Mathematical Aspects of Fluid Mechanics [1 ed.] 9781107609259, 1107609259

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‘Mathematical Aspects

of Fluid Mechanics— edied .

)

James C. Robinson, José L Rodrigo and Witold Sadowski

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London Mathematical Society Lecture Note Series: 402 “

Mathematical Aspects of Fluid Mechanics Edited by JAMES C. ROBINSON University of Warwick

JOSE L. RODRIGO University of Warwick

WITOLD SADOWSKI University of Warsaw, Poland

a] CAMBRIDGE UNIVERSITY

PRESS

CAMBRIDGE

UNIVERSITY

PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi, Mexico City

Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/978 1107609259

© Cambridge University Press 2012 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2012 Printed and bound in the United Kingdom by the MPG Books Group A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data

ISBN 978-1-107-60925-9 Paperback

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So

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Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. a ee

To Tania, Elizabeth, & Dorota

Contents

Preface List of Contributors 1

2

page ix Dal

Towards fluid equations by approximate deconvolution models L.C. Berselli

On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph M. Bultéek, P. Gwiazda, J. Malek, K.R. Rajagopal, & A. Swierczewska-Gwiazda

if

23

3

A continuous model for turbulent energy cascade A. Cheskidov, R. Shuydkoy, & S. Friedlander

52

4

Remarks on complex fluid models P. Constantin

70

5

A naive parametrization for the vortex-sheet problem A. Castro, D. Cordoba, & F. Gancedo

6

Sharp and almost-sharp fronts for the SQG equation

88 116

C.L. Fefferman a

8

Feedback stabilization for the Navier—Stokes equations: theory and calculations A.V. Fursikov & A.A. Kornev

130

Interacting vortex pairs in inviscid and viscous

ube}

planar flows T. Gallay vil

Vili

Contents

Stretching and folding diagnostics in solutions of the three-dimensional Euler and Navier—Stokes equations J.D. Gibbon & D.D. Holm 10

11

12

Exploring symmetry plane conditions Euler solutions R.M. Kerr & M.D. Bustamante

201

in numerical 221

On the decay of solutions of the Navier—Stokes system with potential forces I. Kukavica

235

Leray—Hopf solutions to Navier-Stokes with weakly converging initial data G. Seregin

251

equations

Preface

This volume is the result of a workshop, “Partial Differential Equations and Fluid Mechanics”, which took place in the Mathematics Institute at the University of Warwick, June 15th—19th, 2010. Several of the speakers agreed to write review papers related to their contributions to the workshop, while others have written more traditional research papers. We believe that this volume therefore provides an accessible summary of a wide range of active research topics, along with some exciting new results, and we hope that it will prove a useful resource for both graduate students new to the area and to more established researchers. We would like to express their gratitude to the following sponsors of the workshop: the London Mathematical Society, EPSRC (via a confer-

ence grant EP/1001050/1), and the Warwick Mathematics Department. JCR is currently supported by an EPSRC Leadership Fellowship (grant EP /G007470/1). Finally it is a pleasure to thank Yvonne Collins and Hazel Higgens from the Warwick Mathematics Research Centre for their assistance during the organization of the workshop.

James

C. Robinson

José L. Rodrigo Witold Sadowski

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List of Contributors

Those contributors who presented their work at the Warwick meeting are indicated by a star in the following list. We have also included the addresses of the editors.

Luigi C. Berselli* Dipartimento di Matematica Applicata “U. Dini”, Universita di Pisa,

Via F. Buonarroti 1/c, Pisa. Italy. [email protected]

Miroslav Bulitek Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovska 83, 186 75 Prague 8. Czech Republic. [email protected]

Miguel D. Bustamante School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4. Ireland. [email protected]

Angel Castro Departement de Mathematiques et Applications, Ecole Normale Superieure, 45, Rue d’Ulm, 75005, Paris. France. [email protected]

xl

List of Contributors

xii

‘ Alexey Cheskidov * Department of Mathematics, Stat. and Comp. Sci., University of Illinois, Chicago, IL 60607. USA. [email protected]

Peter Constantin * Department of Mathematics, Princeton University, Fine Hall, Wasington Rd, Princeton, NJ 08544. USA. [email protected]

Diego Cérdoba* Instituto de Ciencias Matematicas,

Consejo Superior de Investigaciones Cientificas, Serrano 123, 28006 Madrid. Spain. [email protected]

Charles L. Fefferman* Department of Mathematics, Princeton University, Princeton, NJ, 08544. USA. cf@math. princeton.

edu

Susan Friedlander * Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089. USA. [email protected]

Andrei V. Fursikov * Department of Mechanics and Mathematics, Moscow State University, 119991 Moscow. Russia. [email protected]

Thierry Gallay * Université de Grenoble I,

Institut Fourier, UMR CNRS 5582, B.P. 74, F-38402 Saint-Martin-d’Heéres. France. [email protected]

List of Contributors ve

Francisco Gancedo Departamento de Analalisis Matematico, Universidad de Sevilla,

C\Tara s/n, Campus Reina Mercedes, 41002 Sevilla. Spain. [email protected] John D. Gibbon* Department of Mathematics, Imperial College London, London, SW7 2AZ. UK. [email protected]

Piotr Gwiazda Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw. Poland. [email protected]

Darryl D. Holm* Department of Mathematics, Imperial College London, London, SW7 2AZ. UK. [email protected]

Robert M. Kerr* Mathematics Institute, University of Warwick, Coventry, CV4 7AL. UK. Robert [email protected]

Andrei A. Kornev Department of Mechanics and Mathematics, Moscow State University, 119991 Moscow. Russia. [email protected]

xill

xiv

List of Contributors

Igor Kukavica * Department of Mathematics, University of Southern California, Los Angeles, CA 90089. USA. [email protected]

Josef Malek * Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovska 83, 186 75 Prague 8. Czech Republic. [email protected]

Kumbakonam R. Rajagopal Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843. USA. [email protected]

James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL. UK. [email protected]

Jose L. Rodrigo Mathematics Institute, University of Warwick, Coventry, CV4 7AL. UK. [email protected]

Witold Sadowski Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa. Poland. [email protected]

Gregory Seregin* Mathematical Institute, Oxford University, 24-29 St Giles’, Oxford, OX1 3LB. UK. [email protected]

List of Contributors

Roman Shvydkoy * Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637. USA. [email protected]

Agnieszka Swierczewska-Gwiazda Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw. Poland. [email protected]

XV

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Towards fluid equations by approximate deconvolution models Luigi C. Berselli Dipartimento di Matematica Applicata “U. Dini” Universita di Pisa, Via F. Buonarroti 1/c, Pisa. Italy. [email protected]

Abstract We review a selection of recent results linking approzimate deconvolution operators with the rigorous approximation of the Navier-Stokes equations and their averages.

1.1

Introduction

When studying existence, uniqueness, and other analytical properties of solutions to partial differential equations in many different situations one needs, as a general tool, suitable smoothing/approximation operators. These tools are used to construct approximate initial data and/or approximate equations and are needed, for instance, to show existence through approximation by smooth solutions or to make rigorous calculations that will be otherwise just formal (for example in the study of energy equalities). In the case of the incompressible Navier-Stokes

equations (NSE) (NSE)

uz + (u-V)u—vAu+

Vp =0,

(1815

with V -u = 0, these tools have been used, for instance, in the classical paper by Leray (1934), where the convective term (u- V) u is approxi-

mated by (p.*u-V) u, with {p-}-s0 a classical family of Friedrichs mollifiers. Other applications of smoothing techniques occur also in the study

of singular limits, as in Beirao da Veiga (1993) and Majda (1984). See also Kazhikhov (2006), where he studies approximate sequences weakly converging in stronger topologies, with applications to the study of com-

pressible fluids. Published in Mathematical

Aspects of Fluid Mechanics,

edited by J.C. Robinson,

J.L. Rodrigo, & W. Sadowski. ©Cambridge University Press 2012.

2

Berselli

Recently these tools have also been linked with a particular class of Large Eddy Simulation (LES) models and in particular with some aspects of the mathematical theory of alpha-models. We recall that LES models provide families of approximate systems which are computationally easier to study than the full NSE, see Sagaut (2001), Geurts (2003), ‘Lesieur, Métais, & Comte (2005), and Berselli, Iliescu, & Layton (2006) for an introduction to some of the aspects of LES. These models, which are designed to be numerical methods for the practical computation of averages of the velocity, are derived by means of different physical, analytical, and numerical insights. In particular, we will review results on approximate deconvolution alpha-methods, pointing out that their mathematical analysis has only recently been addressed. We do not treat any questions about modelling or numerical testing, but we just make a review of some recent results for high accuracy models. These models with high accuracy are obtained by introducing approximate deconvolution into some well-known classical models; we will consider them as mathematical methods to produce smooth and stable approximations to the fluid motion, with conserved physical quantities.

We will restrict to the space-periodic case with x €]0,L[° (this is the only setting in which calculations have a sufficient level of mathematical rigour) and we will consider the differential filter associated with the Helmholtz operator A := I — a?A, for some a > 0. To this end, given a field u, we define the averaged field @ as the solution of

Au =u — a’ At =u, with periodic boundary conditions. It is easy to show that in this setting, if V-u=0, then V -@ = 0, and 7@ is also a solution of the Helmholtz— Stokes system with zero pressure. In the sequel we will denote by - the quantity obtained by application of the filter, that is, the inverse of A, i.e. u := A-‘u. Moreover, if H* denotes the usual Sobolev space of periodic

functions with norm || - ||,, and H* Cc (H*)% denotes the subspace of divergence-free fields with zero mean-value, then when u € H®

1. 2.

w@e€ H°*? and |[@— ull) < ca2*-*) for s’ < s, and u—>uin

H’,asa>0t.

We will apply the ideas of approximate deconvolution to the following three systems (Leray-a, Navier—-Stokes-a, and Layton & Lewandowski

1: Towards fluid equations by ADM

or simplified Bardina model)

(L-a)

a

z

wit (w-V)w—vAw+ V¢ =0, wi — W x (V x w) —vAw+ Vq = 0,

(NS-a) (LL)

wit+(w:-V)w—vAw+

Vq=0.

(1.2)

(1.3) (1.4)

These models are supplemented with V -w = 0, initial datum, and periodic boundary conditions. The external force is set to zero to avoid inessential complications. Here w denotes a field which is formally closer and closer to the solutions of the NSE as the parameter a becomes smaller and smaller, hence when the averaging defined through A~! approaches the identity I. Remark 1.1.1 In part of the literature a different notation is used with functions u and v such that v := u—a?Au. Up to some changes in the pressure, the three models without deconvolution (1.2), (1.3), and

(1.4) are also denoted by

(L-a)

vit (u- V)v—vAv+ Vq=0,

(NS-a)

(LL)

vz —ux

(V x v) —vAv+Vq =),

vit (u-V)u—vAv+ Vq=0.

This notation reflects also a different way in which estimates are written, since they can be stated in terms of u or of v. Moreover, in several cases the parameter a is replaced by 6, in analogy with the classical notation used in early studies of LES. We cannot review all the literature concerning these models and we just cite the most theoretical references treating them: therein the interested reader can find further results, related also with numerical computations. The mathematical analysis for the Leray-a model (which is the natural adaptation of Leray’s approach in the periodic case) can be

found in Cheskidov et al. (2005), while a modified method with regularized convective term

(w- V)W has been studied in Iyin, Lunasin,

& Titi (2006). Concerning the NS-a model, also known as the viscous Camassa—Holm

equation, most of the results are proved in Foias, Holm,

& Titi (2002). A modification with nonlinearity given by —w x (V x w) and known as NS-w can be found in Layton et al. (2010). The latter (LL) model is studied in Layton & Lewandowski (2006a). It is also known as the simplified Bardina model and has additionally been analysed in Cao, Lunasin, & Titi (2006). Observe too that this model is the same as that

4

Berselli

of Stolz & Adams when the deconvolution parameter vanishes: see Stolz & Adams, 1999, Stolz, Adams, & Kleiser, 2001, Adams & Stolz, 2001. These three models have been derived by different approaches: for instance smoothing, clear balance of generalized energies (and models helicity for NS-a without viscosity), and scale similarity, topics which -we do not address here. For the interested reader comparison of the conserved quantities can be found in Rebholz (2007); see also Olson & Titi (2007). The introduction of Approximate Deconvolution Models (ADM) in LES can be understood in the light of the following observation: after having solved one of the above systems, the resulting field w is smooth, and unique. It also solves a system (formally) close to that solved by u, where u is a solution of the NSE with the same initial datum. On the other hand, when a — 0* it is possible to prove the convergence w — u, where u is a Leray—Hopf weak solution. (This result is proved in the above references and one needs some care to handle this limit rigorously.) Nevertheless, this latter result does not seem to have a terrific impact in applications, since the radius of the filter a > 0 is related to the mesh size h of the numerical method used to simulate fluid motion. (In our setting this is not the radius, but the name is used by analogy to that related with filtering by convolution with functions of compact support or with Gaussian fields.) The parameter a is related (and it should be of the same order as h) to the smallest persistent scale. Using Fourier

series expansions, if u(x) = )>. A,

as

N — +00.

Having such a family of operators, the Leray-a model can be replaced by the model with higher accuracy

w:+ (Dvnw:V)w-—vAw-+ Vq =0. The convective field Dyw is closer to the field w than the previous one Dow := W present in (1.2). Most likely the properties of the model with

(AD) are much better than those of Leray-a, which can be seen as a zeroth-order deconvolution model. In several cases the introduction of a deconvolution operator is suggested, as in Stolz & Adams (1999) with N ~ 5, and the practical use in computations is to fix N and a, tuning the parameters to optimize the performance of the code versus the numerical instabilities. On the other hand, from the theoretical point of view we would like to have mathematical support for this modelling idea. In order to justify the implementation of deconvolution models, we will study their limiting behaviour, trying to obtain some insight from the analytical results. A

Berselli

6

review of ADM

(2012).

can also be found in the book by Layton & Rebholz

;

Plan of the paper. We will review some of the PDE results concerning the application of AD to the three aforementioned models. In particular, ‘in Section 1.2 we introduce the basics of deconvolution operators and in Section 1.3 we introduce ADM. Next, we will skip all modelling and numerical testing and in Sections 1.4—1.5 we will focus only on the energy spectra and on the rigorous mathematical analysis results which can be obtained by considering the limits a — 0+ for fixed N € N, and N -— +00 for fixed a > 0.

1.2 The approximate

deconvolution

Once a filter is defined, it is computationally relevant to have an approximate way to invert it. Approximation is needed since, in principle, the filtering operator defined by G = A~! is not invertible, or the inverse is not bounded, or it is not possible to invert it stably, due to a small di-

visor problem, as with the Gaussian filter (Berselli et al. (2006) §7). For these reasons one wishes to have some kind of a “best approximation for its inversion” or equivalently an approximate solution to the problem: given @ find u such that

Av

=.

(1.5)

The classical example coming from signal theory is that of a signal filtered by some transmitting/recording device, where the challenge is to reconstruct, in a satisfactory way, the original signal. Early results on deconvolution have been obtained by Wiener (1949), even if the ideas are older and some delay in their diffusion has been caused by the book being classified during World War II. Another field in which one generally uses approximate deconvolution is that of inverse problems as in image reconstruction and, in fact, the first example we will consider comes from this field.

Some deconvolution operators. We review some of the classical deconvolution operators, and we specify them in the case of the Helmholtz operator with periodic boundary conditions, in order to better compare their properties.

1: Towards fluid equations by ADM

t

van Cittert. The van Cittert (1931) algorithm (common in image reconstruction) is a Richardson iteration for equation (1.5) that works as follows: given ug := U, set

uny+1 := uy + (U— Atuy).

The operator Dy is defined by Dy@ := uy. The van Cittert algorithm is based on successive applications of the filter (in fact u,; = 2u — @, un

=

3u — 3u + 4B, and so on). One

of the main

properties of this

algorithm, in the case of the Helmholtz operators is the following, which is based on a representationb via a truncated von Neumann series, see

Dunca & Epshteyn (2006), Stolz et al. (2001), and Berselli et al. (2006) 8 8. Lemma

1.2.1

Let A~+ be defined through the Helmholtz filter. Then,

for any w € L? it follows that w—

Dyw=

lea

ae

Ae TA

ey

This identity can be used to estimate the residual stress in a precise way for different LES models, see Layton & Lewandowski (2006b). One of the interesting features of the van Cittert operator is that it can be applied also in more complicated situations (boundary value problems) even if its properties are slightly different in that setting (in particular the problem of commutation with first order differential operators arises). By specifying the operator in the periodic setting we can write

its symbol as follows: DvCyu(k) := DvCy(k)u(k) with

Broth) = >> ( 0 by

I-(I

Ay jl

To compare with the van Cittert operator we write the explicit expression for the symbol of the Yosida approximation in Fourier variables

DY,,(k) := (1 + a?k?) ——__.. i M+ Trae

a

Tikhonov

and

1

Tikhonov—Lavrentiev.

The

classical

Tikhonov

method, see Tikhonov & Arsenin (1977), is based on the solution of a least squares method, with a regularization parameter pp > 0. The approximate solution of Gu = @ is given by u,, which solves

u,, := argmin ica — al? + plu], Ub

:

where we denote by ||. || the L?-norm. In case of a symmetric and positive-definite operator one can employ the Lavrentiev adaption and the Tikhonov—Lavrentiev regularization is given by the solution of the following minimization problem

ne

Tel

res

u,, := argmin 5 (Gu = @, u) + |jall?|; in

2

2

by differentiation it follows that

U1, = (jel Ay

ets

Ore

1, in the remaining part of the

spectrum (but still within the inertial range) we have the steeper decay E(k) ~ (N + 1)~?/3q-8/3-13/3, This also produces the following estimate for the number of degrees of freedom ;:

Nee

9

3/2

(=) (N + 1)9/4Re9/8. a

1: Towards fluid equations by ADM

al

Deconvolution NS-a. The same reasoning has also~been applied to the Deconvolution NS-a model in Rebholz & Sussman (2010), extending

to this case the estimate in Foias et al. (2002) for the case N = 0. By the balance law in (1.10) we have that the energy and dissipation spectra have the following expressions:

Eo,n(k) = S> Dy(k)|W(k)? + 0?|k]?Dw(k)[W(k)/?, \k|=k V ee, 73 ea —Aw: w: Dywdz,

Ea,N

from which we get that the average velocity of eddies of size k satisfies

(the transport is made by the field il

Dyw) 1/2

U, = (a [ pve:

Dywex)

2k

~ (

1/2

Pni(s)Fo,w(s) |

(1 + a282)

k

kil? Dy (k)t/? Be wk) 7?

(1 + a2k?)1/2

‘

and by inserting this into the same machinery one obtains

Ea,n(k) ~ y

ei (1 + a2k?)1/3 SS 5/3 (Dn(k))3/8

Observe that for small values of ak this is the same estimate as for the NS-a model, while there is a damping factor (N + 1)/3 for ak > 1, due to the properties of the deconvolution operator. Finally, recalling that the estimates are correct in the inertial range, that is when

he Bi N(R) WwW

Wn;

> WwW

qN;

74

weakly in L?(0,T;H?) strongly in L?(0,T;H'),

L©(0,T;H°), V10

G

This space equipped with the norm

lullpv = |lully := inf {rBae

| W(rA*u) dz < i} G

is a Banach space, see Rao & Ren (1991). An N-function y satisfies the A» condition if there exist C, > 0 and

C2 > 0 such that for all s € Re we have

w(2s) < Cyw(s) + Co. Next, if we assume that w satisfies the Ay condition then LY(G) is separable and moreover

(L¥(G))* = LY’ (G). Otherwise,

for any N-function ~, we know that the predual space to

L”’(G) is the closure of bounded functions with respect to the norm in LY(G). We denote this space by E¥(G), and recall that this is a separable space.

