Nonlinear Evolution Equations 9782759834488, 9782759834495, 9787030595621

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Boling GUO, Fei CHEN, Jing SHAO and Ting LUO

Nonlinear Evolution Equations

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EDP Sciences – ISBN(print): 978-2-7598-3448-8 – ISBN(ebook): 978-2-7598-3449-5 DOI: 10.1051/978-2-7598-3448-8 All rights relative to translation, adaptation and reproduction by any means whatsoever are reserved, worldwide. In accordance with the terms of paragraphs 2 and 3 of Article 41 of the French Act dated March 11, 1957, “copies or reproductions reserved strictly for private use and not intended for collective use” and, on the other hand, analyses and short quotations for example or illustrative purposes, are allowed. Otherwise, “any representation or reproduction – whether in full or in part – without the consent of the author or of his successors or assigns, is unlawful” (Article 40, paragraph 1). Any representation or reproduction, by any means whatsoever, will therefore be deemed an infringement of copyright punishable under Articles 425 and following of the French Penal Code. The printed edition is not for sale in Chinese mainland. Customers in Chinese mainland please order the print book from Science Press. ISBN of the China edition: Science Press 978-7-03-059562-1 Ó Science Press, EDP Sciences, 2023

Preface With the study of soliton and chaos in modern physics, a large number of nonlinear evolution equations with nonlinear dispersion or dissipation have continuously emerged, including KdV equation, nonlinear Schrödinger equation, sine-Gordon equation, Zakharov equations, Landau-Lifshitz equations, Boussinesq equation etc., which have soliton solutions, Navier-Stokes equations and Newton-Boussinesq equations which describe phenomenons of turbulence and chaos, nonlinear Schrödinger equation and Zakharov equations with dissipation and damping etc. These equations are closely connected with physical problems, and their research contents are constantly enriched and developed. For example, in addition to the existence, uniqueness and regularity of classical solutions, the long time behavior of which is also studied, including decay, scattering and stability of solutions with respect to space and time, and blow up phenomenon (collapse) which may occur in finite time. There have been a lot of good work on the study of these problems, and many special estimation methods have been formed. On the basis of letting readers understand the physical background of these nonlinear evolution equations, the purpose of this book is to use a relatively simple and clear method and a relatively short space, to introduce some typical methods in the research of these equations, the interesting research contents and some important results obtained. For infinite dimensional dynamical systems, we mainly introduce the basic concepts and research methods of global attractor, inertial manifold, approximate inertial manifold, inertial set, nonlinear Galerkin method etc., in order to lead readers to have a rough, but a clear understanding about the overview of infinite dimensional dynamical systems. The research contents of nonlinear evolution equations are very rich and extensive. Various research methods and results emerge in an endless stream. Limited to author’s existing level and ability, this book is unavoidable to have many inappropriate, incomplete, and even wrong places. I would like to have further criticism and corrections.

Guo Boling 28, June, 2019

Contents Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The wave equation under weak nonlinear action and KdV equation . . . . . . 2 1.2 Zakharov equations and the solitons in plasma . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Landau-Lifshitz equations and the magnetized motion . . . . . . . . . . . . . . . . . . 19 1.4 Boussinesq equation, Toda Lattice and Born-Infeld equation . . . . . . . . . . . 22 1.5 2D K-P equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter 2 The Properties of the Solutions for Some Nonlinear Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1 The smooth solution for the initial-boundary value problem of nonlinear Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 The existence of the weak solution for the initial-boundary value problem of generalized Landau-Lifshitz equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2.1 The basic estimates of the linear parabolic equations . . . . . . . . . . . . . . . . . 34 2.2.2 The existence of the spin equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.3 The existence of the solution to the initial-boundary value problem of the

generalized Landau-Lifshitz equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 The large time behavior for generalized KdV equation . . . . . . . . . . . . . . . . . 42 2.4 The decay estimates for the weak solution of Navier-Stokes equations . . 60 2.5 The “blowing up” phenomenon for the Cauchy problem of nonlinear Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.6 The “blow up” problem for the solutions of some semi-linear parabolic and hyperbolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.7 The smoothness of the weak solutions for Benjamin-Ono equation . . . . . . 93 Chapter 3 Some Results for the Studies of Some Nonlinear Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.1 Nonlinear wave equations and nonlinear Schrödinger equations . . . . . . . . 105 3.2 KdV equation, etc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.3 Landau-Lifshitz equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Chapter 4 Similarity Solution and the Painlevé Property for Some Nonlinear Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.1 Classical infinitesimal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.2 Structure of Lie algebra for infinitesimal operator . . . . . . . . . . . . . . . . . . . . . 156 4.3 Nonclassical infinitesimal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

vi

Contents

4.4 A direct method for solving similarity solutions . . . . . . . . . . . . . . . . . . . . . . . 163 4.5 The Painlevé properties for some PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Chapter 5 Infinite Dimensional Dynamical Systems . . . . . . . . . . . . . . . . . . . 182 5.1 Infinite dimensional dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.2 Some problems for infinite dimensional dynamical systems . . . . . . . . . . . . 187 5.3 Global attractor and its Hausdorff, fractal dimensions . . . . . . . . . . . . . . . . . 196 5.4 Global attractor and the bounds of Hausdorff dimensions for weak damped KdV equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.4.1 Uniform a priori estimation with respect to t . . . . . . . . . . . . . . . . . . . . . . . 207 5.5 Global attractor and the bounds of Hausdorff dimensions for weak damped nonlinear Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.5.1 Uniform a priori estimation with respect to t . . . . . . . . . . . . . . . . . . . . . . . 218 5.5.2 Transforming to Cauchy problem of the operator . . . . . . . . . . . . . . . . . . . 221 5.5.3 The existence of bounded absorbing set of H 1 modular . . . . . . . . . . . . . . 224 5.5.4 The existence of bounded absorbing set of H 2 modular . . . . . . . . . . . . . . 225 5.5.5 Nonlinear semi-group and long-time behavior . . . . . . . . . . . . . . . . . . . . . . . 228 5.5.6 The dimension of invariant set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 5.6 Global attractor and the bounds of Hausdorff, fractal dimensions for damped nonlinear wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 5.6.1 Linear wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 5.6.2 Nonlinear wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 5.6.3 The maximal attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5.6.4 Dimension of the maximal attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.6.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 5.6.6 Non-autonomous system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 5.7 Inertial manifold for one class of nonlinear evolution equations . . . . . . . . 269 5.8 Approximate inertial manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 5.9 Nonlinear Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 5.10 Inertial set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Chapter 6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 6.1 Basic notation and functional space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 6.2 Sobolev embedding theorem and interpolation formula . . . . . . . . . . . . . . . . 348 6.3 Fixed point theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Bibliography i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Index

Chapter 1 Physical Backgrounds for Some Nonlinear Evolution Equations In 1834, the famous British scientist J. Scott. Russell [211] found solitary wave phenomenon. With the development of modern physics and mathematics, people’s interest in this phenomenon has increased in the past more than 40 years. Now, both numerical calculation and theoretical analysis have proved that a large number of nonlinear evolution equations have soliton solutions. The solitary waves have a very peculiar property that they maintain stable waveforms under interaction, which is similar to the property of particle collisions. Accordingly, Kruskal and N. J. Zabusky [242] named them as “solitons”. The solitary waves can not only be observed in nature, but also be produced in the laboratory. With the development of the study for the soliton problem, a large number of nonlinear evolution equations with soliton solutions have occurred in fluid physics, solid state physics, elementary particle physics, laser, plasma physics, superconductor physics, condensed matter physics, biophysics and many other fields. For example, the KdV equation deduced by shallow water wave in 1895 has been obtained in a series of problems such as ion acoustic wave, cold plasma magnetic fluid wave, nonlinear lattice and so on. This group of nonlinear evolution equations with soliton solutions are: nonlinear Schrödinger equation, sine-Gordon equation, Landau-Lifshitz equations, Boussinesq equation, two-dimensional Kadomtsev-Petviashvili equation and so on. These equations have many features in common, such as: they can be solved by the inverse scattering method, have Bäcklund transformation and Darboux transformation, and posses an infinite number of conservation laws and extended structure etc. Since these nonlinear evolution equations belonging to integrable systems are closely related to physical problems, in this chapter, we will briefly introduce the physical background of some important equations. This is very helpful for the qualitative research of partial differential equations.

2

1.1

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

The wave equation under weak nonlinear action and KdV equation

As we know, KdV equation was first established by D. J. Korteweg and G. de Vries [181] in 1895, when they study water wave under the assumptions of long-wave approximation, small and finite amplitude. Now we begin to deduce the equation from water wave, then from other media under weak nonlinear action. We consider the inviscid incompressible fluid (water) with constant gravity field. The space coordinate system is (x1 , x2 , y), the components of the velocity field u are (u1 , u2 , v), the gravitational acceleration takes the negative direction of the axis y. So, there are equations ∇ · u = 0, (1.1.1) 1 ∂u + (u · ∇)u = ∇p − gj. ∂t ρ

(1.1.2)

Now let’s consider the irrotational motion, that is rot u = 0. Therefore, there exists the velocity potential u = ∇φ. By   1 2 |u| = (u · ∇)u − rot u × u = (u · ∇)u (1.1.3) ∇ 2 and integrating on both sides of (1.1.2), one has p − p0 1 = B(t) − φt − (∇φ)2 − gy, p0 2 where B(t) is any function, p0 is any constant. Let Z φ′ = φ − B(t)dt, then u = ∇φ,

1 p − p0 = −φ′t − (∇φ′ )2 − gy. p0 2

(1.1.4)

Here and in the sequel, denote φ′ by φ. According to (1.1.1), we get ∇ · u = 0 =⇒ ∇2 φ = 0.

(1.1.5)

Assume that the surface equation of water is f (x1 , x2 , y, t) = 0.

(1.1.6)

Just because the particle can not pass through the surface, the velocity of fluid which is orthogonal to that surface must equal the normal speed of the surface. The

1.1

The wave equation under weak nonlinear action and KdV equation

3

−ft normal speed of (1.1.6) is q , the normal speed of the velocity of fluid 2 fx1 + fx22 + fy2 u1 fx + u2 fx2 + vfy is q 1 , the qual condition is fx21 + fx22 + fy2 ft + u1 fx1 + u2 fx2 + vfy = 0.

(1.1.7)

Especially, when y = η(x1 , x2 , t), f (x1 , x2 , y, t) ≡ η(x1 , x2 , t) − y, (1.1.7) shows that (1.1.8)

ηt + u1 ηx1 + u2 ηx2 = v.

In addition, on the free surface, p = p0 (ignore the movement of the air), then we obtain  ηt + φx1 ηx1 + φx2 ηx2 = φy ,
φt + 1 φ2 + φ2 + φ2 + gη = 0. x1 x2 y 2

(η = η(x1 , x2 , t))

(1.1.9)

Where u1 = φx1 , u2 = φx2 , v = φy . By the fixed boundary condition, the normal speed of the fluid must be 0, n · ∇φ = 0. Especially, in the bottom y = −h0 (x1 , x2 ), one has φy + φx1 h0x1 + φx2 h0x2 = 0. For horizontal bottom, there exist φy = 0, y = −h0 . Therefore, the formulation of the question is just like the following: look for the velocity potential φ and the surface η, satisfying ∇2 φ = 0;  ηt + φx1 ηx1 + φx2 ηx2 = φy ,
φt + 1 φ2 + φ2 + φ2 + gη = 0; x1 x2 y 2 φy = 0, y = −h0 .

(1.1.10) (y = η(x1 , x2 , t))

(1.1.11) (1.1.12)

For simplicity, in the following, we consider one-dimensional case, that is η = η(x, t). Let y be the height from the horizontal ground, here, φy = 0, y = 0. We introduce a h2 two parameters: α = , β = 20 , where a is the amplitude of the wave, l is the h0 l lt′ glφ′ 2 wavelength, y = h0 + η. Let x = lx′ , y = h0 y ′ , t = , η = aη ′ , φ = , c0 = 8h0 . c0 c0

4

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

Here we ignore the mark “′ ”, by (1.1.5), (1.1.11), (1.1.12), we get βφxx + φyy = 0,

0 < y < 1 + αη;

φy = 0,

y = 0;  1   ηt + αφx ηx − φy = 0,  β y = 1 + αη. 1 1α 2   η + φt + αφ2x + φy = 0, 2 2β

(1.1.13) (1.1.14) (1.1.15)

Assume that the formal solution of (1.1.14) and (1.1.15) is φ=

∞ X m=0

(−1)m

y 2m ∂ 2m f 2m β , (2m)! ∂x2m

(1.1.16)

where f = f0 (x, t). Substituting (1.1.16) into the first equation of (1.1.15), one has   (1 + αη)2 ηt + α fx − (1 + αη)ηx fxx β − fxxx β + · · · ηx 2 1 +(1 + αη)fxx − (1 + αη)3 fxxxx β + o(β 2 ) = 0, 3! that is,   1 α 3 2 ηt + {(1 + αη)fx }x − (1 + αη) fxxxx + (1 + αη) fxxx ηx β + o(β 2 ) = 0. 6 2 (1.1.17) Similarly, Substituting (1.1.16) into the second equation of (1.1.15), one has 1 1 2 }β + o(β 2 ) = 0. η + ft + αfx2 − (1 + αη)2 {fxxt + αfx fxx − α′ fxx 2 2

(1.1.18)

In (1.1.17) and (1.1.18), if ignore the first degree term of β, and differentiate (1.1.18) with respect to x, then ( ηt + {(1 + αη)w}x = 0, (w = fx ) (1.1.19) wt + αwwx + ηx = 0. If reserve the first degree term of β, then  1  ηt + {(1 + αη)w}x − βwxxx + o(αβ, β 2 ) = 0, 6  wt + αwwx + ηx − 1 βwxxt + o(αβ, β 2 ) = 0. 2

(1.1.20)

If ignore the first and more degree terms of α, β in (1.1.20), then, when w = η, the two equations of (1.1.20) change into the same equation ηt + ηx = 0. So, w can be unfolded about α, β: w = η + αA + βB + o(α2 + β 2 ),

1.1

The wave equation under weak nonlinear action and KdV equation

5

where A, B are functions of η and the derivatives of η to x. By (1.1.20), one has   1 ηt + ηx + α(Ax + 2ηηx ) + β Bx − ηxxx + o(α2 + β 2 ) = 0, 6   1 ηt + ηx + α (At + ηηx ) + β Bt − ηxxt + o(α2 + β 2 ) = 0. 2 Since ηt = −ηx + o(α, β), the derivative with respect to t in the first order term can 1 1 be changed into that with respect to x, especially, when A = − η 2 , B = ηxx , the 4 3 two equations above coincide with each other, and 1 3 ηt + ηx + αηηx + βηxxx + o(α2 + β 2 ) = 0, 2 6

(1.1.21)

and

1 1 w = η − αη 2 + βηxx + o(α2 + β 2 ). 4 3 If ignore quadratic term in (1.1.21), then we can get the classical KdV equation 1 3 ηt + ηx + αηηx + βηxxx = 0. 2 6

(1.1.22)

Since ηt = −ηx + o(α, β), change ηxxx into −ηxxt in (1.1.22), then one has 3 1 ηt + ηx + αηηx − βηxxt = 0, (1.1.23) 2 6 which is the BBM equation. Next, we deduce a kind of rather extensively wave equations under weak nonlinear action. At last, they can be summed up as KdV equation or Burgers equation  nt + (nu)x = 0, (1.1.24)       (nu)t + (nu2 + P )x = 0, (1.1.25)   P = P (f, n, u, fi , ni , ui , fij , nij , uij , · · · ),     F (f, n, u, fi , ni , ui , fij , nij , uij , · · · ) = 0,

(1.1.26) (1.1.27)

where n, u, f are state variables, n represents the number density of the particle, u represents the velocity of the particle, i, j represent the derivatives with respect to the space variable x and the time variable t respectively, P represents the function of state variables (n, u, f ) and their derivatives. If f represents function of the parameters, then P is a function of n, u and their derivatives. (1.1.24) is the mass conservation equation and (1.1.25) is the momentum conservation equation. In the following, we introduce some examples: (1) Gas dynamics: f is the pressure p, P =

1 (p − µux ), m

F = p − Aρr ,

mn = ρ,

(1.1.28)

6

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

where ρ is the density, µ is the viscous coefficient. (2) Shallow water wave: η is the depth of the water h, at the moment, there are only two variables h and u, P =

1 2 1 3 gh − h (uxt + uuxx + u2x ). 2 3

(1.1.29)

(3) Magnetohydrodynamic wave in cold plasmas: f is the magnetic intensity B, 1 P = B2, 2   Bx F ≡B−n− = 0. (1.1.30) n x (4) Ion acoustic wave in cold plasmas: f is the electrostatic potential, ψ is the wave function, 1 P = eψ − ψx2 , 2

F ≡ n − eψ + ψxx = 0.

(1.1.31)

In the state of local thermodynamics equilibrium, if all the derivatives in P and F vanish, then P = P (f, n),

F (f, n) = 0.

(1.1.32)

By (1.1.25), one has nut + nuux + Px = 0,

Px =

∂P ∂n ∂P ∂f + . ∂f ∂x ∂n ∂x

∂F ∂f ∂F ∂n ∂f + = 0, eliminating , we get ∂f ∂x ∂n ∂x ∂x   a2 Fn 2 ut + uux + nx = 0, a = Pn − Pf . n Ff

Then by applying

If a2 > 0, then  nt + (nu)x = 0, 2 u + uu + a n = 0. t x x n

(1.1.33)

dx (1.1.33) are hyperbolic equations, the characteristic line of which is = u ± a, dt where a is the wave velocity. By small perturbations from the homogeneous state, we get the wave equation utt − a20 uxx = 0,

1.1

The wave equation under weak nonlinear action and KdV equation

7

where a0 is the homogeneous wave velocity. In the following derivation of KdV equation and Burgers equation, the effects of the nonlinear terms with small perturbations must be considered, that is the effect of the derivatives of P and F . Let us do the following transformation ( ξ = εα (x − a0 t), (1.1.34) τ = εα+1 t, where ε < 1 is the amplitude of the initial perturbations; the exponent α > 0 is undetermined, and a0 is a kind of wave velocity which is a constant. By (1.1.34), (1.1.24) and (1.1.25), one has εnτ + (u − a0 )nξ + nuξ = 0, εuτ + (u − a0 )uξ + n

−1

(1.1.35) (1.1.36)

Pξ = 0.

The state variables (n, f, u) must be performed asymptotic expansions with respect to ε near the equilibrium A = (n, f, u) = (n0 , f0 , 0): n = n0 + εn(1) + ε2 n(2) + · · · , f = f0 + εf (1) + ε2 f (2) + · · · , u = 0 + εu(1) + ε2 u(2) + · · · . P and F are also performed expansions to get P = P0 + Pf0 (f − f0 ) + Pn0 (n − n0 ) + Pu0 (u − u0 ) + o(ε2 ), F = F0 + Ff0 (f − f0 ) + Fn0 (n − n0 ) + Fu0 (u − u0 ) + o(ε2 ). Because of the Galilean invariance of the equation, Pu0 = Fu0 = 0, then by P (1) = Pf0 f (1) + Pn0 n(1) ,

Ff0

∂n(1) ∂f (1) + Fn0 = 0, ∂ξ ∂ξ

we obtain ∂P (1) ∂f (1) ∂n(1) = Pf0 + Pn0 ∂ξ ∂ξ ∂ξ     (1) Fn0 ∂n ∂n(1) = Pn0 − Pf0 = a20 . Ff0 ∂ξ ∂ξ If the second order terms in the above expansion are considered, one can deduce that (2)

Pξ

(2)

(1)

(1)

(1)

= a20 nξ + Annξ + εα−1 Bnξξ + ε2α−1 Cnξξξ ,

(1.1.37)

8

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

where the constants a0 , A, B, C in the above examples are given in the following Table 1.1. Table 1.1

The constants a0 , A, B, C in the equations (1.1.37)

Gas dynamics Water waves Magnetohydrodynamic wave Ion acoustic wave

a0

A

B

C

2hT /m 8 h0 B0 1

0 0 1 0

−u a0 0 0 0

0 8 h30 /3 1 1

From (1.1.35), (1.1.36), comparing the first order terms of ε, one has  2 a (1) (1) (1) (1) a0 nξ = n0 uξ , a0 uξ = nξ . n0 Integrating with respect to ξ and applying the boundary conditions, we have a0 n(1) = n0 u(1) . Substituting it into (1.1.35), (1.1.36) and taking the second order approximation, we conclude that (1)

(1)

(1)

(2)

(1) n(1) nξ + n0 uξ + n(2) uξ − a0 nξ = 0, τ +u

that is n(1) τ +2

a0 (1) (1) (2) (2) n nξ − a0 nξ + n0 uξ = 0 n0

and A (1) (1) B (1) C (1) a2 (2) a0 (1) (1) (2) nτ + n nξ + εα−1 nξξ + ε2α−1 nξξξ + 0 nξ + u(1) uξ − a0 uξ = 0. n0 n0 n0 n0 n   a2 (2) a0 a0 (1) (1) (2) (2) By applying −a0 uξ + 0 nξ = nτ + 2 n(1) nξ , we can eliminate nξ n0 n0 n0 (2) and uξ to deduce the equation of n(1)  n(1) τ +

A 3 a0 + 2a0 2 n0

 (1)

n(1) nξ + εα−1

B (1) C (1) nξξ + ε2α−1 n = 0. 2a0 2a0 ξξξ

(1.1.38)

In (1.1.38), if B ̸= 0 (dissipative, B < 0), α = 1, C = 0, then it turns into Burgers 1 equation; if B = 0 (dispersive), α = , then it becomes KdV equation 2   A 3 a0 C (1) (1) n(1) + n(1) nξ + n = 0. (1.1.39) τ + 2a0 2 n0 2a0 ξξξ

1.1

The wave equation under weak nonlinear action and KdV equation

9

Next, we give a brief introduction of the soliton solutions of KdV equation ut + u ux + µ uxxx = 0,

(1.1.40)

where µ can be positive or negative. If µ < 0, via the transformations u → −u, x → −x, t → t, then (1.1.40) transforms into ut + u ux − µ uxxx = 0.

(1.1.41)

Thus, we set µ > 0. Making u(x, t) = u(ξ), ξ = x − D t, D = const, substituting them into (1.1.41), and integrating twice with respect to ξ, we have  3µ

du dξ

2 = −u3 + 3D u2 + 6A u + 6B = f (u),

(1.1.42)

with A and B being integral constants. The solution of (1.1.42) is real only if f ⩾ 0 and µ > 0. If f (u) only has one real root, then it must be unbounded. Now, we assume that function f (u) has three real roots, i.e., f (u) = −(u − c1 )(u − c2 )(u − c3 ), 1 1 c1 < c2 < c3 . We conclude that D = (c1 + c2 + c3 ), A = (c1 c2 + c2 c3 + c3 c1 ), 3 6 1 B = c1 c2 c3 . The general form of f (u) can be expressed by curve A in Figure 1.1. 6 The exact solution for Eq. (1.1.42) can be represented as Jacobi elliptic function   r  c3 − c1 1 u = u(x, t) = c2 + (c3 − c2 )cn 2 x − (c1 + c2 + c3 )t , k , (1.1.43) 12µ 3 with k 2 = (c3 − cr 2 )/(c3 − c1 ). Eq. (1.1.43) is usually called as cnoidal wave, whose 3µ period is Tp = 4k , since the real period of function cn is 2k, with k being c3 − c1 the elliptic integral of the first kind.

Figure 1.1

Under k = 0, cn (ξ, 0) = cos ξ, the oscillation solution for Eq. (1.1.42) reads as  r   c3 − c1 1 u = c¯ + a cos 2 x − (c1 + c2 + c3 )t , (1.1.44) 12µ 3

10

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

c2 + c3 c3 − c2 ,a= . 2 2 At the case of k = 1, cn (ξ, 1) = sech ξ, corresponds to the curve B in Figure 1.1, whose period becomes infinite, as c2 → c1 , i.e., the common soliton solution for Eq. (1.1.40) is obtained r   c3 − c1 1 u = c1 + (c3 − c1 ) sech2 x − (2c1 + c3 )t . (1.1.45) 12µ 3 with c¯ =

If c1 = u∞ , c3 − c1 = a, then (1.1.45) transforms into r   a h a i u = u∞ + a sech2 x − u∞ + t , 12µ 3

(1.1.46)

where u∞ is the homogeneous state at infinity, a denotes the amplitude of soliton. From Solution (1.1.46), we can see that the velocity of the solitary wave comparing to the homogeneous state is proportional to amplitude, while the width is inversely proportional to the square root of amplitude, and the amplitude is independent of the homogeneous state. If u∞ = 0, µ = 1, then we get from (1.1.46) that √ 2 D (x − D t). (1.1.47) u(x, t) = 3D sech 2

1.2

Zakharov equations and the solitons in plasma

Since the late 60s, many articles have studied the propagation of solitary waves in plasma. In the laser target, people observed the density depression forming near the critical surface, the propagation of the vorticity solitary waves caused by collapse, the solitons that occur when a laser beam naturally focuses in a nonlinear medium, the Langmuir solitons caused by high-frequency electric field and optical solitons produced by high-frequency horizontal field, etc. Due to the continuous improvement of experimental technology, in the interaction between plasma and laser, more and more significant phenomenons of solitons have been observed. From equations of two-fluid mechanics, we can get the important Zakharov equations [243], nonlinear Schrödinger equation, Ion acoustic soliton, Langmuir soliton, Optical soliton, etc. The equations of two-fluid mechanics are composed of the fluid mechanic equations of electron and ion. The ion equations are ∂ni + ∇ · (ni vi ) = 0, ∂t     ∂vi vi × B ni M + vi ∇vi = −Ti ∇ni + ni e E + ; ∂t c

(1.2.1) (1.2.2)

1.2

Zakharov equations and the solitons in plasma

11

the electronic equations are ∂ne + ∇ · (ne ve ) = 0, ∂t     ve × B ∂ve + ve ∇ve = −Te ∇ne + ne e E + ; ne m ∂t c

(1.2.3) (1.2.4)

the Maxwell equations are 1 ∂B = −∇ × E, c ∂t

(1.2.5)

4πe 1 ∂E =∇×B− (ni vi − ne ve ), c ∂t c

(1.2.6)

∇ · B = 0,

(1.2.7)

where ni , ne represent the number densities of the ion and electron, respectively. vi , ve represent the velocities of the ion and electron. M , m represent the masses of the ion and electron. Ti , Te represent the temperatures of the ion and electron. e is the electron charge, E represents the electric field intensity, while B represents ∂ ∂ ∂ +j +k . the magnetic field intensity. c represents the speed of light. ∇ = i ∂x ∂y ∂z Because the velocity of ion is small, we can ignore the magnetic field’s effect acting vi × B on it, that means, taking = 0 in (1.2.2). Since the size of the problem we c considered is very small, only being a few microns, here, we choose Ti , Te to be constants. Just like the usual method, the motion of the plasma can be divided into two parts: low frequency and high frequency. Ions only do high-frequency motion. And the electron and field quantities are divided into low-frequency (the subscript is l) and high-frequency (the subscript is h) parts, that is ne = nl + nh ,

E = El + Eh ,

B = Bl + Bh .

(1.2.8)

Let’s introduce the average amount of the physical quantities with respect to time 1 f (x, t) = T

Z

t+ T2

f (x, t)dt, t− T2

then fx (x, t) = [f (x, t)]x . Since

12

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

! Z T 1 t+ 2 f (x, t)dt T t− T2      1 T T = f x, t + − f x, t − T 2 2 Z t+ T2 1 ft (x, τ )dτ = ft (x, t), = T t− T2

d d f (x, t) = dt dt

and assume f (x, t) = f (x, t), that means the average amount is independent of t. Now, let’s do high-frequency averaging for physical quantities. That is, choose the frequency w to satisfy 1 1 1 > > . T wl Tw T wh So, one has fh (x, t) = 0,

nl (x, t) = nl (x, t)

nh (x, t) = 0,

n l − ne = nh .

For the ion equations, because there’s only the low-frequency oscillation, no changes occur by doing high-frequency averaging while it only needs to change the electric field intensity E into El , so we get ∂ni + ∇ · (ni vi ) = 0, ∂t   ∂vi ni M + vi ∇vi = −Ti ∇ni + ni eEl . ∂t

(1.2.9) (1.2.10)

Take the high-frequency averaging for the electronic equations, that is, ∂ne + ∇ · (ne ve ) = 0, ∂t     ve × B ∂ve m + ve ∇ve = −Te ∇ln ne − e E + . ∂t c

(1.2.11) (1.2.12)

Submitting (1.2.8) into (1.2.11) and (1.2.12), respectively, and applying some properties of the above high-frequency averaging, we obtain ∂nl + ∇ · (nl vl + nh vh ) = 0, (1.2.13) ∂t   ∂vl vl × Bh vl × Bl m + vl ∇vl + vh ∇vh = −Tl ∇ln nl − eEl − e −e . (1.2.14) ∂t c c

1.2

Zakharov equations and the solitons in plasma

13

Considering that m is small, and that vi and vl are near, we can ignore the term mvl in (1.2.14), again let Bl = 0, so one can get 1 ∂El 4πe =− (ni vi − nl vl − nh vh ) , c ∂t c ∇ · El = 4πe(ni − nl ) + f (x),

(1.2.15)

− ∆φ = 4πe(ni − nl ) + f (x),

(1.2.16)

or

where, El = −∆φ, f (x) is any function of the variable x. For simplicity, choose f (x) = 0. The equations (1.2.3)∼(1.2.7) minus equations after averaging, then by  assuming vl = 0, ignoring the terms ∇ · (nh vh − nh vh ), m vh ∇vh − vh ∇vh etc., nh and the approximation that ≪ 1, we achieve at the two-fluid coupled equations nl ∂ni + ∇ · (ni vi ) = 0, ∂t   ∂vi ni M + vi ∇vi = −Ti ∇ni + ni eEl , ∂t

(1.2.17) (1.2.18)

∇ · El = 4πe(ni − nl ), (1.2.19) m ∇nl (1.2.20) eEl = −Te − ∇vh2 , nl 2 ∂nh + ∇ · (nl vh ) = 0, (1.2.21) ∂t   ∂vh nh m = −Te ∇ − eEh , (1.2.22) ∂t nl ∂ 2 Eh − c2 ∇2 Eh + c2 ∇ (∇ · Eh ) − vl2 ∇ (∇ · Eh ) + vl2 · (∇ · Eh ) · ∇ln nl ∂t2  ∂ln nl −4πne e2 mc2  ∂Eh ∂lnnl = Eh − ∇ (∇ · vh ) − ∇2 vh , (1.2.23) − ∂t ∂t m 2 ∂t Te where c is the velocity of light, ve2 = is the square of electrical thermal motion m velocity. Now let   nh ∇nh (1.2.24) ∇ ∼ . nl nl So (1.2.22) is similar to m

∂vh ∇nh = −Te − eEh . ∂t ne

(1.2.25)

14

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

 If

k 2 λ2D

≪1

k is the number of the

waves, λ2D

 Te = , then 4πn0 e2

Te ∇nh /ne ≈ k 2 λ2D ≪ 1. eEh So, it can be deduced by ignoring the term −Te vh =

eεh (x, t) · e−iwp t + c.c, imwp

where Eh = εh (x, t)e−iwp t + c.c, wp = term of previous term. In (1.2.20), − Assume

∇nh that ne (1.2.26)

4πn0 e2 , where c.c is the complex conjugate m

m m e2 ∇vh2 = ∇Eh2 . 2 2 m2 wp2

∂ ln nl = 0 in (1.2.23), then we get the simplified two-fluid equations ∂t

∂ni + ∇ · (ni vi ) = 0, ∂t   ∂vi ni M + vi ∇vi = −Ti ∇ni + ni eEl , ∂t ∇ · El = 4πe(ni − nl ), El = −

e Te ∇nl − ∇Eh2 , nl 2mwp2

(1.2.27) (1.2.28) (1.2.29)

2

∂ 2 Eh 4πne e2 2 2 2 2 − c ∇ E + c ∇ (∇ · E ) − v ∇ (∇ · E ) = − · Eh . h h h l ∂t2 m

(1.2.30) (1.2.31)

The equations (1.2.27)∼(1.2.31) are closed, that is, there are five unknown functions and functional vectors: ni , nl , vi , El , Eh , and the number of the equations (include the vector equations) is also five. There are at least three waves: ion acoustic wave, plasma wave and optical wave. Each wave has a nonlinear term which produces space condensation: that of ion acoustic wave is a transport term vi ∇vi , that of plasma and optical waves are the nonlinear terms in the equation (1.2.31). All of the three kinds of waves have dispersion terms: a charge separation term ∇ · El for ion acoustic wave, vl2 ∇2 Eh for plasma wave and c2 ∇2 Eh for optical wave. Due to the interaction of the nonlinear terms and dispersion terms leading to some kind of balance, sound, plasma, and optical solitons are formed. From these equations, one can get Zakharov equation and nonlinear Schrödinger equation, etc.

1.2

Zakharov equations and the solitons in plasma

15

Let the high-frequency oscillation be 0, then the equations (1.1.27)∼(1.1.31) change into ∂ni + ∇ · (ni vi ) = 0, ∂t   ∂vi ni M + vi ∇vi = −Ti ∇ni − ni e∇φ, ∂t ∇2 φ = 4πe(nl − ni ),   eφ nl = n0 exp , Te

(1.2.32) (1.2.33) (1.2.34) (1.2.35)

where Eh = 0, El = −∇φ are longitudinal waves. In the one-dimensional case, let Ti = 0, vi = vi , and nondimensionalize the above equations, then it deduces that ∂(ni vi ) ∂ni + = 0, ∂t ∂x ∂vi ∂φ ∂vi + vi =− , ∂t ∂x ∂x ∂2φ = eφ − ni . ∂x2 Let ξ = x − Dt, and assume that ni → 1, vi → 0, φ → 0, as |x| → ∞, then ni (D − vi ) = D, 1 1 (D − vi )2 = D2 − φ, 2 2 d2 φ = eφ − ni (ξ). dξ 2

(1.2.36) (1.2.37) (1.2.38) we have (1.2.39) (1.2.40) (1.2.41)

By ni = p

D D2

− 2φ

,

(1.2.42)

we get d2 φ D = eφ − p = F (φ) = G′ (φ), 2 dξ 2 D − 2φ where G(φ) = eφ + D Assume that δD = D − 1 ≪ 1, then  2 1 dφ = 2 dξ

p D2 − 2φ − (D2 + 1).

(1.2.43)

(1.2.44)

by (1.2.43), we obtain

1 2 φ (3δD − φ), 3 "r # δD 2 φ(ξ) = 3δD sech (x − Dt) , 2

(1.2.45) (1.2.46)

16

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

r where the peak value of the solitary wave is 3δD, the width is

2 . δD

Let 1

ξ = ε 2 (x − t),

3

τ = ε 2 t,

(1.2.47)

then the ion acoustic wave equations (1.2.36)∼(1.2.38) change into ∂ni ∂ni ∂ni vi − + = 0, ∂τ ∂ξ ∂ξ ∂vi ∂vi ∂φ ∂vi ε − + vi =− , ∂τ ∂ξ ∂ξ ∂ξ 2 ∂ φ ε 2 = eφ − ni . ∂ξ ε

(1.2.48) (1.2.49) (1.2.50)

Extend ni , vi , φ with respect to ε: ni = 1 + εn(1) + ε2 n(2) + · · · , vi = εv (1) + ε2 v (2) + · · · , φi = εφ(1) + ε2 φ(2) + · · · , and it deduces by substituting those into (1.2.48)∼(1.2.50) that n(1) = v (1) = φ(1) ,


2 φ(1) ∂ 2 φ(1) (2) =φ + − n(2) , ∂ξ 2 2 ∂n(1) ∂n(2) ∂v (2) ∂n(1) v (1) − + + = 0, ∂τ ∂ξ ∂ξ ∂ξ ∂v (1) ∂v (2) ∂v (1) ∂φ(2) − + v (1) =− . ∂τ ∂ξ ∂ξ ∂ξ

(1.2.51) (1.2.52) (1.2.53) (1.2.54)

Differentiating (1.2.52) with respect to ξ, summing up (1.2.53) and (1.2.54), we achieve at ∂n(1) ∂n(1) 1 ∂ 3 n(1) + n(1) + = 0, ∂τ ∂ξ 2 ∂ξ 3 which is the KdV equation. From (1.2.27)∼(1.2.31), let nl = ni , and ni (t) = n0 (t) + n(x, t),

(1.2.55)

1.2

Zakharov equations and the solitons in plasma

17

where n(x, t) is small, and linearize (1.2.27), (1.2.28) and (1.2.31), then one has ∂n + ∇ · n0 v = 0, ∂t ∂v n0 e2 ∇Eh2 , n0 M = −Ti ∇ni − Te ∇ni − ∂t 2mwp2 ∂ 2 Eh 4π(n0 + n) − vl2 ∇2 Eh = − Eh . ∂t2 m

(1.2.56) (1.2.57) (1.2.58)

Here we only consider longitudinal wave, that means, in (1.2.31), by −c2 ∇2 Eh + c2 ∇(∇ · Eh ) = 0, we get (1.2.58). Denote Eh = E, then by differentiating (1.2.56) with respect to t, and taking the divergence to (1.2.57), one has ∂2n E2 − c2s ∇2 n = ∇2 , 2 ∂t 8πM   ∂2E n 2 2 2 − ve ∇ E = −wp 1 + E, ∂t2 n0 where c2s =

(1.2.59) (1.2.60)

Te 4πn0 e2 Ti + Te ≈ , wp2 = . Rewrite E as M M m E(x, t) = ε(x, t)eiwp t + c.c,

(1.2.61)

where the high-frequency part of E is included in e−iwp t , ε(x, t) is slowly varying amplitude. Ignore the term εtt , we get ∂2n |ε|2 2 2 − c ∇ n = ∇ , s ∂2t 2πM ∂ε n − 2iwp − vl2 ∇2 ε = −wp2 ε, ∂t n0

(1.2.62) (1.2.63)

which are the famous Zakharov equations. They change by nondimensionalization into  2 ∂ n   2 − ∆n = ∆|ε|2 , ∂t   i ∂ε + ∆ε − nε = 0. ∂t

(1.2.64) (1.2.65)

Let ε = φ(x − ct)e−ipt+iqx+iθ , where p, q, θ, c are all real constants.

n = n(x − ct),

(1.2.66)

By substituting (1.2.66) into (1.2.64),

18

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

(1.2.65), it’s easy to get their soliton solution ) (    2 m2 1 t+iθ i 2c −i c4 − 4(1−c  m m 2 1 )   sech , (x − ct − x0 ) · e ε(x, t) = p 2) 2 2(1 − c 2(1 − c )     m21 m1 2  n(x, t) = − sech (x − ct − x ) , 0 2(1 − c2 )2 2(1 − c2 ) (1.2.67) c where we have assumed q = (even if the coefficient of φ′ is 0). Here there are 4 2 parameters, x0 stands for the position of the wave packet at time t = 0, θ stands for the initial phase position, and the main parameters are m1 and c, where c means the velocity of the soliton. From (1.2.67), we see that c2 must be less than 1, that m21 means the soliton must be subsonic. is the sunk depth of the density, 2(1 − c2 )2 2(1 − c2 ) is the sunk width. The depth is inversely proportional to the square of m1 the width. When c2 ≪ 1, by static approximation, it deduces from (1.2.64) that

n = −|ε|2 .

(1.2.68)

Substitute that into (1.2.65), then it becomes the nonlinear Schrödinger equation iεt + εxx + |ε|2 ε = 0.

(1.2.69)

If the wave packet is sufficiently narrow in the wave number k spaces, then it deduces equations iεt + ∇2 ε + nε = 0,  2  ∂ − ∆ n = ∆|ε|2 . 2n ≡ ∂t2

(1.2.70) (1.2.71)

At this moment, one can get whister soliton. The symbol before the interaction term of (1.2.70) is opposite to that of Zakharov equations, which leads to that the movement of this soliton is supersonic, and the soliton has a density peak which is different from the density pit of Langmuir equation. When c → 1, that is, when it is near the sound velocity, we can find from the expression of the soliton (1.2.67) that all of the density fluctuation of plasma, energy and the width of soliton etc. tend to zero. Therefore, some physicists propose to use Boussinesq equation or KdV equation to instead of acoustic wave equation. In this case, the density fluctuation autonomously satisfies the equation ntt − nxx − β n2


xx

− αnxxxx = |ε|2xx ,

(1.2.72)

1.3

Landau-Lifshitz equations and the magnetized motion

19

or nt + nx + β n2


x

+ 2nxxx = −|ε|2x .

(1.2.73)

For the following coupled equations iεt + εxx − nε = 0,

(1.2.74)


δ ntt − nxx − nxxxx − δ n2 xx = |ε|2xx , 3

(1.2.75)

4 ml , we can find the soliton solution of (1.2.74), (1.2.75) 3 mi      εls (x, t) = A tanh{B(x − vt − x0 )} sech{B(x−vt−x0 )} exp i 1 vx − Ωt − θ , 2  n (x, t) = 6λ sech2 {B(x − vt − x )},

where δ =

ls

0

(1.2.76) v2 2 where A = 48λ δ, λ = Ω − , v < 1. 4 ∂2ε ∂2ε If change the dispersion term ve2 2 to c2 2 , the equations (1.2.59), (1.2.60) ∂x ∂x turn into ¯2 ∂2 E ∂2n ∂2n − c2s = , (1.2.77) 2 2 2 ∂t ∂x ∂x 8πM 2

2

  2 ∂2E n 2 ∂ E 2 −c = −wp 1 + E, ∂t2 ∂x2 n0

(1.2.78)

then one can get optical soliton. In fact, let E(x, t) = ε(x, t)e−iwp t + c.c, and ignore the terms including εtt (x, t), then we obtain  2 ∂ n ∂2n ∂ 2 |ε|2   , (1.2.79) =  2 − c2s ∂t ∂x2 ∂x2 8πM  ∂2ε ∂ε n   −2iwp − c2 2 = −wp2 ε, (1.2.80) ∂t ∂x n0 which and Zakharov equation only differ in that ve2 is changed into c2 . So, the width of optical soliton is c/ve times wider than that of Langmuir soliton.

1.3

Landau-Lifshitz equations and the magnetized motion

In 1935, L. D. Landau and E. M. Lifshitz [188] proposed the following magnetization equations when they studied the motion of the magnetic domain wall ∂z = λ1 z × H l − λ2 z × (z × H l ), ∂t

(1.3.1)

20

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

where λ1 , λ2 are physical constants and λ2 ⩾ 0; z(x, t) = (z1 (x, t), z2 (x, t), z3 (x, t)) is a 3D vector valued function, which means the magnetization; “×” represents the cross product of vectors in R3 . H l is the effective magnetization, when there exists an electromagnetic field, it holds that H l = ∆z + H,

(1.3.2)

where H(x, t) stands for the magnetic field intensity, by coupling with the Maxwell equation, it turns into 4π 1 ∂E + J, c ∂t c 1 ∂ ∇×E =− (H + 4πz), c ∂t ∇ · (H + 4πz) = 0, ∇ · E = 0, ∇×H =

(1.3.3) (1.3.4) (1.3.5) (1.3.6)

J = σE.

If ignore the electromagnetic field, then the Landau-Lifshitz equations change into ∂z = λ1 z × ∆z − λ2 z × (z × ∆z), ∂t

(1.3.7)

where the first term represents the motion of magnetization z around the effective magnetic field H e , the second term is the Gilbert dissipative term. Since the LandauLifshitz equations are important equations of motion in ferromagnetic medium, in recent decays, many mathematicians and physicists have paid more attention to them. In 1979, Lakshmanan et al. studied the soliton solution of the following one-dimensional Landau-Lifshitz equations without external magnetic field and dissipation zt = z × zxx ,

(1.3.8)

and they get the following spin wave solutions in the form of z = z(x − ct), z = a cos α + {b cos(kx − wt) + c sin(kx − wt)} sin α,

(1.3.9)

where a, b, c are the constant unit vectors which are perpendicular to each other in R3 , α, k are any constants, w = k 2 cos α. Do polar coordinate transform to (1.3.8)   u = sin θ cos φ,   (1.3.10) v = sin θ sin φ,   w = cos θ,

1.3

Landau-Lifshitz equations and the magnetized motion

21

where z = (u, v, w), |z| = 1. Then (1.3.8) changes into θt = −2θx φx cos θ − φxx sin θ, φt sin θ = θxx − (φx )2 sin θ cos θ.

(1.3.11)

From (1.3.11), one can get the following special soliton solution    2 1    cos θ = tanh c(x − ct) ,  2 2    1 1  −1  φ = tan c(x − ct) + cx, tanh 2 2

(1.3.12)

where c is the constant wave velocity. Nakamuka, Tjon and Fogedby et al. successively studied the Landau-Lipshitz equations with external magnetic field zt = z × zxx + z × h,

(1.3.13)

where the constant vector h = (0, 0, h), h = ̸ 0, and they get the following spin wave solutions of (1.3.13) in the form of z = z(x − ct), q q (1.3.14) zt = ( 1 − s20 cos ξ, 1 − s20 sin ξ, s0 ), where ξ = kx − wt + φ, s0 , h, w, φ are all constants. Do polar coordinate transform (1.3.10) to (1.3.13), then one can get the soliton solution of (1.3.13)   x − ct − x0 −2 , cos θ = 1 − A cos h Γ    1 2 x − ct − x0 −1 φ = φ0 + c(x − ct − x0 ) + tan tanh , 2 cΓ Γ where the constants x0 , φ0 are determined by the initial data, and the constants A, Γ satisfy A=2−

c2 , 2h

 Γ=

h−

 c 2 − 12 2

.

(1.3.15)

L. A. Takhtajon, H. C. Fogedby et al. studied the soliton solution of (1.3.13) by using inverse scattering method, and Takhtajon obtained N -soliton solution of (1.3.13). Zakharov proved the standard equivalence between the Landau-Lifshitz equation and the nonlinear Schrödinger equation. Thereafter, there are a series of meaningful research work for, such as, the integrability, infinite conservation laws and Hamilton results of the Landau- Lifshitz equations, especially, the relationship with the nonlinear schrödinger equation.

22

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

From the angle of partial differential equation, it’s very interesting to study the Landau-Lifshitz equation (1.3.8) can be rewritten into zt = A (z)zxx , where



 u   z =  v , w



0  A (z) =  w −v

(1.3.16)

−w 0 u

 v  −u  . 0

(1.3.17)

It’s not difficult to prove that the coefficient matrix A (z) has the following properties (i) zero definite, that is ξ · A (z)ξ = 0,

where ξ, z ∈ R3 .

(ii) singular, that is det A (z) = 0,

where z ∈ R3 .

In fact, A (z) is a three-order symplectic matrix. Therefore, (1.3.16) is strongly degenerate, strongly coupled quasi linear parabolic equations. Since 1982, we carried out a systematic study on several kinds of problems for determining solution to the Landau-Lipshitz equations (where include the one-dimensional and high-dimensional space cases).

1.4

Boussinesq equation, Toda Lattice and Born-Infeld equation

E. Femi et al. examined a mechanical system formed by 64 particles with equal mass connected by nonlinear springs. And they considered the following equations respectively h i  2 2  x ¨i = (xi+1 + xi−1 − 2xi ) + α (xi+1 − xi ) − (xi − xi−1 ) ,    h i 3 3 (1.4.1) x ¨ = (x + x − 2x ) + β (x − x ) , − (x − x ) i i+1 i−1 i i+1 i i i−1     x ¨i = δ1 (xi+1 − xi ) − δ2 (xi − xi−1 ) + c, where i = 1, 2, · · · , 64, with the periodic initial value xi (0) = sin

iπ , 64

x˙i (0) = 0,

(1.4.2)

where xi (t) represents the displacement of the ith particle from the equilibrium state, α, β, δ1 , δ2 and c are some fixed constants. They distribute all kinds of energy, and let

1.4

Boussinesq equation, Toda Lattice and Born-Infeld equation

ak =

X

xi sin

i

ikπ , 64

23

(1.4.3)

then the energy satisfies  1 (xi+1 − xi )2 + (xi − xi−1 )2  Exkin , + Expot = x˙ 2i + i i 2 2 (1.4.4)  E kin + E pot = 1 a˙ 2 + 2a2 sin kπ . k ak k ak 2 128 Classical statistical mechanics holds that any weak nonlinear action will lead to balance of energy, but the actual calculation result is quite surprising, and it does not tend to be thermalization. At that moment, Femi considered only in the frequency space, and failed to find the soliton solution. Later, Toda investigated the nonlinear vibration of the lattice, approximated this case, and obtained the soliton solution which gave a correct answer to Femi’s problem. We consider the crystal as a chain pulled by springs with mass, for example, if rn (t) indicates the departure of the nth spring from its equilibrium position, then we can get the ordinary differential difference equations (equations of motion) m¨ rn = 2f (rn ) − f (rn+1 ) − f (rn−1 ),

(1.4.5)

where n = 0, ±1, ±2, · · · ; f (r) represents the force of the spring; Toda assumed that f (r) = −α(1 − e−βr ), when f (r) is a linear function, for instance, f (r) = −γr, then m¨ rn = γ (rn+1 + rn−1 − 2rn ) .

(1.4.6)

Assume that its traveling wave solution is rn = a cos θ,

θ = wt − pn,

then substituting it into (1.4.6) leads to the dispersion relation p mw2 = 4 sin2 . γ 2 In the nonlinear case, Toda’s method is that: let s˙ n = f (rn ), then (1.4.5) turns into the following equivalent equations s˙ n = f (rn ),

mr˙n = 2sn − sn+1 − sn−1 .

Since
f (r) = −a 1 − e−βr , s¨n = f ′ (rn )r˙n = −αβe−βrn r˙n = −β (α + s˙ n ) r˙n , then

24

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

m s¨n = sn+1 + sn−1 − 2sn . β α + s˙ n

(1.4.7)

Considering the traveling wave solution to (1.4.7) sn = s(θ),

θ = wt − pn,

(1.4.8)

s(θ) satisfies the ordinary differential difference equation mw2 s′′n · = s(θ + p) + s(θ − p) − 2s(θ). β α + ws′′ By the following equation d2n (θ + p) − d2n (θ − p) = −2k 2

d dp



ρn (θ)cn (θ)ρ2n (θ) 1 − k 2 ρ2n (θ)

(1.4.9)  ,

where cn , ρn , dn are the Jacobi elliptic functions, k is the norm of those functions. Let Z ξ E(ξ) = d2n (z)dz, 0

and integral (1.4.9) with respect to p, then E(θ + p) − E(θ − p) − 2E(θ) =

E ′′ (θ) , q + E ′ (θ)

q=

1 ρ2n p

− 1,

So sn = s(θ) = bz(2kθ), E(K) , which is a 2k-periodic Jacobi ζ function. K 1  − 21  mα 2 1 E b= −1+ , θ = wt − pn, β ρ2n 2kp k

where z(θ) = E(θ) − θ

w=

1 2k



αβ m

 12 

1 E −1+ ρ2n 2kp k

− 12 .

When k → 0, sn = When k → 1, sn =

bk 2 sin 2πθ 4

(the linear case),

  1 m ¯
2 1 b sech2 (kn − ¯bt) + δ , 4 αβ 2

(1.4.10)

1.4

Boussinesq equation, Toda Lattice and Born-Infeld equation

25

4αβ k where ¯b = sech2 , δ is a parameter. Then the soliton solution (1.4.10) is m 2 obtained. For differential difference equation (1.4.5), one can get the following Boussinesq equation [20] by passing to the limit with respect to difference utt − uxx + λ1 u2


xx

+ λ2 uxxxx = 0,

(1.4.11)

where λ1 , λ2 are real constants. In the approximation of long wavelengths and small amplitudes, we can get the KdV equation ut + uux + µuxxx = 0,

(1.4.12)

where µ is a real constant. M. Born and M. L. Infeld got the following Born-Infeld equation when they do nonlinear correction of Maxwell equation

1 − φ2t φxx + 2φx φt φxt − 1 + φ2x φtt = 0.

(1.4.13)

In 1966, Barbaskov and Chernikov got the soliton solution of the above equation. When 1 + φ2x − φ2t > 0, one can get (1.4.13) is a hyperbolic equation. Let ξ = x − t, η = x + t, u = φξ , v = φη , then by (1.4.13), we get the equivalent equations ( uη − uξ = 0, (1.4.14) v 2 uξ − (1 + 2uv)uη + u2 vη = 0. ∂(u, v) ̸ 0, then uξ = Jηv , uη = −Jξv , vξ = −Jηu , vη = Jξu . = ∂(ξ, η) So, by (1.4.14), we have ( ξv − ηu = 0,

Assume J =

v 2 ηv + (1 + 2uv)ξv + u2 ξu = 0,

(1.4.15)

which leads to u2 ξuu + (1 + 2uv)ξuv + v 2 ξvv + 2uξu + 2vξv = 0.

(1.4.16)

The characteristic equation of (1.4.16) is u2 dv 2 − (1 + 2uv)dudv + v 2 du2 = 0. √ Let r =

1 + 4uv − 1 ,s= 2v

√ 1 + 4uv − 1 , (u, v) → (r, s), then 2u

r2 ξr + ηr = 0,

ξs + s2 ηs = 0.

(1.4.17)

26

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

(1.4.17) eliminating η leads to (1.4.18)

ξrs = 0. The general solutions of (1.4.18) are

Z

x − t = ξ = F (r) −

Z

x + t = η = G(s) −

s2 G′ (s)ds,

(1.4.19)

r2 F ′ (r)dr,

(1.4.20)

where F (r), G(s) are any functions. Since s r ξr + ηr = rF ′ (r), φr = uξr + vηr = 1 − rs 1 − rs then one can get

Z φ=

rF ′ (r)dr +

Z

sG′ (s)ds.

Let F (r) = ρ, G(s) = σ, r = Φ′1 (ρ), s = Φ′2 (σ), then we obtain φ = Φ1 (ρ) + Φ2 (σ). From (1.4.19) and (1.4.20), one has Z σ 2 x−t=ρ− [Φ′2 (σ)] dσ, −∞

Z

+∞

x+t=σ−

[Φ′1 (ρ)] dρ. 2

ρ

Let Φ1 (ρ), Φ2 (ρ) be local and not equal to 0, when −1 < ρ < 0, 0 < σ < 1, φ = Φ1 (x − t) + Φ2 (x + t),

t < 0,

(1.4.21)

the wave Φ1 gets in from x = −∞, while the wave Φ2 gets in from x = +∞, then as t → ∞, there holds Z +∞ Z +∞ 2 2 φ = Φ1 (x − t + (Φ′2 (σ)) dσ) + Φ2 (x + t − (Φ′1 (ρ)) dρ). (1.4.22) −∞

1.5

−∞

2D K-P equation

In 1970, B. B. Kadomtsev and V. I. Petviashvili [174] put forward the 2D KdV equation, that is, by considering a small disturbance of another space variable y of ∂φ the KdV equation, which means that add a small disturbance , and considering ∂η the dispersive relationship, we have ut + uux + uxxx = φy , φx = ∓

c ∂u , 2 ∂y

(1.5.1) (1.5.2)

1.5

2D K-P equation

27

where ∓ represent negative, positive dispersive, respectively, c is the propagation velocity of perturbation wave. Differentiating (1.5.1) with respect to x, differentiating (1.5.2) with respect to y and eliminating φxy lead to c (ut + uux + uxxx )x ± uyy = 0, 2

(1.5.3)

which is the K-P equation (see [22, 179, 213]). Obviously, if u does not depend on y, it is just the KdV equation, and if u does not depend on t, change y into t, actually, it is just the Boussinesq equation (see [157, 158, 161]). Consider the nonlinear equations ( Φx = M Φ + AΦy , (1.5.4) Φt = N Φ + BΦy + CΦyy , where



! M=  N =

0 1 k−u 0

,

0 1 A= −√ 3

√ 3wy + ux 1 −(2u + 4λ)(λ − u) + uxx − √ uy 3

√

0 0

 ,

−(2u + 4λ) 3wy − ux − 2f

 ,

(1.5.5)

Z w=

udx,

whose integral condition Φxt = Φtx is just the K-P equation. ! Φ2 Φ1 Let Φ = be a column vector solution to (1.5.4), σ = , then σ satisfies Φ1 Φ2  Z 1  2  σy dx, σx =λ − u − σ − √    3   Z Z   4 1 1 σy dx+ √ (λ−u) σy dx− √ uy σt = − (2u+4λ)(λ−u−σ 2 )− √ (1.5.6) 3 3 3   !  Z 2 Z   Z   4 4   √ − 2u φ − σ dx + σ dx − σ + σ σ dx σ.  x y yy y y  3 3 Then by calculation, we can prove that u e = u + 2σx is a Bäcklund transformation of K-P equation, which means u e still satisfies K-P equation.

28

Chapter 1

Physical Backgrounds for Some Nonlinear Evolution Equations

By the method of R. Hirota, and applying operator D  n  m ∂ ∂ ∂ ∂ m m ′ ′ Dt Dx a(x, t)b(x, t) = · a(x, t)b(x , t ) − ′ − t=t′ ∂t ∂t ∂x ∂x′ ′ x=x

and its properties, K-P equation can be rewritten into (Dt Dx + Dy2 + Dx4 )f · f = 0,

(1.5.7)

where u = 2(log f )xx . Then we can obtain the N -soliton solution N =3, f =1 + exp(η1 ) + exp(η2 ) + exp(η3 ) + exp(A12 + η1 + η2 ) + exp(A13 + η1 + η3 ) + exp(A23 + η2 + η3 ) + exp(A12 + A13 + A23 + η1 + η2 + η3 ),

(1.5.8)

where ηi = pi x + qi y − Ωi t,

pi Ωi − p4i + qi2 = 0,

(1.5.9)

and exp(Aij ) = − =

(pi − pj )(Ωi − Ωj ) − (pi − pj )4 + (qi − qj )2 , (pi + pj )(Ωi + Ωj ) − (pi + pj )4 + (qi + qj )2

3(pi − pj )2 + [(qi /pi ) − (qj /pj )]2 , 3(pi + pj )2 + [(qi /pi ) − (qj /pj )]2

(1.5.10)

where i, j = 1, 2, 3, pi and qi are all any constants. N -soliton solution has the following expression: ∂2 log f, ∂x2   (N ) N X X X f= exp  Aij µi µj + ηi µi  .

u=2

µ=0,1

i>j

i=1

(1.5.11)

Chapter 2 The Properties of the Solutions for Some Nonlinear Evolution Equations With the development of theory of soliton, people pay more attention to a large number of nonlinear evolution equations with soliton solutions. For example: KdV equation [37, 71, 81, 83, 118, 122], nonlinear Schrödinger equation [23, 62, 68–70], nonlinear Klein-Gordon equation [60, 124, 125, 160, 195, 234], Zakharov equations [27, 74, 93, 100, 159], Landau-Lifshitz equations [110, 120, 121, 147], Benjamin-Ono equation [119, 153, 233, 252], and so on. These equations belong to the integrable system or close to the integrable system, and besides this important characteristic of the solitons, there are also other obvious physical features: the unity of dispersion and nonlinearity; possessing some kind of volatility, while their solutions have some kind of smoothness; some kind of decay property to the solutions as t → ∞ (or x → ∞), and scattering property. Moreover, people have found that the decay and the smoothness of the solution are closely related. Since these equations are closely related to the latest research on physics, the theoretical study of the method of solution and the property has far gone beyond the traditional method of research. For example, there is an accurate and very important method named scattering inversion method, which opens up a new way for the theoretical study of partial differential equations, Bäcklund transformation which is a method to get special solution, and extended structural method established by external differential form of differential geometric and Lie groups, etc. Meanwhile, in terms of the theoretical research of this kind of nonlinear partial differential equations, it is impossible to copy some traditional methods in the past. For example, for KdV equation, nonlinear Schrödinger equation etc., although their solutions have good smoothness, they do not have extremum principle. Therefore, the integral estimation method must be adopted. But what is different from the usual is that the integral estimation must fully utilize a variety of forms of conservation laws. That is to establish the a priori estimates and the integral inequalities of the coupling solution. As P. D. Lax points out: “For KdV equation, the mean characteristic of which is that it has infinite conservation laws.” The qualitative study of these equations is mainly focused on: global existence and

30 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

uniqueness of the solution under weak conditions; accurate estimations for decay rate of the solution, as t → ∞; partial regularity of the solution when t > 0 and blow up phenomenon of the solution in a variety of norms under certain conditions. In chapter 3, some research results of several kinds of nonlinear evolution equations will be listed. In this chapter, we’ll take several classical equations as examples, and give a brief introduction to the classical methods of the above research contents, so that one can understand the scope and depth of the research on these problems and master the main methods.

2.1

The smooth solution for the initial-boundary value problem of nonlinear Schrödinger equation

We consider the initial-boundary value problem dinger equation  2   iut − ∆u + f (|u| )u = 0, u(x, 0) = u0 (x),   u|∂Ω = 0,

of the following nonlinear Schröx ∈ Ω, t > 0,

(2.1.1)

x ∈ Ω,

(2.1.2)

t > 0,

(2.1.3)

where Ω is a bounded domain in R2 , with smooth boundary ∂Ω. Assume f : R+ → R is a C 2 function on (0, ∞), and satisfies p

0 < f (s) < c1 s 2 , ′

1 2

|f (s)s | ⩽ c2 s ′′

|f (s)s| ⩽ c3 s

p−1 2

p−2 2

,

,

∀s ∈ R+ ,

(2.1.4)

∀s ∈ R ,

(2.1.5)

∀s ∈ R ,

(2.1.6)

+

+

where p ∈ [1, ∞), ci is a positive constant (i = 1, 2, 3). We’ll prove that when p ∈ [2, 3), there exists a global smooth solution. When p = 2, Brezis-Gallouet have obtained related results [14]. Denote H by the function space H = {u|u ∈ H 3 (Ω) ∩ H01 (Ω), ∆u ∈ H01 (Ω)}. Theorem 2.1.1 Let 2 ⩽ p < 3. For any u0 ∈ H , the problem (2.1.1)∼(2.1.3) has a unique global smooth solution u(x, t) satisfying u(x, t) ∈ L∞ (0, T ; H ),

ut (x, t) ∈ L∞ (0, T ; H01 (Ω)).

Proof We can use Galerkin approximation to construct our global approximate solutions. In order to prove the convergence of these approximate solutions, the key ingredient is to get the a priori estimates of the approximate solutions. To this

The smooth solution for the initial-boundary value problem of nonlinear· · ·

2.1

31

end, we only need to establish the a priori estimates of the solutions to the problem (2.1.1)∼(2.1.3), since the estimates for Galerkin approximate solutions are the same. Lemma 2.1.2 Let u = u(x, t) be the smooth solution to the problem (2.1.1)∼ (2.1.3), then Z Z |u(x, t)|2 dx = |u0 (x)|2 dx, (2.1.7) Ω

Z

Ω

Z

Z

|∇u(x, t)|2 dx + Ω

Ω

Z |∇u0 (x)|2 dx +

F (|u(x, t)|2 )dx = Ω

F (|u0 (x)|2 )dx, Ω

(2.1.8) where

Z F (s) =

s

f (σ)dσ.

(2.1.9)

0

Proof Multiplying (2.1.1) with u ¯(x, t) (that is the complex conjugation of u(x, t)), by integrating with respect to x ∈ Ω, taking imaginary part, one has (2.1.7). Multiplying (2.1.1) with u ¯t , by integrating with respect to x ∈ Ω, taking real part, one has (2.1.8). Here we have finished the proof of Lemma 2.1.2. We deduce by Lemma 2.1.2 that sup ∥u(t)∥H 1 (Ω) ⩽ c.

(2.1.10)

0⩽t 0, we have sup ∥∇ut ∥ ⩽ c(T ). 0⩽t⩽T

Proof By Lemma 2.1.4 and Lemma 2.1.5, one has Z Z  ′ |∇ut (x, t)|2 dx + f (|u|2 )|u|2 + f (|u|2 ) |ut |2 (x, t)dx Ω Ω Z
+ f ′ (|u|2 ) u2 u ¯2t + u ¯2 u2t dx Z Ω Z  ′ ⩽ |∇ut (x, 0)|2 dx + f (|u0 |2 )|u0 |2 + f (|u0 |2 ) |ut (x, 0)|2 dx Ω Ω Z
′ 2 2 + f (|u0 | ) u0 u ¯t (x, 0)2 + u ¯20 ut (x, 0)2 dx Ω

) r Z t(   p−1
1 +c 1 + log c 1 + ∥∇ut (s)∥ 2 ∥∇ut (s)∥2 + 1 ds. 0

From (2.1.1), we deduce that Z Z Z

|∇ut (x, 0)|2 dx ⩽ |∇∆u0 |2 dx + |∇ f (|u0 |2 )u0 |2 dx ⩽ c ∥u0 ∥H 3 (Ω) Ω

and Z 

Ω

Ω





f ′ |u0 |2 |u0 |2 + f |u0 |2 |ut (x, 0)|2 dx ⩽ c ∥u∥L∞ (Ω) ∥ut ∥2 ⩽ c ∥u0 ∥H 2 (Ω) ,

Ω

where c(·) represents variable positive constants, which only depends on those in parentheses. So, we get Z Z  ′

 ′

f |u|2 |u|2 + f |u|2 |ut |2 (x, t)dx + f |u|2 u2 u ¯2t + u ¯2 u ¯2t dx Ω

⩽c

Ω

Z

Z

 13 Z  32 3p 3 |u| dx |ut | dx

|u|p |ut |2 dx ⩽ c Ω

Ω

Ω

1 ⩽c∥∇ut ∥ + c ⩽ ∥∇ut ∥2 + c, 2 which leads to r Z t( 4 3

∥∇ut ∥2 ⩽ c + c

1+



log 1 + ∥∇ut ∥

1 2



)p−1


∥∇ut (s)∥3 + 1 ds,

0

therefore, when 0 ⩽

p−1 ⩽ 1, we obtain 2 sup ∥∇ut (t)∥2 ⩽ c(T ), 0⩽t⩽T

∀T > 0.

34 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Here we have finished the proof of Lemma 2.1.6. From (2.1.1), one has sup ∥u(t)∥H 3 (Ω) ⩽ c(T ). 0⩽t⩽T

Here we have obtained the estimates needed, so, the proof of Theorem 2.1.1 is finished. Some results of nonlinear Schrödinger equation can refer to [14,72,73,76,79,84,85, 89–92,94–96,99,102,131,134,136,141,150,151,156,166–169,176,178,235,236] and the references therein. As for some coupled equations of nonlinear Schrödinger equation and other equation, for example: KdV equation, Klein-Gorden equation, Boussinesq equation, BBM equation etc., see [77–88, 124, 125, 129, 140] and the references therein. For attractors of nonlinear Schrödinger equations, see [115, 116, 124] and so on.

2.2

The existence of the weak solution for the initial-boundary value problem of generalized Landau-Lifshitz equations

We consider the following initial-boundary value problem of generalized LandauLifshitz Equations  z = z × zxx + f (x, t, z), 0 < x < l, t > 0, (2.2.1)   t zx (0, t) = zx (l, t) = 0, t ⩾ 0, (2.2.2)   z(x, 0) = z0 (x), 0 ⩽ x ⩽ l, (2.2.3) where z = (u, v, w) is an unknown 3D vector valued function, f (x, t, z) is a 3D vector valued function with respect to x, t, z. “×” means the cross product of two 3D vectors, z0 (x) is a 3D vector valued initial function. (2.2.1) can be regarded as strongly degenerate quasilinear parabolic equations. So, the coefficient matrix of the derivative of the second order is a symplectic matrix, which is zero definite and its determinant is strange. By vanishing viscosity method and Leray-Schauder fixed point principle, we establish the existence of weak solution to the initial-boundary value problem (2.2.1)∼(2.2.3). 2.2.1

The basic estimates of the linear parabolic equations

We consider the initial-boundary value problem of linear parabolic equations  u − A(x, t)uxx + B(x, t)ux + C(x, t)u = f (x, t), (2.2.4)   t ux (0, t) = ux (l, t) = 0, (2.2.5)   u(x, 0) = u0 (x), (2.2.6) where u(x, t) and u0 (x) are two N D vector valued functions.

The existence of the weak solution for the initial-boundary value problem of· · ·

2.2

35

Lemma 2.2.1 Assume linear parabolic equations (2.2.4) and the initial function u0 (x) satisfy the following conditions: (1) A(x, t) is an N × N positive definite matrix in QT = {0 ⩽ x ⩽ l, 0 ⩽ t ⩽ T }; (2) A(x, t), B(x, t) and C(x, t) are N × N bounded measurable matrix; (3) f (x, t) is an N D vector valued function which is square integrable in QT ; (4) u0 (x) ∈ W21 (0, l) is an N D vector valued function which is suitable for the boundary condition, then there exists a unique vector valued solution to the initialboundary value problem (2.2.4)∼(2.2.6), u(x, t) ∈ L∞ (0, T ; W21 (0, l)) ∩ W2

(2,1)

(QT ),

(2.2.7)

satisfying   sup ∥u(·, t)∥w21 (0,l) + ∥ut ∥L2 (QT ) + +∥uxx ∥L2 (QT ) ⩽ k ∥u0 ∥w21 (0,l) + ∥f0 ∥L2 (QT ) ,

0⩽t⩽T

(2.2.8) where the constant k only depends on the norms of the coefficients A, B, C of (2.2.4). Proof Taking the quantity product between (2.2.4) and u, uxx , respectively, one can get (u, ut ) − (u, Auxx ) + (u, Bux ) + (u, Cu) = (u, f ) , (uxx , ut ) − (uxx , Auxx ) + (uxx , Bux ) + (uxx , Cu) = (uxx , f ) . Since Z l

Z (uxx (x, t), ut (x, t)) dx =

(ux (x, t), ut (x, t)) |x=l x=0

0

−

l

(ux (x, t), uxt (x, t)) dx, 0

and applying the boundary condition (2.2.5), we find the first term on the right-hand side of above equation is zero. Then, one has Z l 1 d ∥ux (x, t)∥2L2 (0,l) . (uxx (x, t), ut (x, t)) dx = − 2 dt 0 By taking the difference between two equations on rectangular area Qτ = {0 ⩽ x ⩽ l, 0 ⩽ t ⩽ τ }, (0 < τ ⩽ T ), and by integrating the resultant equation, we obtain ZZ 2 2 ∥u(·, t)∥w1 (0,l) − ∥u0 ∥w1 (0,l) + 2 (uxx , Auxx ) dxdt 2

ZZ

2

Qτ

ZZ

(uxx , Bux + Cu − f ) dxdt + 2

=2 Qτ

(u, Auxx − Bux − Cu + f ) dxdt. Qτ

36 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

According to the conditions of Lemma 2.2.1, it’s easy to deduce (2.2.8). we can get the existence of solution to the initial-boundary value problem by using parameter continuation method, while we can get the uniqueness of solution directly by (2.2.8), this finishes the proof of Lemma 2.2.1. 2.2.2

The existence of the spin equations

We call the equations with small diffusion terms as spin equations zt = εzxx + z × zxx + f (x, t, z),

ε > 0,

(2.2.9)

which are obviously parabolic equations. Now we consider the initial-boundary value problem (2.2.2), (2.2.3) of the quasilinear parabolic equations (2.2.9). Let 3D vector valued functional space B = L∞ (QT ) be the base space of fixed point, we define a functional mapping Tλ : B → B with parameter 0 ⩽ λ ⩽ 1 which maps the base space to itself, such that: for every u ∈ B, the image z = Tλ (u) is a 3D vector valued solution to the initial-boundary value problem (1.3.2), (1.3.3) of the linear parabolic equations zt = εzxx + λu × zxx + f (x, t, u),

(2.2.10)

where 0 ⩽ λ ⩽ 1. By Lemma 2.2.1, z = Tλ (u) is uniquely determined by the initialboundary value problem (2.2.10), (2.2.2), (2.2.3), and belongs to vector functional space
(2,1) G = L∞ 0, T ; W21 (0, l) ∩ W2 (QT ) . Obviously, for any λ, functional operator Tλ is fully continuous; moreover, for any bounded set M in the state space B, operator Tλ is uniformly continuous for 0 ⩽ λ ⩽ 1. In order to use the fixed point theorem to prove the global existence of the 3D vector valued generalized solution to the initial-boundary value problem of generalized spin equations (2.2.10), we need to get the uniform a priori estimates with respect to parameter 0 ⩽ λ ⩽ 1 for all possible fixed points of the mapping Tλ . For this aim, we assume that: (i) 3D vector valued function f (x, t, z) is continuously differentiable with respect to x and z. 3×3 Jacobi matrix of derivatives fz (x, t, z) is semi bounded, that is, for any 3D vector ξ ∈ R3 , there exists a constant b, such that ξfz (x, t, z)ξ ⩽ b|ξ|2 , where (x, t, z) ∈ QT × R3 . And f0 (x, t) = f (x, t, 0) ∈ L2 (QT );

2.2

The existence of the weak solution for the initial-boundary value problem of· · ·

37

(ii) 3D vector valued function z0 (x) ∈ W21 (0, l) is suitable for boundary conditions; (iii) for (x, t, z) ∈ QT × R3 , it holds that |fx (x, t, z)| ⩽ C(x, t)|z|3 + d(x, t),

(2.2.11)

where C(x, t) ∈ L∞ (QT ), d(x, t) ∈ L2 (QT ). By the quantity product of zt = εzxx + λz × zxx + λf (x, t, z)

(2.2.12)

z · zt = εz · zxx + λz · (z × zxx ) + λz · f (x, t, z).

(2.2.13)

and z(x, t), we have

From the assumption (i), we obtain Z z · f (x, t, z) = z ·



1

fz (x, t, τ z)dτ

z + z · f0 (x, t) ⩽ (b + δ)|z|2 +

0

1 |f0 (x, t)|2 , 4δ

where δ > 0, by integral of (2.2.13) on rectangular domain Qτ (0 < τ ⩽ T ), one has Z τ λ ∥z(·, t)∥2L2 (0,l) − ∥z0 (x)∥2L2 (0,l) ⩽ 2λ(b + δ) ∥z(·, t)∥2L2 (0,l) dt + ∥f0 ∥2L2 (QT ) , 2δ 0 then  ∥z(·, t)∥2L2 (0,l) ⩽

∥z0 ∥2L2 (0,l) +

 λ ∥f0 ∥2L2 (QT ) e2λ(b+δ)t , 2δ

(2.2.14)

where 0 ⩽ λ ⩽ 1, 0 ⩽ t ⩽ T , δ > 0, so sup ∥z(·, t)∥L2 (0,l) ⩽ k1 ,

(2.2.15)

0⩽t⩽T

where k1 is a constant independent of ε, λ and t. Now, by the quantity product of (2.2.13) and zxx (x, t), we get zxx · zt = εzxx · zxx + λzxx · (z × zxx ) + λzxx · f (x, t, z).

(2.2.16)

For the left side of (2.2.16), we achieve at Z

l

(zxx · zt )dx =zx (l, t) · zt (l, t) − zx (0, t) · zt (0, t) − 0

=−

1 d ∥zx (·, t)∥2L2 (0,l) , 2 dt

1 d 2 ∥zx (·, t)∥L 2 (0,l) 2 dt (2.2.17)

38 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

for the last term of the right-hand side of (2.2.16), Z l (zxx · f (x, t, z)) dx =zx (l, t) · f (l, t, z(l, t)) 0

Z

l

− zx (0, t) · f (0, t, z(0, t)) − Z =−

(zx · Dx f )dx 0

l

(zx · Dx f )dx.

(2.2.18)

0

By assumption (i), we deduce ZZ ZZ ZZ (zx · Dx f )dxdt = zx · fz (x, t, z)zx dxdt + zx · fx (x, t, z)dxdt Qτ

Qτ



1 ⩽ b+ 2

Z

Qτ τ

∥zx (·, t)∥2L2 (0,l) dt 0

1 + 2

ZZ |fx (x, t, z)|2 dxdt, Qτ

by assumption (ii) and interpolation formula, we get ZZ ZZ 1 2 2 |fx | dxdt ⩽ ∥C∥L∞ (QT ) |z|2 dxdt + ∥d∥2L2 (Qτ ) 2 Qτ

and

Qτ

Z 0

l

|z(x, t)|6 dx ⩽ c1 ∥z(·, t)∥4L2 (0,l) ∥z(·, t)∥2W 1 (0,l) , 2

then (2.2.18) can be changed into ZZ Z (zxx · f (x, t, z)) dxdt ⩽ c2 Qτ

τ

∥zx (·, t)∥2L2 (0,l) dt + c3 ,

(2.2.19)

0

 1 + c1 ∥C∥2L∞ (QT ) k14 , c3 = ∥d∥2L2 (QT ) . where 0 < τ ⩽ T , c2 = b + 2 By integrating (2.2.16) on rectangular domain Qτ , one can get the estimate 

sup ∥zx (·, t)∥L2 (0,l) ⩽ k2 ,

(2.2.20)

0⩽t⩽T

where k2 is independent of λ, ε. When the norms ∥C∥L∞ (QT ) and ∥d∥L2 (QT ) are independent of l > 0, which is the width of QT , k2 is also independent of l. Therefore, there is the following theorem: Theorem 2.2.2 Under the conditions (i)∼(iii), the initial-boundary value problem (2.2.2), (2.2.3) of the spin equations (2.2.9) have a unique 3D vector valued generalized global solution z(x, t) ∈ L∞ (0, T ; W21 (0, l)) ∩ W2

(2,1)

(QT ).

2.2

The existence of the weak solution for the initial-boundary value problem of· · ·

39

When b < 0, take δ > 0, such that b + δ < 0. In the case of T = ∞, denote conditions (i)∼(iii) by (i∞ ), (ii∞ ), (iii∞ ). By (2.2.14), we have the following theorem: Theorem 2.2.3 Under the conditions (i∞ )∼ (iii∞ ), if b < 0, then for the initialboundary value problem (2.2.2), (2.2.3) of the spin equations (2.2.9), the norm of the generalized global solution z(x, t) (obtained in Theorem 2.2.2) with respect to x tends to 0, as t → ∞, that is lim ∥z(·, t)∥L2 (0,l) = 0.

(2.2.21)

T →∞

2.2.3

The existence of the solution to the initial-boundary value problem of the generalized Landau-Lifshitz equations

Now we consider the initial-boundary value problem (2.2.2), (2.2.3) of the generalized Landau-Lifshitz equations (2.2.1). In the above proofs of Theorems 2.2.2 and 2.2.3, we see that the a priori estimates of the approximate solution are independent of λ. Lemma 2.2.4 estimates

The solution zε (x, t) obtained in Theorem 2.2.2 satisfies the


∥zε (·, t)∥L2 (0,l) ⩽ k3 ∥z0 ∥L2 (0,l) + ∥f0 ∥L2 (0,l) e(b+δ)t ,

(2.2.22)

and sup ∥zε (·, t)∥W21 (0,l) ⩽ k4 ,

(2.2.23)

0⩽t⩽T

where 0 ⩽ t ⩽ T, δ > 0, k3 and k4 are independent of ε. When ∥z0 ∥W21 (0,l) , ∥f0 ∥L2 (QT ) , ∥C∥L∞ (QT ) , ∥d∥L2 (QT ) are independent of the width l > 0 of the rectangular domain QT , k4 is independent of l > 0. Lemma 2.2.5 estimates

The solution zε (x, t) obtained in Theorem 2.2.2 satisfies the sup ∥zεt (·, t)∥H −1 (0,l) ⩽ k5 ,

(2.2.24)

0⩽t⩽T

where k5 is independent of ε. When ∥z0 ∥W21 (0,l) , ∥f0 ∥L2 (QT ) , ∥C∥L∞ (QT ) , ∥d∥L2 (QT ) are independent of l > 0, k5 is independent of l. Proof For any ψ(x) ∈ H01 (0, l), we have Z l Z l ψ(x)zεt (x, t)dx = − εψx (x)zεx (x, t)dx 0

0

Z −

Z

l

ψx (x)[zε (x, t) × zεx (x, t)]dx + 0

l

ψ(x)f (x, t, zε (x, t))dx. 0

40 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

From (2.2.23), we get

Z l ψ(x)zεt (x, t)dx ⩽ C4 ∥ψ∥H 1 (0,l) , 0

where c4 is independent of ε and l. According to the definition of the negative order Hilbert space, we get Lemma 2.2.5. Lemma 2.2.6 For the above approximate solution, we have the following estimate ∥zε ∥C (1/2,1/4) (QT ) ⩽ (1 + l)k6 ,

(2.2.25)

where k6 is independent of ε. When ∥z0 ∥W21 (0,l) , ∥f0 ∥L2 (QT ) , ∥C∥L∞ (QT ) , ∥d∥L2 (QT ) are independent of l, k6 is independent of l. Proof Let

Z Wε (t, x) =

x

zε (ξ, t)dξ, 0

then

Z

x

Wεt =

zεt (ξ, t)dξ,

Wεxt (x, t) = zεt (x, t),

0

Wεx (x, t) = zε (x, t),

Wεxx (x, t) = zεx (x, t).

For any ψ(x) ∈ H01 (0, l), it holds that Z Z l l ′ ψ (x)Wεt (x, t)dx = ψ(x)zεx (x, t)dx ⩽ c4 ∥ψ∥H 1 (0,l) 0 0 ⩽ c4 (1 + l)∥ψ ′ (x)∥L2 (0,t) , which leads to sup ∥Wεt (·, t)∥L2 (0,l) ⩽ (1 + l)k5 . 0⩽t⩽T

Obviously, we get sup ∥Wε (·, t)∥L2 (0,l) ⩽ lk4 . 0⩽t⩽T

Therefore, for any given l > 0, {Wε (x, t)} is uniformly bounded with respect to 1 (0, T ; L2 (0, l)). So, ε > 0 in space L∞ (0, T ; W21 (0, l)) ∩ W∞ |zε (x, t2 ) − zε (x, t1 )| =|Wεx (x, t2 ) − Wεx (x, t1 )| 1

3

4 ⩽c5 ∥Wε (·, t2 ) − Wε (·, t1 )∥L4 2 (0,l) ∥Wε (·, t2 ) − Wε (·, t1 )∥W 2 (0,l) 2

⩽c6 |t2 − t1 |

1 4

1 4

3 4

sup ∥Wεt (·, t)∥L2 (0,l) sup ∥Wε (·, t)∥W 2 (0,l) ,

0⩽t⩽T

0⩽t⩽T

2

2.2

The existence of the weak solution for the initial-boundary value problem of· · ·

41

which finishes the proof of Lemma 2.2.6. Now, we consider the limit process ε → 0. For the initial-boundary value problem (2.2.2), (2.2.3) of the generalized Landau-Lifshitz equations (2.2.1), we can give the definition of “weak solution”: Definition 2.2.7 The 3D vector valued function z(x, t) ∈ L2 (0, T ; W21 (0, l)) ∩ C(QT ) is named as the weak solution to the initial-boundary value problem (2.2.2), (2.2.3) of the generalized Landau-Lifshitz equations (2.2.1), if for any text function φ(x) ∈ Φ = {φ|φ ∈ C 1 (QT ), φ(x, T ) ≡ 0}, it holds that ZZ Z l [φt z − φx (z × zx ) + φf (x, t, z)]dxdt + φ(x, 0)z0 (x)dx = 0. (2.2.26) 0

QT

For the approximate solution zε (x, t), it obviously holds that ZZ Z l [φt zε − εφx zεx − φx (zε × zεx ) + φx f (x, t, zε )]dxdt + φ(x, 0)z0 (x)dx = 0, 0

QT

(2.2.27) where φ ∈ Φ. From Lemmas 2.2.4∼2.2.6, The set of 3D vector valued approximate solutions {zε (x, t)} of the initial-boundary value problem (2.2.9), (2.2.2), (2.2.3) is uniformly bounded with respect to ε > 0 in the space L∞ (0, T ; W21 (0, l)) ∩ 1 1 C ( 2 , 4 ) (QT ), where l is a fixed constant. We can choose a subsequence {zεi (x, t)} of the sequence {zε (x, t)}, such that there exists a 3D vector valued function z(x, t) on QT , moreover, {zεi (x, t)} uniformly tend to z(x, t) on QT , and {f (x, t, zεi (x, t))} uniformly tend to f (x, t, z(x, t)) on QT , {zεi x (x, t)} weakly tend to zx (x, t). There1 1 fore, z(x, t) ∈ L∞ (0, T ; W21 (0, l)) ∩ C ( 2 , 4 ) (QT ). In order to prove the convergence of the third term of (2.2.27)ε , we estimate ZZ ZZ φx (zεi × zεi x )dxdt − φx (z × zx )dxdt QT

ZZ

QT

ZZ

φx [(zεi − z) × zεi x ]dxdt +

= QT

φx [z × (zεi x − zx )dxdt. QT

Since {zεi x (x, t)} weakly tend to zx (x, t), the second integral on the right-hand side tends to 0, for the first integral, when εi → 0, ZZ φx [(zεi − z) × zεi x ]dxdt ⩽ ∥φx ∥L2 (QT ) ∥zεi − z∥L∞ (QT ) → 0, QT

so, when εi → 0, (2.2.27)ε tends to (2.2.26), which shows that the 3D vector valued limiting function z(x, t) is a weak solution to the initial-boundary value problem (2.2.2), (2.2.3) of the generalized Landau-Lifshitz equations.

42 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Theorem 2.2.8 Assume the generalized Landau-Lifshitz equations (2.2.1) and the vector valued initial function z0 (x) satisfy conditions (i)∼(iii), then the initialboundary value problem (2.2.1)∼(2.2.3) has at least one global solution z(x, t) ∈ L∞ (0, T ; W21 (0, l)) ∩ C ( 2 , 4 ) (QT ). 1 1

Theorem 2.2.9 When b < 0 and T = ∞, the norm of the 3D vector valued solution z(x, t) obtained in Theorem 2.2.8 with respect to x tends to 0, as t → ∞, that is lim ∥z(·, t)∥L2 (0,l) = 0.

t→∞

2.3

The large time behavior for generalized KdV equation

Consider the initial value problem of the following generalized KdV equation [16] ut + f (u)x + δ(Hu)x + εBu = 0,

x ∈ R, t > 0,

(2.3.1)

u|t=0 = u0 (x)

x ∈ R,

(2.3.2)

where B = Ds , that is fractional derivative of s order satisfying Ds (u)(x) = (2π)−1 Z · |ξ|s u ˆ(ξ)eixξ dξ. H is a differential operator satisfying Z Hu =

p(ξ)ˆ u(ξ)eixξ dξ,

(2.3.3)

where p(ξ) is a positive even function, which is of polynomial growth. We deduce that Z Z (Hu)x udx = c iξp(ξ)|ˆ u(ξ)|2 dξ = 0. Let f (0) = 0, f ′ (0) = 0. There are three cases will be discussed: the first case, having both dispersion and dissipation; the second case, generalized Burgers equation; the third case, generalized KdV equation with only dispersion effect. Theorem 2.3.1 (δ ̸= 0, ε ̸= 0) (i) If f (u) satisfies ′

|f ′ (u)| ⩽ c(|u|p + 1),

p′ < 2(s − 1), s > 1,

and u0 (x) ∈ H s (R), then s

lim |D 2 u(t)|2 = lim |u(t)|∞ = 0.

t→∞

t→∞

(2.3.4)

2.3

The large time behavior for generalized KdV equation

43

(ii) Besides satisfying conditions in (i), let f (u) ∈ C m+1 , m ∈ N, s ⩾ 2 and u0 (x) ∈ H max (m+1,s) (R), then ∥u∥m+1 ⩽ c,

lim |Dm u(t)|∞ = 0,

(2.3.5)

|u| ≪ 1, p > 2s + 1,

(2.3.6)

t→∞

where c is a positive constant. (iii) If f (u) satisfies |f ′ (u)| ⩽ c|u|p , and |u0 |1 < ∞, then |u|∞ = O((1 + t)− 3 ),

|u(t)|2 = O((1 + t)− 2s ), t → ∞.

1

1

(2.3.7)

Proof (i) Multiplying (2.3.1) with u and integrating the result equation with respect to x, then s 1 d |u(t)|22 + ε|D 2 u(t)|22 = 0. 2 dt

(2.3.8)

Integrating (2.3.8) with respect to t, we get Z |u(T )|22 + 2ε

T

s

|D 2 u(t)|22 dt ⩽ c,

∀T ⩾ 0.

