−1 < ξ K = M and we introduce accordingly the distances (note the convention ξ−1 = ξ K−1 = 0) δ κ = ξ κ+ 12 − ξ κ− 12 ,

δk =

and

1 (δ 1 + δ k− 12 ) 2 k+ 2

for κ ∈ 𝕀K and k ∈ 𝕀+K , respectively. The piecewise constant density function u ∈ Pδ⃗ (Ω) corresponding to a vector x⃗ ∈ ℝK−1 is now given by 1/2

u(x) = z κ

for x κ− 1 < x < x κ+ 1 , 2

2

with

zκ =

δκ . x κ+ 1 − x κ− 1 2

2

The Wasserstein-like metric (and its corresponding norm) needs to be adapted as well: the scalar product ⟨⋅, ⋅⟩ δ is replaced by ⃗ δ⃗ = ∑ δ k 𝑣k w k ⟨v,⃗ w⟩ k∈𝕀+K

and

󵄩󵄩 ⃗ 󵄩󵄩 󵄩󵄩v󵄩󵄩δ = ⟨v,⃗ v⟩⃗ δ⃗ .

Hence the metric gradient ∇δ⃗ f(x)⃗ ∈ ℝK−1 of a function f : xδ⃗ → ℝ at x⃗ ∈ xδ⃗ is given by ⃗ k= [∇δ⃗ f(x)]

1 ∂ x f(x)⃗ . δk k

Otherwise, we proceed as before: the entropy is discretized by restriction, and the discretized information functional is the self-dissipation of the discretized entropy. Explicitly, the resulting fully discrete gradient flow equation attains the form x⃗ n∆ − x⃗ n−1 ∆ = −∇δ⃗ EVδ (x⃗ n∆ ) τ

(5.1)

14.5.2 Implementation Starting from the initial condition x⃗ 0∆ , the fully discrete solution is calculated inductively by solving the implicit Euler scheme (5.1) for x⃗ n∆ , given x⃗ n−1 ∆ . In each time step, a damped Newton iteration is performed, with the solution from the previous time step as initial guess.

14.5.3 Numerical experiments In the following numerical experiments, we fix Ω = (0, 1).

394 | 14 Convergence of a scheme for a thin-film equation 0.07

0.07

0.06

0.06

0.05

0.05

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0

0 0

0.2

0.4

Space x

0.6

0.8

1

0.07

0.07

0.06

0.06

0.05

0.05

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0

0

0.2

0.4

0

0.2

0.4

Space x

0.6

0.8

1

0.6

0.8

1

0 0

0.2

0.4

Space x

0.6

0.8

1

Space x

Fig. 1: Evolution of a discrete solution u ∆ , evaluated at different times t = 0, 0.002, 0.012, 0.04 (from top left to bottom right).

14.5.3.1 Evolution of discrete solutions In a paper of Gruen and Beck [2], the authors analyzed, among other things, the behavior of equation (1.1) on the bounded domain (0, 1) with Neumann-boundary conditions and the initial datum u 0ε (x) = (x − 0.5)4 + ε,

x ∈ (0, 1),

with massM = 0.0135 ,

(5.2)

with ε = 10−3 . This case is interesting insofar as the observed film seems to rip at time t = 0.012. Figure 1 shows the evolution of u ∆ for K = 400 and τ = 10−7 at times t = 0, 0.0022, 0.012, 0.04, the associated particle flow is shown in Figure 2 (left). 14.5.3.2 Rate of convergence For the analysis of the scheme’s convergence with initial datum u 0ε with ε = 10−3 from (5.2), we fix τ = 10−7 and calculate solutions u ∆ to our scheme with K = 25, 50, 100, 200, 400. A reference solution u ∆̃ is obtained by solving (5.1) on a much ̃ = (K −1 ; τ ref ) with Kref = 1600 and τ = 5⋅10−8 . In Figure 2 (right), finer grid, which is ∆ ref 1 we plot the L (Ω), L2 (Ω), and L∞ (Ω)-norms of the differences |u ∆ (t, ⋅) − u ∆̃ (t, ⋅)| at time t = 10−4 . It is clearly seen that the errors decay with an almost perfect rate of δ2 ∝ K −2 . 14.5.3.3 Comparison with a standard numerical scheme For an alternative verification of our scheme’s quality, we use a reference solution that ist calculated by means of a structurally different discretization of (1.1). Specifically,

14.5 Numerical results |

10 -3

error at t = 10-5

0.04

Time t

0.03

0.02

395

L1-error L2-error L -error K -2

10 -4

10 -5

10 -6

0.01

10 -7

0 0

0.2

0.4

0.6

0.8

1

Space x

25

50

100

200

400

Number of grid points K

Fig. 2: Left: Associated particle flow of u ∆ for initial datum (5.2). Right: Rate of convergence, using K = 25, 50, 100, 200, 400, and τ = 10−7 . The errors are evaluated at time t = 10−4 .

we employ a finite-difference approximation with step sizes τref and href = (b − a)/Kref in the t- and x-directions, respectively. More precisely, with t n := nτref , and with x k for k = 0, . . . , Kref being the Kref + 1 equidistant grid points in [a, b], the numerical approximation u nk ≈ u(t n ; x k ) of (1.1) is obtained – inductively with respect to n – for given vector u n−1 = (u 0n−1 , . . . , u n−1 Kref ) by solving the fully implicit difference equation u n − u n−1 = u n ⋅ D4ref u n + D1ref u n ⋅ D3ref u n , τref

(5.3)

i are standard finite difference approximations of where u n = (u 0n , . . . , u nKref ) and Dref the ith derivative with equidistant steps href . The product “⋅” of two vectors in (5.3) shall be understood to act component-by-component. The boundary conditions (1.2) are enforced using values u nk at “ghost points” in the obvious way, that is n u −1 = u 0n ,

n u −2 = u 1n ,

u nKref +1 = u nKref ,

u nKref +2 = u nKref −1 .

To produce reference solutions for the examples discussed below, the scheme above is implemented with Kref = 6400 spatial grid points and a time step τref = 5 ⋅ 10−8 . At a given time T = Nτ ref , the respective reference profile x 󳨃→ u ref (T, x) is defined via piecewise linear interpolation of the respective values u nk . Before comparing the advantages and disadvantages of our scheme and the reference scheme in (5.3), let us repeat the experiment of Section 14.5.3.2: Using u 0ε from (5.2) with ε = 10−1 instead of ε = 10−3 , we plot the L1 (Ω), L2 (Ω), and L∞ (Ω)-norms of the differences |u ∆ (t, ⋅) − u ref (t, ⋅)| at time t = 10−4 in Figure 3 (left). Obviously, the new experiment confirms the rate of convergence δ2 ∝ K −2 gained in Section 14.5.3.2. The following remarks concerning both schemes can now be made: – Computational cost: Using Newton’s method to solve both approximations (5.1) and (5.3), the finite-difference scheme is more efficient, unsurprisingly. The reason

396 | 14 Convergence of a scheme for a thin-film equation

−4

x 10

−3

10

 = 10−1



L − error −2 K

10

−5

10



ref

||u (T) − u (T)||

L

p

−4

−6

10

−7

10

25

50

100

200

Number of grid points K

400

relative gain/loss of mass

1

L − error 2 L − error

 = 10−3 3

 = 10−5

2

1

0 0

0.2

0.4

Time t

0.6

0.8

1 −4

x 10

Fig. 3: Left: Rate of convergence, using K = 25, 50, 100, 200, 400 and τ = 10−7 . The discrete solutions are compared with a reference solution of the scheme in (5.3), and the errors are evaluated at time t = 10−4 . Right: Loss of mass preservation using a standard finite-difference scheme and u 0ε of (5.2) with ε = 10−1 , 10−3 and 10−5 .