Finally, we formulate the Young and Hdlder inequalities for Orlicz spaces and V-functions, see e.g. Rao & Ren (1991): the Young inequality reads

la- b| < Y(a) + y*(d)

for all a,b@Ro**: sym *

the Hélder inequality states that if wu€ L¥(G) and v € LY (G), where u and v are real- or matrix-valued functions, then u-v € L(G) and

[ weds < autho

wr

G

In different parts of the paper we will assume that w satisfies one or both of the following assumptions

(H1) the function w satisfies the A> condition, (H2) the function y* satisfies the A» condition.

2: On flows of implicitly constituted fluids

3l

Recall, see Rao & Ren (1991) for detailed proofs, that if (H1) holds then there are positive constants c3,c4 > 0 and r < oo such that

(a) < c3la|” + e4

for all a € R2X¢ sym

ELE)

and similarly (H2) implies that there are positive constants c,,c. and q > 1 such that

(a) > exlal? — co

for alla € R22 sym

*

(H2*)

Note that (H1*) and (H2*) form exactly the condition (2.7). In the rest of the paper the letters r,q are always related to (H1*) and (H2*)

respectively, provided that either (H1) or (H2) holds.

2.3

Selections

Let A be a maximal monotone graph satisfying (A1)—(A4). Consider a mapping S, : Rese =} ieee assigning to each B € Rene exactly one value S.(B) € R&X! so that (B,S,(B)) € A. Such a mapping S, is called a selection of A. Obviously, each such selection S, defined on R2X? sym is

monotone and due to (A4) satisfies

(A4*)

S,(D)-D > c.(p(D) + ¥*(S,.(D)) — d, for all D e ROW?

It is also not difficult to observe, see also Alberti & Ambrosio (1999), that the following condition replaces (A3)

(A3*) For (D,S) € R@X?: sym° if (S—S,(B),D—B) 20

forallBe Ree then (D,S) € A.

In general, S, does not have to be a Borel function. On the other hand, there is a selection S$, that is a Borel function, see e.g. Aubin

& Frankowska

(2009), and only such a selection is considered in the

remaining parts of the paper.

2.4 Convergence tools In this section, we formulate simple criteria that allow us to verify that

S and D(v) fulfill the nonlinear constitutive equation (2.2), which is equivalent to (D(v),S) € A. The first criterion concerns in particular the case when y and 7)* fulfil (H1) and (H2). The second criterion

Bultéek, Gwiazda, Mdlek, Rajagopal, & Swierczewska-Gwiazda

32

concerns the general (non-reflexive) case. We state the first lemma since it is sufficient for many applications and its proef is simpler than the proof for the general case.

Lemma 2.4.1

Let A be a maximal monotone y-graph satisfying (A1)-

(A4). Assume that there are sequences {S"}7°., and {D"}%°, defined ~ on a measurable set G C R* such that

(D",S")'e A D” —D

See

a.e. in G, weakly in LY (G),

(2.9) (2.10)

weakly in LY (G),

(2.11)

and

lim sup n-rco

JG

S27 2D"dz & i)S\; Dida: G

(2.12)

Then (D,S) € A a.e. in G. Proof

We first observe that (2.9)-(2.12) imply that

lim sup ‘b(S" —S,(D)) «(D” —D) dz oo in (2.20), we conclude from (2.15)—(2.17) that Ls Daz>

| s- Bdz++ [s.B)

B) dz

and consequently

i (S.(B) —S)-(B—D)dz>0 G

forall

BE L°(G).

(2.21)

34

Buliéek, Gwiazda, Médlek, Rajagopal, & Swierczewska-Gwiazda

Next, we use a generalization of the Minty method (see Lions (1969), for example) to prove (2.18). For any j > 0 we define the set G;= {zé€G:

|D(z)| 0 and W € L®(G) is arbitrary (note that B is bounded). Doing so, we obtain

| (S.(D + hW) ~$)-Waz> 5 | (S,(0)-D—S-D) dz. (2.23) G;

h Java

Next, using (2.15)—(2.16), and the Hodlder inequality we obtain

| |S.(0) -D —S-D| dz < co. & Consequently, we appeal to the Lebesgue Dominated Convergence The-

orem and (2.22) to let i > oo on the right-hand side of (2.23) and conclude that

lim 4t—70O

(S,(0)-D+S-D) dz =0, G\Gi

which directly implies that

JG;

(S,(D + hW) —S)-Wdz>0

forall j EN.

(2.24)

Our goal is to let h > 04 and to show (2.18). Using the definition of G; it is easy to see that for a subsequence

S.(D+hW) =S D+hW—->D

(D+ AW,S,(D + AW)) EA

weakly in L?(G;), strongly in L*(G;),

ae. in G;.

Therefore, we can use Lemma 2.4.1 (note that (2.12) is a direct consequence of the above convergence results) with LY = L? and we observe that (D,S)EA

ae. in Gy.

(2.25)

2: On flows of implicitly constituted fluids

35

Moreover, since D is bounded in G; then also S is boundéd in G j due to

(2.25). Finally, taking the limit h > 04 in (2.24), we have

| (S—S)-Wdz>0

for allWe L™(G;).

G;

Setting W := — eae lis ys) yields

ih IS 2S\dz R?xRx Re such that

=divS--Vp=— divF.,

divv=0,

(D(v),SyeA_ ww =O

in Q, on O22.

, (2.26)

Even for such a relatively simple problem, difficulty may occur due to the fact that we work with general N-functions w that may not satisfy

(H1) or (H2). To be more precise we will not be able to construct the pressure in general and therefore we project (2.26); onto the space of divergence-free functions and omit the construction of the pressure. Hence, we are led to the following definition.

Definition

2.5.1

Let 2 Cc R®@ be an open bounded set with Lips-

chitz boundary, let A satisfy the assumptions

(A1)—(A4), and take

36

Bultéek, Gwiazda, Mdlek, Rajagopal,

& Swierczewska-Gwiazda

F € LY (Q). We say that a pair (v,S) is a weak solution to problem

(2.26) if

‘

v € Woaiy, D(v) € LY(Q), S € LY (Q), iiS -D(w)dz = ifF -D(w) da for all w € CoUiy; Q Q (D(v),S) € A ae. in 2.

(220)

We recall here the notation for the function spaces used in Defini-

tion 2.5.1. For any p € [1,co], we set

Wo’? (Q)4 = {v : 2 4 R44; € Wo'?(Q) for all i= 1,...,d},

Cae = {v € DIO)": div = 0},

and define

Waa

dy © We (Oyj divvetO},

SSSeq5

alle

LG an t= Wo ary

The main theorem of this section is the following. Theorem

2.5.2

Let the assumptions of Definition 2.5.1 be fulfilled.

Assume that in addition w satisfies (H2). Then there exists a weak solution to problem (2.26). In addition, if w satisfies (H1) then there exists p € L1(Q) such that ij$:D(w0)dr = / F-D(w)+p div w dz Q 2

for allw€ Watteo (2.28)

Proof Let S, be a selection from the graph A having all the properties discussed in Section 2.3 and let S§. denote its mollification:

Si(@) = G+ eV) =f SulC"(E- Cae, sym

where p*(€) =

0 and p € C§° (R22) is a mollification kernel, i.e. a radially symmetric function with support in the unit ball B(0,1) c Sop and Jpaxe pd€= 1. It is not difficult to observe, using the convexity of w and we and by means of Jensen’s inequality, that the

approximation Sf satisfies a condition analogous to (A4). Let {w;}2, be a basis of W*4(Q)4n Wai 3 wean We look for n a vector-valued function v°" := S*"_, c;’"w; such that the coefficients

c;"" solve the following system of n equations (Galerkin approximation)

[sO

")) ™)). Daw) de = fF -D(w) ae,

b= 1, ay Me, (te)

2: On flows of implicitly constituted fluids

37

The existence of a solution to (2.29) can be shown by «using a variant of the Brouwer fixed-point theorem and the a priori estimates shown

below. To obtain them, we multiply the i-th equation in (2.29) by c°" and sum the results over i =1,...,n. We obtain

i S*(D(v")) -D(v©") dx = | F-D(v@") de. Q

(2.30)

Q

Next, using (A4*) and the Young inequality we conclude from (2.30) that .

[ vO) + 8S) Obtaining

the

limit

as

«

—

ax< CF) —Cn(B) f ID(w) — CleFD(WE") ~ Oa. Hence, using the strong convergence (2.32) we see that the right hand side of (2.35) tends to zero as € + 0 and we get

lim inf(S;(D(v""")) — §S,(B)) - (D(v*") — B) > 0

a.e. in Q,

which due to the strong convergence of D(v®") and weak-* convergence of S&(D(v*'")) yields

(S" —S,(B))-(D(v”) —B) >0 By (A3*), (2.36) implies that

for all

Be Ro and a.e. in 2. (2.36)

|

(D(v"),S")EA-

aeinQ.

Obtaining the limit as n — oo. We multiply the 7-th equation in (2.34) by c”, sum the results over i = 1,...,n and obtain the energy equality

Q

a

Dy

tie ae ifF -D(v”) da. Q

(2.37)

By (A4) this relation implies

[Q vow) + v6")de< CF) IReX4find sym set, for any A c R¢X¢ sym? vo : OQ4 R¢ sym x REX4

42

Bultéek, Gwiazda, Malek, Rajagopal, & Swierczewska-Gwiazda

(v,p,S):Q > R?x R x R&X¢ such that Ov — divS + Vp = —divF,

|

div v = 0,

(D(v),S)EA vi O v(0,-)=vo

in Q,

inQ, sont (OpT poet,

(2.45)

inf.

Before defining what we mean by a weak solution of (2.45), we discuss briefly the possibility of introducing a globally integrable pressure. As in the preceding section we solve first (2.45); projected onto the space of divergence-free functions. Then a natural question is whether one can

find an integrable pressure p € L'(Q) such that (2.45) holds without the projections. For the steady case we know (as we have seen) that such

a formulation is available provided that (H1)—(H2) hold. However, the same result cannot be proved for the unsteady case if v is a weak solution.

Indeed, assuming (H1) and (H2), one can find a pressure p of the form p=pi t+ Ope,

with p; € L'(Q), but with po € L™(Q) only. Thus, we see that p is only a distribution. This difficulty can be overcome in several ways. First, one can modify the problem and replace the Dirichlet boundary conditions by Navier’s boundary conditions (2.8). If this is the case, one can conclude that p2 = 0 and consequently that p € L1(Q) provided

that 2 € C!', we refer the reader to Bulféek, Malek, & Rajagopal (2007) for details. With Dirichlet data the situation is more delicate. Roughly speaking, one can show that the function space for pg is of the same

character as the function space for v. Thus, to obtain pg € L?(Q) one needs to show that Ov € L?(Q), that is, one needs to improve the regularity of the solution. For such an improvement, one has to assume better data and/or additional structure on A. To be more precise, one has to either assume that there exists a potential 77 : Re — R, such

that if (S,D(v)) € A then S € Op7(D) and that D(vo) € LY(Q), or replace the implicit form by an explicit one of the type $(D) with S smooth and require that the initial datum vo is smooth enough. Having such difficulties with introducing the pressure, we omit it in In the first case, the desired estimate on the time derivative is obtained by multiplying the equation by 0:v. Consequently, the existence of a potential 7 to S$ is required and we need to control So 7(D(vo)) da. In the second case, the estimate is achieved by applying the time derivative to the equations and multiplying the result by O;v. In this case no potential structure of S is needed but we have to assume that S is a sufficiently smooth function of D.

2: On flows of implicitly constituted fluids

43

what follows and formulate the definitions and all results without the pressure, considering the spaces of divergence-free functions and their duals and using them as test functions in the weak formulation of the balance equations. j In what follows we use the standard notation for Bochner spaces. In order to provide a proper meaning to O:v in a proper dual space we introduce the Gelfand triple V, H, V* as follows Vii= Mee (O)e a Woe M Lite

A := Li div:

Then V is a separable reflexive Banach space. Since then H -— V*. Thus, for any u,w € V we identify (u, W)VaV

==

V 4

H densely,

(u, V)H

and we also find that the following “integration by parts” formula

(Aaw(t), w(t)) = 5 0 be arbitrary, A satisfy the assumptions (A1)—(A4),

vo € L2 4, and F € L¥’ (Q). We say that a pair (v,S) is a weak solution to the problem (2.45) if

Vv €Cw(0,T;H)NL(0,T;Wosiv), D(v) € L*(Q), Se L* (Q), T / (Orv, w)v=,y dt + / S -D(w) dz dt = | F-D(w)daxdt Q

0

(2.46)

Q

for all w € L™©(0,T;V),

lim ||v(t) — voll? = 0,

(2.47)

t—04.

and (D(v),S) € A ae. in Q. Next, we formulate the main theorem of the paper for ~ satisfying

(H2). For 7 satisfying only (H1), see the remark in section 2.5. Let the assumptions of Definition 2.7.1 be fulfilled. Theorem 2.7.2 Let w in addition satisfy (H2). Then there exists a weak solution to

problem

(2.45).

44

Buliéek, Gwiazda, Médlek, Rajagopal, & Swierczewska-Gwiazda

Proof The proof is very similar to the proof of Theorem 2.5.2, where the steady case was treated. Here, however, other-difficulties appear due to the presence of the time derivative. Hence, assume that {w;}?2; is a basis of V that is orthogonal in L?. Due to the separability of V such a basis surely exists. Next, we look for a Galerkin approximation of v of the form v°" := 7", c;'"(t)w; that solves the following system of n ordinary differential equations thOwe”

Wj

dx +f Sr IDiv

Q

|

: D(w;) ar = ‘iFs D(w;) dz,

2

:

for all t € (0,7) and alli = 1,...,n, Vo NU,

(2.48)

ya ve.

Here, S£ denotes a mollification of the selection S$, (see the proof of Theorem 2.5.2) and P” denotes the orthogonal projection of H on the

span{w},..., Wn}. Note that ||P"vo — voll 3 0 as n — oo. Using the standard Carathéodory theory it is not difficult to obtain

a solution to (2.48) at least for some.short time interval [0,7*). This solution can, however, be extended onto the whole time interval [0,7] provided that we show a uniform bound on v°” that does not depend on the length of the time interval. Moreover, following the proof of Theorem 2.5.2 it is also not difficult to let ¢ + 04 and to conclude that there

exist (v”,S") such that v™ := "577, c?(t)wi, (D(v"),S*) © A’almost everywhere in @ and that i Onwv™ - ww, dx + i S32 Q Q

D(a.) dr = i F -D(w;) da, Q

for all t € (0,7) and alli =1,...,n, yO

Obtaining the limit For any t € (0,7) we i-th equation in (2.49) integrate over (0,t) to 1

(2.49)

= FPvig.

5llv" lla +

as n > oo. First, we derive uniform estimates. denote Q; := (0,t) x 2. Then we multiply the by c?(t), sum the results over i = 1,...,n and get the energy identity

S” -_D(v") dzdr =

F -D(v”) dx dr +5IlP"vollo-

2: On flows of implicitly constituted fluids

Ad

Consequently, using (A4), the Young inequality, and the assumptions on Vo and F we find that

sup Iv"(OIE+[vow )) + u*(S") dndt

t€(0,T

0 be arbitrary, A satisfy the assumptions (A1)—(A4), Vo € Le ai and F € L’(Q). We say that a pair (v,S) is a weak solution to problem (2.64) if

Mec

0

batt (OW...)

Dae

i (Q),

Se L” (QO),

ge

é (O:v, w)y«y dt + f (S—v@v)-D(w) dxdt 0

Q

= | F-D(w) dadt, for all w € L~(0,T;V), Q :

eee

a

(D(v),S) EA

men

ella

a.e. in Q.

Next, we formulate the existence theorem for the full problem. As discussed in the preceding section, we need to consider several restrictions for the w function. First, as in Section 2.6 we need to be able to handle the convective term, therefore we need to assume that (H2*) holds. Theorem

2.8.2

Let the assumptions of Definition 2.8.1 be fulfilled.

Let w in addition be of the form w(s) = w(\s|), satisfy (H1), (H2), and suppose that =

q

2d

Gp oe

Then there exists a weak solution to problem (2.64). We do not prove Theorem 2.8.2 here. The same result was proved in

Buliéek et al. (2012) for Navier’s boundary conditions. However, combining the method developed in Buliéek et al. (2012) with those described in Wolf (2007) one can prove an identical result also for homogeneous Dirichlet data.

50

Buliéek, Gwiazda, Mdlek, Rajagopal, & Swierczewska-Gwiazda

Acknowledgments The contribution of MB and AS-G to this work was supported by the Jindéich Neéas Center for Mathematical Modeling, the project LC06052 financed by MSMT. JM’s contribution to this work is a part of the research project MSM

0021620839 financed by MSMT;

JM also thanks

the Czech Science Foundation, the project GACR 201/09/0917 for its support. AS-G has been also supported by National Science Centre. PG is the coordinator of the International Ph.D. Projects Programme of Foundation for Polish Science operated within the Innovative Economy Operational Programme 2007-2013 (Ph.D. Programme: Mathematical

Methods in Natural Sciences). KRR thanks the National Science Foundation for its support of his work.

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Diening, L., Malek, J., & Steinhauer, M. (2008) On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM Control Optim. Calc. Var. 14(2), 211-232. Diening, L., Ruzicka, M., & Wolf, J. (2010). Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Sc. Norm. Super.

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2: On flows of implicitly constituted fluids

51

Frehse, J., Malek, J., & Steinhauer, M. (2003) On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34(5), 1064-1083 (electronic). Gwiazda, P., Swierczewska-Gwiazda, A., & Wrdblewska, A. (2012) Generalized Stokes system in Orlicz spaces. Discrete Contin. Dyn. Syst. Ser. A. 32 2125-2146. Lions, J.-L. (1969) Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Paris. Malek, J. & Rajagopal, K.R. (2005) Mathematical issues concerning the Navier-Stokes equations and some of their generalizations. In Dafermos, C. & Feireisl, E. (eds.) Handbook of Differential Equations. Elsevier B.V. 2, 371-459. Malek, J. & Rajagopal, K.R. (2010) Compressible generalized Newtonian fluids. Z. Angew. Math. Phys. 61, 1097-1110. Rajagopal, K.R. (2003) On implicit constitutive theories. Appl. Math. 48, 279-319. Rajagopal, K.R. (2006) On implicit constitutive theories for fluids. J. Fluid Mech. 550, 243-249. Rajagopal, K.R. (2007) The elasticity of elasticity. Z. Angew. Math. Phys. 58(2), 309-317. Rajagopal, K.R. & Srinivasa, A.R. (2008) On the thermodynamics of fluids defined by implicit constitutive relations. Z. Angew. Math. Phys. 59(4), (a t29% Rajagopal, K.R. & Srinivasa, A.R. (2009) On a class of non-dissipative materials that are not hyperelastic. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sct. 465(2102), 493-500. Rajagopal, K.R. & Walton, J.R. (2011) Modeling fracture in the context of a strain-limiting theory of elasticity: a single anti-plane shear crack. Int. J. Fracture 169, 39-48. Rao, M.M. & Ren, Z.D. (1991) Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker Inc., New York.