(2.3.9)

0

Similarly, one has Z s 1 d ε s 2 1 2 s 2 2 |D u|2 + ε|D u|2 ⩽ |D u|2 + |f ′ (u)ux |2 dx, 2 dt 2 2ε Z  2   ′ ′ s d |D 2 u|22 + ε|Ds u|22 ⩽ c |u|p + 1 |ux |2 dx ⩽ c |ux |2p +2 + |ux |22 , dt where we have used ′ |u|2p ∞

Z = sup y

p′

y

−∞

2uux dx

′

⩽ c|ux |p2 .

Applying Gagliardo-Nirenberg inequality and Young’s inequality [1] to estimate |ux |2 ,   1 ′ 1− 1 p+2 |ux |p2 +2 ⩽ c |Ds u|2s |u|2 s ⩽ η|Ds u|22 + c(η). By choosing η to be small enough, we have s d |D 2 u(t)|22 ⩽ c, dt

44 Chapter 2

Z

∞

and

s

The Properties of the Solutions for Some Nonlinear Evolution Equations s

|D 2 u|22 < ∞. |D 2 u(t)|22 is a positive continuous function with respect to t,

0

therefore s

lim |D 2 u(t)|22 = 0.

t→∞

By Sobolev embedding theorem, H 2 +α → L∞ , α > 0, s > 1, we deduce that 1

lim |u(t)|∞ = 0.

t→∞

(ii) Differentiating (2.3.1) m times, multiplying it with Dm u, then integrating the result equation, one has s Z m+ ε 1 d m 2 |D u|2 + ε|D 2 u|22 = Dm f (u) · Dm+1 u dx ⩽ |Dm+1 u|22 + c|Dm u|22 , 2 dt 4 (2.3.10) where we have used Gagliardo-Nirenberg inequality and the estimate |Dm f (u)|2 ⩽ c(f, m, |u|∞ )|Dm u|2 . Multiplying (2.3.10) with positive constant k, adding the result equation to (2.3.8), s and choosing k > 0 to be small enough and m ⩾ , it holds that 2 Z  d (1 + k|ξ|2m )|ˆ u(ξ, t)|2 dξ ⩽ 0. dt Which leads to the boundedness of ∥u∥H m . In order to prove (ii) of the theorem, change m into m + 1, and use interpolating inequalities, then (ii) is proved. (iii) Consider the fundamental solution to the linearized equation of (2.3.1) Z s −1 G(x, t) = (2π) e−tε|ξ| −itδξp(ξ)+ixξ dξ. (2.3.11) 1

1

It’s easy to find that G(x, t) t s and |G(t)|2 t 2s are bounded. The solution u of (2.3.1) can be represented by the following integral equation Z t u(t) = G(t − T ) ∗ u(T ) − G(t − τ ) ∗ f (u(τ ))x dτ, (2.3.12) T

where “∗” means convolution of two functions, t ⩾ T ⩾ 0. By the basic inequalities of convection, we have Z t |u(t)|∞ ⩽ |G(t − τ )|2 |u(T )|2 + |G(t − τ )|∞ |f (u(τ ))x |1 dτ. (2.3.13) T

2.3

The large time behavior for generalized KdV equation

45

When T is large enough, τ ⩾ T and |u(τ )|∞ is small enough, we get the following estimate from (i) ′ f (u) |u|2 |ux |2 ⩽ c|u|p−1 |f (u)x |1 ⩽ ∞ , u ∞ where c tends to 0, as T → ∞, just like in (i). And from (2.3.13), one has Z t 1 1 − 2s |u(t)|∞ ⩽ C|u(T )|2 (t − T ) +c (t − s)− s |u(τ )|p−1 ∞ dτ T

or W (t) ⩽ c + cW (t)p−1 ,

(2.3.14)

where 1

W (t) = sup (1 + τ − T ) 2s |u(τ )|∞

(p > 2s).

T ⩽τ 0) (i) If f (u) ∈ C 1 , u0 (x) ∈ L1 ∩ H s , then |u(t)|2 = O((1 + t)− 2s ), 1

t → ∞.

(2.3.15)

(ii) If f (u) ∈ C m+1 , u0 (x) ∈ L1 ∩ H m , m ∈ N, s ⩾ 2, then ∥u(t)∥m = O((1 + t)− 2s ), 1

m−1 X

|Dj u(t)|∞ = O((1 + t)− 2s ). 1

(2.3.16)

j=0

(iii) If |f ′ (u)| ⩽ c|u|p , |u| ≪ 1, p > 2(s − 1), then |u(t)|∞ = O((1 + t)− s ). 1

(2.3.17)

46 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Proof (i) By (2.3.8), we have Z Z Z d |u(ξ)|2 dξ = −2ε |ξ|s |ˆ u(ξ)|2 dξ ⩽ −2ε |ξ|s |ˆ u(ξ)|2 dξ dt A Z −1 2 ⩽ −(1 + t) |ˆ u(ξ)| dξ, A

where A = A(t) = {ξ| |ξ| > [2ε(1 + t)− s ]}. 1

By (2.3.18), we get   Z Z Z 1 d 2 2 (1 + t) |ˆ u| dξ ⩽ |ˆ u| dξ ⩽ c dξ ⩽ c[2ε(1 + t)]− s . dt R R\A R\A Then by integrating with respect to t, one can get (2.3.15). Where |ˆ u(t)| ⩽ sup |u(t)|1 < ∞. t⩾0

s , similar to (ii) of Theorem 2.3.1, we have 2 Z Z d (1 + k|ξ|2m )|ˆ u(ξ)|2 dξ ⩽ − ε |ξ|s (1 + k|ξ|2m )|ˆ u(ξ)|2 dξ dt Z −1 ⩽ − (1 + t) (1 + k|ξ|2m )|ˆ u(ξ)|2 dξ,

(ii) Assume m ⩾

A

where, similar to (i), A = A(t) = {ξ| |ξ| > [ε(1 + t)]− s }. 1

Therefore,   Z Z 1 d (1 + t) (1 + k|ξ|2m )|ˆ u(ξ)|2 dξ ⩽ c dξ ⩽ c(1 + t)− s , dt R\A from which, we deduce that ∥u∥m = O((1 + t)− 2s ). 1

Other results can be obtained by Sobolev inequality. (iii) The fundamental solution to the linearized equation (2.3.1) Z 2 −1 G(x, t) = (2π) e−tε|ξ| +ixξ dξ

(2.3.18)

2.3

The large time behavior for generalized KdV equation

47

1

has the following properties: G(x, t)t s is bounded, |G(t)|1 = const. By (2.3.1), we have   Z t   t t ∗u − G(t − τ ) ∗ f (u(τ ))x dτ. u(x) = G 2 2 t/2 By applying the results of (i) and (ii), we get |u(t)|∞ ⩽ct− s + c 1

− 1s

⩽ct

Z

t

t/2 t

Z +c

(t − τ )− s |u(τ )|p−1 ∞ |u(τ )|2 |u(τ )x |2 dτ 1

(t − τ )− s (1 + τ )−(p+1)/2s dτ. 1

t/2

p+1 ⩾ 1, then (iii) has been proved. Applying iterative principle, the assumption 2s p > 2(s − 1) is enough: |u(t)|∞ ⩽ ct−r , r ⩾ (2s)−1 , in each step the index improves at least (p + 2) · (2s)−1 − 1 > 0, until to r ⩽ s−1 . Here we have finished the proof of Theorem 2.3.2. Now consider the third case of (2.3.1), that is the model with only dispersion. Here, it’s more difficult to get the decay estimate, because that its energy is conserved, that is |u(t)|2 =const. If its soliton solution will not decay as t → ∞, it’s also very difficult to get the decay estimate of the solution to the linearized equation. So, we let p(ξ) = |ξ|r−1 , r ⩾ 3, and consider the case of small initial data, If

ut + f (u)x + δ(Dr−1 u)x = 0.

(2.3.19)

Theorem 2.3.3 (δ ̸= 0), ε = 0 (i) If |f (u)| ⩽ c(1 + |u|d ),

d < 2r − 1,

|f ′ (u)| ⩽ c|u|p ,

|u| ≪ 1, p > r + 1

(2.3.20)

and the initial data is small enough: ∥u0 ∥1 + ∥u0 ∥ r−1 ⩽ η, 2

(2.3.21)

where η depends on r and f, then |u(t)|∞ = O((1 + t)− r ), 1

|t| → ∞

(2.3.22)

and there exists solutions u+ and u− of the linearized equation vt + δ(Dr−1 v)x = 0

(2.3.23)

48 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

such that 1

t r |u(t) − u± (t)|2 → 0,

t → ±∞.

(2.3.24)

(ii) The Lq decay estimate which is better than ordinary simple interpolation is |u(t)|2(p+1) = O((1 + |t|)−(1− p+1 ) ), r

1

1

p > [r + (r2 + 4r) 2 ]/2.

(2.3.25)

Proof (i) the fundamental solution to the linearized equation of (2.3.19) is Z r−1 G(x, t) = (2π)−1 e−itδξ|ξ| +ixξ dξ, (2.3.26) which can be represented as G(x, t) = t− r Φr (xt− r ), 1

where Φr (x) = (2π)−1

Z

e−iδξ|ξ|

r−1

+ixξ

1

(2.3.27)

dξ, r ⩾ 2 is bounded for real x. That is

obtained by the asymptotic estimate of generalized Airy function. Now consider the case when t → ∞, (2.3.19) can be written into Z t u(x, t) = G(t) ∗ u0 − G(t − τ ) ∗ f (u(τ ))x dτ. (2.3.28) 0

So we can get Z

t

|u(t)|∞ ⩽|G(t)|∞ |u0 |1 + |G(t − τ )|∞ |f (u(τ ))x |1 dτ 0 Z t 1 1 (t − τ )− r |u(τ )|p−1 ⩽c|u0 |1 t− r + c ∞ dτ.

(2.3.29)

0

Note that

p(ξ) ⩾ c|ξ|λ , |ξ| is large enough, λ ⩾ 2, |f (u)| ⩽ c(1 + |u|d ), ∀u, d < 2λ + 1, f (u) ⩽ c|u|, |u| ≪ 1, 1

then when u0 ∈ D(H 2 ), we can deduce that |u(t)x |2 is bounded; and when ∥u0 ∥ λ 2 is small enough, |u(t)|∞ is small enough. Similar to the proof of (iii) of Theorem 2.3.1, let 1

W (t) = sup (1 + τ ) r |u(τ )|∞ , 0⩽t⩽τ

then by (2.3.29), one has W (t) ⩽ c|u0 |1 + cW (t)p−1 ,

(2.3.30)

W (0) = |u0 |∞ .

(2.3.31)

2.3

The large time behavior for generalized KdV equation

49

By |u0 |1 + ∥u0 ∥ r−1 ⩽ η, we can deduce that W (t) is bounded, so, 2

|u(t)|∞ = O((1 + |t|)− r ). 1

At last, noting that u(t) asymptotically equals Z ∞ u+ (t) =G(t) ∗ u0 − G(t − τ ) ∗ f (u(τ ))x dτ 0 Z ∞ =u(t) − G(t − τ ) ∗ f (u(τ ))x dτ, t

which converges in L2 norm, and Z

∞

|u(t) − u+ (t)|2 ⩽ Z t

|f (u(τ ))x |2 dτ

∞

⩽c

p

τ − r dτ,

p > r + 1,

t 1

which leads to t r |u(t) − u+ (t)|2 → 0, t → +∞. Similarly, we can prove 1

t r |u(t) − u− (t)|2 → 0,

t → −∞.

(ii) The convolution operator Gt u = G(t) ∗ u has the following properties Gt : L2 → L2 ,

|Gt |2,2 = 1.

Gt : L1 → L∞ ,

|Gt |1,∞ ⩽ ct− r . ′ 1 1 1 Gt : Lq → Lq , 2 < q < ∞, + ′ = 1, |Gt |q′ ,q ⩽ ct( q −1)/r . p p   2 1 1 1− , q′ = 1 + . From the integral expression, Denote q = 2(p + 1), a = r q 2p + 1 we have Z t |u(t)|q ⩽|Gt u0 |q + |Gt−τ (f (u(τ ))x )|q dτ 0 Z t ⩽ct−a |u0 |q + c (t − τ )−a |u(τ )|pq dτ, (2.3.32) 1

0

where |f (u)x |q′ ⩽ |f ′ (u)|q/p |ux |2 ⩽ c|u|pq |ux |2 ⩽ c|u|pq . Let W (t) = sup (1 + τ )a |u(τ )|q , then by (2.3.32), we get 0⩽τ ⩽t

W (t) ⩽ c|u0 |q + cW (t)p

(a, p > 1).

50 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Therefore, W (t) is bounded, which leads to |u(t)|2(p+1) = O(1 + |t|)−(1− p+1 )r . 1

Here we have finished the proof of Theorem 2.3.3. In order to study the decay rate in norm of high order derivatives of the solution to the generalized KdV equation as t → ∞, we consider the Cauchy problem of the following KdV equation    α  ∂ uλ ∂ ∂2 ∂u + + − 2 u = 0, x ∈ R, t > 0, (2.3.33) ∂t ∂x λ ∂x ∂x x ∈ R,

u|t=0 = u0 (x),

(2.3.34)

where λ > 1, α > 0. Firstly, we consider the linear problem ∂t u + ∂x(−∂x2 )α u = 0,

u(x, 0) = u0 (x),

(2.3.35)

the solution of which is u(x, t) =

Z

2 t(2α−1)−1

where

Z g(x) =



+∞

g



t(2α−1)−1

−∞

∞

x − x′

u0 (x′ )dx′ ,


cos ξ 2α+1 + xξ dξ.

(2.3.36)

(2.3.37)

0

Assume α > 0, so that the integral (2.3.37) exists. When α = 1, g(x) = 3− 3 πAi (3− 3 x), 1

1

(2.3.38)

where Ai (z) is the well known Airy function. g(x) = O(x− 4 ) 1

(x → −∞),

(2.3.39)

g(x) → +∞ is exponentially decaying, we have Theorem 2.3.4 Let β = 2α + 1, α > 1. (i) When x → −∞, " # β   β−1 √ π |x| π g(x) ∼ cos (1 − β) + + O(|x|−(β−2)/2(β−1) ). β−2   2(β−1) β 4 β(β − 1) |x| 2 β (2.3.40)

2.3

The large time behavior for generalized KdV equation

51

(ii) When x → +∞, if β = 2n(n = 1, 2, · · · ), g(x) ∼ (−1)n+1

∞ X

(−1)k

k=0

[2n(2k + 1)]! (2k + 1)! x2n(2k+1)+1

(β = 2n; n = 1, 2, · · · ); (2.3.41)

if β = 2n + 1(n = 1, 2, · · · ), (2n + 1)− 4

1

g(x) ∼π

2n

1 2

· x−(

2n−1 4n

)

n X

exp

j=1

    1 1 2n i(2j − 1) 2j − 1 π − 2n 1+ 2n × − (2n + 2) ·x exp π + π− ; 2n + 1 2n 4n 4 (2.3.42) if β is not an integer, g(x) ∼ −

β! cos

π  β 2 ,

xβ+1

from (2.3.40)∼(2.3.43), we can deduce that: if α ⩾ Proof Rewrite (2.3.37) into

Z

∞

1 , g(x) ∈ L∞ (R). 2

exp[i(xξ + ξ β )]dξ.

g(x) = Re

(2.3.43)

(2.3.44)

0

(i) The state of g(x) as x → −∞, let  s=

|α| β

β  β−1

,

(2.3.45)

1

and let ξ = (1 + τ )s β , then (2.3.44) is changed into 1

g(x) = s β ReG(s), where

Z

(2.3.46)

∞

G(s) =

exp[isp(τ )]dτ, −1

p(τ ) = (1 + τ )β − β(1 + τ ). Denote G(s) = G1 (s) + G2 (s), where Z G1 (s) =

(2.3.47)

∞

exp[isp(τ )]dτ,

0

Z G2 (s) =

0

exp[isp(τ )]dτ. −1

(2.3.48)

52 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

We first deal with G1 (s). Applying the stationary phase method, and noting p(τ ) defined by (2.3.47) satisfying the conditions of Theorem 13.1 in P101 of P. J. Olver [201](the constants α, p, u, Q and λ mentioned in this theorem are taken separately (β − 1) to be 0, 1 − β, β , τ and 1), we deduce that 2   1 1 i π exp[is(1 − β)] − 21 G1 (s) = p e 4 ) (s → +∞). 1 + o(s  2 2 2 1 β(β − 1)s 2 For G2 (s), by integral transform τ = −τ ′ in (2.3.48) and Theorem 13.1 in the book of Olver, we have G2 (s) = G1 (s) + o(s− 2 )

(s → +∞).

G(s) = 2G1 (s) + o(s− 2 )

(s → +∞).

1

Therefore, 1

Then by the definition of s in (2.3.45), we get (2.3.40). (ii) The state of g(x) as x → +∞. Define β

s = x β−1 .

(2.3.49)

ξ = τ s1/β ,

(2.3.50)

g(x) = s1/β ReG(s),

(2.3.51)

By the transform

(2.3.44) can be rewritten into

where

Z G(s) =

∞

exp[isp(τ )]dτ,

p(τ ) = τ + τ β .

(2.3.52)

0

We find that p(τ ) is a monotone increasing function with respect to τ ∈ [0, ∞), and singularly map the domain [0, ∞) into itself. So, there exists the inverse function of p(τ ), denoted by h(σ), that is τ = h(σ). Since β > 1, p′ (τ ) exists for any τ ∈ [0, ∞), and p′ (τ ) ̸= 0, which keeps the existence of h′ (σ) for any σ ∈ [0, ∞). Therefore, we can change the integral variable, σ = p(τ ), then (2.3.52) is changed into Z ∞ G(s) = exp[isσ]h′ (σ)dσ. (2.3.53) 0

We use the following result to study the asymptotic state of G(s) as s → ∞.

2.3

The large time behavior for generalized KdV equation

53

Let v(ξ) ∈ C m−1 ([0, ∞)), such that lim v (k) (ξ) = 0, 0 < k ⩽ m − 1, then for a ξ→∞

real constant r, it holds that Z ∞ Z ∞ m−1 X v (k) (0) exp(irξ) m exp[irξ] v(ξ)dξ = + v (ξ)dξ, (2.3.54) k+1 (−ir) (−ir)m 0 0 k=0 R∞ where the integration on the right-hand side of (2.3.54) exists. If 0 |v (m) (ξ)|dξ < ∞, then the integration on the right-hand side of (2.3.54) is O(r−m ), r → ∞. If m = ∞, then Z ∞ ∞ X v (k) (0) exp[irξ] v(ξ)dξ ∼ , r → ∞. (2.3.55) (−ir)k+1 0 k=0

Now we consider the function h(σ). h(σ) is infinitely differentiable for any σ ∈ (0, ∞), which likes p(τ ) for τ ∈ (0, ∞). Further more, we have lim h(k) (σ) = 0

σ→∞

(k = 1, 2, · · · ).

(2.3.56)

1 p′ (τ )

(2.3.57)

In fact, since  h(k) (σ) =

1 d p′ (τ ) dτ

k−1

(k = 1, 2, · · · ),

while p′ (τ ) = O(τ β−1 ), τ → ∞, β > 1, so, (2.3.56) holds. Now let β > 1 be an integer, then h(k) (0) (k = 1, 2, · · · ) exists. And by (2.3.55), we get G(s) ∼

∞ X h(k+1) (σ) k=0

(−is)k+1

(s → ∞).

(2.3.58) β

Since only ReG(s) makes a contribution to g(x), we have (s = x β−1 ) 1

g(x) ∼ s β

∞ X h(2k) (0) k=1

(−is)2k

(x → +∞).

(2.3.59)

When β = 2n (n = 1, 2, · · · ), by Lagrange-Bürmann formula, one has h(σ) =

∞ X (2nj)! σ j(2n−1)+1 (−1)j . j! [j(2n − 1) + 1]! j=0

(2.3.60)

Then by (2.3.59) and (2.3.60), we get (2.3.41). When β = 2n + 1 (n = 1, 2, · · · ), h(σ) is an odd function of σ, so, h(2k) (0) = 0, k = 1, 2, · · · . In this case, the summation symbol on the right-hand side of (2.3.59) dose not appear, we obtain g(x) = o(x−µ ) (∀µ > 0).

(2.3.61)

54 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

The asymptotic state of g(x) as x → +∞ can be obtained by steepest descent method. Here g(x) can be represented as 1 β1 e s · G(s), 2

g(x) = e where G(s) =

Z

(2.3.62)

+∞

exp [isp(τ )]dτ , s, τ are still defined by (2.3.49), (2.3.50), and −∞

p(τ ) is still determined by (2.3.52). Firstly, we note that on upper half plane of τ , along the ray arg τ = θk = (4k + 1)π/(4n + 2)

(k = 0, 1, 2, · · · , n),

exp [isp(τ )] tends to 0 exponentially, the stationary points of p(τ ) are the roots of the following equation p′ (τ ) = 1 + (2n + 1)τ 2n = 0, which on the upper half plane of τ are τj∗ = (2n + 1)− 2n exp [iθj∗ ], 1

θj∗ = (2j − 1)π/2n.

Noting 0 < θ0 < θ1∗ < θ1 < θ2∗ < · · · < θn∗ < θn < π,

(2.3.63)

e now change integral path, such that G(s) turns into e G(s) =

n Z X k=1

exp [isp(τ )]dτ,

(2.3.64)

Γk

where Γk is the steepest descent path which passes τk∗ . When τ → ∞, it asymptotically tends to rays arg τ = θτ −1 and arg τ = θτ . For n = 5, see Figure 2.1.

Figure 2.1

2.3

The large time behavior for generalized KdV equation

55

So, we get p(τk∗ ) =

2n τ ∗, 2n + 1 k

p′′ (τk∗ ) = −

2n τk∗

(k = 1, 2, · · · , n).

Therefore  n  X π|τ ∗ | 2 1

G(s) ∼

k

k=1

ns

   2n θk∗ π ∗ exp i τ s+ − 2n + 1 k 2 4

(s → ∞).

(2.3.65)

Then it’s easy to get the asymptotic form (2.3.42) of g(x), as x → +∞. At last, assume β is not an integer. Let β = m + 1 − δ, m = 1, 2, · · · , 0 < δ < 1. Since p(τ ) is continuously differentiable of m order on [0, ∞), we can deduce that h′ (σ) is differentiable of m − 1 order on [0, ∞), p′ (τ ) = 1 + βτ β−1 ,

p′′ (τ ) = β(β − 1)τ β−2 ,

and h′ (0) = 1, h(k) (0) = 0, 2 ⩽ k ⩽ m. So, by (2.3.54), one has Z ∞ exp[isσ] (m+1) 1 + h (σ)dσ. G(s) = −is (−is)m 0

(2.3.66)

By (2.3.57), for k ⩾ 2, we have h(k) (σ) ∼ −β(β − 1) · · · (β − k + 1)σ β−k

(σ → 0),

which deduce that the integral of G on the right-hand side of (2.3.62) satisfies the conditions of Theorem 13.1 in [201], so G(s) ∼

1 exp[iπ(β − m)/2] Γ(β − m) − β(β − 1) · · · (β − m) β−m −is (is)m s

(s → ∞), (2.3.67)

substituting (2.3.67) into (2.3.51), we obtain (2.3.43). Here we have finished the proof of Theorem 2.3.4. Corollary 2.3.5 Assume that u0 (x) ∈ Lp (R) ∩ H 1 (R), then for α ⩾ estimate of the solution of equation (2.3.35) 1

2

|u(t)|Lq ⩽ (1 + t) 2α+1 (1− q ) (|u0 |Lp + |u0 |H 1 ) , where 1 1 + = 1, p q

1 ⩽ p ⩽ 2.

1 , we get 2 (2.3.68)

56 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Proof By Theorem 2.3.4, it holds that |u(t)|L∞ ⩽ ct− 2α+1 |u0 |L1 1

(t > 0).

Further, it’s easy to prove |u(t)|L2 = |u0 (x)|L2 . By the interpolation between L2 and L∞ , we get (2.3.68). Here we have finished the proof of Corollary 2.3.5. Now we consider the estimate of the nonlinear problem. 3 As we all know, if u0 (x) ∈ H s (R), s > , then the initial value problem (2.3.33), 2 (2.3.34) has a unique global solution u ∈ L∞ ([0, T ]; H s (R)). Proposition 2.3.6 For u0 (x) ∈ H 3 (R), then for the solution to the initial value problem (2.3.33), (2.3.34), we have the following estimate:  Z t  |u(t)|H 3 ⩽ |u0 |H 3 exp c |u(τ )|λ−1 dτ (λ > 1). (2.3.69) W 2,2λ 0

Proof By inner product between (2.3.33) and ∂x6 u, we have Z Z 1 d |∂x3 u|2 dx + ∂x3 u∂x3 (uλ−1 ∂x u)dx = 0. 2 dt

(2.3.70)

We decompose the second term on the left side of (2.3.70) into four terms, then we get the following terms: (i) Z λ−1 2 2 uλ−2 ux (∂x3 u)2 dx ⩽ |u|λ−2 q > 1, (2.3.71) L∞ |∂x u|L∞ |u|H 3 ⩽ c|u|W 2,q |u|H 3 , (ii) Z uλ−2 (∂x2 u)2 ∂x3 udx ⩽ |u|λ−2 |∂x2 u|2L2λ |∂x3 u|L2 , L2λ

(2.3.72)

where we have used Hölder’s inequality. By Sobolev inequality |∂x2 u|L2λ ⩽ |∂x2 u|H 1

(λ > 1),

(2.3.73)

so, Z uλ−2 (∂x2 u)2 ∂x3 udx ⩽ c|u|λ−1 |u|2H 3 . W 2,2λ

(2.3.74)

2.3

The large time behavior for generalized KdV equation

57

(iii) Z uλ−3 (∂x u)2 ∂x2 u∂x3 udx ⩽ c|∂x3 u|L2 |∂x2 u|L2λ |∂x u|2 2λ |u|λ−3 . L L2λ

(2.3.75)

(iv) Z ∂x3 uuλ−4 (∂x u)4 dx ⩽ c|∂x3 u|L2 |u|λ−4 |∂ u|4 , L2λ x L2λ

(2.3.76)

therefore, we have d |u|H 3 . |u|H 3 ⩽ c|u|λ−1 W 2,2λ dt Applying Gronwall inequality, we get (2.3.69). Here we have finished the proof of Proposition 2.3.6.  1 1 2 5 2 2 Proposition 2.3.7 Assume that α ⩾ , λ > α + + α + 3α + , then 2 3 4 there exists constant δ > 0, such that for the initial data u0 ∈ W 2,p (R) ∩ H 3 (R),

p=

2λ , 2λ − 1

satisfying |u0 |W 2,p ∩ |u0 |H 3 < δ, the function determined by the solution u(t) to the initial value problem (2.3.33), (2.3.34) 1

M (t) = sup (1 + s)(1− λ )/(2α+1) |u(s)|W 2,2λ

(2.3.77)

0⩽s⩽t

is bounded for any t ∈ [0, T ], and independent of T . Proof The solution to (2.3.33) can be rewritten into Z



t

G(t − s)∂x

u(x, t) = G(t)u0 (x) + 0

uλ λ



where the operator G(t) represents fundamental solution where Z g(x) = 0

∞


cos ξ 2α+1 + xξ dξ.

(2.3.78)

(x, s)ds, 1 t(2α+1)−1

 g

x t(2α+1)−1



,

58 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

By (2.3.68), we have

  1 |u(t)|W 2,2λ ⩽c(1 + t)−(1− λ )/(2α+1) |u0 |H 3 + |u0 | 2, 2λ−1 2λ W Z t 1 −(1− λ )/(2α+1) λ−1 + (t − s) |u ∂x u| 2, 2λ−1 2λ ds. W

0

(2.3.79)

By Hölder’s inequality, we get |uλ−1 ∂x u|

W

2,

2λ 2λ−1

|u|H 3 . ⩽ c|u|λ−1 W 2,2λ

By (2.3.69), we obtain

Z

|u(s)|H 3 ⩽ c|u0 |H 3 exp

(2.3.80)

s

f (τ )M (τ )λ−1 dτ,

(2.3.81)

0

where

Z f (s) =

s

(λ−1)τ

(1 + τ )− λ2α+1 dτ.

(2.3.82)

0

Substituting (2.3.80), (2.3.81) into (2.3.79), one has   M (t) ⩽ c |u0 |H 3 + |u0 | 2, 2λ−1 + c|u0 |H 3 exp[f (t)M (t)λ−1 ]M (t)λ−1 h(t), 2λ W

λ

(2.3.83) where

Z h(t) = 0

t

1

(1 + t)(1− λ )/(2α+1) (t −

1 s)(1− λ )/(2α+1)

(1 + s)−(λ−1)

2

/λ(2α+1)

.

(2.3.84)

under the condition that (λ − 1)2 > 1, λ(2α + 1)

(2.3.85)

the functions f (t) and h(t) are uniformly bounded, then (2.3.83) can be rewritten into M (t) ⩽ cδ{1 + c1 M (t)λ−1 exp[c2 M (t)λ−1 ]},

(2.3.86)

where δ = |u0 |H 3 + |u0 | 2, 2λ−1 2λ . W Now we consider the following function φ(m) = cδ[1 + c1 mλ−1 exp(c2 mλ−1 )] − m.

(2.3.87)

When δ is small enough, φ(m) has a zero point m1 . If δ < m1 , then M (0) < m1 , then we deduce that M (t) ⩽ m1

(∀t > 0),

which has finished the proof of Proposition 2.3.7.

(2.3.88)

2.3

The large time behavior for generalized KdV equation

59

 1 1 3 5 2 2 Theorem 2.3.8 Assume that α ⩾ , λ > α + , and there α + 3α + 2 2 4 2,p 3 exists a constant δ > 0, if the initial date u0 ∈ W (R) ∩ H (R) satisfying p=

2λ , 2λ − 1

|u0 |W 2,p + |u0 |H 3 < δ,

then there exists a unique solution u ∈ L∞ (R+ , H 3 (R)) to the problem (2.3.33), (2.3.34), satisfying −(1−

|u(t)|W 2,2λ ⩽ c(1 + t)

1 )/(2α+1) λ .

(2.3.89)

Further more, the solution to the problem (2.3.33), (2.3.34) asymptotically tends to that of the free problem, that is, there exist solutions u± to the linearized problem (2.3.35), such that |u(t) − u± (t)|H 2 → 0. (t → ±∞)

(2.3.90)

Proof For the solution to the problem (2.3.33), (2.3.34), we have estimate: |u(t)|H 3 ⩽ c|u0 |H 3 exp[cM (T )] ⩽ c|u0 |H 3 exp[cM0 ] ⩽ k, Let

Z u± (t) = u(t) +

∞

 G(t − s)∂x

t

uλ λ

∀t ∈ [0, T ].

(2.3.91)

 (s)ds,

(2.3.92)

the integral on the right-hand side of which exists, because Z ±∞  λ Z ±∞ u 3 G(t − s)∂ ⩽c |u(s)|λ−1 (s)ds x W 2,q |u(s)|H ds 2 λ t t H Z ±∞ 2 λ−1 ⩽ckM0 (1 + s)−(λ−1) /λ(2α+1) ds. t

Which shows that u± (t) has been defined, that is, when t → ±∞, it holds that |u(t) − u± (t)|H 2 → 0. It’s easy to prove that u± satisfy the linearized equation (2.3.35), then we have finished the proof of Theorem 2.3.8. Remark 2.3.9 The decay estimates in Lq norm of the solutions obtained in Theorem 2.3.4 (α = 1) and Theorem 2.3.3 (r = 3) are uniform. We refer to the paper [222] for the long time behavior of generalized KdV equations. For the well-posedness of generalized KdV equations, see [71,83,118,122,250]. As for some coupled equations, e.g. the coupled equations of KdV and nonlinear Schrödinger equation, see [77, 116, 129, 140], and KdV-Burgers equations, see [111], and so on.

60 Chapter 2

2.4

The Properties of the Solutions for Some Nonlinear Evolution Equations

The decay estimates for the weak solution of NavierStokes equations

Consider the decay in L2 norm of the solution to the Cauchy problem of the nD Navier-Stokes equations (n ⩾ 3)   ui + u · ∇ui − ∆ui + ∇i p = f i , i = 1, 2, · · · , n,   t (2.4.1) ∇ · u = 0,   u(x, 0) = u (x), n x∈R , 0 where f = (f 1 , f 2 , · · · , f n ) has suitable decay rate in L2 norm. Denote Z |g(·, t)|p =

Rn

 p1 |g(x, t)|p dx

(p ⩾ 1),

and let f = 0. Theorem 2.4.1 Assume that u : Rn × R+ → Rn , p : Rn × R+ → R are smooth function, u rapidly tends to 0, as |x| → ∞. Let u and p satisfy   ui + u · ∇ui − ∆ui + ∇i p = 0, i = 1, 2, · · · , n,   t (2.4.2) ∇ · u = 0,   u(x, 0) = u (x), 0

and u0 (x) ∈ L2 (Rn ) ∩ L1 (Rn ), then |u(·, t)|22 ⩽ c (t + 1)

−n 2 +1

,

(2.4.3)

where the constant c only depends on the L1 and L2 norms of u0 , and n. Proof Multiplying (2.4.2) with u and integrating on Rn , one has Z Z d |u|2 dx = −2 |∇u|2 dx. dt Rn Rn Applying Plancherel theorem to (2.4.4), we have Z Z d 2 |ˆ u| dξ = −2 |ξ|2 |ˆ u|2 dξ, dt Rn Rn

(2.4.4)

(2.4.5)

where u ˆ(ξ) is the Fourier transform. The integral on the right-hand side of (2.4.5) is written in two parts Z Z Z d |ˆ u|2 dξ = −2 |ξ|2 |ˆ u|2 dξ − 2 |ξ|2 |ˆ u|2 dξ, dt Rn s(t)c s(t)

2.4

The decay estimates for the weak solution of Navier-Stokes equations

61

where s(t) is a ball of Rn , the center of which is original point, and the radius is  r(t) =

n (2t + 1)

 12 (2.4.6)

.

So, d dt

Z

n |ˆ u| dξ ⩽ − (t + 1) Rn

Z

Z |ˆ u| dξ − 2

2

=−

n (t + 1)

|ξ|2 |ˆ u|2 dξ

2

s(t)c

Z

Rn

Z

|ˆ u|2 dξ + s(t)

s(t)



 n − 2|ξ|2 |ˆ u|2 dξ, (t + 1)

then, d dt

Z

n |ˆ u| dξ + (t + 1) Rn

Z

2

n |ˆ u| dξ ⩽ (t + 1) Rn

Z |ˆ u|2 dξ.

2

(2.4.7)

s(t)

Next, we’ll get the following estimate |ˆ u(ξ, t)| ⩽ c|ξ|−1

(ξ ∈ s(t)),

(2.4.8)

where constant c only depends on the L1 and L2 norms of u0 . By (2.4.7) and (2.4.8), we have Z Z Z n c d |ˆ u|2 dξ + |ˆ u|2 dξ ⩽ |ξ|−2 dξ. dt Rn (t + 1) Rn (t + 1) s(t) Multiplying the above inequality with integral factor (t + 1)n , we get   Z Z d |ξ|−2 dξ. (t + 1)n |ˆ u|2 dξ ⩽ c(t + 1)n−1 dt n s(t) R The right-hand side of the above inequality can be represented as Z r cW0 (t + 1)n−1 rn−1 · r−2 dr, 0

where W0 is the volume of nD unit ball, r(t) is the radius determined by (2.4.6), therefore,   n2 −1   Z n d cW0 n n 2 n−1 (t + 1) |ˆ u| dξ ⩽ (t + 1) ⩽ c(t + 1) 2 . dt n−2 2(t + 1) Rn Integrating both sides of the above inequality with respect to t, one has Z Z   n (t + 1)n |ˆ u|2 dξ ⩽ |ˆ u(ξ, 0)|2 dξ + c (t + 1) 2 +1 − 1 . Rn

Rn

62 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Noting u0 ∈ L2 , we obtain by Plancherel theorem that Z n |ˆ u|2 dξ ⩽ c(1 + t)− 2 +1 . Rn

In order to finish the proof of theorem, we need to establish the inequality (2.4.8). Take Fourier transform to (2.4.2), then we have u ˆt + |ξ|2 u ˆ = G(ξ, t),

(2.4.9)

where G(ξ, t) = −F (u, ∇u) − ξF (p). Here F means Fourier transform. Multiplying (2.4.9) with e|ξ| t , one has 2

2 d |ξ|2 t [e u ˆ] = e|ξ| t G(ξ, t). dt

Integrating with t, we get e

|ξ|2 t

Z

t

e|ξ| s G(ξ, s)ds. 2

u ˆ=u ˆ0 + 0

Therefore, |ˆ u(ξ, t)| ⩽ e−|ξ| t u ˆ0 (ξ) + 2

Z

t

e−|ξ|

2

(t−s)

|G(ξ, s)|ds.

(2.4.10)

0

Next, we establish the estimate |G(ξ, t)| ⩽ c|ξ|.

(2.4.11)

By (2.4.10) and (2.4.11), we obtain |ˆ u(ξ, t)| ⩽ e−|ξ| t |ˆ u0 (ξ)| + c 2

Z

t

e−|ξ|

2

(t−s)

|ξ|ds.

(2.4.12)

0

Since u0 ∈ L1 , the Fourier transform belongs to L∞ , that is |ˆ u0 (ξ)| ⩽ c (∀ξ ∈ R),

(2.4.13)

where c is a determined constant. From (2.4.13) and (2.4.12), we achieve at |ˆ u(ξ, t)| ⩽ ce−|ξ| t + 2

2 c (1 − e−|ξ| t ), |ξ|

which leads to (2.4.8). Since the radius of s(t) is uniformly bounded with respect to t, (2.4.8) holds for any compact set K.

2.4

The decay estimates for the weak solution of Navier-Stokes equations

63

In order to finish the proof, we need to establish inequality (2.4.11). To this end, we analysis every term of G(ξ, t). For convective term, it holds that XZ Z ixξ |uj ui | |ξj |dx, div(u ⊗ u)e dξ ⩽ |F (u, ∇u)| = n n R

i,j

R

where we have used integration by parts and ∇ · u = 0. Since |uj ui (·, t)|1 ⩽ |uj (·, t)|2 |ui (·, t)|2 ⩽ |u0 |22 , we get |F (u, ∇u)| ⩽ c|ξ|.

(2.4.14)

Corresponding estimate of pressure can be deduced by equation X ∂ ∆p = − (ui uj ). ∂x ∂x i j i,j

(2.4.15)

Taking Fourier transform to (2.4.15), we obtain X |ξ|2 F (p) = ξi ξj F (ui uj ). i,j

F (ui uj ) ∈ L∞ leads to |ˆ p| ⩽ c.

(2.4.16)

Combining (2.4.14) and (2.4.16), one has |G(ξ, t)| ⩽ c|ξ|, which finished the proof of Theorem 2.4.1. Now we consider the case of f = ̸ 0. Theorem 2.4.2 Assume that u : Rn × R+ → Rn , p : Rn × R+ → R are smooth functions, and u vanishes at ∞, u and p satisfy   u + u · ∇u − ∆u + ∇p = f,   t (2.4.17) ∇ · u = 0,   u(x, 0) = u (x), x ∈ Rn . 0

If u0 ∈ L1 (Rn ) ∩ L2 (Rn ), f ∈ W −1,1 (Rn ), ∇f = 0, and |f (·, t)|2 ⩽ K(1 + t)− 2 , n

then |u(·, t)|22 ⩽ c(1 + t)− 2 , n

where the constant c depends on n, k, ∥u0 ∥L1 , ∥u0 ∥L2 .

(2.4.18)

64 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Proof The energy inequality (2.4.4) is changed into Z Z Z d 2 2 |u| dx = −2 u · f dx. |∇u| dx + dt Rn Rn Rn Integrating (2.4.19) with respect to t, one has Z Z tZ Z Z tZ 2 2 2 |∇u| dxds = |u0 | dx + |u| dx + 2 Rn

Rn

0

Rn

0

Rn

(2.4.19)

u · f dxds.

(2.4.20)

Firstly, we prove u(·, t) ∈ L2 (Rn ). Applying Schwarz inequality and the assumption of f, the last term on the right-hand side of (2.4.20) turns into Z tZ Z t u · f dxds ⩽ |f (·, s)|2 |u(·, s)|2 ds. (2.4.21) 0

Rn

0

Let α(T ) = sup |u(·, t)|2 , then we get by (2.4.20) and (2.4.21) that 0⩽t⩽T

Z α(T ) ⩽ c + α(T ) 2

T

|f (·, s)|2 ds ⩽ c + α(T )k(T + 1)− 2 +1 , n

0

so, α(T ) ⩽ c,

(2.4.22)

where constant c only depends on L2 norm of u0 and k. Repeating the proof of Theorem 2.4.1, similar to inequality (2.4.7), one has Z Z Z Z n n d 2 2 2 |ˆ u| dξ + |ˆ u| dξ ⩽ |ˆ u| dξ + u · f dx, dt Rn t + 1 Rn t + 1 s(t) Rn where s(t) is the same as the definition in Theorem 2.4.1. By (2.4.22) and the assumption of f , we obtain Z Z Z n n n d |ˆ u|2 dξ + |ˆ u|2 dξ ⩽ |ˆ u|2 dξ + c(t + 1)− 2 . dt Rn t + 1 Rn t + 1 s(t) The same as the proof of Theorem 2.4.1, we need to estimate |ˆ u(ξ, t)| ⩽ c|ξ −1

(ξ ∈ S).

For this purpose, we need to prove |fˆ(ξ, t)| ⩽ c|ξ|, which is deduced by f ∈ W −1,1 . Therefore, the decay estimate in L2 norm of the solution to (2.4.17) can be deduced by a similar method in Theorem 2.4.1. Here we have finished the proof of Theorem 2.4.2. Assume that the solution of (2.4.17) satisfies u ∈ L1 (Rn ), n ⩾ 1, then

2.4

The decay estimates for the weak solution of Navier-Stokes equations

65

Corollary 2.4.3 Assume u0 , u and p satisfy the conditions in Theorem 2.4.1, f satisfies ∇ · f = 0, f ∈ W −1,1 , and |f (·, t)| ⩽ k1 (t + 1)−( 2 +1) . n

Let |u(·, t)|1 ⩽ k2 , then |u(·, t)|2 ⩽ c(t + 1)− 2 , n

where constant c only depends on k1 , k2 and the L1 , L2 norms of u0 . Proof Repeat the proof of Theorem 2.4.2, and apply |ˆ u(ξ, t)| ⩽ k2 , we can finish the proof of Corollary 2.4.3. Now we consider the decay in L2 norm of the Leray-Hopf weak solution to 3D Navier-Stokes equations   u + u · ∇u − ∆u + ∇p = f,   t (2.4.23) ∇ · u = 0,   u(x, 0) = u (x), 0

where f = (f 1 , f 2 , f 3 ) satisfies suitable conditions of decay. Denote
1 R H01 (R3 ) = H01 is a closure of C0∞ (R3 ) in norm of R3 |∇u|2 dx 2 , H −1 is the dual space of H01 , V = C0∞ (R3 ) ∩ {u : ∇ · u = 0}, H is a closure of V in norm of L2 (R3 ), V is a closure of V in norm of H01 (R3 ), V ′ is the dual space of V . Firstly, we consider the case f = 0. We have the following results: Theorem 2.4.4 Assume u0 ∈ H ∩ L1 (R3 ), then there exists a Leray-Hopf weak solution u(t, x) to 3D Navier-Stokes equations (2.4.23) (f = 0) with the initial data u0 , such that |u(·, t)| ⩽ c(t + 1)− 4 , 1

(2.4.24)

where constant c only depends on the L1 and L2 norms of u0 . Theorem 2.4.5 Assume u0 ∈ H ∩ L1 (R3 ), uN and pN satisfy d uN + ψδ (uN )∇uN − ∆uN + ∇pN = 0, dt

(2.4.25)

66 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

where ψδ (u) is a smoothing function of u, which will defined later, then |uN (·, t)|2 ⩽ c(t + 1)− 4 , 1

(2.4.26)

where constant c only depends on the L1 and L2 norms of u0 . The proof of Theorems 2.4.4 and 2.4.5 need the following lemma: Lemma 2.4.6 Assume f ∈ L2 (0, T ; V ′ ), u ∈ L2 (0, T ; V ), p is a distribution, and ut − ∆u + ∇p = f

(2.4.27)

is satisfied on D = R3 × (0, T ) in the sense of distribution, then ut ∈ L2 (0, T ; V ′ ), Z Z d |u|2 dx = 2 (ut , u)dx, dt R3 R3 holds in the sense of distribution. Remove a zero measure set outside, then u ∈ C(0, T ; H). The solution to (2.4.27) with initial data in H is unique in L2 (0, T ; V ). Lemma 2.4.7 Assume f ∈ L2 (0, T ; V ′ ), u0 ∈ H, w ∈ C ∞ (R3 ), ∇w = 0, then there exist a unique function u and a distribution p such that u ∈ C(0, T ; H) ∩ L2 (0, T ; V ), ut + w · ∇u − ∆u + ∇p = f holds on D in the sense of distribution, and u(0) = u0 (x). Corollary 2.4.8 Under the assumption of Lemma 2.4.7, it holds that w · ∇u ∈ L2 (0, T ; V ′ ), ut ∈ L2 (0, T ; V ′ ). In order to define smoothing function ψδ (x), we first define function ψ(x, t) : ZZ ψ(x, t) ∈ C ∞ , ψ ⩾ 0, ψdxdt = 1, supp ψ ⊂ {(x, t) : |x|2 < t, 1 < t < 2}.

2.4

The decay estimates for the weak solution of Navier-Stokes equations

67

If u ∈ L2 (0, T ; V ), let u e(x, t) =

( u(x, t),

if (x, t) ∈ R3 × R+ ,

(2.4.28)

others.

0,

Define ψδ (u)(x, t) = δ

−4



ZZ ψ

y t , δ δ

 u(x − y, t − τ )dydτ,

where δ = T /N , the value of ψδ (u) at time t only depends on that of u at time z ∈ (t − 2δ, t − δ). Lemma 2.4.9 For any u ∈ L∞ (0, T ; H) ∩ L2 (0, T ; V ), there exist ∇ · ψδ (u) = 0, Z Z sup |ψδ (u)|2 (x, t)dx ⩽ C ess sup 0⩽t⩽T

R3

0 0, (2.5.1) (2.5.2)

u(x, 0) = u0 (x),

where g > 0, p > 1. As we all know, the smooth solution to the problem (2.5.1), (2.5.2) satisfies the following conservation laws Z ∥u(t)∥2 =

 21 = const = ∥φ∥2 , |u(x, t)| dx 2

(2.5.3)

  2g |∇u|2 − |u|p+1 dx = const = E0 . p+1 Rn

(2.5.4)

R3

Z

Now, we estimate ∥u(t)∥p+1 . By Sobolev inequality, we have ∥u(t)∥p+1 ⩽ const∥u(t)∥1−θ ∥∇u(t)∥θ2 , 2

(2.5.5)

72 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

1 where =θ p+1



1 1 − 2 n

 +

1−θ , that is 2 θ = n(p − 1)/2(p + 1).

2n , when n = 1, 2, p can be any constant. Since ∥u(t)∥2 is n−2 uniformly bounded, then by (2.5.5), one has When n ⩾ 3, p + 1 ⩽

n(p−1) 2

∥u(t)∥p+1 p+1 ⩽ const∥∇u(t)∥2

(2.5.6)

.

Now, let |E0 | < ∞. We get by (2.5.4) that n(p−1) 2

∥∇u(t)∥22 ⩽ |E0 | + gconst∥∇u(t)∥2

.

(2.5.7)

4 n (p − 1) < 2, that is p < + 1, by the above inequality, we know that ∥∇u∥2 2 n is uniformly bounded, so, we can get the global smooth solution. For large initial data and large p, the “blow up” phenomenon may occur. In the case of E0 ⩽ 0, 4 p > 1 + , we prove that ∥∇u(t)∥2 will blow up in finite time, for some kind of n initial data φ. To this end, we consider the Cauchy problem ( iut = ∆u + F (|u|2 )u, x ∈ Rn , t > 0, (2.5.8) When

x ∈ Rn .

u(x, 0) = φ(x)

(2.5.9)

Lemma 2.5.1 Assume that u(x, t) is the smooth solution to the Cauchy problem (2.5.8), (2.5.9) on 0 ⩽ t < t1 , then we have the following integral equation (i) ∥u(t)∥2 = ∥φ∥2 , Z (ii) [|∇u|2 − G(|u|2 )]dx = const = E0 , (iii) (iv)

d dt

Z

Z |x|2 |u|2 dx = −4Im

d Im dt

where G(u) = Rn .

Z

r¯ uur dx, r = |x|,

Z r¯ uur dx = −2

Ru 0

Z |∇u|2 dx + n

[|u|2 F (|u|2 ) − G(|u|2 )]dx,

F (s)ds, u ¯ represents the conjugation of u, all the integrals are on

Proof Firstly, multiplying (2.5.8) with 2¯ u, taking the imaginary part, we have ∂ 2 |u| = ∇ · (2Im¯ u∇u), ∂t

2.5

The “blowing up” phenomenon for the Cauchy problem of nonlinear Schrödinger· · · 73

so, (i) holds. Multiplying (2.5.8) with |x|2 , integrating on Rn , and integrating by parts for the right-hand side, we obtain (iii). Multiplying (2.5.8) with 2¯ ut , integrating on Rn , and integrating by parts, we get (ii). In order to get (iv) , multiplying (2.5.8) with 2r¯ ur , then we achieve at 2ir¯ ur ut = 2r¯ ur ∆u + 2F (|u|2 )ru¯ ur . Integrating on Rn , and taking the real part, we have I = II + III, where " Z # X I = Re i xk (¯ uxk ut − uxk u ¯t )dx , Z II = 2Re

k

r¯ ur ∆udx, Z

III = 2Re

F (|u|2 )ru¯ ur dx.

We deduce by integrating by parts and direct calculation that Z II = (n − 2) |∇u|2 dx, Z III = −n G(|u|2 )dx. At last, I can be rewritten into    Z X  ∂ ∂ (¯ uxk u) − (u¯ ut ) dx I =Re i xk ∂t ∂xk  Z   Z  d = Re i ru¯ ur dx + nRe i u¯ ut dx . dt We get by (2.5.8) that I=

Z Z d Im r¯ uur dx + 2 |∇u|2 dx dt Z  2  −n |u| F (|u|2 ) − G(|u|2 ) dx.

From which we get (iv). Here we have finished the proof of Lemma 2.5.1. Theorem 2.5.2 Let u be the classical solution to the Cauchy problem (2.5.8), (2.5.9) and φ be Schwartz downhill function classes. Assume that

74 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

(i) E0 ⩽Z0, (ii) I = rφφ ¯ r dx > 0, (iii) there exist constants cn > 1 +

2 , such that n

sF (s) ⩾ cn G(s)

(∀s ⩾ 0),

then, there exists a finite time T , such that lim ∥∇u(t)∥2 = +∞.

t→T −

R Proof Denote y = Im r¯ uur dx. The assumption (ii) leads to y(0) > 0. By (iv) of Lemma 2.5.1, we have Z Z y(t) ˙ = −2 |∇u|2 dx + n [|u|2 F (|u|2 ) − G(|u|2 )]dx. By the assumption (iii), we obtain Z Z y(t) ˙ ⩾ −2 |∇u|2 dx + n(cn − 1) G(|u|2 )dx.

(2.5.10)

The last term on the right-hand side of the inequality (2.5.10) is replaced by (ii) of Lemma 2.5.1, which leads to Z  Z y(t) ˙ ⩾ − 2 |∇u|2 dx + n(cn − 1) |∇u|2 dx − E0 Z =[n(cn − 1) − 2] |∇u|2 dx − n(cn − 1)E0 . (2.5.11) By assumption (i), we know E0 ⩽ 0. Denote kn = n(cn − 1) − 2. By assumption (iii), we have kn > 0, then we deduce by (2.5.11) that y(t) ˙ ⩾ kn ∥∇u∥22 .

(2.5.12)

Since y(0) > 0, kn > 0, the function y(t) is increasing, therefore, y(t) > 0. By (iii) of Lemma 2.5.1, one has Z Z d 2 2 r |u| dx = −4 Im r¯ uur dx = −4y(t) ⩽ 0, dt so, Z

Z r |u| dx ⩽ 2

2

r2 |φ|2 dx ≡ d20 < ∞.

2.5

The “blowing up” phenomenon for the Cauchy problem of nonlinear Schrödinger· · · 75

By Schwarz inequality, there exists Z |y(t)| = y(t) ⩽ c

r |u| dx 2

2

 21 Z

|ur | dr 2

 12

⩽ d0 ∥∇u∥2 .

Then, from (2.5.12), we get the following differential inequalities y(t) ˙ ⩾ From which, on 0 ⩽ t ⩽

kn 2 y (t) and y(0) > 0. d0

d20 , we deduce that kn y(0) y(t) ⩾

d20

y(0)d20 . − kn y(0)t

We have estimate ∥∇u(t)∥2 ⩾ y(0)d20 /(d20 − kn y(0)t), so. lim ∥∇u(t)∥2 = +∞,

t→T −

where T − = d20 /kn y(0), Theorem 2.5.2 is obtained. Corollary 2.5.3 Besides the assumptions (i)∼(iii) in Theorem 2.5.2, add another condition: there exists a constant σ > 0, such that s−1 G(s) ⩽ const · sσ

(∀s > 0),

(2.5.13)

then ∥u(t)∥∞ will blow up in finite time. Proof By assumption (i), E0 ⩽ 0, then we get according to (ii) in Lemma 2.5.1 that Z G(|u|2 ) 2 2 ∥∇u(t)∥2 ⩽ · |u|2 dx ⩽ const∥u(t)∥2σ ∞ ∥φ∥2 , |u|2 so, both ∥u(t)∥∞ and ∥∇u(t)∥2 will blow up. Here we have finished the proof of Corollary 2.5.3. Remark 2.5.4 Consider the initial function φ with the following expression φ(x) = exp(i|x|2 )ψ(x), where ψ(x) is any Schwartz fast decreasing real-valued function Z Z Im rφφ ¯ r dx = 2 r2 |ψ|2 dx. Therefore, (ii) in Theorem 2.5.2 holds.

76 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

At last, we consider the special form of (2.5.8), F (s) = s that is

Here, G(|u|2 ) =

(

(p > 1),

iut = ∆u + |u|p−1 u (where x ∈ Rn , t > 0),

(2.5.14)

u(x, 0) = φ(x).

(2.5.15)

R |u|2 0

p−1 2

s

p−1 2

ds =

requests that s·s holds for some cn > 1 +

p−1 2

2 |u|p+1 . The condition (iii) of Theorem 2.5.2 p+1 

⩾ cn

p−1 2 s 2 p+1

 (∀s > 0)

2 p+1 . Here, we take cn = , such that n 2 p+1 2 = cn > 1 + . 2 n

So we only need to choose p > 1 +

4 , then we get that (iii) of Theorem 2.5.2 holds. n

Therefore, we have Corollary 2.5.5 For the solution of Cauchy problem (2.5.14), (2.5.15), if it satisfies  Z  2 |φ|p+1 dx ⩽ 0, (i) E0 = |∇φ|2 − p+1 Z (ii) Im rφφ ¯ r dx > 0, 4 , n then ∥∇u(t)∥2 and ∥u(t)∥∞ will blow up in finite time. (iii) p > 1 +

Corollary 2.5.6 Consider Cauchy problem (2.5.14), (2.5.15), and the assumptions (i)∼(iii) in Corollary 2.5.5 are satisfied, then (i) for any q ⩾ p + 1, ∥u∥q → +∞, (ii) Let q satisfy

t → T − < ∞.

n n+2 (p − 1) < q < p + 1, when n ⩾ 3, p < , then 2 n−2 ∥u(t)∥q → +∞,

t → T − < ∞.

2.5

The “blowing up” phenomenon for the Cauchy problem of nonlinear Schrödinger· · · 77

Proof In order to prove (i), noting E0 ⩽ 0, for q > p + 1, ∥∇u(t)∥22 ⩽

2 2 ∥u(t)∥p+1 [∥u(t)∥θ2 ∥u(t)∥1−θ ]p+1 , q p+1 ⩽ p+1 p+1

where 1 θ 1−θ = + . p+1 2 q Since ∥∇u(t)∥ blows up, ∥u(t)∥p+1 and ∥u(t)∥q blow up. In order to prove (ii), by Sobolev inequality, one has ∥u(t)∥p+1 ⩽ const∥∇u(t)∥θ2 ∥u(t)∥1−θ , 2 where 1 =θ p+1



1 1 − 2 n

 +

1−θ , q

which is θ=

2n(p + 1 − q) , (p + 1)[2n − q(n − 2)]

q < p + 1,

(2.5.16)

now, E0 ⩽ 0, by (ii) in Lemma 2.5.1, one has p+1

2 ∥∇u(t)∥2 ⩽ const∥u(t)∥p+1 .

So, we deduce from (2.5.15) that (p+1)θ

∥u(t)∥p+1 ⩽ const t∥u(t)∥1−θ ∥u(t)∥p+12 . q If (p + 1)θ < 1, 2

(2.5.17)

then ∥u(t)∥q will blow up. Applying the expression of θ (2.5.16), we know that (2.5.17) equals q>

n (p − 1), 2

then the proof is finished. We refer to the papers [62, 84, 85, 102] for the “blow up” problem of nonlinear Schrödinger equations and the references therein.

78 Chapter 2

2.6

The Properties of the Solutions for Some Nonlinear Evolution Equations

The “blow up” problem for the solutions of some semilinear parabolic and hyperbolic equations

Now, we use the method of convex function to study the “blow up” phenomenon of the solution to some semilinear parabolic and hyperbolic equations. Lemma 2.6.1 Assume when t > 0, positive second order continuously differentiable function ψ(t) satisfies inequality ψ ′′ (t)ψ ′ (t) − (1 + α)(ψ ′ )2 ⩾ −2C1 ψψ ′ − C2 ψ 2 ,

(2.6.1)

where α > 0, C1 , C2 ⩾ 0, then when ψ(0) > 0, ψ ′ (0) > −r2 α−1 ψ(0), C1 + C2 > 0, it holds that ψ(t) → ∞, 1 r1 ψ(0) + αψ ′ (0) , t → t1 ⩽ t2 = p 2 ln 2 C1 + C22 r2 ψ(0) + αψ ′ (0) p p where r1 = −C1 + C12 + αC2 , r2 = −C1 − C12 + αC2 . If ψ(0) > 0, ψ ′ (0) > 0, C1 = C2 = 0, then ψ(t) → ∞,

t → t1 ⩽ t2 =

ψ(0) . αψ ′ (0)

Proof Let Φ(t) = ψ −α (t), then ψ ′ (t) , ψ 1+α (t) ψ ′′ (t)ψ(t) − (1 + α)ψ ′′ (t)2 Φ′′ (t) = −α . ψ 2+α (t) Φ′ (t) = −α

From(2.6.1) we have Φ′′ (t) + 2C1 Φ′ (t) − C2 Φ(t) ≡ f (t) ⩽ 0.

(2.6.2)

For C1 + C2 > 0, we can get the solution to (2.6.2) Z t 1 Φ(t) = β1 er1 t + β2 er2 t + f (τ )[er1 (t−τ ) − er2 (t−τ ) ]dτ ⩽ β1 er1 t + β2 er2 t , r1 − r2 0 (2.6.3) where β1 , β2 are determined by the following algebraic equations: ( β1 + β2 = Φ(0), β1 r1 + β2 r2 = Φ′ (0).

2.6

The “blow up” problem for the solutions of some semi-linear parabolic hyperbolic· · · 79

so, β1 = (r1 − r2 )−1 [Φ′ (0) − r2 Φ(0)] = −(r1 − r2 )−1 [αψ ′ (0) + r2 ψ(0)]ψ −α−1 (0) < 0, β2 = (r1 − r2 )−1 [αψ(0) + r2 ψ(0)]ψ −α−1 (0) > 0. From which, when t1 ⩽ t2 , t → t1 , Φ(t) → 0, that is, ψ(t) → ∞. When C1 = C2 = 0, the result of this lemma will directly deduced by (2.6.2). Here, we finish the proof of Lemma 2.6.1. Consider the Cauchy problem of the following parabolic equation in Hilbert space H: put = −Au + Bu + F (t, u),

(2.6.4)

where the operators p, A, B : H → H. A is a linear symmetric operator, p is a positive operator, A is a negative operator. F (t, u) is a nonlinear operator. Since F is Frechet differentiable with respect to fixed t, there exists a linear functional G(t, u), such that d G(t, u(τ )) = (F (t, u(τ )), uτ (τ )) . dt Moreover, assume that G(t, u(t)) smoothly depends on t, it holds the following relation d G(t, u(t)) = (F (t, u(t)), ut (t)) + Gt (t, u(t)), dt where u(t) smoothly depends on t. We are concerned about the state of the solution to (2.6.4) as t increases. For simplicity, we assume that the classical solution to (2.6.4) exists, let ∥p− 2 Bu∥ ⩽ M1 ∥A 2 u∥ + M2 ∥p 2 u∥ 1

1

1

(2.6.5)

and (F (t, u), u) ⩾ 2(1 + α1 )G(t, u),

α1 > 0.

(2.6.6)

Moreover, Gt (t, u) ⩾ M3 (F (t, u), u) ,

M3 = where β ∈ (0, α1 ), ε ∈ (0, 1).

1 + α1 1 + ε M12 · · , α1 − β 1 − ε 4

(2.6.7)

80 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Theorem 2.6.2 Let p, A, B, F and the solution u(t) to (2.6.4) satisfy conditions 1 mentioned before, and u0 = u(0), B0 = ∥p 2 u0 ∥ > 0 (u0 ̸= 0), A0 ⩾

(1 + α)2 δ B0 , 4α(α1 + 1)

(2.6.8)

where 1 1 M3 A0 = − ∥A 2 u0 ∥2 − B0 + G(0, u0 ), 2 2 q p α = −1 + 1 + β, α1 = −γ2 /α, γ1,2 = −c1 ± c21 + αc2 , M12 + M2 , c2 = 4(1 + α1 )M4 , 4α1  2  M1 M 2 (1 + α1 ) 1 + ε M4 = M3 + M2 + 2 , 4ε 4 α1 − β ε c1 =

then 1

∥p 2 u∥ → ∞, 1 4α2 (α1 + 1)A0 + γ1 (1 + α)2 B0 t → t1 ⩽ t2 = p 2 . 2 c1 + αc2 4α2 (α1 + 1)A0 + γ2 (1 + α)2 B0 When B ≡ 0, M1 = M2 = M3 = M4 = c1 = c2 = γ1,2 = δ = 0, condition (2.6.7) has the following form Gt (t, u) > 0, and condition (2.6.8) turns into 1 1 − ∥A 2 u0 ∥2 + G(0, u0 ) > 0. 2 Proof By the inner product between (2.6.4) and u, one has 1 1 1 (∥p 2 u∥2 )t = −∥A 2 u∥2 + (Bu, u) + (F (t, u), u). 2 By the inner product between (2.6.4) and ut , we get

(put , u) =

(2.6.9)

1

∥p 2 ut ∥2 = − (Au, ut ) + (Bu, ut ) + (F (t, u), ut ) =−

1 1 d d ∥A 2 u∥2 + (Bu, ut ) + G(t, u) − Gt (t, u). 2 dt dt

(2.6.10)

1 1 Let j(t) = − ∥A 2 u∥2 + G(t, u), we deduce by (2.6.5), (2.6.6) and (2.6.9) that 2 1 1 1 1 d 1 2 ∥p 2 u∥ ⩾ − ∥A 2 u∥ − ∥p− 2 Bu∥∥p 2 u∥ + 2(α1 + 1)G(t, u) 2 dt 1 1 1 1 ⩾2(α1 + 1)j(t) + α1 ∥A 2 u∥ − M1 ∥A 2 u∥∥p 2 u∥ − M2 ∥p 2 u∥2 1

⩾2(α1 + 1)j(t) − c1 ∥p 2 u∥2 ,

(2.6.11)

2.6

The “blow up” problem for the solutions of some semi-linear parabolic hyperbolic· · · 81

M12 + M2 . On the other hand, one can get the change of j(t) with 4α1 respect to t by (2.6.10) and (2.6.5): where c1 =

1 1 1 dj(t) ⩾∥p 2 ut ∥2 − ∥p− 2 Bu∥∥p 2 ut ∥ + Gt (t, u) dt 1 1 1 ⩾(1 − ε1 )∥p 2 ut ∥2 − ∥p− 2 Bu∥2 + ∥Gt (t, u)∥ 4ε1 1 1 1 M2 M2 1 ⩾(1 − ε1 )∥p 2 ut ∥2 − 1 (1 + ε1 )∥A 2 u∥2 − 2 (1 + )∥p 2 u∥2 + Gt (t, u), 4ε1 4ε1 ε1 (2.6.12)

where ε1 > 0. Integrating (2.6.9) with respect to t, one has: (∀ε3 > 0) Z t Z t Z t 1 1 1 1 1 2 2 2 2 2 2 ∥p ut ∥ + ∥A u∥ dτ ⩽ ∥p u0 ∥ + (Bu, u)dτ + (F , u)dτ 2 2 0 0 0 Z t Z t 1 1 1 2 − 21 2 2 ⩽ ∥p u0 ∥ + ∥p Bu∥∥p u∥dτ + (F , u)dτ 2 0 0 Z t Z 1 M2 t 1 2 1 1 ⩽ ∥p 2 u0 ∥2 + ε3 ∥A 2 u∥2 dτ + 1 ∥p 2 u∥ dτ 2 4ε3 0 0 Z t Z t 1 + M2 ∥p 2 u∥2 dτ + (F , u)dτ, 0

0

which leads to  2 Z t Z t Z t 1 1 1 1 M1 2 2 2 2 2 2 + M2 (F , u)dτ. (1 − ε3 ) ∥A u∥ dτ ⩽ ∥p u0 ∥ + ∥p u∥ dτ + 2 4ε3 0 0 0 (2.6.13) Integrating (2.6.12) with respect to t from 0 to t, and applying (2.6.13), we have: (ε3 ∈ (0, 1)) Z t 1 j(t) ⩾j(0) + (1 − ε1 ) ∥p 2 ut ∥2 dτ 0   2    M12 M1 M2 1 − (1 + ε3 ) + M2 + 2 1 + 4ε1 (1 − ε3 ) 4ε3 4ε1 ε2 Z t 2 1 1 M1 × ∥p 2 u∥2 dτ − (1 + ε1 )∥p 2 M0 ∥2 8ε (1 − ε ) 1 1 0  Z t M12 + Gt (t, u) − (1 + ε1 )(F , u) dτ. (2.6.14) 4ε1 (1 − ε3 ) 0 Choosing ε1 = (α1 − β)/(1 + α1 ), ε2 = ε3 = ε, ε ∈ (0, 1), β ∈ (0, α1 ), then by (2.6.14), we can get the estimate Z t Z t 1 1 1+β j(t) ⩾ ∥p 2 uτ ∥2 dτ − M4 ∥p 2 u∥2 dτ + A0 , (2.6.15) 1 + α1 0 0

82 Chapter 2

where

The Properties of the Solutions for Some Nonlinear Evolution Equations



 M12 M 2 1 + α1 1 + ε + M2 + 2 · , 4ε 4 (α1 − β) ε M3 1 ∥p 2 u0 ∥2 . A0 = j(0) − 2 M4 = M3

From (2.6.11) and (2.6.15), we obtain   Z t Z t 1 1 1 d 1 2 2 2 2 2 2 ∥p u∥ ⩾ 4(1+β) ∥p uτ ∥ dτ +4(α1 + 1) A0 −M4 ∥p u∥ dτ −2c1 ∥p 2 u∥2 . dt 0 0 (2.6.16) Rt 1 Let ψ(t) = 0 ∥p 2 u(τ )∥2 dτ + c3 , where constant c3 and u0 are to be determined to satisfy conditions in Lemma 2.6.1. Let Z t Z t 1 1 ∥p 2 u∥2 dτ = A1 , ∥p 2 uτ ∥2 dτ = A2 , 0

0

then: (∀ε4 > 0)  2 Z t 1 2 2 ∥p u(t)∥ = ∥p u0 ∥ + 2 (pu, uτ )dτ 1 2

4

0

2 p ⩽ ∥p u0 ∥2 + 2 A1 A2   1 1 ⩽ 1+ ∥p 2 u0 ∥4 + 4(1 + ε4 )A1 A2 . ε4 

1 2

From the above inequality and (2.6.16), one has ψ ′′ (t)ψ(t) − (1 + α)ψ ′ (t)2 ⩾4(1 + β)A2 (A1 + C3 )[4(α1 + 1)(A0 − M4 ψ + M4 C3 ) − 2C1 ψ ′ ]ψ   1 1 − (1 + α) 1 + ∥p 2 u0 ∥4 − 4(1 + α)(1 + ε4 )A1 A2 . (2.6.17) ε4 Choosing ε4 = α, (1 + α) = (1 + β), then by (2.6.17), we have ψ ′′ (t)ψ(t) − (1 + α)ψ ′ (t)2 ⩾ − 2C1 ψ ′ (t)ψ(t) − 4(α1 + 1)M4 ψ 2 (t)   (1 + α)2 1 + 4(α1 + 1)(A0 + M4 C3 )C3 − ∥p 2 u0 ∥4 . 2 (2.6.18) In order to apply Lemma 2.6.1, the following conditions must be satisfied  (1 + α)2 1  4(α1 + 1)(A0 + M4 C3 )C3 − ∥p 2 u0 ∥4 ⩾ 0, (2.6.19) α  ′ ψ (0) > δψ(0), δ = −α−1 γ2 , δ ⩾ 0. (2.6.20)

2.6

The “blow up” problem for the solutions of some semi-linear parabolic hyperbolic· · · 83

Let ψ ′ (0) = ∥p 2 u0 ∥2 , A0 > 0, then when 1

1

C3 =

∥p 2 u0 ∥4 (1 + α)2 · , 4α(α1 + 1) A0

(2.6.19) holds. In order to satisfy (2.6.20), from condition (2.6.8), we get 1

δψ(0) = δC3 =

1 (1 + α)2 δ ∥p 2 u0 ∥4 · < ∥p 2 u0 ∥2 = ψ ′ (0), 4α(α1 + 1) A0

So, the proof of this theorem is finished. Theorem 2.6.3 Assume that the conditions in Theorem 2.6.2 are satisfied, where condition (2.6.7) is changed into Gt (t, u) ⩾ 0,

2α1 λG(t, u) ⩾ M3 (F (t, u), u),

(2.6.21)

where λ > 0, and assume B is a linear operator, then the result of Theorem 2.6.2 holds. Proof Let v(t) = u(t)e−λt , λ > 0, from (2.6.4), we know that v(t) satisfy the following equation e + Bv e + F (t, v), P vt = −Av

(2.6.22)

f(t, v) = e−λt F (t, eλt v). It’s easy to verify that e = A + λP , B e = B, F where A conditions in Theorem 2.6.2 are satisfied. Obviously,   e v = ∥A 12 v∥2 + λ∥P 12 v∥2 . e > 0, ∥A e 12 ∥2 = Av, A From which it deduces that there exist constants M1 and M2 such that condition e is satisfied. Let G(t, e v) = e−2λt G(t, eλt v), then (2.6.5) of B 
 d e f(t, v), vτ , G(t, v(τ )) = e−2λt F (t, eλt v(τ )), eλt vτ (τ ) = F dt

(2.6.23)

which needs 

 f(t, v), v ⩾ 2(1 + α1 )G(t, e v), F

(2.6.24)

so, the assumption (2.6.6) is satisfied. In order to prove the following inequality holds   f(t, v), v , e t (t, v) ⩾ M3 F G

84 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

we get by applying (2.6.22) and (2.6.23) that

e v) =e−2λt Gt (t, eλt v) − 2λe−2λt G(t, eλt v) + e−2λt F t, eλt v(t) , λeλt v G(t,   f(t, v), v e v) + λ F ⩾ − 2λG(t, e v) ⩾2λα1 G(t,   f(t, v), v , ⩾M3 F therefore, equation (2.6.22) satisfies all conditions in Theorem 2.6.2. As for the initial condition, since v(0) = u(0), we can get v(t). So, Theorem 2.6.3 holds. As an example of Theorem 2.6.3, we study the initial value problem of the following parabolic equation  n n X X ∂   ut = (ai,j (x)uxi ) + ai (x, t)uxj + a0 (x, t)u + Cum , (2.6.25) ∂x i i,j=1 i=1   u|∂Ω = 0, u|t=0 = u0 (x), x ∈ Ω ⊂ Rn . (2.6.26) Let the Hilbert space H be L2 (Ω), and satisfy the following inequality on QT = Ω × [0, T ], ν

n X

ξj2 ⩽ ai,j ξi ξj ⩽ µ

i=1

n X

ξi2 ,

aij = aji ,

|ai | = µi ,

|a0 − ν| ⩽ µ1 ,

i=1

|aijt ξi ξj | ⩽ µ2

n X

ξj2 .

j=1





∂ ∂ aij + νI is a self conjugate positive operator with ∂xi ∂xi the boundary condition (2.6.26), and denote it by A(t). One has ! Z n X 1 (Au, u) = ∥A 2 u∥2 ⩾ ν∥u∥21 = ν u2xi + u2 dx

It’s easy to get that −

Ω

and

Z |(At u, u)| =

aijt uxi uxj dx ⩽ Ω

i=1

1 µ2 ∥A 2 u∥, ν

that is ∥(At u, u)∥ ⩽ M1′ ∥A 2 u∥2 + M2′ ∥P 2 u∥2 , 1

where M1 =

p

µν −1 . Denote

n X i=1

ai

1

∂ − νI by B, then ∂xi

2.6

The “blow up” problem for the solutions of some semi-linear parabolic hyperbolic· · · 85

∥Bu∥2 ⩽ (n + 1)µ1 ∥u∥21 ⩽

1 (n + 1)µ1 ∥A 2 u∥2 , ν

which means that condition (2.6.5) is satisfied, where M1 = C R m+1 u dx. And F (t, u) = Cum , So, G(t, u) = m+1 Ω Z 1 (F (t, u), u) = C um+1 dx = G(t, u), m+1 Ω

p

(n + 1)µ1 ν −1 ,

then condition (2.6.6) is satisfied, where 2(1 + α1 ) = m + 1. Condition (2.6.21) is saM3 tisfied, where λ = , M3 is determined by (2.6.7). Then all of assumptions (m + 1)2α1 in Theorem 2.6.3 are satisfied. So, when C > 0, m > 1, solution u(t) of problem (2.6.1), (2.6.2) will blow up in finite time. Now consider the case of hyperbolic equation. Assume that there exists the following equation P utt = −Au + Bu − P ut + F (t, u),

(2.6.27)

where P, A are linear symmetric operators, P > 0, A > 0, the operator B may be nonlinear, and they satisfy ∥P − 2 Bu∥ ⩽ M1 ∥A 2 u∥, 1

1

(2.6.28)

where a > 0 is a real constant, the nonlinear term F (t, u) has potential G(t, u), and satisfies (F (t, u), u) ⩾ 2(1 + 2α)G(t, u)

(α > 0).

(2.6.29)

Moreover, assume that G(t, u) satisfies Gt (t, u) ⩾ M1 G(t, u).

(2.6.30)

Theorem 2.6.4 Assume all the assumptions mentioned before are satisfied, and the initial data of (2.6.27) satisfies the following inequality   q 1 α ′ 2 2 2(P u(0), ut (0)) ≡ ψ (0) > a + a + 2M1 ∥P 2 u(0)∥2 4   q α 2 2 ≡ a + a + 2M1 ψ(0), (2.6.31) 4 1 1 1 1 − ∥A 2 u(0)∥2 − ∥P 2 ut (0)∥2 + G(0, u(0)) > 0, 2 2

(2.6.32)

86 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

then 1

∥P 2 u(t)∥ → ∞,

t → t1 ⩽ t2 ,

 p 1 2 + 2M 2 ψ(0) + αψ ′ (0) a − a 1 2 . ln 4  t2 = p  2 a + 2M12 − 1 a + pa2 + 2M 2 ψ(0) + αψ ′ (0) 1 4 −

(2.6.33)

1

Proof Let ψ(t) = ∥P 2 u(t)∥2 , it’s easy to see that ψ(t) satisfies conditions in Lemma 2.6.1, actually,  1  1 1 ψ ′ (t) = 2 P 2 u, P 2 u , ψ ′′ (t) = 2∥P 2 u∥2 + 2(Ptt , u),   1 2  1 1 1 2 2 ψ (t)ψ − (1 + α)ψ (t) =4(1 + α) ∥P 2 ut ∥ + ∥P 2 u∥ − P 2 u, P 2 ut h i 1 + 2ψ (P utt , u) − (1 + 2α)∥P 2 ut ∥2 ′′

′

2

⩾2ψy,

(2.6.34)

where 1

y(t) = (P utt , u) − (1 + 2α)∥P 2 ut ∥2 . Apply (2.6.27)∼(2.6.29) to estimate the lower bound of y(t): y(t) = − ∥A 2 u∥2 + (P − 2 Bu, P 2 u) − a(P ut , u) + (F (t, u), u) − (1 + 2α)∥P 2 ut ∥2   1 1 1 1 ⩾2(1 + 2α) − ∥A 2 u∥2 − ∥P 2 ut ∥2 + G(t, u) 2 2 1 1 1 1 a d (2.6.35) + 2α∥A 2 u∥2 − M1 ∥A 2 u∥∥P 2 u∥ − ∥P 2 u∥2 , 2 dt 1

1

and let

1

1

1 1 1 1 j(t) = − ∥A 2 u∥2 − ∥P 2 ut ∥2 + G(t, u), 2 2

we get estimate by (2.6.35): (∀ε1 > 0) 1

1

y(t) ⩾ 2(1 + 2α)j(t) + 2α∥A 2 u∥2 − ε1 ∥A 2 u∥2 −

1 1 M12 a d ∥P 2 u∥2 − ∥P 2 u∥2 . 4ε1 2 dt

Take ε1 = 2α, then y(t) ⩾2(1 + 2α)j(t) −

1 1 M12 a d ∥P 2 u∥2 − ∥P 2 u∥2 . γα 2 dt

(2.6.36)

2.6

The “blow up” problem for the solutions of some semi-linear parabolic hyperbolic· · · 87

On the other hand, multiplying (2.6.27) with ut , and applying (2.6.5), one has   1 1 1 1 1 1 2 d 1 2 2 2 − ∥P u∥ − ∥A u∥ + G(t, u) = − (P − 2 Bu, P 2 ut ) + a∥P 2 ut ∥2 + G(t, u) dt 2 2 1

1

⩾ − M1 ∥A 2 u∥∥P 2 ut ∥ + Gt (t, u) 1 1 M1 M1 ⩾− ∥A 2 u∥2 − ∥P 2 ut ∥2 + Gt (t, u). 2 2 (2.6.37) By (2.6.37) and (2.6.30), we obtain d j(t) ⩾ M1 j(t) + Gt (t, u) − M1 G(t, u) ⩾ M1 j(t). dt

(2.6.38)

Since j(0) > 0, then j(t) > 0. So, (2.6.34) and (2.6.36) lead to y(t) ⩾ −

α M12 ψ(t) − ψ ′ (t) 8α 2

and ψ ′′ ψ − (1 + α)ψ ′ (t)2 ⩾ −

M1 2 α ′ ψ − ψ ψ. 8α 2

Noting assumption (2.6.33), we find that ψ(t) satisfies all the conditions in Lemma 2.6.1, which finishes the proof of Theorem 2.6.4. Remark 2.6.5 Conditions (2.6.29) and (2.6.30) don’t need (F (t, u), u) and G(t, u) to be positive, and constant M1 in condition (2.6.30) can be arbitrarily large. Theorem 2.6.4 can be extended to a more general case: the coefficient a of operators A, B and P ut can depend on t, but A(t) and a(t) should satisfy the following conditions |(At u, u)| ⩽ M1∗ ∥A 2 u∥2 , 1

0 ⩽ a(t) ⩽ a.

(2.6.39)

If (2.6.37) is changed into 1 1 1 dj(t) ⩾ − M1 ∥A 2 u∥∥P 2 ut ∥ − M1∗ ∥A 2 u∥2 + G(t, u) dt   2 1 1 M1 ⩾ − ε1 ∥P 2 ut ∥2 − + M1∗ ∥A 2 u∥2 + Gt (t, u), 4ε2

take ε2 =

M12 + M1∗ , that is 4ε2 ε2 =

M1∗ +

p M1∗ + M12 , 2

then, the inequality (2.6.38) holds. Where M1 is replaced by M1′ = M1∗ + and M1 in (2.6.30) is replaced by M1′ , then Theorem 2.6.4 holds.

(2.6.40)

p M1∗ + M12

88 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Theorem 2.6.6 Assume that all the conditions in Theorem 2.6.4 are satisfied, M1 in (2.6.30) is replaced by M1′ . The coefficients of operators A, B and P ut depend on t, At (t) and a(t) satisfy (2.6.39), then the results of Theorem 2.6.4 holds. Now, we study the following hyperbolic equation P utt =

n n X
X ∂ aij (x, t)uxj + ai (x, t)uxi + ai (x, t)u + F (t, u), ∂xi i=1 i,j=1

(2.6.41)

where P is any positive operator, the nonlinear term F (t, u) satisfies conditions (2.6.29), (2.6.30). Now, we prove (2.6.30) can be relaxed to Gt (t, u) ⩾ 0.

(2.6.42)

Let v(t) = u(t)e−λt , then v(t) satisfies e + Bv e + Cv e t + Fe(t, v), P vtt = Av

(2.6.43)

e = A + λ2 P , B e = B, C e = −e where A aP , e a = a + 2λ, Fe = e−λt F (t, eλt v). For P , f(t, v), its potential G(t, e B, e (2.6.28) is satisfied. For the nonlinear term F e v) = A, −2λt λt e G(t, e v) satisfies (2.6.39). As shown before,   f(t, v), v . e t (t, v) = e−2λt Gt (t, eλt v) − 2λG(t, e v) + λ F G If G satisfies conditions (2.6.29) and (2.6.42), then e t (t, v) ⩾ 4αλG(t, e v). G   M1 M′ Choose λ = in Theorem 2.6.6, λ ⩾ 1 , then (2.6.30) is satisfied. To make 4α 4α the solution v(t) of (2.6.43) to satisfy conditions in Theorem 2.6.4, v(0) and vt (0) are needed to satisfy (2.6.31) and (2.6.32). Since v(0) = u(0) and vt (0) = ut (0) − λu(0), so it needs that u(0) and ut (0) satisfy 1

2 (P (0), ut (0)) > (2λ + M2 )∥P 2 u(0)∥2 ,

(2.6.44)

1 1 1 1 1 − ∥A 2 u(0)∥2 − ∥P 2 ut (0) − λP 2 u(0)∥2 + G(0, u(0)) ⩾ 0, (2.6.45) 2 2 √ α where M2 = M1 2. Condition (2.6.45) can be replaced by a simpler condition: 4 1 1 1 1 1 λ − ∥A 2 u(0)∥2 − ∥P 2 ut (0)∥2 + (λ + M2 )∥P 2 u(0)∥2 + G(0, u(0)) ⩾ 0. (2.6.46) 2 2 2

Then we can get the following theorem

2.6

The “blow up” problem for the solutions of some semi-linear parabolic hyperbolic· · · 89

Theorem 2.6.7 Besides (2.6.30), we assume that all the conditions in Theorems M1 2.6.4 or 2.6.6 are satisfied, and (2.6.42), (2.6.44), (2.6.45) or (2.6.46) hold, λ > 4α   M′ in Theorem 2.6.6, λ > 1 , then 4α 1

∥P 2 u(t)∥ → ∞,

t → t1 ⩽ t2 ,

(2.6.47)

√ √ 2 M1 2ψ(0) + 4αψ ′ (0) √ t2 = ln . M1 −M1 2ψ(0) + 4αψ ′ (0) Now we give some examples. Example 2.6.1 Consider the following generalized Boussinesq equation utt = uxx − C1 uxxxx + C2 (um )xx

(C1 ⩾ 0).

(2.6.48)

When m = 2k + 1 and k ⩾ 1, C2 < 0; when m = 2k and k ⩾ 1, C2 ⩾ 0 (if m is an integer, then um can be replaced by |u|m−1 u, m > 1, C2 is negative). Study the initial-boundary value problem ut |t=0 = u1 (x),

u|t=0 = u0 (x),

0 ⩽ x ⩽ 1,

(2.6.49)

u|x=0 = u|x=1 = 0,

(2.6.50)

uxx |x=0 = uxx |x=1 = 0.

(2.6.51)

∂2 satisfies (2.6.50), which is a positive definite ∂x2 operator. Apply P to be its inverse operator, that is P (vxx ) = u, P (vxxxx ) = −uxx . Where v satisfies (2.6.50), u satisfies (2.6.51), the problem (2.6.48), (2.6.50), (2.6.51) can be written into the following form

We have known that the operator −

P utt = −u + C1 uxx − C2 um ,

(2.6.52)

here Theorem 2.6.6 can be directly used to equation (2.6.52). Where A = −C1

∂2 + I, ∂x2

B = 0,

a = 0,

F (t, u) = −C2 um .

∂2 is a positive one, nonnegative definite, so, operator P is non∂x2 negative. If C1 < 0, C2 = 0, then the problem (2.6.48)∼(2.6.51) don’t have smooth solution.

Since operator −

Example 2.6.2 The equations utt = ±(um x )x

(2.6.53)

90 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

have infinite conservation laws for any m, Z k [utt ∓ (um x )x ] ut dx = 0, and

k = 1, 2, · · ·

Z n [utt ∓ (um x )x ] ux dx = 0.

Three of them are  Z  1 2 1 I1 = ut ± um+1 dx, 2 m+1 x  Z  2 1 3 ut ± ut um+1 dx, I2 = x 3 m+1  Z  1 4 1 6 2m+3 I3 = ut ± u2s um+1 + u dx. x 4 m+1 2(2m + 1)(m + 1)(2m + 3) x Equation (2.6.52) has the following form ! u v

= t

1 2

! 0 1 −1 0



 δH  δu     δH  , δv

Z  H(u, v) = −

 2 m+1 2 u + v dx, m+1 x

where v = ut . For equation (2.6.53), when m ⩾ 2 is an even number, for arbitrary symbol, or when m > 2 is an odd number, for negative symbol, whether Cauchy problem or bounary value problem, the right-hand side will blow up. Actually, at present, the equation has the following form P utt = −Au − Bu − αP ut + Fˆ (t, u),

(2.6.54)

∂ ∂ where P = I, A = 0, B = 0, Fˆ (t,u) = D ∗ F (Du), D = , D ∗ = − , F (t,v) = ∂x ∂x ∓v m . For (2.6.54), when Fˆ (t,u) = D ∗ F (t,Du),

(2.6.55)

where D ∗ is the conjugate operator of D, D is an arbitrary linear operator, F (t,v) satisfies conditions in Theorem 2.6.7, then the result of Theorem 2.6.7 still holds. The following Zabusky equation
utt = −uxxxx − uxx ∓ u2x x (2.6.56)

2.6

The “blow up” problem for the solutions of some semi-linear parabolic hyperbolic· · · 91

satisfies all the conditions in Theorem 2.6.7 and (2.6.45), therefore, still satisfies the result of Theorem 2.6.7, where P = I,

A=

∂4 + I, ∂x4

B=−

∂2 + I, ∂x2

D=

∂ , ∂x

D∗ = −

∂ . ∂x

For equation utt = −uxxxx − uxx − u + u3 ,

(2.6.57)

Theorem 2.6.7 holds, where P = I,

A=

∂4 + I, ∂x4

B=−

∂2 , ∂x2

F (t, u) = u3 .

Example 2.6.3 FPU problem: 2 2 ξ¨n = ξn+1 − 2ξn + ξn−1 + (ξn+1 − ξn ) − (ξn − ξn−1 ) ,

(2.6.58)

which can be attributed to the following equation: utt = uxx + u2x


x

(2.6.59)

,

which has the form of (2.6.54), where ∂2 ∂ , B = 0, D = , ∂x2 ∂x Fˆ = D ∗ F (Du) , F (t, v) = −v 2 . A=−

For equation (2.6.59), by equality Z
[utt − uxx − u2x x ]ukt dx = 0,

k = 1, 2, · · · ,

one can get infinite conservation laws. Example 2.6.4 Toda Lattice equation −1 ξ¨n = eξn+1 −ξn − eξn ξn−1 ,

(2.6.60)

which can be attributed to the following equation: utt = (eux )x ,

(2.6.61)

∂ its nonlinear term Fˆ = D ∗ F (Du) , D = , F (v) = −ev . The function F (v) ∂x does not satisfy conditions in above theorem, so, the above theorem can not be applied to it. But one can use other method to study its “blow up” phenomenon.