–

–

⃗ whereas the for this is the complicate structure of the Jacobian matrix of ∇δ EVδ (x), Jacobian matrix of the right-hand side of (5.3) is easy and quick to implement. Numerical experiments show that the finite-difference scheme can be approximately five-times faster than our scheme, using the same values for K and τ. Conservation of mass: It is generally known that standard numerical schemes as the finite-difference approximation in (5.3) do not preserve mass. Depending on the initial datum, the loss or gain of mass can decrease quickly with time and causes inaccurate solutions. In Figure 3 (right), we plot the relative change of mass 1 |M ∫Ω u ref (t, x) dx − 1| with M = ∫Ω u 0ε (x) dx for ε = 10−1 , 10−3 , 10−5 and t ∈ [0, 10−4 ]. One can observe that the preservation of mass of solutions to the finitedifference scheme is seriously harmed in case of smaller choices of ε. This is why we used ε = 10−1 in the second experiment for the rate of convergence, since smaller values for ε produce reference solutions whose change of mass yield to significant distortions of the L p -errors. Conservation of positivity: In general, one can expect positivity of the discrete solution to the finite-difference scheme starting with a sufficiently positive initial function. This situation changes dramatically if one considers initial densities with regions of small values or even zero values. Take, for example, the initial datum u 0ε in (5.2) with ε = 0. Then the solution to the scheme in (5.3) – again using Kref = 3200 spatial grid points and a time step τref = 5 ⋅ 10−8 – contains negative values after the very first time iteration and finally loses any physical meaning after some more iterations. In contrast, our scheme can still handle the case when

A Appendix | 397

ε = 0 in (5.2), although one usually has to assume strict positivity for initial values in our approach. Conclusively, our scheme has a major advantage in comparison with standard numerical solvers if one is interested in a stable and structure-preserving discretization for (1.1). Moreover, the slightly plus of the finite-difference scheme and of similiar approximations discussed in the first point – less computational cost – is invalidated by the fact that one needs much finer discretization parameters compared to our structurepreserving scheme to gain solutions with an adequate physical meaning.

A Appendix Lemma A.1 (Gargliardo–Nirenberg inequality). For each f ∈ H 1 (Ω), one has 2/3

1/3

‖f‖C1/6 (Ω) ≤ (9/2)1/3 ‖f‖H 1 (Ω) ‖f‖L2 (Ω) .

(A.1)

Proof. Assume first that f ≥ 0. Then, for arbitrary x < y, the fundamental theorem of calculus and Hölder’s inequality imply that y

3 󵄨 3 1/2 󵄨󵄨 󵄨󵄨f(x)3/2 − f(y)3/2 󵄨󵄨󵄨 ≤ ∫ 1 ⋅ f(z)1/2 |f 󸀠 (z)| dz ≤ |x − y|1/4 ‖f‖L2 (Ω)‖f 󸀠 ‖L2 (Ω) . 2 2 x

Since f ≥ 0, we can further estimate 1/3 1/3 󵄨2/3 󵄨 |f(x) − f(y)| ≤ 󵄨󵄨󵄨f(x)3/2 − f(y)3/2 󵄨󵄨󵄨 ≤ (3/2)2/3 |x − y|1/6 ‖f‖L2 (Ω) ‖f‖H 1 (Ω) .

This shows (A.1) for nonnegative functions f . A general f can be written in the form f = f+ − f− , where f± ≥ 0. By the triangle inequality, and since ‖f± ‖H 1 (Ω) ≤ ‖f‖H 1 (Ω) , 1/3

1/3

‖f‖C1/6 (Ω) ≤ ‖f+ ‖C1/6 (Ω) + ‖f− ‖C1/6(Ω) ≤ 2(3/2)2/3 ‖f‖L2 (Ω) ‖f‖H 1 (Ω) . This proves the claim. ⃗ one obtains Lemma A.2. For each p ≥ 1 and x⃗ ∈ xδ with z⃗ = z δ [x], ∑ ( 1/2

κ∈𝕀K

δ p ) = ∑ (x κ+ 1 − x κ− 1 )p ≤ (x K − x0 )p . 2 2 zκ 1/2

(A.2)

κ∈𝕀K

Proof. The first equality is simply the definition (1.11) of z κ . Since trivially x κ+ 1 −x κ− 1 ≤ 2

1/2

x K − x0 for each κ ∈ 𝕀K , and since p − 1 ≥ 0, it follows that ∑ (x κ+ 12 − x κ− 12 )p ≤ (x K − x0 )p−1 ∑ (x κ+ 12 − x κ− 12 ) = (x K − x0 )p . 1/2

κ∈𝕀K

1/2

κ∈𝕀K

2

398 | 14 Convergence of a scheme for a thin-film equation

Bibliography [1] [2] [3] [4] [5] [6] [7] [8]

[9] [10] [11]

[12] [13] [14] [15]

[16]

[17] [18] [19] [20] [21]

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, J. Phys.: Condens. Matter 17:291–307, 2015. J.-D. Benamou, G. Carlier, Q. Mérigot and E. Oudet, Discretization of functionals involving the monge-ampzere operator, arXiv preprint arXiv:1408.4536, 2014. F. Bernis, Integral inequalities with applications to nonlinear degenerate parabolic equations, in Nonlinear problems in applied mathematics, SIAM, Philadelphia, PA, 1996, pp. 57–65. F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations 83:179–206, 1990. M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations 3:417–440, 1998. A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal. 46:691–721, 2008. C. J. Budd, G. J. Collins, W. Z. Huang and R. D. Russell, Self-similar numerical solutions of the porous-medium equation using moving mesh methods, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357:1047–1077, 1999. M. Burger, J. A. Carrillo and M.-T. Wolfram, A mixed finite element method for nonlinear diffusion equations, Kinet. Relat. Models 3:59–83, 2010. E. A. Carlen and S. Ulusoy, Asymptotic equipartition and long time behavior of solutions of a thin-film equation, J. Differential Equations 241:279–292, 2007. J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM J. Sci. Comput., 31:4305– 4329, 2009/10. J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin film equation, Comm. Math. Phys. 225:551–571, 2002. J. A. Carrillo and M.-T. Wolfram, A finite element method for nonlinear continuity equations in lagrangian coordinates. Working paper. F. Cavalli and G. Naldi, A Wasserstein approach to the numerical solution of the onedimensional Cahn–Hilliard equation, Kinet. Relat. Models 3:123–142, 2010. R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: global entropy estimates, existence and qualitative behavior of solutions, SIAM J. Math. Anal. 29:321– 342, 1998 (electronic). J. Denzler and R. J. McCann, Nonlinear diffusion from a delocalized source: affine selfsimilarity, time reversal, & nonradial focusing geometries, Ann. Inst. H. Poincaré Anal. Non Linéaire 25:865–888, 2008. B. Düring, D. Matthes and J. P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B 14:935–959, 2010. L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the HeleShaw flow, Calc. Var. Partial Differential Equations 13:377–403, 2001. U. Gianazza, G. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal. 194:133–220, 2009. L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput. 28:1203–1227, 2006 (electronic). G. Grün, On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions, Math. Comp. 72:1251–1279, 2003 (electronic).

