Nonlinear Partial Differential Equations and Their Applications (Studies in Mathematics and its Applications, Vol 31)

NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS Coll~ge de France S e m i n a r Volume XIV STUDIES IN...

Author: D. Cioranescu | J.L. Lions

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NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS Coll~ge de France S e m i n a r Volume XIV

STUDIES IN MATHEMATICS AND ITS APPLICATIONS

VOLUME

31

Editors: Minnesota P . G . C I A R L E T , Paris P.L. L I O N S , Paris H.A. VAN DER V O R S T , Utrecht D.N. A R N O L D ,

Editors Emeriti: Paris G. P A P A N I C O L A O U , New York H. F U J I T A , Tokyo H.B. K E L L E R , Pasadena J.L. L I O N S *

ELSEVIER AMSTERDAM - BOSTON - LONDON - NEW YORK - OXFORD - PARIS SAN D I E G O - SAN F R A N C I S C O - S I N G A P O R E - S Y D N E Y - TOKYO

NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS Coll~ge de France Seminar Volume XIV Editors DOINA CIORANESCU C e n t r e N a t i o n a l de la R e c h e r c h e S c i e n t i f i q u e L a b o r a t o i r e J.L. L i o n s U n i v e r s i t 6 P i e r r e et M a r i e C u r i e Paris, F r a n c e and JACQUES-LOUIS LIONS * C o l l ~ g e de F r a n c e Paris, F r a n c e

2002 ELSEVIER AMSTERDAM - BOSTON - LONDON - NEW YORK - OXFORD - PARIS SAN D I E G O - SAN F R A N C I S C O - S I N G A P O R E - S Y D N E Y - TOKYO

E L S E V I E R SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands 9 2002 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science via their homepage (http://www.elsevier.com) by selecting 'Customer support' and then 'Permissions'. Alternatively you can send an e-mail to: [email protected], or fax to: (+44) 1865 853333. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier Science Global Rights Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2002 Library o f Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data A catalogue record from the British Library has been applied for.

ISBN: 0-444-51103-2 ISSN: 0168-2024 The paper used in this publication meets the requirements o f ANSI/NISO Z39.48-1992 (Permanence o f Paper). Printed in The Netherlands.

To the memory of Jacques-Louis Lions

This Page Intentionally Left Blank

Contents Preface

ix

An introduction to critical points for integral functionals D. Arcoya and L. Boccardo A semigroup formulation for electromagnetic waves in dispersive dielectric media H.T. Banks and M. W. Buksas

13

Limite non visqueuse pour les fluides incompressibles axisym~triques J. Ben Ameur and R. Danchin

29

Global properties of some nonlinear parabolic equations M. Ben-Artzi A model for two coupled turbulent flows. Part I: analysis of the system C. Bernardi, T. Chac6n Rebollo, R. Lewandowski and F. Murat D~termination de conditions aux limites en mer ouverte par une m~thode de contrSle optimal F. Bosseur and P. Orenga

10

11

12

13

57

69

103

Effective diffusion in vanishing viscosity F. Campillo and A. Piatnitski

133

Vibration of a thin plate with a "rough" surface G. Chechkin and D. Cioranescu

147

Anisotropy and dispersion in rotating fluids J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

171

Integral equations and saddle point formulation for scattering problems F. Collino and B. Despres

193

Existence and uniqueness of a strong solution for nonhomogeneous micropolar fluids C. Conca, R. Gormaz, E. Ortega and M. Rojas

213

Homogenization of Dirichlet minimum problems with conductor type periodically distributed constraints R. De Arcangelis

243

Transport of trapped particles in a surface potential P. Degond

273

14

Diffusive energy balance models in climatology J.I. Diaz

15

Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems M. EUer, V. Isakov, G. Nakamura and D. Tataru

16

On the unstable spectrum of the Euler equation S. Friedlander

17

D~composition en profils des solutions de l'~quation des ondes semi lin~aire critique ~ l'ext~rieur d'un obstacle strictement convexe I. Gallagher and P. Gdrard

18

19

297

329 351

367

Upwind discretizations of a steady grade-two fluid model in two dimensions V. Girault and L.R. Scott

393

Stability of thin layer approximation of electromagnetic waves scattering by linear and non linear coatings H. Haddar and P. Joly

415

20

Remarques sur la limite a --~ 0 pour les fluides de grade 2 D. Iftimie

21

Remarks on the Kompaneets equation, a simplified model of the Fokker-Planck equation O. Kavian

469

Singular perturbations without limit in the energy space. Convergence and computation of the associated layers D. LeguiUon, E. Sanchez-Palencia and C. de Souza

489

22

23

Optimal design of gradient fields with applications to electrostatics R. Lipton and A.P. Velo

24

A blackbox reduced-basis output bound method for noncoercive linear problems Y. Maday, A. T Patera and D.V. Rovas

457

509

533

25

Simulation of flow in a glass tank V. Nefedov and R.M.M. Mattheij

26

Control localized on thin structures for semilinear parabolic equations P.A. Nguyen and J.-P. Raymond

591

Stabilit~ des ondes de choc de viscosit~ qui peuvent ~tre caract~ristiques D. Serre

647

27

571

Preface This volume is the 14 th and last one of the series "Nonlinear Partial Differential Equations and their Applications. Coll~ge de France Seminar", which published the texts of the lectures given at the seminars organized by Jacques-Louis Lions, from his election at the Coll~ge de France in 1973 until his retirement in 1998. It was one of the foremost seminars in nonlinear PDE's and their applications during that period. It is unfortunate that because of his untimely death, on May 17, 2001, Jacques-Louis Lions will not see its publication. This volume is dedicated to his memory.


Studies in Mathematics and its Applications, Vol. 31

D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chap ter 1 A N I N T R O D U C T I O N TO CRITICAL P O I N T S FOR INTEGRAL FUNCTIONALS

D. ARCOYA AND L. BOCCARDO

1. Introduction The study of minima of functionals defined in spaces of functions may be considered one of the keystones of the mathematical analysis. Remind the efforts by the great mathematicians of the last and present century to develop sufficient conditions on the functional for the existence of minimum. This theory is deeply related with the existence of solutions of boundary value problems. Indeed, this connection is estabilished by the well-known Euler-Lagrange equations associated to the functional. However, there exist boundary value problems for which the associated functional is indefinite, i.e. it is unbounded from below and from above. This means that it has not global extrema and so we have to search the solutions of the problem among the critical points, i.e. the points for which the derivative of the functional vanishes.

From the abstract point of view there is a difference between the study of minima and of critical points. Indeed, for the existence of minima we need only assumptions on the functional. On the contrary, we point out that the results of existence of critical points involve additional hypotheses on the regularity of the functional to assure the existence of a derivative in some sense. This may explain why the theory of mimima handles classes of functionals with more general hypotheses of smoothness than the critical point theory. In some papers [4], [5], [6], we overcame this difference by developing a critical point theory for nondifferentiable functionals. We observe explicitely that our model functionals does not involve similar functions to the modulus. In fact, the nondifferentiability of the considered functionals is due to the introduction of some smooth Carathdodory function A(x, u) (as smooth as you_ want). Specifically, we consider here functionals or defined in W~

2

An introduction to critical points for integral functionals

(~"~ C I ~ N

open, N > 1) by

J(v) = f~ A(x, v)lVvl dx - / ~ F(x, v +) dx, v ~_ Wlo'2(a),

(1)

with 0 < a 0,

and a'(x,z) - 0, z < 0. (a3) Either

a(x, z) is increasing and concave with respect to z _> 0,

(6)

a(x, z) is decreasing and convex with respect to z :> 0.

(7)

or

Let X - W 1'2(~), endowed with the usual norm I[" I]; Y - W1'2(~t) N

L2/(2-q)(~t), endowed with the norm [[-I[Y - ] ] " ]]2/(2-q). By (al) and (a2) the functional J is continuous on X and satisfies (H). We point out t h a t X=Yonlyforq____~ a(x, Un)lVTk(?.t)lq-2VTk(u).VWn,kdx +f

,~l>k

a(x,

Un)lVuniq-2VUn

9VTk(u)dx

and the right hand side converges to zero. Moreover,

a'z(X,~)Wn,klVu~lqdx 0 and thus u r 0. m

4. Main examples The abstract theorem (with X = w l ' 2 ( f t ) and Y = W01'2(ft)N L ~ ( f t ) ) of the Section 2 is applied now to obtain nonnegative critical points of the functional J " W~'2(ft) ,, ~ IR t2 {+cxD} defined by

J(v) = /aA(x,v)[Vv[edx - faF(x,v+)dx, v c Wl'2(~),

(9)

i.e. nonnegative solutions of the b o u n d a r y value problem:

tt E 1wlO'2' (ft) n L ~2(ft), , -div(

A(x, u) Vu)

+

-~Az(x , u)]Vu !