92 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Z 

 1 2 ux I0 = u +e dx which is a conservation integral has been known, others 2 t can be obtained by the following equality Z [uxx − (eux )x ]ekux dx = 0, k = 1, 2, · · · . For example  Z  1 ut eux + u3t dx, 6  Z  1 1 I2 = ut e2ux + u3t eux + u5t dx. 3 60 I1 =

In order to prove the blow up phenomenon of the solution to (2.6.61), rewrite (2.6.61) into utt =

1 2ux
e = e2ux uxx . x 2

(2.6.62)

Let ux = p, ut = q, then (2.6.62) can be written into qt = e2p · px ,

p t = qx .

Introduce new transformations of variables α = α(x, t), β = β(x, t), such that αt = ep αx ,

βt = −ep βx .

then (2.6.62) can be written into the following equations ep pα = qα ,

ep pβ = −qβ .

So, function w = ep and q satisfy the linear equations wα = qα ,

wβ = −qβ .

Then one get wave equations wαβ = 0,

qαβ = 0.

(2.6.63)

It’s easy to see that, for finite α, β and some initial value problems of (2.6.62), w(α, β) may be negative, so, w = ep satisfies p → −∞ at some points on the plane (x, t), that means it will blow up. We refer to [63, 86, 171] for “blow up” problem of nonlinear wave equations and [105] for that of symmetric regularized wave equations.

2.7

The smoothness of the weak solutions for Benjamin-Ono equation

2.7

93

The smoothness of the weak solutions for Benjamin-Ono equation

We consider the partial regularity of the weak solutions for Benjamin-Ono equations with the initial data being in H n (R)(n = 2, 3) with respect to the space variables. We focus on the following initial value problem of Benjamin-Ono equations ut + uux − Huxx = 0,

x ∈ R, t > 0,

(2.7.1)

u(x, 0) = u0 (x),

x ∈ R,

(2.7.2)

where H is a Hilbert operator (Hf ) (x) = p · V

1 π

Z

+∞

−∞

f (y) dy = F −1 (−isgn(ξ)fˆ(ξ)), x−y

where fˆ means Fourier transform of f , F −1 represents inverse Fourier transform. For Hilbert operator, there exist the following classical properties: (∀u, v ∈ L2 (R)) H 2 u = u, Z +∞ Z uHvdx = − −∞ Z +∞

(2.7.3) +∞

vHudx,

(2.7.4)

−∞

Z

+∞

H(u)H(v)dx = −∞

uvdx,

(2.7.5)

−∞

H(u, v) = uHv + vHu + H(Hu · Hv).

(2.7.6)

In addition, we have known that H is a bounded operator from Lp (R) to Lp (R) (p > 1) and from H s (R) to H s (R) (∀s). Let
s
s J s = I − ∂x2 2 , I s = −∂x2 2 , I = I ′ = H∂x . Let A and B be two operators, [A, B] = AB − BA be a commutator, especially, [J s , f ]g = J s (f g) − f J s g. The following lemma can be found in [26]. Lemma 2.7.1 Let A : R → R be a C ∞ function, and A′ ∈ C0∞ , then the commutator [H, A]∂x Maps L2 (R) to L2 (R), and ∥[H, A]fx ∥0 ⩽ C∥f ∥0 , where constant C only depends on A′ .

∀f ∈ L2 (R)

(2.7.7)

94 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Remark 2.7.2 By equality [H∂x , A]f = [H, A]fx + H(A′ f ), we can deduce that [H∂x , A] is bounded in L2 (R). Lemma 2.7.3 For any s ∈ R, there exists constant C > 0, such that for any ϕ ∈ S(Rn ) (Schwartz function class) and f ∈ H s−1 (Rn ), it holds that ∥[J s , ϕ]f ∥0 ⩽ C∥ϕ∥|s|+n+2 ∥f ∥s−1 .

(2.7.8)

n , 2 then the correspondence relation (f, g) → f g is a continuous bilinear form from H a (Rn ) × H b (Rn ) to H c (Rn ). Lemma 2.7.4 Let a, b, c ∈ R such that a ⩾ c, b ⩾ c, a+b ⩾ 0, and a+b−c >

Lemma 2.7.5 Let s ⩾ 1, then there exists a constant C only depending on s, such that ∥[J s , ϕ]u∥0 ⩽ C∥ϕ′ ∥s ∥u∥s−1 ,

∀ϕ ∈ S(R).

(2.7.9)

Proof Let v = [J s , ϕ]u, then vˆ represents Z +∞ s
s ˆ vˆ(x) = C (1 + x2 ) 2 − (1 + y 2 ) 2 ϕ(y)dy. −∞

There exists a constant C, such that Z +∞ s−1 ˆ − y)| |ˆ |ˆ v (x)| ⩽C (1 + x2 ) 2 |x − y| |ϕ(x u(y)|dy −∞

Z

+∞

(1 + y 2 )

+C −∞

s−1 2

ˆ − y)| |ˆ |x − y| |ϕ(x u(y)|dy.

ˆ Let h(x) = |x||ϕ(x)|, w ˆ = |ˆ u(x)|, then ∥hw∥s−1 ⩽ ∥h∥s ∥w∥s−1 . Noting ∥h∥s = ∥ϕ′ ∥s , ∥w∥s−1 = ∥u∥s−1 , such that ∥hJ s−1 w∥0 ⩽ |h|∞ ∥J s−1 w∥0 ⩽ ∥h∥s ∥w∥s−1 = ∥ϕ′ ∥s ∥u∥s−1 . If s > 1, we get ∥v∥0 ⩽ c∥ϕ′ ∥s ∥u∥s−1 . Here we have finished the proof of Lemma 2.7.5. n

Theorem 2.7.6 For any u0 ∈ H 2 (where n = 2, 3), there exists solution u to the initial value problem (2.7.1), (2.7.2), such that for any T > 0, it holds that n+1

u ∈ L∞ (R+ , H 2 (R)) ∩ L2 (0, T ; Hloc2 (R)). n

(2.7.10)

2.7

The smoothness of the weak solutions for Benjamin-Ono equation

95

Proof Let ϕ : R → R be a strictly increasing bounded smooth function, and ϕ′ (x) ∈ C0∞ (R). {u0j } represents a sequence of functions in H ∞ (R), such that n u0j → u0 , in H 2 (R). Let uj be the smooth solution to (2.7.1) with initial data u0j . From [2], We know that ∥uj (·, t)∥ n2 is uniformly bounded with respect to t, and the bound only depends on ∥u0j ∥ n2 . Therefore, ∥uj (·, t)∥ n2 is uniformly bounded n+1

with respect to j and t. The uniform boundedness of uj in L2 (0, T ; Hloc2 (R)) with respect to j will be obtained later. We have omitted the subscript j in the following proof. Firstly, we consider the case when n = 2. Differentiating (2.7.1) with respect to x, multiplying it with ϕux , and integrating it on R, we get by integrating by parts that Z Z +∞ Z +∞ 1 d +∞ 2 ′ ϕux dx + ϕ ux Huxx dx + ϕuxx Hϕxx dx 2 dt −∞ −∞ −∞ Z +∞ =− ϕux (uux )x dx. (2.7.11) −∞

Applying H∂x to (2.7.1), multiplying it with ϕHux , and integrating it on R, we get by integrating by parts that Z Z +∞ Z +∞ 1 d +∞ ϕ(Huxx )2 dx − ϕ′ Hux uxx dx − ϕuxx Huxx dx 2 dt −∞ −∞ −∞ Z +∞ =− ϕHux H(uux )x dx. (2.7.12) −∞

By summing up (2.7.11) and (2.7.12), one has Z Z +∞ Z +∞ 1 d +∞ ϕ[u2x + (Hux )2 ]dx + ϕ′ ux Huxx dx − ϕ′ Hux uxx dx 2 dt −∞ −∞ −∞ Z +∞ Z +∞ =− ϕux (uux )x dx − ϕHux H(uux )x dx −∞ Z +∞

=−2 Z

−∞ +∞

−∞ Z +∞

ϕux (uux )x dx +

ϕ′ uu2x dx −

=4

−∞ Z +∞

+ −∞

1 3

Z

+∞

−∞

Hux [H, ϕ](uux )x dx

ϕ′′′ u3 dx

−∞

Hux [H, ϕ](uux )x dx −

Z

+∞

−∞

ϕu2 uxxx dx.

(2.7.13)

By integrating by parts several times, the last term on the right-hand side of (2.7.13) equals Z +∞ Z Z +∞ Z +∞ 1 +∞ ′′′ 3 − ϕu2 uxxx dx = ϕ u dx − 3 ϕ′ uu2x dx − ϕu3x dx. (2.7.14) 2 −∞ −∞ −∞ −∞

96 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

On the other hand, Z

Z

+∞

−∞

ϕu2 uxxx dx = −

+∞

−∞

ϕ′ u2 H(uux )dx − 2

Z

+∞

−∞

ϕuux H(uux )dx

Z d +∞ 2 ϕu Hux dx dt −∞ Z Z Z +∞ 2 +∞ ′ 3 1 d +∞ 4 2 − ϕ u Hux dx − ϕu dx + ϕ′ u5 dx 3 −∞ 6 dt −∞ 15 −∞ Z +∞ Z +∞ + ϕ′ u(Hux )2 dx + ϕux (Hux )2 dx. (2.7.15) +

−∞

−∞

By the properties (2.7.3)∼(2.7.6) of Hilbert transform, it holds that Z

Z

+∞

−∞

ϕux (Hux )2 dx =

Z

+∞

−∞ Z +∞

= −∞ Z +∞

Z

−2

ϕu3x dx − 2 ϕu3x dx − 2

= −∞

u3x dx + 2

+∞

ϕux H(ux Hux )dx

−∞ Z +∞

−∞ Z +∞ −∞

H(ϕux )ux Hux dx [H, ϕ]ux ux Hux dx

+∞

−∞

ϕux (Hux )2 dx.

(2.7.16)

So, (2.7.15) turns into Z − =−

+∞

−∞ Z +∞

ϕu2 uxxx dx ϕ′ u2 H(uux )dx − 2

−∞ Z +∞

Z

+∞

−∞

ϕuux H(uux )dx +

d dt

Z

+∞

−∞

ϕu2 Hux dx

Z Z +∞ 1 d +∞ 4 2 ′ 3 ϕ u Hux dx − ϕu dx + ϕ′ u5 dx 6 dt −∞ 15 −∞ −∞ Z Z +∞ 1 +∞ 3 2 +∞ ϕ′ u(Hux )2 dx + ϕux dx − [H, ϕ]ux · ux Hux dx. (2.7.17) 3 −∞ 3 −∞ −∞

2 − 3 Z + Z

+∞

−∞

ϕu3x dx can be eliminated by (2.7.14) and (2.7.17), and the expression of

2.7

The smoothness of the weak solutions for Benjamin-Ono equation

Z −

97

+∞

−∞

ϕu2 uxxx dx can be obtained. Then we get by (2.7.13) that 1 d 2 dt Z −

Z

ϕ −∞ +∞

− +

1 2 1 4 Z

u2x

 Z +∞ 3 2 1 4 + (Hux ) − u Hux + u dx + ϕ′ ux Huxx dx 2 4 −∞ 2

ϕ′ Hux uxx dx

−∞ Z +∞

1 =− 4



+∞

−∞ Z +∞ −∞ Z +∞ −∞ +∞

+ −∞

1 ϕ u dx + 10 ′′′ 3

Z

+∞

−∞

3 ϕ u dx − 4 ′ 5

Z

+∞

−∞

ϕ′ u2 H(uux )dx

ϕ′ u3 Hux dx ϕ′ uu3x dx +

3 4

Z

+∞

−∞

ϕ′ u(Hux )2 dx −

1 Hux [H, ϕ](uux )x dx − 2

Z

3 2

Z

+∞

−∞

ϕuux H(uux )dx

+∞

−∞

[H, ϕ]ux · ux Hux dx.

(2.7.18)

By Lemma 2.7.1, one has Z +∞ Hux [H, ϕ](uux )x dx ⩽ C∥ux ∥0 ∥uux ∥0 ⩽ C (∥u0 ∥1 ) . −∞

By Sobolev inequality 1

1

∥g∥∞ ⩽ ∥g∥02 ∥gx ∥02 ,

(2.7.19)

and apply Lemma 2.7.1 again, we achieve at Z 1 +∞ 1 − [H, ϕ]ux · ux Hux dx ⩽ |[H, ϕ]ux |∞ ∥ux ∥20 2 −∞ 2 1 1 1 ⩽ ∥[H, ϕ]ux ∥02 ∥∂x [H, ϕ]ux ∥02 ∥ux ∥20 2 5 1 C ⩽ ∥u0 ∥ 2 ∥ux ∥02 2 ⩽C(∥u0 ∥1 ). Therefore, the right-hand side of (2.7.18) is controlled by C(∥u0 ∥1 ). The last two terms on the left side of (2.7.18) can be rewritten into Z

+∞

−∞

ϕ′ ux Iuxx dx +

Z

+∞

−∞

ϕ′ Hux IHux dx,

(2.7.20)

where I = I ′ = H∂x . For (2.7.20), we only need to estimate one of them, and similar processing can be made to the other one. The first term of (2.7.20) can be rewritten

98 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

into Z

+∞

−∞

Z

′

ϕ ux Iux dx =

+∞

Z

′

ϕ ux Jux dx +

−∞ +∞

Z

ϕ

=

′



Z

+∞

−∞

2

J ux

−∞

+

1 2

+∞

−∞

Z

ϕ′ ux (I − J)ux dxx +∞

dx +

ux [J, ϕ′ ]ux dx −

ϕ′ ux (I − J)ux dx

−∞ Z +∞ −∞

ux [J 2 , ϕ′ ]J 2 ux dx. 1

1

(2.7.21)

Since f (x) = |x| − (1 + x2 ) 2 ∈ L∞ (R), the second term on the right-hand side of (2.7.21) can be controlled by C∥ux ∥20 ⩽ C(∥u0 ∥1 ). According to Lemma 2.7.3, the last two terms on the right-hand side of (2.7.21) are also bounded. So, from (2.7.18), we deduce that   Z 3 1 1 d +∞ ϕ u2x + (Hux )2 − u2 Hux + u4 dx 2 dt −∞ 2 4  Z +∞     2 2 1 1 + ϕ′ J 2 ux + J 2 Hux dx 1

−∞

⩽C(∥u0 ∥1 ).

(2.7.22)

Integrating (2.7.22) with respect to t, t ∈ [0, t], 0 < t ⩽ T , there holds Z tZ

+∞

ϕ′



−∞

0

1 ⩽CT − 2

Z

2

1

J 2 ux

+∞

−∞

 1 2  + J 2 Hux dxdt

  t 3 2 1 4 2 2 ϕ ux + (Hux ) − u Hux + u dx 0 . 2 4

Since ϕ and ∥u∥1 are uniformly bounded with respect to t and j, it deduces that: (j is large enough) Z tZ 0

+∞

ϕ′



−∞

1

J 2 ux

2

 1 2  + J 2 Hux dxdt ⩽ C,

where constant C only depends on T , ϕ and ∥u0 ∥1 . By choosing a suitable ϕ, we have Z TZ R  2 1 J 2 ux dxdt ⩽ C, 0

−R

where constant C only depends on T , R and ∥u0 ∥1 . Therefore, Theorem 2.7.6 holds when n = 2. 3 Now we consider the case when n = 3. Applying J 2 to (2.7.1), multiplying it

2.7

The smoothness of the weak solutions for Benjamin-Ono equation

99

3

with ϕJ 2 u, and integrating it on R, then we obtain by integrating by parts that 1 d 2 dt Z =−

Z

+∞

Z  3 2 ϕ J 2 u dx +

−∞ +∞

+∞

−∞

ϕ′ J 2 uJ 2 Hux dx + 3

3

Z

+∞

3

3

ϕJ 2 ux J 2 Hux dx

−∞

3

ϕJ 3 uJ 2 (uux )dx.

−∞

(2.7.23)

3

3

Applying J 2 H to (2.7.1), multiplying it with ϕJ 2 Hu, and integrating it on R, then we obtain by integrating by parts that 1 d 2 dt Z =−

Z

Z  3 2 ϕ J 2 Hu dx −

+∞

+∞

ϕ′ J 2 HuJ 2 ux dx − 3

−∞

−∞ +∞

3 2

3

Z

+∞

3

3

ϕJ 2 Hux J 2 ux dx

−∞

3 2

(2.7.24)

ϕJ HuJ H(uux )dx.

−∞

Summing up (2.7.23) with (2.7.24), one has 1 d 2 dt Z −

Z

ϕ −∞ +∞

−∞ Z +∞

=−



+∞

Z 2  3 2  2 dx + J u + J Hu 3 2

−∞

−∞ Z +∞ −∞ +∞

−∞ Z +∞

−2

3

3

ϕ′ J HuJ ux dx Z

3

3

3 2

3 2

+∞

−∞

3

3

ϕJ 2 HuJ 2 H(uux )dx

−∞ Z +∞

ϕJ uJ (uux )dx +

Z −2

ϕ′ J 2 uJ 2 Hux dx

3 2

3 2

ϕJ 2 uJ 2 (uux )dx −

=−2

+∞

−∞

3

3

J 2 Hu[H, ϕ]J 2 (uux )dx

ϕuux Judx Z ϕux (I − J)uux dx + 2

+∞

−∞

(2.7.25)

ϕuux Huxx dx.

The last term on the right-hand side of (2.7.25) can be rewritten into Z

Z

+∞

2 −∞

ϕuux Huxxx dx = − 2

+∞

−∞ Z +∞

+2 −∞ Z +∞

+2 −∞

ϕ′ u2 u2x dx

d +2 dt

Z

ϕuu2x dx

−∞ Z +∞

ϕ′ uux Huxx dx + 2 ϕuuxx Huxx dx,

+∞

−∞

ϕuu3x dx (2.7.26)

100 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

and the last term on the right-hand side of (2.7.26) satisfies Z +∞ Z +∞ Z +∞ 2 ϕuuxx Huxx dx = − 2 ϕ′ uHux uxx dx − 2 ϕ′ uHuxx H(uux )dx −∞

+

1 3 Z

−∞ +∞

Z

Z

ϕ′ (Hux )3 dx − 2

−∞ +∞

−∞ +∞

−∞

ϕux Hux H(uux )dx

Z d +∞ ϕu(Hux )2 dx + ϕuux (Hux ) dx + dt −∞ −∞ Z +∞ Z +∞ + ϕu2x Hux dx + ϕu2x Huxx dx 2

−∞ Z +∞

−2 where Z +∞ −∞

Z ϕu2x Huxx dx = − 2 d − dt

ϕ′ uux Huxx dx +

−∞ Z +∞ −∞

(2.7.27)

ϕuHuxx H(uux )dx,

−∞

+∞

−∞

Z ϕuu2x dx

−

Z

+∞

−∞

ϕ′ u2 u2x dx Z

+∞

−∞

ϕuu3x dx

−2

+∞

−∞

ϕuuxx Huxx dx.

Z +∞ Substituting the above equality into (2.7.27), the expression of 2 ϕuuxx Huxx dx −∞ Z +∞ can be obtained, so does that of 2 ϕuux dxHuxxx . Then (2.7.25) turns into Z

1 d 2 dt Z +



+∞

ϕ −∞ +∞

ϕ′ J 2 uJ 2 Hux dx − 3

−∞ Z +∞

3 =− 2 + +

−∞

 2  3 2 3 2 2 J 2 u + J 2 Hu − 3uux − u (Hux ) dx

1 2 1 6 Z

−∞ Z +∞ −∞ Z +∞ −∞ +∞

3

ϕ′ u2 u2x dx

Z −2

ϕ′ (Hux )3 dx −

−∞ Z +∞

−2 Z

−∞ +∞

+ −∞

+∞

−∞ +∞

−∞

ϕuux (Hux )2 dx +

ϕ′ uux Huxx dx −

+

Z

Z

1 2

3

3

3

Z

ϕuux Judx − Z

+∞

−∞

+∞

−∞

ϕ′ uHux H(uux )dx

ϕ′ u2x Hux dx +

3 2

Z

+∞

−∞

ϕuu3x dx

+∞

−∞ Z +∞ −∞

ϕuux (I − J)uxx dx − 3

ϕ′ J 2 HuJ 2 ux dx

ϕux Hux H(uux )dx ϕ′ uHux uxx dx Z

+∞

−∞

J 2 Hu[H, ϕ]J 2 (uux )dx + 2

ϕuHuxx H(uux )dx Z

+∞

−∞

3

3

J 2 u[J 2 , ϕ]uux dx.

(2.7.28)

2.7

The smoothness of the weak solutions for Benjamin-Ono equation

101

The first to the third  terms  on the right-hand side of (2.7.28) are obviously controlled by C∥ux ∥02 ⩽ C ∥u0 ∥ 23 . The fifth term on the right-hand side of (2.7.28) is also bounded,   C|ux |3L3 ⩽ C∥ux ∥31 ⩽ C∥u∥31 ⩽ C ∥u0 ∥ 32 , 6

6

where we have used H n/2(n+2) (R) → Ln+2 (R). Additionally, we have Z +∞ Z +∞   ϕ′ uux Huxx dx − ϕ′ uHux uxx dx ⩽ C∥u∥ 32 ∥ϕ′ uux ∥ 21 + ∥ϕ′ uHux ∥ 21 . −∞

−∞

By Lemma 2.7.4, these two terms can be controlled by     C∥u∥ 32 ∥ϕ′ u∥1 ∥ux ∥ 21 + ∥ϕ′ u∥1 ∥Hux ∥ 12 ⩽ C∥u∥23 ⩽ C ∥u0 ∥ 32 , 2

and by Lemma 2.7.4 again, we obtain Z +∞ Z +∞ −2 ϕuux (I − J)uxx dx − ϕuH(uux )Huxx dx −∞ −∞ h i ⩽C∥u∥ 32 ∥ϕuux ∥ 12 + ∥ϕuH(uux )∥ 12 h i ⩽C∥u∥ 32 ∥ϕu∥1 ∥ux ∥ 12 + ∥ϕu∥1 ∥H(uux )∥ 12 , then

Z ∥ϕu∥21 =

Z

+∞

+∞

ϕ2 u2 dx + −∞

−∞

(ϕ′ u + ϕux ) dx ⩽ C∥u∥21 . 2

So, Z

Z

+∞

−2

ϕuux (I − J)uxx dx − −∞   ⩽C∥u∥23 ⩽ C ∥u0 ∥ 32 .

+∞

−∞

ϕuH(uux )Huxx dx

2

By Lemma 2.7.1, one has Z +∞ 3 3 3 J 2 Hu[H, ϕ]J 2 (uux )dx ⩽C∥u∥ 32 ∥[H, ϕ]J 2 (uux )∥0 −∞

  ⩽C∥u∥ 23 ∥u2 ∥ 23 ⩽ C∥u∥23 ⩽ C ∥u0 ∥ 32 . 2

By Lemma 2.7.5, it holds that Z +∞ 3 3 3 3 2 J 2 u[J 2 , ϕ]uux dx ⩽C∥J 2 u∥0 ∥[J 2 , ϕ]uux ∥0 −∞

  1 ⩽C∥u∥ 32 ∥ϕ′ ∥ 32 ∥J 2 (uux )∥0 ⩽ C∥u∥23 ⩽ C ∥u0 ∥ 23 . 2

102 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

Then, it deduces from (2.7.28) that   Z 2  3 2 3 1 d +∞ ϕ J 2 u + J 2 Hu − 3uu2x − u(Hux )2 dx 2 dt −∞ Z +∞ Z +∞   3 3 3 ′ 32 2 ϕ J uJ Hux dx − ϕ′ J 2 HuJ 2 ux dx ⩽ C ∥u0 ∥ 23 . + −∞

(2.7.29)

−∞

The last two terms on the left side of (2.7.29) can be estimated similarly to the case when n = 2:   Z 2  3 2 3 1 d +∞ ϕ J 2 u + J 2 Hu − 3uu2x − u(Hux )2 dx 2 dt −∞ Z +∞     + ϕ′ (J 2 u)2 + (J 2 Hu)2 dx ⩽ C ∥u0 ∥ 32 , −∞

since ϕ and ∥u∥ 23 are uniformly bounded with respect to t, by integrating the above inequality with t, one has: (t ∈ [0, t], 0 ⩽ t ⩽ T ) Z T Z +∞ h
2
2 i ϕ′ J 2 u + J 2 Hu dxdt ⩽ C, 0

−∞

where C only depends on T , ϕ and ∥u0 ∥ 23 (j is large enough). Then, it deduces by choosing a suitable ϕ that Z TZ R
2 J 2 u dxdt ⩽ C, 0

−R

here, we finish the proof of Theorem 2.7.6. Now we consider the initial value problem of long wave equation with medium depth: 1 ut + uux + ux − T uxx = 0, δ where 1 T f (x) = − 2δ

Z

+∞

−∞

u(x, 0) = u0 (x),

 π(x − y) coth u(y)dy. 2s

(2.7.30)



(2.7.31)

(2.7.30) is summed up as KdV equation, when it describes the movement of shallow water wave (δ → 0), and it is summed up as Benjamin-Ono equation, when it describes the movement of deep water (δ → ∞). Obviously, there exists the following lemma: Lemma 2.7.7 For any δ > 0, ∀ξ ∈ R, it holds that 1 1 − + 2π|ξ| ⩽ 2πξ coth(2πδξ) ⩽ + 2π|ξ|. δ δ

2.7

The smoothness of the weak solutions for Benjamin-Ono equation

103

Remark 2.7.8 From Lemma 2.7.7, the difference between the operator H∂x2 and 1 the operator T ∂x2 + ∂x is a zero order quasi differential operator, which is bounded δ in H s (R). 3 and 0 < T < ∞. If u ∈ C(0, T ; H s (R)) 2 is the solution of problem (2.7.30) with initial data u0 (x) ∈ H s (R), then Theorem 2.7.9 (i) Assume that s >

s+

u ∈ L (0, T ; Hloc 2

1 2 (R)).

(2.7.32)

n

(ii) Assume that u0 ∈ H 2 , n = 2 or 3, then there exists a solution to problem (2.7.30) u ∈ L∞ (0, T ; H 2 (R)) ∩ L2 (0, T ; Hloc n

(n+1)/2

(R)).

(2.7.33)

Proof Equation (2.7.30) can be rewritten into ut + uux − Huxx = Lu,

(2.7.34)

1 L = T ∂x2 + ∂x − H∂x2 . δ

(2.7.35)

where

By Remark 2.7.8, we know that L is a bounded operator on H s (R). (i) Let ϕ be the one given in Theorem 2.7.6, it holds that Z Z +∞  1 d +∞  s 2 ϕ (J u) + (J s Hu)2 dx + ϕ′ J s uJ s Hux dx 2 dt −∞ −∞ Z +∞ − ϕ′ J s HuJ s ux dx −∞ Z +∞

=−2

Z s

Z

−∞ +∞

+∞

s

Z

−∞ +∞

ϕJ s uJ s Ludx +

+

−∞ Z +∞

=−2 Z

−∞ +∞

−∞

ϕJ s HuJ s HLudx −∞

ϕJ s u[J s , u]ux dx +

Z

+∞

ϕ′ u(J s u)2 dx +

−∞

Z

s

+

J s Hu[H, ϕ]J s (uux )dx

ϕJ uJ (uux )dx +

+∞

−∞ +∞

s

J Hu[H, ϕ]J (uux )dx +

Z

s

ϕux (J s u)2 dx Z

+∞

s

ϕJ s Hu J s HLudx.

ϕJ u J Ludx + −∞

−∞

(2.7.36) By Lemma 2.7.1, Lemma 2.7.3 and the boundedness of L, we know that the terms on the right-hand side of (2.7.36) are controlled by C∥u0 ∥2s .

104 Chapter 2

The Properties of the Solutions for Some Nonlinear Evolution Equations

(ii) Similar to the proof of Theorem 2.7.6, compared to (2.7.18) and (2.7.28), the terms including the operator L are added, but these terms can be controlled by ∥u0 ∥ n2 (n = 2 or 3), so Theorem 2.7.9 holds. Smith equation can be written into ut + uux + Ls ux = 0,

(2.7.37)

where Ls u ˆ(ξ) = Ps (ξ)ˆ u(ξ),

Ps (ξ) = 2π

 p 1 + ξ2 − 1 .

Similar to (2.7.34), (2.7.37) can be rewritten into ut + uux − Huxx = Ks u.

(2.7.38)

π is bounded, as |ξ| → ∞, therefore, 2π|ξ| − Ps (ξ) is bounded. |ξ| Ks is a zero order quasi differential operator, which is a bounded operator in H s (R). So, we can use the method of long wave equation with medium depth to deal with the local smoothness of weak solution to the initial value problem of Smith equation. We refer to [233] for smoothing properties of some weak solutions to BenjaminOno equation. As for some other results, see [119,153,252] and the references therein. Noting that 2π +

Chapter 3 Some Results for the Studies of Some Nonlinear Evolution Equations In recent years, a systematic and deep research has been done to some nonlinear evolution equations, such as: nonlinear wave equation, nonlinear Schrödinger equation, KdV equation etc., which includes existence, uniqueness, regularity, scattering of the global solution, and blow up problem of the solution. A series of good results have been obtained. Limited to space, in this chapter, we focus on some interesting and classical results. The contents of this chapter may be referred to [106], [224] as well as their references.

3.1

Nonlinear wave equations and nonlinear Schrödinger equations

Consider the following nonlinear wave equation utt − ∆u + f (u) = 0,

(3.1.1)

and nonlinear Schrödinger equations iut − ∆u + f (u) = 0.

(3.1.2)

Here, for NLW equation (3.1.1), let f (s) : R → R be a real-valued function, f (0) = 0, F ′ (s) = f (s), F (0) = 0. For NLS equation (3.1.2), let f (s) : C → C be a complexvalued function, f (0) = 0, f (u) = g(|u|2 )u, where g(s) is a real-valued function, F (u) = G(|u|2 )/2, G′ (s) = g(s), s ∈ R, G(0) = 0. It’s easy to get that the energy E(u(t)) of NLW equation (3.1.1) satisfies  Z  1 2 1 2 E(u(t)) = u + |∇u| + F (u) dx, (3.1.3) 2 t 2 and that of NLS equation (3.1.2) satisfies  Z  1 E(u(t)) = |∇u|2 + F (u) dx. 2

(3.1.4)

106

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

Let us define two Hilbert spaces. Firstly, the energy spaces are X = H 1 × L2 ,

for NLW (3.1.1),

1

for NLS (3.1.2).

X=H , Secondly, we define X2 = H 2 × H 1 , 2

for NLW (3.1.1), for NLS (3.1.2).

X2 = H ,

We use the following notations: |u|p represents the norm of Lp (Rn ) (1 ⩽ p ⩽ ∞), the norm of the Sobolev space W k,p (Rn ) is X Z |u|pk,p = |∂ α u(x)|p dx, |α|⩽k

and H k (Rn ) = W k,2 (Rn ). Z ∥u∥rr,p

+∞

Z

r/p |u(x, t)| dx dt p

= −∞

means the norm of the space Lr (Lp ) = Lr (R; Lp (Rn )). If X is a Banach space, then C(X) is the set of functions which is strongly continuous with respect to t valued in X. We have Theorem 3.1.1 (weak solution) equation (3.1.2), if

Considering NLW equation (3.1.1) and NLS

F (u) ⩾ −C|u|2

(3.1.5)

for some constant C, and |F (u)/f (u)| → ∞,

|u| → ∞,

(3.1.6)

then for any initial energy E(u(0)) < ∞, u(0) ∈ L2 , there exists a weakly continuous solution u : R → X (NLW or NLS), such that E(u(t)) ⩽ E(u(0)),

∀t ∈ R.

(3.1.7)

The uniqueness of the weak solution remains to be solved. (3.1.5) is not a good condition, while condition (3.1.6) requires the growth of F (u) with respect to u is slower than the index, as |u| → ∞. For NLS equation, we have |u(t)|2 ⩽ |u(0)|2 . When condition (3.1.6) is slightly strengthened, we can get that the equality holds.

3.1

Nonlinear wave equations and nonlinear Schrödinger equation

107

Theorem 3.1.1 can also make other assumptions. For example, for NLS equation, we replace condition (3.1.5) with the following weaker condition F (u) ⩾ −C|u|2 − C|u|q+1 ,

q 0,

(3.1.10)

where 1 2, ∥∇N u∥r,p+1 ⩽ CT ∥u∥p−1 ∞,p+1 ∥∇u∥r,p+1 . By H 1 ⊂ Lp+1 , one has ∥N u∥F ⩽ CT ∥u∥p−1 E ∥u∥F .

3.1

Nonlinear wave equations and nonlinear Schrödinger equation

111

On the other hand, from (3.1.17), we obtain Z

!1/r′

T

′

⩽ CT ∥u∥p−1 ∞,p+1 ∥u∥r ′ ,p+1 ,

|u(t)|pr p+1 dt

∥N u∥∞,2 ⩽ CT ∥f (u)∥r′ ,(p+1)′ ⩽ CT 0

in which r′ can be changed into a larger one r. Similarly, the differential operator satisfies ∥∇N u∥∞,2 ⩽ CT ∥u∥p−1 ∞,p+1 ∥∇u∥r,p+1 , so, ∥N u∥E ⩽ CT ∥u∥p−1 E ∥u∥F . Let G = C(L2 ) ∩ Lr (Lp+1 ), then N u − N v satisfies the following estimate p−1

∥N u − N v∥G ⩽ CT (∥u∥E∩F + ∥v∥E∩F )

∥u − v∥G ,

(3.1.18)

here, G → 0, as T → 0. By the interpolation between ∥u∥2+ n4 ,2+ n4 ⩽ C∥φ∥2

and ∥u∥2 = ∥φ∥2 ,

we obtain ∥u∥r,p+1 ⩽ C|φ|2 , where

  1 1 2 =n − r 2 p+1

1⩽p⩽1+

or

r=

4 , n−2

(3.1.19)

4(p + 1) . n(p − 1)

By (3.1.19), we can get ∥u0 ∥E∩F ⩽ C|φ|1,2 . Let R = 2C|φ|1,2 , then for T being small enough, the map u → u0 + N u is compressible on the complete metric space {u ∈ G : ∥u∥E∩F ⩽ R}, therefore this map has a unique fixed point, and time T depends on the H 1 norm of φ. Since the H 1 norm of u(t) is bounded, we can repeat the above process, with the initial time T, 2T, · · · , till to (0, ∞). The proof of theorem 3.1.1 Let NLS equation satisfy conditions (3.1.6) and (3.1.8). For u being large enough, we take the method of cutting off the nonlinear term f (u). Assume that for u being bounded, fj (u) → f (u) uniformly, and fj (u) satisfies (3.1.10) and (3.1.11), and the conditions (3.1.6) and (3.1.8) hold uniformly with respect to fj . For fixed j, applying Theorem 3.1.2, there exists solution uj ∈ C(X) of the equation i∂t uj − ∆uj + fj (uj ) = 0,

(3.1.20)

112

Chapter 3

satisfying Z 

Some Results for the Studies of Some Nonlinear Evolution Equations

  Z  1 1 |∇uj |2 + Fj (uj ) dx = |∇u0 |2 + Fj (u0 ) dx ⩽ const, 2 2

which is independent of t and j. Z

(3.1.21)

Z |uj |2 dx =

|u0 |2 dx.

As before, by the assumption (3.1.19), one deduces that the sequence uj is uniformly bounded in H 1 with respect to t and j. So, by weak compactness, there exists a subsequence (still denote uj ) such that uj → u,

weakly ∗ in L∞ (H 1 ),

which proves the weak convergence of the linear part of (3.1.20). The difficulty is to prove the nonlinear term fj (uj ) converges to f (u) in some sense. We’ll prove that it converges for almost everywhere. Because H 1 (B) compactly embeds into L2 (B), where B ⊂ Rn is any bounded set, we also need the compactness with respect to R time. We deduce by (3.1.6) and the boundedness of Fj (uj )dx that fj (uj ) are bounded in L∞ (L1 + L2 ). One can deduce that i∂t uj = ∆uj − f (uj ) is bounded in L∞ (H −1 +L1 ). By Aubin compactness theorem and diagonal selection method, there exists a subsequence uj → u,

a.e.,

therefore fj (uj ) → f (u),

a.e..

In order to get the solution of partial differential equation, it needs that the convergence holds at least in the sense of distribution. That can be gotten by almost R everywhere convergence, applying Egorov lemma, the boundedness of Fj (uj )dx and the assumption (3.1.6). According to fj (uj ) → f (u), strongly in L1 (B), where B ⊂ Rn+1 is a bounded set, one gets that lim uj = u satisfying NLS equation. By j→∞

applying weak limit and Fatou lemma, the equality (3.1.21) leads to the inequality E(u(t)) ⩽ E(u0 ), and by the continuity of u, we know that it satisfies the initial data. Now let’s give a simple proof for NLW equation. Firstly, consider the proof of Theorem 3.1.1 with the assumption (3.1.9). At this moment, F (u) ⩾ 0, then one can deduce the boundedness of |∇u|2 and |∂t u|2 by the energy conservation (3.1.3). So,

3.1

Nonlinear wave equations and nonlinear Schrödinger equation

113

|u|2 is also bounded in finite time. Just like the proof of NLS equation, let’s use fj to approach f , which leads to fj (uj ) → f (u) for almost everywhere. If remove the assumption (3.1.6), it’ll need new methods. For uj , applying the following equality  Z  Z  2  d2 1 2 u dx = ujt − |∇u|2 − uj f (uj ) dx, j 2 dt 2 ZZ one can get the boundedness of

uj f (uj )dxdt in finite time, and with Egorov

lemma, we can deduce that fj (u) → f (u) in L1loc , and other proofs are as before. The proof of Theorems 3.1.2 and 3.1.3 of NLW equation are more difficult than that of NLS equation. The difficulty comes from the Lp+1 − Lq estimate of the 4 ; while this restrict is linear wave operator, where q = (1 − p−1 )−1 , p ⩽ 1 + n−1 not needed for NLS equation. Now, applying the assumption (3.1.11) to prove the uniqueness of the solution to NLW equation. In fact, let u and v be two solutions of NLW equation, then it deduces that Z t p−1 |u(t) − v(t)|q ⩽ C (t − τ )−1+2/(n+1) (|u(τ )|γ,q + |v(τ )|γ,q ) |u(τ ) − v(τ )|q dτ, 0

(3.1.22) n 4 ,γ= . Applying Hölder’s inequality, and taking the Lα n−1 2(n + 1) norm of the first term of the integral on the right-hand side of (3.1.22), the Lβ norm of the second term, the L∞ norm of the third term, where α−1 + β−1 = 1, β =  (n − 2 −1 β(p−1) γ,q 2)[2(n + 2)(p − 1)] , we can prove that u, v ∈ L (W ), α 1 − < 1, n+1 therefore,   |u(t) − v(t)|q ⩽ CT sup |u(τ ) − v(τ )|q . where q = 2+

τ

When T → 0, CT → 0, the proof of the uniqueness of the solution is finished. The uniqueness may also hold for the case that the nonlinear term f (u) is a function of arbitrary growth. For NLW equation, assume f (0) = 0, f is non-decreasing, and f (u) is bounded with respect to u < 0, n = 3. For example, f (u) = eu −1. Then there exists a unique smooth solution for suitable initial data. In fact, f (u) ⩾ −A, ∀u ∈ R, let u, v, w respectively satisfy equations
∂t2 − ∆ u + f (u) = 0,


∂t2 − ∆ v = A,


∂t2 − ∆ w + f (v) = 0,

(3.1.23)

and assume that u, v, w have the same initial data, then u, v, w satisfy the integral

114

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

equations Z

t

R(t − τ ) ∗ f (u(τ ))dτ,

(3.1.24)

R(t − τ ) ∗ Adτ,

(3.1.25)

u(t) = u0 (t) − 0 Z t

v(t) = u0 (t) + 0

Z

t

R(t − τ ) ∗ f (v(τ ))dτ,

w(t) = u0 (t) −

(3.1.26)

0

where R ⩾ 0 is the Riemann function of wave equation. By (3.1.24), (3.1.25) and the definition of A, we have v ⩽ u. Then f (u) ⩽ f (v). By (3.1.24) and (3.1.26), it holds that u ⩾ w. So, w ⩽ u ⩽ v for any x ∈ R3 , t ∈ R+ . Therefore, u is pointwise bounded, then the uniqueness and regularity are easy to be obtained. For the following NLS equation iut = −∆u + λ|u|p−1 u,

t ∈ R, x ∈ Rn ,

(3.1.27) (3.1.28)

u(t0 ) = u0 (x),

where t0 ∈ R λ ∈ R, we can prove the existence of the L2 norm of the solution, so there exists the following theorem: 4 , then for any u0 ∈ L2 (Rn ) and t0 ∈ R, n there exists a unique global solution u(t) of (3.1.27), (3.1.28), such that Theorem 3.1.4

Let 1 < p < 1 +

u(t) ∈ C(R; L2 (Rn )) ∩ Lrloc (R; Lp+1 (Rn )), Z t u(t) = U (t − t0 )u0 − i U (t − τ )f (u(τ ))dτ,

(3.1.29) t ∈ R,

(3.1.30)

t0

∥u(t)∥L2 (Rn ) = ∥u0 ∥L2 (Rn ) ,

t ∈ R,

(3.1.31)

where r=

4(p + 1) , n(p + 1)

U (t) = ei∆t ,

f (z) = λ|z|p−1 z

(z ∈ C).

(3.1.32)

The integral is Bochner integral on H −1 (Rn ), further more, let j = 1, 2, · · · and u0 satisfy u0j , u0 ∈ L2 (Rn ), such that u0j → u0 , in L2 (Rn ) (as j → ∞). Assume uj (t) and u(t) are solutions with the initial data uj (t0 ) = u0 and u(t0 ) = u0 respectively, then for any t ∈ R, it holds that uj (t) → u(t),

in L2 (Rn ),

as j → ∞.

(3.1.33)

Consider the global solution to the initial-boundary value problem of NLS equation in exterior domain. Let n ⩾ 2, Ω be the exterior domain in Rn with smooth

3.1

Nonlinear wave equations and nonlinear Schrödinger equation

boundary ∂Ω. Consider the following initial-boundary value problem  2   iut − ∆u + f (|u| )u = 0, x ∈ Ω, t > 0, u(x, 0) = u0 (x), x ∈ Ω,   u|∂Ω = 0, t > 0.

115

(3.1.34) (3.1.35) (3.1.36)

Here f (·) is a real-valued function, u0 is a given complex-valued function, there exists the following theorem: Theorem 3.1.5 Let n be an integer, n ⩽ 3. If the following conditions are satisfied: (i) the supplementary set Ωc of Ω is star shaped, that means Ωc is a bounded domain including the origin, for any y ∈ ∂Ωc , it holds that y · ν(y) < 0, where ν(y) represents inner normal unit vector at y ∈ ∂Ωc ; (ii) there exists a real number γ ⩽ 2, such that W (w) = (2 + n)F (w) − nf (w)w ⩽ γF (w), Z w where F (w) = f (σ)dσ;

∀w ⩾ 0,

0

(iii) γ < 0, F (w) ⩾ 0, ∀ w ⩾ 0; (iv) 0 ⩽ γ < 2, −C1 w

γ+2 2


n ⩽ F (w) ⩽ C2 w n−2 + w ,

∀w ⩾ 0,

4 ; n (v) γ = 2,

where γ ⩾

−C1 w

n+2 n

⩽ F (w) ⩽ C2 w

n+2 n

∀w ⩾ 0;

;

(vi) f (s) ∈ C2 , s ∈ [0, ∞), for p2 ⩾ p1 > 2, it holds that |f (w)| ⩽ C1 wp1 /2 + C2 wp2 /2 , |f ′ (w)| ⩽ C3 w

p1 −2 2

|f ′′ (w)w| ⩽ C5 w

+ C4 w

p1 −2 2

∀w ⩾ 0,

p2 −2 2

+ C6 w

∀w ⩾ 0,

,

p2 −2 2

;

(vii) when γ < 2,  p1 > max p2 >

2 , 2−γ

 4 ,2 , 2−γ

n = 3, n = 1, 2,

116

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

and when γ = 2,  p1 > max p1 >

 4 ,2 , 2−k

n = 3,

2 , 2−k

n = 1, 2,

where 0 < k < 2, which only depends on ∥u0 ∥2 . Then there exists a positive constant δ, such that for any u0 ∈ H 2 ∩H01 , |x|u0 ∈ L2 , when ∥u0 ∥2 < δ, there exists a unique global solution to the initial-boundary value problem (3.1.34)∼(3.1.36), u ∈ C([0, ∞); H 2 ∩ H01 ) ∩ C 1 ([0, +∞); L2 ). Theorem 3.1.6 Let n ⩾ 3, k be an integer, k ⩾ (ii) given in Theorem 3.1.5, and for p2 ⩾ p1 ⩾ 1, m

|f (m) (w)w 2 | ⩽ C1 w

p1 2

+ C2 w

p2 2

,

n , f ∈ C 2k satisfy condition 2

∀w ⩾ 0, m = 0, 1, · · · , 2k.

Again assume that the conditions (iii), (iv), (v) in Theorem 3.1.5 are satisfied, and when γ < 2, it holds that   2 2n + 2∗ (4k − n) max , 1 , p1 > 2∗ (4k − n) 2−γ when γ = 2, p1 >

2n + 2∗ (4k − n) max 2∗ (4k − n)



 2 ,1 , 2−k

where 2∗ = 2n/(n − 2), if n ⩾ 3; 2∗ is any real number, if n = 2; 2∗ = ∞, if n = 1. Then there exists a positive constant δ, such that for any u0 ∈ H01 ∩ H 2k , ∆j u0 ∈ H01 (where j = 1, 2, · · · , k − 1), |x|u0 ∈ L2 and ∥u0 ∥2k < δ, there exists a unique global solution u to the initial-boundary value problem (3.1.34)∼(3.1.36), such that u ∈ C([0, +∞); H 2k ∩ H01 ) ∩ C 1 ([0, +∞); H 2(k−1) ∩ H01 ). For high-dimensional NLS equation, by the embedding relation between Besov space and Sobolev space, we can do detailed a priori estimates such that the global smooth solution can be obtained. Consider the following nonlinear Schrödinger equation iut − ∆u + f (|u|2 )u = 0,

(3.1.37)

u(0, x) = u0 (x),

(3.1.38)

where x = (x1 , x2 , · · · , xn ) ∈ Rn , t ∈ R.

3.1

Nonlinear wave equations and nonlinear Schrödinger equation

117

Theorem 3.1.7 Let 6 ⩽ n ⩽ 9, ρ satisfy 1

and

Z

s

f (σ)dσ ⩾ −C1 s(1+r)/2 ,

F (s) =

∀s ∈ R+ ,

0

4 . Then for any u0 ∈ W k,2 (Rn ), there n exists a unique solution u to the initial value problem of (3.1.37), (3.1.38), such that where C1 is a constant, r ∈ R, 1 ⩽ r < 1 +

u ∈ C 0 ([0, ∞); W k,2 (Rn )) ∩ C 1 ([0, ∞); W k−2,2 (Rn )). n + 2, then under the assumption in Theorem 3.1.7, u 2 is a global smooth solution of the initial value problems of (3.1.37), (3.1.38). Corollary 3.1.8 If k >

For 1 ⩽ n ⩽ 6, the global smooth solution of (3.1.37) and (3.1.38) has been obtained; by applying the estimates of the derivatives with respect to time t in Besov space, we can extend the result of theorem 3.1.7 to the case 9 ⩽ n ⩽ 11. For high-dimensional NLW equation, by similar method, similar results can be obtained. Now, we consider the blow up problem of solutions to NLS equation and NLW equation. Starting from NLS equation (3.1.2) with initial data u(x, 0) = φ(x). We have the following theorem: Theorem 3.1.9 Let uf (u) ⩽

  4 2+ F (u), n

∀u ∈ R.

(3.1.39)

If E(φ) < 0, then there is not any global smooth solution to NLS equation (3.1.2) (“smooth solution” means it is fully differentiable and small enough at infinity). Proof Applying quasi conformal equality,  Z  Z d 1 |xu − 2it∇u|2 + 4t2 F (u) dx = −2t k(u)dx, dt 2

118

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

where k(u) = nf (u)u − 2(n + 2)F (u), we have Z Z  2 2  d 2 r |u| + 4tx · Im (u∇¯ u) + 8t e dx = −4t k(u)dx, dt

(3.1.40)

R 1 where e = |∇u|2 + F (u), E(u) = edx = E(φ). Then applying the equality of 2 expansion transformation, Z Z d [x · Im (u · ∇¯ u)] dx = −4E − k(u)dx, (3.1.41) dt by (3.1.40), (3.1.41), one has   Z  Z Z tZ d2 d 2 2 2 −4t k(u)dx . r |u| dx − 8t E − 4t k(u)dxdt = dt2 dt 0 The above equality can be rewritten into Z Z d2 2 2 r |u| dx = 16E + 4 k(u)dx. dt2

(3.1.42)

The condition (3.1.39) means k(u) ⩽ 0, so, Z r2 |u|2 dx ⩽ 16Et2 + C1 t + C0 , for large enough t which is negative (E < 0), which is impossible, therefore, Theorem 3.1.9 is proved. Let’s compare condition (3.1.39) and condition (3.1.8) of the existence theorem. 4 4 For f (u) = −|u|p−1 u, p < 1 + is needed for condition (3.1.8), while p ⩾ 1 + is n n needed for condition (3.1.39). For NLW equation, there exists the following theorem. Theorem 3.1.10 Let uf (u) ⩽ (2 + ε)F (u),

∀u ∈ R,

(3.1.43)

for some ε > 0, if the initial data satisfies E(φ) < 0, then there is not any global solution to NLW equation (3.1.1). In detail, if u ∈ C([0, T ]; X), F (u), uf (u) ∈ L1 ((0, T ) × Rn ), E < 0 and (3.1.43) holds, then T < ∞. For example, f (u) = m2 u − |u|p−1 u, m > 0, then for any p > 1, (3.1.43) holds;
R 2 if |u|p+1 dx is controlled by ut + |∇u|2 + m2 u2 dx, then E is negative. In Theorem 3.1.10, the assumption E < 0 can be changed into the following weaker condition: s Z Z if E > 0, assume uut dx|t=0 > 2E u2 dx|t=0 . (3.1.44) R

3.1

Nonlinear wave equations and nonlinear Schrödinger equation

119

The proof of theorem 3.1.10 By expansion transformation, we get the equality

 Z  Z  2  d2 1 2 u dx = ut − |∇u|2 − uf (u) dx. 2 dt 2

Which leads to

Z Z d2 I 2 = (2 + 2α) ut dx + 2α |∇u|2 dx dt2 Z + [(2 + 4α)F (u) − uf (u)]dx − (2 + 4α)E,

(3.1.45)

Z

ε 1 2 u dx. Choosing α = , noting the last two terms of the right-hand 2 4 side of (3.1.45) being positive, multiplying (3.1.2) with I, and ignoring the gradient term, it holds that Z Z ′′ 2 I[I + (2 + 4α)E] > (1 + α) ut dx · u2 dx > (1 + α)(I ′ )2 . where I(t) =

Let H(t) = I(t) − E(t + τ )2 , one has ′′

HH > (1 + α)(H ′ )2 . Choose τ to be large enough, such that H ′ (0) > 0, therefore, J = H −α satisfies J ′′ (t) < 0,

J(0) = 0,

J ′ (0) < 0,

which leads to J(t) < J(0) + tJ ′ (0). Then there exists T > 0, such that J(T ) = 0. So, when t → T ,

Z u2 dx → ∞, the

proof of Theorem 3.1.10 is finished. Generally speaking, NLW equation with small initial data (small L∞ norm), not satisfies E < 0, since the nonlinear term can be controlled by the linear term. For example, f ′ (0) ̸= 0. So, this kind of solution will not blow up. But if f ′ (0) = 0, then the solution with small initial data may still blow up. We have the following theorem. Theorem 3.1.11 Consider the following equation utt − ∆u − |u|p = 0.

(3.1.46)

If 1 < p ⩽ γ(n − 1), then there is not any global solution to (3.1.46) with small initial data u(0) ∈ Cc∞ (Rn ), where function γ(n) is " # 12 2 1 1 1 2 1 + + . γ(n) = + + 2 n 2 n n

120

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

√ In fact, when n = 3, γ(n − 1) = γ(2) = 1 + 2, F. John proved that all solutions with initial data in Cc∞ will blow up; for n = 1, the condition becomes 1 < p < ∞; √ for n = 2, the condition is 1 < p ⩽ (3+ 17)/2; for n = 4, the condition is 1 < p ⩽ 2. Proof Let n = 3, the initial data u(x, 0) ∈ Cc∞ , satisfy Z Z u(x, 0)dx > 0, ut (x, 0)dx > 0. Assume the support of the initial data is {|x| ⩽ k}, let J(t) = equation (3.1.46) and Hölder’s inequality, we get Z J ′′ = |u|p dx ⩾ J p C(1 + t)−3(p−1) .

R

udx, by integrating

(3.1.47)

On the other hand, from (3.1.46), since R ⩾ 0(n = 3), we have Z t u(t) = u0 (t) + R(t − τ ) ∗ |u(τ )|p dτ ⩾ u0 (t), 0

where u0 (t) satisfies (∂t2 − ∆)u0 = 0,

u0 (0) = u(0),

∂t u0 (0) = ∂t u(0).

u0 is the solution of the free wave equation, satisfying Z u0 dx = C1 + C2 t. Let C1 > 0, C2 > 0, u0 (x, t) be supported in {t − k ⩽ |x| ⩽ t + k} = A, so, by Hölder’s inequality, one has Z 1/p Z Z C0 + C1 t = u0 dx ⩽ udx ⩽ C(1 + t)2(p−1)/p |u|p dx . A

A

From (3.1.47), we obtain ′′

J =

Z |u|p dx ⩾ C(1 + t)2−p .

Integral the above equation twice with respect to t, and noting J(0) > 0, J ′ (0) > 0, we deduce that J ⩾ C(1 + t)4−p ,

J ′ > 0.

(3.1.48)

By (3.1.47), (3.1.48) and ε > 0, one has J ′′ ⩾J 1+ε J p−1−ε C(1 + t)−3(p−1) ⩾J 1+ε C(1 + t)(4−p)(p−1−ε) (1 + t)−3(p−1) =J 1+ε C(1 + t)−µ ,

(3.1.49)

3.2

KdV equation, etc.

121

√ where µ = (p − 1)2 + ε(4 − p). According to 1 < p < 1 + 2, (3.1.49) and ε being small enough, there exists 0 < µ < 2. And multiplying (3.1.49) by J ′ > 0, we have  ′ (J ′ )2 ⩾ J 2+ε C(1 + t)−µ + positive terms. Therefore,  1 µ ε J ′ ⩾ J 2+ε C(1 + t)−µ 2 = J 1+ 2 C(1 + t)− 2 . Then

  − 2ε Z t ε −µ J(t) ⩾ C0 − C (1 + τ ) 2 dτ . 2 0

From the above inequality, we know that, when t → T , Z t µ C0 − C (1 + τ )− 2 dτ → 0, J(t) → ∞, 0

then the proof of Theorem 3.1.11 is finished. Concerning the study of nonlinear wave equation we refer to the [24,63,74,78,80, 86,87,135,171,187,214], which include high-dimensional case. For systematic survey on the nonlinear wave equation, we refer to the books [48, 106, 205, 218, 224] . As to the nonlinear Schrödinger equation, see [23, 32, 62, 68].

3.2

KdV equation, etc.

1. Consider generalized KdV equation, ∂t u + D3 u + a(u)Du = 0,

(3.2.1)

x ∈ R.

(3.2.2)

with initial data u(x, 0) = u0 (x),

Introduce potential function V (x), V (0) = V ′ (0) = 0, a = −V ′′ (0). Firstly, we consider the uniqueness of the L2 solution. Theorem 3.2.1 Let V (ρ) satisfy the condition |V ′′ (ρ)| ⩽ C|ρ|p ,

0 0 (may T = ∞), then the solution of (3.2.1) with the initial data u(0) = u0 (x) is unique in the following sense, 2 u ∈ (L∞ loc ∩ Lw )([0, T ]; L ), β

1 ([0, T ]; L2 ), (1 + x+ ) 2 u ∈ Lqloc 1

1 hα2 0 u ∈ Lqloc ([0, T ]; Lr ),

α0 > 0,

(3.2.4)

122

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

where 2 1 1 =β= − , q1 p 4 1 1 p = − , r 2 4 3 1 1 p 1 < − = + , q 4 2r 2 8 x+ = max{x, 0}, hα0 (x) is a nonnegative continuous function, satisfying hα0 (x) = eα0 x ,

x ⩽ 0,

α0 > 0,

0 ⩽ hα0 (y) ⩽ hα0 (x) ⩽ exp[α0 (x − y)]hα0 (y),

∀x ⩾ y.

Theorem 3.2.2 Let V (ρ) satisfy (3.2.3), where p ⩾ 2, u0 ∈ L2 , T > 0, then the solution to the initial value problem (3.2.1), (3.2.2) is unique in the following sense, 1

1 χ+ (t + x) u |u|p ∈ Lqloc ([0, T ); L1 ), q1 > 4, 1 χ− (|u|p ) ∈ l∞ (Lqloc ([0, T ); L1 )),

q1 > 4,

χ+ u ∈ Lqloc ([0, T ); L1 ),

q>

4 , 3

χ− eαx/2 u ∈ l∞ (Lqloc ([0, T ); L∞ )),

q>

4 , α > 0, 3

where χ+ means eigenfunction on R+ , χ− means eigenfunction on R− , the norm of the space ls (Lq (I, Lr )) is

∥v; ls (Lq (I, Lr ))∥ =

 " X Z 

j∈z

Z dt

|χj v|r dx

I

1  rq # qs  s 

< ∞.

Now, we consider the uniqueness of the H 1 solution. Theorem 3.2.3 Let V ′′ (ρ) be an absolutely continuous function, V ′′ (0) = 0, V ′′′ be locally bounded, u0 ∈ H 1 , T > 0, then the solution to the initial value problems (3.2.1), (3.2.2) is unique in the following sense 1 u ∈ (L∞ loc ∩ Cw )([0, T ); H ),

χ+ Du ∈ L1loc ([0, T ); L∞ ).

For the global existence of the smooth solution to the initial-boundary value

3.2

KdV equation, etc.

123

problem of the following KdV equation ut + ux + uux + uxxx = 0,

x, t > 0,

(3.2.5)

u(x, 0) = f (x),

x ⩾ 0,

(3.2.6)

u(0, t) = g(t),

t ⩾ 0,

(3.2.7)

there is the following theorem. k+1 Theorem 3.2.4 Let k be a positive integer, f ∈ H 3k+1 (R+ ), g ∈ Hloc (R+ ) and satisfy k + 1 compatibility conditions

g (j) (0) = φ(j) (0),

0 ⩽ j ⩽ k,

where φ(j) (x) is defined by φ(0)(x) = f (x), ( (j+1)

φ

(x) = −

φ(j) x

+

φ(j) xxx

" j 1 X + 2 i=0

! j i

# ) (i)

φ φ

(j−i)

, x

then there exists a unique smooth solution u to the problem (3.2.5)∼(3.2.7) + 3k+1 u ∈ L∞ (R+ )). loc (R ; H

(3.2.8)

When k > 1, then one can get the classical solution to (3.2.5)∼(3.2.7) in R+ × R+ . 2. Consider the initial value problem of Hirota equation iψt + αψxx + iβψxxx + iγ |ψ|2 ψ ψ(x, 0) = ψ0 (x), where i =

x ∈ R,


x

+ δ|ψ|2 ψ = 0,

t ⩾ 0,

(3.2.9) (3.2.10)

√ −1, α, β, γ, δ are real constants. There’s the following result.

Theorem 3.2.5 (the existence and uniqueness) Assume that α, β, γ, δ are real constants, βγ ̸= 0, ψ0 (x) ∈ H k (R), then for any positive T > 0, there exists a unique smooth solution ψ(x, t) ∈ W (k, T ) to the problem (3.2.9)∼(3.2.10), where W (k, T ) = {u|∂ts u ∈ L∞ (0, T ; H k−3s (R)),

0⩽s⩽

k , k ⩾ 3}. 3

(3.2.11)

Theorem 3.2.6(convergence) Under the assumption in Theorem 3.2.5, we have (i) If ψδ represents the global solution to the problem (3.2.9)∼(3.2.10), then there exists a function ψ ∈ W (k, T ), such that ψδ → ψ

weakly ∗

in W (k, T ),

as δ → 0,

(3.2.12)

124

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

where ψ is the solution to the following DNLS-KdV equation with the initial data (3.2.10) i∂t ψ + αDx2 ψ + iβ∂x3 ψ + iγ∂x (|ψ|2 ψ) = 0.

(3.2.13)

(ii) If ψαδ represents the global solution to the problem (3.2.9)∼(3.2.10), then there exists a function ψ ∈ W (k, T ), such that ψαδ → ψ

weakly ∗

as α, δ → 0,

in W (k, T ),

where ψ is the unique smooth solution to the MKdV equation with the initial data (3.2.10). (iii) Let α = ̸ 0, γ/β = δ/α, if ψβγ represents the global solution to the problem (3.2.9)∼(3.2.10), then there exists a function ψ ∈ W ∗ (k, T ), such that ψβγ → ψ

weakly ∗

in W ∗ (k, T ),

as β, γ → 0,

where ψ is the unique smooth solution to the nonlinear Schrödinger equation i∂t ψ + αDx2 ψ + δ|ψ|2 ψ = 0 with the initial data (3.2.10), where   k W ∗ (k, T ) = u|∂ts u ∈ L∞ (0, T ; H k−2s (R)), 0 ⩽ s ⩽ , k ⩾ 3 . 2

(3.2.14)

(3.2.15)

Theorem 3.2.7 (the smoothness and decay estimate) Under the conditions given in Theorem 3.2.5, ψ(t, x) represents the global smooth solution to the problems (3.2.9)∼(3.2.10), then we have k+1 k+1−3r (i) ∂tr ψ ∈ L2 (0, T ; Hloc (R)), r ⩽ ; 3 k−j (ii) if γ0 ⩾ is a real number, 0 ⩽ j ⩽ k − 3, and |x|γ0 ∂xj ψ0 ∈ L2 (R), then 2 ∂xj ψ(x, t) = O(|x|−(1− 2(k−j) )γ0 ), 1

|x| → +∞.

3. Consider the initial value problem of the following generalized mixed nonlinear Schrödigner equation: ut = iαuxx + βu2 u ¯x + γ|u|2 u + ig(|u|2 )u, u(x, 0) = u0 (x),

(x, t) ∈ R × R+ .

(3.2.16)

It includes the following physical models. (i) Ablowitz equation iut = uxx − 4iu2 u ¯x + 8|u|4 u.

(3.2.17)

3.2

KdV equation, etc.

125

(ii) Chen-Lee-Lin equation iut + uxx + iα|u|2 ux = 0.

(3.2.18)

(iii) Gerdjikov-Ivanov equation iut + uxx − β|u|2 u − 2δ 2 |u|4 u − 2iδu2 u ¯x = 0.

(3.2.19)

(iv) Kundu equation
iut + uxx + iα |u|2 u x + β|u|2 u + 8(4δ + α)|u|4 u + 4iδ(|u|2x )u = 0.

(3.2.20)

Theorem 3.2.8 Assume α, β, γ are real constants, α = ̸ 0, the real-valued function g(v) and the initial data u0 (x) satisfy the following conditions: (A1) u0 (x) ∈ H s (R), g(v) ∈ C s (R+ ), s ⩾ 3 is an integer; (A2) there exist two real constants M ∈ R, δ ∈ (0, 2), such that sgn(α)g(v) ⩽ M (1 + v 2 ),

∥u0 ∥2 ⩽ M1 ,

(3.2.21)

or sgn(α)g(v) ⩽ M (1 + v 2−δ ) + M0 v 2 ,

(3.2.22)

where the constant M0 < (β + γ)(5β − 3γ)/16|α|, M1 only depends on α, β, γ, then for any T > 0, there is a unique smooth solution to the initial value problem (3.2.15), (3.2.16) \

[s/2] s u(x, t) ∈ B∞ (T ) =

r W∞ (0, T ; H s−2r (R)).

r=0

Theorem 3.2.9 Let α, β, γ be real constants, α ̸= 0. Assume u0 (x) ∈ H s (R), |x|ρ Dj u0 (x) ∈ L2 (R), j = 0, 1, · · · , k,   1 s , s , k ⩽ s − 2, s ⩾ 3. If u(x, t) ∈ B∞ (T ) is a solution to the problem where ρ ∈ 2 (3.2.16), then |Dm u(x, t)| = O(|x|−ρm ),

|x| → ∞,

where t ∈ [0, T ], 0 ⩽ m ⩽ s − 1,

ρm

  1    s−m−  2ρ . = min ρ,   s−k  

(3.2.23)

126

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

4. Consider the mixed initial-boundary value problem of the high-dimensional nonlinear Schrödinger equations with damping effect:   n X ∂u ∂u ∂ =γ aij (x) + b(x)q(|u|2 )u + c(x, t)u = f (x, t), ∂t ∂x ∂x i j i,j=1 u(x, 0) = u0 (x), x ∈ Ω, t > 0,   X ∂u cos(n, xj ) + h(x)u = 0, aij (x) ∂xi ∂Ω

(3.2.24) (3.2.25) (3.2.26)

where u = (u1 (x, t), u2 (x, t), · · · , uN (x, t)) is unknown complex vector valued function, Ω ⊂ Rn is a bounded domain with smooth boundary ∂Ω ∈ C 2 , n represents the unite normal vector of ∂Ω, c(x, t) = (cij (x, t)) is an N × N matrix of complex functions, f (x, t) = (f1 (x, t), f2 (x, t), · · · , fN (x, t)) is a known complex vector valued function, aij (x)(i, j = 1, 2, · · · , n) are real functions, h(x), b(x) and q(x) satisfy the following assumptions. n X (I) aij (x)ξi ξj ⩾ a0 |ξ|2 , ξ = (ξ1 , ξ2 , · · · , ξn ) ∈ Rn , a0 > 0, aij = aji , aij ∈ i,j

¯ C 1 (Ω). ∂cij (x, t) ∈ L∞ (QT )(i, j = 1, · · · , N ), QT = Ω × (0, T ). ∂t ¯ q(s) ∈ C 1 (R+ ), b(x) ∈ C 0 (Ω). ¯ (III) b(x) ⩾ 0, h(x) ⩾ 0, q(s) ⩾ 0, h(x) ∈ C 0 (Ω), (IV) f (x, t), ft (x, t) ∈ L∞ (0, T ; L2 (Ω)), u0 (x) ∈ H 2 (Ω), γ = γ0 + iγ1 , γ0 ⩾ 0, |γ| > 0. (II) cij (x, t),

¯ u0 ∈ H 2 (Ω), f (x, 0) ∈ L2 (Ω), q(s) ∈ Theorem 3.2.10 Assume aij ∈ C 0 (Ω), + 2 2 C (R ), q(|u0 | )u0 ∈ L (Ω), and ∂cij (x, t) ⩽ M1 (where i, j = 1, 2, · · · , n), ∥f ∥ ⩽ M2 , (i) |cij (x, t)| ⩽ M1 , ∂t ∥ft ∥ ⩽ M2 , M1 , M2 are constants; 1

(ii) aij = aji (where i, j = 1, 2, · · · , n),

n X

aij (x)ξi ξj ⩾ a0 |ξ|2 , ∀ξ ∈ Rn ;

i,j=1

(iii) γ = γ0 + iγ1 , γ0 ⩾ 0, 0 ⩽ b(x), h(x) ⩽ M (M =Const); p (iv) q(s) ⩾ 0, q ′ (s) ⩾ 0, ∀s ∈ [0, ∞), q(s) ⩽ A0 s 2 + B0 (p > 0), where a0 is a positive constant, A0 , B0 are also constants, then the problems (3.2.24)∼ (3.2.26) have at least one weak solution u(x, t) satisfying u(x, t) ∈ L∞ (0, T ; H 1 (Ω)) ∩ W 1,∞ (0, T ; L2 (Ω)).

3.2

KdV equation, etc.

127

¯ u0 (x) ∈ H 2 (Ω), f (x, 0) ∈ L2 (Ω), Theorem 3.2.11 Assume aij ∈ C 1 (Ω), 2 2 1 + q(|u| )u0 ∈ L (Ω), q(s) ∈ C (R ), and n X (i) aij = aji (where i, j = 1, 2, · · · , n), 0 < a1 ⩽ aij ⩽ a2 , aij (x)ξi ξj ⩾ a0

n X

i,j=1

|ξi |2 , ∀ξ = (ξ1 , ξ2 , · · · , ξn ) ∈ Rn .

∂cij (x, t) ⩽ M1 , ∥f (·, t)∥ ⩽ M2 , ∥ft (x, t)∥L2 (Q ) ⩽ M2 (ii) |cij (x, t)| ⩽ M1 , T ∂t (where M1 , M2 are constants). (iii) q(s) ⩾ 0, q ′ (s) ⩾ 0, p |q(s)| ⩽ A0 s 2 + B0 , j=1

where  A0 , B0are positive constants, when 1 ⩽ n < 3, p ∈ (0, ∞); when n ⩾ 3, 4 p ∈ 0, . n−2 (iv) 0 ⩽ b(x), h(x) ⩽ M (where M is a positive constant), γ = γ0 + iγ1 ,

γ0 ⩾ 0,

|γ| > 0.

Then the problems (3.2.24)∼(3.2.26) has at least one generalized solution u(x, t), u(x, t) ∈ L∞ (0, T ; H 2 (Ω)) ∩ W 1,∞ (0, T ; L2 (Ω)). 5. Considering the initial value problem of the following nonlinear dispersion equation: ∂t u + β∂x3 u + δ∂x2 H(u) + γ∂x H(u) + ∂x g(x) = 0, u(x, 0) = φ(x),

(x, t) ∈ R × R+ ,

where β, δ, γ are real constants. 1 H(u) = p · n

Z

∞

−∞

u(y, t) dy y−x

(3.2.27) (3.2.28)

(3.2.29)

is the Hilbert transform, p· stands for the Cauchy principal value, it includes the following models with physical meaning. (i) Benjamin-Ono-KdV equation ∂t u + β∂x3 u + δ∂x2 H(u) + ∂x g(u) = 0.

(3.2.30)

(ii) Generalized KdV equation ∂t u + β∂x3 u + ∂x g(u) = 0.

(3.2.31)

(iii) Benjamin-Ono equation ∂t u + δ∂x2 H(u) + u∂x u = 0.

(3.2.32)

128

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

Theorem 3.2.12 Assume β, δ, γ are constants, β = ̸ 0, γ ⩽ 0, the initial-valued function φ(x) ∈ H k (R), g(·) ∈ C k (R), where k ⩾ 3, if there exists a constant c, such that |g ′ (v)| ⩽ c (1 + |v|η ) , where 0 < η ⩽ 2, then the Cauchy problem   (3.2.27), (3.2.28) has a unique classical k r k−3r . solution u ∈ W∞ (0, T ; H (R)), r ⩽ 3 r Theorem 3.2.13 Assume that uδγ (x, t) ∈ W∞ (0, T ; H k−3r (R)) is the solution to the Cauchy problem (3.2.27), (3.2.28), then there exist three functions uδ , uγ and u, such that (i) lim uδγ (x, t) = uγ (x, t), where δ→0

k

uγ (x, t) ∈

[3] \

r W∞ (0, T ; H k−3r (R))

r=0

is the unique global smooth solution to the Cauchy problem (3.2.33), (3.2.34) ∂t u + β∂x3 u + γ∂x H(u) + ∂x g(u) = 0,

(3.2.33)

u(x, 0) = φ(x).

(3.2.34)

(ii) lim uδγ (x, t) = uδ (x, t), where γ→0

k

uδ (x, t) ∈

[3] \

r W∞ (0, T ; H k−3r (R))

r=0

is the unique global smooth solution to the Cauchy problem (3.2.30), (3.2.28). (iii) lim uδγ (x, t) = lim uδ (x, t) = lim uγ (x, t) = u(x, t), δ→0 γ→0

where u(x, t) ∈

δ→0

[k/3] T r=0

γ→0

r W∞ (0, T ; H k−3r (R)) is the unique global smooth solution to

(3.2.31), (3.2.28). 6. Consider the initial value problem of Zakharov equations in innonuniform medium i∆εt + ∆2 ε − ∇ · (n∇ε) = 0, (3.2.35)

ε|t=0 = ε0 (x),

nt = ∆Φ,

(3.2.36)

Φt = n + |∇ε|2 ,

(3.2.37)

n|t=0 = n0 (x),

Φ|t=0 = Φ0 (x),

x ∈ R2 ,

(3.2.38)

3.2

KdV equation, etc.

129

where ε(x, t) = (ε1 (x, t), · · · , εN (x, t)) is an unknown complex-valued function, n(x, t), Φ(x, t) are unknown and Φ(x, t) is the low frequency  real-valued  function, √ ∂2 ∂ ∂ ∂2 + . potential, i = −1, ∇ = , ,∆= 2 ∂x1 ∂x2 ∂x1 ∂x22 Theorem 3.2.14 Suppose that (i) ε0 (x) ∈ H m+3 (R2 ), n0 (x) ∈ H m+1 (R2 ), Φ0 (x) ∈ H m+2 (R2 ), m ⩾ 0; 2 ∥ Ψ(x) ∥2 , 3 where Ψ(x) is the fundamental solution of the following equation (ii) ∥ ∇ε0 (x) ∥2 ⩽

∆Ψ − Ψ + Ψ3 = 0, then ε(x, t), n(x, t), Φ(x, t) is the unique global smooth solution of the initial problems (3.2.35) ∼ (3.2.38), for ε(x, t) ∈ L∞ (0, T ; H m+3 (R2 )), εt (x, t) ∈ L∞ (0, T ; H m+1 (R2 )), n(x, t) ∈ L∞ (0, T ; H m+1 (R2 )), nt (x, t) ∈ L∞ (0, T ; H m (R2 )), Φ(x, t) ∈ L∞ (0, T ; H m+2 (R2 )), Φt (x, t) ∈ L∞ (0, T ; H m+1 (R2 )). Consider the following initial-boundary value problems i∆εt + ∆2 ε − ∇(n · ∇ε) = 0, ntt − ∆n = ∆(|ε|2 ), ε|t=0 = ε0 (x), ε|∂Ω = 0,

x ∈ Ω,

t > 0,

n|t=0 = n0 (x), nt |t=0 = n1 (x), ∂ 2 ε = 0, n|∂Ω = 0, t > 0, ∂v 2 ∂Ω

(3.2.39) (3.2.40) (3.2.41) (3.2.42)

where Ω ⊂ R2 is the bounded domain with smooth boundary condition ∂Ω ∈ C 2 , and v is unit exterior normal of ∂Ω. The following theorem is obtained. Theorem 3.2.15 Suppose that ε0 (x) ∈ H 4 (Ω)∩H01 (Ω), n0 (x) ∈ H 2 (Ω)∩H01 (Ω), and ∥∇ε0 (x)∥ ⩽ λ, where λ is the constant, then the unique global smooth solution of the initial-boundary value problems (3.2.39) ∼ (3.2.42) exist and satisfy the following:

130

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

ε(x, t) ∈ L∞ (0, T ; H 4 (Ω) ∩ H01 (Ω)), εt (x, t) ∈ L∞ (0, T ; H 2 (Ω) ∩ H01 (Ω)), n(x, t) ∈ L∞ (0, T ; H 2 (Ω) ∩ H01 (Ω)), nt (x, t) ∈ L∞ (0, T ; H01 (Ω)), ntt (x, t) ∈ L∞ (0, T ; L2 (Ω)). 7. Consider the initial value problem of three-dimensional nonlinear Schrödinger Boussinesq equations iεt + ∆ε − nε − β|ε|2 ε = 0, (3.2.43) nt = ∆φ,

(3.2.44)

φt = n + f (n) + µnt − λ∆n + |ε|2 ,

(3.2.45)

ε|t=0 = ε0 (x), n|t=0 = n0 (x), φ|t=0 = φ0 (x), (3.2.46) √ where x ∈ R3 , t ∈ R+ , µ > 0, λ > 0, β > 2, i = −1, ε(x, t) = (ε1 (x, t), · · · , εN (x, t)) is an unknown complex function, η(x, t), φ(x, t) are unknown real-valued function, f (s)(s ∈ R1 ), η0 (x), φ0 (x) are the known real-valued functions and ε0 (x) is the complex initial-valued function, ∆≡

∂2 ∂2 ∂2 + + . 2 2 ∂x1 ∂x2 ∂x23

Theorem 3.2.16 Suppose that ′ Rn 11 3 , A > 0; (i) f (n) ∈ C , 0 f (s)ds ⩾ 0, f (n) ⩽ A|n|q−1 , 1 ⩽ q ⩽ 3 (ii) µ > 0, λ > 0, β > 2; (iii) ε0 (x) ∈ H 4 (R3 ), n0 (x) ∈ H 4 (R3 ), φ0 (x) ∈ H 4 (R3 ), then ε(x, t), n(x, t), φ(x, t) is the global solution of the initial value problems (3.2.43) ∼ (3.2.46), for ε(x, t) ∈ L∞ (0, T ; H 4 (R3 )), εt (x, t) ∈ L∞ (0, T ; H 2 (R3 )), εtt (x, t) ∈ L∞ (0, T ; L2 (R3 )), n(x, t) ∈ L∞ (0, T ; H 4 (R3 )), nt (x, t) ∈ L∞ (0, T ; H 2 (R3 )), ntt (x, t) ∈ L∞ (0, T ; H 1 (R3 )), φ(x, t) ∈ L∞ (0, T ; H 4 (R3 )), φt (x, t) ∈ L∞ (0, T ; H 2 (R3 )), φtt (x, t) ∈ L∞ (0, T ; H 1 (R3 )).

3.2

KdV equation, etc.

131

Theorem 3.2.17 If it satisfies the conditions of the Theorem 3.2.16, and supposing (i) f (n) ∈ C 2k−1 , k ⩾ 2; (ii) ε0 (x) ∈ H 2k (R3 ), n0 (x) ∈ H 2k (R3 ), Φ0 (x) ∈ H 2k (R3 ), then the global smooth solution of the initial value problems (3.2.43) ∼ (3.2.46) exist, and satisfy the following: ε(x, t) ∈ L∞ (0, T ; H 2k (R3 )), Dtj ε(x, t) ∈ L∞ (0, T ; H 2(k−j) (R3 )), ∞

0 ⩽ j ⩽ k;

n(x, t) ∈ L (0, T ; H (R )), 2k

3

Dtj n(x, t) ∈ L∞ (0, T ; H 2k−(j+1) (R3 )), ∞

0 < j ⩽ k;

φ(x, t) ∈ L (0, T ; H (R )), 2k

3

Dtj φ(x, t) ∈ L∞ (0, T ; H 2k−(j+1) (R3 )),

0 < j ⩽ k.

Theorem 3.2.18 Suppose that (i) f (n) ∈ C 2 ; (ii) ε0 (x) ∈ H 2 (R3 ), n0 (x) ∈ H 2 (R3 ), φ0 (x) ∈ H 2 (R3 ), then ε(x, t), n(x, t), φ(x, t) ∈ L∞ (0, T ; C 2 ) is the unique global smooth solution of the initial value problems (3.2.43) ∼ (3.2.46). 8. Consider the initial value problem of coupled nonlinear wave equation ut = uxxx + 6uux + 2vvx ,

(3.2.47)

vt = 2(uv)x ,

(3.2.48)

u|t=0 = u0 (x),

v|t=0 = v0 (x).

(3.2.49)

Theorem 3.2.19 Suppose that (u0 (x), v0 (x)) ∈ H k (R), k ⩾ 4, then the initial value problem (3.2.47) ∼ (3.2.49) exists the unique smooth solution \ r (u, v) ∈ W∞ (0, T ; H s (R)), (3.2.50) s+3r⩽k r where W∞ (0, T ; H k (R)) is the Sobolev space; Dts Dxl f (x, t) ∈ L∞ (0, T ; L2 (R)), for 0 ⩽ s ⩽ r, 0 ⩽ l ⩽ k.

9. Consider the initial value problem of nonlinear singular integro-ordinary equation in deep water ut + 2uux + Huxx + b(x, t)ux + c(x, t)u = f (x, t), u|t=0 = u0 (x),

x ∈ R,

t > 0,

(3.2.51) (3.2.52)

132

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

where H is the Hilbert transformation Hu(x, t) =

1 P· π

Z

∞

−∞

u(y, t) dy, y−x

(3.2.53)

P · represents the integral Cauchy principal value. If b(x, t) = c(x, t) = f (x, t) = 0, (3.2.51) is ut + 2uux + Huxx = 0. (3.2.54) This is the famous Benjamin-Ono equation for deep water wave with the soliton solution. Theorem 3.2.20 If b(x, t) ∈ W∞ (Q∗T ), Q∗T = {x ∈ R, 0 ⩽ t ⩽ T }, c(x, t) ∈ (2,0) (2,0) W∞ (Q∗T ), f (x, t) ∈ W2 (Q∗T ), and φ(x) ∈ H 2 (R), then the initial value problem (3.2.51), (3.2.52) exists the global generalized solution u(x, t), \ (1) u(x, t) ∈ L∞ (0, T ; H 2 (R)) W∞ (0, T ; L2 (R)) = Z. (3.2.55) (2,1)

Theorem 3.2.21 If b(x, t) ∈ W∞ (Q∗T ), c(x, t) ∈ L∞ (Q∗T ), then the initial value problem (3.2.51), (3.2.52) exists the unique global generalized solution u(x, t) ∈ Z. (1,0)

Theorem 3.2.22 If φ(x) ∈ H M , M ⩾ 2, then the initial value problem (3.2.54), (3.2.52) of Benjamin-Ono equation exists the unique global smooth solution u(x, t), [M 2 ]

u(x, t) ∈

\

W∞,loc (R+ , H M −2k (R)), (k)

(3.2.56)

k=0 + 2 where derivative uxr ts (x, t) ∈ L∞ loc (R , L (R)), 0 ⩽ 2s + r ⩽ M .

This subsection contains some results on the existence, uniqueness, regularity of the global solution to some well-known models. Except for the reference mentioned in the previous chapters, for the wellposedness of these evolution equations, we refer to the papers [34,97,101,137–139,145,177,198,202–204,214,221,244,246,253,255] and the references cited there for more details. The stability of the solitary wave solution to the evolution equations possessing symmetric structure, like KdV equation, refers to [65, 66].

3.3

Landau-Lifshitz equations

1. Consider the initial value problem of one dimensional Landau-Lifshitz equation Z t = Z × Z xx , Z(x, 0) = Z 0 (x),

x ∈ R, t > 0,

|Z 0 (x)| = 1,

x ∈ R.

(3.3.1) (3.3.2)

3.3

Landau-Lifshitz equations

133

Theorem 3.3.1 Suppose that the initial value function DZ 0 (x) ∈ H k (R), k ⩾ 3, satisfies |Z 0 (x)| = 1, ∀x ∈ R, then the initial value problem (3.3.1), (3.3.2) exists the unique global smooth solution Z(x, t), k

DZ(x, t) ∈

[2] X

s W∞ (0, T ; H k−2s (R)).

(3.3.3)

s=0

2. Consider the linear initial-boundary value problem of one-dimensional LandauLifshitz equation Z t = Z × Z xx + f (x, t, Z),

0 < x < l, t > 0.

(3.3.4)

t ⩾ 0,

(3.3.5)

The first boundary condition Z(0, t) = Z(l, t), the second boundary condition Z x (0, t) = Z x (l, t) = 0,

t ⩾ 0,

(3.3.6)

and mixed boundary condition Z(0, t) = 0,

Z x (l, t) = 0,

t ⩾ 0,

(3.3.7)

Z x (0, t) = 0,

Z(l, t) = 0,

t ⩾ 0,

(3.3.8)

or and the initial condition Z(x, 0) = Z 0 (x),

0 ⩽ x ⩽ l.

(3.3.9)

We make the following assumptions. (I) f (x, t, Z) is a continuously differentiable function for x and Z, and 3×3 order ∂f (x, t, Z) is semi-bounded. That is, there is a constant Jacobi derivative matrix ∂z b, such that ξ · fZ (x, t, Z)ξ ⩽ b|ξ|2 , ξ ∈ R3 , (3.3.10) and f0 (x, t) = f (x, t, 0) ∈ L2 (QT ), QT = {0 ⩽ x ⩽ l, 0 ⩽ t ⩽ T }. (II) f (x, t, Z) satisfies |fx (x, t, Z)| ⩽ c(x, t)|Z|3 + d(x, t),

(x, t, Z) ∈ QT × R3 ,

where c(x, t) ∈ L∞ (QT ), d(x, t) ∈ L2 (QT ). (1) (III) Z 0 (x, t) ∈ W2 (0, l), and satisfies the boundary conditions.

(3.3.11)

134

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

Theorem 3.3.2 Suppose the assumptions (I), (II), (III) are satisfied, then Z(x, t) exists that is the global weak solution of the second initial-boundary value problem (3.3.4), (3.3.6) and (3.3.9) of the generalized Landau-Lifshitz equation, \ 1 1 (1) Z(x, t) ∈ L∞ (0, T ; W2 (0, l)) C ( 2 , 4 ) (QT ). (3.3.12) Theorem 3.3.3 If the conditions of the Theorem 3.3.2 are satisfied, and supT 1 1 (1) posing f (x, t, 0) ≡ 0, then Z(x, t) ∈ L∞ (0, T ; W2 (0, l)) C ( 2 , 4 ) (QT ) exists that is the global weak solution of the first initial boundary value problem (3.3.4), (3.3.5) and (3.3.9) of the generalized Landau-Lifshitz equation, and that is also the global weak solution of the mixed initial-boundary value problem (3.3.4), (3.3.7) (or (3.3.8)), and (3.3.9). Theorem 3.3.4 If b < 0, then Z(x, t) that is the solution all of the above initial-boundary value problem satisfies lim ∥Z(·, t)∥L2 (0,l) = 0.

t→∞

(3.3.13)

Now we consider the first boundary problem Z(0, t) = 0,

Z(x, 0) = φ(x),

0 ⩽ x < ∞,

(3.3.14)

0 ⩽ x < ∞,

(3.3.15)

and the second boundary problem Z x (0, t) = 0,

Z(x, 0) = φ(x),

in the half-unbounded domain Q∗T = {0 ⩽ x < ∞, 0 ⩽ t ⩽ T }. Theorem 3.3.5 If f (x, t, Z) and Z 0 (x, t) satisfy the conditions of (I), (II) and (III) in Q∗T , and also satisfy one of the following conditions: ¯ t), (x, t, Z) ∈ Q∗ × R3 , (IV∗1 ) f (x, t, Z) ⩽ c¯(x, t)F (Z) + d(x, (3.3.16) T where F (Z) is the continuous function in Z ∈ R3 , c(x, t) and d(x, t) ∈ L∞ (0, T ; L2 (R+ )), or (IV2∗ ) f (x, t, Z) ⩽ c(x, t)|Z|l + d(x, t), (3.3.17) where l ⩾ 0, c(x, t) ∈ L∞ (Q∗T ), d(x, t) ∈ L∞ (0, T ; Ls (R+ )), 1 < s ⩽ 2. T (1,1) Then Z(x, t) ∈ L∞ (0, T ; W (1) (R+ )) Cloc2 4 (Q∗T ) is existed that is the global weak solution of the second boundary condition (3.3.15) of the generalized LandauLifshitz equation (3.3.4). 3. Consider the nonlinear initial-boundary value problem of one-dimensional Landau-Lifshitz equation Z t = Z × Z xx + f (x, t, Z),

0 < x < l, t > 0,

(3.3.18)

3.3

Landau-Lifshitz equations

135

( Z x (0, t) = grad ψ0 (t, Z(0, t)),

t ⩾ 0,

− Z x (l, t) = grad ψ1 (t, Z(l, t)), Z(x, 0) = φ(x),

t ⩾ 0,

0 ⩽ x ⩽ l,

(3.3.19) (3.3.20)

where ψ0 (t, Z) and ψ1 (t, Z) are quantity functions, grad represents a gradient operator for Z. We give the following assumptions. (A) The functions ψ0 (t, Z) and ψ1 (t, Z) have the continuous derivative for t, and have the second order continuous mixed derivative for t ∈ [0, T ] and Z ∈ R3 . 3×3 Hessian matrix H0 (t, Z) = grad grad ψ0 (t, Z) and H1 (t, Z) = grad grad ψ1 (t, Z) are nonnegative definite, and grad ψ0 (t, 0) = grad ψ1 (t, 0) = 0. (B) The matrix fZ (x, t, Z) is semi-bounded, i.e., there is a constant b, such that ξ · fZ (x, t, Z)ξ ⩽ b|ξ|2 . (C) φ(x) ∈ H 2 (0, l), and satisfies the nonlinear boundary conditions at the endpoints of interval [0, l]. Theorem 3.3.6 If the conditions of (A), (B) and (C) are satisfied, then the nonlinear initial-boundary value problem (3.3.18) ∼ (3.3.20) of the generalized LandauLifshitz equation exists the global weak solution \ 1 1 Z(x, t) ∈ L∞ (0, T ; H 1 (0, l)) C ( 2 , 4 ) (QT ). Consider the following mixed-boundary value problem Z x (0, t) = grad ψ0 (t, Z(0, t)),

(3.3.21)

Z(l, t) = 0,

(3.3.22)

Z(x, 0) = φ(x).