Bibliography

| 399

[22] G. Grün, Droplet spreading under weak slippage – existence for the Cauchy problem, Comm. Partial Differential Equations 29:1697–1744, 2004. [23] G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Numer. Math. 87:113–152, 2000. [24] D. Kinderlehrer and N. J. Walkington, Approximation of parabolic equations using the Wasserstein metric, M2AN Math. Model. Numer. Anal. 33:837–852, 1999. [25] R. S. Laugesen, New dissipated energies for the thin fluid film equation, Commun. Pure Appl. Anal. 4:613–634, 2005. [26] R. C. MacCamy and E. Socolovsky, A numerical procedure for the porous media equation, Comput. Math. Appl. 11:315–319, 1985. Hyperbolic partial differential equations, II. [27] D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations 34:1352–1397, 2009. [28] D. Matthes and H. Osberger, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM Math. Model. Numer. Anal. 48:697–726, 2014. [29] D. Matthes and H. Osberger, A convergent Lagrangian discretization for a nonlinear fourth order equation, Accepted at Found. Comput. Math., 2015. [30] A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69:931–980, 1997. [31] H. Osberger, Long-time behaviour of a fully discrete lagrangian scheme for a family of fourth order, arXiv preprint arXiv:1501.04800, 2015. [32] G. Russo, Deterministic diffusion of particles, Comm. Pure Appl. Math. 43:697–733, 1990. [33] C. Villani, Topics in optimal transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. [34] L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes forlubrication-type equations, SIAM J. Numer. Anal. 37:523–555, 2000.

F. Al Reda and B. Maury

15 Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance Abstract: This paper establishes a link between some space discretization strategies of the finite volume type for the Fokker–Planck equation in general meshes (Voronoï tesselations) and gradient flows on the underlying networks of cells, in the framework of discrete Wasserstein distances on graphs recently proposed by Maas [6]. Keywords: Gradient flows, Wasserstein distance, finite volumes discretization, Fokker–Planck equation, heat equation, resistive network, Markov kernel AMS Subject Classification: 60J60, 94C15, 35K05, 35K08, 65N08, 60J27, 60J28.

15.1 Introduction We aim here at identifying gradient flow structures in some space-discretization schemes of the Fokker–Planck equation on general meshes, in the spirit of the approaches proposed recently in [1, 2] for cartesian discretizations. Since the core of the paper consists in building links between macroscopic notions / properties and their discrete counterparts, in a context where two reference measures are present (uniform Lebesgue measure and stationary measure associated to an attractive potential), let us start by fixing some principles in terms of notation. Probability measures will be denoted by the letter p (we shall use the same letter to denote their density with respect to the underlying Lebesgue measure, or its discrete counterpart), stationary measures (with respect to some evolution process) by π, and relative densities with ̃, π ̃, respect to π by ρ. All discrete notions will be singled out by a tilda sign, e.g., p etc. . . . The space variable will be denoted by r, while x and y will be used to denote discrete vertices. Since the seminal work of Jordan et al. [3] in 1998, it is known that the Fokker– Planck (FP) equation in a domain Ω: ∂ t p − ∆p − ∇ ⋅ (p∇Φ) = 0 ,

F. Al Reda, Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay cedex, France B. Maury, Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay cedex, France

15.1 Introduction

|

401

with appropriate no-flux boundary conditions, can be interpreted in the Wasserstein space as the gradient flow for p H(p) = ∫ p log ( ) dr π

(= ∫ ρ log (ρ) dπ with ρ = p/π) ,

Ω

Ω

which is the relative entropy with respect to the stationary measure π = e−Φ , up to a normalization constant. This property is schematized in the diagram below (see Figure 1, blocks A − B − C, on the top), and we refer the reader to [4, 5] for a thorough description of the underlying theory. At the discrete level, a similar framework has been proposed in [6, 7]. The Euclidean domain is replaced by a network N, defined by its (finite) set of vertices V and a Markovian kernel (K(x, y)) x,y∈V ,

with K(x, y) ≥ 0 , ∑ K(x, y) = 1 ∀x ∈ V . y∈V

̃ , it verifies π ̃ =tK π ̃ . It is unique as soon as K The stationary measure is denoted by π is irreducible, i.e., ∀ x, y ∈ V there exists a path (x0 = x, x1 , x2 , . . . , x m = y) such that ̃ (x) > 0 for all x ∈ V. We say that K(x, x1 ) × K(x1 , x2 ) × ⋅ ⋅ ⋅ × K(x m−1 , y) > 0, and then π ̃ (x)K(x, y) = K(y, x)π ̃ (y) for all x, y in V (detailed balance equation). K is reversible if π The discrete counterpart of the FP equation is the heat-flow equation ∂ t ρ̃ + (I − K)ρ̃ = 0 ,

(1.1)

̃ on V with respect to π ̃ . Note that the where ρ̃ is the density of a probability measure p ̃ straight discrete counterpart of FP equation would be an equation of the measure p ̃ (x) = 1, but we shall follow [6] in favoring densiitself, with K replaced by tK, and ∑ p ̃ , i.e., densities ρ̃ verifying ∑ ρ̃ (x)π ̃ (x) = 1. It has been established ties with respect to π ̃ in [6] that (1.1), for an appropriate metric W2 which is the discrete counterpart of the standard Wasserstein distance, is a gradient flow of the discrete relative entropy ̃ ρ̃ ) = ∑ ρ̃ (x) log(ρ̃ (x)) π ̃ (x) H(

(1.2)

x∈V

̃2 (see Section 15.2 for detailed definiwith respect to the Wasserstein-like metric W tions). This discrete setting is also schematized in Figure 1 (blocks à − B̃ − C,̃ on the bottom). Although it was not the original purpose in [6], a connection can be made between the two settings by means of discretization strategies. As detailed in the next section, an Euclidean domain Ω can be partitioned into cells (e.g., Voronoï cells associated to a collection of points in the domain, see Figure 2). Now consider the network associated to those cells (one may consider that the vertices are the centroids of the cells). To any measure μ on Ω one can associate a discrete measure that is, for each vertex associated to cell K, the measure μ(K). As detailed in [11], a link can be made between the

402 | 15 Finite volume discretization and Wasserstein gradient flows

Fig. 1: Continuous setting versus discrete setting.

Wasserstein distance on the Euclidean domain and the discrete Wasserstein distance on the network, at least in the case of a regular decomposition (cartesian grid). This link will be described more precisely in the next section, it is indicated by the arrow 2 in Figure 1 that relates blocks B and B.̃ Besides, integrating any function of the density at the continuous level has a discrete counterpart (we are especially interested in entropy-like functionals), it consists in summing up the corresponding values for the discrete densities built as described above. This approach can be seen as a quadrature formula to compute the approximation of an integral, for which convergence properties can be expected as the cell decomposition is refined. It is indicated by the arrow 3 in Figure 1 that relates blocks C and C.̃ The core of the present article is an attempt toward closing the diagram by expliciting the link between blocks A and à (arrow 1). More precisely, we aim at showing that, in the context of finite volume methods, some space discretization strategies of the FP equation lead to Ordinary Differential Equation that are consistent with the gradient flow structure on the underlying network. Note that this interpretation of finite volume discretization schemes as gradient flows has already been addressed in two recent papers. In [2], the authors use this gradient flow structure to characterize the long time behavior of discrete solutions to a fourth order equation. In [1], a finite volume scheme is studied in the discrete Wasserstein setting, and a new type of convergence proof is proposed in this context. In both cited papers, the space discretization is regular (i.e., one-dimensional for the second one, and d-dimensional with a carte-

15.2 Preliminaries

|

403

Ω

Fig. 2: From the Euclidean domain to the associated network.

sian grid for the first one). We aim here at showing that an extension to nonregular space-discretization is not out of reach. In particular, we show that finite volume discretization strategies for very general meshes lead to problems that can be interpreted as gradient flows for a discrete Wasserstein-like metric, with a functional that can be seen as an approximation of its continuous counterpart. Let us make it clear, though, that no discrete-to-continuous convergence result is known for the Wasserstein distance for nonregular meshes. The outline of the paper is as follows. In Section 15.2 we recall the main obtained result on the FP equation and its gradient flow formulation, we define the Wassersteinlike distance of Maas and state his first result in terms of gradient flows using this distance. Then we describe the Gromov–Hausdorff convergence in the special case of the ̃ to d-dimensional torus and show the convergence of the discrete relative entropy H its continuous counterpart H. Section 15.3 proposes a finite volume discretization of the FP equation in space and the analysis of the Markov chain deduced from this discretization and seen as an ODE in time. We show that this ODE is the gradient flow of ̃ and we finalize the paper with some conclusive remarks the discrete relative entropy H and perspectives.