=

Fu(x , u) - f(x, u)

}

(P)

8


where f 9f~ x IR , IR is a C a r a t h 6 o d o r y function w i t h subcritical growth. It is clear t h a t for a solution u of ( P ) we are m e a n i n g u e W01'2(f~) cl L ~ ( f ~ )

/

fa A(x, u ) V u V v d x + ~1 fa A" (x,u)lVul2vdx - fa f(x, u) vdx

/

for every v C W 1'2 (f~) C'l L ~ ( a ) . T h e h y p o t h e s e s t h a t we a s s u m e on the C a r a t h 6 o d o r y coefficient A " f~ x IR --, IR are t h e following: (A1) T h e r e exists c~ > 0 such t h a t

c~ < A(x, z), for a l m o s t every x c f~ and z >_ 0. (A2) T h e r e exists R1 > 0 such t h a t for every z _> R1. (A3) T h e r e exist m > 2 a n d (m-2)2

Ct 1 ~>

Atz(X, z) >_ 0 for almost every x c f~,

0 such t h a t

A(x' z) - 2zA1(x'

_> c~

for almost every x E f~, z _> 0. Notice t h a t all a s s u m p t i o n s on A(x, z) are for z _> 0. In fact, since we are looking for n o n n e g a t i v e solutions of ( P ) we can a s s u m e w i t h o u t loss of generality t h a t A(x, z) is even on z. On the o t h e r h a n d , we will a s s u m e the following conditions on f(x, z)" ( f l ) T h e r e exist C1, C2 > 0 such t h a t

If(x, z)[ ~ witha+l 0 such t h a t

o < .~F(~. z) _< zf(x. ~). for almost x C f~ and every z _> R2 (m is t h e s a m e as in (A3)). (f3)

f(x, Iz[) -- o([z]) at z - - 0 , uniformly in x C ft.

D. Arcoya and L. Boccardo

Theorem

4.1

9

-- Assume (Al-3), (f1-3) and that A(x,z)

lim z --+ + c ~

= O, unif. in x E ~.

(10)

Z a

Then the problem (P) has, at least, one nonnegative and nontrivial solution. R e m a r k s 4.2. 1. The above theorem is essentially contained in [5]. However, in t h a t paper it is assumed in addition t h a t A(x, z) is bounded from above and its derivative A'z(X , z) with respect to z is also bounded. In [7], we have seen t h a t these additional hypotheses are not necessary for the existence. 2. The general case of fianctionals

/~

fl(x, v, Vv) dx - J~ F(x, v +) dx, v E W0:'P(~), (p > 1)

could be also handled as in [5]. For simplicity reasons, we just present here the case p = 2, fl(x, v, Vv) - A(x, v)IVvl 2. 3. Some remarks about the meaning of (A3) and (f2) m a y be found in [5, L e m m a 3.2 and Remarks 3.1]. I

Proof of Theorem 4.1. For every n E IN, let h~ be a nondecreasing C 1 function in [0, oo) satisfying hn(s) = s, Vs E [0, n -

1],

h,~(s) n.

Consider the coefficients A n ( x , z ) =- hn(A(x,z)), x E t2, z E JR. Clearly, An satisfies (A1-3) and, in addition, it is bounded from above with bounded ! derivative A n ( x , z) (with respect to z). In this way, if we define the func12 tionals g~ : W 0' (t2) > IR by setting

&(~) = j~ A~(., ~)lWl ~d.

: s

s+l

~+) d~, ~ c wl'~(a),

then using ( f : - 2 ) and (A3), it can be seen in a similar way to the one in Section 2 t h a t Jn satisfies (C). Indeed, we have

10


L e m m a 4.3 - (Compactness condition) Assume (A1-3) and (fl-2). Then the functional J~ satisfies (C). Using in addition (f3) and following the ideas of [2], it is easily seen that Jn satisfies the geometrical hypotheses of Theorem 2.2. Consequently, by it, there exists a nontrivial and nonnegative solution un of the problem u. ~

Wo~'2(~t) n

/

L ~ (gt),

1 (X , u~ )lW.I = f(x, u+). - d i v ( A n ( x , Un) r u n ) + ~A~

/

(11)

with critical level n =__ inf

max Jn(7(t)), -~EF rE[0,1]

Jn(un)--c

where F - { 7 " [ 0 , 1] ~ Wol'2(f~)n L~ W 1,2(~t) n L ~ (Ft) is such that J~ (e~) < O. An(x, z) < A(x,z) and (10), we observe that

Jn(t~Pl) ~ J(t~91) =

l ts +1 [c~

= 0,7(1) = en}, en e Taking into account that

A(x,t~l) [~7r [2 d x ts-1

--

1 s+l

/

~+ldx]

< O,

for all t C [to, co) if to > 0 is large enough. This allows us to choose e~ - top1 (independent of n c IN). On the other hand, by the Mountain Pass geometry of J1 there exist 5, r > 0 such that Jn(V)

~ J l ( v ) ~ (~, VlI~II _< ~,

(i.e., roughly speaking, v - 0 is a strict local minimum of Jn, uniformly in n E IN). This implies that Cn ~ 5.

(12)

We claim that {un} is bounded in w l ' 2 ( ~ ) . An (x, z) _~ so that u ~: 0. Thus u is the weak nontrivial (and nonnegative) solution we were looking for. II Remark 4.4. We conclude by noting that in [7] the reader can find more existence results for nonlinearities f(x, z) which are different combinations of concave and convex functions in the quasilinear spirit of [1], [8]. A c k n o w l e d g m e n t . This paper was partially presented by the second author at the Coll@e de France Seminar (24.3.1995). Both authors would like to thank to the organizers of the Seminar for having given the opportunity of presenting their work.

References [1] Ambrosetti, A., Brezis, H. and Cerami, G., Combined effects of concave arid convex nonlinearities in some elliptic problems. J. Funct. Anal. 122 (1994), 519-543. [2] Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349-381. [3] Arcoya, D. and Boccardo, L., Nontrivial solutions to some nonlinear equations via minimization. Variational Methods in Nonlinear Analysis, edited by A. Ambrosetti and K.C. Chang, 49-53, Gordon and Breach Publishers, 1995. [4] Arcoya, D. and Boccardo, L., A min-max theorem for multiple integrals of the Calculus of Variations and applications. Rend. Mat. Acc. Lincei, s. 9, v. 6, 29-35 (1995).

12


[5] Arcoya, D. and Boccardo, L., Critical points for multiple integrals of Calculus of Variations. Arch. Rat. Mech. Anal. 134, 3(1996), 249-274. [6] Arcoya, D. and Boccardo, L., Some remarks on critical point theory for nondifferentiable functionals, to appear in NoDEA. [7] Arcoya, D. and Orsina, L., Landesman-Lazer conditions and quasilinear elliptic equations, Nonlinear Anal. TMA. 28 (1997), 1623-1632. [8] Boccardo, L., Escobedo, M. and Peral, I., A Dirichlet problem involving critical exponent. Nonlinear Anal. TMA., 24 (1995), 1639-1648. [9] Boccardo, L., Murat, F. and Puel, J.P., Existence de solutions faibles pour pour des ~quations quasilin~aires s croissance quadratique. Res. Notes in Mathematics 84, Pitman, 1983, 19-73. [10] Canino, A. and Degiovanni, M., Nonsmooth critical point theory and quasilinear elliptic equations, in Top. methods in differential equations and inclusions, Kluwer Academic Publisher, 1995. [11] Dacorogna, B., Direct Methods in the Calculus of Variations. SpringerVerlag, 1989. [12] Degiovanni, M. and Marzocchi, M., A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl. 167 (1994), 73-100. [13] Pellacci, B. Critical points for non diferentiable functionals, Boll. U.M.I 11-B (1997), 733-749. [14] Stampacchia, G., Equations elliptiques du second ordre ~ coefficients discontinus. Les Presses de L'Universit~ du Montreal, 1966. David Arcoya Departamento de An~lisis Matem~otico Universidad de Granada 18071-Granada Spain E-mail: [email protected] Lucio Boccardo Dipartimento di Matematica Universit~ di Roma 1 Piazza A. Moro 2 00185 Roma Italy E-mail: [email protected]


D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 2 A SEMIGROUP FORMULATION FOR ELECTRMAGNETIC WAVES IN DISPERSIVE DIELECTRIC MEDIA

H.T. BANKS AND M.W. BUKSAS

1. Introduction In a forthcoming monograph [2] we have developed a theoretical and computational framework for electromagnetic interrogation of dispersive dielectric media. In that work we show that one can take a time domain variational or weak formulation of Maxwell's equations in dispersive materials and, in the context of inverse problems, use partially reflected polarized microwave pulses to determine both dielectric material properties and geometry of bodies (specifically for plane waves inpinging on slab geometries in paradyms which approximate far field interrogation). This is done in configurations involving either supraconductive reflecting back boundaries or acoustically generated virtual reflectors. The propagation and reflection of electromagnetic waves in dispersive .dielectric media is, of itself, an interesting topic of investigation. As we point out in the next section (and demonstrate computationally in [2]), the underlying dynamical systems are not typical of either standard parabolic or standard hyperbolic (even with the usual dissipation) systems and are hence of mathematical interest. In this short note, we consider the Maxwell system for rather general dispersive dielectric media and show that such systems, under appropriate conditions on the polarization law, generate Co semigroup solutions. These results are presented in the context of the 1dimensional interrogating systems developed in detail in [2] and we invite interested readers to consult that reference for more detailed discussions and development of the underlying model employed here.