(3.3.23)

We replace the condition (A) by the following condition (A′ ). (A′ ) Hessian matrix H0 (t, Z) = grad grad ψ0 (t, Z) is nonnegative definite which is made up of these functions ψ0t (t, Z) ∈ C 0 , ψ0 Z t ∈ C 0 , ψ0zz ∈ C 0 , and ψ0 (t, Z). Theorem 3.3.7 If it satisfies the conditions of (A′ ), (B), (C), and f(l,t,0)=0, then the mixed-nonlinear boundary value problem (3.3.18), (3.3.21) ∼ (3.3.23) of the Landau-Lifshitz equation exists the global weak solution Z(x, t), \ 1 1 Z(x, t) ∈ L∞ (0, T ; H 1 (0, l)) C ( 2 , 4 ) (QT ). Consider the mixed-nonlinear boundary value problem in the semi-unbounded domain Q∗T = {x ∈ R+ , 0 ⩽ t ⩽ T }, Z x (0, t) = grad ψ0 (t, Z(0, t)), Z(x, 0) = φ(x),

t ⩾ 0,

0 ⩽ x < ∞.

(3.3.24) (3.3.25)

136

Chapter 3

Some Results for the Studies of Some Nonlinear Evolution Equations

Theorem 3.3.8 Suppose that it satisfies the conditions of (A′ ), and (B∗ ) Jacobi derivative matrix fZ is semi-bounded in Q∗T × R3 , i.e., there is a constant b, such that ξ · fZ (x, t, Z)ξ ⩽ b|ξ|2 , ∀ξ ∈ R3 , (x, t, Z) ∈ Q∗T × R3 , (C∗ ) φ(x) ∈ H 1 (R+ ) and satisfies the boundary conditions (3.3.24) at the point x = 0, (D∗ ) f (x, t, Z) , fx (x, t, Z) , ft (x, t, Z) ⩽ a(x, t)F (Z)+b(x, t) or f (x, t, Z) , fx (x, t, Z) , ft (x, t, Z) ⩽ c(x, t)|Z|k + d(x, t), where a(x, t), b(x, t), d(x, t) ∈ L∞ (0, T ; L2 (R+ )), c(x, t) ∈ L∞ (Q∗T ), k ⩾ 0, F (Z) ∈ C 3 . Then the mixed-nonlinear boundary value problem (3.3.18), (3.3.24), (3.3.25) of the Landau-Lifshitz equation exists the global weak solution Z(x, t) in the semiunbounded domain Q∗T , and Z(x, t) ∈ L∞ (0, T ; H 1 (R+ ))

\

( 1 , 41 )

Cloc2

(Q∗T ).

4. Consider the initial-boundary value problem of m-dimensional (m ⩾ 2) Landau-Lifshitz equations Z t = Z × ∆Z + f (x, t, Z), Z(x, t) = 0,

x ∈ Ω, t > 0,

x ∈ ∂Ω,

Z(x, 0) = φ(x),

0 ⩽ t ⩽ T, x ∈ Ω,

(3.3.26) (3.3.27) (3.3.28)

where Ω ⊂ Rn is bounded domain, and have smooth boundary ∂Ω ∈ C 2 . Suppose that the following conditions is satisfied. ∂f (x, t, Z) is semi-bounded for Z, that is, (Im ) 3 × 3 Jacobi derivative matrix ∂Z there is a constant b, such that ξ · fZ ξ ⩽ b|ξ|2 ,

∀ξ ∈ R3 .

(IIm ) Equations is homogeneous, i.e., f (x, t, 0) ≡ 0, and  |f (x, t, Z)| ⩽ A|Z|l + B,

2  |∇f (x, t, Z)| ⩽ A|Z|1+ m + B,

(IIIm ) φ(x) ∈ H01 (Ω).

4 (m ⩾ 2), m−2 A > 0, B > 0.

2⩽l⩽

(3.3.29)

3.3

Landau-Lifshitz equations

137

Theorem 3.3.9 If it satisfies the conditions of (Im ), (IIm ) and (IIIm ), then the initial-boundary value problems (3.3.26) ∼ (3.3.28) of the Landau-Lifshitz equations exist the global weak solution ∞

Z(x, t) ∈ L

(0, T ; H01 (Ω))

\

)

(

C

0, 3+[1m ] 2

(0, T ; L2 (Ω)).

Theorem 3.3.10 If f (x, t, Z) ∈ C 2 , x ∈ Rm , Z ∈ R3 , Jacobi derivative matrix is semi-bounded, then the initial-boundary value problems (3.3.26) ∼ (3.3.28) exist the unique smooth solution Z(x, t) ∈ C (3,1) (QT ). Theorem 3.3.11 Suppose that it satisfies the following conditions (i) Z · f (x, t, Z) ⩾ c0 |Z|2+δ , (x, t) ∈ QT , Z ∈ R3 ,

(3.3.30)

for c0 > 0, δ > 0. (ii) The initial function ∥φ(x)∥L2 > 0, (2,1)

then the generalized solution Z(x, t) ∈ W2 (QT ) of the initial-boundary value problems (3.3.26) ∼ (3.3.28) blow up in the limited time, i.e., ∃t0 > 0, s.t., lim ∥Z(·, t)∥Lp (Ω) = +∞,

t→t0

2 ⩽ p < ∞.

(3.3.31)

5. Consider the geometrical generalization of Landau-Lifshitz equations, that is the case is extended from Z ∈ R3 to Z ∈ RN (N > 3). Now, we consider the generalization of Landau-Lifshitz equations, Z t =∗ [Z ∧ g1 (Z) ∧ · · · ∧ gn−2 (Z) ∧ ∆Z] + f (x, t, Z),

(3.3.32)

where Z = (Z1 , Z2 , · · · , ZN ) is N -dimensional vector-valued function (N ⩾ 2), x = (x1 , x2 , · · · , xn ) ∈ Rn , t ∈ R+ , ∧ is exterior product, ∆ is N -dimensional Laplacian operator, ∗ is Hodge star operator, gk (Z) is N -dimensional vector-valued function (k = 1, 2, · · · , n − 2; n ⩾ 2), Z ∈ RN , f (x, t, Z) is N -dimensional vectorvalued function, x ∈ Rn , x ∈ R+ , Z ∈ RN . Equations (3.3.32) can be rewritten as follows Z t = A (Z)∆Z + f (x, t, Z),

(3.3.33)

where A (Z) is N × N matrix, and Z ∈ RN . Equations (3.3.32) or (3.3.33) are the nonlinear Schrödinger equations with component form for N = 2.

138

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Some Results for the Studies of Some Nonlinear Evolution Equations

Equations (3.3.32) are Z t = g1 (z) × ∆Z + f (x, t, Z),

(3.3.34)

where × is exterior product of two 3-dimensional vectors for N = 3. If g(Z) ≡ Z, we get Landau-Lifshitz equations. Generally, coefficient matrix A (Z) of second derivative term in equation (3.3.33) is the anti-symmetric and null-defined. If n ⩾ 3, det |A (Z)| = 0, so equations (3.3.32) or (3.3.33) are strongly coupling, strongly degeneracy, and quasi-linearity. If n = 1, for the cauchy problem of equations (3.3.32) and various types of initial boundary value problems mentioned earlier, we can prove the existence of its the global weak solution Z(x, t) ∈ L∞ (0, T ; H 1 ). If n ⩾ 2, for the initial-boundary value problems of the following equations, Z t =∗ (Z ∧ α2 ∧ · · · ∧ αn−2 ∧ ∆Z) + f (x, t, Z), Z(x, t) = 0,

x ∈ ∂Ω, 0 ⩽ t < ∞,

Z(x, 0) = φ(x),

x ∈ Ω,

(3.3.35) (3.3.36) (3.3.37)

we get the existence of their global weak solution + 1 Z(x, t) ∈ L∞ loc (R ; H0 (Ω)) ∩ C

(0, 2+[1 n ] ) 2

(R+ ; L2 (Ω)),

where Z(x, t) = (z1 (x, t), z2 (x, t), · · · , zN (x, t)), ak is the linearly independent constant vector in RN (k = 2, 3, · · · , n − 2). 6. Landau-Lifshitz equations and harmonic map in Riemannian manifold. Suppose that (M, γ) and (N, g) are two compact Riemannian manifolds, dim M = n. Firstly, we consider the situation M is without boundary, N = S 2 . Consider the following Landau-Lifshitz equations (M → S 2 ) Z t = −α1 Z × (Z × ∆M Z) + α2 Z × ∆M Z,

(3.3.38)

where α1 > 0, α2 is constant, and ∆M is Laplace-Beltrami operator.   1 ∂ ∂ αβ √ ∆M i = √ γ γ α γ ∂xβ ∂x = γ αβ in local coordinates (x1 , x2 , · · · , xn ).

∂2 ∂xα ∂y β

− Γkαβ

∂ , ∂xk

(3.3.39)

3.3

Landau-Lifshitz equations

139

The heat flow equations of harmonic mapping M → S 2 is Z t = ∆M Z + |∇Z|2 Z, where |∇Z|2 =

XX αβ

γ αβ

i

(3.3.40)

∂ui ∂ui . ∂xα ∂xβ

(3.3.41)

We obtain the connection between the solution of Landau-Lifshitz equations and harmonic map, and get the following results. (i) In the classical sense, there are equivalent that the solution of equations (3.3.38) with initial value Z 0 (x), |Z 0 (x)|2 = 1,

x ∈ M,

(3.3.42)

and the solution of equations   1 ∂ αβ √ ∂Z Z t = α1 √ γ α + α1 |∇Z|2 Z γ γ ∂xβ ∂x   1 ∂ αβ √ ∂Z γ α +α2 Z × √ γ γ ∂xβ ∂x

(3.3.43)

with the same initial value. So the heat flow equations of Landau-Lifshitz equations and harmonic map are equivalence for α1 = 1, α2 = 0. (ii) In the classical sense, Z : M → S 2 is the harmonic map, if and only if Z satisfies Landau-Lifshitz equations (3.3.38) and Z t (x, t) = 0, t ⩾ 0. Let V (M T ; S 2 ) = {Z : M × [τ, T ] → S 2 |Z be measurable R RT R ess sup0⩽t⩽T M |∇Z(·, t)|2 dM + τ M (|∇2 Z|2 + |Z t |2 )dM dt < ∞}, then we have the following result (iii). (iii) If Z 1 , Z 2 ∈ V (M T ; S 2 ) are the solutions of Landau-Lifshitz equations with initial value problem Z 1 (x, 0) = Z 2 (x, 0) = Z 0 (x), then Z 1 (x, t) = Z 2 (x, t). (iv) Except for limited points, for any initial value Z 0 (x) ∈ H 1,2 (M, S 2 ), the equation (3.3.43) exists the unique regular solution in M × [0, +∞). lim

sup ER (Z(·, t), xk ) > ε1 ,

T →T k T 0, then problems (3.3.47) and (3.3.49) exist the global weak solution Z(x, t), Z(x, t) ∈ L∞ (0, T ; H 1 (M )) ∩ C ( 2 , r∗ ) (0, T ; Lr (M )), 1

1

2n n 1 1 , }, + ∗ = 1. l(n − 2) n − 1 r r The initial value problem of the Landau-Lifshitz equations are considered in this subsection not only on the classical sense but also on the Riemann manifold. Due to the space limitation, the details refer to [29, 117, 133, 142, 152, 163]. The well-posedness of the system of ferromagnetic chain, which is a general case of the Landau-Lifshitz equations, refers to [245, 247–249, 254, 256]. where r = min{

Chapter 4 Similarity Solution and the Painlevé Property for Some Nonlinear Evolution Equations It is well known that when all active singular point of an ordinary differential equation are single-valued, i.e., there are simple singular points, then this kind of ordinary differential equations are called as having the Painlevé property. Experience shows when ordinary differential equations has the Painlevé property, then they are integrable. In 1981, the famous experts in soliton, Ablowitz et al. [4] showed that if a partial differential equation can be solved by scattering inversion method, then corresponding to this partial differential equation, the ordinary differential equation obtained by using a similar method has the Painlevé property. They still conjecture that if the obtained ordinary differential equation by using similarity method such as infinitesimal transformation, or transformations of the appropriate independent variable and dependent variable, has a Painlevé property, then the partial differential equation is integrable. Hence, the relationship between integrability of given systems and the Painlevé property, as well as the infinitesimal transformation different from the dimensional method to solve the similarity solution, has become one of the important topics in partial differential equations. Similar to ordinary differential equation, if the solution of a given partial differential equation is single-valued about the activity singular manifold, then this partial differential equation has a Painlevé property, by which we can investigate Bäcklund transformation, soliton solution and integrability of this equation. In this chapter, the classical and non-classical infinitesimal transformations, Lie algebra structure of infinitesimal operator will be studied by some concrete examples of partial differential equations, and the similarity equations and similarity solutions are obtained, see [13, 19, 33, 128] and references therein. On the other hand, taking KdV equation, nonlinear Schrödinger equation and Boussinesq equation as examples, we prove that these equations have Painlevé properties, and further obtain their Bäcklund transformation, one can see [239, 240].

142

Chapter 4

4.1

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

Classical infinitesimal transformations

Second-order partial differential equation H(uxx , uxt , utt , ux , ut , u, x, t) = 0

(4.1.1)

have m boundary conditions Bβ (ux , ut , u, x, t) = 0,

(4.1.2)

(x, t) is in boundary curve ωβ (x, t) = 0,

β = 1, 2, · · · , m,

(4.1.3)

where u(x, t) is dependent variable, and x, t are independent variables. The above problem for determining solutions (4.1.1), (4.1.2), and (4.1.3) is denoted for the system S. Consider the following one parameter (ε) Lie transformation group   u∗ = u∗ (x, t, u; ε),   (4.1.4) x∗ = x∗ (x, t, u; ε),   t∗ = t∗ (x, t, u; ε). If u = θ(x, t) is a solution of the system S, we use v instead of u, x∗ = x∗ (x, t, θ(x, t); ε) instead of x, and t = t∗ (x, t, θ(x, t); ε) instead of t in the system S. This resulting system is denoted as S ∗ . Suppose that v = θ(x∗ , t∗ ) is a solution of the system S ∗ . Definition 4.1.1 If u = θ(x, t) is a solution of the system S, then V = u∗ (x, t, θ(x, t); ε) is also a solution of the system S ∗ . The system S is invariant in transformation (4.1.4). So we obtain θ(x, t) that is satisfied the functional equation with one parameter ε, θ(x∗ (x, t, θ(x, t); ε), t∗ (x, t, θ(x, t); ε)) = u∗ (x, t, θ(x, t); ε).

(4.1.5)

(4.1.4) expand respect to variables ε = 0, and obtain the following infinitesimal term (η, ζ, τ )(o(ε)),   u∗ = u + εη(x, t, u) + o(ε2 ),   (4.1.6) x∗ = x + εζ(x, t, u) + o(ε2 ),   t∗ = t + ετ (x, t, u) + o(ε2 ). Using (4.1.6), functional differential equation is expanded respect to variables ε = 0, θ(x + εζ(x, t, θ) + o(ε2 ), t + ετ (x, t, θ) + o(ε2 )) = θ(x, t) + εη(x, t, θ) + o(ε3 ). (4.1.7)

4.1

Classical infinitesimal transformations

143

o(ε) term of functional differential equation (4.1.7) results as follows one order partial differential equations ((η, ζ, τ )) of θ(x, t), ζ(x, t, θ)θx + τ (x, t, θ)θt = η(x, t, θ).

(4.1.8)

Equation (4.1.8) is the invariant curved surface condition. General solution of equation (4.1.8) is obtained the following characteristic equation dx dt dθ = = . ξ(x, t, θ) τ (x, t, θ) η(x, t, θ)

(4.1.9)

In principle, general solution of equation (4.1.9) is obtained, and have two parameters, where one of them is an independent variable ζ(x, t, θ), which is called similarity variations, and another is dependent variable f (ζ). We get the similar form θ = F (x, t, f (ζ)),

(4.1.10)

substitute (4.1.10) into (4.1.1), and get f (ζ) which is satisfied the differential equations. Specifically, from the integral of the first equation (4.1.9), we get ζ(x, t) = const (⇒ x = g(ζ, t)).

(4.1.11)

It defines similar curved line in plane (x, t). From the second equation of (4.1.9), dt dθ = , τ (g(ζ, t), t, θ) τ (g(ζ, t), t, θ)

(4.1.12)

G(t, ζ(x, t), θ) = const = f (ζ).

(4.1.13)

we get the solution

That is, (4.1.10) holds θ = F (x, t, f (ζ(x, t))). In search of infinitesimal transformations, we must calculate some derivative transformations, such that u = θ(x, t) is some solution of the system S, and v = u∗ (x, t; θ(x, t); ε) is a corresponding solution of the system S ∗ , for example
∂ ∂x = ∗ x∗ − εζ(x, t, θ) + o(ε2 ) ∗ ∂x ∂x   ∂ζ ∂θ ∂x ∂ζ ∂x =1 − ε + + o(ε2 ) ∂x ∂x∗ ∂u ∂x ∂x∗   ∂ζ ∂ζ =1 − ε + θx + o(ε2 ). ∂x ∂u

(4.1.14)

144

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

Similarly, we have    ∂ζ ∂ζ ∂x   = −ε + θt + o(ε2 ),  ∗  ∂t ∂t ∂u       ∂t ∂τ ∂τ =1−ε + θt + o(ε2 ),  ∂t∗ ∂t ∂u       ∂τ ∂τ ∂t    ∗ = −ε + θt + o(ε2 ), ∂x ∂t ∂u

(4.1.15)

and u∗ = u∗ (x, t, θ(x, t); ε) = θ(x, t) + εη(x, t, θ) + o(ε2 ). Thus, we get the transform of one order and two order derivative transformation ∂u∗ ∂(θ(x, t) + εη(x, t, θ)) = + o(ε2 ) ∂x∗ ∂x∗ ∂(θ(x, t) + εη(x, t, θ)) ∂x = ∂x ∂x∗ ∂t + ∗ θt + o(ε2 ). ∂x

(4.1.16)

Substituting (4.1.14), (4.1.15) to (4.1.16), we have    ∂u∗ ∂η ∂η ∂ζ = θ + ε + − θx x ∂x∗ ∂x ∂u ∂x  ∂τ ∂ζ 2 ∂τ − θt − θx − θx θt ∂x ∂u ∂u +o(ε2 ).

ηx and ηt are the infinitesimal of ∂η + ηx = ∂x



(4.1.17)

∂u∗ ∂u∗ and , then we get ∂x∗ ∂t∗

 ∂τ ∂η ∂ζ ∂ζ 2 ∂τ − θx − θt − θ − θx θt . ∂u ∂x ∂x ∂u x ∂u

(4.1.18)

Similarly, we have ∂η ηt = + ∂t



 ∂η ∂τ ∂ζ ∂τ 2 ∂ζ − θt − θx − θ − θx θt , ∂u ∂t ∂t ∂u t ∂u

(4.1.19)

4.1

Classical infinitesimal transformations

  ∂2η ∂2η ∂2τ ∂2ζ ηxx = 2 + 2 θ − θt − x ∂x ∂x∂u ∂x2 ∂x2  2  ∂2ζ ∂ η ∂2τ 2 − 2 + θ − 2 θx θt x ∂u2 ∂x∂u ∂x∂u   ∂2ζ 3 ∂2τ 2 ∂η ∂ζ − 2 θx − θ θt + −2 θxx ∂u ∂u2 x ∂u ∂x ∂ζ ∂τ −2 θxt − 3 θxx θx ∂x ∂u ∂τ ∂τ − θxx θt − 2 θxt θx , ∂u ∂u  2  ∂2η ∂ η ∂2τ ∂2ζ ηtt = 2 + 2 − 2 θt − 2 θx ∂t ∂t∂u ∂t ∂t  2  2 ∂ η ∂2τ ∂ ζ + − θx θt θ2 − 2 ∂u2 ∂t∂u t ∂t∂u   ∂2τ 3 ∂2ζ 2 ∂η ∂τ − 2 θt − θ θx + −2 θtt ∂u ∂u2 t ∂u ∂t ∂ζ ∂τ −2 θxt − 3 θtt θt ∂t ∂u ∂ζ ∂ζ − θtt θx − 2 θxt θt , ∂u ∂u

145

(4.1.20)

(4.1.21)

and  2  ∂ η ∂2τ ∂2η + − θt ηxt = ∂x∂t ∂x∂u ∂t∂x   2 ∂ η ∂2ζ ∂2τ 2 − θx − θ + ∂t∂u ∂t∂x ∂x∂u t  2  ∂ η ∂2τ ∂2ζ 2 ∂2ζ + − θ θ − θ − x t ∂u2 ∂x∂u ∂u∂t ∂t∂u x ∂2τ ∂2ζ ∂τ θtt − 2 θx θt2 − θt θx2 − 2 ∂u ∂u  ∂x  ∂η ∂ζ ∂τ ∂ζ + − − θxt − θxx ∂u ∂x ∂t ∂t ∂τ ∂ζ −2 θt θxt − 2 θx θxt ∂u ∂u ∂τ ∂ζ − θx θtt − θt θxx . ∂u ∂u

(4.1.22)

Equation (4.1.1) is invariant in infinitesimal transformations (4.1.4), so we have H(u∗x∗ x∗ , u∗x∗ t∗ , u∗t∗ t∗ , u∗x∗ , u∗t∗ , u∗ , x∗ , t∗ ) = 0.

(4.1.23)

146

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

Therefore, equation (4.1.1) is invariant in infinitesimal transformations (4.1.4), the necessary and sufficient conditions are that infinitesimal term (o(ε)) ≡ 0 in equation (4.1.23). We treat the equation (4.1.23) as an equation of eight variables with its own infinitesimal items. So equation (4.1.1) is invariant in the transformations (4.1.4). Given solution u = θ(x, t) of equation (4.1.1) satisfies the following equation ηxx

∂H ∂H ∂H ∂H ∂H ∂H ∂H ∂H +ζ +τ = 0. (4.1.24) + ηxt + ηtt + ηx + ηt +η ∂uxx ∂uxt ∂utt ∂ux ∂ut ∂u ∂x ∂t

In search of infinitesimal transformations (4.1.4), we can get it from equation (4.1.24). At this point, the equation contains derivatives of various θ, and its coefficient depends on (θ, x, t) and unknown (η, ζ, τ ). Merger of similar items, coefficient of various derivatives of θ is 0. That is, we get the determining equation of infinitesimal transformations. Therefore, we get (η, ζ, τ ). Example 4.1.1 Consider the similar solution of heat conduction equation uxx − ut = 0. Using Lie group transformation of infinitesimal (η, ζ, τ ),   u∗ = u∗ (x, t, u; ε) = u + εη(x, t, u) + o(ε2 ),   x∗ = x∗ (x, t, u; ε) = x + εζ(x, t, u) + o(ε2 ),   t∗ = t∗ (x, t, u; ε) = t + ετ (x, t, u) + o(ε2 ).

(4.1.25)

(4.1.26)

Equation (4.1.25) is invariant in the transformation of (4.1.26). From (4.1.19) and (4.1.20), u = θ(x, t) satisfies u∗xx∗ − u∗t∗ = θxx − θt + ε(ηxx − ηt ) + o(ε2 ), where

 ηxx − ηt =

   ∂2η ∂2η ∂2ζ ∂η ∂ζ + 2 − θx − + ∂x2 ∂t ∂x∂u ∂x2 ∂t    2  ∂τ ∂η ∂2τ ∂ η ∂2ζ + − − θ + − 2 θ2 t ∂t ∂u ∂x2 ∂u2 ∂x∂u x   ∂ζ ∂2τ ∂τ 2 + −2 θx θt + θ ∂u ∂x∂u ∂u t   ∂2ζ ∂2τ 2 ∂η ∂ζ − 2 θx3 − θ θ + − 2 θxx t ∂u ∂u2 x ∂u ∂x ∂τ ∂ζ ∂τ θxx θt −2 θxt − 3 θxx θx − ∂x ∂u ∂u ∂τ −2 θxt θx . ∂u

(4.1.27)

4.1

Classical infinitesimal transformations

147

u = θ(x, t) satisfies (4.1.25), and substituting θt to θxx in the above equation. This equation reduces to the following simplified system  2    ∂η ∂ζ ∂ η ∂2η ∂2ζ − + ηxx − ηt = + 2 − θx ∂x2 ∂t ∂x∂u ∂x2 ∂t   2   ∂ η ∂2ζ ∂τ ∂ζ ∂2τ θ + − 2 + −2 − θ2 t ∂t ∂x ∂x2 ∂u2 ∂x∂u x   ∂2τ ∂2ζ 3 ∂ζ −2 θx θt − θ + −2 ∂u ∂x∂u ∂u2 x ∂2τ ∂τ ∂τ − 2 θx2 θt − 2 θxt − 2 θxt θx . (4.1.28) ∂u ∂x ∂u Equation (4.1.25) is invariant in transformation (4.1.26), the necessary and sufficient condition is that ηxx − ηt = 0, ∀x, t, u = θ(x, t). Suppose that the same derivative terms’ coefficient of θ is 0 in (4.1.28), that is, let the coefficients of θx , θtx , θt , θx θtx , θi θx , θt2 be 0. Items that do not contain a derivative θ are preserved in (4.1.28), then we get solutions ζ, η, τ that are satisfied partial differential equations. Therefore, we have ∂τ θx θtx : = 0, ∂u ∂ζ (4.1.29) = 0, θt θx : ∂u ∂2η θx2 : = 0, ∂u2   η = f (x, t)u + g(x, t),   ζ = X(x, t),   τ = T (x, t),

(4.1.30)

where f , g, X, and T are arbitrary functions of x, t. Let coefficient of θtx , θx and θt , and reserved item are 0, we get T = T (t), 2

′ ∂X − T (t) = 0, ∂x

(4.1.31) (4.1.32)

∂X ∂2X ∂f − +2 = 0, 2 ∂t ∂x ∂x

(4.1.33)

∂2f ∂f − = 0, 2 ∂x ∂t

(4.1.34)

∂2g ∂g − = 0. ∂x2 ∂t

(4.1.35)

148

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

Notice that g(x, t) is an arbitrary solution to the equation (4.1.25), we consider the case of g(x, t) = 0, and solve the equations (4.1.31)∼(4.1.34). The group transformations of heat conduction equation is obtained    ζ = k + δt + βx + γxt,   τ = α + 2βt + γt2 , (4.1.36)   2     x t δx   + + λ u, − η = −γ 4 2 2 where α, β, γ, δ, k, and λ are six arbitrary constants. From characteristic equation dt du dx = = , ζ τ η

(4.1.37)

we have similar variable, similar layout and similar solution, where ζ, τ and η are defined by (4.1.36). And then we discuss the following four situations. kγ − δβ kβ − δα x − (At + B) (i) β 2 ̸= αγ, ζ = p , where A = ,B= . Similar 2 2 αγ − β αγ − β 2 α + 2βt + γt layout of solution is ρ γt + β − C u = θ(x, t) = F (ζ) 1 (α + 2βt + γt2 ) 4 γt + β + C   Aζ p t exp − (A2 + γζ 2 ) + α + 2βt + γt2 , (4.1.38) 4 2   p 1 β 1 2 +λ+ (δ − A2 C 2 ) . Substituting (4.1.38) where C = β 2 − αγ, ρ = 2C 2 4γ into heat conduction equation (4.1.25), we get the following differential equation of F (ζ). 1



F ′′ + βζF ′ + (Dζ 2 + E)F = 0,   αγ A2 C 2 δ2 β where D = ,E= − λ+ = −2Cρ + . 2 4γ 4 2 2 √ − βξ4 Let z = ζ C, F (ζ) = G(z)e , (4.1.39) implies that d2 G + dz 2



 1 1 + ν − z 2 G = 0, 2 4

(4.1.39)

(4.1.40)

β 2 − 1 = −2ρ − 1 . This equation is standard hypergeometric type, its where ν = C 2 2 solution can be expressed as parabolic cylinder functions. Dν (z), Dν (−z), D−ν−1 (iz), E−

4.1

Classical infinitesimal transformations

149

D−ν−1 (−iz), any two of them are linearly independent solutions. They have many co-known properties.    1  2 Γ γ z2  1 1 1 2 − 4   2  Dν (z) = 2 e , F1 − ν, ; z  1 1 2 2 2 Γ − ν 2 2    1   Γ − 1 1 1 3 1 2  2  , F1 +2− 2 z  − ν; ; z  . 1 2 2 2 2 Γ − ν 2 Solutions are expressed as orthogonal Hermite polynomial Dn (z) = e−

z2 4

Hen = (−1)n e He0 = 1,

Hen (z),

z2 2

He1 = z,

dn − z 2 e 2, dz n He2 = z 2 − 1, · · ·

for integeres ν = n = 0, 1, 2, · · · , and then D−1 (z) = e

z2 4

 1  1 (2π)− 2 erf c 2− 2 z , · · · ,

where series of Dν (z) is  z2 3   z ν e− 4 (1 + o(z −2 )), | arg z| < π,   4   1   2 (2π) 2 νπi −ν−1 z2  ν − z4 2   z e e z e4 (1 + o(z )) −   Γ(−ν)    π 5 Dν (z) = · (1 + o(z −2 )), π > arg z > ,  4 4   1   2 z (2π) 2 −2πi −ν−1 z2    z ν e− 4 (1 + o(z −2 )) − e z e4   Γ(−ν)     π 5   · (1 + o(z −2 )), − > arg z > − π, 4 4 for small z, and |z| ≫ |ν|. Parabolic cylinder functions are entire functions of z. (ii) β 2 = αγ, γ ̸= 0,   1 k − δβ . ζ = x+δ+ 2(t + β) t + β Similar layout of solution is   F (ζ) L2 M Lζ ζ 2 (t + β) θ= √ exp + − − , 12(t + β)3 t+β 2(t + β) 4 t+β

(4.1.41)

150

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

k − δβ where L = and M = − 2



 β δ2 +λ+ . F (ζ) satisfies the equation 2 4

d2 F − (2ζ − M )F = 0. dζ 2

(4.1.42)

1

Let z = L 3 ζ, F (ζ) = G(z), (4.1.42) implies that d2 G − (z − ν)G = 0, dz 2 where

(4.1.43)

M

ν=

2 . L3 Airy function Ai (z − ν) and Bi (z − ν) are linearly independent solutions of 2 3 (4.1.43), and are defined by 3−1 order Bessel function. Let ω = z 2 , then 3 r z −1 Ai (z) = π K 1 (ω), 3 3  1√  Ai (−z) = z J 13 (ω) + J− 13 (ω) , 3 r   z I− 13 (ω) + I 13 (ω) , Bi (z) = 3 r   z Bi (−z) = J− 13 (ω) − J 13 (ω) . 3

(iii) β 2 = αγ, β = γ = 0, α ̸= 0, ζ =x−

k δt2 − t. 2α α

Similar layout of solution is  2  δ kδt2 δ u = θ(x, t) = F (ζ) exp − t3 − + λt − ζt , 12 4 2

(4.1.44)

where F (ζ) satisfies the equation d2 F dF +k + 2 dζ dζ



 δ ζ − λ F = 0. 2

(4.1.45)

  31 δ k Let z = − ζ, F (ζ) = G(z)e− 2 ζ , δ ̸= 0, then (4.1.45) implies that 2 d2 G − (z − ν)G = 0, dz 2

(4.1.46)

4.1

Classical infinitesimal transformations

where

 ν=−

151

   23 k2 2 . +λ 4 δ

(4.1.46) is the same as equation (4.1.45). If k = 0 and using classical method of separation of variables, then   √ √ u = eλt A cos −λx + B sin −λx . (iv) α = β = γ = 0, δ ̸= 0, ζ = t. At this time, similar layout of solution is x

u = θ(x, t) = F (t)e 4(t+k) (λ−x) ,

(4.1.47)

where F (t) satisfies the one order equation λ2 − 8(t + k) dF = F (t), dt 16(t + k)2 therefore, 2

(x− 2 ) P θ(x, t) = √ e− 4(t+k) . t+k λ

Example 4.1.2 equation

Consider the similar solution of nonlinear heat conduction ∂ ∂x

  ∂u ∂u , k(u) = ∂x ∂t

(4.1.48)

with the initial-boundary value conditions u(x, 0) = 0, u(0, t) = f (t),

x > 0, t ⩾ 0.

(4.1.49) (4.1.50)

We consider invariant property of the problem for determining solution (4.1.48) ∼ (4.1.50) in infinitesimal transformations. Firstly consider invariant property of (4.1.48). Suppose that Lie group transformation of infinitesimal is   u∗ = u + εη(x, t, u) + o(ε2 ),   (4.1.51) t∗ = t + ετ (x, t, u) + o(ε2 ),   x∗ = x + εζ(x, t, u) + o(ε2 ), then it is easy to verify that τu = ξu = τx = 0,

(4.1.52)

for any k(u). Reserved determining equation is ζt + 2K ′ (u)ηx − K(u)ζxx + 2K(u)ηxu = 0,

(4.1.53)

152

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

K(u)τ ′ (t) + ηK ′ (u) − 2K(u)ξx = 0, ′

′

′

′′

′

K (u)τ (t) + K (u)ηu + ηK (u) − 2K (u)ζx + K(u)ηuu = 0, K(u)ηxx − ηt = 0.

(4.1.54) (4.1.55) (4.1.56)

Solving η from (4.1.54), substituting (4.1.56), and then we get ζ=

x ′ τ (t) + ax2 + bx + c, 2

τ = τ (t),

η=

K(u) (4ax + 2b) , K ′ (u)

(4.1.57)

where a, b and c are arbitrary constants. Substituting (4.1.57) into (4.1.55), if one of a and b is not 0, we get  ′′ K = 0, (4.1.58) K′ therefore, K(u) = λ(u + k)ν ,

(4.1.59)

where λ, k and ν are arbitrary constants. Finally, substituting (4.1.57) into (4.1.53), we have   4K(u)K ′′ (u) x ′′ τ (t) + 2aK(u) 7 − = 0. 2 (K ′ (u))2

(4.1.60)

There are three cases of this equation, discussing them separately. (i) For arbitrary K(u), invariant group of three parameters (α, β, γ). ζ = βx + γ,

τ = 2α + 2βt,

η = 0.

(4.1.61)

(ii) Invariant group of four parameters (α, β, γ, δ). K(u) = λ(u + k)ν , ζ = βx + γ + δx, (4.1.62)

τ = 2α + 2βt, η=

2δ (u + k) ν

(Limiting case is K(u) = λeνu ). (iii) Invariant group of five parameters (α, β, γ, δ, ρ), K(u) = λ(u + k)− 3 , 4

  ζ = βx + γ + δx + ρx2 ,    τ = 2α + 2βt,    η = − 3 δ(u + k) − 3ρ(u + k). 2

(4.1.63)

4.1

Classical infinitesimal transformations

153

Considering boundary value condition (4.1.49) and initial-boundary value condition (4.1.50), and using invariant property of t = 0, imply that α = 0. Using invariant property of x = 0, imply that γ = 0. From invariant property of (4.1.50), imply that η(0, t, f (t)) = 2f ′ (t). (4.1.64) Therefore, f (t) = const = 1, for the case (i). Case (ii), f (t) = ctB ,

(4.1.65)

where B and c are arbitrary constants (δ = βνB). Case (iii), f (t) = ctB ,

(4.1.66)

where B and c are arbitrary constants (δ = βνB). In any case, the case (iii) is meaningless, in fact, equation (4.1.48) is meaningless for u = 0. Now the case (i) and (ii) are further analyzed. (i) Arbitrary K(u), f (t) = 1, characteristic equation is

Therefore, similar variable is

dt du dx = = . x 2t 0

(4.1.67)

x ζ=√ t

(4.1.68)

(t = 0, ζ = ∞; x = 0, ζ = 0). Similar layout of solution is (4.1.69)

u = F (ζ).

Substituting (4.1.69) into (4.1.48), we get the following differential equation of F (ζ),   d dF ζ dF K(F ) + · = 0, (4.1.70) dζ dζ 2 dζ where F (ζ) is satisfied the corresponding boundary conditions F (∞) = 0,

F (0) = 1.

(ii) K(u) = λuν , f (t) = ctB . Characteristic equation is dx dt du = = , (1 + νB)x 2t 2Bu imply that similar variable is

x

ζ= t

(νB+1) 2

.

(4.1.71)

(4.1.72)

154

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

(Let νB > −1, t = 0, imply that ζ = ∞; x = 0 imply that ζ = 0.) Corresponding similar layout of solution is u = tB F (ζ). (4.1.73) Substituting (4.1.73) into (4.1.48), F (ζ) we get the following differential equation of F (ζ) ,   d νB + 1 dF ν dF ζ − BF = 0, (4.1.74) λ F + dζ dζ 2 dζ where F (ζ) satisfies the corresponding boundary conditions F (∞) = 0,

(B > 0),

F (0) = C.

(4.1.75)

Notice that the stretching group of one parameter µ, ζ ∗ = µζ,

F ∗ = µ ν F, 2

(4.1.76)

and leave (4.1.74) unchanged. In fact, let 1 z= , ζ

  1 G(z) = F , z

then (4.1.76) and (4.1.77) imply that     d νB + 1 dG dG λz 2 − BG = 0, Gν z 2 − z dz dz 2 dz

(4.1.77)

(4.1.78)

with boundary condition G(0) = 0,

G(∞) = C.

(4.1.79)

G∗ = µ− ν G.

(4.1.80)

(4.1.78) is invariant in the Lie group z ∗ = µz,

2

If G = g(z) is a solution of equation (4.1.78) and initial value condition g(0) = 0,

g ′ (0) = 1,

(4.1.81)

if g(∞) = g∞ ̸= 0, and since (4.1.78) is invariant in the transformation (4.1.80), therefore 2 (4.1.82) G(z) = µ ν g(µz) is also a solution for any µ, and then 2

G(∞) = µ ν g∞ , Choose µ, such that

 µ=

c g∞

G(0) = 0.  ν2 ,

then from (4.1.82), we have G(z) is a solution of the problem (4.1.78) and (4.1.79).

4.1

Classical infinitesimal transformations

155

Example 4.1.3 Consider the Boussinesq equation 1 utt + (u2 )xx + uxxxx = 0. 2

(4.1.83)

Suppose that one parameter Lie group transformation of infinitesimal for (x, t, u) is

x

where functions U , U (4.1.86). If

xx

ζ = x + εX(x, t, u) + o(ε2 ),

(4.1.84)

τ = t + εT (x, t, u) + o(ε2 ),

(4.1.85)

2

η = u + εU (x, t, u) + o(ε ),

(4.1.86)

ηζ = ux + εU x + o(ε2 ),

(4.1.87)

ηζζ = uxx + εU xx + o(ε2 ),

(4.1.88)

ηζζζζ = uxxxx + εU xxxx + o(ε2 ),

(4.1.89)

ητ τ = utt + εU tt + o(ε2 ),

(4.1.90)

, U

xxxx

and U

tt

are determined by equations (4.1.84) ∼

1 ητ τ + (η 2 )ζζ + ηζζζζ = 0, (4.1.91) 2 is invariant in transformations (4.1.84) ∼ (4.1.86), then Boussinesq equation (4.1.83) is invariant. Using (4.1.84) ∼ (4.1.90), and considering first order term of ε, then we have U tt + uU xx + uxx U + 2ux U x + U xxxx = 0. (4.1.92) Similarly, infinitesimal X(x, t, u), T (x, t, u) and U (x, t, u) will be obtained by solving some determining equations. Then we get X = αx + β,

T = 2αt + γ,

U = −2αu,

(4.1.93)

where α, β and γ are arbitrary constants. Similar layout is obtained by solving characteristic equation dx dt du = = . X(x, t, u) T (x, t, u) U (x, t, u) (i) α = 0, this is traveling wave solution, u(x,t) = f (z), z = γx−βt, f (z) satisfies 1 d2 f β 2 f + γ 2 f 2 + γ 4 2 = Az + B, 2 dz

(4.1.94)

where A and B are integral arbitrary constants. The first Painlevé equation d2 w = 6w2 + Z dz 2

(4.1.95)

156

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

Chapter 4

is obtained for γ = 0. (ii) α = ̸ 0, we have g(z) u(x, t) = γ , t+ 2α

β α , z= γ  12 t+ 2α x+

(4.1.96)

where g(z) satisfies z 2 d2 g 7z dg d2 g + + 2g + g + 4 dz 2 4 dz dz 2



dg dz

2 +

d4 g = 0, dz 4

which is obtained by the forth Painlevé equation  2 dw 3 b 1 d2 w + w3 + 4zw2 + 2(z 2 − a)w + . = 2 dz 2w dz 2 w

4.2

(4.1.97)

(4.1.98)

Structure of Lie algebra for infinitesimal operator

Let’s illustrate Lie algebra structure of infinitesimal operator in § 4.1 by group transformation   ζ = k + δt + βx + γxt,     τ = α + 2βt + γt2 , (4.2.1)  2    x t δ  f = −γ + − x + λ,  4 2 2 of heat conduction equation uxx − ut = 0. Definition 4.2.1 For Lie group transformation with one parameter ε.   u∗ = U ∗ (x, t, u; ε) = u + εη(x, t, u) + o(ε2 ),   (4.2.2) x∗ = X ∗ (x, t, u; ε) = x + εζ(x, t) + o(ε2 ),   t∗ = T ∗ (x, t, u; ε) = t + ετ (x, t) + o(ε2 ), we define infinitesimal operator with X = ζ(x, t)

∂ ∂ ∂ + τ (x, t) + η(x, t, u) . ∂x ∂t ∂u

(4.2.3)

For some neighborhood of ε = 0, we have ! ∞ n n X ε X u∗ = eεX u = u n! n=0   ε2 ∂η ∂η ∂η = u + εη(x, t, u) + ζ(x, t) + τ (x, t) + η(x, t, u) + · · · , (4.2.4) 2! ∂x ∂t ∂u

4.2

Structure of Lie algebra for infinitesimal operator

 ∂ζ ∂ζ x =e ζ +τ + ··· , ∂x ∂t   ∂τ ∂τ ε2 ∗ εX ζ +τ + ··· . t = e t = t + ετ (x, t) + 2! ∂x ∂t ∗

εX

ε2 x = x + εζ(x, t) + 2!

157



(4.2.5) (4.2.6)

For any function F (x, t, u) ∈ C ∞ , we get F (x∗ , t∗ , u∗ ) = eεX F (x, t, u).

(4.2.7)

Let Xj (j = 1, 2, · · · , 6) denote the infinitesimal operator in (4.2.1) corresponding to parameters k, α, λ, β, γ and δ respectively, then we have  ∂ ∂ ∂   X = , X2 = , X3 = u ,   1 ∂x ∂t ∂u     ∂ ∂   + 2t , X4 = x ∂x ∂t  2  (4.2.8) ∂ x ∂ t ∂   X = xt + 2t − + u ,  5   ∂x ∂t 4 2 ∂u      X6 = t ∂ − x u ∂ . ∂x 2 ∂u Definition 4.2.2 If each element of Lie algebra L satisfies the following combination law (also known as switching) for the operation ⊗ : (i) If X ∈ L , D ∈ L , then X ⊗ D ∈ L (Closeness); (ii) X ⊗ D = −D ⊗ X ; (iii) If L , X , D and w ∈ L , then we have Jacobi identity X ⊗ (D ⊗ w) + D ⊗ (w ⊗ X ) + w ⊗ (X ⊗ D) = 0, then Lie algebra L is the vector space defined on a domain F . Obviously, vector set with vector multiplication cross ⊗ in three-dimensional Euclidean space R3 is a typical example of Lie algebra. We know that infinitesimal operators with Lie group transformation form a vector space in the real number field or complex number field. In order to make this vector space into a Lie algebra µ, we need to introduce an switching operator [·, ·]. If X , D ∈ µ, then X ⊗ D = [X , D] = X D − DX , where [X , D] is commutant of infinitesimal operators X and D. It is easy to verify that the above combination law (i) ∼ (iii) is satisfied. Commutant table can be set up for every Lie algebra. For infinitesimal operators (4.2.8), we have the following commutant Table 4.1.

158

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

Table 4.1

Commutant Table

X1

X2

X3

X4

X5

X1

0

0

0

X1

X2

0

0

0

2X2

X3 X4

0 −X1

0 0

0 0

X5

−X6 1 X3 2

0 −2X2 1 −X4 + X3 2 −X1

X6 1 X4 − X3 2 0 2X5

0

−2X5

0

0

0

−X6

0

0

X6

X6 1 − X3 2 X1 0 X6

From Table 4.1, we can draw that Lie algebra (4.2.8) is formed by operators X1 , X2 , and X5 , for example, [X1 , X5 ] = X6 , that is, X6 is formed by X1 and X5 ; 1 [X1 , X6 ] = − X3 , that is, X3 is formed by X1 and X6 ; 2 1 [X2 , X5 ] = X4 − X3 , that is, X2 , X5 and X3 are formed by X4 . 2

4.3

Nonclassical infinitesimal transformations

For given partial differential equations Lu = 0, using Lu = 0 and the invariant surface conditions X(x, t, u)

∂u ∂u + T (x, t, u) = U (x, t, u), ∂x ∂t

where infinitesimal transformation with one parameter is   x′ = x + εX(x, t, u) + o(ε2 ),   t′ = t + εT (x, t, u) + o(ε2 ),   u′ = u + εU (x, t, u) + o(ε2 ),

(4.3.1)

(4.3.2)

we obtain non-classical infinitesimal transformation. That is, infinitesimal transformation can not be got by classical infinitesimal transformation in § 4.1. Therefore, we get the new solution. Now Burgers equation Lu = ut + uux − uxx = 0

(4.3.3)

as an example to illustrate the above case. Note that in fact there are only two independence infinitesimals in (4.3.1), so we can suppose that T (x, t, u) = 1. Thus,

4.3

Nonclassical infinitesimal transformations

159

the invariant surface condition (4.3.1) is ut = U − Xux .

(4.3.4)

From (4.3.4), we have utx = Ux + Uu ux − Xuxx − Xx ux − Xu u2x . Since u satisfies equation (4.3.3), we get uxx = ut + uux , and using (4.3.4), we get utx = (Ux − XU ) + (Uu + X 2 − Xx − Xu)ux − Xu (ux )2 .

(4.3.5)

From (4.3.4) and (4.3.3), imply that u′ u′x′ − uux = ε {(uUx ) + (U + uUu − uXx )ux + (−Xu u)(ux )2 } + o(ε2 ), u′t′ − ut = ε {(Ut + U Ux ) + (−XUu − Xt − Xu U )ux +(Xu X)u2x + o(ε2 ), −(u′x′ x′ − uxx ) = ε {(−Uxx − U Uu + 2U Xx ) + (Xxx − 2Uxu − (u − X)(Uu − 2Xx ) + 3Xu U )ux +(2Xxu − Uuu + 3Xν (u − X))u2x + Xuu (ux )3 + o(ε2 ). Using the invariant conditions, we get the coefficients of ux , u2x , u3x and items that do not contain ux are 0. The coefficients of u3x is Xuu = 0, imply that X = C2 (x, t)u + C1 (x, t).

(4.3.6)

The coefficients of u2x is Uuu = 2Xxu + 2uXu − 2XXu . From (1.3.6), we have Uuu = 2((C2 )x + C2 (1 − C2 )u − C2 C1 ), so

1 U = B(x, t)u + D(x, t) + u2 ((C2 )x − C2 C1 ) + u3 C2 (1 − C2 ). 3

(4.3.7)

The coefficients of ux and (ux )0 are 0 respectively, imply that U − Xt − (2Uxu − Xxx ) + uXx − 2XXx = 0,

(4.3.8)

160

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

Ut + uUx − Uxx + 2U Xx = 0.

(4.3.9)

Therefore, for the determination of the solution of Burgers equation, we translate into researching the equations (4.3.6) ∼ (4.3.8). Generally speaking, it is difficult to find general solutions to these equations, so we only consider some special solutions. From the first equation of characteristic equation dt du dx = = , X 1 U

(4.3.10)

we have similar variable η(x, t) = const, imply that X = x(t, η); using the second equation of (4.3.10), integral, we get the similar layout. Simply, let C2 = 0 and from (4.3.6), (4.3.7), then we have X = C1 (t) = A(x, t),

U = B(x, t)u + D(x, t).

(4.3.11)

Substituting (4.3.11) into (4.3.8) and (4.3.9), imply that Bu + D − At − 2Bx + Axx + uAx − 2AAx = 0, Bt u + Dt + u(Bx u + Dx ) − (Bxx u + Dxx ) + 2(Bu + D)Ax = 0.

(4.3.12) (4.3.13)

Because A, B and D have nothing to do with u, let the coefficients of u and u2 in above equation are 0 and from (4.3.12), then we have B = −Ax .

(4.3.14)

Bt + Dx − Bxx + 2BAx = 0,

(4.3.15)

Bx = 0.

(4.3.16)

From (4.3.13), we get

We derive B = B(t) from equation (4.3.16). From (4.3.14), we have A = −B(t)x + E(t).

(4.3.17)

Equation (4.3.15) is converted to Bt + Dx − 2B 2 = 0, and imply that Dx = F (t) = 2(B(t))2 − B ′ (t). And then we get D = F (t)x + G(t). As the corollary of (4.3.16) ∼ (4.3.16), equations (4.3.12) and (4.3.13) are D − At − 2AAx = 0, We will get B, E, F and G.

Dt + 2DAx = 0.

(4.3.18)

4.3

Nonclassical infinitesimal transformations

(i) E = 0, X=A=

x , 2t + m

U =− similar variable is

161

m = const,

u , 2t + m

x η=√ , 2t + m

and similar layout is

√ f (η) = u 2t + m,

f ′′ + f ′ (η − f ) + f = 0.  2  −1 b 2 (ii) E = −RB, G = b (t + d) + c , where R, b, c and d are constants, 2 G′ (x + R), 2G    G′ b U= u+G x+1 , 2G 2 X=A=−

similar variable is η=

t+d , x+R

c = 0,

and similar layout is f (η) =

(t + d)−1 , 1 u− η

η 2 f ′′ + 2ηf ′ + f f ′ = 0. Integral is f = a2 th Solution is −1

u = a2 (t + d)

 th

a

2

2

a2 2



 (a3 − η −1 ) . x+R a3 − t+d

(iii) Without any assumption of E,   1 X = (t + d)−2 x + + N (t + d)−2 , 2 −1

U = −(t + d)

−2

u + (t + d)



 +

x+R . t+d

d, N = const, 1 x+ 2

 .

162

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

Similar variable and similar layout are respectively    1 N x+ + (t + d)−1 2 2 η= , (t + d)   1 f (η) = u − η − N (t + d)−2 (t + d). 2 Solution is Bessel function. For Boussinesq equation, 1 utt + (u2 )xx + uxxxx = 0, 2 using non-classical infinitesimal transformation, we also get similar variable and similar layout: u(x, t) = f (z) − 4λ2 t2 , z = x + λt2 , (4.3.19) where λ is constant, and f (z) satisfies the equation df d3 f +f + 2λf = 8λ2 z + A, dz 3 dz

(4.3.20)

where A is integral constant. Using the transformation f (z) = η(ζ) + 2λζ,

z=ζ−

A δλ2

in (4.3.20), then η(ζ) satisfies the equation   d3 η dη dη + 2λ ζ + 2η = 0. + η dζ 3 dζ dζ

(4.3.21)

(4.3.22)

It is easy to know that the solution of equation (4.3.22) can be converted to the solution of the second type Painlevé equation d2 w = 2w3 + zw + a, dz 2

(4.3.23)

where a is arbitrary constant. It is easy to verify that Boussinesq equation comes down to infinitesimal transformation of similar layout (4.3.19) which is X(x, t, u) = 2λt,

T (x, t, u) = −1,

U (x, t, u) = δλ2 t.

(4.3.24)

We can use non-classical infinitesimal transformation for equations. For example, for boundary layer equation uut + vvx = uxx ,

ut + vx = 0,

(4.3.25)

4.4

A direct method for solving similarity solutions

163

let u′ = u + εU (x, t, u) + o(ε2 ), v ′ = v + εV (x, t, u, v) + o(ε2 ), similarly, we obtain the solution with two invariant surface conditions Xux + ut = U,

Xvx + vt = V.

It should be noted here is U = U (x, t, u) and V = V (x, t, u, v).

4.4

A direct method for solving similarity solutions

For the given partial differential equations, we now introduce a method to seek the solution of the following form u(x, t) = U (x, t, w(z(x, t))).

(4.4.1)

Substituting (4.4.1) into the given partial differential equations, and given some conditions for v and its derivative, we come to determine w(z) which is satisfied the solution of differential equations. The following example to Boussinesq equations, let’s briefly introduce this method. For Boussinesq equation (4.1.83) 1 utt + (u2 )xx + uxxxx = 0, 2 (4.4.1) may take the following special form u(x, t) = α(x, t) + β(x, t)w(z(x, t)),

(4.4.2)

where α(x, t), β(x, t) and z(x, t) are undetermined. Substituting (4.4.2) into (4.1.83), we have 2 βzx4 w′′′′ + (6βzx2 zxx + 4βx zx3 )w′′′ + β(3zxx + 4zx zxxx )
2 2 2 ′′ +12βx zx zxx + 6βxx zx + αβzx + βzt w

+ (βzxxxx + 4βx zxxx + 6βx zxx + 4βxxx zx + 2αx βzx +2αβx zx + αβzxx + 2βt zt + βztt ) w′ +(βxxxx + 2αx βx + αβxx + αxx β + βtt )w +β 2 zx2 ww′′ + β(4βx zx + βzxx )ww′ + β 2 zx2 (w′ )2 +(βx2 + ββxx )w2 + (αtt + ααxx + αx2 + αxxxx ) = 0.

(4.4.3)

164

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Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

4 Remark 4.4.1 The coefficient βzw of w(4) is normalized coefficient, and this requires that the other coefficients have the form βzx4 Γ(z), where Γ is undetermined function of z.

Remark 4.4.2 For simplicity, Γ(z) through the differential, integral and other operations, still denoted as Γ(z). Remark 4.4.3 Without loss of generality, we have three degrees of freedom available for α, β, z and w. (i) If α(x, t) has the form α = α0 (x, t) + β(x, t)Ω(z), then we get Ω ≡ 0. In fact, with w(z) − Ω(z) instead of w(z) can be obtained. (ii) If β(x, t) has the form β = β0 (x, t)Ω(z), then we get Ω ≡ 1. In fact, it can w(z) . be obtained for the replacement w(z) → Ω(z) (iii) If α(x, t)Ω is determined by equation Ω(z) = z0 (x, t), where Ω(z) is invertible function, i.e., we choose Ω(z) = z. In fact, it can be obtained for the replacement z → Ω−1 (z). Now similar solution of Boussinesq equation is determined using the direct method. From the coefficients of ww′′ and (w′ )2 , we have βzx4 Γ(z) = β 2 zx2 , where Γ(z) is undetermined function. Using (ii) of (4.4.3) and choose Γ(z) = 1, we have β = zx2 . From the coefficient of w′′′ , we have βzx4 Γ(z) = 4βx zx3 + 6βzx2 zxx , where Γ(z) is another undetermined function. From (4.4.4), we get zx Γ(z) +

zxx = 0. zx

Integrating the above formula, we can get Γ(z) + lnzx = θ(t), where θ(t) is integral function. Taking exponent, we get zx Γ(z) = θ(t). Integrating it, imply that Γ(z) = xθ(t) + Σ(t),

(4.4.4)

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A direct method for solving similarity solutions

165

where Σ(t) is another integral function. Using (iii) of (4.4.3) and choose Γ(z) = z, we have z = xθ(t) + σ(t), (4.4.5) where θ(t) and σ(t) are undetermined. From (4.4.4) and (4.4.5), we get β = θ2 (t).

(4.4.6)

From the coefficients of w′′ , we have 2 βzx4 Γ(z) = β(3zxx + 4zx zxxx ) + 12βx zx zxx + 6βxx zx2 + β(αzx2 + zt2 ),

where Γ(z) is undetermined function. From (4.4.5) and (4.4.6), the above formula can be simplified as 2  dθ dσ . θ4 Γ(z) = αθ2 + x + dt dt Using (i) of (4.4.3) and choose Γ(z) = 0, we get  2 1 dθ dσ α=− 2 x + . θ (t) dt dt So, the equation (4.4.3) can be simplified as  2  d θ d2 σ d2 θ θ6 (w′′′′ + W w′′ + (w′ )2 ) + θ2 x 2 + 2 w′ + 2θ 2 w dt dt dt   2 ! 2   2 d w 1 dθ dσ 6 dθ dθ dσ − 2 x + + 4 x + = 0. dt θ dt dt θ dt dt dt Let (4.4.7) be differential equation of w(z), for other coefficients, we have  2  d θ d2 σ θ6 γ1 (z) = θ2 x 2 + 2 , dt dt θ6 γ2 (z) = 2θ

d2 θ , dt2

2 !   2 dθ dσ d 1 x + θ6 γ3 (z) = − 2 dt θ dt dt   2 6 dθ dθ dσ + 4 x + , θ dt dt dt

(4.4.7)

(4.4.8) (4.4.9)

(4.4.10)

where γ1 (z), γ2 (z) and γ3 (z) are undetermined functions. Since z = xθ(t) + σ(t) and the right-hand side of (4.4.8) is the linear function of x, imply that γ1 (z) = Az + B, where A and B are constants, and θ4 (A(xθ + σ) + B) = x

d2 θ d2 σ + 2. dt2 dt

(4.4.11)

166

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Compare the same order power of x, we get d2 θ = Aθ5 , dt2

(4.4.12)

d2 σ = θ4 (Aσ + B). dt2

(4.4.13)

and

From (4.4.9) and (4.4.10), it is easily got γ2 (z) = 2A,

γ3 (z) = −2(Az + B)2 .

Therefore, general similar variable and similar layout of Boussinesq equation are 1 u(x, t) = θ w(z) − 2 θ (t) 2

 2 dθ dσ x + , dt dt

(4.4.14) (4.4.15)

z(x, t) = xθ(t) + σ(t),

where θ(t) and σ(t) satisfy equations (4.4.12) and (4.4.13), and w(t) satisfies equation w′′′′ + ww′′ + (w′ )2 + (Az + B)w′ + 2Aw = 2(Az + B)2 .

(4.4.16)

Equation (4.4.16) generally is equivalent to the fourth type Painlevé equation; but when A = 0, it is equivalent to the second type Painlevé equation; but when B = 0, it is equivalent to the first type Painlevé equation with Weierstrass ellipse equation. Next, there are three cases of discussing similar solution separately. (i) A = 0, B = 0. Generally solution of equations (4.4.12) and (4.4.13) are θ(t) = a1 t + a0 ,

σ(t) = b1 t + b0 .

Similar solution of Boussinesq equation is  u(x, t) = (a1 t + a0 )2 w(z) −

a1 x + b1 a1 t + a0

2 ,

(4.4.17)

z = x(a1 t + a0 ) + b1 t + b0 ,

(4.4.18)

1 w′′ + w2 = c1 z + c0 . 2

(4.4.19)

where w(z) satisfies

If choose a1 = 1, a0 = b1 = b0 = 0, we have u(x, t) = t2 w(z) −

x2 , t2

z = xt,

(4.4.20)

4.4

A direct method for solving similarity solutions

167

where w(z) satisfies equation (4.4.19). This is new similar layout of Boussinesq equation corresponding to the first type Painlevé equation. The corresponding transformation of infinitesimal is X = −x,

T = t,

U = 2u + 6

x2 , t2

(4.4.21)

and this is not spacial case of classical infinitesimal transformation of Boussinesq equation. (ii) A = 0, B ̸= 0. Generally solution of equation (4.4.12) and (4.4.13) are θ(t) = a1 t + a0 ,  1 6   Ba−2 1 (a1 t + a0 ) + b1 t + b0 , a1 ̸= 0; 30 σ(t) =   1 Ba2 t2 + b1 t + b0 , a1 = 0. 2 0 (a) a1 = 0, similar solution of Boussinesq equation is u(x, t) = a20 w(z) −

(Ba20 t + b1 )2 , a20

1 z = a0 x + Ba20 t2 + b1 t + b0 , 2

(4.4.22)

(4.4.23)

where w(z) satisfies equation w′′′ + ww′ + Bw = 2B 2 z + C0 .

(4.4.24)

Equation (4.4.24) is equivalent to the second type Painlevé equation. When a0 = 1, b1 = b0 = 0, it comes down to similar solution (4.3.19) obtained by non-classical infinitesimal transformation. (b) a1 ̸= 0, similar solution of Boussinesq equation is 2 1 a21 x + (a1 t + a0 )5 + a1 b1   5 u(x, t) = (a1 t + a0 )2 w(z) −   , a1 (a1 t + a0 ) 

z = x(a1 t + a0 ) +

B (a1 t + a0 )6 + b1 t + b0 , 30a21

(4.4.25)

(4.4.26)

where w(z) satisfies (4.4.24). Let a1 = 1, a0 = b1 = b0 = 0, we have u(x, t) = t2 w(z) −

(x + λt5 )2 , t2

1 z = xt + λt6 , 6

(4.4.27)

168

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Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

where w(z) satisfies (4.4.24)(choose B = 5λ). This is another similar layout of Boussinesq equation. It is corresponding to the second type Painlevé equation (4.3.23), and corresponding to infinitesimal transformation is X = −(x + λt5 ),

T = t,

U = 2u + 2(x + λt5 )(3x − 2λt5 )t2 .

When λ = 0, it is equal to (4.4.20) and (4.4.21). (iii) A = ̸ 0, at this time, let B = 0 in the equation (4.4.13), multiplying (4.4.12) dθ and integrating, and we have by dt  2 dθ 1 (4.4.28) = Aθ6 + A0 , dt 3 θ(t) = C0 (t + t0 )− 2 . 1

(4.4.29)

3 , when A0 = 0, substituting (4.4.29) into (4.4.13), we have σ(t) = C1 (t + 4A 1 3 t0 ) 2 + C2 (t + t0 )− 2 . Let t0 = 0, C0 = 1 and C2 = 0, we get similar solution C04 =

1 u(x, t) = t−1 w(z) − t−2 (x − 3C1 t2 )2 , 4

(4.4.30)

z = xt− 2 + C1 t 2 , 1

3

w(z) satisfies equation 3 3 9 w′′′′ + ww′′ + (w′ )2 + zw′ + w = z 2 . 4 2 8

(4.4.31)

If C1 ̸= 0, this is a new similar transformation, and it corresponds to the fourth type z2 in (4.4.31), then g(z) satisfies equation Painlevé equation. Let w(z) = g(z) + 4 (4.1.98). When A0 ̸= 0, let k2 + 1 , A0 = k 2 , A = 3k 2 √ 1 where k is undetermined constant. When k = (1 ± i 3), using the transformation 2 θ2 (t) = (η 2 (t) − A)−1 , then (4.4.28) implies to 

dη dt

2 = (1 − η 2 )(1 − k 2 η 2 ).

(4.4.32)

The solution of equation (4.4.32) is Jacobi elliptic function Sη (t + t0 , k). Therefore,

4.4

A direct method for solving similarity solutions

 θ(t) =

Sη2 (t

k2 + 1 + t0 , k) − 3k 2

169

− 12 .

k2 + 1 4 d2 σ θ σ, its solution is Equation (4.4.13) becomes 2 = dt 3k 2     2 − k2 1 σ(t) = c t − E(t + t ; k) + D θ(t), 0 3k 2 k2

(4.4.33)

(4.4.34)

where E(t + t0 ; k) is the second type elliptic integration, and we have Z t+t0
E(t + t0 ; k) = 1 − k 2 Sn2 (S; k) dS, 0

where C and D are arbitrary constants. Choose D = 0, we get the similar solution. u(x, t) = (Sn2 (t + t0 ; k) − A)−1 w(z) − C(Sn2 (t + t0 ; k) − A)    2 − k2 −2 − x+C t − k E(t + t0 ; k) · Sn (t + t0 ; k) 3k 2 p · (1 − Sn2 (t + t0 ; k))(1 − k2 Sn2 (t + t0 ; k))
2 ·(Sn2 (t + t0 ; k) − A)−1 ,  z=

 x+C

2 − k2 t − k −2 E(t + t0 ; k) 3k 2



(Sn2 (t + t0 ; k) − A)−2 ,

(4.4.35)

√ 1 3 k2 + 1 1 i where k = ± i ,A= = ∓ √ and w(z) satisfies equation 2 2 2 3k 2 2 3 2

w(4) + ww′′ + (w′ )2 + Azw′ + 2Aw = 2A2 z 2 .

(4.4.36)

This is another new similar transformation, and corresponding to the forth type Painlevé equation. The direct method of solving similar solution can be used as a series of nonlinear evolutionary equation. Here are two more examples, let’s give a brief introduction. Consider the Burgers equation ut + uux + uxx = 0.

(4.4.37)

u(x, t) = α(x, t) + β(x, t)w(z(x, t)),

(4.4.38)

Let Substituting (4.4.38) into (4.4.37), we get βzx2 w′′ + (2βx zx + βzxx + βzt + αβzx )w′ +(βxx + βt + αβx + αx β)w + β 2 zx ww′ + ββx w2 +αxx + αx + ααx = 0.

(4.4.39)

170

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The coefficient of w′′ is normalized coefficient, and in order to get the ordinary differential equation, using the coefficient of ww′ , we get βzx2 Γ(z) = β 2 zx , where Γ(z) is undetermined function. Using (i) of (4.4.3) and choose (4.4.40)

β = zx . Using the coefficient of w2 , we have βzx2 Γ(z) = ββx ,

where Γ(z) is undetermined function. From (4.4.40), integrating twice of above equation, and using (iii) of (4.4.3), we have z = xθ(t) + σ(t),

β = θ(t),

(4.4.41)

where θ(t) and σ(t) are undetermined functions. Equation (4.4.39) can be simplified to    dθ dσ θ3 (w′′ + ww′ ) + θ x + + αθ w′ dt dt   dθ + αx θ w + αxx + αt + ααx = 0. (4.4.42) + dt  dθ dσ x + , dt dt  2 d2 θ dθ θ 2 −2 = A2 θ6 , dt dt

If

1 α=− θ



(4.4.43) (4.4.44)

d2 σ dθ dσ −2 = θ5 (A2 σ + 2B), (4.4.45) 2 dt dt dt where A and B are arbitrary constants, then we get the ordinary differential equation dθ of w(z). Multiplying (4.4.44) by 2θ−2 , and integrating it, we get dt  2 dθ = A2 θ6 + C 2 θ4 , (4.4.46) dt θ

where C is arbitrary constant. Therefore, similar layout and similar variable of Boussinesq equation (4.4.37) are   1 dθ dσ u(x, t) = θ(t)w(z) − x + , z = xθ(t) + σ(t), θ dt dt

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A direct method for solving similarity solutions

171

where θ(t) and σ(t) satisfy (4.4.45) and (4.4.46). There are four cases of discussing separately. (i) A = 0, C = 0. At this time, (4.4.45) and (4.4.45) have solutons θ(t) = θ0 , σ(t) = Bt2 + C1 t + C2 . Let θ0 = 1, i.e., we get the similar solution u(x, t) = w(z) − 2Bt − C1 ,

z = x + Bt2 + C1 t + C2 .

(4.4.47)

1 (ii) A = ̸ 0, C = 0. Let A = − , B = 0, then 2 θ(t) = (t − t0 )− 2 , σ(t) = C3 (t − t0 ) 2 + C4 (t − t0 )− 2 . 1

1

1

Let t0 = 0, C4 = 0, we have u(x, t) = t− 2 w(z) + 1

1 x − C3 , 2t 2

z = t− 2 x + C3 t 2 . 1

(4.4.48)

1

(iii) A = 0, C ̸= 0. Let C = −1, then θ(t) = (t − t0 )−1 , σ(t) = B(t − t0 )−2 + C5 (t − t0 )−1 + C6 . Let t0 = 0, C5 = 0, C6 = 0, then we have u(x, t) = t−1 w(z) +

x 2B + 2 , t t

x B z = + 2. t t

(4.4.49)

(iv) A = ̸ 0, C ̸= 0. Let A2 = −1, B = 0, C 2 = 1, then we have σ(t) = (t2 + 1)− 2 (C7 t + C8 ).

θ(t) = (t2 + 1)− 2 ,

1

1

Let C8 = 0, we get u(x, t) = (t2 + 1)− 2 w(z) + 1

z=

x + C7 t (t2 + 1)

1 2

xt − C7 , t2 + 1

(4.4.50)

.

Classical Lie group infinitesimal transformation of Boussinesq equation is   X = αx + βt + γxt + δ,   (4.4.51) T = 2αT + γt2 + k,   U = −αu + γ(x − tu) + β,

172

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where α, β, γ, δ and k are arbitrary constants. We next consider the direct method of similar solution of KdV equation ut + uux + uxxx = 0.

(4.4.52)

u(x, t) = α(x, t) + β(x, t)w(z(x, t)),

(4.4.53)

Suppose that substituting (4.4.53) into (4.4.52), we have βzx3 w′′′ + (3βx zx2 + 3βzx zxx )w′′ + (3βxx zx + 3βx zxx + βzxxx +βzt + αβzx ) w′ + (βxxx + βt + αβx + αx β)w +β 2 zx ww′ + ββx w2 + αxxx + αt + ααx = 0.

(4.4.54)

The coefficients of w′′′ are normalized coefficients, and in order to get the ordinary differential equation, using the coefficient of ww′ , we get βzx3 Γ(z) = β 2 zx , where Γ(z) is undetermined functions. Using (i) of (4.4.3), we get β = zx2 .

(4.4.55)

From coefficient of w2 , we have βzx3 Γ(z) = ββx , where Γ(x) is undetermined function. From (4.4.55), integrating twice of above equation, and using (iii) of (4.4.3), we have z = xθ(t) + σ(t),

β = θ2 (t),

(4.4.56)

where θ(t) and σ(t) are undetermined functions. Equation (4.4.54) can be simplified to    dθ dσ + αθ w′ θ5 (w′′′ + ww′ ) + θ2 x + dt dt   dθ + 2θ + αx θ2 w + αxxx + αt + ααx = 0. (4.4.57) dt If

1 α=− θ



dθ dσ x + dt dt

dθ = Aθ3 , dt

 ,

(4.4.58) (4.4.59)

4.5

The Painlevé properties for some PDE

173

d2 σ dθ dσ −2 = 2θ6 (A2 σ + B), (4.4.60) dt2 dt dt where B is another arbitrary constant. Similar solution of KdV equation is   1 dθ dσ 2 u(x, t) = θ (t)w(z) − x + , z = xθ(t) + σ(t), θ dt dt θ

where θ(t) and σ(t) satisfy (4.4.59) and (4.4.60). Now, there are two cases of discussing separately, 1 (i) A ̸= 0. Let A = − , B = 0, then we have 3 θ = (t − t0 )− 3 , 1

σ(t) = C1 (t − t0 ) 3 + C2 (t − t0 )− 3 . 2

1

Let t0 = 0, C2 = 0, this imply that u(x, t) = t− 3 w(z) + 2

2 x − C1 , 3t 3

z=

x + C1 t 1

t3

.

(4.4.61)

(ii) A = 0. Let θ = 1, we have σ(t) = Bt2 + C3 t + C4 , and let C4 = 0, we get u(x, t) = w(z) − 2Bt − C3 ,

z = x + Bt2 + C1 t.

(4.4.62)

Classical Lie group infinitesimal transformation of KdV equation is X = αx + βt + γ,

T = 3αt + δ,

U = −2αu + β,

(4.4.63)

where α, β, γ and δ are arbitrary constants.

4.5

The Painlevé properties for some PDE

As we know, the solution of an ordinary differential equation (or system of differential equations) can be considered as an analytic function with complex variable of the time t. If the singularity of solutions depends on the initial conditions, then this kind of singular points are called being movable; while if the singularity comes from the coefficients of the equation, then they are called fixed singularity points. If all the movable singularity points of an ordinary differential equation (or system of differential equations) are single-valued, i.e., they are simple singular points, then the differential equation (or system of differential equations) has Painlevé property. What is the relation between the Painlevé property of system of ordinary differential

174

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

equations and partial differential equation? After researches show that, if a partial differential equation can be solved by inverse scattering method, then the system of ordinary differential equations related to Gelfand-Levitan-Marchenko equation has Painlevé property which is obtained by using similarity transformation. Ablowitz et al. [4] conjecture that, for a given partial differential equation, if all the systems of ordinary differential equations obtained by similarity transformation have Painlevé property, then this partial differential equation is integrable. Therefore, it is necessary to define the Painlevé property of PDE corresponding to that of ordinary differential equations, and establish the relation between Painlevé properties and integrability (Lax pairs, Bäcklund transformation). The research works of the Painlevé properties, Lax pairs, and Bäcklund transformation have been the subject of much attention during the last 50 years; see the seminal papers by Uğurhan Muğan et al. [237], A. Ramani et al. [206], DengShan Wang [238], Matthew Russo et al. [212], Nikolay A.Kudryashovet al. [6], and the references cited therein. Definition 4.5.1 If the solutions of a partial differential equation are singlevalued with respect to movable singularity manifold, then we call it owning Painlevé property. To be more specific, suppose that there is an odd manifold ϕ(z1 , z2 , · · · , zn ) = 0,

(4.5.1)

where ϕ = ϕ(z1 , z2 , · · · , zn ). If u = u(z1 , z2 , · · · , zn ) is a solution of partial differential equation, then when u has the layout u(z1 , z2 , · · · , zn ) = ϕα

∞ X

uj ϕj ,

(4.5.2)

j=0

and uj = uj (z1 , z2 , · · · , zn ) is an analytic function about (z1 , z2 , · · · , zn ) near the manifold (4.5.1), α is negative integer. Then the partial differential equation is said to have the Painlevé property. Next with some specific examples, we illustrate that how to decide the solution layout (4.5.2) for the given partial differential equation. Example 4.5.1 Burgers equation ut + uux = σuxx .

(4.5.3)

Suppose that there is an expansion formula u = ϕα

∞ X j=0

uj ϕj ,

(4.5.4)

4.5

The Painlevé properties for some PDE

175

where ϕ = ϕ(x, t) and uj = uj (x, t) are analytic functions about (x, t) near the manifold M = {(x, t) : ϕ(x, t) = 0}. Substituting (4.5.4) into (4.5.3), and comparing the lowest item, we have α = −1. (4.5.5) Simultaneously, we get the following recurrence formula of uj : uj−2,t + (j − 2)uj−1 ϕt +

j X

uj−m (um−1,x + (m − 1)ϕx um )

m=0

= σ (uj−2,xx + 2(j − 2)uj−1,x ϕx + (j − 2)uj−1,x ϕx
+(j − 2)uj−1 ϕxx + (j − 1)(j − 2)uj ϕ2x .

(4.5.6)

Tidying up the term that contains uj , we have ϕ2x (j − 2)(j + 1)uj = F (uj−1 , · · · , u0 , ϕt , ϕx , ϕxx , · · · ), j = 0, 1, 2, · · · .

(4.5.7)

From (4.5.7), uj is undetermined for J = −1, 2. These values of J are called resonance of recurrence relations. uj is arbitrary on these points. j = −1 corresponds to arbitrary (not defined) odd manifold (ϕ = 0). If j = 2, introduce arbitrary function u2 and consistency condition, this condition for function (ϕ, u0 , u1 ) requires that the right-hand side of (4.5.6) is identically equal to 0. For Burgers equation (4.5.4) and using (4.5.6), we have u0 = −2σϕx ,

(4.5.8)

ϕt + u1 ϕx = σϕxx ,

(4.5.9)

∂x (ϕt + u1 ϕx − σϕxx ) = 0.

(4.5.10)

j = 0, j = 1, j = 2,

Using (4.5.9), and consistency condition (4.5.10) is identically equal to satisfy in j = 2, so Burgers equation has the Painlevé property. Further, if introduce arbitrary function u2 = 0, (4.5.11) and requiring u1 t + u1 u1x = σu1xx ,

(4.5.12)

then we get uj = 0,

j ⩾ 2.

(4.5.13)

At this time, we get the following Bäcklund transformation of Burgers equation u = −2σ

ϕx + u1 , ϕ

(4.5.14)

176

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

where (u, u1 ) satisfies Burgers equation, and ϕt + u1 ϕx = σϕxx .

(4.5.15)

If u1 = 0, we get Cole-Hopf transformation. If u1 = ϕ, we have ϕx + ϕ, ϕ

(4.5.16)

ϕt + ϕϕx = σϕxx .

(4.5.17)

u = −2σ where

Example 4.5.2 KdV equation ut + uux + σuxxx = 0, u = ϕ−2

∞ X

u j ϕj .

(4.5.18) (4.5.19)

j=0

It is easy that they resonate when j = −1, 4, 6. Consistency conditions are met identically when j = 4, 6. Therefore, KdV equation has the Painlev´ e property then

j = 2,

j = 0, u0 = −12σϕ2x ,

(4.5.20)

j = 1,

u1 = 12σϕxx ,

(4.5.21)

ϕx ϕt + ϕ2x u2 + 4σϕx ϕxxx − 3σϕ2xx = 0,

(4.5.22)

ϕxt + ϕxx u2 − ϕ2x u3 + σϕxxxx = 0,

(4.5.23)

j = 3, j = 4,

consistency condition

(4.5.24) ∂ (ϕxt + ϕxx u2 − ϕ2x u3 + σϕxxxx ) = 0. ∂x From (4.5.23), we know that consistency condition (4.5.24) is met identically. When j = 6, consistency condition is satisfied. Let resonance function u4 = u6 = 0,

(4.5.25)

u3 = 0.

(4.5.26)

u2t + u2 u2x + σu2xxx = 0,

(4.5.27)

and further requests It is easy to prove: if then we have uj = 0,

j ⩾ 3.

(4.5.28)

4.5

The Painlevé properties for some PDE

177

Based on the above conclusion, we can get (i) uj = 0, j ⩾ 3, (ii) u0 = −12σϕ2x , u1 = 12σϕxx , (iii)(a) ϕx ϕt + ϕ2x u2 + 4σϕx ϕxxx − 3σϕ2xx (b) ϕxt + ϕxx u2 + σϕxxxx = 0,

            = 0,                       

(iv)u2t + u2 u2x + σu2xxx = 0, ϕ2x ϕxx + 12σ + u2 , ϕ2 ϕ ∂2 or u = 12σ 2 lnϕ + u2 , ∂x (v)u = −12σ

(4.5.29)

where ut + uux + 6uxxx = 0. From (4.5.29), we have Bäcklund transformation of KdV equation. In fact, ϕt is solved in (iiia) of (4.5.29), and then differentiating with respect to x and using (iiib) of (4.5.29), let ϕx = ν 2 , (4.5.30) we have (i) 6σvxx + u2 x = λv,

)

(ii) 2vt + u2 vx + λvx + 2σvxxx = 0.

(4.5.31)

Therefore, Bäcklund transformation of KdV equation is u = 12σ

∂2 lnϕ + u2 , ∂x2

where ϕx = ν 2 , 6σvxx + u2 v = λv, 2vt + u2 vx + λvx + 2σvxxx = 0. u2 and u are solutions of the KdV equation (4.5.18), i.e., u2 t + u2 u2x + σu2xxx = 0, ut + uux + σuxxx = 0. Choosing σ = 1 in (4.5.29), and eliminating u2 in (a) and (b) of (4.5.29), we get φt + {φ; x} = λ, φx

(4.5.32)

178

Chapter 4

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

where ∂ {φ; x} = ∂x



φxx φx

 −

1 φ2xx 2 φ2x

(4.5.33)

is Schwarz derivative of φ. (4.5.32) is invariant in Möbius group φ=

aψ + b . cψ + d

(4.5.34)

v1 , v2

(4.5.35)

Let φ= where (v1 , v2 ) satisfies vxx = av,

(4.5.36)

vt = bvx + cv,

and we get u2 2 1 a = − (u2 + λ), b = − + λ, 6 3 3 Example 4.5.3 Boussinesq equation

c=

ux . 6

1 utt + 2uuxx + 2u2x + uxxxx = 0. 3 Similarly, we get u = ϕ−2

∞ X

(4.5.37)

(4.5.38)

uj ϕ j .

(4.5.39)

j = −1, 4, 5, 6.

(4.5.40)

j=0

Resonance points are From recurrence relations, we have j = 0,

u0 = −2ϕ2x ,

j = 1,

u1 = 2ϕxx , 4 j = 2, ϕ2t − ϕ2xx + ϕx ϕxxx + 2u2 ϕ2x = 0, 3 1 j = 3, ϕtt + ϕxxxx + 2ϕxx u2 − 2ϕ2x u3 = 0, 3 j = 4 (resonance), if consistency condition is satisfied,   ∂2 4 2 2 2 ϕ − ϕ + ϕ ϕ + 2u ϕ = 0, x xxx 2 x t xx ∂x2 3

(4.5.41) (4.5.42) (4.5.43) (4.5.44)

(4.5.45)

then u4 is arbitrary. From (4.5.43), (4.5.45) is satisfied. We can verify that j = 6 (resonance) also satisfies more complicated consistency condition. Let uj = 0,

j = 3, 4, 5, 6,

4.5

The Painlevé properties for some PDE

179

then we can verify that uj = 0,

j ⩾ 3.

(4.5.46)

Therefore, we get that Bäcklund transformation is u=2

∂2 lnφ + u2 , ∂x2

(4.5.47)

where (u, u2 ) satisfies (4.5.38), and 4 ϕ2t − ϕ2νx + ϕx ϕxxx + 2u2 ϕ2x = 0, 3 1 ϕtt + ϕxxxx + 2ϕxx u2 = 0. 3 Example 4.5.4 Nonlinear Schrödinger equation iνt + νxx + 2ν|ν|2 = 0. The equation can be written as the following system of equations ( iUt + Uxx + 2U 2 V = 0, − iVt + Vxx + 2U V 2 = 0,

(4.5.48) (4.5.49)

(4.5.50)

(4.5.51)

where V = U ∗ . Equations’ system (4.5.51) have the Painlevé property, and have expanded form ∞ ∞ X X U = φ−1 Uj φj , V = φ−1 Vj φj . (4.5.52) j=0

j=0

Resonance points are j = −1, 0, 3, 4,

(4.5.53)

and Bäcklund transformation is U=

U0 + U1 , φ

V =

V0 + V1 , φ

where (φ, U0 , V0 , U1 , V1 ) is determined by the following equations:   U0 V0 = −φ2x ,      4φ2x U1 − 2U02 V1 = −iφt v0 − 2φx U0x − φxx v0 ,     2 2    − 2V0 U1 + 4φx V1 = iφt V0 − 2φx V0x − φxx V0 ,  iU0t + U0xx + 2V0 U12 + 4U0 U1 V1 = 0,     − iV0t + V0xx + 2U0 V12 + 4V0 V1 U1 = 0,       iU1t + U1xx + 2U12 V1 = 0,     − iV1t + V1xx + 2U1 V12 = 0.

(4.5.54)

(4.5.55)

180

Similarity Solution and the Painlevé Property for Some Nonlinear· · ·

Chapter 4

From (4.5.55), we have   U0 V0 = −φ2x ,      U0 V1 + V0 U1 = φxx ,      (V0 U0x − U0 V0x )   , U0 V1 − V0 U1 = −iφt + φx    φt (V0 U0x − U0 V0x )  2i + λ, =   φx φ2x    !   2  2   φ 1 φ φ  t xx t  + + 2iλ − λ2 ,  U1 V1 = − 4 φx φx φx and ∂ ∂t



φt φx



∂ + ∂x

3 {φ; x} − 2



φt φx

2

φt λ2 − 2iλ + φx 2

(4.5.56)

! = 0,

(4.5.57)

where λ is an integral constant. Example 4.5.5 Kadomstev-Petriashvili equation is utt + u2x + uuxx + 6uxxxx + uyy = 0.

(4.5.58)

This equation has expanded form u = ϕ−2

∞ X

uj φj ,

(4.5.59)

j = −1, 4, 5, 6.

(4.5.60)

j=0

and resonance points are From recurrence formula, we have

j = 2,

j = 0,

u0 = −12σφ2x ,

(4.5.61)

j = 1,

u1 = 12σφxx ,

(4.5.62)

ϕt ϕx + 4σφx φxxx − 3σϕ2xx + φ2y + u2 φ2x = 0,

j = 3,

ϕxt + σφxxxx + φyy + φxx u2 − φ2x u3 = 0,

(4.5.63) (4.5.64)

j = 4 (resonance), using (4.5.63), we learn that consistency condition ∂2 (ϕt φx + 4σϕx φxxx − 3σφ2xx + φ2y + u2 φ2x ) = 0 ∂x2 is satisfied. Therefore, u4 is arbitrary.

(4.5.65)

4.5

The Painlevé properties for some PDE

181

j = 5 (resonance), using (4.5.64), we learn that consistency condition ∂2 (ϕxt + σϕxxxx + φyy + φxx u2 − φ2x u3 ) = 0 ∂x2

(4.5.66)

is satisfied. Therefore, u5 is arbitrary. In like manner, j = 6 (resonance) also satisfies more complicated consistency condition. Therefore, uj = 0, j ⩾ 4. So, KP equation (4.5.58) has the Painlevé property, and has Bäcklund transformation u = 12σ

∂2 ln φ + u2 , ∂x2

where (u, u2 ) satisfies KP equation (4.5.58).

(4.5.67)

Chapter 5 Infinite Dimensional Dynamical Systems As we know, there has been at least fifty years for the research of finite dimensional dynamical systems. There has been a lot of important results. However, the research of dynamical systems are far from limited to the cases of finite dimensions. The turbulence in fluid dynamics is just a problem of infinite dimensional dynamical systems. If we take the Fourier expansion of unknown functions in system of Newton-Boussinesq fluid mechanical equations of Bénard convection, under the approximation by retaining three motion modes, we can obtain the well known Lorenz model. Hence, it is just an approximation in describing the Newton-Boussinesq fluid mechanical equations of Bénard convection. In the early 1990s, there have been many nonlinear evolution equations which had solitons in physics, they all belong to integrable systems, such as KdV equation, nonlinear Schödinger equation, Zakharov equations, sine-Gordon equation etc., under the effort of dissipation, they evolute from solitons to chaos, which are non-integrable systems, one can see [3,43,182,210]. All these facts show that the research on infinite dimensional dynamical systems are imperative, which are the in-depth studies and extensions of finite dimensional dynamical systems, see [28, 44, 85, 93, 126, 127, 150] and references therein. There are some important characters of infinite dimensional dynamical systems: firstly, they have chaos in existing spaces, i.e., they might generate chaos and turbulence in some regions, while the chaos and turbulence do not appear in other regions. The cross flow is a typical example which will be discussed later. Contract to the finite dimensional dynamical systems, they consider the chaos in time periods only; secondly, they might generate singular sets in some spatial regions. For example, the rotation rot u of velocity u in the fluid motion in three-dimensional spaces might tends to infinite at some regions of the domain Ω. J. Leray predicted the appearance of turbulence at that time in 1932. Hence the researches of infinite dimensional dynamical systems will lay a new road of the researches of turbulence, and this is the reason why so many physicists and mechanicians major in these fields. From the viewpoint of mathematics, on the basis of former works on finite dimensional dynamical systems by S. Smale, J. Moser, Melnikov, B. Mandelbrot put forward the definition of fractal set in 1977, O. A. Ladyzhenskaya [185, 186], M. I.