15.2 Preliminaries We describe in this section with some details the constitutive blocks of the diagram presented in Figure 1.

Blocks A–B–C: Fokker–Planck equation as a gradient flow, continuous setting Let us first recall some basic facts on the Wassertein space of measures and gradient flows therein (we refer to [4, 5, 8] for a detailed presentation of these considerations). Let Ω be a bounded domain. For any two measures p0 and p1 in P(Ω), the (quadratic)

404 | 15 Finite volume discretization and Wasserstein gradient flows

Wasserstein distance between them is defined by 󵄨 󵄨2 W2 (p0 , p1 )2 = inf ∫ 󵄨󵄨󵄨󵄨r󸀠 − r󵄨󵄨󵄨󵄨 dγ(r, r󸀠 ) , γ∈Π Ω×Ω

where Π is the subset of P(Ω× Ω) for all those γ with marginals p0 and p1 , respectively, i.e., ∫ φ(r) dγ(r, r󸀠 ) = ∫ φ(r) dp0 (r) , Ω×Ω

Ω

∫ ψ(r󸀠 ) dγ(r, r󸀠 ) = ∫ ψ(r󸀠 ) dp0 (r󸀠 ) Ω×Ω

Ω

for any continuous functions φ and ψ. An alternative formulation has been proposed by Benamou–Brenier [9], it consists in writing the squared Wasserstein distance as follows (we consider here a convex domain): 1

{ } 󵄨 󵄨2 W2 (p0 , p1 ) = inf {∫ ∫ 󵄨󵄨󵄨∇ψ t (r)󵄨󵄨󵄨 p t (r) dr dt} , p t ,ψ t {0 Ω } 2

(2.1)

where the infimum runs over curves (p t )t∈[0,1] in P(Ω) that join p0 and p1 in the following way (transport of p t by ∇ψ t ): ∂ t p t + ∇ ⋅ (p t ∇ψ t ) = 0 .

(2.2)

Now we aim at defining a notion of gradient for a functional H that is consistent with the Wasserstein framework. The more appropriate notion is that of Fréchet subdifferential in a Wasserstein sense, that can be defined for a wide class of functionals, with very weak smoothness assumptions (see, e.g., [4]). Since this notion does not have any natural counterpart at the discrete level, we shall focus here on a more restrictive definition of the gradient: Definition 2.1. Let H : P(Ω) → ℝ be a functional. We shall say that H admits a gradient w ∈ L2p at p ∈ P(Ω), and then write grad H(p) = w , if, for every measure path t → p t defined in a neighborhood of 0 and satisfying ∂p t + ∇ ⋅ (p t 𝑣t ) = 0 , p0 = p , ∂t where 𝑣t is a L2 vector field, it holds that d H(p t ) − H(p0 ) H(p t )|t=0 = lim = ∫ 𝑣0 ⋅ w dp . t→0 dt t Ω

15.2 Preliminaries

|

405

We may now define the notion of gradient flow in this setting: Definition 2.2. The probability measure path t 󳨃→ p t is said to be a gradient flow for a functional H if p t verifies (in the distributional sense) ∂ t p t + ∇ ⋅ (p t u t ) = 0 , u t = − grad H(p t )

for a.e. t ,

where the gradient is defined according to Definition 2.1. Let us consider the case where H reads H(p) = ∫ f(r) dp(r) + ∫ g(p(r)) dr , Ω

Ω

where f and g are regular functions. Then the transport velocity u t can be identified as u t = −∇f − ∇ (g󸀠 (p t )) . (2.3) Now consider the Fokker–Planck equation on a domain Ω: { { { { { { {

∂p − ∆p − ∇.(p∇Φ) = 0, in Ω ∂t ∂p ∂Φ −p = 0, on ∂Ω. ∂n ∂n

(2.4)

and the relative entropy functional: H(p) = ∫ p(r) log (

p(r) ) dr π(r)

Ω

= − ∫ log(π(r)) dp(r) + ∫ p(r) log(p(r)) dr . Ω

(2.5)

Ω

We obtain from (2.3) u t = −∇f − ∇ (g󸀠 (p t )) =

∇π ∇p t − ∇(1 + log(p t )) = −∇Φ − , π pt

which identifies the FP equation (2.4) as a gradient flow in the Wasserstein sense for the relative entropy functional (2.5). We refer again to [4, 5] for a thorough presentation of these facts.

̃ B– ̃ C:̃ Discrete setting Blocks A– Let V be a finite set in ℝ2 . Definition 2.3. We say that (K(x, y))x,y∈V is an irreducible and reversible Markov kernel on V × V if K satisfies:

406 | 15 Finite volume discretization and Wasserstein gradient flows (1) K(x, y) ≥ 0 ∀x, y ∈ V , ∑ K(x, y) = 1 ∀x ∈ V. y∈V

(2) The irreducibility condition: K(x, y) ≥ 0 ∀x, y ∈ V , ∑y∈V K(x, y) = 1 ∀ x ∈ V. ̃ (x)K(x, y) = K(y, x)π ̃ (y) ∀ x, y ∈ V. (3) The reversibility condition: π ̃ the unique stationary measure Let (K(x, y))x,y∈V be as in the definition. We denote by π of K, such that ̃ (x) = ∑ K(y, x)π ̃ (y) (i.e., π ̃ =t K π ̃) , π ̃ (x) > 0 π

̃ (x) = 1 . ∀x ∈ V , ∑ π

y∈V

x∈V

We define the associated set of probability densities on V by ̃ (x) = 1} . D(V) = {ρ̃ ∈ (ℝ+ )V , ∑ ρ̃ (x)π x∈V

Following [6], we define the discrete gradient, the discrete divergence and the two ̃ resp., as follows: scalar products with respect to a fixed ρ̃ ∈ D(V) and π ̃ : V → ℝ, we define its discrete gradient – Discrete gradient: For a function ψ ̃ : V × V → ℝ by ̃ψ ∇ ̃ y) = ψ(y) ̃ − ψ(x) ̃ ̃ ψ(x, ∇ –

∀x, y ∈ V × V .

̃ : V × V → ℝ, we define its discrete Discrete divergence: For a discrete field u ̃ ⋅ ũ : V → ℝ by divergence ∇ ̃ ⋅ ũ )(x) = (∇

1 ̃ (x, y) − ũ (y, x))K(x, y) ∑ (u 2 y∈V

∀x ∈ V .