Wolf, J. (2007) Existence of weak solutions to the equations of nonstationary motion of non-Newtonian fluids with shear-dependent viscosity. J. Math. Fluid Mech. 9, 104-138.

3

A continuous model for turbulent energy cascade Alexey Cheskidov Department of Mathematics, Stat. and Comp. Sci., University of Illinois, Chicago, IL 60607. USA. [email protected]

Roman Shvydkoy Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637. USA. [email protected]

Susan Friedlander Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089. USA. [email protected]

Abstract In this paper we introduce a new PDE model in frequency space for the inertial energy cascade which reproduces the classical scaling laws of Kolmogorov’s theory of turbulence. Our point of view is based upon studying the energy flux through a continuous range of scales rather than the discrete set of dyadic scales. The resulting model is a variant of the Burgers equation on the half line with a boundary condition that represents a constant energy input at integral scales. The viscous dissipation is modelled via a damping term. We show existence of a unique stationary solution, both in the viscous and inviscid cases, which replicates the classical dissipation anomaly in the limit of vanishing viscosity. A survey of recent developments in the deterministic approach to the laws of turbulence, and in particular, to Onsager’s conjecture is given.

Published in Mathematical Aspects of Fluid Mechanics, edited by J.C. Robinson, J.L. Rodrigo, & W. Sadowski. ©Cambridge University Press 2012.

52

3: A continuous model for turbulent energy cascade

3.1 3.1.1

Motivation

53

for the model -

Onsager and Kolmogorov

The Euler equations for the motion of an incompressible, asd are

fluid

Ou rr +(u-V)u=—Vp,

(3.1)

V-u=0,

(3.2)

where u(z, t) is a divergence-free velocity vector and pressure. We consider the system in three spatial assume that the domain is either periodic or the obtain the energy equation we multiply (3.1) by u

p(z, t) is the internal dimensions and we entire space R*®. To and integrate, using

(3.2) to give ~—

| wdx=

=

[(vu

wae

We define the total energy flux II by

Il = ic -Vju-uda.

(3:3)

For smooth solutions we can integrate by parts and use (3.2) to conclude that Il = 0 and hence energy conservation holds, i.e.

[weer diz = /\u(a,0)|? dx

for t > 0.

(3.4)

However, in the context of turbulent flows in the limit of vanishing viscosity, it is appropriate to consider the Euler equations in the sense of distributions and impose only minimal assumptions on the regularity of the velocity field u. In the absence of sufficient smoothness we cannot integrate by parts in (3.3) or even make sense of (3.3) and ensure that II = 0. Conservation of energy might then be violated. Hence it is of interest to ask what are the minimal regularity assumptions on the velocity that ensure that (3.4) holds. Observing that the integrand in (3.3) is cubic in wu and contains one spatial derivative suggests that if wu has Holder continuity h > 1/3, integration by parts is justified and II = 0. In fact this was the conjecture made many years ago by Onsager in his seminal paper on statistical fluid dynamics, see Onsager (1949). More precisely, he conjectured that (a) every weak solution to the Euler equation with smoothness h > 1 /3 conserves

energy

and

Cheskidov, Shuydkoy, & Friedlander

54

(b) there exists a weak solution with h < 1/3 that does not conserve energy. : of the flow is irregularity the to Such putative energy dissipation due called anomalous or turbulent dissipation. A detailed historical account of Onsager’s theory is given by Eyink & Sreenivasan (2006). All physical fluids are viscous, if only very weakly so. Turbulent fluids are believed to be described by the Navier-Stokes equations

Ot

Vv u=0,

(3.6)

where v, which could be very small, is the coefficient of viscosity and f is an external force which supplies energy into the system. The “classical” Kolmogorov theory of turbulence predicts that energy-dissipative solutions to the Euler equation may arise in the limit of vanishing viscosity

for “generic” viscous flows that are governed by (3.5)—(3.6). In homogeneous, isotropic turbulence the mean kinetic energy per unit mass is

defined by

B= 5(luP), where we denote by (-) the long time average. The energy density spectrum is defined by

Bln) = 5>(lex). Here uc, denotes the filtered velocity field containing all frequencies

below a wave number «. Hence E = f,° E(«)dk. The mean energy dissipation rate per unit mass is defined by

ev = (y|Vu"|?), where wu” is a solution to (3.5)—(3.6). Kolmogorov (1941) predicted that the energy cascade mechanism in fully developed three-dimensional turbulence produces a striking phenomenon, namely the persistence of nonvanishing energy dissipation in the limit of vanishing viscosity, i.e.

lim €

>e>Q0,

(3.7)

v—-

where ¢ tem. The anomaly. support)

is the anomalous dissipation rate for the inviscid Euler syspositivity of the limit in (3.7) is referred to as the dissipation Let us now assume that f = fe, ,; (ie. f has finite Fourier and that solutions to (3.5) tend to a statistically stationary

3: A continuous model for turbulent energy cascade

55

state with uniformly bounded mean energy. We multiply (3.5) by uZ,, and obtain

1, («) = —v(|Vuz,.|*) + (f -u%,).

(3.8)

If & > Kf we have

ee

ee

|.

On the other hand, by Bernstein’s inequality,

v (|u|?) < vm? (|u"|?). Since the energy is uniformly bounded by assumption, we obtain from

(3.8) that

linn) Ly (ie)

v—0

aime

4

v—0

es

Thus in the limit of vanishing viscosity the average solution of the forced Euler equation inherits the anomalous dissipation rate e. As Frisch (1985) describes, a self-similarity hypothesis on the velocity increments in small (spatial) scales implies that the energy spectrum as a function of the wave number « has the power law E(k) ~ Belge Pls

in the “inertial” range & € |[Kf, Ka]. Here ka is the Kolmogorov dissipation wave number given by

ka = (e/v?)/*

and

«= max{|«|: 6% € supp f}.

This power law is known as the K41 turbulence model. Although the 5/3 power law is consistent with much physical data, there are also experiments that indicate turbulent regimes with alternative power laws. In fact, Kolmogorov’s 1941 theory requires that the local velocity fluctuations are uniformly distributed over space. However, in reality dynamical stretching of the vortex filaments in three-dimensional flows leaves some regions of the fluid domain with moderate turbulent activity and other regions with intense activity. This so called spatial intermittency should reasonably be accounted for in the description of the scaling laws. The

expressions for E(«) and «q that incorporate the dimension D of the effective dissipation region are

Ene ke

(3.9)

and

EA

es

(3.10)

56

Cheskidov, Shuydkoy, & Friedlander

where D € [0,3]. Thus the classical K41 model corresponds to

D =

3, i.e. uniform distribution over three-dimensional space, while D = 0 corresponds to a fully intermittent model where energy cascades through

scales and dissipates only on points.

3.1.2

Onsager’s Conjecture and Besov spaces

In the past few years there have been a number of articles that address part (a) of Onsager’s Conjecture. These include articles by Constantin,

BE, & Titi (1994), Eyink (1994), and Duchon & Robert (2000). It was shown that appropriate function spaces to examine the Euler equations in the context of Onsager’s Conjecture are Besov spaces. In such spaces the notion of energy balance when the velocity is “a little smoother” than Hélder h > 1/3 can be made precise. These are the natural spaces to work with in terms of a description of the energy flux phrased by a Littlewood—Paley decomposition which provides detailed information concerning the cascade of energy. Recently Cheskidov et al. (2008) obtained the largest Besov space where conservation of energy is ensured for the Euler equation. We note that to date! there are no examples of Euler flows that possess some smoothness and confirm the second part of Onsager’s Conjecture, although there are examples of “very weak” Euler solutions that violate the energy balance condition (De Lellis & Székelyhidi, 2009; Scheffer, 1993; Shnirelman, 1997) . We recall the definition of a weak solution of the Euler equation.

A vector field u € C,,([0,T]; L7(R*)) is a weak solution of the Euler equations with initial data ug € L?(R*) if for every compactly supported

test function ~ € Cgé°([0, T] x R*) with V.-~ = 0 and for every0

O0,

(3.23)

3: A continuous model for turbulent energy cascade

61

where @ = 2 + 2c, The appropriate range of a is [3, Sly which exactly corresponds to the classical range of the energy density power laws with the spatial intermittency correction.

Remark 3.2.1

Equation (3.23) is a variant of the much studied Bur-

gers equation. The one-dimensional Burgers equation can be viewed as

the most basic nonlinear PDE that has the bilinear structure of the nonlinearity of the Euler and Navier-Stokes equations. It can be invoked as a model for one-dimensional compressible fluids. However there is no clear physical basis for using the Burgers equation in physical space as a model for turbulence. On the other hand, as we have argued, the locality of the energy flux manifested using Littlewood—Paley theory, motivates (3.23) as a PDE model for the turbulent cascade in frequency space. Remark 3.2.2 In the past few decades a number of “toy models” for turbulence have been studied to test Kolmogorov’s theory. In particular,

the derivation of the classical Desnyansky & Novikov (1974) discrete model follows a similar path. The flux there is modelled by taking H; = Qu aj;a;4+1, where a’ represents the total energy in the j-th dyadic shell, while d is an intermittency parameter with the appropriate range of values. The model is thus an infinite system of ODEs given by

4

ant oe v2 a; — PS where a_;

;

ple + 2% asaj41 =

GN

BIE ee

= 0. This model, as well as its inviscid versions, has been

extensively studied by Katz & Pavlovié (2005), Cheskidov & Friedlander (2009), Kiselev & Zlatos (2005), and others. To our knowledge the PDE model we present is the first continuous model.

In Section 3.3 we examine the inviscid (v = 0) form of (3.21) with boundary condition (3.22). We prove that there is a unique fixed point which is a global attractor. Moreover every solution reaches it in finite time. The inviscid equation exhibits anomalous dissipation and the average energy spectrum has the power law ee) Ses

In Section 3.4 we turn to the viscous model v > 0 (3.21), which is in essence a damped Burgers equation. Again the PDE has a unique fixed point and this converges to the inviscid fixed point as v + 0. The viscous fixed point reproduces Kolmogorov’s energy density spectrum in the inertial range and it becomes zero identically after the dissipation wave number Kg. We further consider the Leray regularization of equation

(3.21). We show that all bounded solutions of the regularized equation converge pointwise and in the metric of L?-space to a fixed point which

62

Cheskidov, Shvydkoy,

& Friedlander

in turn converges to the fixed point of equation (3.21) in the limit as the regularization parameter goes to zero. The averdége dissipation rate for the viscous system converges to the anomalous dissipation rate for the inviscid system giving an example of the dissipation anomaly predicted by Kolmogorov.

3.3

Inviscid case

In this section we study the inviscid version of the model (3.21) a=

San"

(KO),

ge be

OK a(1,t) = 1/3, a(k, 0) a

(3.24)

ao(K) a 0,

where a € [5/3, 8/3]. Note that the energy equality

Sof

dt2

ate iidie—
0 represents the energy input rate in this model. The unique fixed point of (3.24) is given by

AK) Ses grel™

(3.27)

We note that the fixed point does not satisfy (3.26). Moreover, it does not satisfy the energy equality since

M8 endAgeil ars mag ey yak A hRrdiges Uses:

(3.28)

The anomalous energy dissipation rate is the difference betweem the energy input rate and the time derivative of the total energy. Thus using (3.25) and (3.28) we observe that the anomalous dissipation rate for the fixed point is exactly the energy input rate ¢. We will show that this also holds for every other solution asymptotically in time. In order to do this we will prove that A° is a global attractor. We use the following change of variables: c= KO.

w(é,t) = ener

ETa(e7ty te 1/3 t),

3: A continuous model for turbulent energy-cascade

63

where y = st. Then (3.24) reduces to the Burgers equation Wi = wwe,

Greet|.

with w(1,t) = 1, w(E,0) = wo(€) > 0. We extend it to the Burgers equation on the whole real line Wt = WWE,

ce = to(£),

EER,

a

where

0

wo(f) = 4 wol€)

2(1 — €). We will show that y,(€,¢) = € +t and consequently w(€,t) = 1. Indeed, let y =£&+s. First, consider the interval s > 1 — €. Then 1 2

fly) = 5 e — f0 wolg)de-E-s +1>FE +0

provided that s 4 t. Since t > 2(1 — €), it follows that

Re ti e™ i Heal and hence on the interval s < 1 — € we also have

eine ieno(6)ag> — f wol6)ae > ME +0) 2

+

0

Cheskidov, Shvydkoy, & Friedlander

64

Therefore, (3.30) implies that w(€,t) = 1 for € € (0,1), ¢ > 2 and hence, returning to the original variables,

‘

a(x,t)=A%(K),

t> 2e*/94.

Hence the average energy spectrum for solutions to the inviscid model is about €2/3«7%.

3.4

Viscous

case

In this section we study the viscous model (3.21)

hae a, = —3an*—(kK 2a) — vK7a, OK a(1,t) = el/9, a(k, 0) =

ao(K), ao E Peta

oe

(3.31)

.

= 0,

where a € [5/3,8/3]. There exists a unique fixed point to (3.31) given by

;

Paice

Ror

|8

=

+

a

Lasky, ee

n*)|

0

K > Kd,

where Kq is Kolmogorov’s dissipation wavenumber described in Subsection 3.1.1. For the model it is explicit and given by

a=

(3 — ws re

[14

"

To see the parallel with the classical expressions for Kg we note that for v small one has

&\4 Kad ™

(5)

5 -

for

a=

3°

and

Ka © =

ps’

for

—

Be

3

In the limit of vanishing viscosity we immediately obtain from (3.32) and (3.27) that A’ (Kk) > A°(k),

3: A continuous model for turbulent energy cascade

65

uniformly on any finite interval [1, Ko] as v + 0. With a little more effort we can show that the convergence also takes place in L?({1,00)). Indeed, / |A”(K) — A°(K)|?da < is 1

| hee OK a, y2

Kd

a eine Since Kg —

ni

Seer ad at

oo we see that the first integral vanishes as

v >

0. The

second integral behaves like v? for a < 7/3, like v? log(v) for a = 7/3, and like v(¢-))/G-%) for 7/3 0. In order to study the time-dependent solutions to the viscous system (3.31) we use the same change of variables as in the previous section

fae ki

miannulest ian!" Ga aalte ader Yhyt),

where y = —/. Then (3.31) reduces to the damped Burgers equation GEA We = Wwe — we?"w,

OE

R-2,

witha-+ic €6,, eT fork -

Now, for h € T and y € B,,, it follows that

|DJa(7) — DJa(y — A)|* < Cr(II(2? — 2°, 2? — "Ile |h/(r — 9!) +A

(63 (2? = 2*))(y) — A(G5(2?Sz

Ga hs);

and using the fact that H is a bounded map from C® into itself one finds

|A(03 (27=2"))(y) — (OZ (27=2"))(y—A)|e =|H (95 (27 —z*))(y)—H (83(27-27) (7A) le

SCO (2? = z*)|In|Al?, from which we obtain

|DJg(y) — DJa(y — h)|« " CR MI" |Al? T=

— 2", 0? —a")Ilr.

5: A naive parametrization for the vortex-sheet problem

105

In an analogous way we may define DJs and split it as follows:

Kr= x f (a= a") 8)~(@? - (9) if

TT

x

Ce OS Os Geis [27(y) — 2#(y — B)

(62 (2? — z1)(y)—62(22 — A) —B)) s 2) =2 BP oP and

ae = / (w"(y — B) — w(7)) (6224 (q) — 0221(y — B))+ Aly, B) a8, with A(y, 8) = |z*(y) — 2*(y— BI? — |e" (7) — 2° (7 — B)|-®. All kernels in the integrals in K7, Kg and Kg have degree 0 so the control of all

these terms is analogous; now we will show in detail the analysis of term K7. We rewrite it as

Kr= 5 f BC, B)C(.8)D(7,8)a8, with

(7,8)

(w kes Ne ae - ( A

CERNE (02.2°(7)

Be

“ Gay

es i

ey)

:

and

D(x, B) =

z

a= (7-8)

Pr’

to get the splitting

K7(y) — Kr(y —h) = 11 + Lo + Ls, where

ty= = f (BU.8)- Boh, 8)CO, BD 8) a8, il

Oe

qT

B(y, B)(C(y, B) — Cy — h, B)) Dy, B) 48,

106

Castro, Cérdoba, & Gancedo

and

tg== | Biy,8)C(,8)(DO,B) - Diy — h, B)) 8. ~ On ae We rewrite the term B as

B(y,B) = ii0q(w* — w*)(7 — 88) ds, and therefore

|B(7,8) — By — h, B)|x < |Joo? — wo"||AI?, for y € B, and h € T. This yields

|Lil« < ||o? —

|| Al? ||8a27 || R?
3 and [ 20=0.

T

Then if there exists a point og where wo(o0) > 0 and wo is not C™

in

do, there is no solution of equation (5.25) in the class C((0,T); H*(T)) with s > - and T > 0. In addition, wp € C™

is not sufficient to obtain

existence.

Remark 5.4.2 In the case of the real line R the equation (5.25) is also ill-posed in H® with s > 3/2, for some non-analytic initial data. For more details see Castro & Cérdoba (2008). Proof

We will proceed by contradiction.

Let us assume

that there exist a solution of equation

(5.25) in the

class C([0,T), H*(T)) with w(o,0) = wo(c). First we note that if the initial data wo are of mean

zero, then the

solution @ will remain of mean zero.

Now, taking the Hilbert transform of equation (5.25) yields il

O,Ha — gw

— WW)

= 0,

where we have used the following properties of the Hilbert transform for a periodic function with mean zero:

108

Castro, Cérdoba, €& Gancedo

e H(Hw) =-w.

e H(wHaw) = 3((Hw)? — w?).

‘

We define the complex valued function z(o,t) := Ha(o,t) — iw(a,t), which satisfies 1 Onz — 5%0 =. 14)

(5.26)

Take P,(u) to be the Green’s function for the Laplacian for the Dirichlet problem in the unit ball,

and Pa(u) will be

Pou) = i P,(u)@(r) dr. 8B(0,1) Therefore

Z(u) = P(Hw—iw)(u),

with

u=re',

is an analytic function on the unit ball. Applying P to the equation (5.26) yields a de

5P(222)

where we can write the right-hand side as i

1

5P (ze) pa gP2P2)o, since both terms have the same restriction to the boundary of the unit ball and are both harmonic. Thus Z(u,t) satisfies the equation 1 Zt

—

5220

ee =0

on

u

€

BO, 1);

hence Zt — siuZZu =0

Z(u,0) = Zo(u) =

on

we B(0,1),

P(Hw — imo) (u).

We will define the complex trajectories X (u, t) by dX (u, : fa = —FiX (u,1)Z(X(u, t,t),

X(u,0)=u,

wu€ B(0,1).

5: A naive parametrization for the vortex-sheet problem

109

For sufficiently small t, by Picard’s Theorem, these trajectories exist and

X(u,t) € B(0,1). Therefore dZ(X (u, t), t) dt

Bo ZX) A SiX(1st)Z(X (uy t),t)Zu(X (a, #),€) ayy Thus, we have

2(X(u,t),t) = Zo(u), and

= —F1X (u,t)Zo(u).

aa Moreover

X(u,t) = uew220), Taking moduli in the last expression we obtain

R(u,t) =|X(u,t)| = ren 2Pore), If we consider a point e!7° = up € OB(0, 1) with wo(a0) > 0, then

R(uo,t)= e200)! 0, and a continuity argument yields

Z(X (a0, t), t) = 20(¢0) = Hwo(00) — iwo(o0), where to simplify we write by X(uo,t) = X(o0,t). Then we have

Xilagat

= ei(7o— 3 20(70)t)

Taking the derivative with respect to 0 of this equation we find that

a

= i(1 ~ 5200(00)t)X (0051).