5.1

Infinite dimensional dynamical systems

183

Vishik, C. Foias [49–55], B. Nicolacnko [196–198], R. Teman [228–231], J. K. Hale, etc. G. R. Sell [218, 219] research some properties of certain nonlinear evolution equations with dissipative effort, the global attractors, existence of inertial manifolds, their Hausdorff dimensions, estimations of upper and lower bounds of fractal dimensions, the dynamical structures of attractors, approximate inertial manifolds, nonlinear Galerkin methods, inertial sets, etc., are widely studied, and many results are obtained. For example, the global attractors of Y. Kuramoto and G. Sirashinsky equations [182, 183], inertial manifolds [15, 38–42, 47, 50, 52, 55], the existence and estimation of dimension of inertial sets [15, 46], the global attractors and estimation of dimension of equations, such as nonlinear Schrödinger equation with dissipative effect, one-dimensional system of Zakharov equations, nonlinear wave equations with dissipative effect, Boehner flows, Brussel oscillator [183], KdV equation with damping [116]. Meanwhile, some examples are also presented which have no global attractors and inertial manifolds. In mathematics, the important mathematical theories of infinite dimensional dynamical systems have been set up, which offer methods of theoretic researches and numerical computation. Among these theories, it is critical to set up the uniformly a priori estimation of t on large scale of solutions of problem for particular determining solutions from the viewpoint of qualitative research of partial differential equations. The infinite dimensional dynamical systems and chaos are essential the qualitative properties as t → ∞. Hence, research on these problems lays some important topics of nonlinear partial differential equations. With the development of huge computers, it is prospected that the research on chaos and turbulence will come into a flourishing stage under the combination of theoretical research and numerical computation. Of course, the infinite dimensional dynamical systems are far more complicated. The understanding of it is very superficial. For example, there are many important theoretical and practical problems in researches such as the global attractors, topological structure of inertial manifolds, chaos of conservative systems. These problems remains to be further studied in the future. In this chapter, we mainly introduce the basic definitions, approximation methods of infinite dimensional dynamical systems and some important research progress.

5.1

Infinite dimensional dynamical systems

We consider the solution of the differential equation du(t) = F (u(t)) dt

(5.1.1)

u(0) = u0

(5.1.2)

with initial condition

184

Chapter 5

Infinite Dimensional Dynamical Systems

and the asymptotic behaviors of u(t) as t → ∞, where the unknown function u = u(t) is in linear space H, which is called phase space, F (u) projects H onto itself. There are two cases to be considered: 1. the case of finite dimensional, where u = u(t) ∈ H = RN ; 2. the case of infinite dimensional, where u = u(t) ∈ H is a certain Hilbert space. Although both of the two cases have many common properties, they have significant differences. Of course, we can consider the finite dimensional dynamical systems as approximations of some finite state of infinite dimensional dynamical systems. In common circumstances, the function F in (5.1.1) depends certain parameter λ, that is F (u) = Fλ (u). All physical quantities and states vary with λ as t → ∞. We give the following statements in detail. (i) As λ is small, say λ < λ1 , (5.1.1) has a unique stationary solution, which is the unique solution u = uS1 of equation Fλ (u) = 0.

(5.1.3)

This stationary solution is stable, it absorbs all trajectories, that is u(t) → uS1 ,

as t → ∞,

where u(t) is a solution of (5.1.1), (5.1.2) with arbitrary u0 . (ii) For larger λ, say λ1 < λ < λ2 , (5.1.3) have some other solutions, this means that (5.1.1) have stationary solutions uS2 , uS3 , · · · . In this case, uS1 loses its stability, the stationary solutions have bifurcations at λ = λ1 , u(t) is convergent to one of the stationary solutions such as u(t) → uS2 ,

as t → ∞,

in this case, the limit depends on u0 , and each stationary solution has its own absorbing region, and absorbing all solutions of (5.1.1), (5.1.2) around it. (iii) When λ gets larger and larger, say λ2 < λ < λ3 , then Hopf bifurcations appear, in this case, the flows are not stationary, we have u(t) → ϕ(t),

as t → ∞,

where ϕ is a time-periodic solution of (5.1.1) with period T , that is dϕ(t) = F (ϕ(t)) for 0 ⩽ t ⩽ T, dt

(5.1.4)

5.1

Infinite dimensional dynamical systems

185

ϕ(t + T ) = ϕ(t) for t ⩾ 0. (5.1.5) Then the subharmonic bifurcations or almost periodic solutions proposed by Feigenbaum might occur (See the paragraph (iv) below). (iv) For more larger λ, say λ3 < λ < λ4 , then invariant tori appear, that is u(t) → ϕ(t),

as t → ∞,

(5.1.6)

where ϕ(t) is almost periodic solution of (5.1.4) of the form ϕ(t) = g(ω1 t, · · · , ωn t),

(5.1.7) 1 are here g is Ti -periodic function with respect to its i−th variables and all ωi = Ti independent rational numbers for i = 1, 2, · · · , n. In this case, the flow looks like chaos, however, the Fourier analysis shows that its character is determined by modes with discrete frequencies ωi , rather than the chaotic status. (v) Finally, λ > λ4 . the chaos appears, u(t) is random with respect to all the times. The Fourier analysis shows that it has a strip of continuous spectrum. Now u(t) → X,

as t → ∞,

(5.1.8)

which is showed in Figure 5.1 i.e., dist(u(t), X) → 0 as t → ∞, where X is the invariant set of phase space H, the semigroup {S(t)}t⩾0 related to (5.1.1), (5.1.2) is invariant, i.e., S(t)X ⊂ X for t ⩾ 0. (5.1.9)

Figure 5.1

The set X may be a fractal set, just as a Cantor set or a multiplication of Cantor sets on certain interval. When t is large, u(t) is on X or wandering aside X. Now we give two practical examples in physics to show the phenomena which caused by infinite dimensional dynamical systems. Example 1.Bénard problems For given a rectangle container, which is filled with fluid, if we keep warming its bottom y = 0, then with the increasing of temperature difference to its top y = β of the container (see Figure 5.2), let’s look at the physical phenomena after convection. This problem can be characterized by the following definite solutions problems of two-dimensional Newton-Boussinesq equations ∂u − (u · ∇)u − ν∆u + ∇p = e2 (T − T1 ), ∂t

(5.1.10)

186

Chapter 5

Infinite Dimensional Dynamical Systems

∂T + (u · ∇)T − k∆T = 0, ∂t

(5.1.11)

divu = 0,

(5.1.12)

u = 0,

for x = 0 and x = a,

T = T0 , y = 0;

T = T1 , y = β,

(5.1.13) (5.1.14)

where u, p, T is the velocity, pressure and temperature, respectively, ν is the viscosity coefficient, k is the heat conduction coefficient, e2 is the unit vector paralleling to y-axis, T0 > T1 , α, β > 0. The experiment results and numerical simulations show T0 − T1 < λ1 , the fluid remains stationary, the equilibrium solution that, when λ = T0 S u1 is the solution of classical heat conduction problem of which temperature is linear related to y-direction; when λ1 < λ < λ2 , the solutions of heat conduction problem turn to unstable, the fluid starts moving till the form of classical circles corresponding to certain stationary solutions uS2 , see Figure 5.3 below, when λ2 < λ < λ3 , the flow generates periodic solution of time, the corresponding boundaries of circles start periodic oscillating; when λ3 < λ < λ4 , the corresponding boundaries of circles oscillate randomly, spectral analysis shows that period-doubling bifurcations appear; and finally, when λ > λ4 , the circles disappear, the whole fluid is chaotic.

Figure 5.2

Figure 5.3

Example 2. The circumfluence of a spherical object.

5.2

Some problems for infinite dimensional dynamical systems

187

Suppose that there is a fluid with velocity U∞ at infinite passes through a spherical object, we can take λ as the Reynolds number λ = Re =

ν|u∞ | R

with ν the dynamical viscosity, R the radius of the sphere. The ν and R are constants in practical experiment, we can increase λ by increasing the velocity U∞ = |U∞ | at infinite. In the first step of the experiment, the fluid is in a state of laminar flow (smaller Reynolds number, Figure 5.4(i)). In the second step of the experiment, the fluid generates stationary von K´ arman vortex (Figure 5.4(ii)). In the third step of the experiment, when Hopf bifurcations appear, no stationary flows exist, the fluid generates periodic flows of time. In this stage, the K´arman vortex moves to the right and disappear, and take approximal periodic motion at the left-side (Figure 5.4(iii)). In the fourth step of the experiment, the K´ arman vortex disappear when U∞ is very large, and the fluid takes quasi-periodic motion. At the last step of the experiment, for more larger U∞ , the fluid is in a stage of turbulence (Figure 5.4(iv)). The whole fluid is un-stationary for all times, which is known as attractor in mathematics.

Figure 5.4

5.2

Some problems for infinite dimensional dynamical systems

In this section, we introduce some problems for infinite dimensional dynamical systems further. In general, they are nonlinear evolution equations with dissipative effect. As the dissipative terms equal to zero, they are integrable systems or nearintegrable systems, which have soliton solutions. Numerical simulation and theoretical analysis show that all these physical systems turn into chaos. 1. System of Zakharov differential equations with damping and inhomogeneous medium. Consider one-dimensional system of Zakharov equation iεt + εxx − nε = 0,

(5.2.1)

188

Chapter 5

Infinite Dimensional Dynamical Systems

ntt − nxx = |ε|2xx ,

(5.2.2)

where ε(x, t) is the electric field intensity with complex-valued, n(x, t) is the density perturbation with real-valued. In 1972, Zakharov [243] raised this system of equations firstly, and found its soliton solutions, which can successfully explain the phenomena of density pit occurring near the critical surface of the laser target. It also appears in DNA model of molecular structure. This system of equations can be obtained by linearizing simplified bi-fluid equations. In 1982, G. D. Doolen et al. [43] computed the one-dimensional Zakharov equation with damping term and driving source (by Fourier transformation): X i[∂t + νe (k) − k 2 ]Eh (t) = nk′ Ek−k′ + Sk (t), (5.2.3) k′

[∂t2 + 2νi (t) + k 2 ]nk (t) = −k 2

X

∗ E−k ′ Ek−k ′ (t),

(5.2.4)

k′

where νe , νi denote the damping ratios of electron and ion, respectively, Sk (t) is the driving source. For (5.2.3) and (5.2.4) with periodic boundary conditions, the computation was conducted using 64 − 1024 modules, the driving source was divided into three separated cases. (i) Coherent source ( 1 νe ω02 , for |k| ⩽ kdr ; Sk (t) = 0, for |k| > kdr , where νe , ω0 , kdr are constants. (ii) Coherent beam ( νe (k) = −νdr , for |k| ⩽ kdr ; Sk (t) = νe (k) = νe = const > 0, for |k| > kdr , where νdr is a constant. (iii) Noise source ( Sk (t) =

1

νe ω02 ξk (t), for |k| ⩽ kdr ; 0, for |k| > kdr ,

where ξk (t) is a known function. After a huge calculation, for five modules, νdr = −0.025,

νe = 0.05,

νi = 0.005k,

kdr = 0.2,

numerical results show that the stage turn total chaos from the coexistence of soliton and chaos.

5.2

Some problems for infinite dimensional dynamical systems

189

Figure 5.5 denotes the contour of equalling |E(x, t)|2 in x, t-plane, with five modules, kdr = 0.2, coherent beam source, νd = −0.025, νe = 0.05, νi = 0.005, L = 64, W = 0.13. Figure 5.6 shows the relationship between energy W and Lyapunov exponent L1 , where noise source is denoted by “×” and beam source by “+” with value kdr = 0.27, and coherent source by “•”.

Figure 5.5

Figure 5.6

In 1989, by adding a different exogenous source, Robinson [208,209] and Newman computed the two-dimensional Zakharov system of equations with damping and inhomogeneous medium ∇(i∂t + ∇2 + iˆ r)ε = ∇(nε) + Sk (t),

(5.2.5)

(∂t2 + 2Cs νˆ∂t − Cs2 ∇2 )n = ∇2 |ε|2 ,

(5.2.6)

numerical results show the existence of chaos, where rˆ is the Langmuir damping operator, νˆ is the ion acoustic damping operator, Cs is ion acoustic velocity, the transfused exogenous term in Fourier space is ( rb Eb , as k = kb ; Sk (t) = 0, as k ̸= kb ,

190

Chapter 5

Infinite Dimensional Dynamical Systems

with Eb a definite physical constant. As Kb and rb taking the following different values (i) kb = 0, rb = −1.7 × 10−4 Wp , (ii) kb = 0.066kb x, rb = −1.7 × 10−4 Wp , (iii) kb = 0.225kb x, rb = −1.7 × 10−4 Wp , with some definite physical constants kb and Wp , we can obtain the following numerical results list as Figures 5.7∼5.9.

Figure 5.7

Langmuir wave of Strong turbulence is injected into single color wave source kb = 0, rb = −1.7 × 10−4 Wp

2. Nonlinear Schordinger equation with damping term i(εt + rˆε) + εxx + (|ε|2 − |ε|20 )ε = 0,

(5.2.7)

where rˆ is a damping operator, F [ˆ rε] = r(k)εk (t), |ε|20 is the average of |ε|2 corresponding to space. Suppose that (5.2.7) has an approximate solution ε(x, t) = ε0 (t) exp[i(k0 x − ω0 t)] + ε1 (t) exp[i(k1 x − ω1 t)] + ε2 (t) exp[i(k2 x − ω2 t)], (5.2.8) where 2k0 = k1 + k2 , ωσ2 = kσ2 for σ = 0, 1, 2. If replacing ε in (5.2.7) by (5.2.8), we can obtain a system of ordinary differential equations of ε0 (t), ε1 (t) and ε2 (t).

5.2

Some problems for infinite dimensional dynamical systems

191

Russell and Ott [210] computed them, numerical results show that there exist period doubling bifurcation, singular attractor and tangent bifurcation from chaos to periodic solution. Another example is nonlinear Schrödinger equation [48] with dissipative term iqt + qxx + |q|2 q = iµ(q + qxx ) − ir|q|2 ,

(5.2.9)

as µ, r ≪ 1, we can find its soliton solution   ivt iv(x − vt) 2t + A0 exp iA0 + 2 4 2 , (5.2.10) qS (x, t) = A0 (x − vt) cosh 2 s 6 where A0 = , v is the propagation velocity of soliton. When µ, r ∼ 1, (1 + 4r/µ) numerical result shows that the chaos appears.

Figure 5.8

Langmuir wave and ionized sound wave Wp t = 2.23 × 105

3. Kuramoto-Sirashinsky equation. Kuramoto [182] in 1978 when considering the dissipative structure of reactiondiffusion system, and Sirashinsky [194] in 1977 when considering the instability analysis of flame combustion and fluid mechanics, obtained the following (KS) equation

192

Chapter 5

Infinite Dimensional Dynamical Systems

(5.2.11) independently for β = 0, ν = 0. Later, these kind of equations are obtained in bifurcation solutions of Navier-Stokes equations [5] and flow of adhesive membrane [220].

Figure 5.9

Langmuir wave of Strong turbulence is injected into single color wave source kb = 0.066kD x, rb = −1.7 × 10−4 Wp

The KS-type equation is of the form 1 φt + φ2x + νφ + αφxx + βφxxx + rφxxxx = 0, 2

(5.2.11)

where α > 0, ν > 0, r > 0. Let φx = u. Differentiating (5.2.11) with respect to x, we have ut + uux + νu + αuxx + βuxxx + ruxxxx = 0, (5.2.12) where r is the higher order adhesive damping, ν is the linear damping, α is the dispersion of color, and β is the coefficient of color dispersion. Let u = eikx+σt .

(5.2.13)

Then we have the relationship of dispersion σ = −ν + αk 2 − rk 4 + iβk 3 .

(5.2.14)

Obviously, when Reσ > 0, u increases as t increasing; while for Reσ < 0, u is damped. We can easily obtain that when σ is a constant and ν > ν0 =

α2 , 4σ

(5.2.15)

the system is linearly stable; otherwise, it is linearly unstable. When ν = 0 and βuxxx ∼ uux ≫ αuxx ∼ ruxxxx , we have the soliton solution 2

u(x, t) = N0 + N sech

"

N (τ ) 12β

 21 

Z t x− 0

1 N0 + N 3

(5.2.16) 

# dt ,

(5.2.17)

5.2

Some problems for infinite dimensional dynamical systems

where N = N (τ ), τ = st, N (t) satisfies the equation    21αβ dN 4r = − N N 2. dt 189β 2 5r

193

(5.2.18)

On the contrary, if βuxxx ∼ uux ≪ αuxx ∼ ruxxxx ,

(5.2.19)

then we have the chaotic solution. In 1987, Nicolaenko et al. [196] considered the periodic problems of the following KS-type equation   1 2 ˜ 3 )x + βφ ˜ =0 (5.2.20) φt + 4φxxxx + α ˜ φxx + φx − δ(φ x 2 with periodic condition φ|t=0 = φ0 (x), (5.2.21)  4 L L where α ˜=δ , δ˜ = 4δ = 10−2 , β˜ = 4β , δ = 0.0025, β = 0.1, L is the 2π 2π parameter of bifurcation. They obtained the following numerical results: φ(x + 2π, t) = φ(x, t),



2

23 < α ˜ ⩽ 43.5,

periodic stable, the solution tends to fixed point, no chaos appears.

43.5 < α ˜ ⩽ 58,

intermittency chaos, homoclinic cycle.

58 < α ˜ ⩽ 73,

periodic stable, the global attractor is fixed point.

73 < α ˜ ⩽ 94,

chaos, turbulence.

94 < α ˜ ⩽ 128, 128 < α ˜ ⩽ 149,

periodic stable. chaos, turbulence.

The different cases of different t for α ˜ = 88 are displayed in Figures 5.10 ∼ 5.14.

Figure 5.10

sture 0 < t < 0.512

194

Chapter 5

Figure 5.11

Figure 5.13

Infinite Dimensional Dynamical Systems

Figure 5.12

1.34 < t < 1.85

0.126 < t < 0.638

Figure 5.14

1.61 < t < 2.00

0.832 < t < 1.340

Figures of bifurcation and chaos is Figure 5.15 with the change of α. In 1989, Holemes et al. [121] considered the bifurcation of periodic problems of the following KS-type equation 1 ut + αuxxxx + uxx + u2x = 0 2

(5.2.22)

with periodic condition u(x + L, t) = u(x, t),

u(x, 0) = u0 (x).

(5.2.23)

Using Galerkin method, i.e., making Fourier expansion of solution u(x, t) =

∞ X

ak (t)φk (x),

(5.2.24)

k=−∞

where φk (x) = ei2πkx/L ,

ak = a∗k ,

substituting (5.2.24) into (5.2.22), they obtained   l2 1X a′l (t) = l2 1 − al (t) + j(1 − j)aj al−j , µ 2 j

(5.2.25)

(5.2.26)

5.2

Some problems for infinite dimensional dynamical systems

where µ =

1 α



L 2π

195

2 .

Figure 5.15

bifurcation and chaos

Under third-order truncation and fourth-order truncation, one can obtain the bifurcation of |a| as µ changing. See Figure 5.16. 4. Sine-Gordon equation with dissipative and force oscillating terms. In many physical problems as Josephson tie, the sine-Gordon equation with dissipative and force oscillating terms φtt − φxx + sin φ + εφt = r sin ωt

(5.2.27)

is deduced. Bishop and Olsen et al. [200] give the following results: when ε, r are smaller, (5.2.27) exists soliton solution, the velocity of the soliton is changed, as the soliton approaching to stationary state, it is oscillating; while as ε, r are large, the chaos appears. See Figure 5.17.

196

Chapter 5

Figure 5.16

Infinite Dimensional Dynamical Systems

bifurcations

where bold line is steady state, shadow line is attractive heteroclinic orbit, dotted line is non-attractive heteroclinic orbit, + is the numbers of positive (instable) eigenvalue

Figure 5.17

5.3

space-time ditribution graph of solution ϕ(x, t) of SG equation with periodic condition, where εα = 0.2, ω = 0.6

Global attractor and its Hausdorff, fractal dimensions

In this section, we will introduce one of the important concepts of global attractor, state the existence theorem of global attractor and give the estimation of Hausdorff and fractal dimensions with it. We refer to [9,10,12,180,228,230] for global attractor and Hausdorff, fractal dimensions. For attractors of Newton-Boussinesq equation, see [108, 143] and so on. Definition 5.3.1 Suppose E is a Banach space, S(t) is a semi-group operator, that is S(t) : E → E, S(t + τ ) = S(t) · S(τ ) for all t, τ ⩾ 0 and S(0) = I (identity operator). If a compact subset A ⊂ E is called a global attractor of semi-group S(t) if it satisfies the following conditions: (i) Invariant property. A is unchangeable under semi-group S(t), that is for all t ⩾ 0, S(t)A = A . (5.3.1)

5.3

Global attractor and its Hausdorff, fractal dimensions

197

(ii) Attractive Property. A absorbs all bounded sets in E, that is for all bounded sets B ⊂ E, dist(S(t)B, A ) = sup inf ∥S(t)x − y∥E → 0, x∈B y∈A

as t → ∞.

(5.3.2)

In particular, all trajectories S(t)u0 originating from u0 converge to A as t → ∞, i.e., dist(S(t)u0 , A ) → 0,

as t → ∞.

(5.3.3)

The structure of global attractor is complicated, it contains the simple equilibrium point u∗ , satisfying F (u∗ ) = 0 of the initial value problem of nonlinear evolution equation du(t) = F (u(t)), (5.3.4) dt u(0) = u0 ,

(5.3.5)

the equilibrium points may be multiple solutions, it also contains time periodic trajectories, trajectories of almost-periodic solutions, fractals and singular attractors, etc. Moreover, it may not be smooth manifold, and the dimension may be non-integer. In order to give the existence theorem of global attractor, we introduce the definition of absorbing set. Definition 5.3.2 Given a bounded set B0 ⊂ E, if there exists a t0 (B0 ) > 0, such that for all bounded set B ⊂ E, S(t)B ⊂ B0 , f or all t ⩾ t0 ,

(5.3.6)

then we call B0 is a absorbing set of E, see Figure 5.18 for details.

Figure 5.18

Theorem 5.3.3 ([9]) Suppose E is a Banach space, {S(t), t ⩾ 0} is a semigroup operator, that is S(t) : E → E, S(t + τ ) = S(t) · S(τ ) for all t, τ ⩾ 0 and S(0) = I, where I is an identity operator. Moreover, suppose semi-group operator S(t) satisfies the following conditions:

198

Chapter 5

Infinite Dimensional Dynamical Systems

(i) S(t) is uniformly bounded on E, that is, for all R > 0, there exists a constant C(R), such that ∥u∥E ⩽ R implies ∥S(t)U ∥E ⩽ C(R),

f or all t ∈ [0, ∞).

(5.3.7)

(ii) There exists a bounded absorbing set B0 in E. (iii) S(t) is a completely continuous operator for t > 0. Then the semi-group operator S(t) has a compact global attractor A . Remark 5.3.4 The conclusion of Theorem 5.3.1 still true if we replace the bounded absorbing set B0 in (ii) by a compact absorbing set B0 , and the completely continuity of S(t) is replaced by the continuity of S(t). Remark 5.3.5 We can prove that the global attractor A in Theorem 5.3.3 is the ω-limit set of absorbing set B0 , that is, A = ω(B0 ) =

\ [

S(t)B0 ,

(5.3.8)

s⩾0 t⩾s

where the closure is taken on the whole E. Another existence theorem of global attractor is given below. Theorem 5.3.6 ([230]) Suppose E is a Banach space, semi-group operator {S(t)} is continuous. There exists an open set U ⊂ E and a bounded set B ⊂ U such that B is absorbed in U . Moreover, if one of the two conditions holds: (i) The operator S(t) is compact uniformly for sufficiently large t, that is, for all bounded set B, there exists a t = t0 (B), such that [

S(t)B

(5.3.9)

t⩾t0

is relative compact in E; or (ii) S(t) = S1 (t) + S2 (t), and the operator S1 (·) is compact uniformly for sufficiently large t, that is, S1 (·) satisfies (5.3.9), the operator S2 (t) : E → E is continuous, and for each bounded set B ⊂ E, rB (t) = sup ∥S2 (t)ϕ∥E → 0,

(5.3.10)

¯ φ∈B

then the ω-limit set A = ω(B) of B is a compact attractor, it absorbs all bounded sets in U . Moreover, it is the maximal bounded attractor in U , and it is connected if U is both convex and connected.

5.3

Global attractor and its Hausdorff, fractal dimensions

199

Hence, in order to prove the existence of GS, we should verify the hypotheses in Theorem 5.3.3 or 5.3.6, among these hypotheses, the following three hypotheses are of great important: (i) The existence of semi-group operator S(t); (ii) There exists a bounded or compact absorbing set; (iii) {S(t), t > 0} is a completely continuous operator or satisfies (5.3.9), or (5.3.10). In order to characterize the geometric properties of global attractor, we give some estimation of its Hausdorff dimension and fractal dimension firstly. Definition 5.3.7 The Hausdorff measure of a set X is defined by µH (x, d) = lim µH (X, d, ε) ε→0

= sup µH (X, d, ε),

(5.3.11)

ε>0

where µH (X, d, ε) = inf

X

rid ,

(5.3.12)

i

the infimum is taken by all ball with radius ri ⩽ ε covering X. If there exists a constant d = dH (X) ∈ [0, ∞] such that µH (X, d) = 0, for d > dH (X), µH (X, d) = ∞, for d < dH (X),

(5.3.13)

then the number dH (X) is called the Hausdorff dimension of set X. Definition 5.3.8 The fractal dimension of set X is defined by dF (X) = lim sup ε>0

log nX (ε) , log 1ε

(5.3.14)

where nX (ε) is the minimal number of balls with radius ri ⩽ ε covering X. It is clear that dF (X) = inf{d > 0, µF (X, d) = 0}, (5.3.15) where µF (X, d) = lim sup εd nX (ε). Since µF (X, d) ⩾ µH (X, d), we have ε→0

dH (X) ⩽ dF (X). Now we consider the initial value problem   du(t) = F (u(t)), t > 0, dt  u(0) = u , 0

(5.3.16)

(5.3.17)

200

Chapter 5

Infinite Dimensional Dynamical Systems

where F (u) is as given function, F (u) : E → E, E is a Bananch space. Suppose for all u0 ∈ E, there exists a global solution u(t) ∈ E. The mapping S(t)u0 : E → E is the semi-group operator of initial value problem (5.3.17). Let F : E → E be Fr´echet differentiable. The initial value problem of linear equation   dU (t) = F ′ (S(t)u ) · U (t), 0 dt (5.3.18)  U (0) = ξ, is solvable for each u0 and ξ ∈ E. Finally, S(t) is differentiable, the derivative L(t, u0 ) is defined by L(t, u0 ) · ξ = U (t) for all ξ ∈ E,

(5.3.19)

and U (t) is a solution of (5.3.18). Since (5.3.18) is the variational equation of (5.3.17), hence all hypotheses above are appearance and easy to verify. For fixed u0 ∈ L2 , suppose that ξ1 , ξ2 , · · · , ξJ are J elements in L2 . Let U1 (t), U2 (t), · · · , UJ (t) be J solutions of linearization equation (5.3.18) with initial value U1 (0) = ξ1 , U2 (0) = ξ2 , · · · , UJ (0) = ξJ . By direct computation, we have d ∥U1 (t) ∧ · · · ∧ UJ (t)∥2L2 − 2tr(F ′ (u(t) · QJ ))∥U1 (t) ∧ · · · ∧ UJ (t)∥2L2 = 0, (5.3.20) dt where F ′ (u(t) = F ′ (S(t)u0 ) is a linear projection U → F ′ (u(t))U , and u(t) = S(t)u0 is a solution of (5.3.17), the symbol ∧ is the outer product, tr is the trace of corresponding operator, QJ is the orthogonal projection from L2 to the subspace spanned by U1 (t), U2 (t), · · · , UJ (t). The J-dimensional volume ∧Jj=1 ξ is ωj (t) = sup sup ∥U1 (t) ∧ · · · ∧ Uj (t)∥2∧jL2 . u0 ∈A ξ∈L2

(5.3.21)

It is easy to verify that ωj (t) is sub-exponent with respect to t, that is, ωj (t + t′ ) ⩽ ωj (t)ωj (t′ ) for all t,

t′ ⩾ 0,

(5.3.22)

thus the limit 1

lim ωj (t) t = Πj for all 1 ⩽ j ⩽ J

(5.3.23)

ΠJ ⩽ exp{qJ }

(5.3.24)

t→∞

exists. By (5.3.20),

5.3

Global attractor and its Hausdorff, fractal dimensions

201

with qJ = lim sup qJ (t),

(5.3.25)

t→∞

 Z t  1 ′ qJ = sup sup tr(F (S(τ )u0 )QJ (τ ))dτ, i = 1, 2, · · · . t 0 u0 ∈A ξi ∈L2 |ξi |⩽1 Definition 5.3.9 A sequence Λ1 , Λ2 , · · · , Λm is defined by Λ1 = Π1 , Λ1 Λ2 = Π2 , · · · , Λ1 · · · Λm = Πm , or

Πm , Πm−1  1t  ωm (t) = lim , t→∞ ωm−1 (t)

Λ1 = Π1 , Λm

Λm =

for m ⩾ 2,

(5.3.26)

for m ⩾ 2.

Then we call Λm is the global Lyapunov exponent on set A, we call µm = log Λm ,

m>1

is the corresponding Lyapunov exponent. By (5.3.24), we have µ1 + µ2 + · · · + µJ < q J .

(5.3.27)

Theorem 5.3.10 Under the solvability hypothesis of initial value problem (5.3.17) and initial value problem (5.3.18), if for some m and t0 > 0, qJ (t) ⩽ −δ < 0 for all t ⩾ t0 ,

(5.3.28)

then the volume element ∥U1 (t) ∧ · · · ∧ UJ (t)∥ΛJ L2 is exponential decay as t → ∞; for u0 ∈ A and ξ1 , ξ2 , · · · , ξJ ∈ L2 , we have ∥U1 (t) ∧ · · · ∧ UJ (t)∥Λm L2 ⩽ ∥U1 (t0 ) ∧ · · · ∧ Um (t0 )∥ΛJ L2 exp(−δ(t − t0 )). If A is a functional invariant set to semi-group S(t), then for some j, qj < 0,

(5.3.29)

we have Πj = Λ1 · · · Λj < 1, µ1 + · · · + µm < 0, which implies further that

Λj < 1,

(5.3.30)

µj < 0.

(5.3.31)

that is,

202

Chapter 5

Infinite Dimensional Dynamical Systems

Theorem 5.3.11([230]) Suppose that the initial value problem (5.3.17) of nonlinear evolution equation exists a global attractor, and the global attractor is bounded in H 1 (Ω). Suppose further that the initial value problem of linear equation of (5.3.18) is solvable, the semi-group operator determined by initial value problem (5.3.17), S(t)u0 is differentiable. For some j, the qj defined by (5.3.25) satisfies (5.3.32)

qj < 0,

then the Hausdorff dimension and fractal dimension of the global attractor A are finite, and the Hausdorff dimension of A ⩽ j, and the fractal dimension of A is less than or equal to   (ql )+ . (5.3.33) j 1 + max 1⩽l⩽j−1 |qj | Let H be a Hilbert space, X ⊂ H be a compact set, S : X → H be a nonlinear continuous projection such that SX = X. (5.3.34) If for each u ∈ X, there exists a linear projection L(u) ∈ L (H), such that sup u, v∈X 0 0, r > 0, b > 1. Now we want to prove that u(t) = (x(t), y(t), z(t)) is bounded as t → ∞, and it exists an absorbing set. By (5.3.38), we have 1 d 2 b2 |u| + σx2 + y 2 + bz 2 = b(r + σ)z ⩽ (b − 1)z 2 + (r + σ)2 , 2 dt 4(b − 1)

5.3

Global attractor and its Hausdorff, fractal dimensions

203

d 2 b2 |u| + 2l|u|2 ⩽ (r + σ)2 , where l = min(1, σ), dt 2(b − 1) |u(t)|2 ⩽ |u(0)|2 e−2lt +

b2 (v + σ)2 (1 − exp(−2t)). 4l(b − 1)

(5.3.39)

By which we have b(r + σ) . lim sup |u(t)| ⩽ ρ0 with ρ0 = p 2 l(b − 1) t→H0

(5.3.40)

Hence there exists an absorbing set B(0, ρ) : the ball with center 0 and radius ρ > ρ0 . In fact, if B0 is a bounded set in R3 with radius R, then for t ⩾ t(B0 ) with 1 R2 t(B0 ) = log 2 , 2l ρ − ρ20 S(t)B0 ⊂ B(0, ρ). Moreover, we can prove that the ball B(0, ρ) is positive invariant, i.e., S(t)B(0, ρ) ⊂ B(0, ρ),

for t ⩾ 0.

By the estimation (5.3.39), we can verify the hypotheses of Theorem 5.3.6 are fulfilled, thus the Lorenz model exists global attractor. Now we turn to estimate the Hausdorff dimension of Lorenz attractor. Rewriting (5.3.38) as u˙ = F (u), (5.3.41) where

 σx − σy   F (u) = F (x, y, z) = −  σx + y + xz  . bz − xy + b(r + σ) 

(5.3.42)

The first variation of (5.3.41) is dU = F ′ (u)U, dt

(5.3.43)

where F ′ (u) is the 3 × 3 matrix −F ′ (u) · U = A1 U + A2 U + B(u)U, with

 σ  A1 =  0 0

 0 0  1 0 , 0 b



0 −σ  A2 =  0 0 0 0

for all u = (x, y, z) ∈ R3 .

 0  0 , 0



0  B(u) =  z −y

 0 0  0 x −x 0

(5.3.44)

204

Chapter 5

Infinite Dimensional Dynamical Systems

Now we consider initial value problems of (5.3.41) with initial condition U (0 ) = ξ,

ξ ∈ R3 ,

(5.3.45)

where ξ = (ξ1 , ξ2 , ξ3 ), U = (U1 , U2 , U3 ). We consider the changes of two and three dimensional volume elements: d |U1 ∧ U2 ∧ U3 | = |U1 ∧ U2 ∧ U3 |tr(F ′ (u)), dt

(5.3.46)

d |U1 ∧ U2 | = |U1 ∧ U2 |tr(F ′ (u) · Q), dt

(5.3.47)

where Q = Q2 (t, u0 , ξ1 , ξ2 ) is the orthogonal projection of R3 on the subspace spanned by U1 (t), U2 (t). By (5.3.44), the trace of F ′ (u) is −(σ + b + 1), thus by (5.3.20), we have |U1 (t) ∧ U2 (t) ∧ U3 (t)| = |ξ1 ∧ ξ2 ∧ ξ3 | · exp(−(σ + b + 1)t),

(5.3.48)

thus the three-dimensional infinitesimal volume element decays as t exponents increasing for Lorenz equation. By (5.3.43), we get sup ξ∈R3 |ξi |⩽1, i=1,2,3

|U1 (t) ∧ U2 (t) ∧ U3 (t)| ⩽ exp(−(σ + b + 1)t),

(5.3.49)

ω3 (t) = e−(σ+b+1)t ,

(5.3.50)

Λ1 Λ2 Λ3 = lim ω3 (t) t = e−(σ+b+1) ,

(5.3.51)

µ1 + µ2 + µ3 = −(σ + b + 1),

(5.3.52)

1

t→∞

where Λ1 , Λ2 , Λ3 and µ1 , µ2 , µ3 are the uniformly Lyapunov exponents and Lyapunov exponents of Lorenz attractor, respectively. In a similar manner, we have Z t  ′ |U1 (t) ∧ U2 (t)| = |ξ1 ∧ ξ2 | exp (tr(F (u(τ ))), Q(τ ))dτ . (5.3.53) 0

If ξ1 ∧ ξ2 ̸= 0, then |U1 (t) ∧ U2 (t)| ̸= 0. Suppose ϕ1 , ϕ2 , ϕ3 are orthogonal bases of R3 such that ϕ1 , ϕ2 are the bases of span by [U1 (t), U2 (t)], then tr(A1 + A2 ) · Q = trA1 · Q ⩾ 1 + b + σ − m, where m = max{1, b, σ}; tr(B(u) · Q) =

2 X ((B(u)φi )φi , i=1

5.3

Global attractor and its Hausdorff, fractal dimensions

205

where φi = (xi , yi , zi ), hence tr(B(u) · Q) =

2 X (zxi yi − xi zi y) = −zx3 y3 + x3 z3 y, i=1

q q p p 1 x23 + y32 + z32 y 2 + z 2 |tr(B(u) · Q)| ⩽ |x3 | y32 + z32 y 2 + z 2 ⩽ 2 1p 2 1 ⩽ y + z 2 ⩽ |u|. 2 2 By (5.3.36), for t large enough t ⩾ t1 (δ), b(r + σ) tr(B(u) · Q) ⩾ − p − δ, 4 L(b − 1) where δ > 0 is a sufficiently small number. Hence

where

sup ξ∈R3 |ξi |⩽1, i=1,2

|U1 (t) ∧ U2 (t)| ⩽ |ξ1 ∧ ξ2 | exp((k2 + δ)t),

(5.3.54)

b(r + σ) k2 = −(σ + b + 1) + m + p , 4 L(b − 1)

(5.3.55)

|U1 (t) ∧ U2 (t)| ⩽ exp((k2 + δ)t) for t ⩾ t1 (δ).

(5.3.56)

Hence ω2 (t) ⩽ exp((k2 + δ)t),

for t ⩾ t1 (δ).

(5.3.57)

Since δ > 0 is a sufficiently small number, letting t → ∞, we have 1

Λ1 Λ2 = lim ω2 (t) t = exp(k2 ),

(5.3.58)

µ1 + µ2 ⩽ k2 .

(5.3.59)

t→∞

Let d = 2 + S, 0 < S < 1, for t ⩾ t1 (δ) and k(δ) = −S(σ + b + 1) + (1 − S)(k2 + δ) < 0,

(5.3.60)

we obtain by (5.3.49) and (5.6.64) that ωd (t) ⩽ W21−S (t)W3S (t) ⩽ exp(k(δ)t) < 1. By Theorem 5.3.12, we get the Hausdorff dimension of Lorenz attractor A ⩽ d. If (5.3.60) holds, i.e., k2 + δ S> , (5.3.61) σ + b + 1 + k2 + δ

206

Chapter 5

Infinite Dimensional Dynamical Systems

then

k2 + δ . σ + b + 1 + k2 + δ Since δ > 0 can be selected sufficiently small, we have dH (A) ⩽ 2 +

dH (A) ⩽ 2 +

(5.3.62)

k2 . σ + b + 1 + k2

(5.3.63)

Hence we obtain the following theorem. Theorem 5.3.13 Suppose (5.3.64)

σ > 0, r > 0, b > 1,

then the Hausdorff dimension of Lorenz attractor A is bounded, its bound is estimated by (5.3.63), where m = max(1, b, σ). If σ = 10, r = 8, b =

8 , which is the type of Lorenz model, then 3 dH (A) ⩽ 2.588.

5.4

Global attractor and the bounds of Hausdorff dimensions for weak damped KdV equation

We consider the existence of global attractor of the following damped KdV equation (5.4.1)

ut + uux + uxxx + νu = f with periodic boundary condition u(x + L, t) = u(x, t),

for all x ∈ R,

t ⩾ 0,

u(x, 0) = u0 (x),

(5.4.2) (5.4.3)

where ν > 0, f (x, t) is a known function. By [56], as u0 (x) and f (x, t) are sufficiently smooth functions, the global solution of (5.4.1)-(5.4.3) exists. When f (x, t) is independent of t, or it is periodic function with respect to t, the solutions of (5.4.1)-(5.4.3) form a semi-group {S(t)}t>0 , that is S(0) = I,

S(t1 + t2 ) = S(t1 )S(t2 ) for all ti ∈ R+ .

(5.4.4)

It is easy to verify that the semi-group operator S(t) : H 2 (Ω) → H 2 (Ω) with Ω = [0, L] is weakly continuous. We first give the uniform priori estimation with respect to t of periodic boundary value problem (5.4.1)-(5.4.3), by which we construct the absorbing set, and prove the existence of global attractor.

5.4

Global attractor and the bounds of Hausdorff dimensions for weak damped KdV · · · 207

5.4.1

Uniform a priori estimation with respect to t

By multiplying (5.4.1) with multipliers M0 (u) = 2u,

(5.4.5)

M1 (u) = 2uxx + u2 ,

(5.4.6)

18 M2 (u) = uxxxx + 6uuxx + 3u2x + u3 , 5 respectively, we get the following energy equations: Z Z
d u2 dx + 2νu2 − 2f u dx = 0, dt  Z  Z
d u3 2 ux − dx + 2ν(u2x − u3 ) + f u2 + 2fx ux dx = 0, dt 3  Z  9 2 u4 d 2 u − 3uux + dx dt 5 xx 4  Z   18 2 2 2 4 + ν u + 6u uxx + 3uux + u 5 xx  18 2 3 + fxx uxx + 6uf uxx + 3f ux + f u dx = 0. 5

(5.4.7)

(5.4.8) (5.4.9)

(5.4.10)

Lemma 5.4.1 Suppose ν > 0, f ∈ H 2 . Then there exists a constant ρ2 = ρ2 (L, ν, ∥f ∥2 ), such that for all R > 0, there exists a T2 (R) such that ∥S(t)u0 ∥2 ⩽ ρ2 f or all u0 ∈ H2 , where ∥v∥m

and ∥u0 ∥2 ⩽ R f or all t ⩾ T2 (R), (5.4.11) Z k 2 m d v P 2k 2 2 = L |v|k , |v|k = k dx. In other words, the closed ball dx k=0

B2 = {v ∈ H 2 , ∥v∥2 ⩽ ρ2 }

(5.4.12)

in H 2 is a bounded absorbing set of semi-group {S(t)}. For each bounded set B ∈ H 2 , there exists a T2 (B), such that S(t)B ⊂ B2 f or all t ⩾ T2 (B).

(5.4.13)

Proof We use (5.4.8) to get the estimation of |S(t)u0 |0 , (5.4.9) to get |S(t)u0 |1 , and (5.4.10) to get |S(t)u0 |2 . Firstly, by (5.4.8), we get d 2 |u| + 2ν|u|20 ⩽ 2|f |0 |u|0 , dt 0

(5.4.14)

|S(t)u0 |0 ⩽ |u0 |0 e−νt + |f |0 (1 − e−νt )/ν,

(5.4.15)

hence

208

Chapter 5

Infinite Dimensional Dynamical Systems

which shows that S(t)u0 is uniformly bounded in L2 , and |S(t)u0 |0 ⩽ 2|f |0 /ν, where T0 (u0 ) =

for all t ⩾ T0 (u0 ),

(5.4.16)

1 ν|u0 |0 log . ν |f |0

(5.4.17)

Secondly, by (5.4.9), we define  Z  u3 2 dx, ϕ(u) = ux − 3

(5.4.18)

then we prove that ϕ(u) has a lower bound 1

ϕ(u) ⩾ In fact, since

10 1 1 2 23 |u|1 − |u|03 − L− 2 |u|30 . 2 4

(5.4.19)

Z u3 dx ⩽ |u|L∞ |u|20 ,

where |v|

L∞

1 2



≡ sup |v(x)| ⩽ |v|0 0⩽x⩽L

1 2|v|1 + |v|0 L

(5.4.20)  12 ,

and hence Z √ 5 5 1
1 u3 dx ⩽ |u| 2 2|u|1 + L−1 |u|0 2 ⩽ 2|u| 2 |u| 2 + L− 21 |u|30 , 0 0 1

(5.4.21)

(5.4.22)

by Young′ s inequality, we have Z 1 10 3 u3 dx ⩽ 1 |u|21 + 2 |u| 3 + L−1 |u|36 . 0 2 4

(5.4.23)

In order to estimate the second term in (5.4.9), we can rewrite it as the form of νϕ(u) + ξ(u), where Z Z 5 ξ(u) = ν|u|21 − ν u3 dx + (f u2 − 2fx ux )dx. (5.4.24) 3 By (5.4.23), we get ν 2 2 3 5ν 10 5νL− 2 3 |u|1 − |u|03 − |u|0 − |f |∞ |u|20 − 2|f |1 |u|1 , 6 4 3 1 1 6 2 3 5ν 10 5νL− 2 3 −ξ(u) ⩽ |f |21 + |f |L∞ |u|20 + |u|03 + |u|0 . (5.4.25) ν 4 3 1

ξ(u) ⩾

1

5.4

Global attractor and the bounds of Hausdorff dimensions for weak damped KdV · · · 209

Rewriting (5.4.9) as the form d ϕ(u) + νϕ(u) = −ξ(u), dt then (5.4.15) and (5.4.25) imply that d ϕ(u) + νϕ(u) ⩽ K1 (u0 )e−νt + K2 , dt where

10 2 3 5ν 5νL− 2 3 |u0 | 3 + |u|0 , 4 3 1

K1 (u0 ) = |f |L∞ |u0 |20 +

(5.4.26) 1

(5.4.27)

1

7 1 2 3 5 10 5 K2 = 6ν + |f | + (5.4.28) |f |03 ν − 3 + L− 2 |f |30 ν −2 . 4 3 Multiplying both sides of (5.4.26) by eνt , and integrating them on [0, t], we get

−1

|f |21

L∞

|f |20 ν −2

ϕ(u(t)) ⩽ (ϕ(u0 ) + K1 (u0 )t)e−νt + K2 (1 − e−νt )ν −1 . By (5.4.23), we get ( ) 1 10 3 23 3 K2 2 − 12 2 3 ϕ(u(t)) ⩽ |u0 |1 + |u0 |0 + L |u|0 + K1 (u0 )t e−νt + . 2 4 ν

(5.4.29)

(5.4.30)

Noticing that the inequality (5.4.19), (5.4.30) implies |u(t)|1 = |S(t)u0 |1 is still bounded as t → ∞, then by (5.4.16), we can select a ρ1 = ρ1 (ν, f, L) and T1 (u0 ), which depends on ∥u0 ∥1 , such that ∥S(t)u0 ∥1 ⩽ ρ1 for all t ⩾ T1 (u0 ). Finally, denoted by 9 ψ(u) = |u|22 + 5

Z 

 u4 2 − 3uux dx, 4

(5.4.31)

(5.4.32)

we represent (5.4.10) as d ψ(u(t)) + ψ(u(t)) = −η(u(t)), (5.4.33) dt 9 the principal term in (5.4.33) is |u|22 , and η(u) is bounded to O(∥u∥1 ), similar 5 estimation implies (5.4.11). Then we obtain the following theorem. Theorem 5.4.2 The set A = ω(B2 ) =

[ [ S⩾0 t⩾S

S(t)B2

(5.4.34)

210

Chapter 5

Infinite Dimensional Dynamical Systems

satisfies (1) A is bounded and weakly closed in H 2 , (2) S(t)A = A , (3) For each bounded set B ⊂ H 2 , S(t)B tends to A in the sense of weak topology in H 2 , where ¯ in (5.4.34) is the closure in the sense of weak topology in H 2 . Now we turn to estimate the dimension of attractor. Consider the first variational equation of (5.4.1) vt + (uv)x + vxxx + νv = 0 (5.4.35) with periodic boundary condition v(x + L, t) = v(x, t),

for all x, t ∈ R,

(5.4.36) (5.4.37)

v(x, 0) = v0 (x), where u = u(t) : S(t)u0 , t ∈ R, u0 ∈ H is a trajectory, and 2

v0 (x) ∈ H 2 .

(5.4.38)

Since u ∈ L∞ ((0, ∞); H 2 ), it is easy to prove that the linear equation (5.4.35)(5.4.37) have a solution v ∈ L∞ ((0, ∞); H 1 ). (5.4.39) We can even prove that the linear map (5.4.40)

(DS(t)u0 )v0 = v(t) is the uniform differentiation in the following sense.

Proposition 5.4.3 For given R and T with 0 < R, T < ∞, there exists a constant C = C(R, T ) such that for u0 , h0 and t satisfying ∥u0 ∥ ⩽ R,

∥u0 + h0 ∥ ⩽ R,

|t| ⩽ T,

(5.4.41)

we have ∥S(t)(u0 + h0 ) − S(t)u0 − (DS(t)u0 )h0 ∥1 ⩽ C∥h0 ∥21 .

(5.4.42)

Now we investigate the change of m-dimensional volume of the linear operator v(t) = DS(t)u0 , u0 ∈ A on H 1 . Taking v01 , v02 , · · · , v0m ∈ H 1 , we investigate the change of their Gram determination ∥v 1 (t) ∧ · · · ∧ v m (t)∥21 =

det ((v i (t), v j (t)))1

1⩽i,j⩽m

(5.4.43)

as t varying, where v i (t) = (DS(t)u0 )v0i with u0 ∈ H 2 , and ((·, ·))1 is the scalar product to norm ∥·∥1 . It is known that (5.4.43) is the square of volume of the polyhedron generated by vectors v 1 (t), v 2 (t), · · · , v m (t), we prove that this determination will decay exponentially for sufficiently large m as t → ∞ below. Firstly, we give some lemmas on the outer algebra in Hilbert space.

Global attractor and the bounds of Hausdorff dimensions for weak damped KdV · · · 211

5.4

Lemma 5.4.4 Suppose ϕ is a sesquilinear symmetric form which is forced and bounded, i.e., there exist positive numbers a and b such that a|η|2 ⩽ ϕ(η, η) ⩽ b|η|2 f or all

η ∈ H,

(5.4.44)

then for all η 1 , η 2 , · · · , η m ∈ H, we have am

det (η i , η j ) ⩽

1⩽i,j⩽m

det

1⩽i,j⩽m

ϕ(η i , η j ) ⩽ bm

det (η i , η j ).

(5.4.45)

1⩽i,j⩽m

Proof Taking η 1 , η 2 , · · · , η m ∈ H, then the determination of their second form ! m m X X i i (x1 , x2 , · · · , xm ) → ϕ xi η , xi η i=1

i=1

is the Gram determination, too, it equals to the products of its m eigenvalues, that is ! m m m Y X X det ϕ(η i , η j ) = maxm mmin ϕ xi η i , xi η i . (5.4.46) ∑ 2 G⊂R xi =1 l=1 dimG=l

1⩽i,j⩽m

i=1

i=1

i=1

Then combining (5.4.44) and (5.4.46) implies the desired result. Lemma 5.4.5 Suppose ψ, ψ2 are two sesquilinear symmetric forms in Rn , and ψ is positive definite. By {ωi }m i=1 , we denote the sequences ψ2 related to ψ, i.e., ωl = maxm min x∈F F ⊂R dimF =l x̸=0

ψ2 (x, x) , ψ(x, x)

f or

l = 1, 2, · · · , m.

Then for all ξ 1 , ξ 2 , · · · , ξ m ∈ Rm , we have m X l=1

det {(1 − δjl )ψ(ξ i , ξ j ) + δjl ψ2 (ξ i , ξ j )} =

1⩽i,j⩽m

m X

(5.4.47)

! ωl

l=1

det

1⩽i,j⩽m

ψ(ξ i , ξ j ). (5.4.48)

Proof Let {θ1 , θ2 , · · · , θm } be a base in Rm such that it makes ψ and ψ2 diagonal, that is ψ(θi , θj ) = δij , ψ2 (θi , θj ) = ωi δij . P Let ξ i = pij θj . Then j

ψ(ξ i , ξ j ) =

X

ψ2 (ξ i , ξ j ) =

pia pja ,

a

X

ωa pia pja ,

a

by which we have (1 − δjl )ψ(ξ i , ξ j ) + δjl ψ2 (ξ i , ξ j ) =

X a

pia pja ((1 − δjl ) + ωa δjl ) =

X a

pia Qlaj ,

212

Chapter 5

Infinite Dimensional Dynamical Systems

where Qlaj = ((1 − δjl ) + ωa δjl )pja . Hence the left-hand side of (5.4.48) turns into m X l=1

det(pQl ) = det p

m X

det Ql = det p·

l=1

m X

det{((1−δjl )+ ωa δjl )δja } = det p·

l=1

m X

ωl ,

l=1

which implies (5.4.48). Now we give two families of quadratic forms q(t; ·) and r(t; ·) on infinite dimensional Hilbert space H for t ∈ [0, T ] with 0 < T ⩽ ∞, and a family of linearly continuous operators {L(t), 0 ⩽ t < T } with L(0) = I, such that for all η0 ∈ H, the function t → q(t; L(t)η0 ) is absolutely continuous from [0, T ] to R, and for almost t ∈ [0, T ], d {q(t; L(t)η0 )} = r(t; L(t)η0 ). dt

(5.4.49)

By | · | and (·, ·), we denote the norm and inner product in H, respectively, by ϕ, ˜ we denote the quadratic form in the sense of polar coordinate of the quadratic form ϕ. Suppose that q(t; ·) is uniformly forced and bounded on [0, T ], i.e., there exist positive constants α(T ) and β = β(T ), such that α|η|2 ⩽ q(t; η) ⩽ β|η|2 ,

for all t ∈ [0, T ].

(5.4.50)

Moreover, suppose that there exist a nonnegative self-adjoint and compact operator K in H, constants C0 and σ ∈ [0, 1], such that for almost t ∈ [0, T ], |r(t; η)| ⩽ C0 |η|2(1−σ) (Kη, η)σ ,

for all η ∈ H.

(5.4.51)

By {Ki }∞ i=1 , we denote the non-increasing sequence of eigenvalues of K, we give the estimation of the following Gram determination Gm (t) =

det (L(t)η0i , L(t)η0j ),

1⩽i,j⩽m

(5.4.52)

where η01 , η02 , · · · , η0m are arbitrary elements in H. Theorem 5.4.6 Under the hypotheses on q(t; ·) and r(t; ·), we have the estimation for all t ∈ [0, T ] that ( ! ) m c0 X σ j i m −m det (L(t)η0 , L(t)η0 ) ⩽ β α exp Kl t det (η0i , η1j ). (5.4.53) 1⩽i,j⩽m 1⩽i,j⩽m α l=1

Proof If η01 , η02 , · · · , η0m are linearly dependent, then Gm (t) = 0 for all t ∈ [0, T ], and (5.4.53) holds obviously, hence we suppose that η01 , η02 , · · · , η0m are linearly independent. Set Hm (t) = det q˜(t; L(t)η0i , L(t)η0j ).

(5.4.54)

5.4

Global attractor and the bounds of Hausdorff dimensions for weak damped KdV · · · 213

By Lemma 5.4.4, for fixed t ∈ [0, T ] and ϕ(η, η) = q˜(t; L(t)η, L(t)η), (5.4.50), we know that (5.4.44) is true. By (5.4.45), we see that for all t ∈ [0, T ], αm Gm (t) ⩽ Hm (t) ⩽ β m Gm (t).

(5.4.55)

Set η i (t) = L(t)η0i and rewrite Hm (t) by Hm (t) =

det

1⩽i,j⩽m

q˜(t; η i (t), η j (t)).

By the classical derivative of a determination, we have   m X d d det (1 − δjl )˜ q (t; η i (t), η j (t)) + δjl q˜(t; η i (t), η j (t)) , Hm (t) = 1⩽i,j⩽m dt dt l=1 (5.4.56) where δjl is the Kronecker symbol. For given t ∈ [0, T ], by (5.4.56) and using Lemma 5.4.5 with ψ = q˜(t; ·) and ψ2 = r˜(t; ·), i.e., we set ! m X i ψ(x, x) = q t; xi L(t)η0 = q(t; L(t)η0 ), x ∈ Rm , i=1 m P

with η0 =

xi η0i . Using (5.4.52), we get

i=1

d {q(t; L(t)η0 )} = r(t; L(t)η0 ) = ψ2 (x, x). dt Hence (5.4.48) and (5.4.56) imply that d Hm (t) = dt

m X

! (5.4.57)

ωl (t) Hm (t).

l=1

By (5.4.47) we get the estimation of ωl (t) as r t; ωl (t) = maxm min x∈F F ⊂R dimF =l x̸=0

q t;

m X i=1 m X

! xi ηi (t) !.

(5.4.58)

xi ηi (t)

i=1

Since η0 =

m P

xi η0i and η(t) = L(t)η0 , and noticing (5.4.50), (5.4.51), we get

i=1

r(t; η(t)) |r(t; η(t))| c0 |η(t)|2(1−σ) |(Kη(t), η(t))|σ c0 ⩽ ⩽ ⩽ q(t; η(t)) q(t; η(t)) α|η(t)|2 α



|(Kη(t), η(t))| |η(t)|2

σ .

214

Chapter 5

Infinite Dimensional Dynamical Systems

By (5.4.58), we get 

 σ1 α (Kη(t), η(t)) ⩽ maxm min , |ωl (t)| x∈F F ⊂R c0 |η(t)|2

(5.4.59)

dimF =l x̸=0

where η(t) =

m P

xi L(t)η0i . Noticing that the linearly independence of η01 , η02 , · · · , η0m

i=1

implies Gm (0) > 0 and (5.4.58), there exists a Tmax ∈ [0, T ] such that for t ∈ [0, Tmax ], Hm (t) > 0. (5.4.60) If Tmax < T , then H(Tmax ) = 0.

(5.4.61)

For t ∈ [0, Tmax ], we get by (5.4.55) and (5.4.60) that Gm (t) > 0, hence for all t ∈ [0, Tmax ], L(t)η01 , L(t)η02 , · · · , L(t)η0m are linearly independent. Since F ⊂ Rm , dimF = l and t ∈ [0, Tmax ], (m ) X i F = xi L(t)η0 , x ∈ F i=1

is an l dimensional subspace in H, hence 

 σ1 α (Kξ, ξ) ⩽ max min |ωl (t)| . F ⊂H ξ∈F c0 |ξ|2

(5.4.62)

dimF =l ξ̸=0

The right-hand side of (5.4.62) is the l-th eigenvalue of K, thus |ωl (t)| ⩽ c0

Kl σ , α

and by (5.4.60) imply that dHm (t) c0 ⩽ dt α

m X

! Kl

σ

Hm (t) for all t ∈ [0, Tmax ].

(5.4.63)

l=1

The integrating the above inequality implies ( ! ) m c0 X σ Hm (t) ⩽ Hm (0) exp Kl t for all t ∈ [0, Tmax ]. α

(5.4.64)

l=1

On the other hand, the right-hand side of (5.4.57) has a lower bound, hence Tmax = T , then we deduce (5.4.53) by (5.4.55)and (5.4.64). This completes the proof.

5.4

Global attractor and the bounds of Hausdorff dimensions for weak damped KdV · · · 215

By Theorem 5.4.6, we can obtain the change of m-dimensional volume of DS(t)u0 for u0 ∈ A in H 1 . Suppose x is an invariant set in H 2 , that is, S(t)x = x for all t ∈ R.

(5.4.65)

Theorem 5.4.7 Suppose x is a bounded invariant set in H 2 , then there exist constants c1 and c2 , such that for each u0 ∈ x, m ⩾ 1 and t ⩾ 0, √ ∥(DS(t)u0 )v01 ∧· · ·∧(DS(t)u0 )v0m ∥1 ⩽ ∥v01 ∧· · ·∧v0m ∥1 cm 1 exp(c2 m−γm )t,

∀v0i ∈ H 1 . (5.4.66)

Proof For convenience, set ω i (t) = v i (t)eγt .

(5.4.67)

Equation (5.4.35) can be simplified to ωt + (uω)x + ωxxx = 0.

(5.4.68)

Since ((uω)x + ωxx )(uω + ωxx ) can be considered as a differential form with respect to x, multiplying (5.4.68) with 2ωxx + 2uω and integrating it, we get Z 2 ωt (ωxx + uω)dx = 0. (5.4.69) Hence Z To estimate

∥ω∥21

=

d dt

Z

Z (ωx2 − uω 2 )dx = −

u2t ω 2 dx.

(5.4.70)

(ω 2 + L2 ωx2 ), multiplying (5.4.68) by 2(1 + µ)ω with µ(t) be

any integrable function on t ∈ [0, L], we get Z Z d 2 (1 + µ)ω dx + 2(1 + µ) [(uω)x + ωxxx ]ωdx dt Z Z d =(1 + µ) ω 2 dx + (1 + µ) ux ω 2 dx = 0. dt

(5.4.71)

Multiplying (5.4.70) by L2 and adding it with (5.4.68), we get d {qµ (t; ω(t))} = rµ (t; ω(t)), dt

(5.4.72)

Z

where qµ (t; η) =

{η 2 + L2 ηx2 + (µ − L2 µ(t))η 2 }dx,

(5.4.73)

216

Chapter 5

Infinite Dimensional Dynamical Systems

Z rµ (t; η) = −

{((1 + µ)ux + L2 ut )η 2 }dx,

(5.4.74)

η ∈ H 1 . By selecting suitable µ, qµ (t; ·) is equivalent to the norm ∥ · ∥1 in H 1 . Select µ = µ(x) = L2 sup |v|L∞ , (5.4.75) v∈x

2

then µ is finite. Since X is bounded in H , we get by (5.4.75) ∥η∥21 ⩽ qµ (t; η) ⩽ (1 + 2µ)∥η∥21 ,

∀ η ∈ H 1 , ∀ t ∈ R.

(5.4.76)

By (5.4.1), the integral containing ut in the right-hand side of (5.4.74) turns to Z Z 2 2 2 −L ut η dx = L (uux + uxxx + γu − f )η 2 dx, (5.4.77) where

Z

Z uxxx η 2 dx ≡ −

Z uxx (η 2 )x = −2

uxx ηx ηdx,

Z  21  1 uxx ηx ηdx ⩽ |u|2 |η|1 |η|L∞ ⩽ |u|2 |η| 2 |η|1 2|η|1 + 1 |η|0 . 0 L Hence the right-hand side of (5.4.70) can be estimated as 3

1

|rµ (t; η)| ⩽ c3 ∥η∥12 ∥η∥02 ,

(5.4.78)

where c3 is a constant, it depends on γ, L, ∥f ∥0 and the radius of ball containing X in H 2 . Now we estimate (5.4.66) by using Theorem 5.4.7. Let H = H 1 , | · | = ∥ · ∥1 , (·, ·) = ((·, ·))1 . Let q = qµ , r = rµ , L(t) = eνt DS(t)u0 , η0i = ui0 . Taking α = 1, β = 1 1 + 2µ in (5.4.50), σ = , c0 = c3 in (5.4.51), and 2 Z ((Kη, η))1 = ηξdx, ∀ η, ξ ∈ H 1 , (5.4.79)  −1 2 2 ∂ where K = 1 + L is the inverse of unbounded operator v → v + L2 vxx in ∂x2 1 H 1 , with eigenvalues , β ∈ Z, then we have, 1 + 4π 2 β 2 m X l=1

1

ωl4 ⩽

2m X l=1

1

ωl4 ⩽ 2

m X √ 1 (1 + 4πl2 )− 4 ⩽ c4 m.

(5.4.80)

l=0

Hence (5.4.53) in Theorem 5.4.6 becomes  √ det(ω i (t), ω j (t))1 ⩽ (1 + 2µ)m exp(c3 c4 mt)

det (v0i , v0j ),

1⩽i,j⩽m

(5.4.81)

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 217

5.5

where ω i (t) = eγt v i (t). Then (5.4.81) implies (5.4.66) with 1

c1 = (1 + 2µ) 2 , c2 = c3 c4 /2. We can easily obtain the following theorem using Theorem 5.4.7. Theorem 5.4.8 The global attractor A of (5.4.1)-(5.4.3) has finite fractal and Hausdorff dimensions in H 1 . We refer to the papers [56,111,116] for the existence of global attractor of damped KdV equation and the references therein.

5.5

Global attractor and the bounds of Hausdorff dimensions for weak damped nonlinear Schrödinger equation

We consider the nonlinear damped Schrödinger equation of the form iut + uxx + g(|u|2 )u + iγu = f

(5.5.1)

u|t=0 = u0

(5.5.2)

with initial condition and one of the boundary conditions (denoted by (I)) Dirichlet boundary condition: u(0, t) = u(L, t) = 0,

∀ t ∈ R.

(5.5.3)

Neumann boundary condition: ∂u ∂u (0, t) = (L, t) = 0, ∂x ∂x Periodic boundary condition: u(x, t) = u(x + L, t),

∀ t ∈ R.

∀ x ∈ R,

∀ t ∈ R.

(5.5.4)

(5.5.5)

The nonlinear function g(u) ∈ C ∞ (u ∈ [0, ∞)) satisfies the increasing condition G+ (s) = 0, s3

(5.5.6)

h(s) − ωG(s) ⩽ 0, s3

(5.5.7)

lim

s→+∞

and there exists ω > 0, such that lim sup s→+∞

where

Z

h(s) = sg(s), G(s) =

s

g(σ)dσ, G+ (s) = max(G(s), 0), G− (s) = max(−G(s), 0). 0

(5.5.8)

218

Chapter 5

5.5.1

Infinite Dimensional Dynamical Systems

Uniform a priori estimation with respect to t

Part A Some integral inequalities By multiplying both sides of (5.5.1) with u ¯, and integrating it on [0, L], we get Z

Z

L

i

Z

L

ut u ¯dx + 0

Z

L

Z

L

|u|2 g(|u|2 )dx + iγ

uxx u ¯dx + 0

0

L

|u|2 dx =

fu ¯dx. (5.5.9)

0

0

Let u satisfy boundary condition (I). Then Z

Z

L

L

uxx u ¯dx = −

|ux |2 dx.

0

0

Taking the real part and image part of (5.5.9), respectively, we get 1 d 2 dt Z

Z

Z

L

|u| dx + γ Z

Z

u¯ ut dx −

Im

|ux | dx + 2

0

0

(5.5.10)

fu ¯dx,

0 L

L

|u| dx = Im 2

0

L

Z

L

2

0

Z

L

L

2

h(|u| )dx = Re

fu ¯dx,

0

(5.5.11)

0

where h is defined in (5.5.8). Next, multiplying both sides of (5.5.1) with u ¯t , integrating it on [0, L], and taking the real part, we get Z Re

Z

L

0

Z

L

Z

L

g(|u|2 )u¯ ut dx − γIm

uxx u ¯t dx + Re

L

fu ¯t dx. (5.5.12)

u¯ ut dx = Re

0

0

0

Since u satisfies boundary condition (I), we have Z

Z

L

L

uxx u ¯t dx = −

ux u ¯xt dx.

0

0

Hence (5.5.12) implies 1 d 2 dt

Z

Z

L

{−|ux | + G(|u| )}dx − γIm 2

2

0

Z

L

L

u¯ ut dx = Re 0

fu ¯t dx.

(5.5.13)

0

Multiplying (5.5.11) with γ and summing it with (5.5.13), we get 1 d 2 dt

Z

L

{|ux |2 − G(|u|2 ) + 2Re(f u ¯)}dx Z

0 L

Z {|ux | − h(|u| ) + Re(f u ¯)}dx = Re 2

+γ

2

0

(5.5.14)

L

ft u ¯dx. 0

Let (·, ·)0 and | · |0 be inner product and norm of L2 (0, L), respectively: Z

L

1

u(x)¯ v (x)dx, |v|0 = {(v, v)0 } 2 .

(u, v)0 = 0

(5.5.15)

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 219

5.5

Rewrite (5.5.10) as the form 1 d 2 |u| + γ|u|20 = Im(f, u)0 . 2 dt 0

(5.5.16)

With respect to (5.5.14), we introduce two functionals Z

L

ϕ(v) = |vx | + 2Re(f, v)0 − 2

G(|v|2 )dx,

(5.5.17)

h(|v|2 )dx,

(5.5.18)

0

Z

L

ψ(v) = |vx |2 + Re(f, v)0 − 0

then (5.5.14) has the form 1 d ϕ(u) + γψ(u) = Re(ft , u)0 . 2 dt

(5.5.19)

Part B Inequalities under hypotheses on g Lemma 5.5.1 Under hypothesis (5.5.6), there exists a constant Cε′ which depends on g and ε, such that for all v, Z L 2ε |vx |20 − G+ (|v|2 )dx ⩾ (1 − 8ε|v|40 )|vx |20 − 2 |v|60 − LCε′ . (5.5.20) L 0 Proof By (5.5.6), we get for any ε > 0, there exists a constant Cε′ ⩾ 0 such that for all s ⩾ 0, G+ (s) ⩽ εs3 + Cε′ . (5.5.21) Hence

Z

Z

L

|vx |20 −

L

G+ (|v|2 )dx ⩾ |vx |2 − ε 0

|v|6 dx − LCε′ .

(5.5.22)

0

Since

 sup |v(x)| ⩽ |v|0 2

0⩽x⩽L

 1 2|v|1 + |v|0 , L

(5.5.23)

we get Z



L

|v| dx ⩽ 6

0

|v|20

sup |v(x)| ⩽ 4

0⩽x⩽L

|v|40

8|vx |20

 2 2 + 2 |v|0 . L

(5.5.24)

Then (5.5.23) and (5.5.24) imply (5.5.20). Lemma 5.5.2 Under hypothesis (5.5.7), for any ε > 0, there exists a constant Cε′′ which depends on g and ε, such that for all v, Z 0

L


2ε h(|v|2 ) − ωG(|v|2 ) dx ⩽ 8ε|v|40 |vx |20 + 2 |v|60 + LCε′′ . L

(5.5.25)

220

Chapter 5

Infinite Dimensional Dynamical Systems

Proof By (5.5.7), for all ε > 0, there exists a constant Cε′′ > 0, such that for all s ⩾ 0, h(s) − ωG(s) ⩽ εs3 + Cε′′ . (5.5.26) Let s = |v|2 . Integrating (5.5.26) on [0, L] and using (5.5.24), we obtain (5.5.25). Part C A priori estimation By(5.5.10), if f ∈ L∞ (R+ ; L2 (0, L)), u0 ∈ L2 (0, L), then u ∈ L∞ (R+ ; L2 (0, L)). Defining |f |0,∞ = ess sup |f (t)|0 , (5.5.27) t⩾0

using Cauchy-Schwarz inequality, we get |f |20,∞ γ 1 d 2 |u|0 + γ|u|20 ⩽ |u|20 + . 2 dt 2 2γ Hence,

|f |20,∞ d 2 |u|0 + γ|u|20 ⩽ . dt γ

(5.5.28)

Integrating with respect to t implies |u(t)|20 ⩽ |u(0)|20 exp (−γt) +

|f |20,∞ (1 − e−γt ). γ2

(5.5.29)

By (5.5.20), we know ϕ(u) is forced if ε > 0 is small enough, by which we can obtain the estimation of u ∈ L∞ (R+ ; H 1 (0, L)). Proposition 5.5.3 Suppose u is a regular solution of (5.5.1), (5.5.2) with boundary condition (I). Under hypotheses(5.5.6) and (5.5.7), there exists a constant ϕ∞ such that for all t ⩾ 0, ϕ(|u(t)|) ⩽ ϕ(u(0))e−γωt + ϕ∞ (1 − e−γωt ). Set e2∞

=

sup |u(t)|20 t⩾0

⩽ max

ε0 =

|u(0)|20 ,

|f |20,∞ γ2

(5.5.30)

! ,

ω e−4 , 32 + 10ω ∞

(5.5.31) (5.5.32)

then

1 2ε0 |ux |20 − 2 e6∞ − LCε′ − 2|f |20,∞ e∞ . 4 L Using(5.5.17), (5.5.18) and (5.5.25), we get ϕ(|u(t)|) ⩾

ωϕ(v) − ψ(v) ⩽ (ω − 1)|vx |20 + 2ωRe(f, v)0 − Re(f, v)0 + 8ε|v|40 |vx |20 +

(5.5.33)

2ε 6 |v| + LCε′′ . L 0 (5.5.34)

5.5

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 221

By (5.5.31), Cauchy-Schwarz inequality and 0 < ω ⩽ 1, we get for all t ⩾ 0, 2ε 6 C + LCε′′ . L2 ∞

(5.5.35)

1 d ϕ(u) + γωϕ|u| = r(ωϕ|u| − ψ|u|) + Re(ft , u)0 . 2 dt

(5.5.36)

ωϕ(u) − ψ(u) ⩽ 3|f |0,∞ e∞ + 8εe4∞ |ux |20 + Rewrite (5.5.19) as

We estimate Re(ft , u)0 firstly, |Re(ft , u)0 | ⩽ |ft |0 |u|0 ⩽ |ft |0,∞ |u|20 + L2 |ux |20


12

,

L2 |ft |20,∞ ωγ |ux |20 + . 4 ωγ

(5.5.37)

2ε 6 |u| − LCε′ + 2Re(f, u)0 . L2 0

(5.5.38)

|Re(ft , u)0 | ⩽ |ft |0,∞ e∞ + By (5.5.17) and (5.5.20), we get ϕ(u) ⩾ (1 − 8ε|u|40 )|ux |20 −

Since 0 < ω ⩽ 1, (5.5.31) and (5.5.35) imply ωϕ(u) − ψ(u) + where ε0 satisfies

3ε0 e6∞ ω ω |ux |20 ⩽ ϕ|u| + + 4|f |0,∞ e∞ + L(Cε′ 0 + Cε′′0 ), (5.5.39) 4 2 L2  ω ω −8 1+ ε0 e4∞ = 0. 4 2

(5.5.40)

For this ε0 , (5.5.38) implies (5.5.33), (5.5.36), (5.5.37) and (5.5.39) imply that 6ε0 e6∞ γ 2L2 dϕ(u) +γωϕ(u) ⩽ +(8γ|f |0,∞ +2|ft |0,∞ )e∞ + |ft |20,∞ +2Lγ(Cε′ 0 +Cε′′0 ), 2 dt L ωγ (5.5.41) hence we obtain (5.5.30) by multiplying right-hand side of (5.5.20) with ϕ∞ = 5.5.2

1 . γω

(5.5.42)

Transforming to Cauchy problem of the operator

Part A Functional form We introduce the Hilbert space H = L2 (0; L). Let unbounded linear operator A on L2 be defined as Av = −vxx

(5.5.43)

222

Chapter 5

Infinite Dimensional Dynamical Systems

with domain  2 H (0; L) ∩ H01 (0; L),      corresponding to boundary condition (5.5.3);    v ∈ H 2 (0; L), v (0) = v (L) = 0 , x x D(A) =  corresponding to boundary condition (5.5.4);     2  v ∈ Hloc (R), v(x + L) = v(x), ∀ x ∈ R ,    corresponding to boundary condition (5.5.5). For each γ > 0, the operator A + γ is an isomorphism from D(A) to H, since the embedding from D(A) to H is compact, hence (A + γ)−1 is a compact operator in H, and it is self-adjoint, hence, a basis of Hilbert space H formed by the eigenvector of A exists. Suppose {λj }∞ j=0 is the nondecreasing sequence of eigenvalues of A: 0 ⩽ λ1 ⩽ λ2 ⩽ · · · ⩽ λj → +∞,

j→∞

and the corresponding normalized eigenfunctions are {ωj }∞ j=0 . s For s ∈ R, let A be the power operator with domain D(As ). For example, 1 V = D(A 2 ), then the corresponding boundary condition is  1 H0 (0; L),      corresponding to boundary condition (5.5.3);    H 1 (0; L), V =  corresponding to boundary condition (5.5.4);    1  {v ∈ Hloc (R), v(x + L) = v(x), ∀x ∈ R},    corresponding to boundary condition (5.5.5); while V ′ = D(A− 2 ) is the dual space of V . Part B Functional form of Cauchy problem Given a function f with 1

f ∈ L∞ loc (R; H),

′ ft ∈ L∞ loc (R; V ),

(5.5.44)

we want to find a function u ∈ C(R; V ),

(5.5.45)

iut − Au + g(|u|2 )u + iγu = f,

(5.5.46)

u(0) = u0 ,

(5.5.47)

such that

the equation (5.5.46) holds in the sense of distribution, it takes value in V ′ , V0 ∈ V is a given element. Part C Existence and uniqueness of solution

5.5

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 223

By Segal theory, Cauchy problem (5.5.45)-(5.5.47) have a unique solution u(t) ∈ [0, T∗ ], where T∗ = +∞ or lim sup |u(t)| = +∞. By Proposition 5.5.3, the latter t→T∗

cannot occur. In fact, we have the following theorem. Theorem 5.5.4 Suppose that g satisfies (5.5.6) and (5.5.7), for given u0 ∈ V , f satisfies (5.5.45). Then initial value problem (5.5.45)-(5.5.47) exist a unique solution, and for t ∈ R, the map u0 → u(t) is continuous. Furthermore, if f ∈ L∞ (R+ ; H),

ft ∈ L∞ (R+ ; V ′ ),

then u ∈ L∞ (R+ ; V ).

(5.5.48)

We prove the second part of this theorem using uniform estimation with respect to t only, and sketch the proof of the first part. Indeed, this classical result can be obtained by the method of J. L. Lions. Firstly, using Galerkin method, we construct finite dimensional approximate solutions of (5.5.45)-(5.5.47). We select a basis {ωj }∞ j=0 , where m = 0, 1, · · · , then the corresponding approximate solutions are {um (t)}m∈M . The equalities (5.5.16) and (5.5.19) hold for um (t), too. Hence we obtain H 1 -modular estimation of {um (t)} with independent to m, since ft ∈ L∞ (R; V ′ ), inequality (5.5.37) should be replaced by |Re⟨ft , u⟩| ⩽ |ft |∗ · |u|20 + L2 |ux |20


21

,

(5.5.49)

where ⟨·, ·⟩ is the pair of V and V ′ ; | · |∗ is the dual modular of the modular in V ′ |V |V = |V |20 + L2 |vx |20


12

.

(5.5.50)

Let m → ∞. Using standard method, we see that u ∈ L∞ loc (R+ ; V ). Using t → −t, (5.5.45)-(5.5.47) can be discussed similar, thus we obtain a solution u ∈ L∞ loc (R; V ) of (5.5.45)-(5.5.47), and u(·, t) is weakly continuous from R to V . The strong continuity can be obtained by bootstrap technique, i.e., (5.5.46) is obtained by improving the smoothness of f˜ in the linear equation iut − Au = f˜ = f − iγu − g(|u|2 )u. Finally, we prove the uniqueness and continuous dependence of solution. Suppose u1 and u2 are two solutions of (5.5.45)-(5.5.47), let ω = u1 − u2 . Then ω satisfies iωt − Aω + iγω = g(|u2 |2 )u2 − g(|u1 |2 )u1 .

(5.5.51)

∞ Since ui ∈ C(R; V ), iωt − Aω ∈ L∞ loc (R; V ), and ω ∈ Lloc (R; V ), similar estimation as before, we have 1 d |ωx |20 = Im⟨iωt − Aω, Aω⟩, (5.5.52) 2 dt

224

Chapter 5

Infinite Dimensional Dynamical Systems

1 d |ω|2 = Im⟨iωt − Aω, ω⟩. (5.5.53) 2 dt 0 Then using (5.5.51)-(5.5.53) and the classical Gr¨onwall inequality, we obtain the estimation of |ω|V . The uniqueness is obvious since ω(0) = 0. 5.5.3

The existence of bounded absorbing set of H 1 modular

Suppose that f ∈ L∞ (R+ ; H),

ft ∈ L∞ (R+ ; V ′ ).

(5.5.54)

By (5.5.29), (5.5.30) and (5.5.33), the solution in Theorem 5.5.4 satisfies u(x, t) ∈ L∞ (R+ ; V ). We have the following proposition. Proposition 5.5.5 Suppose that hypotheses (5.5.6) and (5.5.7) hold, there exists a constant ρ∞,1 , such that for arbitrary R > 0 and v0 ∈ V with |u0 |20 + L2 |u0x |20 ⩽ R2 ,

(5.5.55)

there exists a T1 (R) > 0, such that the solution of (5.5.45)-(5.5.47) satisfies |u(t)|20 + L2 |ux |20 ⩽ (ρ∞,1 )2 ,

∀ t ⩾ T1 (R).

(5.5.56)

Proof By (5.5.29) and (5.5.16), we get |u(t)|20 ⩽ R2 e−γt + Hence, |u(t)|20 ⩽ where T0 (R) = e2∞ by

2 log γ





|f |20,∞ (1 − e−γt ). γ2

2|f |20,∞ , γ2

∀ t ⩾ T0 (R),

(5.5.57)

|f |0,∞ . We consider the case t ⩾ T0 (R) only. Replacing γR

2|f |20,∞ in (5.5.35), similar to (5.5.40), selecting ε1 with independent to R as γ2  ω ω  4|f |40,∞ −8 1+ ε1 = 0, 4 2 γ4

(5.5.58)

replacing (5.5.37) by (5.5.49), we get 48ε1 |f |20,∞ dϕ(t) 4|f |0,∞ + γωϕ(t) ⩽ + 16|f |20,∞ + |ft |∗,∞ 6 2 dt γ L γ 2L2 |ft |2∗,∞ + + 2Lγ(Cε′ 1 + Cε′′1 ) γω =ϕ˜∞ ,

(5.5.59)

5.5

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 225

where |ft |2∗,∞ = ess sup |ft (t)|∗ .

(5.5.60)

t⩾0

By (6.52), as t ⩾ T0 (R), we have ϕ(u(t)) ⩽ ϕ|u|T0 (R)e−γ(t−T0 (R)) +

ϕ˜∞ . γω

(5.5.61)

Since the upper bound of ϕ(u0 (T (R))) depends on R and u0 only, we can select a T (R) ⩾ T0 (R) such that ϕ(u(T0 (R)))e−γω(T (R)−T0 (R)) ⩽

ϕ˜∞ γω

(5.5.62)

with u0 satisfying (5.5.55). Then for t ⩾ T (R), (5.5.61) and (5.5.62) imply ϕ(u(t)) ⩽

2ϕ˜∞ . γω

By (5.5.33), it holds for t ⩾ T (R) ⩾ T0 (R), replacing ε0 by ε1 , and e∞ by

(5.5.63) 2|f |20,∞ , γ2

then we have by (5.5.55) |ux |20

√ 8 2|f |20,∞ 8ϕ˜∞ 64 6 ′ ⩽ + 6 |f |0,∞ + 4LCε1 + = K. γω γ γ

(5.5.64)

Combining with (5.5.57) and (5.5.64), we get (5.5.56), where ρ2∞,1 = 5.5.4

2|f |20,∞ + L2 K. v2

The existence of bounded absorbing set of H 2 modular

Now, we strengthen the hypotheses on f and ft . Let f ∈ L∞ loc (R; H),

ft ∈ L∞ loc (R; H),

(5.5.65)

and u0 ∈ D(A). Then the solution obtained in Theorem 5.5.4 satisfies u ∈ L∞ loc (R; D(A)),

ut ∈ L∞ loc (R; H),

(5.5.66)

and the map u0 → u(t) is continuous in D(A). In fact, let η = ut , differentiating (5.5.46) with respect to t, we get iηt − Aη + {g(|u|2 ) + g ′ (|u|2 )|u|2 }η + g ′ (|u|2 )u2 η¯ + iγη = ft .

(5.5.67)

By (5.5.46), we have η(0) = ut (0) = −iAu0 + ig(|u0 |2 )u0 − γu0 − if (0) ∈ H,

(5.5.68)

226

Chapter 5

Infinite Dimensional Dynamical Systems

where u0 ∈ D(A), f ∈ C(R; H). Taking inner product by (5.5.67) with η then taking imaginary part, we get 1 d 2 |η| + γ|η|20 + Im(g ′ (|u|2 )u2 , η 2 )0 = Im(ft , η)0 . 2 dt 0

(5.5.69)

sup |u(x, t)| < ∞,

(5.5.70)

Since 0⩽x⩽L t⩽T

together with (5.5.65), (5.5.68), (5.5.69) and Gronwall inequality implies ut = η ∈ L∞ loc (R; H).

(5.5.71)

Au = iut + g(|u|2 )u + iγu − f.

(5.5.72)

By (5.5.46), we have By (5.5.65), (5.5.71) and u ∈ L∞ loc (R; V ), we get (5.5.66), the proof of the continuity of map u0 → u(t) in D(A) is similar to that of Theorem 5.5.4. If we suppose f ∈ L∞ (R+ ; H), ft ∈ L∞ (R+ ; H), (5.5.73) then we have the following proposition. Proposition 5.5.6 Suppose g satisfies (5.5.6), (5.5.7), and (5.5.73) holds, then there exists a constant ρ∞,2 such that for arbitrary R > 0 and u0 ∈ D(A) with |u0 |20 + L2 |u0x |20 + L4 |u0xx |20 ⩽ R2 ,

(5.5.74)

there exists a T2 (R) > 0, such that the solution of (5.5.45)-(5.5.47) satisfies |u(t)|20 + L2 |ux |20 + L4 |uxx |20 ⩽ (ρ∞,2 )2 ,

∀ t ⩾ T2 .

(5.5.75)

Proof Suppose (5.5.56) holds. For sufficiently large t, we estimate |uxx |20 . Taking inner product by (5.5.46) with Aut + γAu then taking real part, we get −Re(Au, Aut + γAu)0 + Re(g(|u|2 )u − f, Aut + γAu)0 = 0.

(5.5.76)

Since g(|u|2 )u ∈ V , we have Z 2

Re(g(|u| u, Au))0 =

L

 g(|u|2 )|ux |2 + g ′ (|u|2 )Re(|ux |2 u ¯ + u¯ u2x ) dx.

(5.5.77)

0

Using the L∞ estimation of u and the L1 estimation of |ux |2 , we have |Re(g(|u|2 )u, Au)|0 ⩽ ρ′∞,1 ,

∀ t ⩾ T1 (R),

(5.5.78)

5.5

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 227

where ρ′∞,1 solely depends on ρ∞,1 . Next, we estimate Z

L

Re(g(|u|2 )u, Aut ) = Re 0


¯xt dx. g(|u|2 )u x u

(5.5.79)

The above equation is equal to Z

Z

L

L

g(|u|2 )Re(ux u ¯xt ) + 0

g ′ (|u|2 )Re(u¯ ux )Re(u¯ uxt )dx.

0

We get 1 d 2 dt

Z

L



g(|u|2 )|ux |2 + 2g ′ (|u|2 )Re(u¯ ux )2 dx − R(u),

(5.5.80)

0

where  R(u) = g ′ (|u|2 )(|ux |2 Re(u¯ ut ) + 2Re(u¯ ux )Re(ut u ¯x )) 0  + 2g ′′ (|u|2 )Re(u¯ ut )Re(u¯ ux )2 dx. Z

L

(5.5.81)

(5.5.46) implies that ut = −iAu + h,

(5.5.82)

and (5.5.56) implies that (2)

|h(t)|0 ⩽ ρ∞,1 ,

∀ t ⩾ T1 (R).

(5.5.83)

Substituting (5.5.82) into (5.5.81), using the L∞ estimation of u and (5.5.56), we obtain by (5.5.81) that (Z ) Z L L (3) 2 2 |ux | |Au|dx + |ux | |h|dx . (5.5.84) |R(u)| ⩽ ρ∞,1 0

0

Using inequality   1 sup |ux (x, t)|2 ⩽ |ux (t)|0 2|Au|0 + |ux (t)|0 , L 0⩽x⩽L

(5.5.85)

we get (3)

|R(u)| ⩽ ρ∞,1 |ux |L∞ |ux |0 (|Au|0 + |h|0 ). Hence by (5.5.83) and (5.5.85), for all t ⩾ T1 (R), we get   3 (4) |R(u)| ⩽ ρ∞,1 1 + |Au|02 .

(5.5.86)

228

Chapter 5

Infinite Dimensional Dynamical Systems

Now for (5.5.76), we get   3 1 d (4) ϕ1 (u) + γψ1 (u) ⩽ ρ1∞,1 + (ft , Au)0 + ρ∞,1 1 + |Au|02 , 2 dt

∀ t ⩾ T1 (R), (5.5.87)

where Z ϕ1 (u) =

|Au|20

+ 2(f, Au)0 −

L



g(|u|2 )|ux |2 + 2g ′ (|u|2 )Re(u¯ ux )2 dx,

(5.5.88)

0

ψ1 (u) = |Au|20 + (f, Au)0 .

(5.5.89)

By (5.5.56) and Cauchy-Schwarz inequality, we get (5)

|Au|20 ⩽ 2ϕ1 (u) + 4|f |20,∞ + ρ∞,1 ,

(5.5.90)

ϕ1 (u) − ψ1 (u) ⩽ |f |0,∞ |Au|0 + ρ∞,1 .