̃ (x, y) = −u ̃ (y, x), the discrete diverNote that for an anti-symmetric field u, i.e., u gence reads: ̃ ⋅ ũ )(x) = ∑ (u ̃ (x, y))K(x, y) ∀x ∈ V . (∇ y∈V

–

̃ ϕ ̃ : V → ℝ, we define their scalar product ̃ : For ψ, Scalar product with respect to π ̃ by with respect to π ̃ ϕ⟩⟩ ̃ ̃ ϕ(x) ̃ π ̃ (x) . ⟨⟨ψ, = ∑ ψ(x) ̃ π x∈V

–

Scalar product with respect to ρ̃ : For ũ , 𝑣̃ : V × V → ℝ, we define their scalar product with respect to ρ̃ by ̃ , 𝑣̃⟩̃ρ = ⟨u

1 ̃ (x) , 𝑣(x, y)K(x, y)θ(ρ̃ (x), ρ̃ (y))π ∑ ũ (x, y)̃ 2 x,y∈V

̃ ‖̃ρ the associated norm where θ(⋅, ⋅) is defined by (2.7), and we denote by ‖ u ̃ ‖̃ρ = √⟨̃ ̃ ⟩̃ρ . ‖u u, u Note that the latter is a discrete counterpart of the norm of a velocity field in L2p , with p = ρπ.

15.2 Preliminaries

407

|

We denote by ⟨., .⟩1 the scalar product with respect to the density ρ̃ = 1 defined by ρ̃ (x) = 1, ∀x ∈ V. We can easily check that the integration by parts formula holds in the following sense: ̃ u ̃ ∇ ̃ ψ, ̃ ⋅ ũ ⟩⟩ . ̃ ⟩ = − ⟨⟨ψ, ⟨∇ ̃ 1 π The definition of the discrete transportation metric is inspired by the Benamou– Brenier formulation, it translates equation (2.2) at the discrete level. It is defined as (see [6, 7, 10]): Definition 2.4. For ρ̃ 0 , ρ̃ 1 ∈ D(V), we set 1

{ } ̃ t (y) − ψ ̃ t (x))2 K(x, y)θ(ρ̃ t (x), ρ̃ t (y))π ̃2 (ρ̃ 0 , ρ̃ 1 )2 = inf 1 ∫ ∑ (ψ ̃ (x) dt} W { 2 ̃ ψt ρt ,̃ { 0 x,y∈V } where the infimum runs over all piecewise C1 curves ρ̃ t : [0, 1] → D(V) and all piecẽ t : [0, 1] → ℝV satisfying: wise continuous ψ dρ̃ t ̃ t (y) − ψ ̃ t (x)) K(x, y)θ(ρ̃ t (x), ρ̃ t (y)) = 0 ∀x ∈ V , (x) + ∑ ( ψ dt y∈V

(2.6)

where:

β−α { , { log(β) − log(α) θ(α, β) = ∫ α β dt = { { 0 {α, is the logarithmic mean of α and β. 1

1−t t

if α ≠ β

(2.7)

if α = β

̃2 can also Using the definitions of the discrete gradient and the discrete divergence, W be formulated as follows (see [6, Lemma 3.5]): 1

{ } ̃ t ‖2 dt ̃2 (ρ̃ 0 , ρ̃ 1 )2 = inf ∫ ‖∇ ̃ψ W ̃ { ρt } ̃ ̃ ρt ,ψt {0 } where the infimum runs over all piecewise C1 curves (ρ̃ t )t∈[0,1] joining ρ̃ 0 and ρ̃ 1 in D(V) according to dρ̃ t ̃ ̃ ψ)(x) ̃ ⋅ (Θ(ρ̃ t ) ∙ ∇ = 0, ∀x ∈ V (x) + ∇ dt

(2.8)

where Θ(ρ̃ t ) : V × V → ℝ is defined by: Θ(ρ̃ t )(x, y) = θ(ρ̃ t (x), ρ̃ t (y)) and ∙ denotes the entrywise product of two matrices. The Wasserstein gradient of a functional may now be defined following [6, Prop. 4.2]. ̃ : D(V) → ℝ be a functional. We shall say that H ̃ admits a Definition 2.5. Let H V×V ̃∈ℝ gradient w at ρ̃ ∈ D(V), and then write ̃ H( ̃ ρ̃ ) = w ̃, grad

408 | 15 Finite volume discretization and Wasserstein gradient flows if, for any measure path t → ρ̃ t on D(V) defined in a neighborhood of 0, with dρ̃ t ̃ + ∇ ⋅ (Θ(ρ̃ t ) ∙ 𝑣̃t ) = 0 , ρ̃ 0 = ρ̃ , dt it holds that

d̃ ̃ , 𝑣̃0 ⟩̃ρ . H(ρ̃ t )|t=0 = ⟨ w dt ̃ we can write its gradient flow equation After computing the gradient of a functional H, in D(V). ̃H ̃ : D(V) → ℝ be a functional and grad ̃ be its gradient accordDefinition 2.6. Let H ̃ by ing to Definition (2.5). We define the discrete gradient flow equation of H dρ̃ ̃ H( ̃ ρ̃ ))(x) = 0 ̃ ⋅ (Θ(ρ̃ ) ∙ grad (x) − ∇ dt

∀x ∈ V .

Like in the continuous setting, the gradient (in the previous sense) of a certain class of functionals can be computed explicitly. ̃ be a generalized entropy functional: Proposition 2.7. Let H ̃ : ρ̃ ∈ D(V) 󳨃󳨀→ H( ̃ ρ̃ ) = ∑ f(x)ρ̃ (x)π ̃ (x) + ∑ g(ρ̃ (x))π ̃ (x) ∈ ℝ H x∈V

x∈V

where f, g are differentiable functions, f, g : [0, 1] → ℝ. Then ̃ H( ̃ ρ̃ ) = ∇f ̃ + ∇g ̃ 󸀠 ∘ ρ̃ ) . grad Proof. It is a straightforward application of the definitions above d ̃ dρ̃ t ̃ (x)|t=0 H(ρ̃ t )|t=0 = ∑ (f(x) + g󸀠 (ρ̃ t (x))) (x)π dt dt x∈V ̃ ⋅ (Θ(ρ̃ t ) ∙ 𝑣̃t )(x)π ̃ (x)|t=0 = − ∑ (f(x) + g󸀠 (ρ̃ t (x)))∇ x∈V

̃ ⋅ (Θ(ρ̃ ) ∙ 𝑣0̃ )⟩⟩ = ⟨∇(f ̃ + g󸀠 ∘ ρ̃ ), Θ(ρ̃ ) ∙ 𝑣0̃ ⟩ = − ⟨⟨f + g󸀠 ∘ ρ̃ , ∇ ̃ 1 π ̃ + ∇g ̃ 󸀠 ∘ ρ̃ , 𝑣0̃ ⟩ = ⟨∇f ̃ ρ which concludes the proof.

Heat flow equation as gradient flow of the discrete entropy Note that the heat-flow equation dρ̃ + (I − K)ρ̃ = 0 , dt

(2.9)

15.2 Preliminaries

|

409

where K = (K(x, y))x,y is the Markov matrix, can also be written as dρ̃ (x) − ∇ ⋅ (∇ρ̃ )(x) = 0 dt

∀x ∈ V .

We may now identify the heat-flow equation with the gradient flow for the relative entropy. Theorem 2.8. The gradient flow in D(V) (according to Definition (2.6)) of the discrete relative entropy ̃ ρ̃ ) = ∑ ρ̃ (x) log(ρ̃ (x))π ̃ (x) H( x∈V

is the heat-flow equation (2.9). Proof. For a detailed proof, we refer the reader to [6, Theorem 1.2]. From Proposition 2.7, we have ̃ H( ̃ ρ̃ ) = ∇(1 ̃ + log(ρ̃ )) = ∇(log( ̃ grad ρ̃ )) ̃ is: for g(ρ̃ ) = ρ̃ log ρ̃ and f = 0, and the discrete gradient flow equation of H dρ̃ ̃ ⋅ (Θ(ρ̃ ) ∙ ∇(log( ̃ (x) − ∇ ρ̃ )))(x) = 0 dt

∀x ∈ V .