2. Modeling of dispersiveness in dielectric media We begin with time domain Maxwell's equations in second order form (e.g.,

14

A semigroup formulation for electromagnetic waves

see [2]) for the electric field E = E(t, z) of 1-dimensional polarized waves

+ 115 + -1- & E0

c 2E"= - - -1L

-

E0

(1)

~0

where c = 1 is the speed of light in vacuum, J~ is the conduction current density, J8 is a source current density and P is the electric polarization of the dielectric medium. We assume very general constitutive material laws for the polarization and conductivity given by

P(t, z) - (gp 9 E)(t, z) =

/o

gp(t - s, z)E(s, z)dz

(2)

3~(t, z) -- (gc 9 E)(t, z) - ~o t g~(t - s, z)E(s, z)dz

(3)

where we have tacitly assumed that E(t, z) = 0 for t < 0 and that both gp((,z) and gc((,z) vanish for ( < 0. With these assumptions, the integrals in equations (2), (3) are equivalent to integration over all of (-c~, c~) ~nd thus are indeed convolutions. The displacement susceptibility kernel gp (also referred to as the dielectric response function(DRF)) and the conductivity susceptibility kernel gc introduce nonlocality in time in the polarization and conductivity relationships [1], [15] which is equivalent to frequency dependence of the dielctric permittivity E and conductivity a when using frequency domain approach. We assume that either P or de or both may contain instantaneous (local in time) components by introduction of 5 distributions in the kernels gp and/or gr respectively. A medium is dispersive if the phase velocity of plane waves propagating through it depends on the frequency of the waves [16, Chap.7], [10, Chap.8]. To determine the dispersive nature of a medium described by equations (1)-(3) we seek plane-wave solutions of the homogeneous analogue of (1) of the form E(t, z) = Eoe -i(~t+~z) which travel in the z direction and have wavelength A = 2w/~. The phase velocity Vp of these waves is the speed at which planes of constant phase move through the medium. In this case the argument w t - ~z is constant and dz

Vp : d--t = w/~.

(4)

Seeking plane wave solutions of the form Eoe -i(~t+~z) in (1) is equivalent to seeking solutions of the form Eoe +i~z in the frequency domain version of (1). Thus we use the Fourier transform in (1) and obtain ~d2

iw

eo

EO

H. T. Banks and M. W. Buksas

15

where we have ignored the source term d8 and where the overhat will represent the Fourier transform throughout. Since we see from (2) and (3) t h a t /5 _ t)p/~ and Jr = t)cE:, this can be written

c2/~,, + 002(1 + i~c + ~)__pp)/~_--O. 02s

(6)

s

We note t h a t (6) is the generalized Helmholtz equation [16, p. 271] +

-

0

(7)

with 00 2

= -~(1 ~

~00E0

)

(8)

E0

which has solutions/~(w, z) = Eoe +i~(~)z. It follows that the corresponding time domain solutions are our desired solutions of the form E(t, z) = Eoe -i(wt+i~z) where the wavenumber n = n(w) will in general depend on the frequency w. The equation (8) relating the frequency w and the wavenumber of propagating waves is known as the dispersion equation for the medium. In the case of vacuum or free space where t~p = g~ - 0 so t h a t n - w/c, we obtain the corresponding phase velocity vp = c = the speed of light as expected. More generally the phase velocity in a dielectric medium with conductivity and polarization is given by

=

/V/1 +

+

(9)

In light of (9) and the definition of a dispersive medium, we see t h a t if either [tc/w or ~p depend on w, we will have dispersiveness. Several special cases are worthy of note. For instantaneous conductivity, t h a t is, go(t, z) = aS(t) so t h a t (3) reduces to O h m ' s Law Jc = erE, we see t h a t the term i[t~/eow becomes icr/eow. Thus a medium with simple Ohm's Law conductivity will be dispersive (it is also dissipative in the usual sense since the conductivity term in (1) becomes --r For instantaneous polarization (often assumed in standard EO treatments of the Maxwell theory) we find gp(t,z) = eoXh(t), where X is the dielectric susceptibility constant and hence [lp/eo = X and the medium is not dispersive. One must turn to more complicated (and more realistic) models, such as those of Debye or Lorentz, to have a polarization based contribution to dispersiveness in a medium. For the usual Debye polarization model [11, p.386] one has

gp(t) -- e - t / r

eoo)/~',

t > O,

(10)

16

A s e m i g r o u p formulation for electromagnetic waves

where ~- is a relaxation parameter and es, e~ are familiar dielectric constants. In this case one finds 1 - iwz O p ( W ) - e 0 ( e s - eo~)[ 1 _[_T2~d2]

For the Lorentz model [16, p.496] we have gp(t) - e~

-t/2~ sin~ot,

t > 0,

(11)

Vo where ~0 - v/w02 - 1/4T2. In the frequency domain this yields

4T 2 -

-

and again we have a polarization based dispersive medium. Higher order (the Debye and Lorentz models correspond to first and second order, respectively, differential equation models for the polarization P - see [2] and the references therein) models, as well as combinations of such models also lead to dispersion in a medium. Thus, in summary we see that instantaneous conductivity but not instantaneous polarization yields dispersiveness in a medium. For a polarization contribution to dispersiveness one must include first or higher order polarization models (instantaneous polarization can be correctly viewed as zero order polarization dynamics). For our semigroup presentation in the next section we shall therefore consider the model (1) with instantaneous conductivity and a general (higher order) polarization model given by (2) with gp = g where the D R F g is assumed smooth in time (i.e., without loss of generality we can assume no instantaneous component for g). Such distributed parameter systems are of interest since they are neither simple hyperbolic nor parabolic in nature. For simple Ohm's Law conductivity and instantaneous polarization (or no polarization), the system (1) becomes a well understood dissipative or damped hyperbolic system for which a semigroup formulation can readily be found in the research literature on distributed parameter systems. However, for (1) with polarization based dispersiveness, we obtain a system with behavior of solutions that are neither standard hyperbolic (finite speed of wave propagation along characteristics) nor standard parabolic (infinite speed of propagation of disturbances). Indeed for (1) with either Debye or Lorentz polarization, rather fascinating solutions can be observed. These

H.T. Banks and M. W. Buksas

17

involve the formulation of so-called Brillouin and Sommerfeld precursors where a pulsed excitation (containing waves with a range of frequencies) evolves into waves propagated with different velocities which coalesce into wave "packets" (see Chapter 4 of [2] and [1] and the references therein for discussions of these phenomena). It is of both mathematical and practical interest to know whether these interesting systems can be described in a semigroup context. The potential advantages afforded by a semigroup formulation are widespread since there is a tremendous literature for control, estimation and identification, and stabilization of systems in a semigroup setting. Results for both stochastic and deterministic control methodologies (in both time domain and frequency domain) including open loop and feedback formulations are abundant [12], [3], [4], [11], [19]. In the next section we present a semigroup formulation of the system (1) with simple Ohm's Law conductivity along with general polarization based dispersiveness generated by polarization laws of the form (2). To be more precise, we take (1) for t > 0 and z C (0, 1) with Jc(t, z) = a ( z ) E ( t , z) where a(z) vanishes outside ~ c (0, 1]. The closed region ~t is some dielectric material region (e.g., a slab or several slab-like regions) containing instantaneous conductivity as well as non trivial polarization of the form (2) with gp(t,z) = g ( t , z ) vanishing outside z c ~t. Using this form of conductivity and polarization in (1), we obtain the system 1 1 E ( t , z) + - - ( ~ ( z ) + g(O, z ) ) E ( t , z) + --[~(0, z ) E ( t , z) 6-0

+

6-0

i)(t - s , z ) E ( s , z ) d s

- c2E"(t,z) - -1Js

(t, z).

(12)

~0

With this system we take boundary conditions (see [2] for details) that represent a total absorbing boundary at z = 0 and a supraconductive boundary at z = 1. This can be expressed by E(t, 0) - cE'(t, 0) = 0

(13)

E(t, 1 ) = 0 .

(14)

With the definitions a(t,z)

=

_l~(t,z), •0

=

~(z)=--lt~(0, z) ~0

+

s(t,z) - -1L(t,z),

6-0

EO

we can write equation (12) as + ",/E + ~ E + o~ 9 E - c2E '' = ,7

(15)

18


where a , E is the usual convolution a 9 E ( t , z) -

~0t a ( t -

s, z ) E ( s , z)ds.

(16)

One can use the boundary conditions ( 1 3 ) - (14) to write ( 1 5 ) i n weak or variational form so as to seek solutions t --4 E ( t ) in V - H~(0, 1) = {r C H~(0, 1 ) : r = 0} in a Gelfand triple setting V r H ~-~ V* with pivot space H = L2(0, 1). Under modest regularity assumptions on a, ~, T and fl, one can establish existence, uniqueness and continuous dependence (on initial conditions and input) of solutions. Details are given in Chapter 3 of [2].

3. A semigroup formulation We turn in this section to a semigroup formulation for the dispersive system (12)- (14) or equivalently, (13)- (15), with instantaneous conductivity and general (non instantaneous) polarization. For our development we assume t h a t 7, ~ c L ~ (0, 1) while a E L ~ ( ( 0 , T) • (0, 1)) and a, ~, 7 vanish outside Ft. We moreover assume that a can be written as ~(t, z) = O~l(t)ol2(z) where 0 < ~L _< c~2(z) < C~U on ~t C (0, 1] for positive constants a L , a V , with a2 vanishing outside ft. We assume t h a t t --~ a~ (t) is positive, monotone nonincreasing, and in H 1(0, T) so that &l (t) < 0. This monotonicity assumption is typical of the usual assumptions in displacement susceptibility kernels (e.g., see [9, p.102] or [15]). We shall return to discuss this monotonicity requirement further after our semigroup arguments of this section. We consider the term (16) given by

/0ta ( t -

s)E(s)ds =

i

a(t-

s)E(s)ds

oo

from (15) and note that it can be equivalently written

l

a(t - s)E(s)ds (x)

f

a ( - ~ ) E ( t + ~)d~ oo

a ( - ~ ) E ( t + ~)d~ -

G ( ~ ) E ( t + ~)d~

where G(~) _-- a ( - ~ ) . We denote GI(~) = a l ( - ~ ) so that G(~) = Gl(~)a2. The approximation is valid for r sufficiently large (r = oc is permitted) so t h a t a(t) .~ 0 for t > r. We observe at this point that (~1 (~) _ 0 with G1 (~) > 0 on ( - r , 0].