(6)

(5.5.91)

d γ (7) ϕ1 (u) + γϕ1 (u) ⩽ ϕ1 (u) + ρ∞,1 , dt 2

(5.5.92)

By (5.5.87), we get

here we use the Young inequality (|ft |0,∞ + γ|f |0,∞ )|Au|0 ⩽ 3

(4)

ρ∞,1 |Au|02 ⩽

γ (8) |Au|20 + ρ∞,1 , 4

γ (4) |Au|20 + ρ∞,1 . 4

Then (5.5.92) implies the estimation  2  γ (7) ϕ1 (u(t)) ⩽ ϕ1 (u(T1 (R))) exp − (t − T1 (R)) + ρ∞,1 . 2 γ Hence, for t ⩾ T2 (R), ϕ1 (u(t)) ⩽

4 (7) ρ . γ ∞,1

(5.5.93)

(5.5.94)

Then (5.5.90), (5.5.94) and (5.5.56) imply the estimation (5.5.74). 5.5.5

Nonlinear semi-group and long-time behavior

Suppose f ∈ H, f (t) ≡ f, ∀t ∈ R. By Theorem 5.5.4, we know that the Cauchy problem (5.5.46), (5.5.47) has a unique solution. Set S(t)u0 = u(t),

t ∈ R.

(5.5.95)

We know S(t) forms a semi-group, and it is continuous on D(A); by Proposition 5.5.5 and Proposition 5.5.6, we obtain the following corollary.

5.5

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 229

Corollary 5.5.7 The set B1 = {u ∈ v, |v|20 + L2 |vx |20 ⩽ (ρ∞,1 )2 }

(5.5.96)

is a bounded absorbing set of S(t) in V, and B2 = {v ∈ D(A), |v|20 + L2 |vx |20 + L4 |vxx |20 ⩽ (ρ∞,2 )2 }

(5.5.97)

is a bounded absorbing set of S(t) in D(A). Proposition 5.5.8 The set ω(B2 ) =

\ [

S(t)B2

(5.5.98)

s⩾0 t⩾s

is a nonempty invariant set of S(t) containing in B2 , i.e., S(t)ω(B2 ) = ω(B2 ),

∀ t ∈ R,

(5.5.99)

the closure is taken in the sense of weak topology in D(A). Proof We can obtain the desired result using the two facts: (i) S(t) is weakly continuous in D(A), (5.5.100) (ii) b ∈ ω(B2 ) ⇔ ∃ two sequences tn ∈ R, bn ∈ B2 , such that S(tn )bn convergents weakly in D(A) as tn → ∞, n → ∞.

(5.5.101)

We prove (5.5.100) firstly. It is sufficient to prove that the weak convergence of vn to v in D(A) implies the weak convergence of S(t)vn to S(t)v in D(A). Since the mapping D(A) onto V is compact, vn is strongly convergent to v in V , thus S(t) is continuous on V , and S(t)vn is strongly convergent to S(t)v in V . On the other hand, vn is bounded in D(A), so does S(t)vn . We select a subsequence of S(t)vn such that it is weakly convergent to certain ω, since S(t)vn is strongly convergent to S(t)v in V , hence ω = S(t)v, which shows that all subsequences of S(t)vn are weakly convergent to S(t)v in D(A). Since B2 is absorbing set, let dω be the distance in D(A). It is easy to prove that there exists a ω, such that dω (S(t)bn , ω) → 0 as n → ∞. This completes the proof of (5.5.101). Suppose B2 is a bounded absorbing set in D(A), then ω(B2 ) is the largest attractor. We have the following theorem. Theorem 5.5.9 The set A = ω(B2 )

(5.5.102)

230

Chapter 5

Infinite Dimensional Dynamical Systems

has the properties: A is bounded, and it is weakly closed in D(A), S(t)A = A ,

∀t ∈ R,

for each bounded set B ∈ D(A), lim d (S(t)B, A ) = 0. ω

t→∞

(5.5.103) (5.5.104) (5.5.105)

A is the largest set satisfying (5.5.103)-(5.5.105)in the sense of inclusion, and it is connected in the weak topology of D(A). Proof The properties (5.5.103) and (5.5.104) follow from Proposition 5.5.8. We prove (5.5.105) by contradiction. Suppose that there exist two sequences tn ∈ R, bn ∈ B with S(tn )bn ∈ B2 such that dω (S(tn )bn , A ) ⩾ ε0 > 0,

(5.5.106)

where ε0 is independent of n. Since S(tn )bn is bounded in D(A), there exists a b ∈ B2 such that dω (S(tn )bn , b) → 0, as n → ∞.      tn tn S bn . By (5.5.101), we know b ∈ ω(B2 ) = A , which Let S(tn )bn = S 2 2 contradicts to (5.5.106). The claim of maximal set in theorem is obvious, since its connectedness comes from the convex and connectedness of B2 , it is compact in the weak topology and the following abstract result. Lemma 5.5.10 Let (E , d) be a metric space, F be a nonempty compact and connected set. Suppose (Σ(t))t⩾0 is the semi-group in E which satisfies (i) for all t ∈ R+ , Σ(t) is continuous on E , (ii) for all e ∈ E , the map mapping R+ to E by t → Σ(t)e is continuous, (iii) there exist a compact set K and t0 ∈ R+ , such that Σ(t)F ⊂ K, then the ω-limit set of F to Σ(t) ω(F ) =

∀t ⩾ t0 ,

\ [

Σ(t)F

s⩾0 t⩾s

is a connected and compact nonempty set. Proof Set ε = B2 , d = dω , Σ(t) = S(t), F = K = B2 , t0 = T2 (B2 ). Then (i)-(iii) follow from Lemmas 5.5.1-5.5.10. Since D(A) can be compact embedded in D(As ) for s < 1, and D(As ) can be continuous embedded in H 2s (0, L), thus D(A) can be compact embedded in 1 H 2s (0, L) for s < 1, which implies the convergence in (5.5.103) by selecting s = 2 of H 2s (0, L).

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 231

5.5

Corollary 5.5.11 For given bounded set B ⊂ D(A), S(t)B is strongly convergent to A in the norm of V . Suppose B1 is a bounded set in V, let A∗ =

\ [

S(t)B1 ,

(5.5.107)

s⩾0 t⩾s

where the closure is taken in the sense of weak topology in V . We have Proposition 5.5.12 For each t ∈ R+ , the map S(t) is continuous on bounded set of V in the sense of H modular topology. Proof By (5.5.51) and (5.5.53), we have 1 d |ω|2 + γ|ω|20 = Im 2 dt 0

Z

L

(g(|u2 |2 )u2 − g(|u1 |2 )u1 )(¯ u2 − u ¯1 )dx,

(5.5.108)

0

where ui = S(t)u0i , ω = u2 − u1 . When both u01 and u02 are contained in certain bounded set of V , i.e., |u0i |1 ⩽ R, we know that for all T ∈ (0, ∞), sup |ui (x, t)| are finite for i = 1, 2, the supremum is taken for all x ∈ [0, L], |t| ⩽ T and |u0i | ⩽ R. Hence there exists a constant c(T, R) such that the right-hand side of (5.5.108) is confined to c(T, R)|ω|2 , using Gronwall inequality, we obtain the desired result. Corollary 5.5.13 with respect to S(t).

The A ∗ defined in (5.5.107) is the largest attractor in V

Since A is bounded in D(A), so does in V . By the invariance of A , we have A ⊂ A ∗.

(5.5.109)

Conversely, if A ∗ is bounded in D(A), then A ∗ ⊂ A , and hence, A = A ∗. 5.5.6

The dimension of invariant set

In this subsection, we will prove that the global attractor A in D(A) has finite fractal dimension. Firstly, we prove the m dimensional volume of the differentiation of semi-group S(t) in V is a contraction, here we use a new equivalent modular, rather than the classical modular, by which we can estimate the modulus obtained by different a priori estimation. Let u0 ∈ V , and v(t) be a solution of linear initial value problem ivt − Av + [g(|u|2 ) + |u|2 g ′ (|u|2 )]v + g ′ (|u|2 )u2 v¯ + iγv = 0,

(5.5.110)

232

Chapter 5

Infinite Dimensional Dynamical Systems

v(0) = v0 ,

(5.5.111)

where u(t) = S(t)u0 . We have obtained that u ∈ C(R; V ). It is easy to prove that the initial value problem (5.5.110) and (5.5.111) has a unique solution v ∈ C(R; V ).

(5.5.112)

The equation (5.5.110) is the variant equation of (5.5.46). Now we consider (5.5.46) as a system of equations of real part and imaginary part, the map DS(t)u0 is defined as (DS(t)u0 )v0 = v(t), (5.5.113) i.e., the differentiation of S(t) at u0 , using the classical method, we obtain S(t) is differentiable at u0 . Proposition 5.5.14 Let R, R1 and T be three positive numbers. There exists a constant c = c(R, R1 , T ) such that for all u0 , v0 , t with |u0 |V ⩽ R1 , |v0 |V ⩽ R and |t| ⩽ T , |s(t)(u0 + v0 ) − s(t)u0 − (Ds(t)u0 )v0 |V ⩽ c|v0 |2V .

(5.5.114)

Let ω(x, t) = v(x, t)eγt ,

(5.5.115)

where v(x, t) is a solution of initial value problem (5.5.110) and (5.5.111). We have iωt − Aω + (g(|u|2 ) + g ′ (|u|2 )|u|2 )ω + g ′ (|u|2 )u2 ω = 0, ω(0) = v0 . Taking inner product by (5.5.116) with ω ¯ t then taking real part, we get Z L  1 d |ωx |20 − Re g(|u|2 )ω ω ¯ t + 2g ′ (|u|2 )Re(u¯ ω )u¯ ωt dx = 0. 2 dt 0 Let

Z Φ(t, ω) =

L



|ωx |2 − g(|u|2 )|ω|2 − 2g ′ (|u|2 )Re(u¯ ω ) dx.

(5.5.116) (5.5.117)

(5.5.118)

(5.5.119)

0

Then (5.5.118) can be rewritten as d Φ(t, ω) = V (t, ω), dt where

 ∂  ∂  ′ |ω|2 g(|u|2 ) + 2Re(u¯ ω )2 g (|u|2 ) ∂t ∂t 0   ∂u + 4g ′ (|u|2 )Re(u¯ ω )Re ω dx. ∂t Z

(5.5.120)

L

V (t, ω) = −

(5.5.121)

5.5

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 233

On the other hand, taking inner product by (5.5.116) with ω ¯ t then taking imaginary part, we get Z L 1 d 2 g ′ (|u|2 )Re(u¯ ω )Im(u¯ ω )dx = 0. (5.5.122) |ω| + 2 2 dt 0 0 Set µ ∈ R. Let qµ (t, ω) = Φ(t, ω) + µ|ωx |20 , (5.5.123) Z L Vµ (t, ω) = V (t, ω) − 4µ g ′ (|u|2 )Re(u¯ ω )Im(u¯ ω )dx = 0, (5.5.124) 0

then (5.5.120) and (5.5.122) imply d {qµ (t, ω(t))} = Vµ (t, ω(t)), dt

∀µ ∈ R.

(5.5.125)

Let µ be µ = µ(x) =

1 + sup {|g(σ)| + 2σ|g ′ (σ)|}, L2 0⩽σ⩽x

(5.5.126)

where X is an invariant set, i.e., S(t)X = X, ∀t ∈ R, X is bounded in D(A),

(5.5.127)

|x|∞ = sup sup |v(x)|.

(5.5.128)

v∈X x∈[0,L]

Then we estimate qµ in (5.5.123) as 1 {|ω|20 + L2 |ωx |20 } ⩽ qµ (t, ω) ⩽ µ{|ω|20 + L2 |ωx |2 }, L2

∀ω ∈ V.

(5.5.129)

1

Hence, for fixed t, {qµ (t, ·)} 2 is equivalent to the modular | · |v . On the other hand, there exists a constant c3 = c3 (X) such that 3

1

|Vµ (t, ω)| ⩽ c3 |ω|02 (|ω|20 + L2 |ωx |20 ) 4 ,

∀ω ∈ V.

(5.5.130)

In fact, by (5.5.37), we have iut = Au − g(|u|2 )u − iγu + f. Since u ∈ X, and X is bounded in D(A), thus there exists a constant c4 = c4 (X) such that |ut |0 ⩽ c4 , ∀t ∈ R. (5.5.131) Noted that for all x, t, |u(x, t)| < x∞ , (5.5.121) implies Z |V (t, ω)| ⩽ c5

Z

L

|ω| |ut |dx ⩽ c4 c5 0

! 21

L

|ω| dx

2

4

0 3 2

⩽ c4 c5 |ω|0



1 2|ωx |0 + |ω|0 L

 21 .

(5.5.132)

234

Chapter 5

Infinite Dimensional Dynamical Systems

This together with (5.5.124) implies Z

L

|rµ (t, ω)| ⩽ |r(t, ω)| + 4µc6

|ω|2 dx, 0

which implies (5.5.130). We define the bilinear form in VR × VR as Z L  ¯ − g(|u|2 )η ξ¯ − 2g ′ (|u|2 )Re(u¯ ¯ + µη ξ¯ dx. ϕ(t; η, ξ) = Re ηx ξ¯x + µξη η )Re(uξ) 0

(5.5.133) It is obvious that ϕ(t; ·, ·) = qµ (·, ·). Hence ϕ(t; ·, ·) is an inner product in V , which is continuous and coercive. We introduce the Gram determination Hm (t) =

det (ϕ(t; ω i (t), ω j (t))

(5.5.134)

1⩽i,j⩽m

and Gm (t) = |ω 1 (t) ∧ · · · ∧ ω m (t)|2V =

det

(ω i (t), ω j (t)).

1⩽i, j⩽m

As we know, the determination of the inner products of vectors ξ 1 , · · · , ξ m in a Hilbert space H is the same as the determination of the quadratic form   m m X  X (x1 , · · · , xm ) → ψ xj ξ j , xj ξ j   j=1

j=1

in Rm , and the classical Mini-Max principal implies this determination equals to the product of its m eigenvalues, i.e., det

1⩽i,j⩽m

ψ(ξ i , ξ j ) =

m Y

max

min

m G⊂Rm ∑ x2i =1 l=1 dimG=l

ψ(xj ξ j , xj ξ j ).

(5.5.135)

i=1

x∈G

If there exists another inner product ψ1 in H , which is continuous and coercive, i.e., αψ(ξ, ξ) ⩽ ψ1 (ξ, ξ) ⩽ βψ(ξ, ξ), ∀ξ ∈ H , (5.5.136) then αm

det

1⩽i,j⩽m

ψ(ξ i , ξ j ) ⩽

det

1⩽i,j⩽m

ψ1 (ξ i , ξ j ) ⩽ β m

det

1⩽i,j⩽m

ψ(ξ i , ξ j ).

(5.5.137)

Now we select inner product H = V , by (5.5.129), (5.5.136) follows if β = µ, α = L−2 . Hence by (5.5.137), we have Lemma 5.5.15 Under the hypotheses mentioned above, L−2m Gm (t) ⩽ Hm (t) ⩽ µm Gm (t),

∀t ∈ R.

(5.5.138)

5.5

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 235

Now differentiating Hm (t) with respect to t, by the differentiable rules of determination, we have X d Hm (t) = det ϕ(t; ω i (t), ω j (t))l , 1⩽i,j⩽m dt m

(5.5.139)

l=1

where ϕ(t; ω i (t), ω j (t))l = (1 − δjl )ϕ(t; ω i (t), ω j (t)) + δjl

d {ϕ(t; ω i (t), ω j (t))}. (5.5.140) dt

By (5.5.123) and the equality listed below 4ϕ(t; ω i (t), ω j (t)) = qµ (t, ω i (t) + ω j (t)) − qµ (t, ω i (t) − ω j (t)), we have

d {ϕ(t; ω i (t), ω j (t))} = ρ(t; ω i (t), ω j (t)), dt

(5.5.141)

4ρ(t; η, ξ) = rµ (t, η + ξ) − rµ (t, η − ξ)

(5.5.142)

where is related to sesquilinear form rµ (t, ·). Let ψ(x, y) = ϕ(t; xj ω j , yj ω j ), ψ2 (x, y) = ρ(t; xj ω j , yj ω j ). By Lemma (5.5.10), we have  d Hm (t) = Hm (t) dt

m X

rµ t; maxm min

x∈F F ⊂R l=1 dimF =l x̸=0

 qµ t;

m X

 xj ω j (t)

j=1 m X

.

(5.5.143)

xj ω j (t)

j=1

We now estimate the change of m-dimensional volume formed by DS(t)u0 in V . Theorem 5.5.16 Suppose X is an invariant set, which is bounded in D(A). Then there exist constants c1 and c2 , such that for each v0 ∈ X, m ⩾ 1 and t ⩾ 0, v i (t) = (DS(t)u0 )v0i fulfills |v 1 (t) ∧ · · · ∧ v m (t)|V ⩽ |v01 (t) ∧ · · · ∧ v0m (t)|V cm 1 exp(c2 − γm )t.

(5.5.144)

Proof Let ω(x, t) = v(x, t)eγt . Then |v 1 (t) ∧ · · · ∧ v m (t)|2 = e−2γmt Gm (t),

(5.5.145)

Gm (t) = |ω 1 (t) ∧ · · · ∧ ω m (t)|2V .

(5.5.146)

where

236

Chapter 5

Infinite Dimensional Dynamical Systems

If |v01 ∧ · · · ∧ v0m |2 = 0, then v0i (t) are linearly dependent, so do v i (t), and hence (5.5.144) holds.If |v01 ∧ · · · ∧ v0m |v = ̸ 0, the continuity implies for t small enough, |v 1 (t) ∧ · · · ∧ v m (t)| ̸= 0, thus {ω 1 (t), · · · , ω m (t)}!are linearly independent. Hence m P for any G ⊂ Rm , x ∈ G\{0}, qµ t; xj ω j (t) ̸= 0, we deduce by (5.5.129), j=1

(5.5.130) that

32 X m j j xj ω (t) rµ t; xj ω (t) j=1 c j=1 0   ⩽ 3 3 . α m m 2 X X j qµ t; xj ω j (t) xj ω (t) j=1 j=1 



m X

(5.5.147)

V

Since Hm (t) ⩾ 0, by (5.5.143), we have 32 m X j xj ω (t) m j=1 d c3 Hm (t) X 0 maxm min Hm (t) ⩽ 3 . x̸=0 G⊂R dt α 2 m l=1 dimG=l x∈G X j x ω (t) j j=1

(5.5.148)

V

As F ⊂ Rm is given with dimF = l, for x ∈ F , F (t) spanned by

m P

xj ω j (t), is a

j=1

l-dimensional subspace of V , thus 32 X m j xj ω (t) 3 j=1 |ξ|02 0 min max min 3 . 23 ⩽ F x∈F ⊂V ξ∈F 2 m x̸=0 X dimF =l ξ̸=0 |ξ|V j x ω (t) j j=1

(5.5.149)

1 |ξ|20 = . 2 2 2 |ξ|0 + L |ξx | 1 + L2 λ l

(5.5.150)

V

On the other hand, max min

F ⊂V ξ∈F dimF =l ξ̸=0

Hence (5.5.149) and (5.5.150) imply dHm (t) c3 ⩽ dt α

m X l=1

1 (1 + L2 λl )3/4

! Hm (t),

Global attractor and the bounds of Hausdorff dimensions for weak damped · · · 237

5.5

where α = L−2 . Since λl ∼ c0 L−2 l2 as l → ∞, a constant c2 such that

m P

1 < ∞, there exists 2 λ )3/4 (1 + L l l=1

dHm (t) ⩽ 2c2 Hm (t), dt

(5.5.151)

which further implies Hm (t) ⩽ e2c2 t Hm (0),

∀t ⩾ 0.

(5.5.152)

Then (5.5.145), (5.5.146), (5.5.138) and (5.5.152) imply (5.5.144) with c1 =

1 √ . L µ

Next we estimate the dimension of global attractor A . As we know, the fractal dimension of a metric space E is defined by AF (E ) = lim sup ε→0+

log Nε (E )  , 1 log E

(5.5.153)

where Nε (E ) is the least number of balls with radius ε covering E . As we know, dH (E ) ⩽ dF (E ), where dH (E ) is the Hausdorff dimension of E . Define nonlinear map S on compact subset X of Hilbert space H , and suppose (5.5.154)

SX = X.

For each u ∈ X, there exists a linear operator L(u) ∈ L (H , H ) such that lim

ε→0

sup u,v∈x 0 c2 , there exists a n0 , such that S = S(n0 t0 ) satisfies (5.5.164)

ω ¯ m < 1,

thus (5.5.160) holds, and further (5.5.161) and (5.5.162) hold, too. This completes the proof of Theorem 5.5.18.

5.6

5.6.1

Global attractor and the bounds of Hausdorff, fractal dimensions for damped nonlinear wave equation Linear wave equation

We consider the linear wave equation of the form u ¨(t) + αu(t) ˙ + Au(t) = f,

0 0, f ≡ 0, then |u|21 + |u| ˙ 2 decays exponentially as t → ∞, while f ̸≡ 0 and f ∈ C(R+ ; H), {u, u} ˙ ∈ Cb (R+ ; E0 ). Let α > 0. 2

240

Chapter 5

Infinite Dimensional Dynamical Systems

Proposition 5.6.2 Let f ∈ Cb (R+ ; H), u0 ∈ V1 , u1 ∈ H. Suppose λ1 is the
λ1 first eigenvalue of A. Then for any ε with 0 < ε ⩽ ε0 , ε0 = min α4 , 2α , and for all t > 0, the solution of initial value problem (5.6.1) and (5.6.2) satisfies |u(t)|21 + |u(t) ˙ + εu(t)|2 ⩽ {|u0 |21 + |u1 + εu0 |2 }e−εt +

1 − e−εt 2 |f |L∞ (R+ ,H) , (5.6.5) αε

|u(t)|2 + |u(t) ˙ + εu(t)|2−1 ⩽ {|u0 |2 + |u1 + εu0 |L2−1 }e−εt +

1 − e−εt 2 |f |L∞ (R+ ,V−1 ) . αε (5.6.6)

Proof Let v = u˙ + εu. Then (5.6.1) can be rewritten v˙ + (α − ε)v + (A − ε(α − ε))u = f.

(5.6.7)

Taking inner product by (5.6.1) with u, and (5.6.7) with v, then summing them up, we get 1 d (|u|21 + |v|2 ) + ε|u|21 + (α − ε)|v|2 − ε(α − ε)(u, v) = (f, v). 2 dt Since ε|u|21 + (α − ε)|v|2 − ε(α − ε)(u, v) ε(α − ε) √ |u|1 |v| λ1 r ε ⩾ ε|u|21 + (α − ε)|v|2 − (α − ε)|u|1 |v| 2α ε α ⩾ (|v|2 + |u|21 ) + |v|2 , 2 2 ⩾ ε|u|21 + (α − ε)|v|2 −

(5.6.8)

we get 1 d (|u|21 + |v|2 ) + ε(|u|21 + |v|2 ) + α|v|2 ⩽ 2(f, v) ⩽ α|v|2 + |f |2 . dt α

(5.6.9)

Then (5.6.9) and Gr¨onwall inequality imply (5.6.5). By (5.6.7), taking inner product with respect to v(·, ·)−1 , we get 1 d (|u|2 + |v|2−1 ) + ε|u|2 + (α − ε)|v|2−1 − ε(α − ε)(u, v)−1 = (f, v)−1 , 2 dt by (5.6.8), we can easily obtain ε|u|2 + (α − ε)|v|2−1 − ε(α − ε)(u, v)−1 ⩾

ε α (|u|2 + |v|2−1 ) + |v|2−1 . 2 2

(5.6.10)

Hence d 1 (|u|2 +|v|2−1 )+ε(|u|2 +|v|2−1 )+α|v|2−1 ⩽ 2(f, v)−1 ⩽ 2|f |−1 |v|−1 ⩽ α|v|2−1 + |f |2−1 , dt α which implies (5.6.6).

5.6

Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 241

Remark 5.6.3 Let ϕ = {u, v}, v = u˙ + εu. Then (5.6.1) can be rewritten as system of equations ϕ˙ + Λε ϕ = F, (5.6.11) where F = (0, f ) and Λε =

ε −ε(α − ε) + A

−1 α−ε

! .

(5.6.12)

We can consider (5.6.11) in the product space E1 = V2 × V1 , E0 = V1 × H or E−1 = H × V−1 . For α > 0, 0 < ε ⩽ ε0 , we define the group operator Σε (t) as linear operator ϕ0 ∈ Ei → ϕ(t) = Σε (t)ϕ0 ∈ Ei , where t ∈ R, ϕ(t) is a solution of ϕ˙ + Λε ϕ = 0 with initial condition ϕ(0) = ϕ0 . Proposition 5.6.2 gives that Σε is exponentially decay in E0 and E−1 for 0 < ε ⩽ ε0 : ∥Σε (t)∥L(E0 ) ⩽ e−εt/2 ,

∀ t ⩾ 0,

(5.6.13)

∥Σε (t)∥L(E−1 ) ⩽ e−εt ,

∀ t ⩾ 0.

(5.6.14)

As α > 0, we consider the bounded solution on the real number axil, i.e., given f ∈ Cb (R; H), we seek for solution of (5.6.1) satisfying boundary condition lim sup |(u(t), u(t))| ˙ E0 < +∞.

(5.6.15)

t→−∞

Proposition 5.6.4 {u, u} ˙ ∈ Cb (R; E0 ), and

Let f ∈ Cb (R; H). Then (5.6.1) has a unique solution Z

{u(t), u(t) ˙ + εu(t)} =

t

−∞

Σε (t − τ ){0, f (τ )}dτ,

(5.6.16)

where 0 < ε ⩽ ε0 . Furthermore, if f, f˙ ∈ Cb (R; H), then {u, u} ˙ ∈ Cb (R; E1 ) and {u, ˙ u ¨} ∈ Cb (R; E0 ). Proof For the uniqueness, we need to prove that u satisfying (5.6.1), together with {u, u} ˙ ∈ Cb (R; E0 ), f = 0, implies u ≡ 0. In fact, we can prove u ≡ 0 under weaker hypothesis that {u, u} ˙ ∈ Cb (R; E+ ). For given t ⩾ s, by (5.6.6) and f ≡ 0, we have |u(t)|2 + |u(t) ˙ + εu(t)|2−1 ⩽ {|u(s)|2 + |u(s) ˙ + εu(s)|2−1 }e−ε(t−s) ⩽ ce−ε(t−s) , where c is independent to s. Let s → ∞. Then we have u(t) = 0.

242

Chapter 5

Infinite Dimensional Dynamical Systems

For the existence, set ψ(t, τ ) = Σε (t−τ )(0, f (τ )). For t ∈ R, we know by (5.6.13) that ψ(t, ·) ∈ L1 (−∞, t; E0 ), and 1

|ψ(t, τ )|E0 ⩽ e 2 ε(t−τ ) |f |L∞ (R,H) ,

∀ τ ⩽ t.

Hence we can define ϕ ∈ C(R; E0 ) as Z

t

ϕ(t) =

(5.6.17)

ψ(t, τ )dτ, −∞

and

Z |ϕ|L∞ (R,E0 ) ⩽

t

−∞

ε

e 2 (t−τ ) dτ |f |L∞ (R,H) =

2 |f |L∞ (R,H) . ε

(5.6.18)

On the other hand, ψ(·, τ ) is a solution of the following Cauchy problem ψ(·, τ ) ∈ C(R; E0 ), d ψ(t, τ ) + Λε ψ(t, τ ) = 0, ψ(τ, τ ) = {0, f (τ )}. dt Since the operator Λε maps E0 to E+ = H × V−1 , it is a linear and continuous d operator from E0 to E−1 , so we have dt ψ(t, τ ) ∈ C(R; E−1 ). Let β be the modular of Λε in C(E0 ; E−1 ). We have d ψ(t, τ ) ⩽ β|ψ(t, τ )|L∞ (R;E0 ) ⩽ βe−ε(t−τ )/2 |f |L∞ (R;H) . ∞ dt L (R;E−1 ) Then (5.6.17) implies Z

t

ϕ(t) ˙ = ψ(t, t) + −∞

dψ(t, τ ) dτ = {0, f (t)} − Λε dt

Z

t

−∞

ψ(t, τ )dτ = {0, f (t)} − Λε ϕ(t),

thus ϕ(t) ∈ C 1 (R; E−1 ). Moreover, we get ϕ(t) = {u(t), v(t)}, where v = u˙ + εu satisfies (5.6.11), this completes the proof of (5.6.16). When f, f˙ ∈ Cb (R; H), we have {u, u, ˙ u ¨} ∈ C([−T, T ]; V2 × V1 × H), ∀ T > 0. Since u ¨ ∈ Cb (R; V−1 ) implies that ω = u˙ satisfies {ω, ω} ˙ ∈ Cb (R; E−1 ), and ω ¨ + αω˙ + Aω = f˙.

(5.6.19)

By the proof given above, we know (5.6.1) has a unique solution {ω, ω} ˙ ∈ Cb (R; E0 ), and this solution is unique in Cb (R; E−1 ). Hence u ¨ ∈ Cb (R; H), u˙ ∈ Cb (R; V1 ), then we get Au ∈ Cb (R; H) by (5.6.1). Hence u ∈ Cb (R; V2 ). This completes the proof of this proposition.

Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 243

5.6

Proposition 5.6.5 Let s ∈ R and u0 ∈ Vs+1 ,

u1 ∈ Vs ,

f ∈ L2 (0, T ; Vs ).

(5.6.20)

Then there exists a unique solution u of initial value problem (5.6.1) and (5.6.2) satisfying {u, u} ˙ ∈ C([0, T ]; Vs+1 × Vs ). (5.6.21) Furthermore, we have the following energy equality 1 d (|u|2s+1 + |v|2s ) + ε|u|2s+1 + (α − ε)|v|2s + ε(ε − α)(u, v)s = (f, v)s , 2 dt

(5.6.22)

where v ≡ u˙ + εu. Proof We give the proof as s = 0 only, for the general case s ∈ R, one just need replace V1 , H, V−1 by Vs+1 , Vs , Vs−1 , respectively. We can consider the more special and simple case as α = 0 and ε = 0. The proof of existence needs Galerkin method, and the uniqueness can be obtained by energy equality. 5.6.2

Nonlinear wave equation

Let Ω ⊂ Rn be a connected bounded and open set, ∂Ω ∈ C ∞ . Suppose u = u(x, t) with x ∈ Ω and t ∈ R+ satisfies the nonlinear wave equation ∂u ∂2u +α − ∆u + g(u) = f, 2 ∂t ∂t

(x, t) ∈ Ω × R+ ,

(5.6.23)

initial boundary condition (x, t) ∈ ∂Ω × R+ ,

u(x, t) = 0, u(x, 0) = u0 (x),

∂u (x, 0) = u1 (x), ∂t

(5.6.24)

x ∈ Ω,

(5.6.25)

where f, u0 , u1 are given functions, α > 0. In order to characterize (5.6.23) ∼ (5.6.25) in a functional format, we introduce the following symbols: H = L2 (Ω),

D(A) = H 2 (Ω) ∩ H01 (Ω),

Au = −∆u,

1

V1 = D(A 2 ) = H01 (Ω).

Then (5.6.23) ∼ (5.6.25) can be written as an infinite-dimensional ordinary differential equation: u ¨ + αu˙ + Au + g(u) = f, (5.6.26) u(0) = u0 ,

u(0) ˙ = u1 .

(5.6.27)

244

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Let λ1 be the first eigenvalue of A, just as that in Proposition 5.6.2, let v = u+εu ˙ and ϕ = {u, v}. Then equations (5.6.26) ∼ (5.6.27) can be rewritten as ϕ˙ + Λε ϕ + Γ(ϕ) = F ,

(5.6.28)

ϕ(0) = ϕ0 = {u0 , v0 },

(5.6.29)

where Γ(ϕ) = {0, g(u)},

F = (0, f ),

v0 = u1 + εu0 .

(5.6.30)

Now we add some assumptions on the nonlinear term g(u). Suppose that g ∈ C (R; R), and Z s G(s) = g(σ)dσ. 1

0

We assume that

G(s) ⩾ 0, s2

(5.6.31)

sg(s) − c1 G(s) ⩾ 0, s2

(5.6.32)

lim inf |s|→∞

there exists a constant c1 > 0, such that lim inf |s|→∞

for n ⩾ 2,   0 ⩽ r < ∞,   ′ r |g (s)| ⩽ c(1 + |s| ), with 0 ⩽ r < 2,   r = 0,

for n = 2, for n = 3,

(5.6.33)

for n ⩾ 4.

Definition 5.6.6 If for any bounded set B in Ei , i = 0, 1, there exists a T = T (B), such that for t ⩾ T (B), the solution ϕ(t) of (5.6.28), (5.6.29) with ϕ0 ∈ B satisfies ϕ(t) ∈ Bi , i = 0, 1, then we call the bounded set Bi is an absorbing set of (5.6.28) in Ei . Theorem 5.6.7 Suppose f, u0 , u1 are given functions satisfying f ∈ Cb (R+ ; L2 (Ω)),

u0 ∈ H01 (Ω),

u1 ∈ L2 (Ω),

(5.6.34)

then the problem (5.6.26)∼(5.6.27), or equivalent (5.6.28)∼(5.6.29), has a unique solution u, which satisfies {u, u} ˙ ∈ Cb (R+ ; H01 (Ω) × L2 (Ω)). Moreover, there exists a closed sphere in H01 (Ω) × L2 (Ω), which is an absorbing set of (5.6.28) in H01 (Ω) × L2 (Ω).

5.6

Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 245

Proof By Proposition 5.6.2, initial-boundary value problem (5.6.23) ∼ (5.6.25) has a unique solution {u, u} ˙ ∈ C(R+ ; V1 × H). We prove that {u, u} ˙ ∈ L∞ (R+ ; V1 × H) and there exists an absorbing set in V1 × H. Now we take uniform estimation, by (5.6.31) and (5.6.32), there exist two nonnegative constants k1 and k2 such that Z 1 |u|21 ⩾ −k1 , ∀ u ∈ V1 , (5.6.35) G(u)dx + 8c 1 Ω Z 1 (5.6.36) (u, g(u)) − c1 G(u)dx + c1 ⩾ −k2 , ∀ u ∈ V1 . 8 Ω Z Z 1 ¯ G(u)dx = G(u)dx + Denoted by |u|2 + k1 , then 8c1 1 Ω Ω Z ¯ G(u)dx ⩾ 0, ∀ u ∈ V1 . Ω

Noting that ϕ = {u, v}, v = u˙ + εu, taking inner product by (5.6.28) with ϕ, we get 1 d (|u|21 + |v|2 ) + ε|u|21 + ε(ε − α)(u, v) + (α − ε)|v|2 + (g(u), v) = (f, v). (5.6.37) 2 dt d G(u) + ε(u, g(u)), and dt ε α ε|u|21 + ε(ε − α)(u, v) + (α − ε)|v|2 ⩾ (|u|21 + |v|2 ) + |v|2 , 2 2

Since (g(u), v) = (g(u), u˙ + εu) =

we get by (5.6.37) that   Z 1 d ε α 2 2 |u|1 + |v| + 2 G(u)dx + (|u|21 + |v|2 ) + |v|2 + ε(u, g(u)) ⩽ (f, v). 2 dt 2 2 Ω (5.6.38) Z ¯ By k2 and the definition of G(u)dx, we have Ω

Z

Z 1 1 ¯ G(u)dx − |u|21 − k2 ⩾ c1 G(u)dx − |u|21 − (k2 + c1 k1 ). 8 4 Ω Ω Z Let y = |u|21 + |v|2 + 2 G(u)dx. Then (5.6.38) implies (u, g(u)) ⩾ c1

Ω

1 dy ε α + (|u|21 + |v|2 ) + |v|2 + εc1 2 dt 4 2

Z

ε ¯ G(u)dx ⩽ k3 + |v||f |L∞ (R+ ;H) , 2 Ω   1 where k3 = 2(k2 + c1 k1 ). Let ρ = min , c1 . Then we deduce by the above 2 inequality that dy 1 + ερy ⩽ εk3 + |f |2L∞ (R+ ;H) . dt α

246

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By Gr¨onwall inequality, we obtain y(t) ⩽ y(0)e−ερt +

1 − e−ερt ερ

 εk3 +

 1 2 |f |L∞ (R+ ;H) . α

There exists a nonnegative constant k4 such that Z 1 G(u)dx + |u|21 ⩾ −k4 , 4 Ω

∀ u ∈ V1 .

(5.6.39)

(5.6.40)

Then (5.6.39) and (5.6.40) (v = u˙ + εu) imply   Z 1 2 |u|1 + |u˙ + εu|2 ⩽ k5 + |u0 |21 + |u1 + εu0 |2 + 2 G(u0 )dx e−ερt , 2 Ω where k5 = 2k4 +

k3 1 + |f |2 ∞ . ρ ερα L (R+ ;H)

We see from (5.6.41) that {u, u} ˙ ∈ Cb (R+ ; ε0 ), and for  Z  1 2 2 log |u0 |1 + |u1 + εu0 | + 2 G(u0 )dx , t⩾ ερ Ω

(5.6.41)

(5.6.42)

(5.6.43)

we have ϕ(t) = {u, v} ∈ {ψ ∈ E0 , |ψ|2E0 ⩽ 2ks + 1} := B0 ,

(5.6.44)

which means B0 is an absorbing set of (5.6.28) in E0 . Theorem 5.6.8 Suppose f, u0 , u1 are given functions satisfying f, f˙ ∈ Cb (R+ ; L2 (Ω)),

u0 ∈ H 2 (Ω) ∩ H01 (Ω),

u1 ∈ H01 (Ω),

(5.6.45)

then the solution of problem (5.6.26)∼(5.6.27) satisfies {u, u} ˙ ∈ Cb (R+ ; (H 2 (Ω) ∩ H01 (Ω)) × H01 (Ω)). Furthermore, the closed sphere in {H 2 (Ω) ∩ H01 (Ω)} × H01 (Ω) is an absorbing set of (5.6.28) in {H 2 (Ω) ∩ H01 (Ω)} × H01 (Ω). Proof We first prove the following fact: for all R > 0, there exist σ1 > 0 and C(R), such that for all ξ ∈ D(A) with |ξ|1 ⩽ R, |g(ξ)|1 ⩽ C(R)(1 + |ξ|2 )1−σ1 . (5.6.46) R In fact, for all R > 0, ξ ∈ D(A) = H 2 (Ω) ∩ H01 (Ω) with Ω |∇ξ|2 dx ⩽ R2 , we have Z ∂ξ ∂ξ 2 2 · dx ⩽ |g ′ (ξ))|2L∞ (Ω) |ξ|21 ⩽ C(1 + |ξ|2r |g(ξ)|1 = (g ′ (ξ))2 L∞ (Ω) )R . (5.6.47) ∂x i ∂xi Ω

5.6

Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 247

For n = 1, 2, by Sobolev embedding theorem, we know H 1+δ (Ω) ⊂ L∞ (Ω) for all δ > 0. By interpolation inequality, we have |ξ|L∞ (Ω) ⩽ Cδ |ξ|H 1+δ (Ω) ⩽ Cδ1 |ξ|δ2 |ξ|1−δ , 1 where ξ ∈ H 2 (Ω)∩ H01 (Ω)  and δ ∈ (0, 1). 1 1 1 , r . Then (5.6.47) implies (5.6.46) with σ1 = . For n ⩾ 4, Let δ = min 2 2 2 since |g ′ (ξ)| ⩽ C, thus (5.6.46) holds with σ1 = 1. For n = 3, By Agmon inequality 1

1

|ξ|L∞ (Ω) ⩽ C|ξ|12 |ξ|22 ,

ξ ∈ V2 (n = 3),

we deduce by (5.6.47) that   r r |g(ξ)|1 ⩽ C(R) 1 + R 2 |ξ|22 . r Then (5.6.33) and (5.6.47) imply (5.6.46) with σ = 1 − . 2 Now, we prove Theorem 5.6.8. Taking inner product in E1 by (5.6.28) with ϕ, we get 1 d (|u|22 + |u|21 ) + ε|u|21 + ε(ε − α)(u, v)1 + (α − ε)|v|21 + (g(u), v)1 2 dt d (5.6.48) = ε(f, u) − (f˙, Au) + (f, Au). dt Since 0 < ε ⩽ ε0 , where ε0 fulfills Proposition 5.6.2, we have ε|u|22 + ε(ε − α)(u, v)1 + (α − ε)|v|21 ⩾

ε α (|u|22 + |v|21 ) + |v|21 . 2 2

(5.6.49)

Let R > 0 be selected as |u0 |22 +|u1 |21 ⩽ R2 . By the continuity of embedding E1 ⊂ E0 , R2 we get |u0 |21 + |u1 |2 ⩽ R02 = . By Theorem 5.6.7, we have {u, u} ˙ ∈ Cb (R+ ; E0 ), λ1 thus there exists a C3 (R0 ) such that 2 |u(t)|21 + |u(t)| ˙ ⩽ C3 (R0 ),

∀ t ⩾ 0,

(5.6.50)

which further implies that there exists a T (R0 ) such that |ϕ(t)|2E0 ⩽ 1 + 2k5 ,

∀ t ⩾ T (R0 ).

(5.6.51)

By (5.6.50) and (5.6.46), we get |(g(u), v)1 | ⩽ |g(u)|1 |v|1 ⩽ C4 (R0 )|v|1 (1 + |u|2 )1−σ1 α C4 (R0 )2 (1 + |u|2 )2−2σ1 ⩽ |v|21 + 4 α α ε ⩽ |v|21 + |u|22 + C5 (R0 ). 4 4

(5.6.52)

248

Chapter 5

Infinite Dimensional Dynamical Systems

Then (5.6.50), (5.6.52) and (5.6.48) imply that ε 1 d  2 |u|2 + |v|21 − (f, Au) + (|u|22 + |v|21 ) ⩽ C6 (|f |2 + |f˙|2 ) + G(R0 ). 2 dt 4

(5.6.53)

By (5.6.53) and f, f˙ ∈ L∞ (R+ ; H), we see {u, u} ˙ ∈ Cb (R+ ; D(A) × V1 ). By (5.6.51), we have   ε2 , t0 = T (R0 ). |u(t0 )|21 + |u(t0 )|2 ⩽ 2(1 + 2k5 ) 1 + λ1   ε2 Let R0 = 2(1 + 2k5 ) 1 + (which is independent to (u0 , u1 )) be selected such λ1 that (5.6.53) hold. Similar to the proof of Theorem 5.6.7, we know that ϕ(t) enters the closed sphere in E1 as t large enough. We consider a more general equation than (5.6.26), in which the operator A is a linear self-adjoint operator in a real separated Hilbert space H, D(A) is dense in H, A is a positive operator and A−1 is a compact operator in H. Suppose that the nonlinear operator g(u) is a C 1 operator mapping V1 to H with Frechet derivative g ′ (u), and satisfies the following hypotheses: there exists a δ with 0 < δ ⩽ 1, for all R > 0, there exist constants C2 = C2 (R), k = k(R) such that ( |g(ξ) − g(η)|−1 ⩽ k|ξ − η|, (5.6.54) |g ′ (ξ) − g ′ (η)|L(V1 ,H) ⩽ C2 |ξ − η|δ1 , where ξ, η ∈ V1 , |ξ|1 ⩽ R, |η|1 ⩽ R. There exist G ∈ C 1 (V1 , R) with G(0) = 0 and p ∈ C(V1 , H) such that g(ξ) = G′ (ξ) + p(ξ),

∀ ξ ∈ V1 .

(5.6.55)

There exists a C1 > 0 such that (ξ, g(ξ)) − C1 G(ξ) ⩾0 |ξ|21 |ξ|1 →∞

(5.6.56)

lim inf

and lim inf

|ξ|1 →∞

G(ξ) ⩾ 0. |ξ|21

(5.6.57)

There exist σ > 0 and constant C3 such that ∀ ξ ∈ V1 , |P (ξ)|H ⩽ C3 (1 + |G(ξ)|) 2 −σ . 1

(5.6.58)

g maps D(A) to V1 , and satisfies Lipschitz condition on any bounded set in D(A); for all k > 0, there exist σ1 > 0 and constant C4 such that ∀ ξ ∈ D(A),

|ξ|1 ⩽ k,

|g(ξ)|1 ⩽ C4 (1 + |Aξ|)1−σ1 .

(5.6.59)

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Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 249

There exists an s ∈ [0, 1] such that for all ξ ∈ D(A),  ′ g (ξ) ∈ L(Vs , H), ′  sup |g (ξ)|L(Vs ,H) < ∞, |Aξ|⩽R

∀R > 0.

(5.6.60)

Now we consider the abstract nonlinear wave equation of the form u ¨ + αu˙ + Au + g(u) = f, u(0) = u0 ,

u(0) ˙ = u1 .

(5.6.61) (5.6.62)

The initial value problem (5.6.61) ∼ (5.6.62) can be transformed into a system of differential equations ϕ˙ + Λε ϕ + Γ(ϕ) = F, (5.6.63) ϕ(0) = ϕ0 ,

(5.6.64)

where ϕ = {u, v}, ϕ0 = {u0 , εu0 + u1 }, Γ(ϕ) = {0, g(u)}, F = (0, f ), and ! ε −1 Λε = . ε(ε − α) + A α − ε For the more general initial value problem (5.6.61) ∼ (5.6.62), we have more general theorems than Theorem 5.6.7 and Theorem 5.6.8. Theorem 5.6.9 Let the hypotheses on H, A, g hold. Suppose f, u0 , u1 are given functions satisfying f ∈ Cb (R+ ; H),

u0 ∈ V1 , u1 ∈ H,

(5.6.65)

then the initial value problem (5.6.61)∼(5.6.62) exists a unique solution u, which satisfies {u, u} ˙ ∈ Cb (R+ ; V1 × H). Moreover, there exists a closed sphere in E0 , which is an absorbing set of (5.6.61) in E0 . Theorem 5.6.10 Under hypotheses in Theorem 5.6.9, let f, u0 , u1 be given functions satisfying f, f˙ ∈ Cb (R+ ; H),

u 0 ∈ V2 , u 1 ∈ V1 .

(5.6.66)

Then the solution of initial value problem (5.6.61)∼(5.6.62) satisfies {u, u} ˙ ∈ Cb (R+ ; V2 × V1 ). Moreover, there exists a closed sphere in E1 , which is an absorbing set of (5.6.61) in E1 .

250

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Infinite Dimensional Dynamical Systems

For the existence, uniqueness of initial value problem (5.6.61)∼(5.6.62), we have the following classical result. Proposition 5.6.11 Suppose f, u0 , u1 are given functions satisfying f ∈ C([0, T ]; H),

u0 ∈ V1 , u1 ∈ H.

Then the initial value problem (5.6.61)∼(5.6.62) exists a unique solution u, which satisfies {u, u} ˙ ∈ C([0, T ]; V1 × H). Furthermore, for each t ∈ [0, T ], the map {u0 , u1 } → {u(t), u(t)} ˙ is an isomorphism from E0 to E0 . 5.6.3

The maximal attractor

Now, we consider the Cauchy problem of the form u ¨(t) + αu(t) ˙ + Au(t) + g(u(t)) = f (t),

(5.6.67)

u(0) = u0 , u(0) ˙ = u1 ,

(5.6.68)

where f˙(t) ≡ 0, ∀t ∈ R, H, A and g(u) fulfill the hypothesis (5.6.2). Moreover, we add two further hypotheses. (1) There exists a σ2 > 0, such that for all ξ ∈ V1 , the differentiation g ′ (ξ) ∈ L(H; V−1+σ2 ), and for all R > 0, sup |g ′ (ξ)|L(H;V−1+σ2 ) < +∞.

(5.6.69)

|ξ|1 ⩽R

(2) f, f˙ ∈ Cb (R; H), for any solution ϕ ∈ Cb (R; E0 ) of (5.6.63), ϕ ∈ Cb (R; E1 ) and   |ϕ|L∞ (R;E1 ) ⩽ C3 |f |L∞ (R;H) + |f˙|L∞ (R;H) + |ϕ|L∞ (R;E0 ) . (5.6.70) In this case, (5.6.67), (5.6.68) can be rewritten as ϕ˙ + Λε0 ϕ + Γ(ϕ) = F,

(5.6.71)

ϕ(0) = ϕ0 ,

(5.6.72)

where ϕ = {u, v}, Γ(ϕ) = {0, g(u)}, F = (0, f ), ϕ0 = {u0 , u1 + ε0 u0 }, and !   α λ1 ε0 −1 , . Λε0 = , ε0 = min 4 2α ε0 (ε0 − α) + A α − ε0 For ϕ0 ∈ E0 , the map S(t) : ϕ0 → S(t)ϕ0 = ϕ(t) is a homeomorphism in E0 . Since F is independent to t, the system (5.6.71) is autonomous. Hence {S(t)}t∈R is

5.6

Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 251

a nonlinear semi-group acting on E0 . By Theorem 5.6.10, S(t) is a homeomorphism in E1 . Although S(t) is a homeomorphism in E0 , it is not a compact mapping. However, the lemma listed below shows that S(t) can be represented to sum of two mappings, one of them tends to zero as t → +∞, and the other is a compact mapping. Lemma 5.6.12 For all t ∈ R+ , the mapping S(t) has a decomposition (5.6.73)

S(t) = Σε0 (t) + U (t),

the continuous mapping U (t) : E0 → E0 is uniformly compact, i.e., for any bounded S set B in E0 and for all t ∈ R, S(τ )B is relatively compact in E0 . τ ⩾t

Proof The solution of initial value problem (5.6.71)∼(5.6.72) can be written as Z

t

Σε0 (t − s)(F(s) − Γ(ϕ(s)))ds.

{u(t), u(t) ˙ + εu(t)} = ϕ(t) = Σε0 (t)ϕ0 + 0

Hence,

Z

t

Σε0 (t − s)(F(s) − Γ(ϕ(s)))ds.

U (t)ϕ0 =

(5.6.74)

0

Let {ξ(t), ζ(t)} = U (t)ϕ0 . Then by (5.6.74), we have ˙ + ε0 ξ(t), ζ(t) = ξ(t)

(5.6.75)

¨ + αξ(t) ˙ + Aξ(t) = f − g(u(t)), ξ(t) ξ(0) = 0,

t > 0,

˙ ξ(0) = 0.

(5.6.76) (5.6.77)

By Theorem 5.6.9, for t ⩾ 0, {u(t), u(t)} ˙ is contained in bounded set B of V1 × H. d ′ ˙ is uniformly bounded with respect By (5.6.69), we have (f −g(u)) = −g (u(t))u(t) dt to t in V−1+σ . Hence there exists a C˜ ⩾ 0 such that 2

d (f − g(u)) dt

−1+σ2

˜ ⩽ C,

∀t ⩾ 0.

˙ satisfies By (5.6.75), η(t) = ξ(t)  d η¨(t) + αη(t) ˙ + Aη(t) = (f − g(u(t))), dt  η(0) = 0, η(0) ˙ = f − g(u0 ).

(5.6.78)

∀t > 0,

(5.6.79)

Applying Proposition 5.6.2 to A(σ2 −1)/2 η, we have {η(t), η(t)} ˙ is contained in a bounded set of Vσ2 × Vσ2 −1 . Going back to (5.6.76), since f − g(u(t)) is uniformly

252

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S

˙ {ξ(t), ξ(t)} is bounded in Vσ2 +1 × Vσ2 . The compactness S ˙ ⊂ Vs2 for s1 > s2 implies that {ξ(t), ξ(t)} is a compact set

bounded in H, we get

t⩾0

of the embedding Vs1

t⩾0

in V1 × H. Let B be a bounded set in E0 . Then its ω-limiting set is defined as ω(B) =

\ [

E0

S(t)B

(5.6.80)

,

s⩾0 t⩾s

it is easy to verify that the ω-limiting set has the property: for any ϕ0 ∈ E0 , ϕ0 ∈ E0 belong to ω(B) ⇔ ∃(tn , ϕn ), ϕn ∈ B, lim tn = +∞, lim dE0 (s(tn )ϕn , ϕ) = 0. n→∞

n→∞

(5.6.81) Hence ω(B) is a functional invariant set, i.e., S(t)ω(B) = ω(B),

∀ t ∈ R.

(5.6.82)

Proposition 5.6.13 Suppose that (i) there exists a bounded absorbing set B0 ⊂ E0 of S(t), (ii) for any bounded set B ⊂ E0 , there exists a compact set K ⊂ E0 such that lim sup dE0 (S(t)ϕ, K) = 0,

t→+∞ φ∈B

then U = ω(B0 ) =

T S

S(t)B0

E0

(5.6.83)

is the maximal attractor of S(t) in E0 .

s⩾0 t⩾s

Proof By (i), we know ω(B0 ) ⊂ B0 , thus ω(B0 ) is bounded in E0 . Taking B = ω(B0 ) in (5.6.83), then (5.6.82) implies ω(B0 ) ⊂ K. Hence ω(B0 ) is bounded and compact in E0 . The remainder part need to prove that for any bounded set B in E0 , lim sup dE0 (S(t)ϕ, ω(B0 )) = 0. (5.6.84) t→+∞ φ∈B

We prove it by contradiction. If there exists certain bounded set B in E0 , which (5.6.84) does not hold for this B, then there exists a sequence {(tn , ϕn )} satisfying ϕn ∈ B, tn → +∞ as n → ∞, and dE0 (s(tn )ϕn , ω(B0 )) ⩾ δ > 0.

(5.6.85)

By (5.6.83), dE0 (s(tn )ϕn , K) → 0 as n → ∞, thus there exists a sequence {kn } ⊂ K such that s(tn )ϕn − kn → 0. By the compactness of K, there exists a subsequence {kn′ } such that kn′ → k ∈ K as n′ → +∞. By (i), for n′ sufficiently large, S(tn′ /2)ϕn′ is contained in B0 . Since S(tn′ /2)S(tn′ /2)ϕn′ → k as n′ → +∞, we know k ∈ ω(B0 ) by (5.6.81), which contradicts to (5.6.85). This completes the proof. We can obtain the following theorem by Proposition 5.6.13.

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Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 253

Theorem 5.6.14 There exists a maximal attractor U of S(t) in E0 , i.e., for any bounded set B ⊂ E0 , lim

sup dE0 (S(t)ϕ0 , U ) = 0.

t→+∞ φ0 ∈B

Furthermore, U is bounded in E1 . Proof By Theorem 5.6.9, there exists a bounded absorbing set B0 in E0 , thus hypothesis (i) in Proposition 5.6.13 holds. Now we show (ii) is true, too. In fact, let B be a bounded set in E0 . By Lemma 5.6.12, there exists a compact set K ⊂ E0 such that [ U (t)B0 ⊂ K. (5.6.86) t⩾0

We prove (5.6.87)

lim dE0 (S(t)B, K) = 0

t→+∞

firstly. In fact, let ρ be the radius of sphere containing B ⊂ E0 , ψ0 ∈ B. Then by (5.6.73), we have S(t)ψ0 = Σε0 (t)ψ0 + U (t)ψ0 . (5.6.88) By (5.6.13), we get dE0 (S(t)ψ0 , K) ⩽ |Σε0 (t)ψ0 | ⩽ ρe−ε0 t/2 , thus (5.6.87) follows. Hence, by Proposition 5.6.13, we obtain that U = ω(B0 ) is a maximal attractor of (5.6.71) and (5.6.72) in E0 . As a corollary of (5.6.70), U is bounded in E1 . In fact, let R0 be the radius of sphere containing B0 ⊂ E0 and ϕ0 ∈ U . Since U is invariant with respect to S(t), there exists a trajectory ϕ(t) ∈ U , ϕ(0) = ϕ0 , then (5.6.70) implies ϕ(t) ∈ E1 , ∀t ∈ R, and |ϕ(0)|E1 ⩽ C5 (R) with R = |f | + R0 . This completes the proof. 5.6.4 Dimension of the maximal attractor Suppose S is a continuous mapping: X → Y , such that X = SX

or

X ⊂ SX,

(5.6.89)

and S is uniformly differentiable on X, that is, for all ϕ ∈ X, there exists a linear operator L = L(ϕ) ∈ L (Y ) on Y , such that sup φ,ψ∈X 0 0 such that (5.6.97)

w ¯D < 1, then the Hausdorff dimension of X is finite, and less than or equal to D. (ii) There exists a D > 0 such that D = N + s, N, 0 < s ⩽ 1, (w ¯N +1 )(D−l)/(N +1) w ¯N < 1,

l = 1, 2, · · · , N,

(5.6.98)

then the fractal dimension of X is finite, and less than or equal to D. By taking advantage of the estimate of w ¯D , one could obtain another form of Theorem 5.6.15. Set w ¯m (t) = sup wm (L(t; ϕ0 )), φ0 ∈U

(5.6.99)

5.6

Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 255

According to the chain rule of differentiation, for ϕ0 ∈ U , wm (L(t1 + t2 ; ϕ0 )) = wm (L(t1 ; S(t2 )ϕ0 ) ◦ L(t2 ; ϕ0 )) ⩽ wm (L(t1 ; S(t2 )ϕ0 )wm (L(t2 ; ϕ0 )) ⩽w ¯m (t1 )w ¯m (t2 ). For the above formula, we apply the following inequality wm (L1 · L2 ) ⩽ wm (L1 )wm (L2 ),

∀L1 , L2 ∈ L (Y ),

(5.6.100)

where wm (L) = ∥Λm L∥L (Λm Y ) . Hence, w ¯m (t1 + t2 ) ⩽ w ¯m (t1 )w ¯m (t2 ).

(5.6.101)

This implies that the function t → w ¯m (t) is sub-exponential. Hence, the following limit hold: 1

1

IIm ≡ lim w ¯m (t) t = inf w ¯m (t) t < +∞. t→+∞

t>0

(5.6.102)

Next we estimate IIm . A direct computation infers the following expression of w ¯m (L(t; ϕ0 )) w ¯m (L(t; ϕ0 )) =

1

sup Gram(η 1 (t), · · · , η m (t)) 2 ,

(5.6.103)

|η0i |E0 =1

where Gram(x1 , · · · , xm ) = [(x1 , · · · , xm ), (x1 , · · · , xm )] =

det (xi , xj ),

1⩽i,j⩽m

η i (t) (for 1 ⩽ i ⩽ m) is the solution of the following variational form of (5.6.71) η˙ + Λϵ0 η + Γ′ (ϕ)η = 0

(5.6.104)

η(0) = η0 ,

(5.6.105)

with the initial data

ϕ is a solution of (5.6.71) and (5.6.72). For ϕ0 ∈ E, let Λϵ0 (ϕ) be a linear operator, such that Λε0 (ϕ)ξ = Λε0 ξ + Γ′ (ϕ)ξ,

∀ξ ∈ E0 ,

(5.6.106)

256

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Infinite Dimensional Dynamical Systems

where ϕ is a solution of (5.6.71) and (5.6.72). By the classical derivative rule for determinate, one has d d Gram(η 1 (t), · · · , η m (t)) = [(η 1 , · · · , η m ), (η 1 , · · · , η m )] dt dt m h i X dη k , · · · , ηm ) =2 (η 1 , · · · , η m ), (η 1 , · · · , dt k=1 m h X

i (η 1 , · · · , η m ), (η 1 , · · · , Λε0 (ϕ)η k , · · · , η m ) .

= −2

k=1

Since m h i X (η 1 , · · · , η m ), (η 1 , · · · , Λε0 (ϕ)η k , · · · , η m ) = tr(Λε0 (ϕ)·Qm )·Gram(η 1 , · · · , η m ), k=1

where Qm (t) denotes the orthogonal projection of E0 on the subspace spanned by (η 1 (t), · · · , η m (t)), then d Gram(η 1 (t), · · · , η m (t)) + 2tr(Λε0 (ϕ) · Qm (t)) · Gram(η 1 (t), · · · , η m (t)) = 0. dt (5.6.107) Let Z n o 1 t qm = − lim sup − inf inf tr(Λε0 (ϕ(s))) · Q ds , (5.6.108) φ0 ∈U t 0 rankQ=m t→∞ where Q denotes any orthogonal projection on E0 , and is independent of s. By equation (5.6.107), it is deduced that lim sup Gram(η 1 (t), · · · , η m (t)) t ⩽ e−2qm , 1

(5.6.109)

t→+∞

where we applied |η0i |E0 ⩽ 1,

Gram(η 1 (0), · · · , η m (0)) ⩽ 1.

From equations (5.6.102), (5.6.103) and (5.6.109), it follows that IIm ⩽ e−qm .

(5.6.110)

Let the Lyapunov exponent µi satisfy µ1 = log π1 ,

µi = log πj − log πj−1 ,

j ⩾ 2.

Then we have Theorem 5.6.15 holds. In order to apply Theorem 5.6.15 to estimate the dimension of the maximal attractor of equations (5.6.71) and (5.6.72), we must check the assumptions in Theorem 5.6.15.

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Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 257

Proposition 5.6.16 For any t0 > 0, the mapping S(t0 ) is C 1+δ (where δ is given in (5.6.54)) on a bounded set of E0 . Its derivative at ϕ0 is a linear operator on E0 : ξ → L(t0 , ϕ0 )ξ = η(t0 ), where η(t0 ) is the value of η(t) at t = t0 , and η(t) is the solution of the following linear equation η˙ + Λε0 η + Γ′ (ϕ)η = 0,

(5.6.111)

η(0) = ξ,

(5.6.112)

ϕ is a solution of (5.6.71) and (5.6.72). Proof Given ϕ0 and ϕ˜0 ∈ E0 satisfying |ϕ0 |E0 ⩽ R and |ϕ˜0 |E0 ⩽ R, ϕ(t) and ϕ(t) ˜ are the solutions of (5.6.71) and (5.6.72). According to Theorem 5.6.9, there exists a constant K1 (R), such that |ϕ(t)|E0 ⩽ K1 (R),

|ϕ(t)| ˜ E0 ⩽ K1 (R),

∀t ∈ R+ .

(5.6.113)

Set ψ(t) = ϕ(t) − ϕ(t). ˜ Then, there holds ψ˙ + Λε0 ψ + Γ(ϕ) − Γ(ϕ) ˜ = 0,

(5.6.114)

ψ(0) = ϕ0 − ϕ˜0 .

(5.6.115)

For equation (5.6.114), taking the inner product with ψ infers 1 d |ψ|2 + (Λε ψ, ψ) = (Γ(ϕ) ˜ − Γ(ϕ), ψ). 2 dt E0

(5.6.116)

Since g(ξ) satisfies Lipschitz condition, from (5.6.113), it is deduced that |Γ(ϕ) − Γ(ϕ)| ˜ E0 ⩽ k|ϕ − ϕ| ˜ E0 .

(5.6.117)

From equation (5.6.116) and (Λε0 ψ, ψ) ⩾ 0, it is direct to obtain 1 d |ψ|2 ⩽ k|ϕ − ϕ| ˜ 2E0 = k|ψ|2E0 . 2 dt E0 Hence, |ϕ(t) − ϕ(t)| ˜ ⩽ ekt0 |ϕ0 − ϕ˜0 |,

0 ⩽ t ⩽ t0 .

(5.6.118)

Assume η ∈ (R+ ; E0 ) is the solution of (5.6.111) and (5.6.112). Set θ(t) = ϕ(t) − ϕ(t) ˜ − η(t),

(5.6.119)

where ϕ˜0 = ϕ0 + ξ, ξ ∈ E0 , then θ(t) satisfies θ(0) = 0 and θ˙ + Λε0 θ + Γ′ (ϕ)θ = ζ,

(5.6.120)

258

Chapter 5

Infinite Dimensional Dynamical Systems

where ζ = Γ(ϕ) − Γ(ϕ) ˜ − Γ′ (ϕ)(ϕ − ϕ). ˜

(5.6.121)

Rewrite ζ as follows Z ζ=

1

n o Γ′ (τ ϕ˜ + (1 − τ )ϕ) − Γ′ (ϕ) (ϕ − ϕ) ˜ dτ.

0

Combining (5.6.53) and (5.6.113), there exists a constant C2 = C2 (R), such that |Γ′ (τ ϕ˜ + (1 − τ )ϕ) − Γ′ (ϕ)|L (E0 ) ⩽ C2 τ δ |ϕ − ϕ| ˜ δ. Hence, |ζ|E0 ⩽ C2′ |ϕ − ϕ| ˜ 1+δ .

(5.6.122)

Taking the inner product with θ in E0 , equation (5.6.120) infers that 1 d 2 |θ| + (Λε0 θ, θ) + (Γ′ (ϕ)θ, Q) = (ζ, θ). 2 dt Owing to the existence of a C > 0, such that |Γ′ (ϕ(t))|L (E0 ) ⩽

C , 2

and equation (5.6.118)-(5.6.122), we have d 2 |θ| ⩽ C|θ|2E0 + C2′ e2(1+δ)kt0 |ξ|2+2δ E0 , dt E0

0 ⩽ t ⩽ t0 .

(5.6.123)

By Grönwall’s inequality and θ(0) = 0, it is obtained that |θ(t0 )|2E0 ⩽

eCt0 − 1 ′ 2(1+δ)kt0 2+2δ C2 e |ξ|E0 , C

(5.6.124)

which proves Proposition 5.6.16. Now, attention is turned to check (5.6.90). In fact, given ϕ and ψ ∈ U , denote S = S(t0 ), L(ϕ) = L(t0 , ϕ), then Z 1n o Sψ − Sϕ − L(ϕ)(ψ − ϕ) = L(θψ + (1 − θ)ϕ) − L(ϕ) (ψ − ϕ) dθ. 0

Since U is bounded on E0 , according to Proposition 5.6.16, there exists a C, such that |L(θψ + (1 − θ)ϕ) − L(ϕ)|E0 ⩽ |θψ + (1 − θ)ϕ − ϕ|δE0 ⩽ C|ψ − ϕ|δE0 , which infers

|Sψ − Sϕ − L(ϕ)(ψ − ϕ)|E0 ⩽ |ψ − ϕ|δE0 , |ψ − ϕ|E0

which implies that (5.6.90) holds. We have the following theorem.

5.6

Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 259

Theorem 5.6.17 The maximal attractor of (5.6.71) and (5.6.71) has finite fractal dimension and Hausdorff dimension. Proof Apply Theorem 5.6.9, and define qm as (5.6.108). Let Q be an arbitrary orthogonal projection with rank m in E0 , (ξ i )i∈N be an orthogonal basis in E0 such that span{ξ 1 , ξ 2 , · · · , ξ m } = QE0 . For a given ϕ0 in U , ϕ(t) is a solution of (5.6.71), (5.6.72). From (5.6.106), we have tr(Λε0 (ϕ) · Q) =

m X

{(Λε0 ξ i , ξ i )E0 + (Γ′ (ϕ)ξ i , ξ i )E0 }.

(5.6.125)

i=1

Denote ξ i = {ϕi , ψ i } ∈ V1 × H = E0 . Then, (Λε0 ξ i , ξ i )E0 + (Γ′ (ϕ)ξ i , ξ i )E0 = ε|ϕi |2E0 + ε0 (ε0 − α)(ϕi , ψ i ) + (α − ε0 )|ψ i |2 + (g ′ (u)ϕi , ψ i ).

(5.6.126)

Since U is bounded on E0 , ϕ(t) = {u(t), u(t) ˙ + εu(t)} ∈ U , ∀t ∈ R, by (5.6.60), there exists s ∈ [0, 1], such that sup sup |g ′ (u(t))|L (Vs ,H) ≡ β < +∞.

φ0 ∈U t⩾0

(5.6.127)

From (5.6.8), (5.6.125), (5.6.126) and (5.6.127), we deduce tr(Λε0 (ϕ) · Q) ⩾ ⩾

m n X ε0 i=1 m n X i=1

2

o (|ϕi |21 + |ψ i |2 ) − β|ϕi |s |ψ i |

m β2 X i 2o ε0 i 2 (|ϕ |1 + |ψ i |2 ) − |ϕ |s . 4 ε0 i=1

(5.6.128)

Since {ξ i } are orthogonal on E0 , the first term on the right-hand side of (5.6.128) ε0 β2 equals to m, the second term becomes − tr(M · Q), where M ∈ L (E0 ), 4 ε0 M {ϕ, ψ} = {As−1 ϕ, 0}. Hence, from (5.6.128), it is deduced that tr(Λε0 (ϕ) · Q) ⩾

ε0 β2 m− tr(M · Q). 4 ε0

(5.6.129)

The trace of M · Q can be estimated by eigenvalue, the eigenvalues of M are {λs−1 }j⩾1 ∪ {0}, where 0 ⩽ λ1 ⩽ · · · ⩽ λj ⩽ · · · are the eigenvalues of A, j tr(M · Q) ⩽

m X i=1

λs−1 . i

(5.6.130)

260

Chapter 5

Infinite Dimensional Dynamical Systems

Therefore, (5.6.125) implies tr(Λε0 (ϕ) · Q) ⩾ m

ε

β 2 1 X s−1  · λ . ε0 m i=1 i m

0

4

−

(5.6.131)

Due to the right-hand side of (5.6.131) is independent of ϕ0 ∈ U and t, one has qm

ε

β 2 1 X s−1  − ⩾m · λ . 4 ε0 m i=1 i m

0

From the compactness of A−1 , we deduce lim

i→∞

1 X s−1 λ →0 m i=1 i

(5.6.132)

1 = 0. Hence, by the Cesàro mean λi

m

(m → ∞),

there exists m∗ ⩾ 1, such that ∗

m 1 X s−1 ε0 λ ⩽ . m∗ i=1 i 4

(5.6.133)

From (5.6.132), we discover qm∗ > 0. Consequently, the fractal dimension and the Hausdorff dimension of U are finite. This proves the Theorem 5.6.17. 5.6.5

Application

Now, we list some examples as applications of the theory developed in § 5.6.3 and § 5.6.4. 1. Nonlinear wave equation ∂2 ∂u − ∆u + g(u) = f (x), u(x, t) + α ∂t2 ∂t

Ω × R+ ,

(5.6.134)

with initial-boundary value conditions u(x, t) = 0, u(x, 0) = u0 (x),

∂Ω × R+ , ∂u (x, 0) = u1 (x), ∂t

(5.6.135) (5.6.136)

where α > 0, Ω ⊂ Rn is a bounded region, g(u) ∈ C 1 : R → R and satisfies the assumptions (5.6.31)-(5.6.33), f , u0 and u1 are given in L2 (Ω),H01 (Ω) and L2 (Ω), respectively. When g(u) = sin u, (5.6.134) is the sine-Gordon equation; when g(u) = |u|γ u, γ ⩾ 0(n = 1, 2), 0 ⩽ γ < 2(n = 3), (5.6.134) becomes the nonlinear wave equation in quantum mechanics.

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Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 261

Observing Theorem 5.6.7, gives that the problem (5.6.134)-(5.6.136) has a unique solution {u, u} ˙ ∈ Cb (R+ ; H01 (Ω) × L2 (Ω)). From the results in § 5.6.3 and § 5.6.4, we know that the problem (5.6.134)-(5.6.136) has a maximal attractor U in H01 (Ω) × L2 (Ω). This means that U is a compact set, it attracts all the trajectories of the manifold of the bounded set in H01 (Ω) × L2 (Ω). Moreover, if f, g, Ω ∈ C ∞ , then ¯ × C ∞ (Ω). ¯ U ∈ C ∞ (Ω) Theorem 5.6.18 There exists a bounded set U in {H 2 (Ω) ∩ H01 (Ω)} × H01 (Ω), which depends on g, f, α and Ω, such that (i) U has finite fractal dimension. (ii) for any initial data {u0 , u1 } given in the bounded set of H01 (Ω) × L2 (Ω), the corresponding solution {u, u} ˙ of the initial value problem (5.6.134)-(5.6.136) converges to U in H01 (Ω) × L2 (Ω) when t → ∞. Proof The results mentioned above come from Theorem 5.6.14 and Theorem 5.6.15. We will check all the assumptions in these theorems, i.e. (5.6.53)-(5.6.60), (5.6.69) and (5.6.70). The assumption (5.6.53)-(5.6.59) has been checked in Theorem 5.6.7 and Theorem 5.6.8. For (5.6.60), let ξ ∈ D(A) = H 2 (Ω) ∩ H01 (Ω), the R differentiation of the mapping u → Ω g(u(x)) dx is the operator Z η→ g ′ (ξ)η dx. Ω

The norm of the linear operator in H = L2 (Ω) is bounded by |g ′ (ξ)|L∞ (Ω) . By (5.6.33), (5.6.60) holds for any n ⩾ 4; when n ⩽ 3, (5.6.60) can be derived from H 2 (Ω) ,→ L∞ (Ω). Now, we are going to prove (5.6.69) and (5.6.70). Since the other cases can be proved obviously, we will only consider the case when n = 3. From the classical Sobolev embedding theorem (n = 3)

6

H s (Ω) ,→ L (3−2s) (Ω),

0⩽s


Ω

262

Chapter 5

Infinite Dimensional Dynamical Systems

which demonstrates (5.6.69). As to the proof of (5.6.70), we consider the following linear equation ξ¨ + αξ˙ + Aξ = h,

(5.6.139)

˙ ∈ where h ∈ Cb (R; V−1+δ ), 0 ⩽ δ ⩽ 1. If ξ is a solution of (5.6.139) and {ξ, ξ} Cb (R; H × V−1 ), then ˙ ∈ Cb (R; Vδ × V−1+δ ). {ξ, ξ} (5.6.140) ˙ depends the norm of h in Cb (R; V−1+δ ) only. Now, attention is The norm of {ξ, ξ} given to (5.6.70). Given f in Cb (R; H), f˙ ∈ Cb (R; H) and {u, u} ˙ ∈ Cb (R; V1 × H)

(5.6.141)

is the solution of (5.6.61), then we have u ¨ + αu˙ + Au = f − g(u). Let ξ = u, ˙ then ξ¨ + αξ˙ + Aξ = f˙ − g ′ (u)u. ˙

(5.6.142)

From (5.6.69) and (5.6.144), it follows that function f˙ − g ′ (u)u˙ ∈ Cb (R; V−1+σ2 ),

σ2 =

2−r . 2

(5.6.140) leads to {¨ u, u} ˙ ∈ Cb (R; Vσ2 × V−1+σ2 ). Due to Au = f − g(u) − u ¨ − αu˙ ∈ Cb (R; V−1+σ2 ), one obtains {u, u} ˙ ∈ Cb (R; V1+σ2 × Vσ2 ).

(5.6.143)

and the norm of {u, u} ˙ in Cb (R; V1+σ2 × Vσ2 ) is controlled by the bounded function 3 |f |L∞ (R;H) + |f˙|L∞ (R;H) + |{u, u}| ˙ L∞ (R;E0 ) . If 1 + σ2 > (i.e. 0 ⩽ r < 1), combining 2 (5.6.138) and V1+σ2 ⊂ H 1+σ2 (Ω) infers u ∈ Cb (R; L∞ (Ω)).

(5.6.144)

Then g ′ (u) ∈ Cb (R; L∞ (Ω)), h = f˙ − g ′ (u)u˙ ∈ Cb (R; H). Hence, from (5.6.139) and (5.6.140), we obtain that {¨ u, u} ˙ ∈ Cb (R; V1 × H), which implies that u ∈ Cb (R; V2 ). 3 This proves (5.6.70). If 1 + σ2 ⩽ (i.e. 1 ⩽ r < 2), the bootstrap argument will 2 1 be applied. Assume {u, u} ˙ ∈ Cb (R; V1+δ × Vδ ), 0 ⩽ δ < . Using (5.6.137) and 2

5.6

Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 263 6

V1+δ ⊂ H 1+δ (Ω), it is deduced that u ∈ Cb (R; L 1−2δ (Ω)). Hence, (5.6.33) gives that 6 6 g ′ (u) ∈ Cb (R; L (1−2δ)r (Ω)). On the other hand, u˙ ∈ Cb (R; L 3−δ (Ω)), then g ′ (u)u˙ ∈ Cb (R; Vs ),

s=

δ − (1 − 2δ)r . 2

If s ⩾ 0, h = f˙ − g ′ (u)u˙ ∈ Cb (R; H), the conclusion holds obviously. If s < 0, h ∈ Cb (R; Vs ), by (5.6.140), it follows that {u, u} ˙ ∈ Cb (R; V1+s × Vs ), u ∈ Cb (R; V2+s ). If 3 2 + s ⩾ , (5.6.144) can be derived from (5.6.138), which infers that the conclusion 2 is true. If not, repeating the process. Now, we are going to prove that the process is terminated in finite steps. In fact, assume {u, u} ˙ ∈ Cb (R; V1+δn × Vδn ) after taking n-th step , next step is {u, u} ˙ ∈ Cb (R; V1+δn+1 × Vδn+1 ), where 1 + δn+1 = 2 + Sn+1 =

δn − (1 − 2δn )r , 2

2−r 1 + 2r δn + . 2 2

1 . 2 Therefore, (5.6.138) and (5.6.144) are satisfied. This proves (5.6.70), which implies Theorem 5.6.18 holds. 2. Initial-boundary value problems of some sine-Gordon equations (systems). (i) Initial-boundary value problem of sine-Gordon equation with the second type boundary condition.

Since 1 ⩽ r < 2, δn → +∞, then there exists an integer n0 , such that δn0 >

 utt + αut − ∆u + u + sin u = f (x),     ∂u (x, t) = 0, ∂Ω × R,  ∂ν    u(x, 0) = u0 (x), ut (x, 0) = u1 (x),

(5.6.145a) (5.6.145b) (5.6.145c)

where Ω is a bounded region, ∂Ω is smooth, ν is an outer normal vector to ∂Ω, f ∈ L2 (Ω). Its functional framework is H = L2 (Ω),

o n ∂ξ D(A) = ξ ∈ H 2 (Ω), = 0, x ∈ ∂Ω , ∂ν

Aξ = −∆ξ + ξ, g(ξ) = sin ξ, Z G(ξ) = (cos ξ − 1) dx, p ≡ 0. Ω

Then, this case can be discussed by the theorem mentioned above.

264

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(ii) Sine-Gordon system.  u + u1t − ∆u1 + sin u1 + h(u1 − u2 ) = f1 ,    1tt     u2tt + u2t − ∆u2 + sin u2 + h(u2 − u1 ) = f2 , ui (x, t) = 0, x ∈ ∂Ω, t ⩾ 0, i = 1, 2      ∂u   ui (x, 0) = u0i (x), i (x, 0) = u1i (x), ∂t Let

(5.6.146a) (5.6.146b) (5.6.146c)

x ∈ Ω.

(5.6.146d)

n o H = L2 (Ω)2 , D(A) = (ξ1 , ξ2 ) ∈ H 2 (Ω)2 , ξi = 0, ∂Ω , A(ξ1 , ξ2 ) = (−∆ξ1 , −∆ξ2 ), g(ξ1 , ξ2 ) = (sin ξ1 + h(ξ1 − ξ2 ), sin ξ2 + h(ξ1 − ξ2 )), Z n o h G(ξ1 , ξ2 ) = cos ξ1 + cos ξ2 − 2 + (ξ1 − ξ2 )2 dx, p ≡ 0. 2 Ω (iii) Non-gradient sine-Gordon system. (

u1tt + u1t − ∆u1 + sin(u1 + u2 ) = f1 ,

(5.6.147a)

u2tt + u2t − ∆u2 + sin(u1 − u2 ) = f2 .

(5.6.147b)

Let



λ1 λ1  ξ1 , −∆ξ2 − ξ2 , 2 2 where λ1 is the first eigenvalue of −∆ with Dirichlet boundary condition. λ λ1  1 ξ1 , ξ2 , g(ξ1 , ξ2 ) = 2 2 Z λ1 (ξ 2 + ξ22 ) dx, G(ξ1 , ξ2 ) = 4 Ω 1 A(ξ1 , ξ2 ) =

− ∆ξ1 −

p(ξ1 , ξ2 ) = (sin(ξ1 + ξ2 ), sin(ξ1 − ξ2 )). (iv)

 u + αut + ∆2 u + |u|r u = f,    tt  ∂u u(x, t) = (x, t) = 0, ∂Ω × R+ ,  ∂r    u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ Ω,

where σ > 0, Ω ⊂ R , n ⩽ 3. Let n

H = L2 (Ω), o n ∂ξ D(A) = H 4 (Ω) ∩ H02 (Ω) = ξ ∈ H 4 (Ω), ξ = = 0, ∂Ω ∂ν

(5.6.148a) (5.6.148b) (5.6.148c)

Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 265

5.6

g(ξ) = |ξ|r ξ, (v)

G(ξ) =

Z

1 r+2

|ξ|r+2 dx,

p≡0

Ω

  u + αut + uxxxx + u + g(u) = f,   tt u(x + L, t) = u(x, t), x ∈ R, t ⩾ 0    u(x, 0) = u (x), u (x, 0) = u (x), 0

t

(5.6.149a) (5.6.149b) (5.6.149c)

1

where the nonlinear operator defined as follows: assume ξk is the Fourier coefficient of ξ, i.e.,  X x . (5.6.150) ξ(x) = ξk exp 2iπk L k∈Z

Set g(ξ)k =

X

|l|σ |ξl |δ |k|2σ ξk.

(5.6.151)

l∈Z

Z

1 2

If denote (−∆) by D, then the operator g is in proportion to ξ →

L

(|Dσ ξ|2 )δ D2σ ξ. 0

Introduce 1 G(ξ) = δ+2

X

!δ |l| |ξl | σ

2

(5.6.152)

,

l∈Z

where ξl is given in (5.6.149c). When 0 ⩽ σ < 1, the problem (5.6.148c)-(5.6.149b) can be dealt with the abstract framework, where H = L2 (0, L),

4 D(A) = Hper (0, L),

∂4u + u, p ≡ 0. ∂x4 When 0 ⩽ σ < 1 is replaced by σ = 1, the model is significant. Au =

5.6.6

Non-autonomous system

Suppose f is given in Cb (R; H), f˙ ∈ Cb (R; H). Assume there exists a T > 0, such that f (t + T ) = f (t), ∀t ∈ R. (5.6.153) Introduce a class of operators {S(t, s)} as follows: (t, s ∈ R) Given ϕ0 ∈ E, assume ϕ(t) = S(t, s)ϕ0 ,

t ∈ R, s ∈ R,

(5.6.154)

where ϕ(t) is a solution of the following Cauchy problem ϕ(τ ˙ ) + Λε0 ϕ(τ ) + Γ(ϕ(τ )) = F (τ ),

τ ∈ R,

(5.6.155)

266

Chapter 5

Infinite Dimensional Dynamical Systems

(5.6.156)

ϕ(s) = ϕ0 ,

where F (τ ) = {0, f (t)}. By Proposition 5.6.2, we know that the problem (5.6.155), (5.6.126) is solvable, S(t, s) ∈ C(E0 , E0 ). In addition, by the uniqueness of the solution of (5.6.155), (5.6.126), there holds ∀t ∈ R,

S(t, t) = idE0 ,

S(t1 , t)S(t, t2 ) = S(t1 , t2 ),

(5.6.157)

t1 , t2 , ∀t ∈ R.

(5.6.158)

t, s ∈ R.

(5.6.159)

From the periodic condition, one has S(t + T, s + T ) = S(t, s), Hence, {S(nT, 0)}n∈Z is a discrete group. Set S(nT ) ≡ S(nT, 0),

∀ n ∈ Z.

(5.6.160)

X is an invariant subset of E0 with the mapping {S(nT )}n∈Z , which satisfies S(nT )X = X,

∀ n ∈ Z,

(5.6.161)

The maximal attractor U is a compact, invariant set in E0 , and it attracts every bounded subset in E0 , i.e. for fixed t, s ∈ R, the image of any bounded set with mapping S(t + mT, S) in E0 converges to S(t, 0)U when m → +∞. For a given bounded set B ⊂ E0 , the limit set of w is defined as wp (B) =

\ [

E0

S(nT )B

.

(5.6.162)

m⩾0 n⩾m

It has following properties: ϕ ∈ E0 belongs to wp (B) if and only if there exists a sequence (Nn , ϕn ) such that Nn ∈ Z, ϕn ∈ B, and when n → ∞, Nn → +∞, and dE0 (S(Nn , T )ϕn , ϕ) → 0. We have S(nT )wp (B) = wp (B),

∀n ∈ Z.

(5.6.163)

Theorem 5.6.19 Suppose (i) S(t, 0) has a bounded absorbing set B0 in E0 , (ii) for the bounded set in E0 , there exists a compact set K ⊂ E0 , such that lim sup dE0 (S(t, 0)ϕ, K) = 0.

(5.6.164)

U = wp (B0 )

(5.6.165)

t→+∞ φ∈B

Then, (a) the set

5.6

Global attractor and the bounds of Hausdorff, fractal dimensions for damped · · · 267

is in E1 , and it is the maximal attractor of (5.6.155), (5.6.156) in E0 . For any bounded set in E0 , t, s ∈ R, we have lim

sup dE0 (S(t + mT ; s)ϕ0 , S(t; 0)U ) = 0.

m→+∞ φ∈B

(5.6.166)

(b) The maximal attractor U has finite fractal dimension. Proof From (i), (ii), together with Proposition 5.6.13 and its proof give rises to (a). For (b), according to the analysis applied in (5.6.161) and § 5.6.4, one can obtain the conclusion by replacing the continuous group {S(t)}t∈R by the discrete group {S(nT )}n∈Z and applying t = nT, n → ∞ instead of t → ∞. Now, take the sine-Gordon equation with impressed force for example. utt + αut − uxx + sin u = Γ sin ωt, u(0, t) = u(L, t) = 0,

x ∈ [0, L], t > 0,

t⩾0

(5.6.167) (5.6.168)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x),

x ∈ [0, L],

(5.6.169)

2π . Apply Theorem ω 5.6.19 in the problem (5.6.167)-(5.6.169), one has the following result. where α > 0, L > 0, Γ = ̸ 0, ∞ ̸= 0. Obviously, the period T =

Theorem 5.6.20 There exists a bounded set U in H 2 (0, L) × H 1 (0, L), which depends on α, L, Γ and ω only, such that (i) for any bounded set with initial value {u0 , u1 } ∈ H01 (0, L) × L2 (0, L) and the corresponding solution of problem (5.6.167)-(5.6.169) is {u, ut }, it converges to U in (H01 (0, L) × L2 (0, L)) when t → +∞. (ii) The Hausdorff dimension of U is less than 1+

3L2 , 3ε20

where ε0 = min

α

π2  . 4 2αL2 ,

(5.6.170)

(iii) Its fractal dimension is less than  4L2  2 1+ 2 . 3ε0

(5.6.171)

Proof When g(u) = sin u, Ω = [0, L], the problem (5.6.167)-(5.6.169) is reduced to the problem (5.6.134)-(5.6.136). To prove (i), we will apply Theorem 5.6.19. The assumptions (i), (ii) in Theorem 5.6.19 are hold by § 5.6.5 for autonomous case. As to f (x, t), it satisfies that f (x, t) ∈ Cb (R; H), ft (x, t) ∈ Cb (R+ ; H), H = L2 (0, L) due to f (x, t) = Γ sin ωt. Hence, the maximal attractor exists. Next, we will consider the dimension estimate. Assume λj = j 2

π2 , L2

j = 1, 2, · · ·

(5.6.172)

268

Chapter 5

Infinite Dimensional Dynamical Systems

are the eigenvalues of the operator Au = −uxx with Dirichlet boundary condition. π2 When λ1 = 2 , ε0 is defined as in Proposition 5.6.2 L  α π2  . (5.6.173) , ε0 = min 4 2αL2 Now, consider (5.6.60) when s = 0. By g ′ (ξ)ϕ = ϕ cos ϕ, it is deduced that Z

′

|g (ξ)|L (L2 (0,L)) = ∫ sup L 0

1

ϕ2 (x) cos2 ξx dx ⩽ 1.

φ2 dx=1

0

Therefore, in (5.6.127), β = 1.

(5.6.174)

From (5.6.132), it is obtained that  qm

 m 2 X ε 1 1 L 0 . ⩾ m − · 4 ε0 m j=1 j 2 π 2

Since

(5.6.175)

m ∞ X X 1 1 π2 , ⩽ = 2 2 j j 6 j=1 j=1

then

ε L2  0 − . qm ⩾ m 4 6mε0

(5.6.176)

Hence, if m > 2L2 /3ε30 , combining with qm+1 > 0 gives rise to (5.6.170). As to the fractal dimension, let m0 satisfy m0 − 1
0, ∀v ∈ D(A), v ̸≡ 0. A−1 is compact, the mapping u → Au is an isomorphism from D(A) to H. As denotes the s-th power of A (s ∈ R). V2s = D(As ) is a Hilbert space with the inner product (u, v)2s = (As u, As v), ∀u, v ∈ D(As ). 1

For u ∈ Vs , let |u|s = (u, u)s2 . Since A−1 is compact and self-adjoint, there exists an orthogonal basis {wj } for H obtained by the eigenvector of A, (5.7.3)

Awj = λj wj . The eigenvalues satisfy 0 < λ1 ⩽ λ2 ⩽ · · · ,

λj → +∞, j → ∞.