̃ ̃ ρ̃ ), which concludes the proof. Notice that Θ(ρ̃ ) ∙ ∇(log( ρ̃ )) = ∇(

Link B–B̃ (arrow 2): Link between continuous and discrete Wasserstein metrics A first result of convergence of the discrete transportation metric was proven in [11, Theorem 3.15]. We consider the space P(𝕋d ) of all the probability measures on the d-dimensional torus 𝕋d = ℝd /ℤ d endowed with the L2 -Wasserstein metric and the d-dimensional periodic lattice 𝕋dn = (ℤ/nℤ)d and endow the space of probability densities D(𝕋dn ) with ̃2,n = W ̃2 /n√2d where the Markov the renormalized discrete transportation metric W kernel K is the one of a simple random walk (uniform transition probabilities) and whose stationary measure δ̃ is the uniform measure on 𝕋dn . ̃ we can identify probability measures on 𝕋d with their In this special case of δ, n ̃ So we consider that P(𝕋d ) ≡ D(𝕋d ). probability densities with respect to δ. n n The convergence result is established in the sense of Gromov–Hausdorff that is defined by Definition 2.9. A sequence of compact metric spaces (Xn , d n ) is said to converge in the sense of Gromov–Hausdorff to a compact metric space (X, d), if there exists a sequence of maps f n : X → Xn which are: – ϵ n -isometric, i.e., for all x, y ∈ X, |d n (f n (x), f n (y)) − d(x, y)| ≤ ϵ n

410 | 15 Finite volume discretization and Wasserstein gradient flows

–

ϵ n -surjective, i.e., for all x n ∈ Xn there exists x ∈ X with d(f n (x), x n ) ≤ ϵ n

for some sequence ϵ n → 0. ̃2,n : Now we are ready to state the convergence theorem of the discrete metrics W ̃2,n ) converge to (P(𝕋d ), W2 ) in the sense Theorem 2.10. The metric spaces (P(𝕋dn ), W of Gromov–Hausdorff as n → ∞. Remark 2.11. An informal convergence result can be done for a general stationary measure by discretizing the continuous FP equation with the scheme described in Section 15.3 and writing the corresponding discrete distance which looks almost like a discretization of the continuous Wasserstein distance.

̃ C̃ (arrow 3): Quadrature for the entropy functional Link C– In order to strengthen the relation between the discrete setting and the continuous one, we are going to show the convergence of the discrete relative entropy to its continuous counterpart (C̃ 󳨀→ C in the diagram of Figure 1). We consider a collection of n points V in Ω, and construct a partition (K x )x of the domain which is relative to V, i.e., each cell K x of the partition contains one point of V (which is x) and: Ω = ⋃ K x , K x ∩ K y = 0 ∀x ≠ y ∈ V . x∈V

Let h be the diameter of the partition K x , i.e., h = maxx∈V diam(K x ). For any probability measure p in P(Ω), we define its discrete counterpart by ̃ ∈ P(V) ̃ (x) = ∫ p(r) dr, ∀x ∈ V, p p

(2.10)

Kx

and then we define its discrete density by ̃ (x) p ρ̃ (x) = , ∀x ∈ V, ρ̃ ∈ D(V) . ̃ (x) π Proposition 2.12. Let p, π be two C1 (Ω) densities with respect to the Lebesgue measure, ̃h bounded from below and above, i.e., ∃ m, M > 0 such that 0 < m ≤ p, π ≤ M, and p̃ h , π be their discrete counterpart defined according to (2.10). We denote by ρ̃ h the discrete ̃ h with respect to π ̃ h . Then, the discrete relative entropy: density of p ̃ h (ρ̃ h ) = ∑ ρ̃ h (x) log(ρ̃ h (x))π ̃ h (x) H x∈V h

converges to the continuous relative entropy: H(p) = ∫ p(r) log ( Ω

when h → 0, at the first order in h.

p(r) ) dr π(r)

15.3 Discretization of the Fokker–Planck equation |

411

Proof. We substract the continuous and the discrete quantities: 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ̃ (x) p(r) p 󵄨󵄨 󵄨 h 󵄨 ̃ ρ̃ h )󵄨󵄨 = 󵄨󵄨 ∑ (∫ p(r) log ( 󵄨󵄨 󵄨󵄨H(p) − H( ̃ dr − p (x) log ) )) ( h 󵄨󵄨 󵄨 󵄨 󵄨󵄨󵄨 ̃ h (x) π π(r) 󵄨󵄨 󵄨󵄨x∈V K x 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 ̃ h (x) p p(r) 󵄨 ) − log ( )) dr󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨 ∑ ∫ p(r) (log ( ̃ h (x) π π(r) 󵄨󵄨 󵄨󵄨x∈V 󵄨󵄨 󵄨󵄨 K x 󵄨󵄨 p(r) p 󵄨 ̃ h (x) 󵄨󵄨 󵄨 󵄨󵄨 dr − ≤ ∑ ∫ p(r)C 󵄨󵄨󵄨 󵄨󵄨 π(r) π ̃ h (x) 󵄨󵄨󵄨 x∈V Kx

M where C is a Lipchitz constant for log on [ m M , m ]. Then, by straightforward computations using the boundedness from below and above of the measures we can bound the difference by C󸀠 × h where C󸀠 is a constant depending on m, M and C.

15.3 Discretization of the Fokker–Planck equation We re-write the Fokker–Planck system by replacing ∇Φ by −∇π/π in the first equation, we obtain ∂p ∇π ∂p p − ∇ ⋅ (∇p − p ) = 0, or equivalently: − ∇ ⋅ (π∇ ( )) = 0 ∂t π ∂t π

(3.1)

Finite volume discretization Let V be a collection of points in Ω, and let (K x )x be the associated Voronoi tesselation (Figure 3). We denote by N the dual network of the space discretization (two vertices are connected whenever the corresponding cells are adjacent). Then we discretize in space the FP equation (in its form (3.1)) by a finite volume scheme (see, e.g., [12]). By integrating the equation on K x , we obtain ̃ ̃ dp p dp p (x) − ∫ π∇ ( ) n x dσ = (x) − ∑ ∫ π∇ ( ) n xy dσ = 0 dt π dt π y∼x Γ xy

∂K x

where n x is the outward normal to K x , Γ xy = K x ∩ K y , n xy = n x |Γ xy and y ∼ x means that y is a neighbor of x, y ≠ x. We approximate p ∫ π∇ ( ) n xy dσ π Γ xy

by

̃ (x) |Γ xy | ̃ (y) p p ̃ (x), π ̃ (y)) ( − ) θ(π ̃ (y) π ̃ (x) π |x − y|

412 | 15 Finite volume discretization and Wasserstein gradient flows

Ky y Γxy

Kx

x

Fig. 3: Voronoi cells with the used notations.

(this approximation is inspired from [13] and was used in [1] for the one-dimensional case) and we get the final form of the semi-discretized equation: ̃ (x) |Γ x,y | ̃ (y) p d p ̃ (x) = ∑ ̃ (x), π ̃ (y)) ( − p θ(π ) , ̃ ̃ (x) π (y) π dt − y| |x y∼x or equivalently: |Γ xy | d p̃ (y) ̃ (x) = ∑ ( ̃ (x), π ̃ (y)) ) p θ(π ̃ (y) π dt − y| |x y∼x − (∑ y∼x

|Γ xy | ̃ (x) p ̃ (x), π ̃ (y))) , θ(π ̃ (x) π |x − y|

(3.2)

which can be seen as an evolution equation on the network N.