H.T. Banks and M.W. Buksas

19

As introduced in the previous section, we take V - H ~ ( 0 , 1 ) , H L2(0,1) with Y ~-+ H ~-~ Y*. We shall have use of H - L22(~), the space L2(~t) with weighting function a2, which is readily seen to be equivalent to L2(gt) due to the upper and lower bounds on a2 C L ~ ( ~ ) . We shall denote the restriction of functions r in L2(0, 1) to fl again by r and write' r e L2(gt) or r e L2~ (~t) whenever no confusion will result. Using the above definitions and approximating, we may write (15) as

E,(t) + 7F,(t) +/3E(t) +

f

G(~)E(t + ~)d~ - c2E ''(t) =

if(u).

(17)

r

Using an approach given in [5], [6], [14] and [7] for viscoelastic systems, we define an auxiliary variable w(t) in W - L ~ l ( - r , 0;/~) by w(t)(O) = E(t)-E(t+O),-r_~ 0 < 0. Since G(0, z) > 0 for 0 E ( - r , 0],z C ~ we may take as an inner product for W the weighted L 2 inner product

(rl, W ) w --

f

Gl(O)(U(O),w(O))FidO

=

r

f

~1(0) r

L

a2(z)u(O,z)w(O,z)dz

under which W is a Hilbert Space. We note that by our notational convention explained above, we have w(t) e W for any E ( t , z ) with E(.,.) e L~I ( - r , 0;H). Using a standard shift notation, we may write w(t) = E ( t ) - E ( t + O) = E ( t ) - Et(O) where Et(O) - E ( t + O) for - r _ 0 < 0. Adding and subtracting appropriate terms in (17), we find

E(t) + ~/E(t) + ;3E(t) +

f

G(()E(t)d( r

-c2E"(t) =

if(U)

f

G(() [E(t) - E t ( ( ) ] d ( r

or, equivalently

E(t) + ~/E + (~ + G l l ) E ( t ) -

f

G(~)w(t)(~)d~ - c2E"(t) - i f ( u )

(18)

r

where Gll (z) - f o r G(~)d~ - a2(z) f o r G1 (~)d~ and w(t)(~) - E(t) Et(~). We observe that Gll, like/3, is in L2(f~) and L2(0, 1). For our semigroup formulation, we consider (18) in the state space Z V x H • W - H~(0, 1) • L2(0, 1) x L ~ l ( - r , 0, H) with states (r r 7) ( E ( t ) , E ( t ) , w ( t ) ) - ( E ( t ) , i E ( t ) , E ( t ) - Et(.)). To define an infinitesimal generator, we begin by defining a fundamental set of component operators. Let A c s ]2") be defined by

(19)

20


where 50 is the Dirac operator 50r = r

Then we find

< -- ~iqh, ~>V*,V -- H q- V*,V

is symmetric, V continuous and V coercive (i.e., ~1 (r r for constants Ao and cl > 0). We also define operators B E s 12") and I22 c s

~_ Cllr

~01r

by

Br = - ~ r - ~r

(22)

so that ( - Be, r and, for ~ e W -

- (7r r

+ cr162

(23)

L~, ( - r , O; H), (Kr/)(z)- { 0

z e [0, 1]\ft

f o 0(r162

(24)

z e a.

Since G({, z) = 0 for z e [0, 1]\f~, we abuse notation and write this as R.

-

f c(~),7(~)< r

even though, strictly speaking, ~(~, z) is only defined for z c ft. With these definitions and notations, equation (18) can then be written

as

(E, e>~.~ + < - AE, r +

+ < - ~E, r

( - k ( E - E+), ~ > ~ . ~

- v-,v

or E,(t) -- fiE(t) + BIE(t) + k ( E ( t )

- E t) +

J(u)

in l;*.

(25)

We rewrite equation (25) as a first order system in the state r (E(t),lF(t), w(t)) where w(t) - E ( t ) - E t. To aid in this we introduce another operator D " dom D c W ~ W defined on dom D - {~ c H l ( - r , 0;/t)[~(0) - 0} by D r / ( O ) - N0~ ( 0 ) .

H.T. Banks and M.W. Buksas

21

We then observe that w(t) = E ( t ) - E t satisfies

Thus we may the equation

=

E(t) - E(t + O) - E(t) - DEt(O)

=

# ( t ) + D(E(t) - Et(O)) =/E(t) + Dw(t)(O).

formally rewrite (25) as a first order system and adjoin to it

@(t) -- D w ( t ) + #(t).

(26)

We then obtain the first order system for ((t) given by - .AC(u) + 7 ( u )

where A given by .4-

(

(27)

o i o) A 0

B I

K D

(28)

is defined on dom A -

{ (r r r/) E ZIr E ~J, r/E dom 7), ~ r + Br C 7-/}.

(29)

That is, A| - (r ~ r + Be + ~ , r + ~ ) for ~ - (r r ~) in dom ,4. The forcing function ~- in (27) is given by ~ = col(z,J,t). To argue that Jt is the infinitesimal generator of a C0-semigroup, we actually consider the system (27) on an equivlaent space Z1 = V1 x H x W where V1 is the space Y with equivalent inner product (r162 1 - ~1(r162 where ~1 is the sesquilinear form given in (21). Recall that ~1 is symmetric, V continuous and V coercive so that it is topologically equivalent to the V inner product. We are now ready to prove the following generation theorem. T h e o r e m - Suppose that '7,/3 C L ~ (0, 1), a e L ~ ((0, 1) • (0, 1)) with c~,~, "Y vanishing outside ~. We further assume that c~ can be written a ( t , z ) = c~l(t)a2(z) where a l e HI(0, T) with al(t) > 0, all(t) < 0, and 0 < O~L ~ O~2(Z) ~ OLU f o r positive constants C~L,C~U. Then the operator .A defined by (28), (29) is the infinitesimal generator of a Co-semigroup on Z1 and hence on the equivalent space Z. Proof. To prove this theorem, we use the Lumer-Phillips theorem ( [15, p. 14]). Since Z1 is a Hilbert space, it suffices to argue that for some A0, A - A~Z is dissipative in Z1 and 7 ~ ( ) ~ - A) = Z ~ for some A > 0, where 7 ~ ( ) ~ - A) is the range of A I - .4. We first argue dissipativeness.


22

Let 9 = (r r r/) c dom .4. Then -

(A~I,, ~}z~

(r r

+ (fi'r + B e +

= (r r

+ (fie + Be, r

--

a l (r all))- 0"1 ( d/), r

-

-__

k6]r

2 nt- C 1 [ r

--

Cllr

2 + (k6 -

r + ~ 1 ( r 1 6 2 - (/~'(A - D ) - I A r 1 6 2

-- ) ~ 0 [ r )~0 -

-- k 5 1 r

k5)[r

H"

Thus if we define the sesquilinear form a~(r 0) - ((A 2 - A B - A - / ~ ' ( A -

D)-IA)r

r

we see that for A sufficiently large, a~ is V coercive and hence, by the LaxMilgram lemma [20], it is invertible. It follows immediately that (33) is invertible for r C V. This completes the arguments to prove the Theorem. m Let S(t) denote the semigroup generated by ,4 so that solutions to (27) are given by

((t) -- S(t)(o +

S ( t - s).T'(f)Vf.

(34)

Solutions are clearly continuously dependent on initial data (0 and the nonhomogeneous perturbation 9r. The first component of ((t) is a generalized solution E(t) of (17). One can now argue that the solution agrees with the unique weak solution obtained in Chapter 3 of [2], by using the arguments in Chapter 4.4 of [8]. Briefly, one argues equivalence for sufficiently regular initial data and nonhomogeneous perturbation. Then density along with continuous dependence is used to extend the equivalence to more general data (see [8] for details).


25

4. Concluding remarks In the previous section we presented a semigroup generation theorem under general conditions on the coefficients a,/3, V of (15). The only possibly restrictive condition involved a(t, z) - ~og(t,z) 1 .. = Oll(t)oL2(Z) where it is required that a l (t) > 0, dl (t) _< 0. We consider more closely the condition for some common polarization laws. For Debye polarization in a region ~t, we have a l (t) - ~ogp(t) 1 .. where gp is given in (10). T h a t is,

gp(t) = eo (% - E~) e _ t / ~ 7"

so that

1 .. (es - E ~ ) e_t/~ > 0 OLl(t) = ~ogp(t ) T3 and (t)

-

-

-

0 or (~l(t) ~ 0 SO that our generation theorem does not guarantee a semigroup representation for systems with a Lorentz polarization law. In spite of this, we do believe that the Lorentz law does yield a system with a semigroup representation. We conjecture that the proof of the theorem we present can be modified to weaken the hypothesis on c~ so as to include Lorentz and other oscillatory (even order) polarization models. We are currently pursuing these ideas. In closing we point out that the general class of dielectric response functions consisting of a linear combination of decreasing exponentials (essentially multiple Debye mechanisms) suggested for glasseous materials by Hopkinson [15](see the discussion in [9, p.101-103] are included under the theory presented in this note.