(5.7.4)

By (5.7.3) and (5.7.4), it is direct to have 1

1

|A 2 u| ⩾ λ12 |u|, 1

1

|Ap+ 2 u| ⩾ λ12 |Ap u|,

1

u ∈ D(A 2 ). 1

∀ u ∈ D(Ap+ 2 ), ∀ p.

(5.7.5) (5.7.6)

Let pN be the orthogonal projection of H on the subspace spanned by {w1 , · · · , wN }, (N = 1, 2, · · · ), QN = I − pN . In (5.7.2), the nonlinear terms R(u), B(u, u) are

270

Chapter 5

Infinite Dimensional Dynamical Systems

bilinear operators D(A) × D(A) → H, C is a linear operator from D(A) to H, 1 f ∈ D(A 2 ). Furthermore, assume ∀u, v ∈ D(A);

(B(u, v), v) = 0, 1

1

1

1

1

1

|B(u, v)| ⩽ C1 |u| 2 |A 2 u| 2 |A 2 v| 2 |Av| 2 , 1 2

1 2

1 2

|Cu| ⩽ C2 |A u| |Au| ,

∀ u, v ∈ D(A);

∀ u ∈ D(A),

(5.7.7) (5.7.8) (5.7.9)

where C1 , C2 and the following Ci (i = 3, 4) are positive constants. For B, C, we add extra continuous properties as follows 1

|A 2 B(u, v)| ⩽ C3 |Au||Av|, 1

|A 2 Cu| ⩽ C4 |Au|,

∀ u, v ∈ D(A),

(5.7.10)

∀ u ∈ D(A).

(5.7.11)

∀ u ∈ D(A), α > 0.

(5.7.12)

Finally, assume A + C is positive, i.e. 1

((A + C)u, u) ⩾ α|A 2 u|2 ,

Consider the initial value problem of (5.7.1), which means (5.7.1) satisfies the initial condition u(0) = u0 ∈ H. (5.7.13) Assume the problem (5.7.11), (5.7.13) has the unique solution S(t)u0 , ∀t ∈ R+ . S(t)u0 ∈ D(A), ∀t ∈ R+ . The mapping S(t) possesses the general property of semigroup. Now, we are going to have the uniform a priori estimate of the solution of equation (5.7.1). Hence, the following inequalities and lemmas are necessary. For β > 0, 1 1 + = 1, 1 < p, q < +∞, one has p q X

|xi yi | =

X

|βxi ||β −1 yi | ⩽

 β −q X  β p X |xi |p + |yi |q . p q

(5.7.14)

Lemma 5.7.1 Assume g(t), h(t), y(t) are positive integrable functions, t0 ⩽ t < ∞, satisfying dy ⩽ gy + h, ∀t ⩾ t0 , (5.7.15) dt and Z t+1 Z t+1 Z t+1 g(s) ds ⩽ α1 , h(s) ds ⩽ α2 , y(s) ds ⩽ α3 , ∀ t ⩾ t0 , (5.7.16) t

t

t

where α1 , α2 , α3 are positive constants, then y(t + 1) ⩽ (α3 + α2 ) exp(α1 ),

∀ t ⩾ t0 .

(5.7.17)

5.7

Inertial manifold for one class of nonlinear evolution equations

271

Taking the inner product with u for (5.7.1), which along with (5.7.7) and (5.7.12) gives rise to 1 1 1 d 2 α 1 1 −1 |u| + α|A 2 u|2 ⩽ |(f, u)| ⩽ λ1 2 |f ||A 2 u| ⩽ |A 2 u|2 + |f |2 . 2 dt 2 2λ1 α

Hence, 1 1 d 2 d |u| + αλ1 |u|2 ⩽ |u|2 + α|A 2 u|2 ⩽ |f |2 . (5.7.18) dt dt αλ1 Taking the inner product with ∆u for (5.7.1), which along with (5.7.8), (5.7.9) and (5.7.14) gives rise to 1

1

3

1

1

3

|B(u, u) + (u, Au)| ⩽ C1 |u| 2 |A 2 u||Au| 2 + C2 |A 2 u| 2 |Au| 2 1 1 1 ⩽ 54(C14 |u|2 |A 2 u|4 + C24 |A 2 u|2 ) + |Au|2 . 4 1 Similarly, we can have |(f, Au)| ⩽ |f ||Au| ⩽ |f |2 + |Au|2 . This implies that 4 1 1 1 1 d 1 2 d 1 |A 2 u| + λ1 |A 2 u|2 ⩽ |A 2 u|2 + |Au|2 ⩽ C6 |u|2 |A 2 u|4 + C7 |A 2 u|2 + 2|f |2 . 2 dt dt (5.7.19) 2 Taking the inner product with A u for (5.7.1) implies that 3 1 d |Au|2 + |A 2 u|2 ⩽ |(B(u, u) + Cu, A2 u)| + |(f, Au2 )| 2 dt 1 3 1 3 ⩽ |(A 2 (B(u, u) + Cu), A 2 u)| + |(A 2 f, A 2 u)| 3

3

1

3

⩽ C3 |Au|2 |A 2 u| + C4 |Au||A 2 u| + |A 2 f ||A 2 u| 1 1 3 1 1 3 ⩽ C8 |Au|4 + C9 |Au|2 + |A 2 f |2 + |A 2 u|2 . 2 2 2 2 Hence, 3 1 d d |Au|2 + λ1 |Au|2 ⩽ |Au|2 + |A 2 u|2 ⩽ C8 |Au|4 + C9 |Au|2 + 3|A 2 f |2 . (5.7.20) dt dt Applying (5.7.15) to (5.7.18), it is deduced that u(t) = S(t)u0 ,

|u(t)|2 ⩽ |u(0)|2 exp(−αλ1 t) + ρ20 (1 − exp(−αλ1 t)), where ρ0 =

(5.7.21)

1 |f |. Therefore, |u(t)| is uniformly bounded with respect to t, and 2λ1 lim sup |u(t)|2 ⩽ ρ20 . t→∞

Z

1

|A 2 u|2 ds is uniformly bounded. By

Then, it is inferred from (5.7.18) that (5.7.19), we have

t+1

(5.7.22)

t

1 1 d 1 2 |A 2 u| ⩽ C10 |A 2 u|4 + (C7 − λ1 )|A 2 u|2 + 2|f |2 , dt

272

Chapter 5

Infinite Dimensional Dynamical Systems

where C10 = C6 b20 , |u(t)|2 ⩽ b20 , t ⩾ 0. Then, according to Lemma 5.7.1, let 1

g = C10 |A 2 u|2 ,

1

1

y = |A 2 u|2 ,

h = (C7 − λ1 )|A 2 u|2 + 2|f |2 .

1

We get that |A 2 u|2 is uniformly bounded in H. By (5.7.19), it is found that R t+1 |Au(s)|2 ds is uniformly bounded. Then, (5.7.20) infers that |Au(t)|2 and Rt t+1 3 | t |A 2 u(s)|2 ds are uniformly bounded with respect to t, which implies that lim sup |A 2 u(t)|2 ⩽ ρ21 ,

1

(5.7.23)

lim sup |Au(t)|2 ⩽ ρ22 .

(5.7.24)

t→+∞

t→+∞

By (5.7.22), (5.7.23) and (5.7.24), we know that after a certain moment (t ⩾ t0 > 0) any solution of (5.7.1) is in the following balls, respectively, B0 = {x ∈ H, |x| ⩽ 2ρ0 }, 1

1

B1 = {x ∈ D(A 2 ), |A 2 x| ⩽ 2ρ1 }, B2 = {x ∈ D(A), |Ax| ⩽ 2ρ2 } and A is the w limit set of B2 , A = w(B2 ) =

\ s⩾0

 Cl 

\

 S(t)B2  ,

t⩾s

which is a global attractor of (5.7.1), closure Cl is in H, and A ⊆ B2 ∩ B1 ∩ B0 . We consider the inertial manifold of the cutoff equation of (5.7.1). Assume θ(s) is a smooth function from R+ to [0, 1]: θ(s) = 1, 0 ⩽ s ⩽ 1; θ(s) = 0, s ⩾ 2, |θ′ (s)| ⩽ 2, s ⩾ 0. Fix ρ = 2ρ2 . Define   s , s ⩾ 0. θρ (s) = θ ρ Then the cutoff equation of (5.7.1) is du + Au + θρ (|Au|)R(u) = 0. dt

(5.7.25)

It is direct to prove the existence and uniqueness of the solution of (5.7.25) with initial value u(0) = u0 ∈ H. Obviously, when |Au| ⩽ ρ, θs (|Au|) = 1. (5.7.24) and

5.7

Inertial manifold for one class of nonlinear evolution equations

273

(5.7.1) are congruent. When |Au| ⩾ 2ρ, θρ (|Au|) = 0. Taking the inner product of A2 u for (5.7.24) infers that 3 1 d 1 d |Au|2 + λ1 |Au|2 ⩽ (Au)2 + |A 2 u|2 ⩽ 0. 2 dt 2 dt

Hence the trajectory of u(t) in D(A) converges to the ball with radius ρ3 ⩾ 2ρ exponentially. In addition, R(u) is local Lipschitz continuous with respect to u. However, F (u) = θρ (|u|)R(u) is global Lipschitz continuous. i.e. there exists a K, such that |F (u) − F (v)| ⩽ K|u − v|, ∀ u, v ∈ H. (5.7.26) Definition 5.7.2 The inertial manifold of the semigroup operator {S(t)}t⩾0 is a smooth manifold µ ∈ H (at least Lipschitz) with finite dimension, it satisfies: (i) µ is invariant. i.e., S(t)µ ⊂ µ. (5.7.27) (ii) µ attracts all the solution of (5.7.25), exponentially. i.e. there exist constants k1 > 0, k2 > 0, for u0 ∈ H, dist(S(t)u0 , µ) ⩽ k1 e−k2 t ,

∀ t ⩾ 0.

(5.7.28)

(iii) the attractor A is in µ. Now, we will construct the inertial manifold to show the existence of the inertial manifold. Assume PN is an orthogonal projection of N dimension in H, QN = I − PN . Denote P = PN , Q = QN . Assume u(t) is a solution of (5.7.25). Let p(t) = P u, q(t) = Qu. Then p(t), q(t) satisfy the following conditions on P H and QH: dp + Ap + pF (u) = 0, (5.7.29) dt dq + Aq + QF (u) = 0, dt

(5.7.30)

where F (u) = θρ (|Au|)R(u) and u = p + q. We are looking for the inertial manifold µ, which is obtained by constructing the graph of Lipschitz function Φ : P D(A) → QD(A), i.e. µ = GraphΦ. As an operator, the function Φ can be derived by the fixed point on Fb,l , where Fb,l is a class of functions Φ : P D(A) → QD(A), and it satisfies (5.7.31) |AΦ(p)| ⩽ b, ∀ p ∈ P D(A), |AΦ(p1 ) − AΦ(p2 )| ⩽ l|Ap1 − Ap2 |,

∀ p1 , p2 ∈ P D(A),

supp Φ ⊂ {p ∈ P D(A) | |Ap| ⩽ 4ρ},

(5.7.32)

274

Chapter 5

Infinite Dimensional Dynamical Systems

where b > 0, l > 0. When p = p(t), q(t) = Φ(p(t)) satisfy (5.7.29), (5.7.30), u = p(t)+Φ(p(t)) is the solution of (5.7.25). Suppose Φ is given in Fb,l , p0 ∈ P D(A), then there exists a unique solution p = p(t; p0 , Φ) of the following equation dp + Ap + pF (p + Φ(p)) = 0, dt

p(0) = p0 .

(5.7.33)

Because σ → θρ R(σ+Φ(σ)) is Lipschitz continuous. Similar to (5.7.30), it is deduced from p = p(t; p0 , Φ), t ∈ R that dq + Aq + QF (p + Φ(p)) = 0. dt

(5.7.34)

Since QF (p+Φ(p)) is bounded: R → H, then there exists a unique bounded solution q(t) when t → ∞. This implies Z 0 q(0) = − ρτ AQ QF (p + Φ(p)) dτ, (5.7.35) −∞

where p = p(τ ) = p(τ ; Φ, p0 ). q(0) depends on Φ ∈ Fb,l and p ∈ P D(A). q(0) = q(0; p0 , Φ). The function p0 ∈ P D(A) → q(0; p0 , Φ) ∈ QD(A) is from P D(A) to QD(A), denotes as F Φ. Hence Z 0 T Φ(p0 ) = − eτ AQ QF (u) dτ,

(5.7.36)

−∞

where u = u(τ ) = p(τ ; Φ, p0 ) + Φ(p(τ ; Φ, p0 )). Since q(0) = Φ(p0 ), we are looking for the conditions that N, b, l should meet, such that (i) T maps from Fb,l to itself; (ii) T is compressible in Fb,l . Now, we consider the function Φ: P D(A) → QD(A), (P = PN , Q = QN = I − PN ). Φ ∈ Fb,l , i.e. it satisfies (5.7.31)-(5.7.33). Introduce the distance ∥Φ − Ψ∥ = sup |AΦ(p) − AΨ(p)|,

(5.7.37)

p∈D(A)

then Fb,l is a complete metric space. For Φ ∈ Fb,l , the mapping T in P D(A) is defined by Z 0 T Φ(p0 ) = − eτ AQ QF (u) dτ, p0 ∈ P D(A), (5.7.38) −∞

where u(τ ) = p(τ ; Φ, p0 )+Φ(p(τ ; Φ, p0 )), p(τ ; Φ, p0 ) is a solution of (5.7.29) satisfying p(0; Φ, p0 ) = p0 . Next, we investigate the properties of the operator T .

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275

Lemma 5.7.3 Assume α > 0 and τ < 0, (AQ)α eτ AQ is a linear and continuous operator on QH, its norm (i.e.|(AQ)α eτ AQ |0p ) in L (QH) is bounded by K3 |τ |−α , when − αλ−1 N +1 ⩽ τ < 0; α τ λN +1 λN +1 e , when − ∞ < τ ⩽ −αλ−1 N +1 . ∞ P

Proof Assume v =

(5.7.39)

bj wj is an element on QH. Then

j=N +1

|(AQ)α eτ AQ v|2 =

∞ X

τ λj 2 2 (λα ) bj ⩽ je

j=N +1

sup (λα eτ λ )2

X

λ⩾λN +1

b2j

= sup (λα eτ λ )2 |v|2 . λ⩾λN +1

Therefore, |(AQ)α eτ AQ |0,p ⩽

sup λα eτ λ . λ⩾λN +1

An elementary calculation shows that ( |τ |−α (αe−1 )α , when αλ−1 N +1 ⩽ τ < 0, sup (λα eτ Aλ ) = α τ λN +1 λ e , when τ ⩽ −αλ−1 λ⩾λN +1 N +1 N +1 We get (5.7.39), where K3 = K3 (α) = (αe−1 )α . As a directly corollary of (5.7.39), one has Z 0 |(AQ)α eτ AQ |0p dτ ⩽ (1 − α)−1 eα λ−α−1 N +1 ,

0 < α < 1.

(5.7.40)

−∞

Form (5.7.10) and (5.7.11), it is inferred that 1

1

1

1

|(AQ) 2 R(u)| ⩽|A 2 B(u, u)| + |A 2 Cu| + |A 2 f | 1

=C3 |Au|2 + C4 |Au| + |A 2 f |. When |Au| > 2ρ, θρ (|Au|) = 0, which implies that 1

|(AQ) 2 F (u)| ⩽ K4 ,

(5.7.41)

1

where K4 = 4C3 ρ2 + 2C4 ρ + |A 2 f |, F (u) is determined by (5.7.30). Lemma 5.7.4 Assume p0 ∈ P D(A), T Φ(p0 ) ∈ QD(A), then −1

2 , |AT Φ(p0 )| ⩽ K5 λN +1 5

−1

4 |A 4 T Φ(p0 )| ⩽ K6 λN +1 ,

where K5 and K6 are proper constants, independent of p0 , Φ.

(5.7.42) (5.7.43)

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Proof Note that Qeτ AQ = eτ AQ , it is easy to know T Φ(p0 ) ∈ QP (A). Z 0 Z 0 1 1 τ AQ |AT Φ(p0 )| ⩽ |AQe F (u)| dτ ⩽ |(AQ) 2 eτ AQ |0p |(AQ) 2 F (u)| dτ, −∞

5

Z

|A 4 T Φ(p0 )| ⩽

0

−∞

Z

5

−∞

|(AQ) 4 eτ AQ F (u)| dτ ⩽

0

−∞

3

1

|(AQ) 4 eτ AQ |0p |(AQ) 2 F (u)| dτ.

The inequalities (5.7.43), (5.7.42) can be obtained from (5.7.40), (5.7.41). Take −1

2 b = K5 λN +1 .

(5.7.44)

Then, for Φ ∈ Fb,l , similar to (5.7.31), T Φ satisfies |AT Φ(p0 )| ⩽ b,

∀p0 ∈ P D(A).

(5.7.45)

T Φ is a bounded set in D(A 4 ) since (5.7.43). Then, the compactness of A− 4 infers that the image of T Φ is a compact subset of QD(A), which is independent of Φ. Now, attention is given to the properties of the support set and continuity of T Φ. 1

5

Lemma 5.7.5 For any Φ ∈ Fb,l , the support set of T Φ is contained in {p ∈ P D(A); |Ap| ⩽ 4ρ}. Proof u = p + Φ(p). If |Ap| > 2ρ, then 1

|Au| = (|Ap|2 + |AΦ(p)|2 ) 2 ⩾ |Ap| > 2ρ. Thus, θρ (|Au|) = 0. Assume |Ap0 | > 4ρ, then on a certain interval of t, one has |Ap(t)| > 2ρ. Therefore, (5.7.29) becomes dp + Ap = 0, dt which implies that 3 1 d 1 d|Ap|2 |Ap|2 + λ1 |Ap|2 ⩽ + |A 2 p|2 = 0. 2 dt 2 dt

Hence, for τ < 0, 2ρ < |Ap(0)| ⩽ |Ap(τ )| exp(λ1 τ ) ⩽ |Ap(τ )|. Furthermore, θρ (|Au(τ )|) = 0, ∀τ ⩽ 0. From (5.7.38), it is deduced that T Φ(p0 ) = 0,

∀Φ ∈ Fb,l .

Firstly, the Lipschitz property of the nonlinear term F (u) is proved.

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277

Lemma 5.7.6 Assume p1 , p2 ∈ P D(A), Φ1 , Φ2 ∈ Fb,l , ui = pi + Φ(pi ), then 1

1

|A 2 F (u1 ) − A 2 F (u2 )| ⩽ K7 [(1 + l)|Ap1 − Ap2 | + ∥Φ1 − Φ2 ∥],

(5.7.46)

where K7 is independent of pi or Φi , for i = 1, 2. Proof Firstly, note that by (5.7.10) and (5.7.11), it is deduced that 1

1

1

|A 2 R(u1 ) − A 2 R(u2 )| ⩽|A 2 [B(u1 , u1 ) − B(u1 , u2 ) 1

+ B(u1 , u2 ) − B(u2 , u2 )]| + |A 2 C(u1 − u2 )| ⩽C3 (|Au1 | + |Au2 |)|Au1 − Au2 | + C4 |Au1 − Au2 | and 1

1

|A 2 R(u1 )| ⩽ C3 |Au1 |2 + C4 |Au1 |2 + |A 2 f |. Define G as follows 1

1

1

1

G = A 2 F (u1 ) − A 2 F (u2 ) = θρ (|Au1 |)A 2 R(u1 ) − θρ (|Au2 |)A 2 R(u2 ). There are three cases to be discussed: (i) 2ρ ⩽ |Au1 |, 2ρ ⩽ |Au2 |, (ii) |Au1 | < 2ρ ⩽ |Au2 | or |Au2 | < 2ρ ⩽ |Au1 |, (iii) |Au1 | ⩽ 2ρ, |Au2 | ⩽ 2ρ. When |Au| ⩾ 2ρ, by taking advantage of θρ (|Au1 |) = 0 and |θ′ | ⩽ 2ρ−1 , we get (i) G = 0, (ii) 1

|G| = |θρ (|Au|)A 2 R(u)| 1

1

= |θρ (|Au1 |)A 2 R(u1 ) − θρ (|Au2 |)A 2 R(u1 )| 1

⩽ |θρ (Au1 ) − θρ (Au2 )||A 2 R(u1 )| ⩽ 2ρ−1 ||Au1 | − |Au2 || · (C3 |Au1 |2 + C4 |Au1 | + |A 2 f |) 1

+ [C3 (|Au1 | + |Au2 |) + C4 ]|Au1 − Au2 |. Hence, 1

1

|A 2 F (u1 ) − A 2 F (u2 )| ⩽ K7 |Au1 − Au2 |, where K7 = 2ρ−1 (C3 4ρ2 + C4 2ρ + |A 2 f |) + C3 4ρ + C4 . From 1

u1 − u2 = p1 − p2 + (Φ1 (p1 ) − Φ1 (p2 )) + (Φ1 (p2 ) − Φ2 (p2 )), it is inferred that |Au1 − Au2 | ⩽ (1 + l)|Ap1 − Ap2 | + ∥Φ1 − Φ2 ∥.

(5.7.47)

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Combining with (5.7.47) gives (5.7.46). Now, we propose next to obtain that T is a Lipschitz mapping from Fb,l to Fbl under certain condition, and it is strictly contractive. Firstly, assume Φ is fixed, p01 , p02 ∈ P D(A), p = p1 (t), p = p2 (t) are the solutions of (5.7.29) satisfying the initial condition pi (0) = p0,i for i = 1, 2. Let ∆ = p1 − p2 , then ∆ satisfies the following equation d∆ + A∆ + P F (u1 ) − P F (u2 ) = 0, dt

(5.7.48)

where ui = pi + Φ(pi ), i = 1, 2. For (5.7.48), taking the inner product with A2 ∆ implies that 3 1 3 1 d |A∆|2 + |A 2 ∆|2 = −(A 2 P (F (u1 ) − F (u2 )), A 2 ∆). 2 dt

(5.7.49)

Employing (5.7.46), we can estimate 3 3 1 d |A∆|2 + |A 2 ∆|2 ⩽ K7 (1 + l)|A∆||A 2 ∆|. 2 dt

Then

3 3 d |A∆| ⩾ −|A 2 ∆|2 − K7 (1 + l)|A∆||A 2 ∆|. dt Furthermore, the fact ∆ ∈ P D(A) shows

|A∆|

3

1

2 |A∆|. |A 2 ∆| ⩽ λN

Hence, |A∆| or

1 d 2 |A∆| ⩾ −λN |A∆|2 − K7 (1 + l)λN |A∆|2 , dt

1 d 2 |A∆| + (λN + K7 (1 + l)λN )|A∆| ⩾ 0. dt

(5.7.50)

This infers that 1

2 |A∆(τ )| ⩽ |A∆(0)| exp(−τ (λN + K7 (1 + l)λN )),

τ ⩽ 0.

(5.7.51)

1 2

Lemma 5.7.7 Assume γN = λN +1 − λN − K7 (1 + l)λN > 0, then for Φ ∈ Fb,l and p01 , p02 ∈ P D(A), there holds |AT Φ(p01 ) − AT Φ(p02 )| ⩽ L|Ap01 − Ap02 |, where

−1

2 [1 + (1 − γN αN )−1 ]e− 2 exp L = K7 (1 + l)λN +1

γN = This derives that Φ ∈ Fb,l .

λN , λN +1

1

−1

γ α  N N , 2

2 αN = 1 + K7 (1 + l)λN +1 .

(5.7.52)

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279

Proof Adopting (5.7.38) and (5.7.46) gives that Z |AT Φ(p01 ) − AT Φ(p02 )| ⩽

0

−∞

|AQeτ AQ Q(F (u1 ) − F (u2 ))| dτ Z

⩽K7 (1 + l)

0

−∞

1

|(AQ) 2 eτ AQ |0p |A∆(τ )| dτ,

where ∆ = p1 − p2 . Combining Lemma 5.7.3 and (5.7.51), we find Z

0

1

−∞

Z ⩽

|(AQ) 2 eτ AQ |0p |A∆(τ )| dτ

− 12 λ−1 N +1 −∞ 0

−1

Z +

− 21 λ−1 N +1

1

2 2 λN +1 exp[τ (λN +1 − λN − K7 (1 + l)λN ] dτ

K3

1 2

 1   1 2 ) dτ |τ |− 2 · exp − τ (λN + K7 (1 + l)λN

· |Ap01 − Ap02 |. An elementary computation infers that the right-hand side of the above expression is bounded by γ α  1 − 12 N N λN +1 e− 2 [1 + (1 − γN αN )−1 ] exp |Ap01 − Ap02 |. 2 This proves (5.7.52). Using (5.7.44), (5.7.52) and Lemma 5.7.5, it is deduced that T Φ ∈ Fb,l . So far, we have proved the mapping T : Fb,l → Fb,l . Now, we assert that T is a Lipschitz mapping. To show this fact, one considers two functions Φ1 , Φ2 with the same initial condition. Let pi = p(ti ; Φi , p0 ),

ui = pi + Φi (pi ),

i = 1, 2.

We estimate |AT Φ1 (p0 ) − AT Φ2 (p0 )|. By a similar approach, it is obtained that 1 d 2 |A∆| + λN αN |A∆| ⩾ −K7 λN ∥Φ1 − Φ2 ∥, dt

(5.7.53)

−1

2 where ∆ = p1 − p2 , αN = (1 + K7 (1 + l)λN +1 ). Thus, by ∆(0) = 0 and (5.7.53), it is deduced that 1

2 K7 λN ∥Φ1 − Φ2 ∥ |A∆(τ )| ⩽ (exp(−αN λN τ ) − 1) αN λN

−1

⩽ K7 λN 2 ∥Φ1 − Φ2 ∥ exp(−αN λN τ ),

τ ⩽ 0.

(5.7.54)

280

Chapter 5

Infinite Dimensional Dynamical Systems

As the same as Lemma 5.7.7, combining (5.7.38), (5.7.46) and (5.7.54), gives that |AT Φ1 (p0 ) − AT Φ2 (p0 )| Z 0 |AQeτ AQ (F (u1 ) − F (u2 ))| dτ ⩽ −∞

Z

⩽ K7

0

1

−∞

|(AQ) 2 eτ AQ |0p [(1 + l)|A∆| + ∥Φ1 − Φ2 ∥] dτ Z

⩽ K7 ∥Φ1 − Φ2 ∥ ·(1 +

0

1

−∞

−1 K7 λN 2 (1

|(AQ) 2 eτ AQ |0p

+ l)e(−αN λN τ ) ) dτ.

(5.7.55)

By Lemma 5.7.3, it infers that the integral on the right-hand side of (5.7.55) is bounded by Z −a 1 − 12 −1 − 12 2e− 2 λN +1 + K7 (1 + l)λN 2 λN +1 exp[τ (λN +1 − λN αN )] dτ Z

0

−∞

1

|τ |− 2 exp(−λN αN τ ) dτ −a h 1  γ α i 1 1 −1 −2 N N − 12 − 2 e− 2 (1 + (1 − γN αN )−1 ) exp ⩽2e λN +1 + K7 (1 + l)λN 2 · λN +1 2 +

K3

−1

1

2

−1

2 ⩽2e− 2 λN +1 + λN 2 L. 1

Hence, |AT Φ1 (p0 ) − AT Φ2 (p0 )| ⩽ L′ ∥Φ1 − Φ2 ∥, −1

∀p0 ∈ P D(A),

(5.7.56)

−1

2 + λN 2 L). where L′ = K7 (2e− 2 λN +1 In summary, we are looking for some certain conditions to ensure that the mapping T is from Fb,l to itself, and it is strictly contractive in Fb,l . This requires that the discovery of sufficient condition for λN and λN +1 , such that 1

L ⩽ l,

L′ < 1. 1

2 First of all, note that γN = λN +1 − λN − kl (1 + l)λN > 0 is equivalent to

1 − γN αN > 0,

(5.7.57)

1 > γN αN > 0,

(5.7.58)

or then from (5.7.57), it is derived that −1

2 L ⩽ K7 (1 + l)λN +1 [1 + (1 − γN αN )−1 ].

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Inertial manifold for one class of nonlinear evolution equations

281

In order to have L ⩽ l, we choose a proper N such that the following two inequalities holds: l − 21 K7 (1 + l)λN +1 ⩽ , (5.7.59) 2 l − 12 (5.7.60) K7 (1 + l)λN +1 (1 − γN αN )−1 ⩽ . 2 (5.7.59) can be rewritten as 1

2 K10 ⩽ λN +1 ,

(5.7.61)

where K10 = 2K7 (1 + l)L−1 . Assume that N is chosen, such that (5.7.61) holds. The inequality (5.7.60) can be written as −1

2 K10 λN +1 ⩽ 1 − γN α N ,

(5.7.62)

which is equivalent to −1

−1

1

2 2 K10 λN +1 − 1 + γN + K7 (1 + l)λN +1 γN2 ⩽ 0,

where γN =

(5.7.63)

λN . Assume λN +1 −1

1

−1

2 2 2 = (λN λ−1 γN2 + K10 λN +1 N +1 ) + K10 λN +1 ⩽ 1, 1

(5.7.64)

Apply (5.7.64) twice, one has −1

−1

−1

1

1

2 2 2 K10 λN +1 − 1 + γN + K10 λN +1 γN2 ⩽ K10 λN +1 − 1 + γN2 ⩽ 0.

(5.7.65)

Due to l ⩽ 18 , it is discovered that K7 (1 + l) ⩽ K10 . Thus (5.7.64) infers (5.7.63). Consequenly, (5.7.62) is obtained. Hence, in order to show that the mapping T is from Fb,l to Fb,l , we assume γN > 0 or 1 − γN αN > 0. This assumption is ensured by (5.7.62). (5.7.59), (5.7.62) are the sufficient conditions to ensure the mapping T is from Fb,l to itself, which is guaranteed by (5.7.61), (5.7.64). It is easy to see that these two inequalities are derived by the following inequality 1

1

2 2 K10 ⩽ λN +1 − λN .

(5.7.66)

In order to show that the mapping T on Fb,l is contractive, it is necessary to 1 have L′ < 1. Assume L′ ⩽ , then it is deduced that 2 1

2 K11 ⩽ λN +1 ,

(5.7.67)

where K11 = 2K7 (2e− 2 + L). Thus, under the conditions of (5.7.66) and (5.7.67), the mapping T is from Fb,l to it self, and it is contractive. Hence, T has a fixed point. 1

282

Chapter 5

Infinite Dimensional Dynamical Systems

We are now going to prove M = Graph(Φ) is invariant with S(t). i.e. S(t)M ⊂ M,

∀t ⩾ 0,

(5.7.68)

and show that it attracts all the trajectories, which converge to M exponentially. Firstly, the proof of the invariance of M is given. In fact, Z Φ(p0 ) = −

0

−∞

eτ AQ QF (u(τ, p0 )) dτ,

(5.7.69)

where u(τ, p0 ) = p(τ ; Φ, p0 ) + Φ(p(τ ); Φ, p0 ). Set p0 as p(t) = p(t; Φ, p0 ) in (5.7.69), and note that p(τ ; Φ, p(t; Φ, p0 )) = p(τ + t; Φ, p0 ), which implies that Z Φ(p(t)) = − =−

0

eτ AQ QF (u(τ ), p(t))) dτ −∞ Z t −∞

e−(t−τ )AQ QF (u(τ ), p0 )) dτ,

∀t ∈ R.

(5.7.70)

For the above equation, taking derivative with respect to t, it is obvious that, (p(t), q(t)) are the solutions of (5.7.29),(5.7.30), and u(t) = p(t) + q(t) is the solution of (5.7.25), where q(t) = Φ(p(t)). This indicates that S(t)M ⊆ M , ∀t ⩾ 0. To show M attract all the solutions of (5.7.25) exponentially, we will start with the statement of the squeezing property of equation (5.7.201). Squeezing Property For every T > 0, γ > 0, r > 0, there exist K2 , K3 (it depends on T ,γ,r and constants C1 ∼ C4 , and independent of S(t) and N ), such that for any N ⩾ 1, one of the following inequalities hold: |QN (S(t)u0 − S(t)v0 )| ⩽ γ|pN (S(t)u0 − S(t)v0 )|

(5.7.71)

or |S(t)u0 − S(t)v0 | ⩽ K2 exp(−K3 αλN +1 t)|u0 − v0 |.

(5.7.72)  1 We apply the above result to any t satisfying t0 ⩽ t ⩽ 2t0 , where t0 = log 2, 2K1 K1 is a constant. When |Au0 | ⩽ r, |Av0 | ⩽ r, there holds |S(t)u0 − S(t)v0 | ⩽ exp(K1 t)|u0 − v0 |, γ=



∀t ⩾ t0 ,

1 , N ⩾ N0 , where N0 satisfies 8 λN0 +1 ⩾ (2K3 αt0 )−1 log(2K2 ).

(5.7.73)

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Inertial manifold for one class of nonlinear evolution equations

283

Then, (5.7.71), (5.7.72) turn to be |QN (S(t)u0 − S(t)v0 )| ⩽

1 |PN (S(t)u0 − S(t)v0 )|, 8

|S(t)u0 − S(t)v0 | ⩽

1 |u0 − v0 |, 2

(5.7.74) (5.7.75)

where u0 , v0 ∈ D(A), |Au0 | ⩽ r, |Av0 | ⩽ r, t0 ⩽ t ⩽ 2t0 . Fix r = 4ρ + b, according to (5.7.24), the trajectory of (5.7.25) falls into the ball centered at t with radius 4ρ = 8ρ2 in D(A). Assume |Au0 | ⩽ 4ρ,

|AS(t)u0 | ⩽ 4ρ,

t ⩾ 0.

We start with the following: for any t1 , t0 ⩽ t1 ⩽ 2t0 , there holds dist(S(t1 )u0 , µ) ⩽

1 dist(u0 , M ), 2

where dist(φ, M ) = inf{|φ − v| : v ∈ M }. Hence, choose v0 , such that |u0 − v0 | = dist(u0 , M ). Then v0 = P v0 + Φ(P v0 ). We require |AP v0 | ⩽ 4ρ. Otherwise, if |AP v0 | > 4ρ ⩾ |AP u0 |, then Φ(P v0 ) = 0, v0 = P v0 . In addition, there exists a β, 0 < β ⩽ 1, such that |AP vβ | = 4ρ, where vβ = βP u0 + (1 − β)v0 ∈ P D(A). This implies that Φ(vβ ) = 0. Therefore, vβ ∈ µ, and |vβ − v0 |2 =|vβ − P u0 |2 + |Qu0 |2 =|(1 − β)(v0 − P u0 )|2 + |Qu0 |2 0, such that (i) the mapping T is from Fb,l to Fb,l . (ii) T has a fixed point in Fb,l . (iii) M = Graph(Φ) is an inertial manifold of (5.7.25). (iv) M contains the global attractor of (5.7.1).

(5.7.77)

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Inertial manifold for one class of nonlinear evolution equations

285

Theorem 5.7.9 Suppose equations (5.7.1), (5.7.25) are given in H, where the 1 nonlinear term F (u) = θρ (|Au|)R(u) satisfies (5.7.26). Fix l, 0 < l < . Assume 8 there is a ρ0 such that (5.7.21) holds for every solution of (5.7.1). Then there exist N0 , K12 , K13 (depends on l and the initial data), such that N ⩾ N0 ,

λN +1 ⩾ K12 ,

λN +1 − λN ⩾ K13 .

(5.7.78)

Then the conclusion in Theorem 5.7.8 holds. Proof Assume the nonlinear term F (u) = θρ (u)R(u) satisfies the global Lipschitz condition |F (u) − F (v)| ⩽ K|u − v|, ∀u, v ∈ H. Choose ρ = 2ρ0 . The space Fb,l is constructed by Φ : P H → QH, satisfying |Φ(p)| ⩽ b,

∀p ∈ P D(A).

|Φ(p1 ) − Φ(p2 )| ⩽ l|p1 − p2 |,

∀p1 , p2 ∈ D(A).

sup pΦ ⊆ {p ∈ P D(A) | |p| ⩽ 4ρ}. The operator T is defined by Z T Φ(p0 ) = −

0

eτ AQ QF (u) dτ, −∞

where u = u(τ ) = p(τ ; Φ, p0 ) + Φ(p(τ ; Φ, p0 )). The inequality (5.7.41) becomes |F (u)| ⩽ K4′ , where K4′ depends on R(u), θ and ρ. By Lemma 5.7.3, when α = 0, the inequality (5.7.40) holds and (5.7.42) becomes |T Φ(p0 )| ⩽ K5′ λ−1 N +1 , where K5′ = K ′ . Thus, take b = K5′ λ−1 N +1 . By Lemma 5.7.5, the norm of |Av| is converted to the norm of |v|, the inequality (5.7.46) becomes |F (u1 ) − F (u2 )| ⩽ K7′ [(1 + l)|p1 − p2 | + ∥Φ1 − Φ2 ∥],

(5.7.79)

where ∥Φ∥ = sup{|Φ(p)| : p ∈ P D(A)}. Let ∆ = p1 − p2 , then (5.7.48) holds. For (5.7.48), taking the inner product with ∆, it is deduced that 1 1 d |∆|2 + |A 2 ∆|2 = −(P (F (u1 ) − F (u2 )), ∆). 2 dt

286

Chapter 5

Infinite Dimensional Dynamical Systems

(5.7.79) infers that 1 d 1 |∆|2 + |A 2 ∆|2 ⩽ K7′ (1 + l)|∆|2 . 2 dt This implies that |∆|

1 d |∆| ⩾ −|A 2 ∆|2 − K7′ (1 + l)|∆|2 ⩾ −λN |∆|2 − K7′ (1 + l)|∆|2 . dt

Hence, the inequality (5.7.51) can be replaced by |∆(τ )| ⩽ |∆(0)| exp(−τ (λN + K7 (1 + l))),

τ < 0.

The assumption for γN in Lemma 5.7.7 can be replaced by |T Φ(p01 ) − T Φ(p02 )| ⩽ L|p01 − p02 |, where L = K7′ (1 + l)Γ−1 N . Actually, Z |T Φ(p01 ) − T Φ(p02 )| ⩽

0

−∞

|eτ AQ |0p |F (u1 ) − F (u2 )| dτ

⩽ K7′ (1+l)|p01 −p02 |×

Z

0

−∞

exp(τ (λN +1 −λN −K7′ (1 + l))) dτ

⩽ K7′ (1 + l)(λN +1 − λN − K7′ (1 + l))−1 |p01 − p02 |. Similarly, one has |T Φ1 (p0 ) − T Φ2 (p0 )| ⩽ L′ ∥Φ1 − Φ2 ∥, ′ −1 where L′ = K7′ λ−1 L. N +1 + K7 [λN + K7 (1 + l)] 1 1 Let L ⩽ l, L′ ⩽ . Then when l < , and 2 8

K12 ⩽ λN +1 ,

K13 ⩽ λN +1 − λN ,

(5.7.80)

T maps Fb,l to Fb,l , and it has a fixed point. This proves the theorem. Now, consider the Galerkin approximation equation of (5.7.25) duM + AuM + PM F (uM ) = 0, dt

(5.7.81)

where F (u) = θρ (|Au|)R(u). uM is taken in PM D(A). Regarding to the Galerkin approximation equation (5.7.81), the following theorem holds. Theorem 5.7.10 Suppose the assumptions in Theorem 5.7.8 are true, l > 0 and N satisfy the conditions in Theorem 5.7.8. Then for any M > N , the equation

5.8

Approximate inertial manifold

287

(5.7.81) has an inertial manifold MM . It is constructed by the graph of the Lipschitz function ΦM , ΦM : PM D(A) → QPM D(A) ⊂ QD(A). The Lipschitz constant L of ΦM is the same as the one of Φ : P D(A) → QD(A) in Theorem 5.7.8. Finally, −1

−1

4 λM4+1 , ∥ΦM − Φ∥ ⩽ 2K6 λN +1

where ∥ΦM − Φ∥ ⩽

|AΦM (p) − AΦ(p)|.

sup p∈P D(A)

5.8

Approximate inertial manifold

In the discussion of global attractor and inertial manifold, we can see that the global attractor may not smooth, even though the inertial manifold is smooth, to pursue it one must take the integral equation on the infinity interval, which gives a lot of trouble in computation. Hence, it is nature to look for an approximate, smooth, easy computation manifold, to approach the global attractor and the inertial manifold. This is the approximate inertial manifold development, see [38, 64, 144, 173, 190, 191, 232] and references therein. Next, we will take the two-dimensional NavierStokes equations for example to explain some approximate inertial manifolds and the estimates of its deviation. Assume that the Navier-Stokes equations have the following form du + νAu + B(u, u) = f, dt

(5.8.1)

u(0) = u0 ,

(5.8.2)

where Au = −P ∆u,

∀u ∈ D(A),

B(u, w) = P [(u · ∇)w],

∀u, w ∈ D(A).

P denotes the orthogonal projection of (L2 (Ω))2 on H, H denotes the closure of V in (L2 (Ω))2 . When u|∂Ω = 0, V = {v ∈ {C0∞ (Ω)}2 , divv = 0}, where A is a linear, unbounded, self-joint operator, A−1 is compact, D(A) is dense in H. Hence, H has an orthogonal basis {wj }∞ j=1 , which are the eigenvectors of operator A, Awj = λj wj , j = 1, 2, · · · , and 0 < λ 1 ⩽ λ2 ⩽ · · · . λm satisfies C0 λ1 m ⩽ λm ⩽ C1 λ1 m,

m = 1, 2, · · · ,

(5.8.3)

288

Chapter 5

Infinite Dimensional Dynamical Systems

where C0 , C1 are some fixed constants. Let C0 , C1 , · · · be absolutely positive constants in the following. It is easy to check that B(u, v) satisfies 1

1

1

1

1

|(B(u, v), w)| ⩽ C2 |u| 2 ∥u∥ 2 ∥v∥ 2 |w| 2 ∥w∥ 2 , |(B(u, v), w)| ⩽ C3 ∥u∥L∞ (Ω) ∥v∥|w|,

∀u, v, w ∈ V,

∀u ∈ D(A), ∀v ∈ V, ∀w ∈ H,

(5.8.4) (5.8.5)

Z

where u ∈ H,

|u|2 =

|u(x)|2 dx, Ω

Z u ∈ V,

∥u∥2 =

|∇u(x)|2 dx. Ω

The Brèzis-Gallouët inequality yields ∥u∥∞ L(Ω)

⩽ C4 ∥u∥ 1 + log

!! 12

|Au|

,

1

λ12 ∥u∥

∀u ∈ D(A).

(5.8.6)

Employing (5.8.5), (5.8.6), we can estimate |(B(u, v), w)| ⩽ C5 ∥v∥|w|∥u∥ 1 + 2 log

|(B(u, v), w)| ⩽ C6 ∥v∥|u|∥w∥ 1+2 log

!! 12

|Au|

,

1

λ12 ∥u∥ |Aw|

∀u ∈ D(A), ∀v ∈ V, ∀w ∈ H. (5.8.7)

!! 12

1

λ12 ∥w∥

,

∀u ∈ H, ∀v ∈ V, ∀w ∈ D(A). (5.8.8)

In addition, B(u, v) satisfies the following fundamental identity (B(u, v), w) = −(B(u, w), v),

∀u ∈ H, ∀v, w ∈ D(A).

(∗)

For the solution u(t) of the problem (5.8.1), (5.8.2), it has been proved that there exists a t0 , which depends on u0 , ν, |f | and λ1 , such that |u(t)| ⩽ M0 , ∥u(t)∥ ⩽ M1 ,

∀t ⩾ t0 ,

(5.8.9)

where the constants M0 and M1 depend only on ν, |f | and λ1 . Now, we consider the approximate inertial manifold of the problem (5.8.1), (5.8.2). Assume Pm is the orthogonal projection of H on Hm = span{w1 , · · · , wm }. Qm = I − Pm . Let p = Pm u, q = Qm u, then (5.8.1) is equivalent to dp + νAp + Pm B(p + q, p + q) = Pm f, dt

(5.8.10)

5.8

Approximate inertial manifold

289

dq + νAq + Qm B(p + q, p + q) = Qm f. (5.8.11) dt M is a subset of H. As the inertial manifold of system (5.8.1), it possesses the following properties: (5.8.12) (i) M is a Lipschitz manifold with finite dimension. (ii) The convention of M is positively invariant set. i.e., if u0 ∈ M, then the (5.8.13) solution of (5.8.1), (5.8.2), u(t) ∈ M, ∀ t > 0. (iii) M attracts all the trajectories exponentially, i.e., for any arbitrary solution, u(t) of the problem (5.8.1), (5.8.2), dist(u(t), µ) → 0, t → ∞ exponentially. This (5.8.14) infers the global attractor A ⊂ M. If one requires M is the graph of the Lipschitz function Φ: Hm → Qm H, then the invariant condition (5.8.13) is equivalent to that, for any solutions p(t), q(t) of (5.8.10), (5.8.11) with q(0) = Φ(p(0)), there holds q(t) = Φ(p(t)), ∀t ⩾ 0. Hence, if such a function Φ exists, then the system (5.8.1), (5.8.2) on the inertial manifold M is equivalent to the following ordinary differential system, which is called the inertial form: dp + νAp + Pm B(p + Φ(p), p + Φ(p)) = Pm f, dt

p ∈ Hm .

(5.8.15)

In order to adopt the smooth manifold to approximate the global attractor, we introduce approximate inertial manifold. Obviously, in (5.8.15), if Φ ≡ 0, then the Galerkin approximation is obtained dum + νAum + Pm B(um , um ) = Pm f, dt

um ∈ Hm .

(5.8.16)

Now, consider the analytic manifold with finite dimension. Denote µ0 = Graph(Φ0 ), and let Φ0 (p) = (νA)−1 [Qm f − Qm B(p, p)],

p ∈ Hm ,

(5.8.17)

be a much better approximate manifold of global attractor. Theorem 5.8.1 Assume m is sufficiently large enough, such that λm+1 ⩾

 2C M 2 2 , ν

(5.8.18)

then, for every solution of (5.8.10) and (5.8.11), u(t) = p(t) + q(t) satisfies 2 |q(t)| ⩽ K0 λ−1 m+1 L , 1

−1

1

2 ∥q(t)∥ ⩽ K1 λm+1 L2 ,

(5.8.19) (5.8.20)

290

Chapter 5

Infinite Dimensional Dynamical Systems 1

|Aq(t)| ⩽ K2 L 2 , dq ⩽ K0′ λ−1 L 12 , m+1 dt ∥q(t) − Φm (p(t))∥ ⩽ K1 λ−1 m+1 L,

(5.8.21) (5.8.22) ∀t ⩾ T∗ ,

(5.8.23)

where T∗ > 0 depends on ν, λ1 , |f | and R0 , |u(0)| ⩽ R0 ,   λm L = log + 1, λ1 K0 , K0′ , K1 , K2 are positive constants, depending on ν, λ1 and |f |. Let B = {p ∈ Hm | ∥p∥ ⩽ 2M1 }, B ⊥ = {q ∈ Qm V, ∥q∥ ⩽ 2M1 }, where M1 satisfies (5.8.13). When m large enough, there exists a mapping Φs : B → Qm V , satisfying Φs (p) = (νA)−1 [Qm f − Qm B(p + Φs (p), pΦs (p))],

∀p ∈ B.

(5.8.24)

Let M s = GraphΦs , which is a C-analytic manifold, and it contains all the stationary solution of (5.8.1). Now, we are going to prove the existence of Φs and give its upper bound. Theorem 5.8.2 Assume m is sufficiently large enough, such that   r22 2 , λm+1 ⩾ max 4r2 , 4M12

(5.8.25)

then there exists a unique mapping Φs : B → Qm V , satisfying (5.8.24), and −1

2 ∥Φs (p)∥ ⩽ λm+1 r1 ,

where

(5.8.26) 1

2 |f |, r1 = ν −1 C5 8M12 L 2 + ν −1 C2 8M12 + ν −1 λm+1 1

r2 = ν −1 C5 2M1 L 2 + ν −1 C2 6M1 , λm L = 1 + log . λ1 1

Proof Suppose p ∈ B is fixed, define Tp : B ⊥ → Qm V , such that Tp (q) = (νA)−1 [Qm f − Qm B(p + q, p + q)]. To show that Tp has a unique fixed point, one needs to start with the proof of Tp : B ⊥ → B ⊥ . Let q ∈ B ⊥ , w ∈ H, |w| = 1, then  1  h   i 1 2 A Tp (q), w ⩽ ν −1 B(p + q, p + q), A− 2 Qm w + A−1 Qm f |w| h     i 1 1 ⩽ ν −1 B(p, p + q), A− 2 Qm w + B(q, p + q), A− 2 Qm w + ν −1 λ−1 m+1 |f |.

5.8

Approximate inertial manifold

291

By (5.8.7) and (5.8.4), it is deduced that ! 21  1 1  |Ap| 2 −2 −1 A Tp (q), w ⩽ ν C5 ∥p + q∥ A Qm w ∥p∥ 1 + log 1 ∥p∥λ12 1 1 1 1 1 2 + ν −1 C2 |q| 2 ∥q∥ 2 ∥p + q∥ A− 2 Qm w |w| 2 + (νλm+1 )−1 |f |  1 λm 2 − 21 1 + log ⩽ ν −1 C5 8M12 λm+1 λ1 −1

2 + ν −1 C2 λm+1 8M12 + (νλm+1 )−1 |f |.

Hence,

−1

2 ∥Tp (q)∥ ⩽ λm+1 r.

(5.8.27)

(5.8.25) yields ∥Tp (q)∥ ⩽ 2M1 . Next, we are going to prove that Tp is contractive. Consider ∂ Tp (q)η = (νA)−1 Qm [B(p + q, η) + B(η, p + q)], ∂q

∀η ∈ Qm V.

Assume w ∈ H, |w| = 1, then       A 12 ∂ Tp (q)η, w ⩽ ν −1 B(p, η), A− 21 Qm w + ν −1 B(q, η), A− 21 Qm w ∂q   1 + ν −1 B(η, p + q), A− 2 Qm w 1  |Ap|  12 ⩽ ν −1 C5 ∥η∥ A− 2 Qm w ∥p∥ 1 + log 1 λ12 ∥p∥ 1 1 1 1 1 2 + ν −1 C2 |q| 2 ∥q∥ 2 ∥η∥ A− 2 Qm w |w| 2 1 12 1 1 1 + ν −1 C2 |η| 2 ∥η∥ 2 ∥p + q∥ A− 2 Qm w |w| 2 i 1 h  λm  21 −2 ⩽ ν −1 C5 2M1 1 + log + ν −1 C2 6M1 λm+1 ∥η∥. λ1 Hence,



∂

Tp (q)

∂q

L (Qm V )

−1

2 ⩽ r3 λm+1 .

By (5.8.25), it is inferred from (5.8.28) that

∂

1

Tp (q) ⩽ .

∂q

2 L (Qm V )

(5.8.28)

292

Chapter 5

Infinite Dimensional Dynamical Systems

By the contraction mapping theorem, it is derived that there exists a unique q(p) ∈ B ⊥ , such that q(p) = Tp (q). Let Φs (p) = q(p). (5.8.27) implies (5.8.26) and µs = GraphΦs is a C-analytic manifold, each trajectory u(t) = p(t) + q(t) locates in a small neighborhood of the manifold µs , the global attractor is in this neighborhood as well. Theorem 5.8.3 Assume m is sufficiently large enough, such that (5.8.25) holds, then for each solution u(t) = p(t) + q(t) of the problem (5.8.10), (5.8.11) satisfies ∥q(t) − Φs (p(t))∥ ⩽

2K0′ − 23 1 λ L2 , ν m+1

∀t ⩾ T∗ ,

(5.8.29)

where T∗ and K0′ are the constants in Theorem 5.8.1. Proof Let ∆(t) = q(t) − Φs (p(t)). (5.8.11) and (5.8.17) imply that νA∆ + Qm [B(∆, p + Φs (p)) + B(p + q, ∆)] +

dq = 0. dt

Taking the inner product with ∆ on H, (∗) infers that  dq  ν∥∆∥2 ⩽ |(B(∆, p + Φs (p)), ∆)| + , ∆ . dt By (5.8.4), it is obtained that dq ν∥∆∥2 ⩽ C2 |∆|∥∆∥∥p + Φs (p))∥ + |∆|. dt

(5.8.30)

When t > T∗ , ∥p(t)∥ ⩽ M1 . From Theorem 5.8.2, it is deduced that Φs (p(t)) ⩽ 2M1 . Substituting (5.8.22) into (5.8.30) infers −1

−1

2 2 ν∥∆∥2 ⩽ C2 λm+1 ∥∆∥2 (M1 + 2M1 ) + K0′ λm+1 L 2 ∥∆∥. 1

From (5.8.25), it is obtained that ∥∆∥ ⩽

2K0′ − 32 1 λ L2 . ν m+1

This proves Theorem 5.8.3. Now, consider another approximation with Φs , which could be successive approximation and explicitly solved. Then, we have the following theorem. Theorem 5.8.4 Assume m is sufficiently large enough, such that (5.8.25) holds. Similar to Theorem 5.8.2, define Tp : B ⊥ → B ⊥ , Tp (q) = (νA)−1 [Qm f − Qm B(p + q, p + q)],

∀ q ∈ B⊥.

5.8

Approximate inertial manifold

Denote

(

293

Φs0 (p) = Tp (0), ∀p ∈ B, Φsn+1 (p) = Tp (Φsn (p)), ∀p ∈ B; n = 0, 1, 2, · · · ,

(5.8.31)

then i  n+1 h 1 − 21 − 12 ∥Φs (p) − Φsn (p)∥ ⩽ 2r2 λm+1 λm+1 ν −1 |f | + 4C5 M12 L 2 ,

(5.8.32)

where r2 is defined in Theorem 5.8.2. Proof First of all, note that Φs0 (p) ≡ Φ0 (p), ∀p ∈ B. Combining Theorem 5.8.2, (5.8.28) and (5.8.25), it is easy to get that  n+1 − 12 ∥Φs (p) − Φsn (p)∥ ⩽ 2r2 λm+1 ∥Φs0 (p)∥.

(5.8.33)

Hence, attention is turned to the estimation of ∥Φs0 (p)∥. By (5.8.31), it is inferred that Φs0 (p) = Φ0 (p) = (νA)−1 [Qm f − Qm B(p, p)]. (5.8.34) Thus |AΦs0 (p)| ⩽ ν −1 |f | + ν −1 |B(p, p)|. Then, (5.8.4) implies that |AΦs0 (p)| ⩽ ν −1 |f | + ν −1 C5 ∥p∥2

1 + log

|Ap| 1

∥p∥λ12

! 12 ,

|AΦs0 (p)| ⩽ ν −1 |f | + ν −1 C5 4M12 L 2 . 1

Hence,

i h 1 − 21 ∥Φs0 (p)∥ ⩽ λm+1 ν −1 |f | + 4C5 M12 L 2 ,

(5.8.35)

which get along with (5.8.35) and (5.8.33) gives (5.8.32). Corollary 5.8.5 Assume m is sufficiently large enough, such that (5.8.25) holds, then for each solution u(t) = p(t) + q(t) of the problem (5.8.10), (5.8.11), there holds  n+1 h i 1 2K0 1 − 21 − 12 L 2 + 2 r2 λm+1 λm+1 ν −1 |f | + 4C5 M12 L 2 , ν ∀ t ⩾ T∗ , n = 0, 1, 2, · · · (5.8.36) −3

2 ∥q(t) − Φsn (p(t))∥ ⩽ λm+1

where Φsn is defined by (5.8.31), T∗ , L and K0′ are the same as those defined in Theorem 5.8.1, r2 is the same as in Theorem 5.8.2.

294

Chapter 5

Infinite Dimensional Dynamical Systems

Proof This is a corollary of Theorem 5.8.3 and Theorem 5.8.4. In order to estimate the deviation of the Galerkin approximation method with finite dimension, one must prove the following lemma. Lemma 5.8.6 Assume m is sufficiently large enough, such that (5.8.25) holds, then for all integer k ⩾ m + 1, −1

2 , ∥Qk Φs (p)∥ ⩽ K1 λk+1

where K1 =

(5.8.37)

16C2 M12  λk  12 + 1. 1 + log ν λ1

Proof From (5.8.24), it is deduced that νAQk Φs (p) + Qk B(p + Φs (p), p + Φs (p)) = Qk f. Taking the inner product with Φs (p) on H implies ν∥Qk Φs (p)∥2 ⩽ |(B(p + Φs (p), p + Φs (p)), Qk Φs (p)| + |f ||Qk Φs (p)|, ν∥Qk Φs (p)∥2 ⩽ |(B(p + Pk Φs (p), p + Φs (p)), Qk Φs (p)| + |(B(Qk Φs (p), p + Φs (p)), Qk Φs (p)| + |f ||Qk Φs (p)|. Combining (5.8.4) and (5.8.7) gives  λk  12 ν∥Qk Φs (p)∥2 ⩽ C5 ∥p + Φs (p)∥2 |Qk Φs (p)| 1 + log λ1 + C2 |Qk Φs (p)|∥Qk Φs (p)∥∥p + Φs (p)∥ + |f ||Qk Φs (p)|. By (5.8.26) and the definition of B, one has  √ λk  12 − 21 − 21 − 21 + C2 8M1 λk+1 ν∥Qk Φs (p)∥ ⩽ C5 8M12 λk+1 1 + log ∥Qk Φs (p)∥ + λk+1 |f |. λ1 Then (5.8.25) infers (5.8.37). Let k ⩾ m + 1, m is sufficiently large enough, satisfying (5.8.25). Consider the general k-order Galerkin approximation: duk + νAuk + Pk B(uk , uk ) = Pk f, dt

uk ∈ Hk .

(5.8.38)

By Theorem 5.8.2, it is obvious that the equation (5.8.38) has a unique solution Φs,k : B → Pk Qm V , which satisfies νAΦs,k (p) + Pk Qm B(p + Φs,k (p), p + Φs,k (p)) = Pk Qm f, Note that Φs,k has all the stationary solutions of (5.8.38).

∀ P ∈ B.

(5.8.39)

5.8

Approximate inertial manifold

295

Lemma 5.8.7 Assume m is sufficiently large enough, such that (5.8.25) holds, then for all k ⩾ m + 1, −1

2 ∥Φs (p) − Φs,k (p)∥K3 λk+1 ,

where

P ∈ B,

(5.8.40)

   (2C2 + C3 ) √ λk  12 − 12 K3 = 1 + 2 8M1 λk+1 1 + log K1 , ν λ1

K1 is defined as in Lemma 5.8.6. Proof For p ∈ B, let u = p + Φs (p), v = p + Φs,k (p), ∆ = Pk (u − v), η = Qk (u − v), u − v = ∆ + η. By (5.8.24) and (5.8.39), one has νA∆ + Pk Qm [B(u − v, u) + B(v, u − v)] = 0, νA∆ + Pk Qm [B(∆ + η, u) + B(v, ∆ + η)] = 0. Taking the inner product with ∆ in H, which get along with the identity (∗) gives ν∥∆∥2 ⩽ |(B(∆ + η, u), ∆)| + |(B(v, η), ∆)|. The identity (∗) implies ν∥∆∥2 ⩽ |(B(∆, u), ∆)| + |(B(η, u), ∆)| + |(B(Pk v, ∆), η)| + |(B(Qk v, ∆), η)|. Taking advantage of (5.8.4), (5.8.8) and (5.8.7) infer   λk  21 λk  21 ν∥∆∥2 ⩽ C2 |∆|∥∆∥∥u∥ + C5 |η|∥u∥∥∆∥ 1 + log + C5 ∥v∥∥∆∥|η| 1 + log λ1 λ1 1

1

1

1

+C2 |Qk v| 2 ∥Qk v∥ 2 ∥∆∥|η| 2 ∥η∥ 2 . It is derived from (5.8.26) that ∥u∥ ⩽ (5.8.41), it is deduced that

(5.8.41)

√ √ 8M1 . Similarly, we have ∥v∥ ⩽ 8M1 . By

 1 √ √ λk 2 − 21 − 21 ν∥∆∥ ⩽ C2 λm+1 ∥∆∥ 8M1 + C2 2 8M1 ∥η∥λk+1 1 + log λ1   21 √ λk − 12 + C2 8M1 ∥η∥λk+1 1 + log . λ1 Then (5.8.25) infers that ∥∆∥ ⩽

 2(2C5 + C2 ) √ λk  12 − 12 8M1 ∥η∥λk+1 1 + log . ν λ1

Combining (5.8.37) and (5.8.42), we get (5.8.40).

(5.8.42)

296

Chapter 5

Infinite Dimensional Dynamical Systems

Theorem 5.8.8 Assume m is sufficiently large enough, such that (5.8.25) holds. Given k ⩾ m + 1, for all p ∈ B, define Tp : B ⊥ → B ⊥ , the same as Theorem 5.8.2 Tp (q) = (νA)−1 [Qm f − Qm B(p + q, p + q)],

∀q ∈ B.

Let Φs,k 0 (p) = Pk Tp (0), Φs,k m+1 (p)

= Pk Tp (Φs,k n (p)),

∀p ∈ B,

(5.8.43)

∀p ∈ B; n = 0, 1, 2, · · · .

Then

 i  h 1 − 21 n+1 − 12 ∥Φs,k (p) − Φs,k λm+1 ν −1 |f | + 4C5 M12 L 2 . n (p)∥ ⩽ 2 r2 λk+1

(5.8.44)

Furthermore, for the solution u(t) = p(t) + q(t) of the problems (5.8.10), (5.8.11), we have n+1 h i 1 2K0′ − 32 1  − 12 − 12 ∥q(t) − Φs,k λm+1 ν −1 |f | + 4C5 M12 L 2 λm+1 L 2 + 2r2 λm+1 n (p(t))∥ ⩽ ν −1

2 + K2 λk+1 ,

∀t ⩾ T∗ ,

n = 0, 1, 2, · · · , (5.8.45)

where T∗ , L and K0′ are the same as the one in the Theorem 5.8.1, r2 is the same as the one in Theorem 5.8.2, K3 is same as defined in Lemma 5.8.7. Proof In order to get (5.8.44), we repeat the proof in Theorem 5.8.4, where Φ will be replaced by Φs,k . Then the estimation (5.8.45) is a corollary of (5.8.29), (5.8.40) and (5.8.44). s

5.9

Nonlinear Galerkin method

The general numerical computation on chaos, inertial manifold and global attractor etc. with t → ∞ is improper, because of the deviation in the form of c(h)eT , where c(h) is a small constant depending on h. When T → ∞, the estimate of deviation may be very large. Theoretically, one must construct the estimate of deviation and convergence of approximate solution while t → ∞. A newly numerical method, socalled nonlinear Galerkin method, come to be appeared, and it possesses enormous advantages in the long time numerical computation. Using the nonlinear Galerkin method, some results of nonlinear evolution equations can refer to [107, 108, 130, 170, 192, 193, 229] and the references therein. In this section, we will take a class of nonlinear evolution equations for example to explain this method. Suppose the nonlinear evolution equation has the following form: du = −νAu − R(u), dt

(5.9.1)

5.9

Nonlinear Galerkin method

297

where R(u) = B(u) + Cu − f,

(5.9.2)

where ν > 0 is a viscosity parameter, A is a linear, unbounded, self-joint operator in Hilbert space and it is positive and closed, D(A) is dense in H. Define the power of A as As , s ∈ R, D(As ) has the norm in the form of |As · |, which is a Hilbert space. 1 1 Define V = D(A 2 ), ∥ · ∥ = |A 2 · |. Since A−1 is compact and self-adjoint, there exists an orthogonal basis {wj } of H, which is constructed by the eigenvectors of A: (5.9.3)

Awj = λj wj ,

where 0 < λ1 ⩽ λ2 ⩽ · · · , λj → ∞, j → ∞. The nonlinear term R(u) satisfies (5.9.2), where B(u) = B(u, u), B(·, ·) is a bilinear operator from V × V to V ′ . C is a linear operator from V to H, f ∈ H. Let b denote the trilinear form in V b(u, v, w) = (B(u, v), w)V ′ ,V ,

∀ u, v, w ∈ V.

Assume that b(u, v, w) = −b(u, w, v), 1 2

1 2

∀ u, v, w ∈ V,

1 2

1 2

|b(u, v, w)| ⩽ C1 |u| · ∥u∥ ∥v∥ · |w| · ∥w∥ , |Cu| ⩽ C2 ∥u∥,

∀ u, v, w ∈ V,

∀ u ∈ V,

(5.9.4) (5.9.5) (5.9.6)

where C1 , C2 and Ci (i > 2) are positive constants. Moreover, assume the mapping B is from V × D(A) to B, and 1

1

1

1

|B(u, v)| ⩽ C3 |u| 2 · ∥u∥ 2 · ∥v∥ 2 · |Av| 2 , 1

1

|B(u, v)| ⩽ C4 |u| 2 · |Au| 2 · ∥v∥,

(5.9.7)

∀ u ∈ V, v ∈ D(A).

(5.9.8)

∀ u ∈ D(A),

(5.9.9)

Finally, assume νA + C is positive, i.e., ((νA + C)u, u) ⩾ α∥u∥2 ,

where α > 0. For (5.9.1), consider the Cauchy problem with initial data u(0) = u0 ,

∀u0 ∈ H.

(5.9.10)

One can prove that the problem (5.9.1), (5.9.10) has a unique solution u(t), t > 0, and u ∈ C (R+ ; H) ∩ L2 (0, T ; V ), ∀ T > 0.

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Furthermore, if u0 ∈ V , then r ∈ C 1 (R+ ; H) ∩ L2 (0, T ; D(A)),

∀ T > 0.

Now, we are looking for the approximate solution of the problem (5.9.1), (5.9.10) in the following form m X um (t) = gjm (t)wj , j=1

um : R → wm = span{w1 , · · · , wn }. +

The function um could be solved as well as the unknown function zm , where zm (t) =

2m X

hjm (t)wj ,

j=m+1

zm : R + → w ˜m = span{wm+1 , · · · , w2m }. (um , zm ) satisfies d (um , v) + ν((um , v)) + (Cum , v) + b(um , um , v) + b(zm , um , v) dt + b(um , zm , v) = (f, v), ∀ v ∈ wm , ν((zm , v˜)) + (Czm , v˜) + b(um , um , v˜) = (f, v˜),

∀ v˜ ∈ w ˜m ,

(5.9.11)

(5.9.12)

with um (0) = Pm u0 ,

(5.9.13)

where Pm is the orthogonal projection of H on wm . The system (5.9.11), (5.9.12) is equivalent to an ordinary differential equation of um . In fact, (5.9.12) is linear with respect to zm , which could be written as νAzm + (P2m − Pm )Czm = (P2m − Pm )(f − B(um )).

(5.9.14)

Due to the assumption (5.9.9), which guarantees that the operator νA+(P2m −Pm )C is coercive and invertible on w ˜m , then zm is explicitly solved zm = (νA + (P2m − Pm )C)−1 (P2m − Pm )(f − B(um )).

(5.9.15)

Hence, the system (5.9.11), (5.9.12) is equivalent to an ordinary differential equations dum + νAum + Pm (Cum + B(um ) + B(um , zm )) = Pm f, dt

(5.9.16)

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299

where zm is given in (5.9.15). Obviously, when zm = 0, it becomes the classical Galerkin method. The existence and uniqueness of the solution um (t) on maximum time interval [0, Tm ] to the problem (5.9.16), (5.9.13) could be obtained by the standard ordinary differential equation theory. According to the following a priori estimate, we know Tm = +∞. Therefore, we consider the convergence of the approximate solution when m → ∞. Theorem 5.9.1 Assume (5.9.4), (5.9.9) are satisfied, given u0 in H, then the solution um of the problem (5.9.16), (5.9.13) converges to the solution u of the problem (5.9.1), (5.9.10) when m → ∞, i.e. um → u converges strongly in L2 (0, T ; V ) and Lp (0, T ; H), for all T > 0, 1 ⩽ p < +∞, um → u converges weakly ∗ in L∞ (R+ ; H). (5.9.17) Proof To prove Theorem 5.9.1, we will start with the a priori estimates for the solution of (5.9.11), (5.9.12). Taking v = um in (5.9.11), v˜ = zm in (5.9.12), then adding this two identities together, which get along with (5.9.14) give rise to 1 d |um |2 + ν∥um ∥2 + (Cum , um ) + ν∥zm ∥2 + (Czm , zm ) = (f, um + zm ). (5.9.18) 2 dt By (5.9.9), it is deduced that 1 d |um |2 + α(∥um ∥2 + ∥zm ∥2 ) ⩽ |f | · |um + zm |. 2 dt

(5.9.19)

Owing to 1

1

∥v∥ = |A 2 v| ⩾ λ12 |v|,

∀ v ∈ V,

(5.9.20)

it is derived from (5.9.19) that 1 d −1 |um |2 + α(∥um ∥2 + ∥zm ∥2 ) ⩽ λ1 2 |f | · ∥um + zm ∥ 2 dt α 1 ⩽ (∥um ∥2 + ∥zm ∥2 ) + |f |2 , 2 αλ1 d 2 |um |2 + αλ1 |um |2 ⩽ |f |2 . dt αλ1 Integrating the above inequality infers |um (t)|2 ⩽ |um (0)|2 e−αλ1 t +

2|f |2 (1 − e−αλ1 t ), λ21 α2

(5.9.21)

∀ t ⩾ 0.

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Hence, when m → +∞, the sequence um is bounded in L∞ (R+ ; H).

(5.9.22)

Back to (5.9.21). Taking integral for t on [0, T ] gives ∀ T > 0, when m → ∞, um and zm are bounded in L2 (0, T ; V ).

(5.9.23)

Now, consider the estimate of zm . Taking v˜ = zm in (5.9.12) implies ν∥zm ∥2 + (Czm , zm ) = −b(um , um , zm ) + (f, zm ). From (5.9.9) and (5.9.7), it is inferred that α∥zm ∥2 ⩽ |B(um )| · |zm | + |f | · |zm | 1

1

⩽ C3 |um | 2 ∥um ∥ · |Aum | 2 · |zm | + |f | · |zm |.

(5.9.24)

By um ∈ wm , zm ∈ w ˜m , there holds 1

1

2 |Aum | ⩽ λm ∥um ∥,

2 ∥um ∥ ⩽ λm |um |,

1

1

2 |Azm | ⩾ λm+1 ∥zm ∥,

2 ∥zm ∥ ⩽ λm+1 |zm |.

(5.9.25) (5.9.26)

Combining these inequalities, it is derived by (5.9.24) that αλm+1 |zm |2 ⩽ C3 λm |um |2 · |zm | + |f | · |zm |, αλm+1 |zm | ⩽ C3 λm |um |2 + |f |.

(5.9.27)

From (5.9.22), we have when m → +∞, zm is bounded in L∞ (R+ ; H).

(5.9.28)

By (5.9.25) and (5.9.26), it is derived from (5.9.24) that 1

2 |um | · ∥um ∥ + |f |. αλm+1 |zm | ⩽ C3 λm

Since λ1 ⩽ λm ⩽ λm+1 , then − 21

1

2 αλm+1 |zm | ⩽ C3 |um | · ∥um ∥ + λ1

· |f |.

By the above inequality and (5.9.22), (5.9.23), we know 1

2 ∀ T > 0, λm+1 zm is bounded in L2 (0, T ; H), when m → +∞.

Now, consider the estimation of

(5.9.29)

dum . By (5.9.4), (5.9.5), it is deduced that dt 1

1

1

1

∥B(u, v)∥V ′ ⩽ C1 |u| 2 ∥u∥ 2 |v| 2 ∥v∥ 2 ,

∀ u, v ∈ V.

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301

Therefore, by the esitmate of (5.9.22), (5.9.23) and (5.9.28), it is derived that B(um ), B(zm , um ) and B(um , zm ) are bounded in L2 (0, T ; V ′ ). Hence, from the differential equation (5.9.14), there holds ∀ T > 0,

dum is bounded in L2 (0, T ; V ′ ), when m → ∞. dt

(5.9.30)

Attention is turned to the convergence of the approximate solution um when m → ∞. Due to λm → +∞ when m → ∞, (5.9.29) implies ∀ T > 0, zm → 0 converges strongly in L2 (0, T ; H), when m → +∞.

(5.9.31)

Hence, from (5.9.23) and (5.9.28), there holds ∀ T > 0, zm * 0 converges weakly in L2 (0, T ; V ); converges weakly ∗ in L∞ (R+ , H).

(5.9.32)

Now, consider the convergence of the sequence um . The estimation of (5.9.22), (5.9.23) and (5.9.30) infers that there exists u∗ and subsequence m′ → +∞, such that um′ * u∗ converges weakly in L2 (0, T ; V ), ∀ T > 0; converges weakly ∗ in L∞ (R+ , H), when m′ → +∞, du∗ dum * converges weakly in L2 (0, T ; V ′ ), ∀ T > 0, when m′ → +∞. dt dt (5.9.33) By the classical compactness principle, (5.9.33) infers ∀ T > 0, um′ → u∗ converges strongly in L2 (0, T ; H), when m′ → +∞.

(5.9.34)

By (5.9.31)-(5.9.34), we can take the limit for (5.9.11), where the only trouble is the bilinear term. Let v ∈ wm be fixed, and m′ ⩾ m. From (5.9.4), it is known that b(um′ , um′ , v) = −b(um′ , v, um′ ). Based on (5.9.7) and the fact that b(·, v, ·) is a bilinear continuous form from V × H to R, it is derived from (5.9.33), (5.9.34) that b(um′ , v, um′ ) → b(u∗ , v, u∗ ) converges strongly in L′ (0, T ), ∀ T > 0, m′ → +∞. Hence, b(um′ , um′ , v) → b(u∗ , u∗ , v) converges strongly in L′ (0, T ), ∀ T > 0, m′ → +∞.

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Similarly, there holds b(zm′ , um′ , v) → b(0, u∗ , v) = 0,

m′ → +∞,

b(um′ , zm′ , v) → b(u∗ , 0, v) = 0,

m′ → +∞,

strongly in L′ (0, T ), ∀ T > 0. Hence, we get the limit function u∗ , satisfying d ∗ (u , v) + ν((u∗ , v)) + (Cu∗ , v) + b(u∗ , u∗ , v) = (f, v), dt

(5.9.35)

for all v ∈ wm . By the continuity, it is true ∀ v ∈ V . Furthermore, (5.9.33) implies that um′ (0) * u∗ (0) converges weakly in H.

(5.9.36)

Owing to um′ (0) = Pm′ u0 , it is derived from (5.9.36) that u∗ (0) = u0 .

(5.9.37)

By (5.9.35), (5.9.36), it is known that u∗ is a solution of problem (5.9.1), (5.9.10), therefore, u∗ = u. According to (5.9.33), the sequence um converges to u. In order to finish the proof of Theorem 5.9.1, one must verify the strong convergence in (5.9.17). Hence, we introduce Xm

1 = |um (T ) − u(T )|2 + 2

Z

T

{ ν∥um − u∥2 + (C(um − u), um − u) 0

+ ν∥zm ∥2 + (Czm , zm )} dt. To prove the theorem, it is sufficient to show (5.9.38)

lim Xm = 0.

m→+∞

In fact, by (5.9.9), (5.9.38), it is known that um converges to u in L2 (0, T ; V ), and um (t) → u(t) converges strongly in H, ∀ t ⩾ 0.

(5.9.39)

Taking advantage of (5.9.39), (5.9.22) and Lebesgue dominated convergence theorem implies that um converges strongly to u in Lp (0, T ; H)(∀ p ∈ [1, ∞)). In addition, notice that except for the strong convergence result for um in (5.9.17), (5.9.38) infers zm → 0 converges strongly in L2 (0, T ; V ),

∀ t ⩾ 0, m → +∞.

(5.9.40)

Now, we are going to prove (5.9.38). Integrating (5.9.18) from 0 to T shows

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Nonlinear Galerkin method

303

Z T 1 |um (T )|2 + {ν∥um ∥2 + (Cum , um ) + ν∥zm ∥2 + (Czm , zm )} dt 2 0 Z T 1 2 = |um (0)| + (f, um + zm ) dt. 2 0 Xm can be rewritten as 1 1 Xm = − (um (T ), u(T )) + |u(T )|2 + |um (0)|2 + 2 2

Z

T



− 2ν((um , u)) + ν∥u∥2 (−Cu, um − u) − (Cum , u) + (f, um + zm ) dt. (5.9.41) 0

Using (5.9.31), (5.9.33) and taking limit for (5.9.41), it is deduced that 1 1 lim Xm = − |u(T )|2 + |u0 |2 + m→+∞ 2 2

Z

T



− ν∥u∥2 − (Cu, u) + (f, u) dt.

0

Replacing u, u by u∗ , v in (5.9.35), the limit mentioned above is 0, i.e. (5.9.38) is obtained, which proves the theorem. In order to improve the convergence of the nonlinear Galerkin method under a strong topologic, we prove the following theorem. Theorem 5.9.2 Assume (5.9.4)-(5.9.9) are satisfied, given u0 ∈ V, the solution um of the problem (5.9.16), (5.9.13) converges to the solution of the problem (5.9.1), (5.9.10) when m → +∞ : um → u converges strongly in L2 (0, T ; D(A)) and Lp (0, T ; V ), T > 0, 1 ⩽ p < ∞, ∞

(5.9.42)

um → u converges weakly ∗ in L (R ; V ). +

Proof The detailed proof depends on the a priori estimates of um and zm . Firstly, show that um ∈ L∞ (R+ ; V ). Let v = Aum in (5.9.11), v˜ = Azm in (5.9.12). Adding the corresponding terms together implies that 1 d ∥um ∥2 + ν|Aum |2 + ν|Azm |2 2 dt = (f, A(um + zm )) − (Cum , Aum ) − (Czm , Azm ) − b(um , um , Aum ) − b(zm , um , Aum ) − b(um , zm , Aum ) − b(um , um , Azm ).

(5.9.43)

To estimate the right-hand side of (5.9.43), the following result holds: |(f, A(um + zm ))| ⩽

ν 6 (|Aum |2 + |Azm |2 ) + |f |2 . 12 ν

(5.9.44)

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Chapter 5

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From (5.9.6), it is deduced that |(Cum , Aum )| ⩽ C2 ∥um ∥ · |Aum | ⩽

ν 3C22 |Aum |2 + ∥um ∥2 , 12 ν

(5.9.45)

|(Czm , Azm )| ⩽ C2 ∥zm ∥ · |Azm | ⩽

3C22 ν |Azm |2 + ∥zm ∥2 . 12 ν

(5.9.46)

To estimate the bound of trilinear term by (5.9.7) gives |b(um , um , Aum )| ⩽ |B(um , um )| · |Aum | 1

3

⩽ C3 |um | 2 · ∥um ∥ · |Aum | 2 C5 ν |Aum |2 + |um |2 · ∥um ∥4 , ⩽ 12 ν

(5.9.47)

where C5 is an absolutely constant. Similarly, |b(zm , um , Aum )| ⩽

ν C5 |Aum |2 + |zm |2 ∥zm ∥2 · ∥um ∥2 , 12 ν 1

1

1

1

|b(um , zm , Aum )| ⩽ C3 |um | 2 ∥um ∥ 2 ∥zm ∥ 2 |Azm | 2 · |Aum | ν C6 ⩽ |Aum |2 + |um | · ∥um ∥ · ∥zm ∥ · |Azm | 12 ν ν ν C7 ⩽ |Aum |2 + |Azm |2 + 2 |um |2 ∥um ∥2 ∥zm ∥2 . 12 12 ν

(5.9.48)

The last term on the right-hand side of (5.9.48) can be estimated as follows 1

1

|b(um , zm , Aum )| ⩽ C3 |um | 2 · ∥um ∥ · |Aum | 2 · |Azm | ν ν C7 |Aum |2 + |Azm |2 + 2 |um |2 · ∥um ∥4 . ⩽ 12 12 ν

(5.9.49)

Substituting all the inequalities into (5.9.43) infers d ∥um ∥2 + ν|Aum |2 + ν|Azm |2 dt 12 2 6C22 ⩽ |f | + ∥zm ∥2 + C8 ∥um ∥2 ν ν · (1 + |um |2 ∥um ∥2 + |zm |2 ∥zm ∥2 + |um |2 ∥zm ∥2 ),

(5.9.50)

where C8 = C8 (ν) depends on ν. (5.9.50) can be written as the following differential inequality dym ⩽ gm ym + hm , (5.9.51) dt

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Nonlinear Galerkin method

305

where ym (t) = ∥um (t)∥2 , 12 2 6C22 |f | + ∥zm ∥2 , ν ν gm (t) = C8 (1 + |um |2 ∥um (t)∥2 + |zm (t)|2 ∥zm ∥2 + |um (t)|2 ∥zm (t)∥2 ).

hm (t) =

Taking the integral with (5.9.51) infers Z t  Z t Z t  ym (t) ⩽ ym (0) exp gm (s) ds + hm (s) exp gm (σ) dσ ds, 0

0

(5.9.52)

∀ t ⩾ 0,

s

(5.9.53) which get along with (5.9.22), (5.9.23) and (5.9.28) gives that um is bounded in L∞ (0, T ; V ) for all T > 0. Moreover, the boundedness of ∥um ∥2 on R+ can be obtained by the following uniformly Galerkin inequality: Lemma 5.9.3 Assume that g(t), h(t) and y(t) are locally integrable, positive functions on (t0 , +∞) and satisfy dy dy ∈ L1loc ([t0 , ∞]), ⩽ gy + h, ∀ t ⩾ t0 . dt dt Z t+1 Z t+1 Z t+1 g(s) ds ⩽ a1 , h(s) ds ⩽ a2 , y(s) ds ⩽ a3 , t

t

(5.9.54) t ⩾ t0 ,

t

where ai (i = 1, 2, 3) are positive constants, then y(t) ⩽ (a3 + a2 ) exp(a1 ),

∀ t ⩾ t0 + 1.

(5.9.55)

Back to (5.9.51), it is discovered that the assumptions in Lemma 5.9.3 hold due to the a priori estimate. Since um , zm are bounded in L∞ (R+ ; H), taking the integral of (5.9.21) with respect to t from t to t + 1 infers Z t+1 (∥um ∥2 + ∥zm ∥2 ) ds ⩽ C8′ , (5.9.56) t

C8′

where is independent of m. Therefore, in (5.9.52), ym , gm and hm satisfy (5.9.54), where a1 , a2 and a3 are independent of m. Hence, (5.9.55) yields ym (t) = ∥um (t)∥2 ⩽ C9 ,

∀ t ⩾ 1,

(5.9.57)

where C9 = C9 (ν) is independent of m. In consequence, (5.9.57) gives that when t ⩾ 1, ∥um (t)∥ is uniformly bounded, and (5.9.53) implies that ∥um (t)∥ is uniformly bounded for t ∈ [0, 1]. Hence, um (t) is uniformly bounded in L∞ (R+ ; H),

m → +∞.

(5.9.58)

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Taking intergal for (5.9.50) infers that ∀ T > 0, um and zm are bounded in L2 (0, T ; D(A)), m → +∞.

(5.9.59)

Next, the estimate on the sequence zm is given, hopefully one can have the similar result to (5.9.59). Taking v˜ = Azm in (5.9.12) implies that ν|Azm |2 = − (Czm , Azm ) − b(um , um , Azm ) + (f, Azm ) 1

1

⩽ C2 ∥zm ∥ · |Azm | + C3 |um | 2 ∥um ∥ · |Aum | 2 |Azm | + |f | · |Azm |, 1

1

ν|Azm | ⩽ C2 ∥zm ∥ + C3 |um | 2 ∥um ∥|Aum | 2 + |f |. Then, by (5.9.25), (5.9.26), it is deduced that 1

1

1

3

2 4 |um | 2 ∥um ∥ 2 + |f |. νλm+1 ∥zm ∥ ⩽ C2 ∥zm ∥ + C3 λm

For large m, there holds ∥zm ∥ ⩽

1 1 2

νλm+1 − C2

1

1

3

4 |um | 2 · ∥um ∥ 2 + |f |). (C3 λm

Combining this inequality with (5.9.58) implies that zm → 0 converges strongly in L∞ (R+ ; V ), m → +∞.

(5.9.60)

dum . From (5.9.7) and (5.9.58)-(5.9.60), it is obdt tained that B(um ), B(zm , um ) and B(um , zm ) are uniformly bounded in L4 (0, T ; H) with respect to m, and Aum is uniformly bounded in L2 (0, T ; H). Hence, (5.9.16) infers that dum ∀ T > 0, is bounded in L2 (0, T ; H), m → +∞. (5.9.61) dt The convergence of (5.9.42) is based on the estimation of (5.9.58)-(5.9.61). Firstly, combining the previous convergent results (5.9.17), when m → +∞, we have Finally, we need to estimate

um * u converges weakly in L2 (0, T ; D(A)), ∀ T > 0, um * u converges weakly ∗ in L∞ (R+ ; V ), du dum * converges weakly in L2 (0, T ; H), ∀ T > 0, dt dt zm * 0 converges weakly in L2 (0, T ; D(A)), ∀ T > 0.

(5.9.62) (5.9.63) (5.9.64) (5.9.65)

These infer the weak convergence of (5.9.47). In order to have the strong convergence of (5.9.42), set Z T 1 2 (|Aum − Au|2 + |Azm |2 ) dt. Ym = ∥um (T ) − u(T )∥ + ν 2 0

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307

It is sufficient to show that lim Ym = 0.

m→∞

In fact, the strong convergence in Lp (0, T ; V ) is based on the estimation of (5.9.58) and the Lebesgue dominated convergence theorem. Integrating (5.9.43) from 0 to T shows Z T 1 1 2 ∥um (T )∥ + ν (|Aum |2 + |Azm |2 ) ds = zm + ∥u0m ∥2 , 2 2 0 Z

T

{(f, A(um + zm )) − (Cum , Aum ) − (Czm , Azm )

zm = 0

− b(um , um , Aum ) − b(zm , um , Aum ) − b(um , zm , Aum ) − b(um , um , Azm )} ds.

(5.9.66)

Hence, Ym can be rewritten in the following form Ym

1 1 = −((um (T ), u(T )))+ ∥u(T )∥2 + ∥u0m ∥2 +ν 2 2

Z

T

(−2(Aum , Au)+|Au|2 ) ds+zm . 0

In addition, from (5.9.62) and (5.9.64), it is deduced that Z lim {−(um (T ), u(T ))} − 2ν

m→+∞

Z

T

(Aum , Au) ds = −∥u(T )∥2 − 2ν 0

T

|Au|2 ds. 0

Secondly, taking advantage of the weak convergence of um and zm in L2 (0, T ; D(A)), their strong convergence in L2 (0, T ; V ) and the boundedness in L∞ (0, T ; V ), one has lim zm . Especially, in (5.9.66), the method to find the limits of different trilinear m→+∞

terms are similar. Here, only the case for the first term is considered. We know that Z

Z

T

T

b(um , um , Aum ) dt − 0

Z

b(u, u, Au) dt 0

T

b(um − u, um , Aum ) ds

= 0

Z

T

b(u, um − u, Aum ) ds

+ 0

Z

T

b(u, u, A(um − u)) ds.

+

(5.9.67)

0

For the first term on the right-hand side of (5.9.67), from (5.9.8), (5.9.58) and Hölder’s inequality, it is deduced that

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Chapter 5

Z

T

Z b(um − u, um , Aum ) ds ⩽ C4

0

Z

T

1

0 T

1

1

|um − u| 2 · |A(um − u)| 2 · |Aum | ds 0

Z Z

·

1

|um − u| 2 ∥um ∥ · |A(um − u)| 2 · |Aum | ds

⩽C ⩽C

Infinite Dimensional Dynamical Systems

 14  Z |um − u| ds

T

T

2

0 T

 41 |A(um − u)|2 ds

0

 |Aum |2 ds .

0

By (5.9.17) and (5.9.59), we know that these terms converge to 0 when m → ∞. For the second term, using (5.9.7) gives that Z

T

Z b(u, um − u, Aum ) ds ⩽ C3

0

Z

T

1

1

1

1

∥um − u∥ 2 · |A(um − u)| 2 · |Aum | ds 0

·

1

0 T

⩽C ⩽C

1

|u| 2 ∥u∥ 2 ∥um − u∥ 2 · |A(um − u)| 2 · |Aum | ds

Z Z

T

0 T

 14  Z ∥um − u∥ ds 2

 21 |Aum |2 ds .