Using the chain rule we obtain

47

WA

_

_ %0a(9o

(X (00,#),HiX (00,8) = Gq(X(004).) = GT eae) )’ X(00,t) = R(o0, t)eiOloo.t)

Taking two derivatives with respect to O d7Z

2000(90)

(1- $20 0(00)t)?

Castro, Cérdoba, & Gancedo

110

For the n-th derivative we have au

X (oo,

t),t)

aon (X(70, #)-#)

"20

(gy)

= ——————_—

(0 — 4z04(o0)t)"#

;

(44

+

2?

“lower terms”.

We observe that (1 — $20o(0)t) #0 for t small enough. Then if wo is not C® in oo this is a contradiction since Z(u,t) is analytic on X (a0, t) for all t > 0.

In addition, if @o(oo) > 0 and 4#2(o9) =0 do”

for all n but wo is not

constant on any neighbourhood of a9, we can conclude that

dSZ 6

(*' (00:#); X?(a0,t))

=

0,

where Gz denotes the imaginary part of z. Continuing this process we obtain that all derivatives satisfy

d"$Z den

(X*(a0,t),

X* (00, t)) UE

The imaginary part $Z(21,272,t) is analytic on (x1,22) = (X1(o0,t), X*(o0,t)) for all t > 0, thus $Z(z1, 72) is constant over the circumference, R = R(oo,t), and this is a contradiction if wo is not constant. O

5.5 Appendix Here we extend the property of the continuity of the pressure known for Darcy’s flow (see Cordoba & Gancedo, 2007 and Cordoba et al., 2011). While writing this paper we learned of the paper by Shvydkoy (2009) who also obtained this fact for more general cases in a different way.

Proposition 5.5.1 Let us consider a weak solution (v,p) satisfying (5.8)-(5.9), where curly = w is given by (5.10). Then we have the identity p (z(a, t), t) = p’(z(a, t), t),

where p’(z(a,t),t) denotes the limit pressure obtained approaching the boundary in the normal direction inside 0). Proof

We shall show that the Laplacian of the pressure satisfies

Ap(z,t) = F(a,t) + f(a,t)d(x — z(a,t)), where F is regular in (/(t) and discontinuous on z(a,t). The amplitude f of the delta function is regular. The inverse of the Laplacian by means

5: A naive parametrization for the vortex-sheet problem

Lit

of the Newtonian potential gives the continuity of the pressure on the free boundary (see Cordoba et al., 2011). Here we shall give the argument for a closed curve, the proof for the other cases being analogous. The expression for the conjugate of the velocity in complex variables is 1

1

v(z,t) | —————~ (z, t) = —-PV Onl / aa (es ae t) da, which for z 4 z(a,t) allows us to show that

O20(Z,

maPY | eae5 dai aa!eee mot) 4 QT

Zz — z(a,t))? Oaz(a, t)

Therefore

3 O25)

: =ae saPy |ac ore

vee tape

for a regular parametrization with 0,z(a,t)

da

(5:20)

# 0. In a similar way

ee cde 202.8) =SEPV foo | ay Oalgyy 52algq)(as8) da. (5.28) These identities allow us to obtain the values of Vv/(z(a,t),¢) and V7vI (z(a,t),t). As for the velocity, the limits are different, but we can compute the values. To get the above formula for the pressure we take the weak-type identity (5.8) with (z,t) = VA(z,t). We can then compute the Laplacian of the pressure in a weak sense using

fe iB [ [ varacae=— ff v 0 JR? 0 JR?

Vadear—

-{ via) R2

fe ff v-(v- V?A) da dt 0 JR2

VACe,0) de == ti + Lo + 13.

Then

by the incompressibility condition. We define

OL(t) = {a € OI(t) : dist(«, AN" (t)) > €},

AN?(t)) > e}. 02 (t) = {a € 2?(t) : dist(x,

Castro, Cordoba, & Gancedo

112

We decompose Iz as J3 + Js + Js + Je, where

i -| [ (v1)?02, Ada dt, 0 JR?

T Ja =-f [ V1 202,07, 0 JR?

z -| ‘iV1V20,z, Or, dz dt, 0 JR?

dz dt,

f J6 -[ if(v2)?02, A da dt. 0 JR?

Using the sets 02 (t) and the identity (5.27) we get

= — lim (is‘ Bic

i, 0

O21(t)

(v1)°02 Adrat+ |eh (v1 (o1)202,Adee) 22(t)

201 02,0102, A dx dt

JR?

T

pr

:

+ | i ((v? (2(t), a,t))?— (vt (z(t), ))”) }

xX Oz, A(z(a, t), t)Oa 22(a, t) da dt = Kk, + Ko. The term ky trivializes because of a subtle integration by parts and identity (5.28) giving

=-

ff x(v0? vuyetal(Oz, 01)”)A da dt R2

< onl PONGD, ava 0

-—T

for

f(a, t) = 2(v?(z(a, t),t)A2, v7(z(a, t),t) —v}(z(a, t), t)Oz, v}(z(a, t), t))Oez2(a, t).

The first term in Ky is part of F(«,t) and the second of f(a, t). We can rewrite Ky» as T

pr

Oo 21

Ky 9 = —2 [ aks B R, usr

eee

5: A naive parametrization for the vortex-sheet problem

We continue with J,

-

oP: Jy =

/ Hh(v2025V1 =e V1 Ox, V2)Ox, A dx dt

= qe (v?v2)(zz(a,t), t) — (vj v9)(z(a, t), t))

x On, A(2(a1, t), t)Oa21 (a, t) dev de =

“

K3 AF Kae

We deal with the term K3 We can rewrite Ky as

in a similar way as with Ky.

On ZY

[wBR, aia ee5 gs m5 On, A(Z)Oa21 da dt.

--[f

|Oa2|?

For Js we split

T Js = d ‘t(v20z, U1 Se V1 Oz, V2)Ox, A Ax dt R2

ahea (vjv3)( CR

ia be (viv2)(z(a, t), t)) x Oz, A(z(a, Gh t)0o22(a, t) da dé

=F Ks ar Kg.

We proceed for Ks in a similar manner as with K,. We obtain for Kg the expression Ke

lak ke BR, a, 22. 7 +@mBR2

On Zi

Baz]2 nz Ore A(#)0a22 da dt.

With Jg one finds

is J6 ay [ 20202,U20z,r

0

dx dt

JR? T

pr

t),#))?— (v3(2(@,#), #))?) = [ ((v3 (z(a, X On, A(z(Q, t), t)Ovza(a, t) da dt = K7+ Kg.

For A7 we proceed as before. We obtain for Kg the expression eA: Kg

=

-2

|

0

On 22 awBR2—

SI

|Ovz|?

0

A(z)Ou21 da dt.

.

113

Castro, Cérdoba, & Gancedo

114

We now sum as follows Ko + K4 + Ke + Kg = Lz, then

In=-

oil

ca, t)

[ ie TE OR ieee

tc

f

L, ns

Ox 2(a2, t)

xXOq2z(a, t) - VA(z(q, t), t) da dt. An integration by parts in the variable a in Lz gives the last term of O f(a,t). The formula for the Laplacian of p is found. Acknowledgements The authors were partially supported by grant MTM 2008-03754 of the MCINN (Spain) and grant StG-203138CDSIF of the ERC.

References Bardos, C. & Lannes, D. (2011) Mathematics for 2d interfaces, to appear in

Panorama et Synthese. Caflisch, R. & Orellana, O. (1989) Singular solutions and ill-posedness for the evolution of vortex sheets. STAM J. Math. Anal. 20 no. 2, 293-307.

Castro, A. & Cérdoba, D. (2008) Global existence, ill-posedness and singularities for a non local flux. Adv. in Math. 219, no. 6, 1916-1936. Cheskidov, A., Constantin, P., Friedlander, S., & Shvydkoy, R. (2008) Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21, no. 6, 1233-1252. Constantin, P., E, W., & Titi, E. (1994) Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys. 165, no. 1, 207-209. Cérdoba, A., Cordoba, D., & Gancedo, F. (2011) Interface evolution: the Hele— Shaw and Muskat problems. Annals of Math 173, no. 1, 477-544. Cordoba, D. & Gancedo, F. (2007) Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Comm. Math. Phys.

273, no. 2, 445-471. Delort, J.M. (1991) Existence de nappes de tourbillon en dimension deux. J. Am. Math. Soc. 4, 553-586. DiPerna, R.J. & Majda, A.J. (1987) Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108,

667-689. Duchon, J. & Robert, R. (1988) Global vortex sheet solutions of Euler equations in the plane. J. Diff. Eqns. 73, 215-224. Ebin, D.G. (1988) Ill-posedness of the Rayleigh-Taylor and Helmholtz problems for incompressible fluids. Comm. Partial Differential Equations 13, no. 10, 1265-1295.

5: A naive parametrization for the vortex-sheet problem

Kamotski,

V. & Lebeau,

Asymptot.

115

G. (2005) On 2D Rayleigh—Taylor instabilities.

Anal. 42, 1-27.

Lebeau, G. (2002) Régularité du probléme de Kelvin-Helmholtz pur l’équation

d’Euler 2d. ESAIM: COCV 08, 801-825.

'

De Lellis, C. & Székelyhidi, L. (2009) The Euler equations as a differential inclusion. Annals of Math. 170, no. 3, 1417-1436. Lopes Filho, M.C., Nussenzveig Lopes, H.J., & Xin, Z. (2001) Existence of vortex sheets with reflection symmetry in two space dimensions. Arch. Rat. Mech. Anal 158, 235-257. Lopes Filho, M.C., Nussenzveig Lopes, H.J., & Shochet, S. (2007) A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution. Trans. Amer. Math. Soc. 359, no. 9, 4125-4142. Majda, A.J. & Bertozzi, A.L. (2002) Vorticity and the Mathematical Theory of Incompressible Fluid Flow. Cambridge University Press, Cambridge. England. Marchioro, C. & Pulvirenti, M. (1994) Mathematical Theory of Incompressible Nonviscous Fluids. Springer-Verlag. Moore, D.W. (1979) The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. London A 365, 105-119. Nirenberg, L. (1972) An abstract form of the nonlinear Cauchy—Kowalewski theorem. J. Differential Geometry, 6, 561-5762. Nishida, T. (1977) A note on a theorem of Nirenberg. J. Differential Geometry 12, 629-633. Scheffer, V. (1993) An inviscid flow with compact support in space-time. J. Geom.

Anal. 3, 343-401.

Shnirelman, A. (1997) On the non-uniqueness of weak solutions of the Euler equations. Comm. Pure Appl. Math. 50, 1261-1286. Shvydkoy, R. (2009) On the energy of inviscid singular flows. J. Math. Anal. Appl. 349, 583-595. Stein, E. (1970) Singular Integrals and Differentiability Properties of Function. Princeton University Press. Princeton, New Jersey. Sulem, C., Sulem, P.L., Bardos, C., & Frisch, U. (1981) Finite time analyticity for the two and three dimensional Kelvin-Helmholtz instability. Comm. Math. Phys. 80, 485-516.

Wu, S. (2006) Mathematical analysis of vortex sheets. Comm. Pure Appl. Math. 59, 1065-1206. Yudovich, V. (1995) Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal, incompressible fluid. Math. Res. Lett. 2, 27-38.

6

Sharp and almost-sharp fronts for the SQG equation Charles L. Fefferman Department of Mathematics, Princeton University, Princeton, NJ 08544. USA. cf@math. princeton. edu

Abstract In this Quasi-Geostrophic tions called “sharp plain why we care know about them.

expository paper we will describe the Surface (SQG) equation and discuss a family of solufronts” and “almost-sharp fronts”. We will exabout these objects and summarize what we

6.1

Introduction

The SQG equation deals with an incompressible fluid in two space di-

mensions. It is convenient to work on the cylinder

C = (R/27Z) x R.

The velocity at position (11,72) € C and time t is given by “= Viw(21, x,t),

(6.1)

where 7 is the stream function and V+ = (—0,,~, 02,1). Equation (6.1) tells us that the fluid is incompressible. The (scalar) temperature 0(21,22,t) is carried along by the fluid. Thus (Op + u- Vz)O(x21, £2,t) = 0.

(6.2)

To close up the system, we assume that the stream function w arises from the temperature 6 by the equation

wy = (-A)-”98,

(6.3)

which on C means that

(e,A= ihK(x —y)6(y,t)dy, Published in Mathematical Aspects of Fluid Mechanics, edited by J.C. Robinson, J.L. Rodrigo, & W. Sadowski. ©@Cambridge University Press 2012.

116

6:Fronts for SQG

117

where K(x) behaves like the i inverse of the distance from «to the origin. We prescribe an initial temperature distribution 60(%1,%2), and we

want to solve equations (6.1)—(6.3) with the initial condition

6=%

att=0.

(6.4)

The equations (6.1)—(6.3) with initial condition (6.4) are called SQG. Let me explain why we care about SQG. The problem arose first in weather prediction. People hoped that SQG would explain how warm and cold fronts form. An initially smooth 09 might, in finite time, acquire a jump discontinuity along a smooth curve. The smooth curve (the

“front” )would evolve in time. Important work by Constantin, Majda, & Tabak (1994a,b) ruled out this scenario, but showed that SQG is an excellent two-dimensional

model for the three-dimensional

(incompressible) Euler equations. In

particular, lines of constant temperature @ for SQG correspond to vortex lines for 3D Euler. Important work of Kiselev, Nazarov, & Volberg (2007) and Caffarelli & Vasseur (2010) treats SQG with a dissipative term. We do not discuss this work here. Next, let us discuss “fronts”. A “sharp front” is specified by a smooth curve I(t) —

io

p(a1, t)} GX,

(6.5)

that evolves with the time t. The curve I(t) separates C into an “upper

component”

= {x2 > y(x1,t)}

(6.6)

and a lower component

=

ey

Oat)

We take (say) 1

for (%1,%2) € Ci, and

O aaa tot) = 5 1

OG, 12,0) = =5

for (1,22) eS,

(6.7) (6.8)

We hope that @ satisfies equations (6.1)—(6.3) in some sense to be discussed later. Such sharp discontinuities do not arise in weather. A more realistic scenario is an “almost-sharp front”, defined as follows.

118

Fefferman

Again, we fix a smooth curve I(t) evolving in time as in (6.5). We are interested in functions 6(a1,#2,t) such that 0 0 is a small parameter. In that (small) annular region we expect that @(a1,22,t) will be as smooth as possible for each fixed t. To explain what we mean by the phrase “as smooth as possible” we introduce a special coordinate system. We define 20

L(t) =

arclength of I(t) = | (1 + (02, 9(21,t))?)” dary 0

and first parametrize I(t) by “normalized arclength” 1

* > £@) [a + (Oy P(yr,t))?)? dyn.

(6.9)

Thus, for fixed t, a function I(t) becomes a function of s, periodic of period 1. (Without the factor L~* in (6.9) the period would depend on time.) For a given point z € I'(t) with normalized arclength s, we write N(z) to denote the unit normal to I(t) at z, pointing into C,. Starting from z, and moving a distance 6-€ along the N(s)-direction, we arrive at a point in our annular region of I(t). For fixed t, we use (s,€) as coordinates on the annular neighbourhood, in place of (x), x2). This coordinate system

depends on 6, and on the time-varying curve '(t). Thus we may express Ofity,.Co,t) = {1(5..6,%)

(6.10)

for a function Q(s,€,t), with period 1 in the s variable. We demand that

Ose t

1

3

for, £21k

(6.11)

for € < —-1,

(6.12)

and that 1

Caseyt) = es

in accordance with the conditions we imposed on 6.

To say that @ is “as smooth as possible” means that,

for each t, N(s,€,t)

€ C°(R/Z x R), and the C®

seminorms of {2 are bounded uniformly in 6 > 0,

(6.13)

which may be taken arbitrarily small.

It is perhaps easiest to think of families of functions 0, 2 depending on a small parameter 6.

6: Fronts for SQG

119

Note that we do not yet discuss the time dependence of (0 $465) which is rather subtle. Note also that our assumption on the C™ seminorms of 22 implies the estimate [oe

oagcca, Oe 6.

for each a,

(6.14)

but that (6.13) is much stronger than (6.14). An “almost-sharp” front of “thickness” 6 is a function 6(@, £2,t) that

can be written in the form (6.10)-(6.13). Now that we have defined a “sharp front” and an “almost-sharp” front we can state our main problems.

6.2 Main problems Q1. Which sharp fronts are (in some sense) solutions to SQG? Q2. Construct a large class of almost sharp fronts that solve SQG. Q3. Is there a limiting equation for 2 in (6.10) if @ satisfies SQG and 6

tends to zero? What can we say about the evolution of I(t) in this situation? The above questions are natural in view of the connection of SQG with warm and cold fronts. Moreover, these questions are analogous to very important and difficult questions about the 3D Euler equations. In fact, sharp-front SQG solutions are analogous to vortex lines, i.e. solutions to 3D Euler whose vorticity is concentrated on a curve I(t) in R° that evolves as time varies. Similarly, almost-sharp fronts are analogous to “vortex tubes”, i.e. 3D Euler solutions whose vorticity is concentrated

on a thin tube of thickness 6, centered about a curve I(t) that evolves in time. A standard conjecture for the motion of a vortex tube under 3D Euler is that a point z € I'(¢) moves with velocity proportional to the curvature

of I(t) at z, in the direction of the binormal to I(t) at z (see Majda & Bertozzi, 2002). This conjecture may be true, but the only known evidence for it is far from rigorous. It is not known what is happening inside a vortex tube. The search for almost-sharp front solutions of SQG is of interest, in

part because it is a (much) simpler model problem for the very difficult problem of understanding vortex tubes. It is also worth mentioning the connection of SQG with the 2D Euler

120

Fefferman

equation. Indeed if you replace (6.3) by the equation

wy = (—A)~18,


0 satisfy

D(A)N{AEC:ReA=o}

=9.

(7.31)

The case when there are points of 5(A) that are to the left of the line {Re =o} will be interesting for us. Denote by X+(A) the subspace of V2(G) generated by the (generalized) eigenfunctions of the operator A corresponding to all the eigen-

values of A in the set {A € C : ReA < o}. By X7(A*) we denote the analogous subspace corresponding to the adjoint operator A*, and we

denote the orthogonal complement of X+(A”*) in V?(G) by X,(A) = Xz:

Xe =Ve(G) eo Xt (A*).

(7.32)

One can show that the subspaces X{(A) and X, are invariant with respect to the action of the semigroup e~4*, and that X, + X}(A) =

Vy (G). Theorem 7.3.1 Suppose that A is the operator defined in (7.29) and that o > 0 satisfies (7.31). Then for each yo € X, the inequality (7.28) holds. Furthermore, the solution of problem (7.30) with such an initial condition is given by the formula

y(t,-) =e Atyg = (2mi)~ f(a- AD) 6

ag dA

(7.33)

a

Here ¥ is a contour belonging to p(A) := C\ D(A) such that arg \= +0 for X € 7,|A| > N for some 0 € (0,7/2) and for N sufficiently large. Moreover, 7 encloses from the left the part of the spectrum (A) that les to the right of the line {Re\ = o}. The complementary part of the spectrum X(A) lies to the left of the contour y. For a proof see Fursikov (2001a,b), for example.