Semi-discretized equation written with probability densities An equivalent semi-discretized equation of (3.2) is written with the probability densĩ , i.e., p ̃ (x) = ρ̃ (x)π ̃ (x): ties ρ̃ with respect to π |Γ xy | d ̃ (x), π ̃ (y))ρ̃ (y)) ρ̃ (x) = ∑ ( θ(π ̃ (x) dt − y| π |x y∼x − (∑ y∼x

|Γ xy | ̃ (x), π ̃ (y))) ρ̃ (x) . θ(π ̃ (x) |x − y| π

(3.3)

Now, equation (3.3) can be written as d ρ̃ = Q ρ̃ , dt

(3.4)

15.3 Discretization of the Fokker–Planck equation |

with

̃ (x), π ̃ (y)) |Γ xy |θ(π { , { { ̃ − y| π (x) |x { { { { ̃ (x), π ̃ (y)) Q(x, y) = {− ∑ |Γ xy |θ(π , { { ̃ (x) |x − y| π { y∼x { { { {0

413

if x ∼ y if x = y otherwise

on the network N. Equation (3.4) is not exactly of the heat flow type (2.9), since Q is not of the form K − I, where K would be a stochastic matrix. Yet, as pointed out in [14], a connection can be made between the two settings: For a matrix Q as above, we set q x = ∑ Q(x, y)

and

qmax = max q x . x

y∼x

We then define the matrix K as follows: Q(x, y) { { { qmax , K(x, y) = { { { qmax − q x , { qmax

if x ≠ y if x = y .

Proposition 3.1. The matrix K resulting from the space discretization of the FP equation (3.1) as described above is an irreducible and reversible Markov Kernel that admits ̃ as stationary measure. π Proof. The matrix K has the following properties: (i) K(x, y) = Q(x, y)/qmax ≥ 0 for x ≠ y, K(x, x) = (qmax − q x )/qmax ≥ 0, and ∑ K(x, y) = ∑ y∼x

y∈V

=∑ y∼x

Q(x, y) qmax − q x + qmax qmax Q(x, y) Q(x, y) +1− ∑ =1. qmax y∼x qmax

(ii) K(x, y) ≠ 0 for x ∼ y and the network is strongly connected, we deduce that K is irreducible and then has a unique stationary measure. ̃ satisfies the detailed balance equation for all x, y ∈ V: (iii) π ̃ (x), π ̃ (y)) ̃ (x) |Γ xy |θ(π π Q(x, y) = ̃ qmax qmax |x − y| π (x) ̃ (x), π ̃ (y)) ̃ (y) |Γ xy |θ(π π Q(y, x) ̃ (y)K(y, x) ̃ (y) =π =π = ̃ (y) qmax qmax |x − y| π

̃ (x)K(x, y) = π ̃ (x) π

so K is reversible, and we have ̃ (y)K(y, x) = ∑ π ̃ (x)K(x, y) = π ̃ (x) ∑π y∈V

y∈V

̃ is the unique stationary measure of K. which proves that π

414 | 15 Finite volume discretization and Wasserstein gradient flows

By definition of K, we have 1 Q = (K − I) , q max so that the solution to (3.4), that is the space-discretized solution, is the solution to the heat-flow equation d ρ̃ + (I − K)ρ̃ = 0 , dt up to an affine time renormalization. Now recall that the continuous FP equation is the gradient flow in the Wasserstein sense (see Definition (2.1)) for the relative entropy (2.5). We have the following discrete counterpart of this property for the finite volume discretization scheme (3.4), that is a direct consequence of the previous developments: Proposition 3.2. The space discretized scheme (3.4) is a gradient flow for the discrete relative entropy (1.2), up to an affine time renormalization, with respect to the discrete ̃2 (see Definition 2.4). Wasserstein distance W

15.4 Conclusive remarks, perspectives We described in this paper how some space discretization finite volume schemes, possibly on unstructured meshes, can be proved to be deeply respectfull of the underlying gradient flow structure. Given a PDE that is the Wasserstein gradient flow of some functional, the ODE resulting from space discretization can be identified as a gradient flow for a discrete functional that is an approximation of the continuous one, in the Wasserstein space of measures defined on the underlying network, the vertices of which are the finite volume cells. This overall consistency with respect to Wasserstein metric, that is expressed by Figure 1, can be used to improve the numerical analysis of a scheme, e.g., by characterizing its long-time behavior (as in [2] in the case of a cartesian mesh). Note that the approach is currently limited to the semi-discretized scheme. Let us add that the considered scheme treats the advection in a diffusive manner, and as such it is intrinsically of the centered type, so that stability issues can be expected. In particular, an Euler Explicit scheme is likely to lead to unconditional unstability. Implicit time-stepping may, in the contrary, provide some stability. Note that Implicit Euler time-stepping applied to (3.4) leads to a problem that is formally very similar to the so-called JKO scheme applied at the discrete level to compute the gradient flow. Implicit schemes are then likely to recover some properties of the JKO one. Let us finally stress that Figure 1 is not fully realized. Indeed, the arrow 2 between blocks B and B󸀠 , which expresses a link between the Wasserstein distance in a domain, and the discrete Wasserstein distance on the network obtained by space discretization, is not covered by a full theory. The only known convergence results ([11]) concern cartesian grids, in the case without potential. In the presence of a nonconstant potential, the framework that has been presented may appear puzzling, because the discrete

Bibliography | 415

Wasserstein distance involves the stationary measure (nonuniform in general), which depends on the potential Φ, whereas its continuous counterpart pertains to the flat domain, and therefore does not depend on Φ. This apparent paradox is due to the ̃2 is defined for densities with respect to fact that, at the discrete level, the distance W ̃ . Comparing both distances would amount to consider two the stationary measure π ̃ 0 and p ̃ 1 , together with probabilities p0 and p1 , compute their discrete counterparts p ̃2 (ρ̃ 0 , ρ̃ 1 ), and check that the ̃ , then ρ̃ 0 = p ̃ 0 /π ̃ , ρ̃ 1 = p ̃ 1 /π ̃ , and finally estimate W π latter converges to W2 (p0 , p1 ) when the discretization is refined. Although not coṽ2 “sees” the measure π, ered by any theoretical result, and in spite of the fact that W while W2 does not, such a property can be expected, because π is involved twice in the ̃ , and then, in a hidden way, discretization process: firstly by computation of ρ̃ from p ̃ through the definition of W2 . One can check in very simple situations that both effect ̃2 (ρ̃ 0 , ρ̃ 1 ) upon π asymptottend to compensate each other, i.e., the dependence of W ically vanishes. It can also be seen in the very definition of the distance itself: each ̃ (x) is involved, it is multiplied by a quantity of the type θ(ρ̃ (x), ρ̃ (y)), where x time π and y are connected. In the context of finite volume schemes, when the discretization is refined, x and y get closer, so that this quantity is asymptotically close to ρ̃ (x), and ̃ (x), which does no longer depend on the stationfinally the real dependence is upon p ary measure. As for non-cartesian meshes, the analogy that we established advocate for a convergence of the discrete Wasserstein metric toward the continuous one, but it remains to be rigorously proven. Acknowledgment: The authors would like to thank F. Santambrogio for interesting discussions and suggestions. Both authors are partially supported by ANR Project Isotace (ANR-12-MONU-0013).