T

 14 |A(um − u)|2 ds

0

0

According to (5.9.17) and (5.9.59), these terms tend to 0 when m → +∞. Finally, the last term in (5.9.67) is linear with respect to um . From (5.9.62), it is easy to know that it converges to 0 when m → +∞. Hence, we prove that Z

Z

T

b(um , um , Aum ) ds → 0

T

b(u, u, Au) ds,

m → +∞.

0

The proofs of the rest terms of (5.9.66) are similar. Finally, we obtain that lim Ym

m→+∞

Z T 1 1 2 2 |Au|2 ds = − ∥u(T )∥ + ∥u0 ∥ − v 2 2 0 Z T + {(f, Au) − (Cu, Au) − b(u, u, Au)} ds 0

= 0. This proves the strong convergence of (5.9.42), which then demonstrates Theorem 5.9.2. Now, consider another nonlinear Galerkin method for the problem (5.9.1) and (5.9.10). Given u0 ∈ V , take the eigenvectors of the operator A as the basis of V . Assume

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309

the approximate solution um has the following form um (t) =

m X

gjm (t)wj ,

j=1

um : R+ → span{w1 , · · · , wm } = Wm . Introduce the unknown function zm , zm (t) =

2m X

hjm (t)wj ,

j=m+1

zm : R + → w ˜m = span{wm+1 , · · · , w2m }. Let um , zm satisfy the following system: d (um , v) + ν((um , v)) + (Cum , v) dt + b(um , um , v) + b(zm , um , v) + b(um , zm , v) + b(zm , zm , v) = (f, v),

∀ v ∈ Wm ,

(5.9.68)

∀ v˜ ∈ w ˜m ,

(5.9.69)

ν((zm , v˜)) + (Czm , v˜) + b(um , um , v˜) + b(zm , um , v˜) + b(um , zm , v˜) = (f, v˜),

(5.9.70)

um (0) = Pm u0 . (5.9.69) can be written as νAzm + (P2m − Pm )Czm + (P2m − Pm )(B(zm , um ) + B(um , zm )) = (P2m − Pm )(f − B(um )).

(5.9.71)

Denote the linear operator of zm on the left-hand side of (5.9.69) by D(um ). To show that the solution {um , zm } of the problem (5.9.68)-(5.9.70) exists in a small region, we must prove that D(um ) is invertible on w ˜m . We get (D(um )˜ v , v˜) = ν∥˜ v ∥2 + (C v˜, v˜) + b(˜ v , um , v˜) ⩾ α∥˜ v ∥2 − C1 |˜ v | · ∥˜ v ∥ · ∥um ∥ −1

2 )∥um ∥. ⩾ ∥˜ v ∥2 (α − C1 λm+1

(5.9.72)

Choose m sufficiently large enough, such that −1

2 α − C1 λm+1 ∥u0 ∥ ⩾

α , 2

(5.9.73)

310

Chapter 5

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then, by the existence theorem of ordinary differential equation, system (5.9.68)(5.9.70) has the solution (um (t), zm (t)), t ∈ [0, Tm ]. On this interval, system (5.9.68), (5.9.69) is equivalent to the ordinary differential system of um dum + νAum + Pm (Cum + B(um + zm )) = Pm f, dt

(5.9.74)

zm = D(um )−1 {(P2m − Pm )(f − B(um ))}. The condition (5.9.73) is satisfied. When m → ∞, λm → ∞, we will show that (at least m is sufficiently large enough) Tm = +∞ in the following, i.e. the solution of (5.9.74) is defined in R+ . Furthermore, we will prove that the solution of (5.9.74) tends to the solution of (5.9.1) when m → +∞. We have the following theorem. Theorem 5.9.4 Assume (5.9.4)-(5.9.9) hold. Given u0 in V, then (i) there exists a constant k = k(u0 ), such that if m satisfies −1

2 ⩾ α − C1 kλm+1

α , 2

(5.9.75)

then the system (5.9.74), (5.9.70) has the solution um defined on R+ ; (ii) the solution um of (5.9.74), (5.9.70) converges to the solution u of (5.9.1), (5.9.10) when m → +∞ : um → u converges strongly in L2 (0, T ; D(A)) and Lp (0, T ; V ), ∀ T > 0, 1 ⩽ p < +∞, and converges weakly ∗ in L∞ (R+ ; V ).

(5.9.76)

Proof The constant k in (5.9.75) will be determined later. Now assume (5.9.73) is true, then (5.9.68)-(5.9.70) has the solution {um , zm } on the interval (0, Tm ). We obtain some priori estimates of {um , zm } on (0, Tm ), which are similar to Theorem 5.9.1 and Theorem 5.9.2. (i) A priori estimate (I). Let v = um in (5.9.68), taking v˜ = zm in (5.9.69), adding these two identities, which get along with (5.9.4) infers 1 d |um |2 + ν∥um ∥2 + (Cum , um ) + ν∥zm ∥2 + (Czm , zm ) = (f, um + zm ). (5.9.77) 2 dt Hence, similar to (5.9.22), (5.9.23) derived from (5.9.18), there holds um is uniformly bounded in L∞ (0, T ; H) with respect to m,

(5.9.78)

um , zm are uniformly bounded in L2 (0, T ; H) with respect to m, ∀ T ∈ (0, Tm ); if Tm < +∞, T = Tm .

(5.9.79)

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311

(ii) A priori estimate (II). Taking v = Aum in (5.9.68), v˜ = Azm in (5.9.69), adding the corresponding two identities, we get 1 d ∥um ∥2 + ν|Aum |2 + ν|Azm |2 = (f, A(um + zm )) − (Cum , Aum ) − (Czm , Azm ) 2 dt − b(um , um , Aum ) − b(zm , um , Aum ) − b(um , zm , Aum ) − b(zm , zm , Aum ) − b(um , um , Azm ) − b(zm , um , Azm ) − b(um , zm , Azm ).

(5.9.80)

For the right-hand side of (5.9.80), it has already been proved that some terms are bounded. By (5.9.7), (5.9.25) and (5.9.26), it is deduced that 1

1

1

3

|b(zm , um , Aum )| ⩽ C3 |zm | 2 · ∥zm ∥ 2 ∥um ∥ 2 |Aum | 2  λ  43 m ⩽ C3 · ∥um ∥2 · |Azm | λm+1 ⩽ C3 ∥um ∥2 · |Azm | ν C10 ⩽ |Azm |2 + ∥um ∥4 , 24 ν 1

(5.9.81)

1

|b(zm , zm , Aum )| ⩽ C3 |zm | 2 · ∥zm ∥ · |Azm | 2 · |Aum |  λ  43 m ⩽ C3 · ∥um ∥2 · |Azm | λm+1 ⩽ C3 ∥um ∥2 · |Azm | C10 ν |Azm |2 + ∥um ∥2 · ∥zm ∥2 , ⩽ 24 ν 1

1

1

(5.9.82)

1

|b(zm , um , Azm )| ⩽ C3 |zm | 2 · ∥zm ∥ 2 · ∥um ∥ 2 · |Aum | 2 · |Azm |  λ  14 m ⩽ C3 · ∥um ∥ · ∥zm ∥ · |Azm | λm+1 ν C10 ⩽ |Azm |2 + ∥um ∥2 · ∥zm ∥2 . 24 ν

(5.9.83)

Finally, the last term on the right-hand side of (5.9.80) possesses the following estimate 1

1

1

3

|b(um , zm , Azm )| ⩽ C3 |um | 2 · ∥um ∥ 2 · ∥zm ∥ 2 · |Azm ∥ 2 ν C11 ⩽ |Azm |2 + |um |2 · ∥um ∥2 · ∥zm ∥2 . 24 ν

312

Chapter 5

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Combining (5.9.44)-(5.9.49), (5.9.81)-(5.9.83) gives the following differential inequality  d 12 2 6C22 ∥um ∥2 + ν|Aum |2 + ν|Azm | ⩽ |f | + ∥zm ∥2 + C12 ∥um ∥2 · 1 + ∥um ∥2 dt ν ν  + |um |2 · ∥um ∥2 + ∥zm ∥2 + |um | · ∥zm ∥2 ,

(5.9.84) where C12 = C12 (ν) depends on ν. (5.9.84) implies that d 12 2 6C22 ∥um ∥2 ⩽ |f | + ∥zm ∥2 + C12 ∥um ∥2 dt ν ν  · 1 + ∥um ∥2 + |um |2 · ∥um ∥2 + ∥zm ∥2 + |um | · ∥zm ∥2 .

(5.9.85)

Integrating (5.9.85) from 0 to T , which get along with (5.9.78), (5.9.79) gives rise to um is uniformly bounded in L∞ (0, T ; V ) with respect to m, 0 < T < Tm . If Tm < +∞, T = Tm .

(5.9.86)

Therefore, if Tm < +∞, (5.9.85) provides the upper bound for ∥um (t)∥, 0 ⩽ t ⩽ Tm . Applying similar approach, one may get the upper bound for ∥um (t)∥ when Tm = +∞. In fact, from (5.9.77), it is derived that Z t+1 Z t+1 ∥um ∥2 ds + ∥zm ∥2 ds ⩽ C13 , ∀ t ⩾ 0, (5.9.87) t

t

where C13 is independent of m. Then, (5.9.87), (5.9.78) imply that (5.9.85) satisfies the assumption in Lemma 5.9.3 (uniformly Grönwall lemma). (5.9.55) infers ∥um (t)∥, for t ⩾ 1, which is independent of m. Since (5.9.85) gives the bound of ∥um (t)∥ for 0 ⩽ t ⩽ 1, we get um is uniformly bounded in L∞ (0, Tm ; V ) with respect to m.

(5.9.88)

Integrating (5.9.84) with respect to t from 0 to T , one has um , zm are uniformly bounded in L2 (0, T ; D(A)) with respect to m. 0 < T < Tm , if Tm < +∞, T = Tm ,

(5.9.89)

(iii) A priori estimate (III). Taking v˜ = Azm in (5.9.69) implies ν|Azm |2 = − (Czm , Azm ) − b(um , um , Azm ) − b(zm , um , Azm ) − b(um , zm , Azm ) + (f, Azm ) 1

1

1

3

⩽ C2 ∥zm ∥|Azm | + C3 |um | 2 ∥um ∥|Aum | 2 · |Azm | + C4 |zm | 2 · |Aum | 2 · ∥um ∥ 1

1

1

3

+ C3 |um | 2 ∥um ∥ 2 ∥zm ∥ 2 |Azm | 2 + |f | · |Azm |.

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Nonlinear Galerkin method

313

Dividing by |Azm | for the above formula, which get along with (5.9.28), (5.9.29) and (5.9.88) gives rise to −1

−1

−1

1

2 4 2 4 |Azm | + C3 λm |Azm | + C3 λm+1 |Azm | + |f |, ν|Azm | = C2 λm+1 + C4 λm+1

Hence

−1

−1

−1

t ∈ [0, Tm ).

1

2 2 4 4 {ν − C2 λm+1 − C4 λm+1 − C3 λm+1 }|Azm | ⩽ C3 λm + |f |.

When m is sufficiently large enough, one has 1 ν 4 |Azm | ⩽ C3 λm + |f |. 2

(5.9.90)

1

2 Owing to |Azm | ⩾ λm+1 ∥zm ∥, it is derived from (5.9.90) that

zm * 0 converges weakly ∗ in L∞ (0, Tm ; V ),

m → +∞.

(5.9.91)

(iv) Take the limit. First of all, verify the solution um of (5.9.74) for sufficiently large m defined in R+ . In fact, it is known from (5.9.88) that there exists a constant k (independent of m), such that ∥um (t)∥ ⩽ k, Hence, if m satisfies

0 ⩽ t < Tm . −1

2 α − c1 kλm+1 ⩾

(5.9.92)

α , 2

then by (5.9.72), it is deduced that (D(um )˜ v , v˜) ⩾

α ∥˜ v ∥2 , 2

t ∈ [0, Tm ), v˜ ∈ w ˜m .

Therefore, the operator D(um ) is uniformly coercive on (0, Tm ). This implies that Tm = +∞, which proves (i) in Theorem 5.9.4. Assume (5.9.75) is true, where k is given in (5.9.92), then the estimates (5.9.89), (5.9.91) hold when Tm = +∞. These estimates are similar to (5.9.58)-(5.9.60). Therefore, we can take the limit m → +∞. Then, um converges to the solution u of the problem (5.9.1), (5.9.10) in the sense of (5.9.17), (5.9.42), which proves (5.9.76). In consequence, Theorem 5.9.4 has been demonstrated. Now, consider some numerical computational formats approximated by some nonlinear evolution equations. For discrete space, there are spectral method, pseudospectral method, finite element method and finite difference method. There are two formats with discrete-time: partial or full display. Assume Vh denotes the vector space with finite dimension, which has two kinds of dot products and their norms: ((·, ·))h , ∥·∥h ; (·, ·)h , |·|h . Vh is an approximation to the classical Sobolev space, ∥·∥h denotes discrete Sobolev norm, |·|h is discrete L2 norm,

314

Chapter 5

Infinite Dimensional Dynamical Systems

Vh ∈ {Vl }, l ∈ H , when h tends to 0, {Vl } approximates to infinite-dimensional space V . Suppose Ci are absolutely positive constants, which are independent of h, Si = Si (h) are constants depends on h, in general, when h → 0, it tends to 0. Assume |uh |h ⩽ C1 ∥uh ∥h , S1 (h)∥uh ∥h ⩽ |uh |h , ∀ uh ∈ Vh . (5.9.93) The bilinear continuous form ah (·, ·) in Vh satisfies |ah (uh , vh )| ⩽ C2 ∥uh ∥h ∥vh ∥h ,

∀ uh , vh ∈ Vh .

(5.9.94)

The trilinear continuous form bh (·, ·, ·) in Vh satisfies bh (uh , vh , vh ) = 0, 1

∀ uh , vh ∈ Vh . 1

1

(5.9.95) 1

|bh (uh , vh , wh )| ⩽ C3 |uh |h2 ∥uh ∥h2 ∥vh ∥h |wh |h2 ∥wh ∥h2 .

(5.9.96)

The bilinear continuous form dh (·, ·) in Vh satisfies |dh (uh , vh )| ⩽ C4 ∥uh ∥h · |vh |h ,

∀ uh , vh ∈ Vh ,

ah (uh , uh ) + dh (uh , uh ) ⩾ C5 ∥uh ∥2 ,

∀ uh ∈ Vh ,

(5.9.97) (5.9.98)

Now, consider the following initial value problem: Looking for function uh : R+ → Vh , such that d (uh , vh )h + ah (uh , vh ) + bh (uh , uh , vh ) + dh (uh , vh ) = (fh , vh ), dt uh (0) = u0h ,

∀ vh ∈ Vh , (5.9.99) (5.9.100)

where u0h is given in Vh , fh ∈ L∞ (R+ ; Vh ). Because of Vh has finite dimension, it is deduced from (5.9.94)-(5.9.98) that the initial value problem (5.9.99)-(5.9.100) has a unique solution uh uh ∈ L∞ (R+ ; Vh ) (5.9.101) and uh is uniformly bounded with respect to h. There are many equations with physical backgrounds can be converted into the form of (5.9.99). Conditions (5.9.94)-(5.9.98) are satisfied. Assume Vh = Vh2 ⊕ Wh ,

(5.9.102)

where Vh2 ⊂ Vh = Vh1 , the elements of Vh2 are yh , y˜h , · · · , and the elements of Wh are zh , zˆh , · · · . For any uh ∈ Vh , it can be uniquely expressed as uh = yh + z h ,

yh ∈ Vh2 , zh ∈ Wh .

(5.9.103)

5.9

Nonlinear Galerkin method

315

Now, consider the following assumption for the decomposition (5.9.102). |((yh , zh ))| ⩽ (1 − δ)∥yh ∥h ∥zh ∥h ,

∀ yh ∈ Vh2 ,

∀ zh ∈ Wh ,

(5.9.104)

where δ ∈ (0, 1), which is independent of h; and assume that |zh |h ⩽ S2 (h)∥zh ∥h ,

∀ zh ∈ W h ,

(5.9.105)

where S2 (h) → 0, when h → 0. Now, give three crucial cases for the decomposition form of (5.9.102). (i) Discrete spectral: Vh is a subspace of the Hilbert space V , its inner product is ((·, ·)), norm is ∥ · ∥, ah (uh , vh ) is a bilinear, continuous, symmetric coercive form on V restricted on Vh , V is continuously embedded and dense in another Hilbert space H, its inner product is (·, ·), and norm is | · |, then ∥uh ∥h = ∥uh ∥,

|uh |h = |uh |,

∀ uh ∈ Vh .

The unbounded, self-adjoint operator A, connecting with a, V, H, has an orthogonal basis in H and V wj , j ∈ N , D(A) ⊂ V . Awj = λj wj ,

0 < λ1 ⩽ λ2 ⩽ · · · ,

λj → ∞, j → ∞.

(5.9.106)

1 , Vh = span{w1 , · · · , wm1 }, (5.9.100) can be regarded m1 as a Galerkin approximation of infinite-dimensional problem in V . Consider the decomposition of (5.9.102). Assume there is another integer m2 ∈ N, m2 < m1 . Denote Given m = m1 ∈ N , h =

Vh2 = span{w1 , · · · , wm2 },

Wh = span{wm2 +1 , · · · , wm1 }.

Vh2 and Wh are orthogonal in Vh (dot product ((·, ·))h = ((·, ·))), (5.9.104) satisfies δ = 1. As to (5.9.105), for all zh ∈ Wh ,

zh =

m1 X

ξj wj ,

j=m2 +1

there holds |zh |2h = ∥

m1 X j=m2 +1

ξj wj ∥2 =

m1 X

|ξj |2 λj ⩾ λm2 +1

j=m2 +1

m1 X

|ξj |2 = λm2 +1 |zh |2h .

j=m2 +1

Hence, S2 = (λm2 +1 )− 2 . 1

(5.9.107)

When it is a continuous problem, a and V are connected with an elliptical boundary value problem, with boundary condition in space, (5.9.99) is the continuous problem for the spectral or pseudospectral approximation.

316

Chapter 5

Infinite Dimensional Dynamical Systems

(ii) Finite element. We only consider the simplest case: piecewise linear element in one dimensional. The space Vh is a subspace of V = H01 (Ω), Ω = (0, L), L > 0. Introduce the inner product Z L duh dvh ((uh , vh ))h = ((uh , vh )) = dx. dx dx 0 Dot product (·, ·)h in L2 (0, L), Z (uh , vh )h = (uh , vh ) =

L

uh vh dx. 0

1 1 , h2 = , N ∈ N, then Vh is a real, continuous function space 2N N on (0, L), it equals to 0 at 0 and L, and it is linear on the interval (jh, (j + 1)h], (j = 0, 1, · · · , 2N − 1). Define Vh = V2h with the same approach, but V2h is linear on the interval [2jh, 2(j + 1)h], j = 0, 1, · · · , N − 1. The base node of Vh is constructed by the functions wj,h of Vh . Denote h1 = h =

wj,h (jh) = 1;

i, j = 1, · · · , 2N − 1, i ̸= j.

wj,h (ih) = 0,

Similarly, the base node of V2h is grouped by the functions wj,2h of V2h , wj,2h (2jh) = 1; i, j = 1, · · · , N − 1, i ̸= j.

w2j,h (2ih) = 0,

The basis of Vh is combined by the basis of V2h and the basis of Wh : Wj,h , j = 2i+1, i = 0, 1, · · · , N − 1. i.e. uh ∈ Vh could be expanded as follows uh =

2N −1 X

uh (jh)wj,h ,

(5.9.108)

u ¯h ((2i + 1)h)w2i+1,h ,

(5.9.109)

j=1

and could be decomposed as uh =

N −1 X

uh (2jh)wj,2h +

j=1

N −1 X i=0

where u ¯h ((2i + 1)h) is the increment of uh , 1 u ¯h ((2i + 1)h) = uh ((2i + 1)h) − (uh (2ih) + uh (2i + 2)h). 2

(5.9.110)

Notice that the summation of the first part in (5.9.109) is corresponding to yh ∈ Vh2 ⊂ Vh , the summation of the second part is corresponding to zh ∈ Wh , if uh (jh) = u(jh), j = 0, 1, · · · , 2N , then u ¯h ((2i + 1)h) =

h2 ′′ u ((2i + 1)h) + o(h3 ), 2

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Nonlinear Galerkin method

317

yh and uh have the same order, but zh has a factor h2 . It is easy to verify that for any function yh ∈ Vh2 , it is orthogonal to any function zh ∈ Wh in V , hence, (5.9.104) holds, δ = 1. (5.9.105) is satisfied easily, we have √ S2 (h) = h/ 3.

(5.9.111)

(5.9.93) can be verified by a general method. For example, for S1 (h), denote ξi = uh (ih), Z L 2N −1 1 X duh 2 (ξi+1 − ξi )2 . dx = dx 2 i=0 0 Similarly,

Z

L

(uh )2 dx = 0

2N −1 h X 2 2 (ξ + ξi+1 + ξi ξi+1 ). 3 i=0 i

h Hence, Sl (h) = √ . 2 3 (iii) Finite difference method. Assume V = H01 (0, L); h = L/2N , N ∈ N. Vh is a step function defined as constant on [jh, (j + vh)) (j = 0, 1, · · · , 2N − 1), and 0 on [0, h] and [L − h, L], Vh = span{wj,h }, ( wj,h =

1, 0,

uh =

[jh, (j + 1)h); elsewhere, j = 1, 2, · · · , 2N − 2. 2N −2 X

uh (jh)Wj,h ,

∀ uh ∈ Vh .

j=1

{Wj,h } is an natural base of Vh , define the dot product as Z

L−h

∇h uh ∇h vh dx,

((uh , vh ))h = 0

Z (uh , vh )h =

L

uh vh dx, 0

where ∇h is a forward difference operator (∇h ϕ)(x) =

ϕ(x + h) − ϕ(x) . h

Let h2 = 2h, define Vh2 = V2h similarly, its basis is constructed by Wj,2h (j = 1, · · · , N −2), V2h ⊂ Vh . In the decomposition (5.9.102), define Wh = span{W2i+1,h },

318

Chapter 5

Infinite Dimensional Dynamical Systems

j = 0, 1, · · · , N − 2. Any function uh ∈ Vh can be written as uh = yh + zh , yh =

N −2 X

yh ∈ V2h ,

uh (2jh)Wj,2h ,

(5.9.112)

j=1

zh =

N −2 X

zh ((2i + 1)h)W2i+1,h + zh ((2N − 2)h)W2N −2,h .

i=0

When i = 0, 1, · · · , N − 2, zh ((2i + 1)h) = u ¯h ((2i + 1)h) = uh ((2i + 1)h) − uh (2ih), zh ((2N − 2)h) = uh ((2N − 2)h).

(5.9.113) (5.9.114)

Obviously, if uh is the restriction of the smooth function u ∈ H01 (0, L) on Vh , then yh and uh has the same order, zh has a factor of h. One can demonstrate (5.9.104) and (5.9.105) by the following lemma. Lemma 5.9.5 The following Cauchy-Schwarz inequality holds r 2 ∥yh ∥h ∥zh ∥h , ∀ y ∈ V2h , ∀ zh ∈ Wh . (5.9.115) |((yh , zh ))h | ⩽ 3 Proof We must prove r  Z L−h Z L−h  12  21  Z L−h 2 |∇h zh |2 dx . ∇h yh ∇h zh dx ⩽ |∇h yh |2 dx · 3 0 0 0 (5.9.116) It is sufficient to show that (5.9.116) is true on each coarse-mesh interval (2jh, 2(j + 1)h), j = 1, 2, · · · , N − 3, instead of (0, L). Take ( m1 , x ∈ [2jh, 2(j + 1)h); yh = m2 , x ∈ [2(j + 1)h, (2j + 3)h),   0, x ∈ [2jh, (2j + 1)h);    P , x ∈ [(2j + 1)h, 2(j + 1)h); 1 zh =  0, x ∈ [2(j + 1)h, (2j + 3)h);    P2 , x ∈ [(2j + 3)h, 2(j + 2)h). As shown in Figure 5.19.

Figure 5.19

5.9

Nonlinear Galerkin method

319

On interval [2jh, (2j + 1)h), ∇h yh = 0,

∇h zh =

P1 . h

On [(2j + 1)h, 2(j + 1)h), ∇h yh = Z

2(j+1)h

m2 − m1 , 2

∇h zh = −

P1 . h

1 (m2 − m1 )P1 h Z  21 1  2(j+1)h ⩽√ |∇h yh |2 dx 2 2jh  Z 2(j+1)h  21 · |∇h zh |2 dx

∇h yh ∇h zh dx = − 2jh

2jh

1 = |m2 − m1 | · |P1 |. h

(5.9.117)

Now, consider the intervals with end points j = 0, j = N − 2, N − 1. The difference is that for j = 0, m1 = 0. Thus, (5.9.117) still holds. Next, consider the intervals [L − 4h, L − 2h) and (L − 2h, L − h). On [L − 2h, L − h], zh is not equal to 0 (for example zh = P2 ), however on [L − h, L), zh = 0. Hence Z

L−h

1 ∇h yh ∇h zh dx = − m1 (P2 − P1 ) h L−4h r 1 1 2 |m1 |((P2 − P1 )2 + P12 + P22 ) 2 ⩽ h 3 r  Z L−h  21  Z L−h  12 2 2 = |∇h yh | dx · |∇h zh |2 dx . 3 L−4h L−4h r 1 2 Owing to √ < , (5.9.115) is obtained. 3 2 Lemma 5.9.6 For any function in Wh , there holds the following strong, discrete Poincaré inequality |zh |h ⩽ S2 (h)∥zh ∥h ,

∀ zh ∈ Wh , S2 (h) = h.

(5.9.118)

Proof As the same as Lemma 5.9.5, it is sufficient to show the similar inequalities hold on [2jh, 2(j + 1)h), j = 0, 1, · · · , N − 1, i.e., Z

Z

2(j+1)h

2(j+1)h

zh2 dx ⩽ S22 (h) 2jh

|∇h zh |2 dx. 2jh

(5.9.119)

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Similarly, it is known that the integral on right-hand side of (5.9.119) equals to 2P12 /h, j = 0, 1, · · · , N − 3; and the left-side integral equals to P12 h. Hence, for the √ S2 (h) = h/ 2 in (5.9.119), the integral of zh2 on [L − 4h, L) equals to h(P12 + P22 ), 1 however, the integral of |∇h zh |2 equals to ((P2 − P1 )2 + P12 + P22 ), we obtain the h inequality (5.9.119), S2 (h) = h, which implies (5.9.118) holds. The proof of (5.9.93) is standard. Now, attention is given to the second inequality 2N P−2 of (5.9.93). Let ξj = uh (jh), uh = ξj Wj,h , and j=1

|uh |2h = h

2N −2 X

ξj2 ,

j=1

2N −2 2N −2 2N −2 2 X 2 4 X 2 4 1 X 2 2 (ξj+1 − ξj ) ⩽ (ξ + ξj ) = ξ = |uh |2h , = h j=1 h j=1 j+1 h j=1 j h

∥uh ∥2h

since ξ0 = ξ2N −1 = 0. Hence, (5.9.120)

S1 (h) = h/2. Lemma 5.9.7 There holds the following identities |yh |h = |yh |2h ,

∥yh ∥2h = 2∥yh ∥22h ,

∀ yh ∈ V2h = Vh2 .

(5.9.121)

Proof It is obviously that the first identity in (5.9.121) holds due to | · |h and | · |2h are L2 norm. To show the second identity in (5.9.121), similar to Lemma 5.9.5, since ∇2h yh = (m2 − m1 )/2h, [2jh, 2(j + 1)h), and ( ∇ h yh = then

Z

0, [2jh, (2j + 1)h); m2 − m1 , [(2j + 1)h, 2(j + 1)h), h

2(j+1)h

|∇2h yh |2 dx = 2jh

(m2 − m1 )2 1 = 2h 2

Z

2(j+1)h

|∇h yh |2 dx. 2jh

(5.9.121) is obtained by taking the summation of j = 0, 1, · · · , N − 2. Remark 5.9.8 The space V2h plays the same role as Vh , then the assumptions similar to (5.9.93)-(5.9.98) are true. Especially, the second assumption in (5.9.93) turns to be S1 (2h)∥yh ∥2h ⩽ |yh |2h , ∀ yh ∈ Vh2 = V2h . (5.9.122)

5.9

Nonlinear Galerkin method

321

Combining with (5.9.120), (5.9.121), the inequality (5.9.122) becomes S¯1 (h)∥yh ∥h ⩽ |yh |h ,

∀ yh2 ∈ Vh = V2h .

¯ S(h) = S1 (2h)/2.

(5.9.123)

For other forms of discretization (spectral and finite element), ∥yh ∥h2 = ∥yh ∥h , |yh |h2 = |yh |h , ∀yh ∈ Vh . Thus, (5.9.123) still holds, but S¯1 (h) = S(h2 ). Now, consider the discretization of time. Format I The initial value u0h in (5.9.101) can be decomposed as yh0 ∈ Vh2 , zh0 ∈ Wh .

u0h = yh0 + zh0 ,

We are sure that yhn ∈ Vh2 , zhn ∈ Wh are cyclic sequences as follows: Suppose yhn , zhn are given. Define yhn+1 ∈ V2h and zhn+1 ∈ Wh in the following: 1 n+1 (y − yhn , yˆh )h + ah (yhn+1 + zhn+1 , yˆh ) k h + dh (yhn+1 + zhn+1 , yˆh ) + bh (yhn , yhn , yˆh ) + bh (yhn , zhn , yˆh ) + bh (zhn , yhn , yˆh ) = (fhn , yhn )h ,

∀ yˆh ∈ Vh2 ,

(5.9.124)

1 n+1 (z − zhn , zˆh )h + ah (yhn+1 + zhn , zˆh ) k h + dh (yhn+1 + zhn+1 , zˆh ) + bh (yhn , yhn , zˆh ) = (fhn , zˆhn )h ,

∀ zˆh ∈ Wh .

(5.9.125)

Here k = ∆t is the time step, fhn is the average of fh : fhn

1 = k

Z

(n+1)k

f (t) dt.

(5.9.126)

nk

If f is smooth, then take fhn = fh (nk), (5.9.124)-(5.9.125) is a linear system of yhn+1 , zhn+1 . By (5.9.98) and Lax-Milgram theorem, it is deduced that yhn+1 , zhn+1 exist uniquely. Format I′ A little different with Format I. Consider b as an implicit function of z in (5.9.124), then (5.9.124) is replaced by 1 n+1 (y − yhn , yˆh )h + ah (yhn+1 + zhn+1 , yˆh ) + dh (yhn+1 + zhn+1 , yˆh ) k h + bh (yhn , yhn , yˆh ) + bh (yhn , zhn+1 , yˆh ) + bh (zhn+1 , yhn , yˆh ) = (fhn , yˆh )h ,

∀ yˆh ∈ Vh2 ,

(5.9.127)

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Format II Assume yhn and zhn are given, define yhn+1 and zhn+1 as follows 1 n+1 (y − yhn , yˆh )h + ah (yhn + zhn+1 , yˆh ) + dh (yhn + zhn+1 , yˆh ) k h + bh (yhn , yhn , yˆh ) + bh (yhn , zhn+1 , yˆh ) + bh (zhn+1 , yhn , yˆh ) = (fhn , yˆh )h ,

(5.9.128)

∀ yˆh ∈ Vh2 ,

1 n+1 (z − zhn , zˆh )h + ah (yhn + zhn+1 , zˆh ) k h + dh (yhn + zhn+1 , zˆh ) + bh (yhn , yhn , zˆh ) = (fhn , zˆh )h ,

(5.9.129)

∀ zˆh ∈ Wh .

zhn+1

Actually, one can solve (5.9.129), then find yhn+1 from (5.9.127). Format III This format is different with format II, which neglects zhn+1 − zhn . For the computation of zhn+1 , it can be determined by solving the following equation ah (yhn + zhn+1 , zˆh ) + dh (yhn + zhn+1 , zˆh ) + bh (yhn , yhn , zˆh ) = (fhn , zˆh ),

∀ zˆh ∈ Wh , (5.9.130)

The solvability of zhn+1 in (5.9.130) can be obtained by (5.9.98) and Lax-Milgram theorem. Now, two examples are given. Example 5.9.1 (Burgers equation) Assume Ω = (0, L), L > 0. Let V = = L2 (0, L). For v > 0, f is given, the equation is

H01 (0, L), H

ut − vuxx + uux = f,

Ω × R+ ,

u(0, t) = u(L, t) = 0,

u(x, 0) = u0 (x).

The variational form of this equation is u : R+ → H01 (Ω) = V, such that d (u, v) + v((u, v)) + b(u, u, v) = (f, v), dt where

Z (ϕ, ψ) =

Z

L

ϕψ dx, 0

((ϕ, ψ)) = 0

L

∀ v ∈ V,

dϕ dψ dx, dx dx

(5.9.131)

∀ ϕ, ψ.

b is an anti-symmetric nonlinear term, Z 1 T b(ϕ, ψ, θ) = ϕ(ψx θ − ψθx ) dx. 3 0 Applying the spectral method, finite element method, finite difference method to discrete, the (5.9.99) is derived, d = 0, in case of (5.9.94)-(5.9.98) are satisfied, C2 = C5 = v, C4 = 0, C3 are some proper constants.

5.10

Inertial set

323

Example 5.9.2(Navier-Stokes equations)

Assume Ω ⊂ R2 ,

∂u 1 − v∆u + (u, 0)u + (divu)u = f, ∂t 2 u = 0, ∂Ω,

x ∈ Ω, t ⩾ 0.

u(x, 0) = u0 (x).

Convert this into the format of (5.9.130) by define u : R+ → V = H01 (Ω), Z (ϕ, ψ) =

ϕψ dx,

((ϕ, ψ)) =

Ω

2 Z X i,j=1

0

L

dϕi dψi dx, dxj dxj

  2 Z dψj 1 X dθj ϕi b(ϕ, ψ, θ) = θ j − ψj dx. 2 i,j=1 Ω dxi dxi Applying the spectral method, finite element method, finite difference method to discrete, the (5.9.99) is derived, d = 0, in case of (5.9.94)-(5.9.98) are satisfied, C2 = C5 = v, C4 = 0, C3 are some proper constants.

5.10

Inertial set

In this section, we introduce some basic results on inertial set (refer to P. W. Bates and S. Zheng [15], A. Eden, C. Foias, B. Nicolaenko and R. Temam [46]). Assume X is a compact, connected, subset of H, S is a Lipschitz continuous map from X to itself. Assume the Lipschitz constant for S on X is LipX (S) = L.

(5.10.1)

If S is restricted to X, then it has a global attractor A . It is compact and connected, which is given by the following form A =

∞ \

S n X.

(5.10.2)

n=1

As is known to all, A attracts all trajectories, when h → ∞, the symmetric Hausdorff distance ρ(S n X, A ) → 0. However, the convergence rate of the attractor is not exponentially controlled in any case. A simple exmaple: suppose H = R, define S : [0, 1] → [0, 1], 1 Sx = , 1+x then A = {0}. The convergence rate of A is a polynomial. To remedy this, we introduce the concept of inertial set.

324

Chapter 5

Infinite Dimensional Dynamical Systems

Definition 5.10.1 A compact set M is called the inertial set of (S, X), if A ⊆ M ⊆ X and (i) SM ⊆ M ; (ii) M has finite fractal dimension dM ; (iii) There exist positive constants C0 and C1 , such that h(S n X, M ) ⩽ C0 exp(−C1 n),

∀n ⩾ 1.

The distance applied here is a standard asymmetric quasi-distance of two sets h(A, B) = max min |a − b|H , a∈A b∈B

where A, B are two compact sets. At all cases, the standard Hausdorff metric is defined as ρ(A, B) = max{h(A, B), h(B, A)}. Now, consider the inertial set of the first order dissipative evolution equation. Suppose the equation has the following form du + Au + R(u) = 0, dt

(5.10.3)

u(0) = u0 .

(5.10.4)

With proper assumption for the operator A, R(u) and initial value u0 , one can not only ensure the existence and uniqueness of the solution, but also have the existence of the compact absorbing set B via the dissipation of the problem. In order to obtain the inertial set, the squeezing property of S is essential. 5.10.2 S possesses the squeezing property in X, if for all δ ∈  Definition  1 0, , there exists an orthogonal projection with rank N0 (δ), such that for any u 4 and v, |Su − Sv|H ⩽ δ|u − v|H , (5.10.5) or |(I − P )(Su − Sv)|H ⩽ |P (Su − Sv)|H .

(5.10.6)

Another expression for the squeezing property: if for u, v ∈ X, |Su − Sv|H >

√ 2|PN0 (Su − Sv)|H ,

(5.10.7)

then |Su − Sv|H < δ|u − v|H ,

(5.10.8)

5.10

Inertial set

325

¯θk R Definition 5.10.3 Let Ek+1;j1 ,··· ,jk be a maximal set in the compact S(B k (aj1 ,··· ,jk ) ∩ S X) satisfying the corn property √ |u − v|H ⩽ 2|PN0 (u − v)|H , (5.10.9) ¯θk R (aj ,··· ,j ) ∩ S k X), aj ,··· ,j ⊂ Ek;j ,··· ,j ), (j1 , · · · , jk ) ∈ where u, v ∈ S(B 1 1 k k k−1  2L1  N0 ¯r (a) denotes a closed ball in H (1, 2, · · · , k0 ), k0 ⩽ +1 , 4δ < θ < 1, B δ centered at a ∈ X with radius r. Let E (k+1) =

k0 [

Ek+1;j1 ,··· ,jk ⊂ S k+1 X.

(5.10.10)

j=1

.. .

jk =1

Theorem 5.10.4

Assume M = A

∞ S ∞ S S

 (E (k) ) , then M is an inertial

j=0 k=1

set of S, and dF (M ) ⩽ max{α(X), N0 },

(5.10.11)

where α(X) = log k0 / log(1 + θ), A is an attractor. Now, given the solution operator S(t) mapping u0 to u(t) and compact, absorbing set B. We obtain the mapping S(t) (where t = t∗ ) from B to itself. Denote the mapping as S∗ = S(t∗ ). (5.10.12) 1 Taking t∗ sufficiently small, such that S∗ possesses squeezing property, where δ < , 8 N0 = N0 (δ). By Theorem 5.10.4, there exists an inertial set M∗ , define [ M= S(t)M∗ . (5.10.13) 0⩽t⩽t∗

If the mapping F : [0, T ] × M∗ → M : F (t, x) = S(t)x

(5.10.14)

is Lipschitz, then M is a compact set with finite fractal dimension. Furthermore, M attracts ({S(t)}t⩾0 , B) exponentially. To explain this fact more in detail, the definition of inertial set is given again. Definition 5.10.5 For a compact set M, if A ⊆ M ⊆ X, and (i) S(t)M ⊆ M, ∀t ⩾ 0; (ii) M has finite fractal dimension dM ; (iii) there exist positive constants a0 and a1 , such that for all t ⩾ 0, dist(S(t)X, M ) ⩽ a0 exp(−a1 t), then the compact set M is an inertial set of ({S(t)}t⩾0 , X).

326

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Infinite Dimensional Dynamical Systems

For the initial value problem (5.10.3), (5.10.4), assume H is a separable Hilbert space, and A is a positive, self-adjoint, linear operator, D(A) ⊂ H, possessing compact inverse A−1 . Furthermore, suppose that the initial value problem (5.10.3), (5.10.4) is solvable by the semigroup of nonlinear operator {S(t)}t⩾0 . Assume S(t) : H → D(A) is continuous,

t > 0.

Assume there exists a compact, invariant, absorbing set B B = {u ∈ H : |u|H ⩽ ρ0 , ∥u∥

1

D(A 2 )

⩽ ρ1 }.

A−1 is compact, denote 1

V = D(A 2 ). Obviously, V is compact embedded in H. For simplicity, let 1

∥u∥ = ∥u∥V = |A 2 u|H ,

|u| = |u|H .

For equation (5.10.3), suppose that R : D(A) → H is continuous.  1i There exists a compact, invariant set X ⊂ B, and real number β ∈ 0, , such that 2 for any u, v ∈ X, one has |R(u) − R(v)| ⩽ C0 |Aβ (u − v)|,

(5.10.15)

where C0 depends on X. Proposition 5.10.6 With the above assumption for equation (5.10.3), there exists time t∗ , such that the discrete operator S∗ = S(t∗ ) satisfies squeezing property, 1 where δ < . 8 Proof First of all, we introduce the projection. Since A is a self-adjoint, positive operator, and possessing compact inverse operator A−1 , then there exists the ∞ complete set {wn }n=1 of eigenvectors corresponding to the eigenvalues {λn }∞ n=1 in H, i.e. , Awn = λwn , ∀n ∈ N, and the eigenvalues satisfy 0 < λ1 ⩽ λ2 ⩽ · · · ⩽ λn ⩽ · · · ⩽ λn→+∞ . Denote Hn = span{w1 , w2 , · · · , wn },

5.10

Inertial set

327

Pn is the orthogonal projection of H on Hn . Hence, Qn = I − Pn is the orthogonal projection of H on the complement of Hn . Assume t∗ is given, now it is going to show the squeezing property, i.e. for any δ > 0, there exists N0 = N0 (δ), such that for u, v ∈ X, by |QN0 (S∗ u − S∗ v)| > |PN0 (S∗ u − S∗ v)|, which derives that |S∗ u − S∗ v| < δ|u − v|.

(5.10.16)

Let W∗ = S∗ u − S∗ v. Consider λ∗ =

∥W∗ ∥2 . |W∗ |2

With the selection of orthogonal projections PN0 and QN0 , it is known that under H and V , PN0 W∗ and QN0 W∗ are orthogonal. Hence λ∗ =

∥PN0 W∗ ∥2 + ∥QN0 W∗ ∥2 1 ∥QN0 W∗ ∥ ∥PN0 W∗ + QN0 W∗ ∥2 . (5.10.17) = > 2 |PN0 W∗ + QN0 W∗ | |PN0 W∗ |2 + |QN0 W∗ |2 2 |QN0 W∗ |

Because of ∥QN0 W∗ ∥2 = (AQN0 W∗ , QN0 W∗ ) ⩾ λN0 +1 |QN0 · W∗ |2 , it is derived that λ∗ >

1 λN +1 . 2 0

(5.10.18)

(5.10.19)

1 λN +1 . Therefore, assume 2 0 u and v are two solutions of equation (5.10.3) satisfying the initial value u0 and v0 . Let W (t) = u(t) − v(t), then W satisfies We are going to derive |W∗ | < δ|u − v| from λ∗ >

dW + AW + R(u) − R(v) = 0, dt

(5.10.20)

W (0) = u0 − v0 = W0 .

(5.10.21)

Firstly, we estimate the Lipschitz constant L of S(t). Taking the inner product of (5.10.20) with W on H yields 1 d |W |2 + ∥W ∥2 + (R(u) − R(v), W ) = 0, 2 dt

(5.10.22)

Estimate the nonlinear term from the assumption (5.10.15) 1

|(R(u) − R(v), W )| ⩽ C0 |Aβ W ||W | ⩽ C0 |W |1−2β |A 2 W |2β |W | ⩽ C0 |W |2(1−β) ∥W ∥2β ,

(5.10.23)

328

Chapter 5

Infinite Dimensional Dynamical Systems

 1i where we applied standard interpolating inequalities, β ∈ 0, . Then using 2 1 1 Young’s inequality p = , q = , which get along with (5.10.22), (5.10.23) β 1−β gives rise to 1 d 1 ∥W ∥2 |W |2 + ∥W ∥2 ⩽ C0 |W |2(1−β) ∥W ∥2β ⩽ C1 |W |2 + . 2 dt 2 2

(5.10.24)

Neglecting ∥W ∥2 on the left-side of (5.10.24) infers d |W |2 ⩽ C1 |W |2 , dt

(5.10.25)

where C1 depends only on C0 and the constant of β. From the Grönwall’s inequality and (5.10.25), it is deduced that L = LipX (S(t)) ⩽ eC1 t . Now, back to search for the projection PN0 , such that it possesses squeezing properties. From (5.10.24), there holds d |W (t)|2 + ∥W ∥2 ⩽ C1 |W |2 . dt Denote λ(t) =

∥W (t)∥2 , |W (t)|2

ξ(t) =

(5.10.26)

W (t) , |W (t)|

which infers that

d |W |2 + (λ(t) − C1 )|W |2 ⩽ 0. dt By Grönwall’s inequality, it is deduced from (5.10.27) that |W |2 ⩽ exp

n

Z

t

−

o λ(τ ) dτ + C1 t |W (0)|2 .

(5.10.27)

(5.10.28)

0

Taking t = t∗ , W (t∗ ) = W∗ gives |S∗ u − S∗ v| = |W∗ | ⩽ δ(t∗ )|u0 − v0 |, where

n δ∗ = δ(t∗ ) = exp

−

1 2

Z

t∗

o λ(τ ) dτ + C1 t∗ .

(5.10.29)

0

1 λN +1 and N0 → +∞, λN0 +1 → ∞. However, 2 0 the behavior of the quotient module λ(τ ), τ < t∗ is unknown. The following lemma provides the control of the quotient module in the past. By (5.10.19), when λ∗ = λ(t∗ ) >

5.10

Inertial set

329

Lemma 5.10.7 Assume λ(t) and ξ(t) are defined in (5.10.4), then λ(t) satisfies the following differential inequality d λ(t) ⩽ C02 λ2β (t). dt If λ(t∗ ) > λ0 , then

Z 0

t0

λ(t) dt ⩾

C2 1 (1 − e−C3 t∗ )λ0 − t∗ , C3 C3

(5.10.30)

(5.10.31)

where C3 and C2 depend only on C0 and β. Proof 1 d 1 λ(t) = [(Wt , AW ) − (Wt , W )λ(t)] 2 dt |W |2 1 (Wt , (A − λ)ξ) = |W | 1 = (−AW − (R(u) − R(v)), (A − λ)ξ), |W |

(5.10.32)

where (5.10.20) is applied to the last identity. (λξ, (A − λ)ξ) = λ(ξ, Aξ) − λ2 |ξ|2 = λ∥ξ∥2 − λ2 =λ

∥W ∥2 − λ2 = 0. |W |2

Hence, |(A − λ)ξ|2 = ((A − λ)ξ, (A − λ)ξ) = (Aξ, (A − λ)ξ) 1 = (AW, (A − λ)ξ). |W |

(5.10.33)

Combining (5.10.33) and (5.10.32) implies that 1 d 1 λ(t) + |(A − λ(t))ξ|2 = (R(u) − R(v), (A − λ)ξ) 2 dt |W | 1 ⩽ |R(u) − R(v)||(A − λ)ξ| |W | C0 β ⩽ |A W ||(A − λ)ξ| |W | C0 ⩽ |W |1−2β ∥W ∥2β |(A − λ)ξ| |W | ⩽ C0 λβ |(A − λ)ξ| ⩽

C02 2β 1 λ + |(A − λ)ξ|2 , 2 2

(5.10.34)

330

Chapter 5

Infinite Dimensional Dynamical Systems

where we applied (5.10.15), (5.10.23) and Young’s inequality p = q = 2. Simplify (5.10.34) infers d λ(t) ⩽ C0 λ2β (t). dt This proves the first part of the lemma. Taking advantage of Young’s inequality, it is derived that d λ(t) ⩽ C3 λ(t) + C2 , (5.10.35) dt 1 where C3 and C2 depend on C0 and β. If β = , then C3 = C02 , C2 = 0. From 2 Grönwall’s inequality, (5.10.35) gives that λ(t) ⩽ eC3 (t−t0 ) λ(t0 ) − (1 − eC3 (t−t0 ) )

C2 . C3

Hence, for 0 ⩽ t0 < t, one obtains the inequality in another direction λ(t0 ) ⩾ eC3 (t−t∗ ) λ(t∗ ) −

C2 C2 > eC3 (t0 −t∗ ) λ0 − . C3 C3

Taking the integral from 0 to t∗ shows Z

t∗

λ(t0 ) dt0 ⩾

0

1 C2 (1 − e−C3 t∗ )λ0 − t∗ . C3 C3

Consider as a simple corollary of Lemma 5.10.7, (5.10.29) can be estimated as follows δ∗ ⩽ exp

n

−

 o 1 λN0 +1  C2 (1 − e−C3 t∗ ) · + + C1 t∗ , C3 2 C3

where we used λ∗ = λ(t∗ ) > chosen as C3 t∗ = 1, then

(5.10.36)

λN0 +1 . Due to C3 depends on β and C0 , t∗ can be 2

δ∗ ⩽ exp

n

1 − λN +1 + 2C3 0

C2 C3

+ C1 o . C3

Finally, if N0 is sufficiently large, such that     1 C2 λN0 +1 > −4C3 ln +4 + C1 , 8 C3 then (5.10.37) implies δ∗ < This demonstrates Proposition 5.10.6.

1 . 8

(5.10.37)

5.10

Inertial set

331

Corollary 5.10.8 With the assumption in Proposition 5.10.6, there exist constants C1 , C2 , C3 , they only depend on C0 and β in (5.10.15), such that if t∗ =

1 , C3

then

C1

L∗ = LipX (S∗ ) ⩽ e C2 .

(5.10.38)

Furthermore, if N0 is sufficiently large, such that   C2 λN0 +1 > 12C3 ln 2 + 4 + C1 , C3 then for any u, v ∈ X, |QN0 (S∗ u − S∗ v)| > |PN0 (S∗ u − S∗ v)|, which infers that for δ∗
0; stream {S(t)}t⩾0 is determined by (5.10.3), then there exists an inertial set M, the estimate of its dimension is dF M ⩽ dF (M∗ ) + 1.

(5.10.39)

Proof As a simple corollary of Theorem 5.10.9 and Corollary 1, we obtain that the mapping S∗ = S(t∗ ) possesses inertial set M∗ on X, such that dist(S∗n X, M∗ ) ⩽ θn R = C4 δ∗n . Furthermore, there holds dF (M∗ ) ⩽ max{N0 , α(X)} ⩽ N0 {1, C5 }.   1  ∗ where C5 = ln + 1 / ln is a constant, which could be estimated by C1 , δ∗ 4δ∗ C2 and C3 . Denote [ M= S(t)M∗ .  2L

0⩽t⩽t∗

332

Chapter 5

Infinite Dimensional Dynamical Systems

Obviously, M ⊆ X, and by the continuity of F (t, X) on [0, T ] × X, it is compact. From A ⊆ M∗ , it is derived that A ⊆ M Notice that M is the image of [0, t∗ ] × M∗ under F , because of Lipschitz function keeps the fractal dimension dF (M ) ⩽ dF ([0, t∗ ] × M∗ ) ⩽ dF (M∗ ) + 1. Now, prove that M is invariant with the effect of stream. Two cases will be considered. If t ∈ (0, t∗ ], from S∗ M∗ = S(t∗ )M∗ ⊂ M∗ it is deduced that [ S(t)M = S(s)M∗ t⩽s⩽t∗ +t

 = 

[

 S(s)M∗ 

t⩽s⩽t∗

⊆M

[

 

[

 



[

[

 S(s)M∗ 

t∗ ⩽s⩽t∗ +t

S(s)M∗  ⊆ M.

(5.10.40)

0⩽s⩽t

For t ⩾ t∗ , denote t = kt∗ + S, k > 0, s ∈ [0, t∗ ], then S(t)M = S(kt∗ )S(s)M ⊆ S k (t∗ )M [ = S∗k M ⊆ S(s)S∗k M∗ ⊆

[

0⩽s⩽t∗

S(s)M∗ = M.

(5.10.41)

0⩽s⩽t∗

Finally, it is about the exponential convergence, for t = kt∗ + s, dist(S(t)X, M ) = dist(S(s)S∗k X, M ) ⩽ L∗ dist(S∗k X, M ) ⩽ L∗ C4 δ∗k t−s/t∗

1

⩽ C6 (δ t∗ )t = C6 δ0t ,

(5.10.42)  1 C 3 where the constant C6 depends on C1 , C2 , C3 and δ0 = δ∗C3 < . This 8 accomplishes the proof of that M is an inertial set for ({S(t)}t⩾0 , X). Now, two specific examples of solving for the inertial set are given. = (L∗ C4 )δ∗

Example 5.10.1 Consider the Kuramoto-Sivashinsky equation ut + uxxxx + uxx − uux = 0. (5.10.43)   L L Assume u(·, t) is defined on − , , and it is odd, L-periodic function, L ⩾ 1. 2 2      L L L L 2 Let H = u ∈ L − , ; u is odd function − , , then the set 2 2 2 2 n  L L o B = u ∈ H | ux ∈ L2 − , , |u|L2 ⩽ ρ0 , |ux |L2 ⩽ ρ1 (5.10.44) 2 2

5.10

Inertial set

333

is attractive for all the bounded set of H, where 5

ρ0 = C 0 L 2 ,

5

(5.10.45)

ρ1 = C1 L 2 ,

C0 and C1 are absolute constants. For u ∈ B, there holds 1

1

1

1

|u|L∞ = |u|L2 2 |ux |L2 2 ⩽ (ρ0 ρ1 ) 2 = (C0 C1 ) 2 L3 . For simplicity, let u′ = ux , then (5.10.43) can be rewritten as ( ut + u′′′′ + u′′ − uu′ = 0, u(0) = u0 , in H.

(5.10.46)

(5.10.47)

We estimate the Lipschitz constant L of the solution operator S(t) on absorbing set. Given the initial values u0 and v0 in B , u(x, t) and v(x, t) are two solutions of (5.10.43), then the difference W (x, t) = u(x, t) − v(x, t) satisfies

where u ¯=

Wt + W ′′′′ + W ′′ − (2W u ¯)′ = 0,

(5.10.48)

W (0) = u0 − v0 ,

(5.10.49)

1 (u + v) ∈ B. Taking the inner product with W for (5.10.48) shows 2

1 d |W (t)|2L2 + |W ′′ (t)|2L2 = |W ′ (t)|2L2 − (¯ u′ (t), W 2 (t))L2 . 2 dt

(5.10.50)

Owing to |W ′ (t)|2L2 ⩽ |W (t)|L2 |W ′′ (t)|L2 ⩽

1 1 |W (t)|2L2 + |W ′′ (t)|L2 2 2

and |(¯ u′ (t), W 2 (t))L2 | = |(¯ u(t), 2W (t)W ′ (t))L2 | ⩽ 2|¯ u(t)|L∞ |W (t)|L2 |W ′ (t)|L2 3

1

⩽ 2|¯ u(t)|L∞ |W (t)|L2 2 |W ′′ (t)|L2 2 4 4 3 1 ⩽ (2) 3 |¯ u(t)|L3 ∞ |W (t)|2L2 + |W ′′ (t)|2L2 . 4 4 It can be simplified that 4 1 d 1 1 |W (t)|2L2 + |W ′′ (t)|2L2 ⩽ |W (t)|2L2 + 2|¯ u(t)|L3 ∞ |W (t)|2L2 . 2 dt 4 2

(5.10.51)

Using |¯ u(t)|2L∞ = |¯ u(t)|L2 |¯ u′ (t)|L2 ⩽ ρ0 ρ1 ⩽ C0 C1 L6 , it is deduced that 2 d 1 |W (t)|2L2 + |W ′′ (t)|2L2 ⩽ (1 + 4(C0 C1 ) 3 L4 )|W (t)|2L2 ⩽ (1 + C2 L4 )|W (t)|2L2 , dt 2 (5.10.52)

334

Chapter 5

Infinite Dimensional Dynamical Systems

1 2 where C2 = 4(C0 C1 ) 3 . In (5.10.52), neglecting |W ′′ (t)|2L2 on the left-hand side, by 2 Grönwall’s inequality, yields |W (t)|2L2 ⩽ exp[(1 + C2 L4 )t]|W (0)|2L2 . Hence, L = LipB (S(t)) ⩽

h1

i (1 + C2 L4 )t .

2 W (t) By λ(t) = |W ′′ (t)|2L2 /|W (t)|2L2 , ξ(t) = , (5.10.53) can be written as |W (t)| i h1 d |W (t)|2L2 + λ(t) − (1 + C2 L4 )t |W (t)|2L2 ⩽ 0. dt 2

(5.10.53)

Therefore, (5.10.28) has the following form |W (t)|2L2 ⩽ δ(t)|W (0)|2L2 , where δ(t) =

n 1 1 exp − 2 2

Z

t

o λ(τ ) dτ + (1 + C2 L4 )t .

(5.10.54)

(5.10.55)

0

In order to show the squeezing property of S∗ = S(t∗ ), denote t∗ = L−4 .

(5.10.56)

By (5.10.19), 1 λN +1 , (5.10.57) 2 0 is the N0 + 1-th eigenvalue of the operator Au = u′′′′ on H. Hence  N + 1 4 0 λN0 +1 = 2π . (5.10.58) L λ∗ = λ(t∗ ) ⩾

where λN0 +1

From (5.10.34), there holds 1 d λ(t) + |(A − λ(t))ξ(t)|2L2 = (ξ ′ (t) − (2¯ uξ)′ , (A − λ(t))ξ(t))L2 . 2 dt

(5.10.59)

It is derived by Schwarz inequality and Young’s inequality that 1 d λ(t) ⩽ |ξ ′ (t) − (2¯ u(t)ξ(t))′ |2L2 dt 2 ⩽ |ξ ′ (t)|2 + 2|¯ u′ (t)ξ(t)|2L2 + 2|¯ u(t)ξ ′ (t)|2L2 ⩽ |ξ(t)|L2 |ξ ′′ (t)|2L2 + 4|¯ u(t)|2L2 |ξ(t)|2L∞ + 4|¯ u(t)|2L∞ |ξ ′ (t)|2L2 ⩽ |ξ ′′ (t)|L2 + 4ρ21 |ξ(t)|L2 |ξ ′ (t)|L2 + 4|¯ u(t)|L2 |¯ u′ (t)|L2 |ξ(t)|L2 |ξ ′′ (t)|L2 1

1

⩽ λ(t) + 4ρ21 |ξ(t)|L2 2 |ξ ′′ (t)|L2 2 + 4ρ0 ρ1 λ 2 (t) 1

1

1

⩽ λ(t) + 4ρ21 λ 4 (t) + 4ρ0 ρ1 λ 2 (t).

(5.10.60)

5.10

Inertial set

335

To make the inequality (5.10.60) be homogenized, we introduce a new parameter β, β = L4 − 1.

(5.10.61)

From Young’s inequality, 1

4ρ21 λ 4 (t) ⩽

1 β λ(t) + 8(ρ81 /β) 3 , 2

(5.10.62)

β 32 λ(t) + ρ20 ρ21 . 2 β Then, combining (5.10.60), (5.10.61) and (5.10.62) gives  ρ8  31 32 d + ρ20 ρ21 . λ(t) ⩽ (β + 1)λ(t) + 8 1 dt β β 1

4ρ0 ρ1 λ 2 (t) ⩽

The Grönwall’s inequality, 0 ⩽ t ⩽ t∗ , shows  i h   ρ8  13 32 λ∗ = λ(t∗ ) ⩽ e(β+1)(t∗ −t) λ(t) + 8 1 + ρ20 ρ21 t∗ . β β

(5.10.63)

(5.10.64)

(5.10.65)

(5.10.57) infers λ(t) ⩾

  ρ8  31  1 32 λN0 +1 e(β+1)(t−t∗ ) − 8 1 + ρ20 ρ21 t∗ . 2 β β

Integrating (5.10.66) from 0 to t∗ implies Z t∗   ρ8  13  16 1 + ρ20 ρ21 t2∗ . λ(t) dt ⩾ (1 − e(β+1)t∗ )λN0 +1 − 4 1 2(β + 1) β β 0

(5.10.66)

(5.10.67)

Hence, by (5.10.55), there holds n 1 − e(β+1)t∗ h  ρ8  13 i o 16 1 exp λN0 +1 + 4 1 + ρ20 ρ21 t∗ + (1 + C3 L4 )t∗ . 2 4(β + 1) β β (5.10.68) Substituting (5.10.45), (5.10.56) and (5.10.61) into (5.10.67), which get along with δ∗ = δ(t∗ ) ⩽

1 − exp(−(β + 1)t∗ ) = 1 − e− 2 > 1

gives rise to

1 , 2

n o 1 1 exp − λ + C . (5.10.69) N +1 3 2 8L4 0 where C3 depends only on C0 and C1 . From (5.10.58), it is obtained that n 2π 4 o 1 (5.10.70) δ∗ ⩽ exp − 8 (N0 + 1)4 + C3 . 2 L  2 ln 2 + C  14 3 Hence, if N0 = C4 L2 , C4 = , then δ∗ < 18 . Combining with the result 2π 4 of the corollary of Theorem 5.10.9, we have δ∗ ⩽

336

Chapter 5

Infinite Dimensional Dynamical Systems

Proposition 5.10.10 For the Kummoto-Skaskinsky equation (5.10.43), there exists an inertial set M0 of the absorbing set B, its fractal dimension can be estimated as follows dF (M0 ) ⩽ C7 L2 , (5.10.71) where C7 depends only on the absolute constants C0 and C1 determined by (5.10.46). Furthermore, there exist positive constants C6 and C9 , such that for all t ⩾ 0, there holds 4 distL2 (S(t)B, M0 ) ⩽ C9 e−C6 L t . (5.10.72) Proof Denote t∗ = L−4 . By (5.10.53), S∗ = S(t∗ ) is a Lipschitz function in B with Lipschitz constant  1 (1 + C2 L4 )t∗ ⩽ eC5 , L∗ = LipB (S(t∗ )) ⩽ exp 2 satisfying squeezing property. From Theorem 5.10.9, it is known that S∗ has an inertial fractal set M∗ ⊆ B, such that   2L  ∗    +1   log  δ∗ , (5.10.73) dF (M∗ ) ⩽ max{α(B), N0 } ⩽ N0 max 1, 1     log   θ∗ L∗ ⩽ ρC3 , N0 = C4 L2 , δ∗ = e−C6 . By (5.10.69), θ∗ = 4δ∗ , the fractal dimension has the following estimate dF (M∗ ) ⩽ C7 L2 , (5.10.74) where C7 depends only on C0 and C1 . By the definition [ M0 = S(t)M∗ , 0⩽t⩽t∗

similar to the proof of Theorem 5.10.9, it is deduced that M0 is an inertial set, and dF (M0 ) ⩽ 2C7 L2 ,

(5.10.75)

(5.10.42) implies 1

distL2 (S(t)B, M0 ) ⩽ C8 L∗ [(δ∗t∗ )]t ⩽ C9 (δ∗L )t ⩽ C9 e−C6 L t . 4

4

(5.10.76)

Concerning the study of Kuramoto-Sivashinsky equations, we refer to the papers [98, 109, 132] by B. Guo and F. Su. Example 5.10.2 The Kolmogorov-Spiegel-Sivashinsky equation (KSS equation) ut + u′′′′ + [(2 − δ(u′ ))2 u′ ]′ + (u′ )2 + αu = 0.

(5.10.77)

5.10

Inertial set

337

u(·, t) is defined on [0, L], the cycle is L; the parameters α and δ are positive, and α < 1, L > 1. Firstly, prove the existence of the absorbing set B = {u ∈ L2 [0, L], |u′ |L2 ⩽ ρ1 , |u′′ |L2 ⩽ ρ2 }, where ρ1 = δ − 2 L 2 , 1

3

ρ2 = C0 ρ1 (1 + δ − 4 + L). 1

(5.10.78)

Consider the following Hilbert space V = {u ∈ H 1 (0, L) : u is a L cycle}.

H = L2 [0, L],

By the local existence theorem, it is known that (5.10.77) has a solution with initial value u0 ∈ V , t ∈ [0, t0 ), t0 > 0. Lemma 5.10.11 Assume u is a solution of (5.10.77) on V × [0, t0 ), then for t ∈ [0, t0 ), |u′ (t)|L2 ⩾ ρ1 , it is derived that d ′ |u (t)|L2 < 0. dt

(5.10.79)

Proof Let v = u′ , then v satisfies vt + v ′′′′ + 2v ′′ + 2vv ′ − 8(v 3 )′′ + αv = 0. Therefore, 1 d 2 |v| 2 + |v ′′ |2L2 + 3δ(v 2 , (v ′ )2 )L2 + α|v|2L2 = 2|v ′ |2L2 . 2 dt L

(5.10.80)

By the standard interpolation, |v ′ |L2 ⩽ |v|L2 |v ′′ |L2 ⩽

1 1 2 |v| 2 + |v ′′ |2L2 . 2 L 2

Hence 1 d 2 (5.10.81) |v| 2 + 3δ(v 2 , (v ′ )2 )L2 + α|v|2L2 ⩽ |v|2L2 . 2 dt L d Since α < 1, for some t ∈ [0, t0 ), |v(t)|L2 ⩾ 0, it is deduced from (5.10.80), dt (5.10.81) that 3δ(v 2 , (v ′ )2 )L2 ⩽ min{2|v ′ |2L2 , (1 − α)|v|2L2 }. (5.10.82) P Let v 2 ∼ m Cm Wm be the Fourier expansion of v 2 on L2 [0, L], where Wm (x) = exp

n 2πi L

o mx .

338

Chapter 5

Infinite Dimensional Dynamical Systems

Rewrite the left-hand side of (5.10.82) 3δ(v 2 , (v ′ )2 )L2 =

3δ 2

Z

L

0

2 X 3 πim ((v 2 )′ )2 dx = δ Cm Wm 2 2 2 m̸=0 L

3δ 4π 2 X 2 2 6δπ 2 X 2 2 2 = = m C |W | m Cm · L 2 m m L 2 L2 L2 m̸=0

m̸=0

6δπ X 2 2 = m Cm . L 2

m̸=0

Combining with (5.10.82) yields 6δπ 2 X 2 2 m Cm ⩽ min{2|v ′ |2L2 , (1 − α)|v|2L2 }. L

(5.10.83)

m̸=0

On the other hand, from |v|2L2 = |v 2 |2L1 = L2 C02 ,  X 1 2  X 2 X 1 2 2 Cm ⩽ m C v(x)2 − |v|2L2 ∞ = 2 m L m2 L m̸=0

m̸=0

 π 2  12  L(1 − α)  12

m̸=0

|v|L2 3 6δπ 2 1  L(1 − α)  12 =√ |v|L2 . δ 18 ⩽

(5.10.84)

Finally, 0 < 2|v ′ |2L2 − 3δ

Z

L

v 2 (v ′ )2 dx ⩽

0

Z

L

(v ′ )2 (2 − 3δv 2 ) dx

0

⩽ |v ′ |2L2 |2 − 3δv 2 |L∞ ⩽ |v ′ |2L2 (2 − 3δ)|v 2 |L∞   1 1 |v|2L2 − v 2 (x) − |v|2L2 ∞ ⩽ |v ′ |2L2 2 − 3δ L L L   L(1 − α)  21  3δ 3δ ⩽ |v ′ |2L2 2 − |v|2L2 + √ |v|L2 . L δ 18 Let f (r) = 2−

3δ 2  δL(1 − α)  12 1 3 r + r, then for r ⩾ δ − 2 L 2 αa , L 2

a ⩾ ln

(5.10.85) 

1−α 18

 /2 ln α,

f (r) ⩽ 0. Hence, for |v|L2 ⩾ δ − 2 L 2 αa , the right-side of (5.10.85) is negative. By contradiction, 3 for t satisfying |v(t)|L2 ⩾ δ −1 L 2 αa , one must have 1

3

d |v(t)|L2 < 0. dt

5.10

Inertial set

339

By the above lemma, it is known that set B0 = {u ∈ H : |u′ |L2 ⩽ ρ1 } is the absorbing set of the solution operator {S(t)}t⩾0 , t < t0 . This infers the existence of global solution. Next, we consider the absorbing set on H 2 (0, L). Lemma 5.10.12 Let B = {u ∈ H | |u′ |L2 ⩽ ρ1 , |u′′ |L2 ⩽ ρ2 },

(5.10.86)

where ρ1 and ρ2 are determined by (5.10.78), then B is the absorbing set of equation (5.10.77) on H. Proof Similar to Lemma 5.10.7, consider v = u′ and w = v ′ = u′′ satisfying the equation. Firstly, by (5.10.80), there holds 1 d 2 |v| + |v ′′ |2 + 3δ|vv ′ |2 + α|v|2 = 2|v ′ |2 ⩽ 2|v||v ′′ |. 2 dt For the above inequality, integrating with respect to t from t0 to t, and taking advantage of |v| ⩽ ρ1 , then yields Z t  |v ′′ (s)|2 + 3δ|vv ′ (s)|2 + α|v(s)|2 ds t0

Z

t

1 |v ′′ (s)| ds + ρ21 2 t0 Z t   12 1 1 ⩽ 2ρ1 (t − t0 ) 2 |v ′′ (s)|2 ds + ρ21 2 t0 Z t 1 1 |v ′′ (s)|2 ds + 2ρ21 (t − t0 ) + ρ21 . ⩽ 2 t0 2 ⩽ 2ρ1

Therefore, Z

t

(|v ′′ (s)|2 + 6δ|vv ′ (s)|2 + 2α|v(s)|2 ) ds ⩽ ρ21 (4(t − t0 ) + 1).

(5.10.87)

t0

For t − t0 ⩽ 1, it is derived from |v ′ |4 ⩽ |v|2 |v ′′ |2 that Z

t

|v ′ |4 ds ⩽ 5ρ41 .

(5.10.88)

t0

Hence, there exists s0 ∈ [t0 , t], such that |v ′ (s0 )| ⩽ 5 4 ρ1 . Secondly, consider w = v ′ = u′′ satisfying the following equation 1

wt + w′′′′ + 2w′′ − 3δ(v 2 w)′ + (2vw)′ + αw = 0. This implies

340

Chapter 5

Infinite Dimensional Dynamical Systems

1 d |w|2 + |w′′ |2 + α|w|2 = 2|w′ |2 − 3δ(v, w3 ) + (w, w2 ) 2 dt Z L 1 ⩽ |w′′ |2 + 2|w|2 + (3δ|v|L∞ + 1) w3 (x) dx 2 0 1 1 1 ′′ 2 2 ⩽ |w | + 2|w| + (3δ|v| 2 |w| 2 + 1)|w|2 |w′ | 2 1 1 5 1 1 ⩽ |w′′ |2 + 2|w|2 + (3δρ12 w 2 + 1)|w| 2 |w′′ | 2 2 1

⩽ |w′′ |2 + 2|w|2 + (3δρ12 ) 3 |w|4 + |w| 3 . 4

10

Rt (5.10.88) infers t0 |w|4 ds ⩽ 5ρ41 . Thus, |w(t)|2 is bounded for t ⩾ t0 by the uniformly Grönwall lemma. The purpose is to find the explicit form of the bound for |w(t)|2 . Consider the function V (v) =

1 1 ′2 |v | − |v|2 + δ|v 2 |2 , 2 4

then, the rate of change along the trajectory {v(t)}t⩾0 is  d dv  V (v(t)) = V˙ (t) = − v ′′ − 2v + δv 3 , dt dt   ′′ 3 ′′ = − v − 2v + δv , (−v − 2v + δv 3 )′′ − 2vv ′ − αv = − |(v ′′ + 2v − δv 3 )′ |2 + 2(v ′′ , vv ′ ) − α|v ′ |2 + 2α|v|2 − δα|v 2 |2 αδ 2 2 |v | + 2(v ′′ , vv ′ ) 2 ⩽ − 2αV + 2(v ′′ , vv ′ ) |v ′′ |2 √ ⩽ − 2αV + √ + 6δ|vv ′ |2 6δ √ 1 ⩽ − 2αV + √ (|v ′′ |2 + 6δ|vv ′ |2 ), 6δ ⩽ − 2αV −

t ⩾ t0 .

Taking integration with respect to t from t0 to t, which get along with (5.10.87), gives rise to 1 V (t) ⩽ exp(2α(t − t0 ))V (t0 ) + exp(−2αt) √ 5ρ21 6δ √ 2 ⩽ V (t0 ) + 5ρ1 / 6δ 1 1 ′ δ |v (t0 )|2 + |v 2 (t0 )|2 + 3ρ21 /δ 2 . (5.10.89) 2 4 1 RL 2 v , applying the Poincaré inOn the other hand, for the function u = v 2 − L 0 equality yields L2 ′ 2 L2 L2 |u|2 ⩽ |u | = |(v 2 )′ |2 = 2 |vv ′ |2 . 2 2 4π 4π π

⩽

5.10

Inertial set

341

1 RL 2 1 Since |u|2 = v 2 (x) − v (y) dy 2 = |v 2 |2 − |v|4 , accordingly, L 0 L L 1 4 L2 |vv ′ |2 . |v| ⩽ L 4π 2 Integrating with respect to t from t0 to t, it is deduced from (5.10.87) and t0 = t − 1 that Z t+1 Z δ t+1 δ L2 ρ2 L2 2 2 δ |v (τ )|L2 dτ ⩽ |v(τ )|4L2 dτ + 2 5ρ21 ⩽ ρ41 + 2 1 . L t 4π L π t |v 2 |2 −

Integrating (5.10.89) from t0 = t to t0 = t + 1 yields √ 5 2 8ρ41 L2 ρ21 3ρ21 2 1 − 21 V (t) ⩽ ρ1 + + + ). 1 ⩽ ρ1 C0 (1 + L + δ 2 2 4L 4π δ2 This infers for t ⩾ t0 , |v ′ (t)|2 ⩽ V (t) + |v(t)|2 ⩽ C02 ρ21 (1 + L + δ − 4 )2 . 1

Now, back to the original equation (5.10.87), we have Au = u′′′′ ,

R(u) = −2u′′ − 3δ(u′ )2 u′′ + (u′ )2 + αu.

The absorbing set is B = {u ∈ H : |u′ | ⩽ ρ1 , |u′′ | ⩽ ρ2 }. Firstly, consider the equation of the difference of two solution. Let w = u1 − u2 , where u1 , u2 are two solutions of (5.10.87), such that w satisfies wt + w′′′′ + 2w′′ − 3δ[(u′1 )2 u′′1 − (u′2 )2 u′′2 ] + [(u′1 )2 − (u′2 )2 ] + αw = 0.

(5.10.90)

u ¯ = u1 + u2 , then 1 d |w|2 + |w′′ |2 + α|w|2 = 3δ(u′′1 u ¯′ w′ , w) + 3δ((u′2 )2 w′′ , w) + (¯ u′ w′ , w) 2 dt ⩽ 3δ|u′′1 |L2 |¯ u′ |L2 |ww′ |L∞ + 3δ|w′′ |L2 |w|L2 |(u′2 )2 |L∞ + |¯ u′ ||w′ ||w|L∞ ⩽ 6δρ2 ρ1 |w|L∞ |w′ |L∞ + 6δ|w′′ ||w||u′2 |L2 |u′′ |L2 + 2ρ1 |w| 2 |w′ | 2 1

3

⩽ 6δρ2 ρ1 |w||w′′ | + 6δρ1 ρ2 |w||w′′ | + 2ρ1 |w| 4 |w′′ | 4 8 1 ⩽ |w′′ |2 + C1 (δ 2 ρ21 ρ22 + ρ15 )|w|2 2 8 1 ′′ ⩽ |w | + C2 (δ 2 ρ21 ρ22 + ρ15 )|w|2 . 2 5

3

342

Chapter 5

Infinite Dimensional Dynamical Systems

By the Grönwall’s inequality, it is inferred that 8

|w(t)| ⩽ exp{C2 [δ 2 ρ21 ρ22 + ρ15 ]t}|w(0)|.

(5.10.91)

Consequently, S(t) is a Lipschitz mapping, the Lipschitz constant can be estimated by (5.10.91). Denote λ(t) = |w′′ (t)|2 /|w(t)|2 ,

ξ(t) = w(t)/|w(t)|2 ,

then

8 1 d |w|2 + (λ(t) − C2 [ρ15 + δ 2 ρ21 ρ22 ])|w|2 ⩽ 0. 2 dt Applying the Grönwall’s inequality implies

|w(t)| ⩽ δ(t)|w(0)|, where δ(t) = −

1 2

Z

t

λ(τ ) dτ + 0

(5.10.92)

C2 85 [ρ + (δρ1 ρ2 )2 ]t. 2 1

(5.10.93)

In order to show the squeezing property of S(t∗ ), asssume t = t∗ , 4 1  2π 1 (N0 + 1) . λ∗ = λ(t∗ ) ⩾ λN0 +1 = 2 2 L

(5.10.94)

Consider the quotient module λ(t) satisfying 1 d 1 λ(t) = (wt , (A − λ)w) 2 dt |w|2   R(u ) − R(u ) 2 1 , (A − λ)ξ = − |(A − λ)ξ|2 + |w| 1 1 1 |R(u2 ) − R(u1 )|2 . ⩽ − |(A − λ)ξ|2 + 2 2 |w|2

(5.10.95)

Now, estimate the difference of nonlinear term |R(u1 ) − R(u2 )| = | − 2w′′ − 3δ[(u′ )2 w′′ + 2v ′′ u ¯′ w′ ] + 2¯ u′ w′ | ⩽ 2|w′′ | + 3δ|(u′ )2 |L∞ |w′′ | + 6δ|v ′′ ||¯ u′ ||w′ |L∞ + 2|¯ u′ ||w′ |L∞ ⩽ 2λ 2 |w| + 3δ|u′ ||u′′ |λ 2 |w| + 6δρ2 ρ1 |w′ | 2 |w′′ | 2 + 2ρ1 |w| 4 |w′′ | 4 1

1

1

1

3

1

1

3

3

⩽ (2 + 3δρ1 ρ2 )λ 2 |w| + 6δρ1 ρ2 λ 8 |w| + 2ρ1 λ 8 |w|. Combining (5.10.96) and (5.10.95), which get along with Young’s inequality, gives rise to λ′ ⩽ C3 δ 2 ρ21 ρ22 λ + C4 (δρ1 ρ2 )2 λ 4 + 4ρ21 λ 4 3

3

⩽ C5 (δρ1 ρ2 )2 λ + C7 [(δρ1 ρ2 )2 + δ − 2 ] 1

⩽ C5 (δρ1 ρ2 )2 λ + C8 (δρ1 ρ2 )2 .

5.10

Inertial set

343

Applying the Grönwall lemma λ(t) ⩽ eβ1 (t−t0 ) λ(t) − (1 − eβ1 (t−t0 ) )(β2 /β1 ),

(5.10.96)

where β1 = C2 (δ1 ρ1 ρ2 )2 , β2 = C8 (δ1 ρ1 ρ2 )2 . For 0 ⩽ t0 ⩽ t, one can solve λ(t0 ) from (5.10.96). Taking advantage of (5.10.94) implies β2 λ(t0 ) ⩾ eβ1 (t1 −t∗ ) λ∗ + (eβ1 (t0 −t∗ ) − 1). β1 Integrating with respect to t0 from t0 = 0 to t0 = t∗ infers Z t∗ β  1 2 (1 − e−β1 t∗ )λ∗ − t∗ . (5.10.97) λ(t0 ) dt0 ⩾ β β 1 1 0 Denote t∗ = β1−1 = C5−1 (δρ1 ρ2 )−2 = −C5−1 L−6 (1 + L + δ − 4 )−2 , 1

(5.10.98)

then, from (5.10.93) it is derived that δ∗ = δ(t∗ ) ⩽ − To guarentee δ∗
0 and nonnegative a, b. 3. Young’s Inequality ab ⩽

1 1 (ε1 a)p + ′ p p



b ε1

 p′

1 1 + = 1, p p′

,

ε1 > 0.

4. Hölder’s Inequality q′ 1/q′ q 1/q  Z Z Z , u(x)v(x) dx ⩽ v(x) dx u(x) dx Ω

Ω

where q ⩾ 1,

Ω

1 1 + ′ = 1. A more general form of Hölder’s Inequality q q Z s 1/λi s Z Y Y |ui |λi dx , ui (x) dx ⩽ Ω Ω i=1

i=1

s 1 P = 1. i=1 λi 5. Grönwall’s Inequality Assume u(x) and h(x) are nonnegative continuous functions on [0, 1], c ⩾ 0 is a constant. If 0 ⩽ x ⩽ 1, Z

where λi ⩾ 1,

x

u(x) ⩽ c +

h(t)u(t) dt, 0

then for 0 ⩽ x ⩽ 1, one has u(x) ⩽ c exp

Z

x

 h(t) dt .

0

6. Jensen’s Inequality Assume Φ : R → R is convex downward, f ∈ L 1 [a, b], then ! Z b Z b 1 1 f (t) dt ⩽ Φ(f (t)) dt. Φ b−a a b−a a

348

Chapter 6

6.2

Appendix

Sobolev embedding theorem and interpolation formula

Assume Ω ⊂ Rn is a bounded region. If there exists a and h, such that for any x ∈ Ω, one can construct a right circular cone Vx with vertex x, apex angle α, height h in Ω, then Ω has the corn property. For example, if ∂Ω ∈ C 1 or Ω is convex, then Ω has the corn property. Theorem 6.2.1 Suppose Ω is a bounded region with the corn property and corn constants α and h. Assume u(x) ∈ C m (Ω) ∩ Wpm (Ω), where p > 1. If m > n/p, then sup |u(x)| ⩽ c∥u∥Wpm (Ω) ,

(6.2.1)

x∈Ω

where the constant c only depends on α, h, m and p. Corollary 6.2.2 Assume m −

n p

> l, where l is a non-negative integer. Then

|Dα u(x)| ⩽ c∥u∥Wpm (Ω) ,

0 ⩽ |α| ⩽ l.

(6.2.2)

Corollary 6.2.3 Assume u(x) ∈ Wpm (Ω) and m − n/p > l, where l is a nonnegative integer. Then u(x) ∈ C l (Ω) (almost everywhere). Theorem 6.2.4 Assume u ∈ C01 (Rn ), p is a real number and greater than n. Then, for any x, y ∈ Rn , n X |u(x) − u(y)| ⩽ c |Di u|0,p = c∥Du∥Lp (Rn ) , |x − y|1−n/p i=1

(6.2.3)

where the constant c depends only on n, p. Theorem 6.2.5 Assume q, r are arbitrary real numbers satisfying 1 ⩽ q, r ⩽ +∞; j, m are arbitrary integers satisfying 0 ⩽ j < m. If u ∈ C0m (Rn ), then 1−α ∥Dj u∥Lp (Ω) ⩽ c∥Dm u∥α Lr (Ω) ∥u∥Lq (Ω) ,

where

(6.2.4)

1 m 1 j 1 = +α − + (1 − α) , p n r n q

and j/m ⩽ α < 1. The constant c depends only on n, m, j, q, r and α. If m − j − integer, then, for j/m ⩽ α < 1, (6.2.5) holds.

(6.2.5) n r

is a nonnegative

6.2

Sobolev embedding theorem and interpolation formula

349

Theorem 6.2.6 (interpolation formula on bounded set Ω ⊂ Rn ) Assume Ω ⊂ R is a bounded region, ∂Ω ∈ C m , u(x) ∈ Wrm (Ω) ∩ Lq (Ω), 0 ⩽ l ⩽ m. If m − l − n/r is not a nonnegative integer, n  n n l ⩽ α ⩽ 1, 1 ⩽ r, q ⩽ +∞, −l =α − m + (1 − α) . m q r q n

If m − l −

n is a nonnegative integer, r l ⩽ α < 1, m

1 < r < ∞,

1 < q < ∞,

then 1−α ∥Dl u∥Lp (Ω) ⩽ c∥u∥α Wrm (Ω) ∥u∥Lq (Ω) ,

(6.2.6)

where the constant c = c(Ω, n, m, j, q, r, α). When p > 0, 1/p Z ; |u|p dx ∥u∥p,Ω = Ω

when p < 0, h= (i) For λ = 0,

hni , |p|

λ=

n − h. |p|

∥u∥p,Ω = sup Dh u(x) ; x∈Ω

(ii) for λ > 0, |Dh u(x) − Dh u(y)| . |x − y|λ x,y∈Ω

∥u∥p,Ω = sup

Theorem 6.2.7(embedding theorem) Suppose there is a bounded set Ω ⊂ Rn , ∂Ω ∈ C m . n (i) If m − > 0, r ∥u∥C (h,α) (Ω) ⩽ c∥u∥Wrm (Ω) , (6.2.7) hni − 1 ⩾ 0. h=m− r hni n If m − is not a postive integer, 1 ⩽ r < ∞, α = 1 − ; r r n If m − is a postive integer, 1 < r ⩽ ∞, 0 ⩽ α < 1, r hni hni If 0 ⩽ α < 1 − , 0 ⩽ h ⩽ m− − 1, the embedding operator Wrm (Ω) ,→ r r C (h,α) (Ω) is completely continuous or the embedding is compact. n (ii) If m − l − = 0, r ∥u∥Wpl (Ω) ⩽ c∥u∥Wrm (Ω) ;

1 ⩽ p < ∞, 1 ⩽ r < ∞.

(6.2.8)

350

Chapter 6

If m − l −

n < 0, r

∥u∥Wpl (Ω) ⩽ c∥u∥Wrm (Ω) ; 1 ⩽ r ⩽ ∞,

If 0 ⩽ l ⩽ m − 1,

1⩽p
0, we have f (u) + λu ̸= 0,

∀u, ∥u∥ = r.

Then, there exists a u0 , ∥u0 ∥ < r, such that f (u0 ) = 0. Theorem 6.3.4 (Schauder fixed-point theorem) Assume C is a bounded, closed, convex set in X, and f : C → C is a compact mapping. Then, there exists x0 ∈ C, such that f (x0 ) = x0 .

6.3

Fixed point theorem

351

Theorem 6.3.5 Assume C is a compact, convex set in X, and f : C → C is a continuous mapping. Then, there exists x0 ∈ C, such that f (x0 ) = x0 . Theorem 6.3.6 Assume C is a self-adjoint Banach space, T : X → X ∗ is a closed, linear mapping, satisfying | < T x, x > | ⩾ c∥x∥2 ,

∀x ∈ X, c > 0.

Then T x = x. Theorem 6.3.7 Assume X is a self-adjoint Banach space, and B :X ×X →C is linear for the first variable, antilinear for the second variable, and has the following properties (i) |B(x, y)| ⩽ c1 ∥x∥∥y∥, ∀x, y ∈ X, (ii)|B(x, y)| ⩾ c2 ∥x∥2 , ∀x, y ∈ X. Then for any x ∈ X, there exists a unique x∗ ∈ X ∗ , such that X ∗ (y) = B(y, x). Theorem 6.3.8 (Leray-Schauder fixed-point theorem) Assume E is a Banach space. If the nonlinear operator equation x − A(x, λ) = 0,

x ∈ E, λ ∈ I = [0, 1],

satisfies the following conditions: (i) for X ∈ I, the operator A : E × I → E is a complete continuous operator; for the bounded set M ⊂ E, the operator A is uniformly continuous with respect to λ, (ii) for all the possible solution x of x − A(x, λ) = 0, for λ ∈ I, the norm ∥x∥E is uniformly bounded in E, (iii) when λ = 0, A(x, 0) = x has unique solution in E, then for all λ ∈ I, the function x − A(x, λ) = 0 has at least one solution.

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Index Absorbing set, 197, 244

Jensen’s inequality, 347

Banach fixed-point theorem, 350 Benjamin-Ono equation, 93 Blow up, 72 Born-Infeld equation, 25 Boussinesq equation, 25 Brouwer fixed-point theorem, 350

Kadomtsev-Petviashvili equation, 27 Korteweg de Vries equation, 2

Cauchy inequality, 347 Decay estimate, 64 Direct method, 169

Landau-Lifshitz equation, 19 Leray-Schauder fixed-point theorem, 351 Lie algebra, 157 Lie group transformation, 156 Lyapunov exponent, 201 Maximal set, 325

Embedding theorem, 349

Navier-Stokes equations, 60

Fractal dimension, 199

Painlevé property, 174 Poincaré fixed-point theorem, 350

Galerkin method, 296 Global attractor, 196 Global Lyapunov exponent, 201 Grönwall’s inequality, 347 Hölder’s inequality, 347 Hausdorff dimension, 199 Hausdorff measure, 199 Inertial manifold, 269, 273 Inertial set, 324, 325 Infinitesimal operator, 156 Infinitesimal transformation, 143

Schauder fixed-point theorem, 350 Schrödinger equation, 30 Similar solution, 169 Smooth solution, 30 Solitons, 10 Squeezing property, 282, 324 Toda lattice, 23 Weak solution, 34 Young’s inequality, 347 Zakharov equation, 10