26


A c k n o w l e d g m e n t . This research was supported in part by the Air Force Office of Scientific Research under grants AFOSR F49620-98-1-0180 and AFOSR F49620-95-1-0447 and the Department of Energy, under contract W-7405-ENG-36. The authors are grateful to Dr. Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB, San Antonio, TX, for his continued encouragement and numerous specific technical disscussions throughout the course of the research reported on here and in [2].

References [1] R. Albanese, J. Penn and R. Medina, Short-rise-time microwave pulse propagation through dispersive biological media, J. of Optical Society of America A, 6:1441-1446, 1989. [2] H.T. Banks, M.W. Buksas, and T. Lin, Electromagnetic Interrogation of Dielectric Materials, SIAM Frontiers in Applied Mathematics, SIAM, Philadelphia, 2000, to appear. [3] A. Bensoussan, G. DaPrato, M.C. Delfour, and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. I, Birkhs Boston, 1992. [4] A. Bensoussan, G. DaPrato, M.C. Delfour, and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. II, Birkhs Boston, 1993. [5] H.T. Banks, R.H. Fabiano and Y. Wang, Estimation of Boltzmann damping coefficients in beam models, In COMCON Conf.on Stabilization of Flexible Structures, 13-35, New York, 1988, Optimization Software, Inc. [6] H.T. Banks, R.H. Fabiano and Y. Wang, Inverse problem techniques for beams with tip body and time hysteresis damping, Mat. Aplic. Comp., 8:101-118, 1989. [7] H.T. Banks, N.G. Medhin and Y. Zhang, A mathematical framework for curved active constrained layer structures: Wellposedness and approximation, Num. Func. Analysis Optim., 17:1-22, 1996. [8] H.T. Banks, R.C. Smith and Y. Wang, Smart Material Structures: Modeling, Estimation and Control, Masson/J. Wiley, Paris/Chichester, 1996. [9] F. Bloom, Ill-Posed Problems for Integrodifferential Equations in Mechanics and Electromagnetic Theory, Vol. 3 of SIAM Studies in Applied Math, SIAM, Philadelphia, 1981. [10] D.K. Cheng, Field and Wave Electromagnetics, Addison Wesley, Reading, MA, 1989. [11] R.F. Curtain and A.J. Pritchard, Infinite-Dimensional Linear Systems Theory, LN in Control and Info. Sci., 8, Springer Verlag, Berlin, 1978. [12] R.F. Curtain and H.J. Zwart, An Introduction to Infinite- Dimensional Linear Systems Theory, Springer Verlag, New York, 1995.


27

[13] R.S. Elliott, Electromagnetics: History, Theory and Applications, IEEE Press, New York, 1993. [14] R.H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations in linear viscoelasticity, SIAM J. Math. Analysis, 21: 374-393, 1990. [15] J. Hopkinson, The residual charge of the leyden jar, Phil. Trans. Roy. Soc. London, 167:599-626, 1877. [16] J.D. Jackson, Classical Electrodynamics, J. Wiley & Sons, New York, 2nd edition, 1975. [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [18] J.R. Reitz, F.J. Melford and R.W. Christy, Foundations of Electromagnetic Theory, Addison Wesley, Reading, MA, 1993. [19] B. van Keulen, H ~ Control for Distributed Parameter Systems : A State Space Approach, Birkhs Boston, 1993. [20] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. H.T. Banks Center for Research in Scientific Computation NC State University Raleigh, NC. 27695-8205 USA E-mail: [email protected] M.W. Buksas Los Alamos National Laboratory T-CNLS MS B258 Los Alamos, NM. 87545 USA E-mail: [email protected]


Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chapter 3 LIMITE NON VISQUEUSE POUR LES FLUIDES INCOMPRESSIBLES AXISYMETRIQUES

J. BEN AMEUR and R. DANCHIN

R ~ s u m ~ . On s'int~resse s la limite non visqueuse des ~quations de NavierStokes incompressibles tridimensionnelles axisym~triques. On suppose que les donn~es initiales ont des propri~t~s de r~gularit6 stratifi~e de type poche de tourbillon. En utilisant la conservation du tourbillon divis~ par la distance l'axe de sym~trie (dans le cas non visqueux), on trouve des r~sultats de convergence pour tout temps similaires ~ ceux de la dimension deux. En particulier, on a convergence forte au sens de la r~gularit~ stratifi~e, et le gradient de la vitesse est born~ ind@endamment de la viscositY. Lorsque les donn~es initiales n'ont pas de r6gularit~ stratifi~e, on donne une majoration de la vitesse de convergence L 2 en fonction de la viscositY, tr~s proche de celle de [4] pour la dimension deux. A b s t r a c t . We are concerned with the inviscid limit for three-dimensional axisymmetric incompressible flows. The initial data are vortex patches or, more generally, have striated regularity. Using the conservation of the vorticity divided by the distance to the axis of symmetry (in the inviscid case), we gather global convergence results similar to those of dimension two, namely, strong convergence for striated regularity and uniform estimates for the gradient of the velocity. When initial data do not have striated vorticity, we give an upper bound depending on the viscosity for the speed of convergence in L 2 norms. This result is similar to the one stated in [4] for two-dimensional fluids.

Introduction Consid~rons le syst~me de Navier-Stokes incompressible en dimension d -- 3:

{ Otv~, + v~, . Vv~, - rave, = -Vp~,, div v, = 0, v.(0) = ~0,

(NS~)

30

Limite non visqueuse pour les fluides incompressibles axisymdtriques

oh ~, la viscosit6, est une constante strictement positive, la vitesse v~(t, x) est un champ de vecteurs sur ]R 3 d6pendant du temps t >_ 0 et la pression p ~ ( t , x ) est un scalaire. La variable d'espace x d6crit ]R 3 entier et on s'int6resse ~ l'6volution du fluide pour tout temps t positif. Formellement, pour v t e n d a n t vers z6ro, on obtient les 6quations d'Euler Otvo + vo . Vv0 = - V p 0 , div v0 = 0, vo(0) = v ~

(NSo)

I1 est bien connu que, pour un champ de vitesse v ~ "un peu mieux que lipschitzien" et u >_ 0, le syst6me ( N S ~ ) est localement bien pos6 et que v~ tend vers v0 fortement lorsque u tend vers z4ro. Notre r6sultat de r6f6rence, dfi ~ T. Kato, sera le suivant (voir [8])" T h 6 o r 6 m e 0.1 - Soit s > 5/2 et v ~ un champ de vecteurs ~ divergence nulle et ~ coefficients dans l'espace de Sobolev H~(]R3). Alors il existe un temps T > 0 tel que pour tout v > O, le syst~me ( N S ~ ) a d m e t t e une unique solution v~ dans C([0, T]; H ' ) n C I ( [ O , T ] ; g ~-2) et tel que de plus, v~ tende v e r s Vo d a n s C([0, T]; g ~) n el([0, T]; H s-2) l o r s q u e u t e n d v e r s O. Sans hypoth6se suppl6mentaire sur la donn6e initiale, la question de l'existence globale d'une solution r6guli6re reste ouverte mais on dispose du crit6re d'explosion suivant (voir [1] et [9])" T h 6 o r ~ m e 0.2 - Soit ~ > 0, s > 5/2 et v~ E C([0, T*[; H s) une solution de ( N S . ) . ' a p p a r t e n a n t pas ~ C([O,T*]; Hs). Soit w~ - rot v . le yecteur tourbillon associd au champ v . . Alors on a

~0T* II.~(t)llL~

dt-

+~.

Ce crit6re d'explosion permet bien 6videmment de retrouver l'existence globale de solutions r6guli6res en dimension deux. Nous nous int6ressons ici s des champs initiaux v ~ axisym6triques, c'ests de la forme

v o = ~o (~, z ) ~ + ~o (~, Z)~z oh nous avons adopt6 un syst6me de coordonn6es cylindriques (r, 9, z) et not6 er-(cosg,

sing, 0),

ee = ( - s i n g ,

cosg, 0),

ez = (0,0,1),

les trois vecteurs de base au point x = (r cos g, r sin 9, z).

J. Ben A m e u r and R. Danchin

31

Pour v ~ suffisamment r~guli~re, (NS~) conserve cette propri~t~ de sym~trie. Le tourbillon w~ se r~duit alors ~ w~ - w~,o(r, z)eo. En identifiant le vecteur tourbillon w~ au scalaire w~,0, on constate que la quantit~ d~f

a~, = w~,/r v~rifie

(Or +v~.V)a~

-

v( ~

+

~

3

= o.

r

Lorsque u - 0, la quantit~ av est visiblement transport~e par le riot r de v~, et si u > 0, l'op~rateur du second ordre apparaissant dans (T~) a le "bon signe". Ceci a permis b. M. Ukhovskfi et V. Yudovitch d ' ~ t a b l i r dans [14] le r~sultat suivant: T h 6 o r ~ m e 0.3 - Soit v ~ E (H 1(IR3))3 un champ de vitesse axisymgtfique divergence nulle. Notons w ~ le tourbillon initial. On suppose que w ~ w ~ E L2(]R3)ALC~(]R3). Alors, pour tout u >_ O, ( N S v ) admet une unique solution vv clans L ~ (JR +; L 2) telle que de plus w~ E L~oc(]R+; L 2 M L ~ ) et w~/r E L~176 L 2 N Lc~). Cette solution reste azisymdtrique pour tout temps. En combinant le th~or~me 0.3 avec les th~or~mes 0.1 et 0.2, il est ais~ de voir que si v ~ E H s (s > 5/2) et w ~ E L2(]R 3) N L~ alors les r~sultats du th~or~me 0.1 sont valables pour t o u t temps. De plus, la solution obtenue reste axisym~trique.