7: Feedback stabilization

141 “

7.3.3

Theorem

on the stabilization of the Oseen

equations Recall that the stabilization problem is to find a control

u € Voo(w) := {w € VE (G) : w(x) =0Vr EG \w}

(7.34)

such that the solution y of (7.26), (7.25) satisfies (7.28) with yo changed

to yo + u. To do this we construct an operator E : Vo (G) > Voh(w) that transforms an arbitrary initial condition yo from (7.25) into a control u such that yo +u € X,. We consider here an analogue of the construction

from Fursikov (2004). It is known that one can choose a basis (di(x),..., dx(ax)) of the space X+(A*) such that restriction (di(x)|,,...,dx(z)|.,) of these elements onto an arbitrary subdomain w € G forms a linearly independent set of vector fields. This property has been proved in Fursikov (2002a,

2004) with the help of Carleman estimates and one abstract result from Fursikov (2001a). We can therefore define the space X, from (7.32) in the following equivalent form,

Xz ={u(x) € Ve(G)

: [@)-a@)ae=0.

Fd bsp Mahy ellaos

G

Theorem 7.3.2 (Fursikov, 2002a & 2004)

There exists a bounded lin-

ear operator

BE: V3(G) + Va(w) such that yo + Eyo € Xo.

Proof Let w; C w be a domain with C® set we consider the Stokes problem

—Aw(x) + Vp(x) =v(z),

divw(z)=0,

boundary Ow. Within this

2£eur;

wlow, =9.

It is well known that for each v € V°(w1) there is a unique solution w of this problem with w € Vj (wi) MNV?(w1). The resolvent operator to this problem we denote as follows: (—7#A);)v = w. The extension of

(—7#A);/v by zero from w to G we also denote by (—7A),,)v. Evidently,

(—#A)z1u € Veb(w1).

We now look for the desired operator EF in the form

Ev(x) = bp oy( #54] (x),

(7.36)

where c; = c;(v) are constants to be determined. Evidently Ev € Vj(G)

Fursikov & Kornev

142

(7.35) and supp Ev C @ . To define the constants c; -we note that by vu+ Eve Xz, if

[ae) G

[Srei(ra)ate i

iS

— [ao -u(a)dx c

(7.37)

for k = 1....,K. As in Fursikov (2002a, 2004) one can prove that this O

system of linear equations has a unique solution.

Thus, by virtue of this theorem, in order to stabilize the problem (7.26), (7.25) one simply has to take u = E'yo.

7.4 Stabilization for the Navier-Stokes

equations

In this section we construct a control for the stabilization of the fully nonlinear problem (7.24), (7.25), which was obtained from the Navier— Stokes system (7.19) by subtracting a steady state from the original variables.

7.4.1

Definition

of the stable invariant

manifold

The natural space for the solution of problem (7.24), (7.27) is

V1

(Qr) = L2(0,T; V?(G) NV (G)) nNH(0,T; VO(G)),

and by virtue of the inclusion C(0,T; Vg (G)) c V1? (Qv7), the natural phase space V for the corresponding dynamical system is V,|(G). The definition of the spaces given around (7.32) and the relations between them imply that

V=Vi4V._,

(7.38)

where V = Vj (G), Vz = X7(A), and VL = X,NV;1(G). It is well known (Ladyzhenskaya, 1963; Temam, 1984) that for each yo € V there exists a unique solution y(t, 7) € YEiOn of problem (7.24), (7.27), where 0 < T\,, — 0 as ||vol] == ||volly + 0. Denote by S(t, yo) the solution operator of the boundary value problem (7.24), (7.27), so that

where y(t,xz) is the solution of (7.24), (7.27). Then S' satisfies the semigroup property S(t2, S(t1, yo)) = S(ti + te, yo),

Wti,te > 0.

7: Keedback stabilization

143

The triple {V, S(t,-),t > 0} is called a (semi)dynamical system with space of states (‘phase space’) V, solution operator $ (t,-), and continuous time t > 0.

We recall now the commonly used concept of the stable invariant manifold W_ = W2(O) defined in a neighbourhood O of the origin. By definition

= {yo €O:

S(t, yo) < O,||S(t,y0) lv < ellyol|ve7’ for all t > 0}, (7.40)

where the quantities c > 0 and o > 0 are fixed and do not depend on yo. The manifold W_(Q) contains all points yo of the neighbourhood O,

such that their trajectories S(t, y,) tend to zero with asymptotic rate not less than e~%*. Using this property one can reduce the solution of

the stabilization problem (7.24), (7.25), (7.28) (with yo changed to yo+u in the last relation) to that of projecting onto W_(O). The stable invariant manifold satisfies the invariance condition

S(t, W_(O)) C W_(O). Moreover, in a neighbourhood O the stable invariant manifold can be defined as a graph in the phase space V = V, + V_ by the formula

ee

VO

ie (ey

vy

fy) yee OV);

(74D)

where O(V_) is a neighbourhood of the origin in the subspace V_, and

f:O(V_) 79 V4

(7.42)

is a certain map satisfying

Ilf@llvs. +0 ee

as

|ly_lly. - 0.

(7.43)

Since W_ is defined by the map y; = f(y_), using the term manifold is quite natural in this case.

The following theorem guarantees the existence of the invariant manifold W_. Theorem

7.4.1

There exists a unique map f : O(V_) >

Vz as in

(7.42) such that the set W_ defined by formula (7.41) is a stable invariant manifold for the family of maps S(t,-) defined in (7.39). Moreover,

IS, yollv < ce" llyolly where the constants

as. t 4 0,

c > 0 and o > 0 do not depend on yo € W_.

(7.44)

Fursikov & Kornev

144

This theorem, along with its method of proof, is well known (see La-

dyzhenskaya & Solonnikov (1973), Marsden & McCracken (1976), Henry

(1981), Babin & Vishik (1992) and references therein). It was shown in Fursikov (2010) that the domain O(V_) of the function f is unbounded with respect to the norm in the space V_.

7.4.2

Feedback operator and stabilization

Here we construct the feedback operator for the Navier-Stokes equations. This operator is the nonlinear analogue of the feedback operator constructed in Theorem 7.3.2 for the Oseen equations.

As in Theorem 7.3.2 we use the domain w € G and the space Voo(w) defined in (7.34), and set O. = {v EV: |lullv < e}. Theorem 7.4.2

Suppose that W_ is the invariant manifold in a neigh-

bourhood of the origin in V = Vj (G) that was constructed in Theorem 7.4.1. Then for sufficiently small ¢ there exists a continuous operator

FO. - Vih(w)

?

(7.45)

such that

v+F(vu)eW Proof

Vue Q,.

(7.46)

We introduce projection operators

Py :VoOVy,

and

PaO Vea ve

(7.47)

for the spaces defined in (7.38) and the notation

Qvu(xz) = v(x) + w(x),

where

w= F(v) € Vib),

(7.48)

and F is the operator we are looking for. By (7.47) and the definition of the invariant manifold W_ (7.41) the desired inclusion Qv € W_ is equivalent to

P,.Qu = f(P_Quv),

(7.49)

where f is the operator defined in (7.42). In addition we have to ensure that the equality

(Qu)(z) =v(z),

rEeG\u,

holds. Recall that in the paragraph before (7.35) we introduced the basis

7: Heedback stabilization

.

145

{d;(z) > J=1,...,K} of X}(A*). It is known (see, for example, Fursikov, 2000a,b) that there exists a basis fe je en nyit==> Lonaiete} rin Vi = XZ(A) that is biorthogonal to the basis {dg} of Xa (AN) ue (€7,¢g) = 0;e orally,

1h) 2). iC where djk is the pete

aise

Thus the map f(u) can be written in the form K

=> STA) ja

and equality (7.49) is equivalent to faut) “CEH

(ih MO — ies ee

wee

eed

(7.50)

G

As with (7.36) we look for the vector field w(x) from (7.48) in the form K

w= —(-#A)5) S— pjdy.

(7.51)

j=l

To find the coefficients (p,...,pxK) =p

we substitute (7.51) into (7.50)

taking into account (7.48). As a result we get

u— Mp = f(v—(p,(-#A),1d)—(e,0-Mp)),

(7.52)

d = (dy(a),...,dx(x)), €= (er(z),..-,ex(a)), and (2d) = DjLy cay. Taking into account the invertibility of the matrix M = ||mj;x||, obtained during the course

of the proof of Theorem

7.3.2, one can apply the

contraction mapping principle to equation (7.52) (Fursikov, 2004). As a result it follows that if ||@|| is sufficiently small, equation (7.52) possesses a unique solution p. The last assumption is satisfied since € in (7.45) is O sufficiently small.

Now by virtue of this theorem one can stabilize problem (7.24), (7.25)

by taking u = F'(yo).

Fursikov & Kornev

146

7.5 Feedback property for a control In this section we disctiss the feedback property that plays a central role in the stabilization problem. We consider here the cases of initial control and a control in the right-hand side, both distributed and impulsive.

7.5.1

Definitions.

The

case of initial control

The important and distinctive property of the control used for stabilization of a solution to an unstable dynamical system is the feedback property. It is precisely this property that allows us to stabilize a system in the unstable situation and, in particular, to create numerical algorithms simulating the original stabilization problem that can be realized in real time, i.e. simultaneously with the functioning of the original stabilization problem. Let

v(t) = fe) + Blu),

vfr-0 = v0,

(7.53)

be a controlled dynamical system, with the state variable u(t) an element of the phase space V, and the control u(t) contained in the space of controls U. The operator B : U —- V is assumed to be continuous with B(0) = 0. Suppose also that @ € V is an unstable steady-state solution of

problem (7.53) without control, i.e. f(¢é) = 0. We consider the problem of stabilizing the dynamical system (7.53) to 6 by means of the control u. In the applied sciences the following non-rigorous but nevertheless very clear definition of ‘feedback control’ is very often used.

Definition 7.5.1 A control u(t) stabilizing the dynamical system (7.53) is called a ‘feedback control’ if it can react on unpredictable fluc-

tuations of the state variable v(t) and dampen them. The most popular mathematical formalization of this notion is as follows.

Definition 7.5.2 The control u(t) is called a ‘feedback control’ if there exists a continuous operator F : V + U such that u(t) = F(v(t)) for each t > 0 and the dynamical system

v'(t) = f(v(t)) + B(F(v(t))), is stable in a neighbourhood initial condition vp.

— ek=0 = v0

of 6 with respect to fluctuations of the

7: Feedback stabilization

.

147

Let us pass from general definitions to the concrete stabilization prob-

lem (7.24), (7.25), with a control u € Vj4(w) in the initial condition. In this problem there are no ‘unpredictable fluctuations’, and so Definition 7.5.1 cannot be applied to this case. On the other hand, Definition 7.5.2 also cannot be applied because the control u depends on t there. But if

we consider the analogue of problem (7.53) with control u independent of time, (7.24), (7.25) will be a particular case of such a problem, and the control constructed in Theorem 7.4.2 for problem (7.24), (7.25) will satisfy the feedback property in the sense of Definition 7.5.2, where the control does not depend on ¢ and the steady-state solution 0 is identically zero. Applying to equation (7.24) the change of dependent variables in (7.23) we can see that the stabilization problem with initial control

(7.19), (7.20) possesses a solution with feedback control as well. In subsection 7.5.3 we introduce more precisely the notion of ‘unpredictable fluctuations’ of the state variable and show that stabilization problem

(7.1)-(7.3) with impulse control (7.4) possesses a solution with feedback control in the sense of Definition 7.5.1, but first we construct, in Section 7.5.2, feedback control for a stabilization problem with distributed control on the right-hand side.

7.5.2

The case of distributed control supported in a subdomain

For simplicity we consider here only the case of the linear Oseen equation. Let us consider the boundary value problem do(t, :)

dt

+ Av(t,-) =u(t,-),

vle=o = Vo,

(7.54)

with vp € V'(G) given, A the operator defined in (7.29), and u(t,x) € L2(R+;Vgb(w)) the control that we are looking for. This control has to be such that i) the solution of problem (7.54) satisfies the estimate

llv(t, )ilvar@ S Cllvollvacaye

(7.55)

where the constant C' = C, does not depend on ||vo||ya(q) and ii) the control is of ‘feedback’ type, i.e. there exists a linear bounded

operator € : Vo (G) + Vob(G) such that u(t,-) = Ev(t,-). Moreover we will look for an operator € in the form

EAA

Fursikov & Kornev

148

where E is the operator defined in (7.36), (7.37), and the magnitude A > 0 will be chosen later.

There exists a control u(t,x) € Lo(R+;Voo(w)) that Theorem 7.5.3 satisfies both conditions i) and ii) above. Proof Recall that the phase space V = Vo. (G) admits the decomposition V = V;+V_ (7.38), and that in V, = Xf (A) one can choose a basis (e,(x),...,e«(ax)) constructed from the (generalized) eigenfunctions of the operator A corresponding to eigenvalues A; with Re A; < a; simi-

larly, in X}(A*) one can choose a basis (d;(x),...,d«(x)) constructed from the (generalized) eigenfunctions of the operator A* corresponding to eigenvalues 4; with Re yj; < o (Fursikov, 2001a,b). These bases are biorthogonal, i.e. they satisfy

(€3,dm)15(G) = 9jm; where 6jm is the Kronecker delta. Therefore v € V, if and only if

i= 5 vje;(c),

where

v; = (v,d;)1,(9)-

(7.56)

j=l We define the desired operator E by the formulas (7.36) and (7.37) from the proof of the Theorem 7.3.2. Comparing (7.37) and (7.56) we see that in fact

u(t,:) = ABu(t,-) =

AEP, v(t,:),

(7.57)

where P, is the projector defined in (7.47). After substitution of (7.57) into (7.54) and applying the projections P, and P_ we obtain, using the notation

v+(t,-)= Pyv(t,-)

and

v_(t,-) = P_v(t,-),

that problem (7.54), (7.57) is equivalent to the coupled equations dv, (t, s)

ade

dv_(t, -)

dt

+ Avy =

AP, Evi

(t,-),

+ Av_ = AP_Eu,(¢,-),

V+|t=0 = Voy = Pyr0,

v_|t=0 = Vo— = P_vp.

(7.58)

7) Feedback stabilization

Using the notation € = (ci,...,cx),

0 = (un

149

50)

ae /di(x)(—#@A 5};)(x)da G = [Vaartante) . Viera)

da) daz,

Wy

and M = (ag) (7.36) as

ey, we can rewrite (7.37) in the form Mé = —3, and

K =~) (M j=l

mA), d;(x),

where

(M71), — Cr

(7.59) Applying the operator P; to (7.59) we get

K P, Eu(x2) = — By3(M~*0);majex(x) eis ==Sar’ Up €n(D) Se Piul(a)s =—

and so (7.58) is equivalent to the problem

DAG Maenw,

Avus(t,: e a dt

| (760)

where Aly, is the restriction of the operator A to V, (recall that Vy is invariant with respect to A), and J is the identity operator. We now

choose A > 0 such that

Reajyh

> o +e

(7.61)

for each eigenvalue , of the operator Aly,, where ¢ > 0 is fixed. Then

(7.60) and (7.61) imply that

lle(é,lv, < CllPyvollv.e OF,

(7.62)

where C = C,1- does not depend on vo, and the solution v_ is defined by the formula t

Eu, v_(t,-) =e “*P_up +f e 4C-7)(P_

(7,-)) dr,

(7.63)

0

where e~4* is the operator defined in (7.33). Fursikov (2001a,b) has shown that

lle74*P_vollv_ < Ce™*"||P_vollv_,

(7.64)

150

Fursikov & Kornev

with the constant C = C, independent of ||P_val|v_Applying (7.62) and (7.64) to (7.63) we obtain t

Ju (6). < Cert P-volv. + CallPseolly, f e-et-en !

—al(t—

—_

+

49)"ar

< e°°"(Ch||P-vollv + C2—=—|IPy20llv,) l-e

—et

(7.65)

< C|lvo ye’,

The bounds in (7.62) and (7.65) imply (7.55) which completes the proof. O 7.5.3

Real processes

A general theory of real processes for stabilization problems subject to boundary control has been developed by Fursikov (2002b,c). Here we recall the main ideas of this theory in the case of impulse control, since in this case the theory is more transparent than for boundary control.

Let us consider the stabilization problem (7.24), (7.25) with control in the initial condition. If the initial condition yo from (7.25) satisfies the bound ||yo||ya¢q@) < € with € small enough, then by Theorem 7.4.2 we can take in (7.25) the feedback control u = F(yo) and obtain

yo +u=yot F(yo) € W_, where W_ is the stable invariant manifold defined in (7.41). This solves the problem from the purely mathematical point of view. However, our aim now is to justify the numerical solution of this problem.

Suppose that we calculate the solution of problem (7.24), (7.25) with initial condition yo + F(yo) at the discrete time instants t, = kr, where k =0,1,2,... and 7 > 0 is fixed. Define S(yo) = S(7, yo), where S(t, yo) is the solution operator (7.39) of problem (7.24), (7.27). Let w* be the result of our calculation at time instant t;,. Since the numerical calculation will not be exact, we have

w* = S(w*-1) 4+ 7y*,

(7.66)

where y* is a numerical error which is unknown to us before time tk, i.e. y* is an ‘unpredictable fluctuation’ (we introduce the multiplier 7

in (7.66) for convenience of normalization). The sequence {w*} defined in (7.66) is called an uncontrolled real process. We suppose that we can estimate the error of our calculation a priori,

lly"llvace) 0 is as in Theorem 7.4.2. It follows from (7.66) that w* ¢ W_ for all k > 1. Therefore, by virtue of the well known structure of the phase flow in a neighbourhood of a steady-state solution, w* moves away from the origin as k > oo and our stabilization construction eolleeess To ensure that our stabilization construction continues to work we have to pass from control in the initial condition to an impulse feed-

back control. In other words (assuming that the calculation of the initial condition wo is exact) we have to change the recurrence relation (7.66) to

@° = yot F(yo), 0” = S(w*-1 + F(w*-!) +7p", ) &=1,2,... (7.68) It follows from (7.68) that for each instant t;, = Tk the real process w* does not belong to the invariant manifold W_, and this is why we apply

at each t;, = Tk the impulse feedback control w* > w*+F(w*) to return to W_. With this approach one can prove the following result.

Theorem 7.5.4 Let F be the feedback map constructed in Theorem 7.4.2. Suppose that the unpredictable fluctuations yp" satisfy (7.67) and

assume that ||yo|lv < € with € defined in Theorem 7.4.2. Then the real process w* constructed by the recurrence relation (7.68) satisfies

Weoia= Cleat lly

o je)

ki co,

(7.69)

where the constant C depends on the operators f and F that appear in

(7.42) and (7.46). This theorem is proved in a similar way to the analogous assertion in

Fursikov (2002b). Note that in contrast to the estimates (7.44), (7.55) proved for feedback initial control in Theorems 7.4.2 and 7.4.1, and for feedback control in the right-hand side in Theorem 7.5.3, the right-hand side of the bound in (7.69) does not tend to zero as k — ov. This is quite natural because, by the definition of a real process, unpredictable fluctuations arise at t, = k7 for each k € N. We should mention that if we assume that the unpredictable fluctua-

tions {y’,i € N} in (7.68) are an independently identically distributed sequence of random vector fields y’ € Vo! (G) then under additional natural assumptions on the random sequence {y'} the random dynamical

system w* defined in (7.68) is ergodic (i.e. it has a unique stationary measure ji) and is exponentially mixing (i.e. the probability distribution of w* tends exponentially to ji in some natural sense). This fact has been

Fursikov & Kornev

152

for established by Duan & Fursikov (2005) for the stabilization problem the Oseen system. é We will not discuss here the theory of real processes in the case of boundary control because this has already been given a detailed exposi. tion in Fursikov (2002b,c).