Bibliography [1] [2] [3] [4] [5] [6] [7]

K. Disser and M. Liero, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, Networks and Heterogeneous Media, AIMS 10(2):233–253, 2015. J. Maas and D. Matthes, Long-time behavior of a finite volume discretization for a fourth order diffusion equation, arXiv:1505.03178. R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM Journal of Mathematical Analysis 29(1):1–17, 1998. L. Ambrosio, N. Gigli and G. Savaré, Gradient flows-In Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zurich, Birkhauser Basel 2008. F. Santambrogio, Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their applications 87, Birkhauser Basel 2015. J. Maas, Gradient flows of the relative entropy for finite Markov chains, Journal of Functional Analysis 261(8):2250–2292, 2011. A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Non-linearity 24(4):1329–1346, 2011.

416 | 15 Finite volume discretization and Wasserstein gradient flows

[8] [9] [10] [11] [12] [13] [14]

C. Villani, Optimal transport, old and new, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag 2013. J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge– Kantorovich mass transfer problem, Numerische Mathematik 84:375–393, 2000. S.-N. Chow, W. Huang, Y. Li and H. Zhou, Fokker–Planck equations for a free energy functional or Markov process on a graph, Arch. Rational Mech. Anal. 203(3):969–1008, 2012. N. Gigli and J. Maas, Gromov–Hausdorff convergence of discrete transportation metrics, SIAM Journal of Mathematical Analysis 45(2):879–899, 2013. R. Eymard, T. Gallouet and R. Herbin, The finite volume method, Handbook of Numerical Analysis. Vol. VII, North Holland, Amsterdam, pp. 713–1022, 2000. A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations 48(1):1–31, 2013. M. Erbar and J. Maas Gradient flow structures for discrete porous medium equations, Discrete and Continuous Dynamical Systems 34(4):1355–1374, 2014.

Index A algorithm 124, 177, 197, 246, 248, 273, 274, 276, 334, 337, 340, 349 annulus 231 B Banach space 22, 23, 42, 318 boundary condition 7, 28, 42, 65–67, 74, 136, 176, 184, 188–190, 199, 201, 305, 324, 338, 356, 361, 394, 395, 401 bounded variation 123, 164, 335 C Carathéodory 261 compression 92 contrast 32, 41, 42, 113, 396 convergence 35, 36, 38, 47, 49, 55, 183, 192, 194, 195, 226, 244, 282, 285, 286, 288, 296, 305, 313, 317, 318, 320–324, 328, 338, 339, 341 convex 41, 42, 124, 176, 204, 206, 210, 218, 221, 228, 232–234, 240, 241, 244, 248, 259, 261–263, 268, 273, 354, 366, 368, 371, 404 correspondence 57, 58 cost functional 45, 46, 53–55, 58, 65 critical point 160, 367, 368 curvature 55, 124, 134, 173, 362 cut locus 6, 9, 10 D deformation 53, 57, 215 differential inclusion 334 diffusion 125, 134, 136, 139, 349, 358, 361, 362 Dirac measure 178, 183 Dirichlet boundary 44, 51 discretization 22, 32, 119, 136, 173, 174, 243–246, 248–250, 252, 260, 273–276, 338, 339, 346, 347, 349 discretization point 194, 197 divergence 7, 44, 51, 81, 83, 89, 98, 99, 108, 117, 275, 298–300, 338, 406, 407 E edge 298, 348 elastic 64, 144, 146, 153 energy 73, 163, 166, 206, 207, 209, 210, 305, 318, 324, 328, 334, 335, 338 energy estimate 363, 370, 381

Euler–Lagrange 18, 367, 368 extremal 273 F final time 43, 46, 51, 52 finite element 273–276, 362 frequency 145, 147 functional 41, 44, 125, 160, 164, 166, 173, 259, 288, 299, 304, 313, 315, 325, 357, 360, 362–366, 377, 393, 402–405, 407, 408, 410, 414 G Gaussian 236, 275 geodesic 127, 328, 329 geometry 65, 66, 74–76, 82, 86, 102, 107, 173, 197, 205, 207, 223, 362 gradient 9, 13, 15–17, 66, 74, 88, 157, 159, 174, 232, 263, 268, 274, 335, 337, 349 group 153 H Hamiltonian 209, 210, 215 Hamiltonian function 215 harmonic 8, 78, 82, 96, 97 I impulse 5 initial value 44, 54, 397 integrable 210, 217, 284, 299, 315 interpolation 50, 186, 309, 360, 363, 368, 375, 395 invariant 74, 89, 103, 115, 153, 237, 285 inverse 46, 167, 177, 229, 276, 344, 362 iterative scheme 28, 29 J Jacobian 396 K Kolmogorov 319, 320 L Laplacian 5, 6, 18 linear system 134 Lipschitz 5, 11, 13, 66, 74, 89, 126, 160, 163, 176, 187, 197, 201, 222, 226, 227, 229, 257, 306

418 | Index

Lipschitz curve 266, 282, 291 lower semicontinuous 229, 283, 286, 287, 297, 309, 328, 331 M mass center 346 metric 124, 128, 259, 287, 306, 350, 401 minimization 42, 135, 209, 289, 296, 309, 367, 378 minimizer 41, 45, 126, 135, 204, 209, 213, 215–218, 225, 229, 231, 309, 348, 360, 366–368, 371 model 173, 207, 210, 212, 231, 257, 259, 261, 263, 264, 269, 275, 276, 304 moment 7, 9, 15, 44, 232, 239, 309, 342, 358 multiplier 125, 135, 136, 139, 142, 335 multiscale 3 N N-body problem 210 necessary condition 92, 143, 224 Neumann boundary condition 28, 29, 44, 85, 109, 262, 274, 275, 324 Newton law 336 noise 119 nonlinear equation 8 nonlocal 361 numerical example 22, 32, 349, 350 numerical method 242 O objective function 144 optical flow 124 optimal control problem 3, 143 optimality condition 43, 45, 47, 60, 232, 241, 245, 259, 263, 264, 275 optimization 3, 42 optimization problem 47, 65, 85, 87, 91, 102, 113, 118, 142, 151, 209, 259 orthogonal projection 46 P parameter 21–24, 31, 32, 38, 42, 43, 47, 48, 125, 127, 134, 137, 168, 169, 190, 191, 214, 242, 267, 282, 283, 341, 342, 359, 360, 366, 367, 370, 397 parametrization 74 partial differential equation 65, 142, 143 PDE 5, 17, 136, 173, 205, 224, 414 perturbation 153, 174

phase 3, 40, 41, 338 Poisson 65, 215, 338 polar coordinate 28, 68, 187, 201 probability density 211, 212, 214, 216, 406, 409, 412 R reconstruction 65, 124, 342 registration 160 regularization 31, 242, 244, 261, 265, 267, 334 remainder 199, 321 restoration 124 rotation 82, 83, 92, 99, 110, 236, 239 S sampling 64 sensitivity 33 smoothing 42, 43, 215 smoothness 11, 35, 150, 362, 404 state constraint 183, 184 state variable 86, 87, 102 stationary point 43, 47, 48 Stokes equation 338 structure 14, 26, 42, 57, 124, 173, 174, 177, 205, 207, 230–232, 237, 238, 260, 281, 350, 400, 402, 414 sufficient condition 49, 233, 262, 285 symmetry 27, 81, 98, 110, 144, 192, 216, 224, 226, 230, 232, 238, 250, 350 symplectic manifold 215 T total variation 125, 164, 363, 372, 373 U unconstrained 340 V variational 42, 49, 54, 65, 66, 143, 145, 173, 174, 210, 212, 313 variational equation 66, 67, 76, 83, 84, 86, 106, 107, 111 variational method 377 vector field 50, 51, 53, 57, 265, 269, 272, 347, 377, 404 velocity 41, 42, 44, 333–342, 344, 346–350, 354, 377, 405, 406 W wave 64, 66, 75, 76, 92, 107, 119