Dans les trois premieres parties de cet article, on s'int~resse plus particuli~ment k des donn~es initiales de type poche de tourbillon. L'~tude de ce genre de structures provient du cas bidimensionnel non visqueux. On parle de poche de tourbillon lorsque west la fonction caract~ristique d'un domaine born~ de ]R 2. Lorsque u - 0, le tourbillon est conserv~ par le riot de la solution. Un r~sultat de Yudovitch (voir [15]) nous assure alors que la structure de poche de tourbillon est stable pour tout temps: le domaine de d~part est simplement transport~ par le riot. Lorsque la fronti~re du domaine initial a une r~gularit~ h61d~rienne C r (r > I), J.-Y. Chemin montre que la solution v0 de (NSo) appartient c~ + Lloc(]R ;Lip) (off Lip d~signe l'espace des fonctions born~es et lipschitziennes), et que la r~gularit~ C r de la poche est pr~serv~e pour tout temps. Ceci r~sulte en fait de r~sultats bien plus g~n~raux de persistance de la r~gularit~ stratifi~e pour les fluides incompressibles (voir [3] et les r~f~rences jointes). Dans [5], le premier auteur s'est int~ress~ k la g~n~ralisation du r~sultat de J.-Y. Chemin pour les fluides faiblement visqueux. On obtient notamment le r~sultat suivant: T h ~ o r ~ m e 0.4 - Soit ~o un ouvert bornd de ]R 2 dont la fronti&re est une courbe simple de classe C T M (E E]0, l[). Soit v ~ le champ de vitesses

32


divergence nulle et s'annulant ~ l'infini, de tourbillon w ~ = leo. Alors, pour tout ~ >_ O, (NS~) admet une unique solution v, dans Lzo~ avec donnde initiale v ~ et ii existe une constante C ne d4pendant que de ~o telle que Vv C ]R +, Vt E ]R +, IlVv.(t)ll~,~ < Ce Ct. Notons ~2t,, le riot de v, ~ 1'instant t et f~t,~ = ~bt,,(f~~ On a les rdsultats suivants de convergence: (i) Pour tout e' < e, O~tt,~ est une courbe simple de classe C 1+~'. Plus prdcisdment, si ~yO E CI+e(S1; ]R 2) est une paramdtrisation rdguli~re de Of~~ et si l'on pose "y~(t) = ~t,~(~~ a/ors ~y~(t) est une paramdtrisation rdguliSre de O~tt,~. De plus ~y~ e Lzo~(~+; C 1+~' ($1,]R2)) uniformgment en v e t ~y~ t .d C 1+'' (SX, lo[ q. . t .d O. (ii) Notons (~tt,v)~-- {x e ]R 2, d(x, [~t,v) > h} et (~t ~ )h = {x C f~t,v, d(x, Of~t,~) > h}. Alors il existe une constante C ne ddpendant que de f~o et telle que pour v, t, h > O, on nit h2 exp(--4(eCt'--l))

]lCd0]]L 2 '

[Iw~(t)- lrh,, IlL2 ((r~F,~)h) _< 2 II

~

min

1, c(vt)X/2e2(~C~-X)h e - a-~', exp(-4(~c'-1))

.

Remarque 0.1. Pour des raisons techniques, nous avons ~t~ amends dans [5] ~ utiliser les espaces de Besov B~,o~ (2 < a < +oo) pour mesurer la r~gularit~ stratifi~e lorsque ~ > 0. L'apparente perte de r~gularit~ dans O~tt,~ n'est en fait due qu'b. l'utilisation d'espaces de HSlder dans l'~nonc~! du th~or~me 0.4. Elle n'a pas lieu pour des poches de tourbillon ~ fronti6re dans ~a~c~ R l+e * I En dimension trois, m6me pour un fluide non visqueux, il n'y a aucune chance pour que la structure de poche de tourbillon stricto sensu soit stable, m6me s temps petit" le tourbillon n'est pas constant le long des lignes de flot. Pour un fluide axisym6trique non visqueux, on sait cependant que la quantit~ a0 est conserv~e. Ceci a permis b. P. Gamblin, X. Saint-Raymond et P. Serfati de prouver des re!sultats globaux de persistance de structures stratifi6es (voir [7], [10] et [11]). Dans le cas visqueux en dimension quelconque et sans hypoth6se particuli~re de sym6trie, on dispose de r6sultats de convergence de (NS~) vers


33

(NSo) en un sens qui p%serve la %gularit~ stratifi~e de type Besov (voir [6]). Ces %sultats ne sont bien stir que locaux en temps. Nous nous proposons de montrer que dans le cas axisym6trique, les %sultats de [6] sont globaux. Lorsque la donn6e initiale est une poche de tourbillon axisymdtrique (i. e. w ~ = r l a o avec t2 ~ domaine axisym~trique fronti~re C1+~), nous prouvons en sus l'analogue du th6or~me 0.4, partie (ii).(~ ceci pros que w v e s t chang~ en w~/r). Enonqons le rdsultat de convergence que nous obtenons pour une telle donn6e initiale. T h ~ o r ~ m e 0.5 - Soit f~o c IR 3 un ouvert bornd ~ symdtrie axiale et frontibre de classe C 1+~ (e E]0, 1[). Soit v ~ l'unique champ de vitesses coefficients H 1, ~ divergence nulle et de tourbillon w ~ = rluo. Alors, pour tout v >_ O, ( N S ~ ) a d m e t une unique solution v , dans Llo~ 1) avec donnde initiMe v ~ et il existe une constante C ne ddpendant que de f~o, telle que Yv C ]R +, Vt c IR +, lIVv~(t)tlLor _< Ce Ct 89log(l+t). On a de plus les rdsultats de convergence suivants.

Soit

(i) Pour tout t >__ 0 et ~' < e, Oat,v est dans C l+e,. { f 0 = 0} une dquation non ddgdndrde de On ~ et ft,v = fo o ~t,-1 . AIors { ft,v = 0} est une dquation non ddgdndrde de Of~t,v, fv C Llo ~ (JR +; CX+e') uniformdment en v et f , tend vers fo dans Llo~176

C 1+~') 1orsque v tend vers O.

(ii) I1 existe une constante C ne ddpendant que de f~o et telle que si 1'on note z(t) d~f exp(--4((1 + t) ct 89 -- 1)), on ait pour tout v, t, h > O,

h2

iI w~(t) r

020

la,,~ [I a) 0

< 211TIIL~ min

/

1, C (v

~l/2e2((l+t)ct 89

h2 z(t))

-1)e-9-~

Dans la derni~re partie de ce travail, on abandonne les hypotheses de r6gularit6 stratifi6e pour les donn~es initiales et on s'int~resse k la vitesse de convergence des solutions v~ de ( N S , ) vers celle v0 de (NSo) ~ v ~ fix6e v~rifiant les hypotheses du th~or~me 0.3. Le gradient de v0 n'est alors pas n6cessairement born6 et peut exploser comme un logarithme au voisinage de

34


certains points. On peut cependant montrer que v~ converge vers v0 fortement dans LtoC~(]R+; L 2) avec une vitesse de convergence qui se comporte comme une puissance de u se d6gradant au cours du temps" T h ~ o r ~ m e 0.6 - Soit v ~ E H 1 un champ de vecteur axisymdtrique ~ divergence nuUe tel que w ~ w ~ c L 2 fq L ~ . Notons v~ la solution de ( N S u ) donnde par le thdor~me 0.3. Alors il existe une constante C universelle telle que pour tout temps T > 0 et viscositg v >_ 0 vdrifiant uTg2(T)
0 et il existe une constante C ne d @ e n d a n t que des donndes initiales et telle que

V~ E ]R +, Vt C ]R +, IIVvv(t)llL~ _~ Ce ct 89log(l+t). Sous l'hypoth~se (7-/1), on a de plus le rdsultat suivant: Notons Xt,v la famille transportde p a r le riot de v~. Alors Xt,~ reste (e, a)-substantielle pour tout temps t ~_ 0 et wv(t) reste dans BEa(Xt,u) uniformdment en v. Enfin, Si e' < e, a/ors X ~ ( x , D)r X~,~ et div X~,~ tendent respectivement vers X ~ ( x , D)r Xo,x et div Xo,~ dans Llo~176 B~'), et X . , ~ ( x , D ) w . tend vers Xo,~(x,D)wo dans L~oc(IR+; B~'-I). Sous l'hypoth~se (7-/2), on a de plus le rdsultat suivant" Pour tout t ~_ 0 et u > O, Et,u d6_~_fCt,u(EO) est une surface compacte de classe B TM, et o~u(t) est dans B~,~.~, . Enfin, Zt,~ tend yers ~t,o au sens suwant. Soit { f o = 0} une dquation de classe B~ +E de p o. Posons ft,, ear f ~ r

Alors {ft,~ = 0} est une dquation de Et,~ et pour tout

e' < e , f~ tend vers fo dans L~o~(IR+; B[ +~') lorsque u tend vers O.