7.6 Description of numerical algorithms The second part of this paper is devoted to a description of numerical schemes to implement the above stabilization construction. The most difficult part of this construction is connected with the calculation of the stable invariant manifold (see (7.41)) and with the calculation of the projection operators onto this set. The presentation below is based on a number of (axiomatic) assumptions, which treat the operator S(-), an abstract analogue of the operator S(T, -) with T > 0 chosen large enough, where S(t,-) is the solution operator of the boundary value problem

(7.24), (7.27). We use the notation S"*1(-) = $(S$"(-)), n=1,2,... Below we study both the stable invariant manifold defined in (7.41) and also the local stable manifold. Observe that the local stable manifold studied below is a much more general object than the stable invariant manifold (7.41), since the latter occurs near a fixed point, while local stable manifolds are associated with time-dependent trajectories.

7.6.1

General

definitions

We describe the construction of our numerical algorithms in terms of (semi)dynamical systems with discrete time in an abstract setting.

Let V be a Banach space with norm || - || and let S(-):

V + V bea

smooth map which, evidently, will satisfy the semigroup property

S*(S(u)) = S*+2(u),

Vit,ig

EN,Wu€ V.

Then {V,S*(-),i = 0,1,2,...} is called a (semi)dynamical system with state space V, solution operator S, and discrete time i = 0, 12 ee The set 14 (20) = {z; = S*(zo), i=0,1,...,7,.. .} is called the trajectory of the point zp) € V. We suppose that the map S is smooth enough that it can be linearized, i.e. S(z;+ u) = S(z) =te L(z)u =) R(z;)[u]

7; Feedback stabilization

153 2

in a neighbourhood O,, (specified below) of each point z; € T',(20), where L(z;) : V — V is a bounded linear operator and

R(%)[u] = S(z + u) — S(%) — L(z)u is a continuous map such that there exist projection operators Px(2;) and magnitudes us@

pw, 7, CsOs 0, so that the following hyperbol-

icity conditions mere 1967, Pesin, 1977) hold in the neighbourhood

Oz, = {u: ||P (2) (zi —u)|| < ry,

(Al)

Py (2)+aa

(A2y

ela Pela)

Pz (zi)|| < c&

= PiGiee)V,

Lis) ( P(4:)Vie P2( aaa)Vs

(A3) ||L(z)wll p; (A5) I| Px (2:41) {R(%)[ui] — R(z:) [ue] }||

and

< 0) (maxf|lui|, lull} )|lua— wall, for all u1,0: 2; + t1,9 € Oz,, where 0 (y) are continuous positive nondecreasing functions of y > 0,

eo? (0).= 0, and i, ri), ow are certain parameters. We suppose also Ciara

Conditions (A1)-(A5) mean that in a neighbourhood of each point z;, there exist subspaces P;(z;)V and P_(z;)V that are expanded and contracted, respectively, by the action of the linear part L(z;) of the map S; the phase space V can be decomposed as their direct sum, y=

Patz) V = PAZ

\V.

Applying L(z;) to this decomposition we obtain an analogous decomposition at the point z;,,. Assuming that the dimension of the subspace

P,(z)V is finite, the stable subspace P_(z;)V_ has finite codimension. These properties are typical of problems in mathematical physics. The Generalized Hadamard—Perron Theorem claims that if the parameters and functions from conditions (A1)—-(A5) satisfy certain relations (i.e. for so called trajectories of hyperbolic type (Anosov, 1967), or for partially nonuniformly hyperbolic trajectories (Pesin, 1977)), then for some r) there exists a stable invariant manifold

W_(S,O) = {m: me

O,,,||S"(m) — $"(z)||

< Cu", n,1 2 0}

in the neighbourhood O = U%2,0-.,. Moreover, in a neighbourhood of

Fursikov & Kornev

154

f (%); each point 2; this manifold can be defined by a map We (S, Oil a: = W_(z; f)

(cf. (7.40), (7.41), and (7.43) above; here we take p = rage

Our goal is to realize an approximate construction of the local stable

manifold W_(zo, f), ie. of the map f©) that determines this manifold.

Note again that W_(zo,f (°)) contains all points belonging to Oz, whose trajectories tend towards the trajectory of zo with some prescribed rate Cy. Since numerical stabilization problems are considered on finite time

intervals, we assume below that conditions (A1)—(A5) are realized only for a finite segment of the trajectory,

P45) See

os ee

ge

(In fact the schemes considered below admit a uniform closure (at least formally) in the case of infinite n and thus provide a constructive proof

of the existence of the local stable manifold

W_ (zo, f).)

It is worth remarking that when one is solving stabilization problems

on V numerically, an evolution operator S,(-) with a “short” discrete time is usually given, so that the rate of evolution when iterating S,

is “slow”. In this situation one has to define S(u) := SN°(u) for some choice of “typical” No € N. For an operator S(t, u) with continuous time

t € |0, co[ corresponding, for instance, to the problem (7.24), (7.27) the passage to discrete time is made by defining S(u) := SN°(u) := S(Nor, u), for some T > 0 which is not necessarily small. This change helps to ensure that conditions (A3) and (A4) hold in the case of nontrivial Jordan boxes and increases the effectiveness of the proposed schemes.

7.6.2 We now

Stable invariant manifold for a fixed point

consider the approximate

construction

manifold in a neighbourhood of a fixed point z for simplicity that zo =

of a stable invariant

= $(zo), assuming

0. In this case all the operators,

functions,

and constants from conditions (Al)-(A5) do not depend on the index i, so we will drop the (z;) on all quantities and use the notation S(u) = Lu+ R(u), Pz, ps, r, and Cy. Note also that the subspaces PV and (P_V)* are finite-dimensional subspaces corresponding to the Jordan boxes of appropriate eigenvalues. So bases of these subspaces can be constructed by solving the so-called problem of spectrum dichotomy for the operators L and L* (taking into account the nontriviality of the Jordan boxes).

7» Feedback stabilization

1535)

By virtue of the conditions (A1)-(A5), the operator S(u) = Lu+R(u) with

u =

v+w,v

€ P.O,

w

€ P_O,

can be written in the form

S(u) = S;(u) + S_(u), where S(u) = PiS(u). Here

Si(u+w)=Livt+ Ry(vt+w) and $_(v+w) =L_w+R_(v+w), where Liu = PyLu and R4(u) = PzR(u). Let us consider the class B, (O).of Lipschitz maps f : PLO > P,O, where O = {wu: ||P_(u)|| 0 such that for ag € O the problem (7.75) possesses a unique solution. For an arbitrary initial approximation ug = ao + lo, uo € O, the method (7.76)

converges geometrically to un € W_(O, fn).

7.6.4

The stable manifold corresponding to a trajectory

Now let us consider the problem of the approximate construction of the stable manifold corresponding to a trajectory. Below we use the following

notation for the operators from conditions (A1)—(A5):

L(z)=L,

R(z;){u = RO(u), l]

and

Pi (z%) = PY.

Assuming that S(zo) # zo, fix a positive integer n > 0 and take a segment of trajectory

D3(20) = {ep = Oe),

de

- ea.

For each i = 0,.... n we consider the class B.,i) (OM) that consists of all continuous maps f(w) : PYOW PMO, with

OO = {uz ||P (u)|| 00, then there exists a local stable manifold W_(S, 0), and the function f°) determined in such a way approximates f ) ina

- neighbourhood O,,. For numerical simulations we take zero as the initial

function, i.e. f((w) = 0. Since f(”) is tangent to the subspace PMY,

this approximation has an error that is O((r™)?).

7.6.5

Projection onto the stable manifold

Let us consider the following problem.

PROBLEM (If). Project the initial condition ag onto the stable manifold W_(zo, f) along a given subspace £L = span(e1,..-, €io)By definition this means that we have to construct u = ag+l, 1 € £, such

that u € W_(zo, f). In other words we have to construct u = ap +1 that satisfies the condition S(u) € W_(z, f). The corresponding equation is

P™ [S(ao +1) — S(z0)] = f (P™ [S(ao +1) — S(z0)]).

(7.80)

To solve this problem we consider the iteration

PO [L (by +1*+1) + RO (bo + L*)]

= f9(PY[LO(by£1) + ROG) +1*)])>

(7.81)

where /* = )*i°,cKe; and bp = ao — zo. Convergence of this scheme has been proved by Kornev (2006). It is reasonable to choose the starting approximation ug = ao + 1° using the condition ug € PQ). Since the function f™ is tangent to the subspace PYYy, and f( is tangent to POY, this approximation

has an error that is O((r))?). The problem of approximate projection to the stable manifold along the subspace £ can be reduced to the solution of the equation

P\” [5 (ag +1”) — S"(zo)] =0,

where

]" = ees i=1

and we must find the unknown coefficients c?. Note that this equation corresponds to the equations for problems (ff) and (lf) with f(™ =0. Theorem

7.6.4

Let the conditions of Theorem 7.6.3 be fulfilled. Sup-

pose that the function f

€ B,a)(O)) is tangent to the subspace PV

7:'Feedback stabilization

|

161

at zero, and that the system of vectors {po [ei] ee where dim POV a to, form a basis for POY. Then there exists an r) > 0 such that for by € OM problem (7.80) possesses a unique solution u € W_(zo, f). For an arbitrary starting approximation up = ap + Io, uo € O Aemtive method (7.81) converges geometrically to wu. Moreover u = ag +1, 1 = A cil; and

|S” (ao + 1) — 8" (z0)|| < Cp”. Note that, as was mentioned above in Section 7.2.2, control in the initial condition is equivalent to impulse control, i.e. it is “instantaneous control”. This is why in problems from applied sciences this kind of control requires some modifications. Within the context of partial differential equations, an appropriate modification is to consider the problem of stabilization by boundary control in the form proposed by Fur-

sikov (2001a—2004) and used for calculations in Chizhonkov (2003, 2004), Chizhonkov & Ivanchikov (2004), Ivanchikov (2006), and Ivanchikov, Kornev, & Ozeritskii (2009). The connection between initial and boundary control was explained above in Section 7.2.3

7.6.6

Calculations

with control in the right-hand side

Another kind of control that can be used in applications is stabilization by control in the right-hand side. Here we describe one method connected with stabilization by this kind of control that differs from that considered above in Section 7.5.2. We formulate the stabilization problem by control

in the right-hand side in the following form (Kornev, 2008). Let

Sp(z + u) = S(z) + Lut RO(u) + S(u, F), where F is the desired control function and the operator S ()(.,-) prescribes the rule of applying this control. The case F' = 0 corresponds to

the solution operator for the problem without control, i.e. So(-) = S(-). Given initial conditions zo,a9 € V and qr > 0, we need to find F ¢ F

such that

PL SB(a0) — S$”(zo)Ill < @, |Fl|

pe,

inf, Q=arl|PO[a0 — oll), ll- ||=H

(7.82)

aa

In this case F gives the subspace of admissible right-hand sides, and the magnitude 0 < gr < 1 sets the stabilization rate along the subspace

Fursikov & Kornev

162

problem is PIM. Note that since the operator S(-) from the initial tee nonlinear, stabilization along the subspace pi )V does not guaran n equatio 0 = Q that case the In stabilization on the whole space V.

(7.82) takes the form || Po” [S%.(ao) — §"(zo)]|| = 0, and the method becomes the Method of Nonlinear Equations (7.73) with respect to F’. We construct an approximate solution of problem (7.82) in the following way. First we write the linearized version of relation (7.82), taking R®(u) =0, S(u,F)» JF, and obtain

Py”[ao — zolll, PS [Lad + LrF ll < ar|| Fe Ff, ||F|| + inf, Ee LoD pe), ZO,

Lp Steppe), 4h (- 1)

LOO

(7.83)

+..;

(n-2) Jin-1) 4

L(r-1) J(n—2) vey FORA)

We denote the solution of this problem by Fyn.g. If one knows finite

bases for the subspaces P{”)V, (P™V)+, and F then (7.83) is reduced to a generalized least squares problem and can be solved by standard methods. Suppose, then, that we have found the function Fyng. Then we calculate h = Laao + Lr equation for F:

Fyn,g and consider the following nonlinear

P.")[S%(ao) — 5" (z0)] = P& (hl, h = Lgao + LrFr2,Q

an /

This problem can be solved with a simple iterative scheme of the type

PY (Lr Fiti + Re(Fe)] = PY? bo], Fo = Fing Having obtained the solution Fgn.g of (7.84), one can use this to stabilize the initial nonlinear

problem.

In this case the control F S",Q

a ides stabilization of the nonlinear problem (7.82) in the subspace p\" /V in the same way that the optimal control Fy» g stabilizes the linear problem (7.83). However, the optimality condition ||Fsn,g|| > inf takes place only approximately. The process of stabilization is realized here for i = 0,1,....,7—1 with

a function Fsng which does not depend on i. If additional stabilization is needed for i = n,n+1,...,2n —1, then the control function can be calculated again with the help of the same algorithm for the next time segment i=n,n+1,..., 2n — 1, and so on.

7:'

Feedback stabilization

163

2

7.7 7.7.1

Results

of numerical

calculations

The physical model and its mathematical

setting

Using the zero-approximation method, Chizhonkov & Ivanchikov (2004)

and Ivanchikov (2006) solved numerically the stabilization problem via boundary control for Couette flow. As far as we know this is the first successful attempt to stabilize by boundary control an unstable solution of the Navier-Stokes equations in the “velocity-pressure” variables that describe a real physical experiment. Here we consider the problem of numerical stabilization for an unstable flow of four-vortex structure. Suppose that we have an experimental plant that is a shallow rectangular horizontal container filled with electrolyte (an aqueous solution of CuSO,). On both opposite inner sides of the container copper electrodes are placed and under the container a system of direct magnets is set up. The electric current passing through the water produces a deflecting Lorentz force. This leads to the appearance of a flow consisting of four vortices that becomes unstable when the current is high.

It is known (see Danilov et al. (1996) and references therein) that for a certain range of the plant’s parameters the movement of the liquid is described with a high level of accuracy by a quasi-two-dimensional Navier— Stokes system. In dimensionless variables “stream function-vorticity”

this system is

Oe = Aw —w — [phy] + AR+ Au, Ria wo,

(0,0) =a,

(7.85)

— Vy Oz.

Here 7)(t, x, y) is an unknown stream function, h(a, y) describes the magnetic field, and u(t, x,y) is an additional controlling field or is equal to zero. The influence of the bottom of the container is reduced in this case to the damping of the horizontal flows by a linear law. We supplement the system with the following initial and boundary conditions:

plea =0, Q=

Adglen=0,

dio =v",

(7.86)

(0, 71] x (0,U2].

In this case Dirichlet conditions are prescribed on the boundary for the lai 1015; east! functions ~ and w. Let 7 = 2.83 - Oto

Ah(z,y) = Xo enn sin( =) sin"), C99 & —37.75, c13 = €31 = 0.01e22,

and Cmn = 90 for other

m,n.

Fursikov & Kornev

164

characterisThese simplifications have an influence on the quantitative Danilov (cf. picture tics of the fluid, but do not change the qualitative ic et al.. 1996: Kornev & Oczeritskii, 2010). In this case the harmon

{m = 2,n = 2} on the right-hand side forms structures in the flow

that are close to those seen experimentally and the other two harmonics produce additional instabilities.

7.7.2

The structure of the phase portrait

For the numerical solution of the system (7.85), (7.86) we apply a Crank— Nicolson finite-difference scheme for approximation in time; the operator A we approximate by A’ on a “cross” five-point stencil; and for the approximation of the operator [-,-] we use an Arakawa scheme. The unknown grid functions yj and wi, provide approximations of the func-

tions w(t, x,y) and w(t,xv,y) at the nodes (nz, ihg, jhy). Let "t+? = S_ (2b) with wp” = {Vi}, i.e. S,(-) is the solution operator for the difference scheme outlined above. We define the operator S' for the corresponding dynamical system as the difference scheme operator

for No steps, i.e. S(-) = SN°(-). All the calculations below were done for hz, hy ~ 0.015 and r ~ 0.001.

For the chosen parameters the difference equation has an unstable

steady-state solution Z;; = S(2,;) with a four-vortex structure (as in the left-hand picture of Figure 7.1), and in its neighbourhood there are stable quasiperiodic oscillations. Taking into account the relative smallness of 13 and c3; we have 2;; © 243, where z;; satisfies the equation (pAr

A= A") zi; = —C22 sin(2mihz) sin(47jh,).

During oscillations there is periodic merging of the two vortices placed on the diagonal (which possess identically directed rotation) to one vortex directed along the diagonal (as in the right-hand picture of Figure rea ( After that this vortex disintegrates into two vortices with the original structure, and the second pair of vortices flows together to one vortex directed along the second diagonal of the container. This process repeats periodically to a high precision. The trajectory with initial condition vi = 2;; is close to a stationary one on a small time segment but later its behaviour oscillates as described above.

Let a := S?°(z), no = 6100, i.e. the trajectory of the point a = {ai;}

advances beyond the trajectory of the point {z:;} in no steps. The functions 2;; and a;; are depicted in Figure 7.1.

7: Feedback stabilization

165

Using the algorithms we have outlined above we can also solve numerically the problem of stabilizing the trajectory of a point a to the trajectory of a point z, or to a steady-state Z. For this we have to con-

struct the linearization

L = L(-)) ... L©) of the operator S$ along the

trajectory of the point z, and then calculate the subspaces PMY. It is convenient to define the unstable subspace Pp? V by its orthogonal complement (PMV)+, which can similarly be constructed by means of the operators L and L*. With the parameters above, No = 610, and n =

1 we have two-dimensional subspaces POV = {e0)

7 (POv)+. If

we suppose that the point z is fixed it allows us to construct POY using

the linearization A(z) of the right-side of equation (7.85) only in z. The subspace (POv)+ can be constructed by means of the operator A*(z).

Figure 7.1 The functions z (on the left) and a (on the right)

7.7.3

Stabilization by control of the initial condition

In this section we describe the stabilization by initial data with the help

of the algorithm (7.80) (details of how this algorithm was realized are given in Kornev, 2006; Kornev & Ozeritskii, 2010). Let the basis {e1, e2} in the space of admissible displacements £ have the formestOre A= Lied, (0 MT aevat 0

: : < ©= otha, (ths,

jn)

Zo;

where 9 := (0.5, 1] x [0,0.5]. In this case the initial function a changes only in the subdomain Q. Note that the choice of the space of admissi-

Fursikov & Kornev

166

0.01 0.00 -0.01

Figure 7.2 The functions / (on the left) and 6% (on the right)

ble displacements is an important problem (see Chizhonkov (2004) and Ivanchikov (2006)). In some sense the subspaces POY and (POvy)+ (or their orthogonal projections onto the preassigned space L) are optimal. The function I for the iteration process with parameters n = 2 and

No = 1220 is depicted on the left of Figure 7.2. The resulting trajectory S*(a +1) tends monotonically to the trajectory S*(z) as k increases to N, where N © 7400. The accuracy function

6X := §N(z)—S%(a+l) is depicted on the right of Figure 7.2. Note that the stabilization decreases the initial error by 17 times, i.e. 6% /5° ~ 0.06. For k > N the trajectory of the point a does not tend to the trajectory of the point z.