Remarque 1.4. L'hypoth~se v ~ E H 1 est automatiquement v~rifi~e d~s que ~o c L p N L ~ pour un p _< 6/5 (c'est le cas par exemple si coo c B ~ ( X ~ est support compact ou dans L 1). Remarque 1.5. Lorsque u = 0, la partie existence et r~gularit~ d'une solution pour le syst~me d'Euler s'~tend aux espaces de HSlder (voir [7], [10] et [11]). Dans le cas u > 0, la preuve du th~or~me 1.1 utilise des estimations uniformes en u pour les solutions d'une ~quation de la chaleur avec terme de convection. De telles estimations ne sont pas connues dans le cadre des espaces de HSlder (voir [5] section 4) mais sont vraies dans les Bar pour l < a < +oo. Remarque 1.6. Si (7-/2) est v~rifi~e, on peut construire une famille (e,a)substantielle X ~ constitute de 6 champs de vecteurs tangents ~ N o et telle que coo e B~ (X ~ (voir [6] partie 5).


39

Indiquons bri~vement pourquoi le (i) du th6or~me 0.5 d6coule du th~orbme 1.1. Supposons que w ~ = r l a o avec t2 ~ ouvert born6 de classe C I+E. Comme les fonctions de C 1+~ ~ support compact sont aussi dans t o u s l e s B I+E, on a w ~ c B~,oao pour tout a > 3/e d'apr~s la proposition 1.2. L'hypoth~se (7-/2) est donc v4rifi4e. On conclut en appliquant le th4or~me 1.1 puis en utilisant r4p4tition que B~ ~-. C e - ~ .

2. Un r~sultat de d4croissance exponentielle Dans cette partie, on 4tudie le syst~me

{(0~ + v. V)a - .(O~a + O~a + }O~a) = O, air=0 = a0~

(T~)

que v~rifie w~/r avec le champ de vitesses v -- v~. Nous nous int~ressons au cas oh a0 est la fonction caract~ristique d'un domaine born6 et off v est lipschitzien. Dans le th4or~me suivant, nous montrons qu'aux 4chelles spatiales grandes devant x/-~, la fonction a(t) est proche de la fonction caract6ristique du domaine transport6 par le riot. T h 4 o r ~ m e 2.1 -- Soit ~ > 0 et v E -"'Llo~(ltt+;Lip(ltt3))__ -^"~" - "~" un champ de vecteurs azisymdtrique ~ divergence nulIe. On suppose que a = a(t, r, z) vdrifie ( I v ) avec ao -- ao(r,z) C L2(]R3). N o t o n s Fo le support de ao, ~2 le flot de v et Ft = Ct(Fo). Posons (Ft)h = {x e ]R 3, d(x, Ft) > h}, (F~) h =

{~ e f~, d(~, OF~) > h} et V(t) - fo

a

[[a(tlllL2((Ft)~)
O, on

h 2

e -2-6~vt exp(-4V(t))

]]aO]lL2.

(2.1)

Dans le cas off ao est la fonction caractdristique d'un domaine bornd Fo, on a de plus

i[a(t) - 1F~ IlL2((FT)h)

(2.2) < 2 i]a01[L2 min

1

1+ C

e-9--6~vt exp(-4v(t))

off C est une constante universelle. Ddmonstration. Elle ressemble beaucoup ~ celle du th~or~me 1.1 de [5]. La principale diffgrence provient du terme du second ordre dans (T~) qui n'est plus un Laplacien "classique".


40

Par une mdthode d'dnergie, on trouve v t e ]R +

,

2 Ilao ILL.-.

Ila(t)ll 2L~ + 2~ "] ~ IlVa(s)lt 2L"- ds < Jo

(2.3)

Soit (I)o C C ~ ( I R 3) axisymdtrique (c'est-5~-dire ne ddpendant que de r et de z). On pose ~O(t,x) = (I)o(r si bien que ~ est conservde par le flot et reste axisymdtrique. Supposons dans un premier temps que v e Lto~176 (S(]R3)) 3) et a e Lzo~176 S(]R3)). D'apr~s (T.), ~ a vdrifie

(Ot+v.V-u(02+02+

3

O~))(~a)=-ua(O2+O2+-O~)~-2uV~.Va.

(2.4)

r

Supposons de plus que 9 est constante hors d'un compact (qui ddpend du temps). Dans ces conditions, ~a(t) est dans H 1. Rappelons par ailleurs que pour une fonction u = u(r, z), le Laplacien se rdduit s Au = O~u + O~u + O~u/r. En prenant le produit scalaire de (2.4) avec ~a(t) au sens L 2 et en se souvenant que div v = 0, on trouve ld

2

/m

2 dt ]l~a[[ L2 --u

3

2 ~ a ( A ( ~ a ) + -O~(~a))r dx a 2 ~ ( A ~ + -O~(~a)) dx - 2u

= -u 3

r

~ a V ~ . Vadx. 3

En intdgrant par parties, on obtient 2l ddt "" " 2 " + u[lV(~a)l] 2L 2 (__ /ma2 (lla2allL,] < u I[aV~]I 2 L 2 -

3

r --Or~2 dx )

(2.5)

Pour d~montrer (2.1), prenons (I) de la forme O(t,x) = e x p r avec r = fo(~2-1(t,x)) et f0 axisymdtrique constante hors d'un compact. On a visiblement IlaVOllL2 0 et a > 0 (convenablement %gularis6e), puis en faisant tendre R vers +oo et en prenant le "meilleur" a. Pour d~montrer (2.2), posons w ( t , x ) = a ( t , x ) - 1g,(x) et ~ ( t , x ) = 9o ( r avec ~o E Co~(Fo) axisym6trique. Remarquons que 1F~ vdrifie (Or + v . V)IF, = 0 et que Supp ~t est compact dans Ft, ce qui implique d2tOrw = ~tOra. On en d6duit donc que

(

Ot-[-v" V - - U Ant---Or r

=-uw

A+-0r r

((~W)

(2.7)

~-2uV~-Vw.

Une m6thode d'~nergie donne 1 d -2- dt

(ll~wll2L~)

2 2 + ~ IlV(+w)llL~ < ~ IlwV+ll L 2 -

-

/~

w2c3~(I)2 3

r

d~.

(2.8)

Soit ho > 0 et Xo E C ~ ( F o ) axisym6trique vatant 1 sur (F~)ho et ~ valeurs dans [0,1]. On impose de plus que IIV)iOlIL~ _ 0, on a done construit une ( N S ~ ) qui correspond & la donn6e initiale v n. 0 la proposition 3.1 s'appliquent, mais les termes donn@ initiale done de n. I1 est en fait classique

[lS

~

a,X 0

__ It ~

a,X 0

solution %guli~re vn,. de Toutes les estimations de de droite d6pendent de la (voir [3] on [5]) que

.

On conclut done au % s u l t a t suivant.

Proposition 3.2 - La solution Vn,v vdrifie routes les e s t i m a t i o n s de la p r o p o s i t i o n 3.1 avec une c o n s t a n t e C i n d d p e n d a n t e de n et de t/. T r o i s i ~ m e ~ t a p e : La convergence de Vn,v vers v~ solution de ( N S v ) v~rifiant les propri~t@ de r6gularit~ stratifi~e voulues se fMt comme dans [5]. On commence par ~tablir que la suite est de Cauchy en petite norme (dans des espaces de H61der ~ indice n~gatif par exernple) puis on interpole avec les estimations uniforrnes de la proposition 3.2. La limite v~ v~rifie de plus toutes les estimations de la proposition 3.1 avec une constante C inddpendante de t/. Q u a t r i & m e ~ t a p e : I1 s'agit d'~tudier la limite non visqueuse de v~. D'apr~s l'estimation (0.2) du th~or~me 0.6, le champ Vn,, tend vers vn,0 dans L1o~(R+; L2). On utilise alors les estimations uniformes des ~tapes 2 et 3,

46

Limite non visqueuse pour les fluides incompressibles axisymbtriques

et les rdsultats de convergence de l'dtape 3 pour obtenir les rdsultats de convergence pour la rdgularitd stratifide. I1 reste k traiter le cas off (T/2) est vdrifide. D'apr~s la remarque 1.6, on salt ddjk que (7-/1) est vdrifide pour une famille X ~ (e,a)-substantielle bien choisie. On dispose donc d'une unique solution globale vdrifiant des propridtds de rdgularitd stratifide. En appliquant k cette solution les rdsultats de la partie 5 de [6], on obtient les propridtds de convergence voulues pour la rdgularitd conormale, m

4. C o n v e r g e n c e tifide

pour des donndes initiales sans rdgularitd stra-

Cette partie est consacrde ~ la preuve du thdor~me 0.6. Sous les hypotheses w ~ ~ E L 2 N L ~ , l'existence de solutions ayant les propri~t~s voulues est assurde par le thdor~me 0.3. En appliquant une ddf

mdthode d'dnergie ~ wv - v ~ - v0, on obtient d IIw~llL= 2 + ~ IlVw~IIL= 2 -< ~ IlWolIL~ ,,~w IIV II ,,L2 + I(t) 21 dt avec

(4.1)

r

I(t) = Jm ~ Iw,(t, ~)121Vvo(t, x)l d~. On constate par un calcul direct exploitant l'incompressibilitd du champ v0 que 11~TVOl]L2= IlwollL2. En revenant ~ (4.1), on trouve done 1 d

2

/2

2 dt [IWu]IL2 --< -4

2

IIcMO[]L2 + I.