7.7.4

Stabilization by control of the right-hand side

We now describe the solution of the stabilization problem for the trajectory {Sj,(a),i = 0,1,...} towards the steady-state solution Zig bY means of the right-hand side F' with the help of the algorithm (7.84), as stabilization to the finite segment of trajectory {S*(z),i = 0,1,.°.,n} for some small n. In our opinion this problem is important from a practical point of view. For details of the implementation of this algorithm see Kornev (2008) and Kornev & Oczeritskii (2010). Let No = 1, ie S(-) = S;(-). The domain of control Q and the form of the control function we keep as above: (0)

F =span(e;,e2),

where fork=1,2

e, = 0

(the, jhy) otherwise.

EQ

7: Feedback stabilization

Figure 7.4 The functions 5

167

(on the left) and SN (a) (on the right)

However, now the correction F’ we are looking for is added in the form

u®(F) = c(k)e, + co(k)eo to the right-hand side of (7.85) and is a piecewise constant

mri

function of time on each segment of stabilization:

kionim-+1),m 0,1).

Let gr = 0.5 and n = 61. On the left of Figure 7.3 the form of the

control function u* of the operator S-(-) on the first segment 0 0. Note that we always have

for

iL Ss) =O

= Q(r)

S

Fe 4

Indeed, if we assume on the contrary that Q(r) + Ar?q'(r) > 0 for all r > 0, then the function h(r) = Q"(r) + Q(r)/(4r?) is nonnegative and satisfies h(r) ~ q(0)/(4r) as r > 0, and h(r) ~ (477) as7 > oo, Thus

Gallay

184

V/rh € L'((0,00)), but if we integrate by parts we obtain

) dr= 0. ve(a"e) + G2 [0 aennyar = fo r 0 ik

co

if

oo

which yields a contradiction. Finally, in addition to (8.22), we also assume that q?/q’ decays rapidly at infinity k a(p)2

for all k € N.

sup uni o |q’(r)|

(8.23)

As was already observed, the asymptotic profile w, is already an approximate solution of (8.19), (8.20) in the sense that, if we substitute (w.,v.) for (w,v) in (8.19), the left-hand side converges to zero as d — oo. Our goal is to construct here more accurate approximations, which take into account the interaction of the vortices. We look for solutions of the form

w=

Ws +w, vs

tu,

(8.24)

where u = K [w] is the velocity field obtained from w via the Biot-Savart law (8.3). The symmetry (8.17) implies that w(x, —Z£2) = w(x1, 22), ui(@1, —£2)

=

—u1(%1, 22),

(8.25)

U2(%1, —©2) = U2(@1, £2), and in agreement with (8.16) and (8.18) we impose

| w(x)dx = ( ee 7) den. R2 R2

(8.26)

Finally, we assume without loss of generality that w has no radially symmetric component,

namely

27

| w(rcos6,rsin@)dé = 0 0

for all r > 0.

(8.27)

We can always realize (8.27) by including, if necessary, the radially symmetric part of w into the asymptotic profile w,. In this respect, it is

important to note that both conditions in (8.22) are open. Inserting (8.24) into (8.19), we obtain for w the equation

Aw +u-Vw+ Ralw] = 0,

(8.28)

where A is the linearized operator defined by

Aw = vu, -Vu+u-Vu,,

(8.29)

8: Interacting vortex pairs

.

185

and Raq|w] is a remainder term which depends on the distance d between the vortex centres

Ralw|(«)= (v.(x + 2x4) u(—ax — 224)

Olw. +w|(e + 24)*) V (w(x)

+ w(x)).

(8.30) In (8.28)-(8.30), it is understood that u = Kw] is the velocity field associated to w. We look for solutions w of (8.28) in the Hilbert space

Xoo (wre LAR?) * | [o(0)Pp(|2l?)da< oo},

(8.31)

R2

equipped with the scalar product

(w1,W2) = ‘iew1(x)we (2x) (|x|?) da, R

and with the associated norm ||w|| = weight p : [0,00)

W1,W2 € X,

(w,w)'/?. Here we define the

+ R, by

The reason for this particular choice is that the linear operator A has nice properties in the space X, see Section 8.2.2 below. In view of (8.23), the asymptotic profile w, and all its moments belong to X: for any k EN, we have

IIo/?* well?

= | Ln|“*q(|e|2)p(|e2)dx

Bf rk g(r)? DUT dr =< YOO, AAS 0 It is easy to verify that the operator A commutes with rotations about the origin in R?, see Lemma 8.2.2 below. It is thus natural to use polar coordinates (r, 9) in the plane, and to decompose our space X as a direct

sum

Lexy (0)

= @ Pa

(8.32)

i

where P,, is the orthogonal projection in X defined by the formula

(P,,w)(r cos 6, r sin @) 27

= eee Ds

| 0

w(r cos 0’, rsin 8") cos(n( — 0’) do",

nen.

186

Gallay

funcIn particular, Xo = PX is the subspace of all radially symmetric the of s function cohtains P,X = X, tions. and for n > 1 the subspace

form w(rcos6,rsin@) = a;(r) cos(n#) + a2(r) sin(n@). With this notation, condition (8.27) means that Pow = 0.

8.2.2

The linearized operator and its right inverse

We now discuss the main properties of the linearized operator A in (8.29). In the particular case where w, is the profile G of vortex (8.7), the operator A was studied in detail in Gallay & (2005) and Maekawa (2011), and we shall obtain here analogous

defined Oseen’s Wayne

results

in a more general situation. From (8.29) we know that A = A; + Ao, where Aiw = v, - Vw and Agu = K[w]- Vw,. As can easily be verified, Ag is compact in X, while A, is unbounded. The maximal domain of A is therefore

Dk) = DU) Sloe xX

Vee Xk

Remarkably this operator is skew-symmetric in X. Lemma

8.2.1

For all w1,w2 € D(A), we have

(Aw,, we) + (wi, Aw2) = 0.

Proof

We shall prove in fact that both operators

A,,Azg are skew-

symmetric. First, since the weight p(|a|”) is radially symmetric, we have

(vs » Var, wo) + (wi, 0» Vwo) = } p(|2|?) vs - V(wiwe) dx = 0, R2

because the velocity field p(|a|?)v.(x) is divergence free. Next, since Vw,.(x) = (2/m)zq'(\z|7), we have 2

(uz: Vw, W2) + (wi, U2:Vws)

= -= [ ((w-11)uoe + (w-up)wr) dtr, R2

see Gallay & Wayne (2005), Lemma 4.8. Combining both equalities we obtain the desired result. O

As in the Gaussian case (Gallay & Wayne, 2005) the operator A is invariant under rotations about the origin in the plane R2. It is thus natural to work in polar coordinates (r,@) and to expand the vorticity w(rcos@,rsin@) as a Fourier series with respect to the angular variable §. In these variables, the action of A can be described fairly explicitly. Let

g(r) =

la

Qnr2 ’

ere ee

:

1

r>0.

(8.33)

8: Interacting vortex pairs

187

Then we have the following result.

Lemma 8.2.2)

Fizn €N. If w= a,(r)sin(nO) then

Aw = nly(r)an(r) — g(r) An(r)] cos(n8), where A,, is the regular solution of the differential equation

a2 —A’(r) - =A,(r) + aa An(?) Earn),

aS

(8.34)

Similarly, ifw = —an(r) cos(n@), then

Aw = nlp(r)an(r) — g(r) An(r)] sin(n8). Proof Ifn=0, namely if w is radially symmetric, it is straightforward to verify that Aw = 0. Thus we assume that n > 1, and that w = an(r) sin(n@). If % denotes the stream function defined by —Aw = w,

we have ~ = A,(r) sin(n@), where A, is the regular solution of (8.34), namely

ALT=

i la (=) san(s)ds + [Ey

The velocity field u = K[w] = —V+

Ug

san(s)es) ;

eee

is thus

=A,r) cos(n@)e, — A’(r) sin(né)ee,

where x er =

Since Aw = v,

ANG

|x|

and

eg =

AG

A

fa].

-Vw +u- Vw., we conclude that

2 2 es Nan (7) cos(nO) + ” An(r) cos(né) Td (r?), ZT i 1

which, in view of (8.33), is the desired result. The case where w = 0 —dn(r) cos(n@) is similar. As an application of Lemma 8.2.2, we can characterize the kernel of the operator A. We already know that Aw = 0 if w is radially symmetric. Moreover, differentiating the identity v.- Vw.» = 0 with respect to x1 and x2, we obtain A(O,w,) = A(O2w.) = 0. As in the Gaussian case

(Maekawa, 2011) we conclude that the following lemma is true. Lemma

8.2.3.

Ker(A) = Xo @ {a101wW« + a202U. |1, 2 € R}.

188

Gallay Since the decomposition (8.32) is invariant under the action of

Proof

A. it is sufficient to characterize the kernel in each subspace X,,. The case n = 0 is trivial, because Xp C Ker(A), so we assume from now

on that n > 1. If w = ap(r)sin(n@) satisfies Aw = 0, we know from Lemma 8.2.2 that ya, — gAn = 0. In view of (8.34), this can be written in the equivalent form —

nr? _ g(r) Ant) -Foy) An(r) —1> An(r) + (7 All

FFAS

a,

A,

—"()h

re

0)

(8.35) 8.35

Now, the second assumption in (8.22) means that 2

4 2|,/

SOO

See

r>o

(7)

=

op

r>0

a

4.

Q(r)

Thus, if n > 2, the “potential” term (n?/r? — g/y) in (8.35) is positive, and since A,(r) + 0 as r > 0 and r — ov, the maximum principle implies that A, = 0, hence also a, = 0. Thus Ker(A)N X, = {0} if im > 2.

In the particular case n = 1 it is easy to verify that Ai(r) = ry(r) is the regular solution of (8.35). Using (8.34) we find.a;(r) = rg(r), so that w = a;(r)sin(@) = —O.w,. Similarly a;(r)cos(@) = —O,w,, hence the kernel of A in X; is spanned by the functions {0,w,, O2w.}.

a

Using the same arguments as in Maekawa (2011) one can show that the operator A is not only skew-symmetric, but also skew-adjoint in X.

This implies that Ker(A) = Ran(A)+, hence

Ran(A) = Ker(A)t. Let

Yeige.s.

few e Xt.

We now show that Ker(A)+ MY Cc Ran(A), and we establish a semiexplicit formula for the inverse of A on that subspace. Proposition 8.2.4

If f © X,NY for some n > 2, there exists a unique

w € XnMD(A) such that Aw = f. Specifically, if f = bn(r) cos(n@), then wW = a,(r)sin(né), where br(r)

ng(r)’

(8.36)

and A, is the regular solution of the differential equation ”

a,

n?

~An(t) — ~An(r) + (S = a) An(r) = at

r > 0..{8.37)

8: Interacting vortex pairs

189

Similarly, if f = by(r) sin(n@), then w = —a,(r) cos(n@).

Proof

Ifw =a,(r)sin(n@), then Aw = n[y(r)an(r) —g(r)An(r)] cos(né)

by Lemma 8.2.2, where A, satisfies (8.34). The equation we have to solve

is therefore n(pa,, —gAn) = bn, which gives (8.36). Moreover, combining (8.36) and (8.34), we obtain (8.37).

Proceeding as in Gallay (2011), Lemma 3.4, we now show that (8.37) has a unique regular solution, and we establish a representation formula. As we observed in the proof of Lemma 8.2.3, the “potential” term (n?/r? — g/g) in (8.35) is positive if n > 2. Let w+, w— be the (unique)

solutions of the homogeneous equation (8.35) such that

wp(r)~r™

asr>0,

end

il(r)~r”

asr—-oo.

By the maximum principle, the functions 4, w_ are strictly monotone

and linearly independent. The Wronskian W = #,w_ — w_w/, satisfies W’'+W/r =0, hence W(r) = 2nk/r for some k > 0, and we also have

w_(r) ~ Kr™

asr—oo

and

wi(r) ~ ar”

Tr

asr—0.

With these notations the unique regular solution of (8.37) has the following expression for r > 0

fT bls) bn(s)

* why(s) Pn(s)

Ant) = 400)|rey mee) St 8 fey me)(8.38) If f = bn(r)cos(n@) € Xn, it is straightforward to verify that the function A, defined by (8.38) is continuous and vanishes at the origin and at infinity. Moreover, we know from (8.33) that y(r) ~ 1/(2mr?) as r — oo. Thus, if we assume that f € X,MY, we see that the function a,

defined by (8.36) satisfies [>~ an(r)*p(r7)r dr < oo. As a consequence, if w = an(r) sin(n6), we conclude that w € X, 9 D(A), and Aw = f by construction. O Remark

8.2.5

If n = 1 the conclusion of Proposition 8.2.4 fails be-

cause 0;w. € X;Ker(A) for j = 1,2. However, if f ¢ Xi NY

satisfies

(f,0;w») = 0 for 7 = 1,2, one can show that there exists a unique Ker(A)~ such that Aw = f. w € X,M D(A)

8.2.3

The perturbation expansion

Equipped with the technical results of the previous section, we now go back to equation (8.28), which we want to solve perturbatively for

large d. This equation can be written

as Aw + Na|w]

=

0, where

Gallay

190

Nalw] = u- Vw + Ra[w]. Before starting the calculations, we briefly explain why we expect to find a unique solutioh under our symmetry assumptions.

First, if w € X satisfies (8.25)-(8.27), then w € Ker(A)+, hence w is uniquely determined by Aw. Indeed, as was already observed, (8.27) means that Pow = 0. Moreover, it follows from (8.25) and (8.26) that

73 (0;Ws,w) = -=/ zjwdar = 0, T

. aes

JiR2

hence w € Ker(A)+ by Lemma 8.2.3. Next, if w and u = K[w] have the symmetries (8.25), it is straightforward to verify that the nonlinearity in (8.28) satisfies

Na[w](x1, —v2) = —Na|w](x1, 22) for all « = (21,22) € R?. This implies that (0,w.,.Na[w]) = 0 and PyNa{w] = 0. Moreover, we have (d2w.,Na|w]) = 0 by construction, because this is the relation we imposed to determine the angular speed

© in (8.20). Thus, we see that Ng{w] € Ker(A)+, and if we can prove in addition that |x|? Ng|w] € X, then Proposition 8.2.4 (and Remark 8.2.5) will imply that Ny{w] € Ran(A). We can therefore hope to find a unique

w € Ker(A)+M D(A) such that Aw + Ng[w] = 0. To begin our perturbative approach, we compute the remainder term (8.30) for w = 0, namely

Ra(x) = Ra{0|(x) = (v.(a+ 24) — O[ws|(a + ta)*) -Vw.(x),

(8.39)

where x € R*. From (8.21), we know that

val) = aC l

Qin), ,

at

4

oo

where G(r) =f alo)as,

g

Ue

By assumption, the term Q( ||?) decays faster than any inverse power of |r| as |x| — oo, hence we can neglect its contribution in our calculations. For any fixed x € R? we thus have

U(r + 224) =

Leg)

1

1

Qn [2xal? + 9gV(@.20a) +O(G5)

where

eee OS

(c+y)~ yt jz + yl? ly?

as do,

8: Interacting vortex pairs

Setting — (rcos@,rsin@), y = 24 Gallay (2011), Lemma 3.2, we find

=

VG,9) = =n"

191

(d,0), and proceeding as in

os sin(né),

“s

;

=

(8.41)

dn

Gf V2(z,y) = F,So (-1)" oa cos(n@). n=

In particular, returning to (8.40) and using definition (8.20), we obtain ~

Ole

2

= = [ale + 22) OL

ede

~s +0(=),

(8.42)

as d — oo. Note that the term V(z,2zq) in (8.40) gives no contribution to the angular velocity Q[w,]. On the other hand, inserting (8.40) and (8.42) into (8.39) and using the expansion (8.41) together with the relation Vw. = —ag(|x|), where g is defined in (8.33), we find that for

£—(rcos0,r sing) re

Ra(x)Niet

a)n — sin(né) + O(se )

as

d— oo.

Motivated by this result, we now construct inductively an approximate

solution of (8.28) of the form

=

£

1

ao)

=

é

1

where each velocity profile u is obtained from w via the Biot-Savart law (8.3). The order @ of the approximation is in principle arbitrary, but the complexity of the calculations increases rapidly with @, and we shall restrict ourselves to ¢ = 4 for simplicity. Of course, we assume that the

symmetry and normalization conditions (8.25)—(8.27) hold at each order of the approximation. In particular, we have

(x)dx = [ tow (2)da = i;ayw'™ iiw'™ R? JR? R2

(x)dx = 0,

(8.44)

for all n € {2,...,¢}. In view of Gallay & Wayne (2002), Appendix B, (x) decays at least as fast as this implies that the velocity field u\")

when || — oo. It follows that the term u(—a — 2xq) in (8.30) is Odes) as d — oo, and will therefore not contribute to w'™ for n < 4. For the

192 same

Gallay reason,
ov. Indeed, the leading term Q[w,] was computed in (8.42) and we know that the contribution of u2(—x — 2xq) is negligible. Moreover,

using (8.40), (8.41), and (8.44), it is easy to verify that

[ (ve)a(a + 20q)w(2)de = O(d-*), R2

as d + oo. Summarizing, we have shown that

Ra|w](x) = (v.(a + 224) — 2[w.](x + ta)*) -V(ws (ax)+ w(x)) + o(=) 1 meg eye (=-V(e, 22a)

=). V(x) + o(=):

(8.45)

as d + oo. Similarly, the quadratic term u- Vw in (8.28) satisfies u-Vw

1

1

= a

Vus'?) 4 O(=)

as

d-—> oo.

(8.46)

It is now a straightforward task to determine the first vorticity pro-

files in the expansion (8.43). From (8.45) and (8.46), we know that the nonlinearity Nalw] = u- Vw + Ralw] in (8.28) satisfies, for x = (rcos6,rsin@) € R?,

Nalw(x) = se) (Ss n(26) — d3sin(30) +0(=) 2 d?

as d— oo. (8.47)

Thus, to ensure that Aw + Na[w] = O(d~*), we must impose

Aus + ao

sin(nd)=0

for n=2,3.

(8.48)

By Proposition 8.2.4, (8.48) has a unique solution w(™ € X,,M D(A) of the form w'(zr) = an(r) cos(n8), n ) : u(r) = = Antn) sin(n@)e, — A’(r) cos(nO)eg,

where a;,(r), An(r) are given by (8.36), (8.37), with

bn(r) = (—1)"r"g(r)/(27).

8.49

fled

8: Interacting vortex pairs

193

As is easily verified, the symmetry conditions (8.25)-(8.27 ) are satisfied by the profiles w”), u\”) for n = 2,3, and the velocity |u'™(a)| decays

like |z|7"—1 as |a| — oo.

Computing the profiles w, u is more cumbersome, but also more representative of what happens in the general case. First of all, the quadratic term (8.46) is no longer negligible, and using (8.49) for n = 2 we find that

u(x) - Vw)(2) = By(r) sin(46), where

Bi(r) = =(Ad(r)aa(r) —Aa(r)ah(r)). Note that, to ensure that u®?) - Vw?) © X, we need an assumption on the second derivative of the function g appearing in (8.21). For instance,

in analogy with (8.23), one can impose k

WW

2

sup ECS

r>o

|q'(r)|

0, we set 2 = a/(md?) and we denote by wo the vorticity distribution

(8.13) where a; = ag = a and m1 = T2 = d/2. For any

196

Gallay

vy > 0 we consider the (unique) solution w”(a,t) of the rotating viscous i

vorticity equation

ee

Ow + (u — Qa): Vw = vAw,

(8.56)

with initial data wo. Up to a rotation of an angle Mt, the vorticity dis-

tribution w”(«,t) coincides with the solution of the nonrotating equation (8.2) with the same initial data, which is studied in Gallay (2011). The advantage of using a rotating frame is that the vortex centres remain fixed, instead of evolving according to the point vortex dynamics

(8.4). As a matter of fact, Theorem 2.1 in Gallay (2011) establishes that w’(-,t) + wo as vy > 0, for any t > 0. To obtain a more precise convergence result, we decompose the solu-

tion of (8.56) into a sum of viscous vortices a L— 2} a L — Xe w" (x,t) = = wi (—.#) ~ = ws ( me

8.57

where ©} = —22 = (d/2,0). As is shown in Gallay (2011), both vorticity profiles wi} (€,t), wy (€,t) can be approximated by the same function

€ER%, £>0,

+(S)RQ, (t) =GO) weoo

(858)

where G is the Gaussian profile (8.7) and the first order correction F, is constructed as follows. Let £ be the Fokker—Planck operator L=

Ag+

1

ae: Ver,

EER’,

and A be the linearized operator (8.29) with w, = G, namely

Aw = v?-Vw+

K[u]- VG.

Here we use the functional setting of Section 8.2 in the particular case where the asymptotic profile w, is the Oseen vortex G. This means that

q(r) = ¢e~"/4 in (8.21), and the assumptions (8.22), (8.23), and (8.50) are Clearly satisfied. With these notations, the profile F, is the unique solution of the linear equation V

Zl -L)F + AR, +A = 0,

(8.59)

where A(é) = HhLG(E), see Gallay (2011, Section 3.3). In polar coordinates € = (rcos6,rsin@), we thus have

: = Tega” —r?/4”/sin(20) = 5t 1r7g(r) sin(26),

co Interacting vortex pairs where g is defined in position (8.32) of the follows that F, € X2 simple equation AFpo

(8.33). In particular, if we function space (8.31), we too. Now, setting v = 0 + A = 0, which coincides

197

use the angular decomsee that A € Xo, and it in (8.59) we obtain the with (848) for i «2:

Since X2M Ker(A) = {0} we conclude that Fy = w'?). It is clear from (8.59) that the actual profile F, is close to F if the viscosity v is small

compared to the circulation a of the vortices. As a matter of fact, it is

shown in Gallay (2011, Lemma 3.5) that

FL — Follx < C

V

y+ta

(8.60)

To formulate our main approximation result, we introduce a function space with a weaker norm than X. Given any § > 0 we denote by Z, the space

Zp = {we L(R?) : |lwlla < co}, where

wll = [_ w(e)PeP dé. Applying Theorem 2.5 of Gallay (2011) to the particular situation considered here we obtain the following result.

Proposition 8.3.1

Fir T > 0 and let w’(x,t) be the solution of the

rotating viscous vorticity equation (8.56) with initial data wo. There exist positive constants K, 6, depending only on the product QT, such that, if w”(a,t) is decomposed as in (8.57), then the vorticity profiles wi(€,t) satisfy

max |lw! (4) — wrop(Olla < K(Fee

(8.61)

ch

for allt € (0,T], where wapp(E,t) is given by (8.58). This result approximate Indeed, if we present case,

can be reformulated in a slightly different way, using the solutions of Euler’s equation constructed in Section 8.2. set € = /vt, and if we remember that w, = G in the we see that the inviscid profile w. defined in (8.54) satisfies

wyal6) = uipplét) + (4) (MO - RO) +0((3)): Thus, combining (8.60) and (8.61), we obtain the following corollary.

Gallay

198 Corollary 8.3.2

Under the assumptions of Proposition 8.3.1, we have

max w(t) ~wyalla 0, we have

1

|| || = 7

Goplltlds (ony! Iz

aa:

We may assume that u is not identically zero.

=

1

a (27)”

P

1

= 12

|| | | heed ail? | dé + (27r)” \el>a

dé

er

< Ca™||ai||70 + oa | | Iz ae \el2a ||? |a|? dé

< Ca?|ltllz~ + < ||Vul. We now choose

a = |\Vulf oer say) 27Zr n+2

2

+2)

and (11.2) is established.

O

Note that the inequality (11.2) implies that

\|Vulli2 >—“=—_—

(11.3)

Clays

as long as u is not identically zero. In the proof of the theorem, we also need the following Gronwall-type lemma.

Lemma

11.2.3.

Let a > 1 and B € R. Assume

x € C'([0,00), [0,00)) satisfies ba

Cexie

2+ —~———;
0. Then

O(t-A-AV/e-D), a(t) = 4 O((logt)-Ve-)), O(1),

Bc Bal eRe

that the function

11: On the decay of’ solutions of the Navier-Stokes system

239

The proof of the lemma is obtained by separation of the variables x and ¢ and integration, and it is thus omitted. Proof of Theorem 11.2.1 The proof consists of a priori estimates on the solution. In order to make the proof rigorous, we apply the inequalities to the Leray regularization and pass to the limit. Since we may assume that uo is not identically zero, Leray’s regularized solution differs from

zero for all time. Therefore, we may assume that u(t) is not identically zero for all t > 0. Case n > 3. Let u be a solution of the Navier-Stokes system. From the energy inequality

d Gllullis +[IVullz2 =0

T= [S))]

IL. and from the inequality (11.3), we get

ello le to dt Le and thus

d Sllullee + lel” ih 4

cya

(11.4)

Also, using the semigroup representation for solutions, we have

t

u = eu — iief—9)4 pV . (u @ u) ds,

(11.5)

0

where P is the Leray projection and V - (u @ u) is the vector with components 0;(uju,;) for 7 =1,...,n. Since

We FyIze < Cllfllz~ we get

IVu(s)[lcole(s)lleas, la lle~ 0, where

Wlvll rere ~anx(a,b)) = Mllullzzqe»yllz3(a,0)> and we obtain

\|a(t)||> < Cll Goll z- + Cll uoll x2 lull 2222 (@@=x(0,t))>

(11.6)

since ||Vul| 7222 (@»x(0,t)) 0. Based on (11.6), we get

ito = {O00 ia O(1), y>

(11.7) DIR

From (11.4) and (11.7), we get

d aglull 2% C(t

aeae

SS

if y < 1/2 and

SIfullee + alas!” 1/2. Using Lemma 11.2.3, we get

na nie u(t) lz2 = (gha ee or"), (since n > 3 and y > 0, we have 2/n—4y/n < 1). Now after iterating for finitely many times, we obtain the desired result ||u(t)||;2 = O(t-"/*) as follows. Let 7, be the m-th iterate, starting with yo = 0. If n = 3 we have yo = 0, 71 = 1/4, y2 = 3/8 (actually we can take y2 = 1/2, but it is better to take a slightly smaller number to avoid logarithms), y3 = 5/8, and y4 = 3/4. The iteration for n = 4 is similar: we have 7, = 1/4,

Y2 = 3/4, and 73= 1. For n > 5, we have 7; = (n — 2)/4 and y2 = n/4. Case n = 2. The proof is completely different from the one for n >

3. Since 86 ||Vu(t)||Z2dt < . we may, by changing the initial time, assume that Jan ||Vu(t)||7..dt < €2, where €9 > 0 is at our disposal, and \|@ol|z~ < 00. Let T > 2. au (11.5), we get for t € [0,T] \|u(t) || n2

0 is arbitrary. Going back to (11.19), we get ||ai||z~ = O(t—!/?) and then (11.17) follows as in the n > 3 case. For strong solutions satisfying ||u(t)||,2 = O(1), (11.17) implies

]Orul|z2 = O42 for all a € NG, see Kukavica (2001).

Ial/2)

11: On the decay, of solutions of the Navier-Stokes system

Proof of Theorem 11.4.1

Fix me

247

{1,... ,n}. First, oa have

O:(ZmuUk) — A(Lmug) =

—0;(LmUjUE) + UmUE — Ok(LmP) + Skemp —2Omux

fork =1,...,n. Therefore, after multiplication by & Uz

(11.20)

and integration

by parts 5% |2, URUR +2 [0, (Bratt)O;Conr ye +

finan

fAx(emp)emin+

~ [044 eR

Beealy

fSemen

— 2 fOmuntmun Ss [Cement + 2¢mUmp+ |u|?)

(iile21)

(no summation convention for index m), whence, using Lemma 11.2.2,

5

lemullze

emul 3

Cline

< C(|lullzs + llpliza)llemullc2 + llullz2-

Using the «- Young inequality, we have for every €

C(\lullZ« + |lpllz2)llemullz2 lltmullan/”

CE)

(mu) II

(em) Ze? (lullZs + UIpllz2 OPO,

whence, choosing € > 0 sufficiently small,

latent [244 5 gg emule

Clieauy 22

< Ol (emu) LO (lleullzs + llpllz2)°O PO

+ Callie

Now, from (11.20), we get

(iieZ2)

(Lm tor)

Lintih =e

(Ss)ds. _ ifeft 94 (_9; (SmUjUk) + UmUk — Op (fmP) + OkmP — 20mUx)

0

In order to estimate the pressure term, we observe that

END)

=O;

5(ten Uatly )

20; (ums) — 20mP,

from where

Zp = RiRj(mits) + 24710; (Ug) + 2A" Omp,

Kukavica

248

in where R; denotes the i-th Riesz transform. Replacing this expression ‘ (11.22), we get nt

LmUk = e'*(LmUoK) — | d,e*—9)4 Ri Ry (@mUjUk)(s) ds

t + | aie? ig (Umuk + Okmp + 2R.R; (tenths) + 2R«Rmp)(s) ds 0

°

.

t

9)*u,(s) ds. 94 (tm ujur)(s) ds — 2 | Ome’ - | A;et0

;

(11.23)

Using the inequality

d

qylamulln2 < C(\lullzs + llpllz2 + |Vullz2), which follows as in (11.21) (however, without integrating by parts in the term —2 f Omup%muz but by using

-2 fAunt

< C||Vullz2||zmullz2

instead) and (11.17) we get ||2,ul||;2 = O(1). From

Nenu) I~ < (enue) lox + f —spleuls)|zallu(s)Ineas +f (lu(s)lRe + Wel) ds-+ f° Srpllale)lle= ds, which follows from (11.23), we obtain ||(z,,u)"||,-~ = O(1), which then by (11.22) and Lemma 11.3.2 implies ||z,u||p2 = O(t-"/*) as desired. O Acknowledgments DMS-1009769.

The work was supported in part by the NSF grant

References Amrouche, C., Girault, V., Schonbek, M.E., & Schonbek, T.P. (2000) Pointwise decay of solutions and of higher derivatives to Navier-Stokes equations. SIAM J. Math. Anal. 31, no. 4, 740-753 (electronic). Bae, H.-O. & Jin, B.J. (2005) Temporal and spatial decays for the Navier— Stokes equations. Proc. Roy. Soc. Edinburgh Sect. A 135, no. 3, 461-477.

11: On the decay of solutions of the Navier-Stoke s system

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Brandolese, L. (2004) Asymptotic behavior of the energy atid pointwise estimates for solutions to the Navier-Stokes equations. Rev. Mat. Iberoamertcana 20, no. 1, 223-256.

Brandolese, L. (2004) Space-time decay of Navier-Stokes flows invariant under rotations. Math. Ann. 329, no. 4, 685-706.

Brandolese, L. & Meyer, Y. (2002) On the instantaneous spreading for the Navier-Stokes system in the whole space. ESAIM

Var. 8, 273-285 (electronic).

Control Optim.

Cale.

Brandolese, L. & Vigneron, F. (2007) New asymptotic profiles of nonstationary solutions of the Navier-Stokes system. J. Math. Pures Appl. (9) 88, no. 1, 64-86. Carpio, A. (1996) Large-time behavior in incompressible Navier-Stokes equations. SIAM J. Math. Anal. 27, no. 2, 449-475. Caffarelli, L., Kohn, R., & Nirenberg, L. (1982) Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math.

35, no. 6, 771-831.

Dobrokhotov, 5.Y. & Shafarevich, A.I. (1994) Some integral identities and remarks on the decay at infinity of the solutions to the Navier-Stokes equations in the entire space. Russian J. Math. Phys. 2, no. 1, 133-135.

Fujigaki, Y. & Miyakawa, T. (2001) Asymptotic profiles of nonstationary incompressible

Navier-Stokes

flows in the whole

space.

SIAM

J. Math.

Anal. 33, no. 3, 523-544 (electronic). Foias, C. & Saut, J.-C. (1984a) Asymptotic behavior, as t —+ +00, of solutions of Navier-Stokes equations and nonlinear spectral manifolds. Indiana Univ. Math. J. 33, no. 3, 459-477. Foias, C. & Saut, J.-C. (1984b) On the smoothness of the nonlinear spectral manifolds associated to the Navier-Stokes equations, Indiana Univ. Math. J. 33, no. 6, 911-926. Foias, C. & Saut, J.-C. (1987) Linearization and normal form of the Navier— Stokes equations with potential forces. Ann. Inst. H. Poincaré Anal. Non Linéaire 4, no. 1, 1-47. Foias, C. & Saut, J.-C. (1991) Asymptotic integration of Navier-Stokes equations with potential forces. I Indiana Univ. Math. J. 40, no. 1, 305-320. Gallay, T. & Wayne, C.E. (2006) Long-time asymptotics of the Navier-Stokes equation in R? and R°®. [Plenary lecture presented at the 76th Annual

- GAMM Conference, Luxembourg, 29 March-1 April 2005], ZAMM Z. Angew. Math. Mech. 86, no. 4, 256-267. Kajikiya, R. & Miyakawa, T. (1986) On L? decay of weak solutions of the Navier-Stokes equations in R”. Math. Z. 192, no. 1, 135-148. in R”™, Kato, T. (1984) Strong L”-solutions of the Navier-Stokes equation 471-480. 4, no. 187, Z. with applications to weak solutions. Math.

Kukavica, I. (2001) Space-time decay for solutions of the Navier-Stokes equations. Indiana

Univ. Math. J. 50, 205-222.

tokes Kukavica, I. (2009) On the weighted decay for solutions of the Navier-S system. Nonlinear Anal. 70, no. 6, 2466-2470.

250

Kukavica

Kukavica, I. & Reis, E. (2011) Asymptotic expansion for solutions of the Navier-Stokes equations with potential forces. J.. Diff. Eq. 250, 607— 622. Kukavica, I. & Torres, J.J. (2006) Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations.

Nonlinearity

19, no. 2, 293-

303. ‘Kukavica, I. & Torres, J.J. (2007) Weighted L” decay for solutions of the Navier-Stokes equations. Comm. Partial Differential Equations 32, no. 46, 819-831. Lemarié-Rieusset, P.G. (2002) Recent developments in the Navier—Stokes problem. Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL. Leray, J. (1934) Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, no. 1, 193-248. Miyakawa, T. (2002) Notes on space-time decay properties of nonstationary incompressible 271-289.

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12

i

Leray—Hopf solutions to Navier-Stokes equations with weakly converging initial data Gregory Seregin Mathematical Institute, Oxford University, 24-29 St Giles’, Oxford, OX1 SLB. UK. [email protected]

Abstract This note addresses the question of convergence of a sequence of weak Leray—Hopf solutions to the initial boundary value problem for the 3D Navier-Stokes equations provided that the corresponding initial data converge weakly to their limit. Under certain rather mild assumptions, it is shown that the limit velocity field is a weak Leray—Hopf solution with the limit initial data.

12.1

Weak

perturbations of zero initial data

Consider the Cauchy problem for the classical Navier-Stokes system:

OU +07 Vou — Av =—Vq,

divy = 0,

(12.1)

in Qoo = 2x]0,0o[, where Q is a domain in R®, with the Dirichlet boundary condition

Vv

=0 OQ

(122)

[0,00

and with the initial condition

vio = 0

(12r3)

in Q. Here v and q stand for the velocity field and for the pressure field, respectively. Assume that

be I(Q), by J.C. Robinson, Published in Mathematical Aspects of Fluid Mechanics, edited 2012. Press University ge ©Cambrid Sadowski. W. & Rodrigo, J.L.

251

Seregin

252

where J(Q) is the Lo-closure of the set

C§4(Q) = {v € Cge(Q) : divu = Oh. It is well known, see Leray (1934), that this problem has at least one weak solution having finite energy and satisfying the energy inequality t

5 [enPae+

fe)

1

ff ivoPar< 5 |ipPae

0 2

(12.4)

Q

for all t > 0, where dz = dxdt. This solution is unique and smooth for

sufficiently small values of t if b € 73(Q), where T3(Q) is the closure of the set CGg(Q) with respect to the Dirichlet integral. However, we do not know whether or not it is globally unique. Now, let us describe a special subclass of weak Leray—Hopf solutions satisfying the energy inequality locally. We use the following notation

Qr =2x]0,T[

and Ly,s(Qr) = L*(0,T; L”(Q)).

Here, we follow the papers of Ladyzhenskaya & Seregin (1999) and Sere-

gin (2002). Theorem 12.1.1 Let 2 be a bounded domain with smooth boundary. There exists at least one pair of functions v and q with the following differentiability properties:

v € Loo(0, 00;J(M)) N La(0, 00; F3(O)); the function

tr [oe - u(x) dx Q is continuous

on (0, oo| for anyueé

XOv € Lg 3(Qr),

xq ELe3s(Qr),

J(Q);

xV*veE Ls 3(Qr),

xVqe Le 3(Qr),

for ail functions x € Cj (IR) such that x > 0 and x = 0 in a neighbourhood of t= 0 and for any T > 0. This pair satisfies equations (12.1) a.e. in Qo, and the energy inequality (12.4) holds for allt > 0.

12: Navier-Stokes with weakly converging initial data The initial data are fulfilled in the strong L2-sense:

Ilv(-,t) — B()|lng

90

as

253

-

tO.

Moreover, for all t > 0, the functions v and q obey the local energy inequality

[[email protected] dz + 2 [|v oleae at Q

Qt

< i{lv]?(Oep + Ay) + (|u|? + 24) 0- Vy} da de! Qt

for all non-negative functions p € C§°(R*), vanishing in a neighbourhood

Of eel

oy i

(he

Now assume that we have a sequence of initial data a‘*) bounded in

L2(Q) and converging to zero in the sense of distributions, i.e.,

sup |la) lo = M > —. inf 57 Let ing to The any k

v‘*) be a solution to the initial data b = first observation to € N, we can find t,

12. (12.7)

the Cauchy problem (12.1)—(12.3) correspondal*), be made is as follows. By Theorem 12.1.1, for > 0 such that

PO) a0 asa 7 for every t €]0, tx]. Now, let us see what happens if k —> oo. By standard arguments, see

Hopf (1951) and Ladyzhenskaya (1970), one can select (if necessary) a subsequence with the following properties. For any i),

yl) + y

(12.8)

weakly-star in L.(0,7; L2(Q)) and strongly in L3(Qx]0, T[),

Vu) + Vu

(12.9)

Seregin

254

weakly in L2(Qx]0,T[), and - w(x) dr > [ue - w(x) dx

[en 2

(12.10)

Q

strongly in C([0,T]}) for each w € Lo(Q). From (12.8)—(12.10), it follows that u has finite energy and satisfies the Navier-Stokes system in the sense of distributions with divergence-

free test functions. It follows from (12.6) and (12.10) that this solution is subject to the initial condition zero. Now, our aim is to ascertain whether u is identically zero or not. To this end, it would be sufficient to show that u obeys the energy inequality (12.4), i.e t

5 lwePac+ ff vu dz dt’ s. Our second step is to show that, for any T > 0,

where W2-1( 1\(Qr) =

{vu aL. (QT)

: On, Vv, V2u