(4.2)

Traitons d'abord le cas tr6s simple oh la solution vo de (NSo) est dens 1 + ;Lip). On a Lzoc(]R I -< [lw~ll2i~ rlVv0 ILL=(4.3) En injectant (4.3) dens (4.2) puis en appliquant le lemme de Gronwall, on en ddduit finalement que

IIw.(t,

L= -< ~ I1~o(~,')11 =L~ef= IIv~o(~')rl~- d+ d~. .)jl~ ~ fo ~

Sans hypothbse particulibre de symdtrie, une mdthode d'dnergie appliqude t~ l'dquation du tourbillon Ot~0 + vo. V~o = (~0" V)vo

J. Ben Ameur and R. Danchin

47

permet d'obtenir

lifo(t, ")IIL~ < I1~~

~s

>,. ~ ~..

(4.4)

Dans le cas axisym~trique, on dispose aussi de l'in6galit6 (A.7) de l'annexe. On en d~duit (0.2). Dor~navant, on ne suppose plus que Vv0 est borne. L'in~galit~ (0.1) se montre en adaptant la preuve de [4]. Fixons un r~el a _> 2. Rappelons que d'apr~s [12] pages 42 et 250, il existe une constante C universelle telle que

IlVvollLo -< Ca II~,ollLo.

(4.5)

En appliquant l'in~galit6 de HSlder pour majorer I, on obtient donc I < Ca

It~ooliLo ilw,,ll~.~-~_%

(4.6)

Sn combinant l'in~galit~ de Sobolev iiztiLo _< 4 IlVzllL~. (voir [2] page 162), l'in6galit~

i-~

IlzllL-~:, _< IlzllL~ ~ Ilzil;,~,

(4.7)

le f a r que IlVv~llL~ = ll~IIL~ et l'estimation (A.7), on obtient

ii~vll 2 ~

2-~-< c(ll ~~ !!~ + t I1~~II ~ Ii ~~ ~ ) ~_ ltw. 11.~.

Par interpolation entre (A.7) et (A.4), on a

II~(t.-)ll~ ~ llv~

(Co + t ll~~

--~ 0og(C0 + t il~~

1-~

En injectant cette derni6re in6galit6 dans (4.6) et en utilisant (4.7), on obtient finalement I(t) 2 puisque

xa,~ < e - ~ ) . Apr~s integration, il vient

( II~~ IIL2 + t ll~~ )2 IIv~ +CIIv~ /otf(Co + ~ll~~

~t xa,~(t) < ~ + 7

avec #(z) = - z log z. Comme la fonction z ~ - z log z est croissante sur ]0, e-l], et que nous avons x5,~, C]0, e - ~ [ sur [0, Ta,~,[, on peut appliquer le lemme d'Osgood l'in~galit~ ci-dessus (voir par exemple [4]). En r e m a r q u a n t que z ~ l o g ( - log z) est une primitive de #-1 sur ]0, e-l], on en d~duit que pour tout t c [0, T~,~[ tel que < e -1 on

a

(_

~t ([I~~

~o

)2)) _ log(_ logx6, (t))

/o'


49

2]

d'ofi

exp(--CllvOllL2tf(Co+TI]c~OllL~r

.t

x~.~(t) < ~ + -ff

ilv01lL, +tlI~~

En faisant tendre 6 vers 0 et en utilisant un argument de bootstrap, on obtient (0.1). II

Appendice On g~n~ralise le lemme 3.1 de [10] au cas de r~gularisations construites partir de troncatures qui ne sont pas ~ support compact. On peut ainsi, dans l'6tape 3 de la partie pr~c6dente, r~gulariser les donn~es initiales l'aide d'une d6composition de Littlewood-Paley radiale et r~utiliser telles quelles les estimations de [5] sur la r6gularit6 stratifi6e. L e m m e A.1 - Soit X E S(]R 3) radiale. On suppose que w est le vecteur tourbillon d'un champ de vitesse v axisymdtrique. Notons ~u (X) = n 3 x ( n x ) , O-)n = ~n $ (M, Ol = o2/r et O~n = COn/r Ofl r ddsigne la distance g l'axe. Alors il existe une constante C ne d @ e n d a n t que de X et telle que

llC~nllL~ ~ C ]lC~ltL, pour

1 1, there

~xi~t~ ~ u , i q ~ c ~ i c ~ 1 ~olutio, to (1.1)- (1.2), ~uch that ~(., t) e C~(R n) for all t > 0 and the mapping UO E C 2 ( ~ n ) ~ ~t e C(~4_ , C ~ ( ~ n ) )

is continuous. Furthermore, principles:

the solution satisfies the following m a x i m u m - m i n i m u m

sup u(x, t) -- sup to(x)

xEIR,~ tE(O,T]

xEIR,~

inf u(x, t) -- inf t o ( x ) VT > O,

~ xEIl~n te(O,T]

[IVu( -, t)llL~(~--)

x EIR"

'

--< IlWolIL~(~), Vt ~ O.

(2.1)

(2.2)

In the proof, one shows that the solution exists in a time interval (0, T], where T depends only on IIVu011L~c(~tn). The inequality (2.2) then allows the continuation of the solution to [T, 2T], .... We remark that to prove (2.2)

M. Ben Artzi

59

the equation (1.1) is differentiated with respect to xj. Denoting uj = ~ we get 0

n

,

,

(2.3)

i=1

where

t#i(x, t) = #pfVui p-2 au e L ~ ( R " • (0, T] However, the solution uj to the linear parabolic equation (2.3) is not twice continuously differentiable hence some care must be taken in deducing (2.2) from the standard (linear) maximum principle. See the Appendix in [3] for details. Naturally, our next goal is to investigate the well-posedness of (1.1) in wider spaces of less regular functions, for instance, Lq(]~ ~) for suitable exponents q (possibly depending on p). To allow such solutions, the equation (1.1) is first cast in the integrM form,

u(x,t) =

J

JJ t

G(x-y,t)uo(y)dy+#

~.,

G(x-y,t-s),Vu(y, s)[Pdyds, (2.4)

0 ~"~

where G(x,t) = (47~t)-n/2 exp(-lx[2/4t) is the heat kernel. Taking Vx of Eq. (2.4) and using norms of the type

sup t~[[Vu(., t)[tL~(~), tE(0,T]

for suitable r, a, as in [29] one obtains the local (in time) existence of solutions to (2.4) in Lq(lt{n), for certain exponents. Then, by using the regularizing effect of the parabolic equation (2.4) (see also [9] for a direct argument) one shows that the solution u(.,t) E C2(R n) for t > 0, hence global existence follows from Theorem 2.1 above. As for uniqueness, we note that the solution was constructed by using "growth norms" of the type

sup t~llVu(., t)tlL-(~n).

te(0,T]

Thus, a contraction argument yields uniqueness using such norms (the "Kato-Fujita condition" [19]). However, an alternative approach as in [13] gives uniqueness for solutions in classes like C([O,T];Lq~Rn))r C((0, T]; Cb(Rn)). The exact exponents are summarized in the following theorem.

Global properties of some nonlinear parabolic equations

60 Theorem

2.2 [11] - For 1 _ m a x ( l , qc), q < oo (but q > 1 if qc = 1). Then, given any uo E Lq(I~ ~) (and a n y # E ]R), the equation (2.4) has a unique, global On time) solution u E C([0, oo), Lq(N~)). In p a r t i c u l a r we note t h a t if p >

:=

n+2 n+l

(2.5)

t h e n q~ > 1 and the e x p o n e n t q = 1 is outside the scope of T h e o r e m 2.2. Indeed, as the following claim shows, one cannot expect, for p > p~, to have solutions u of (2.4) for any u0 E LX (N~), even under the mildest a s s u m p t i o n s on u. In presenting the next claim, t h e r e is no a t t e m p t at achieving m a x i m a l generality. Claim

2.3 [12] - Let p > pc = (n + 2 ) / ( n + 1) and # = 1. Introduce, for 1 n+l 2(n+2

0 0, there is no solution u(x, t) of (2.4) in (x, t) E R n x (0, T], where u0 = v~ and such that u E LP((0, T);

WI'p(]I~n)).

Proof. A s s u m e the existence of a solution u(x, t) with the above properties. Since T

/ ~ . IVu' pdxdt < ~176 0

given s > 0 there exists a sequence tj --. 0 such t h a t

~

IVu(x, tj)lPdx < Et;1,j - 1 , 2 , . . . , n

(2.6)

M. Ben Artzi

61

which implies, by the Sobolev inequality (observe that u _ 0 in view of

(2.4)), JfRn u(x'tj)p*dx

1 = p1 t~ ~"

= /

/

I~l>t~ lvl< 89

89

+ /

I

G(x-y,t)uo(y)dydx

I~l>t~Ivl> 89

G(~,t)d~'lluoliLl(i~.)+ / lyl> 89

uo(y)dy.

(2.11)

62

Global properties of some nonlinear parabofic equations

Since/3 < 1, we have

G((, t)d( = O(t N) as t --. 0, N = 1, 2 , . . . 1~l> 89 and, for u0 = v~

/

uo(y)dy = ( 1 - 2-~t~a))lUO[)Ll(R.),t < 1,

(2.13)

[yl> 89 so, since

II~(',t)llLl( -) /

= II

0[)Li(R-),

we conclude t h a t

~t(x, t)dx = 2-~tZ~[[Uo[ILl(~) 4-O(t N) as t --) O.

Setting t = tj and comparing with (2.10) we get, for j = 1 , 2 , . . . ,

C~I/pt~. ~ 2 -~ tj~ II 0111L (R-) +

O(t )

(2.14)

which is a contradiction by the choice of 5, since ~ < 89can be chosen such t h a t ~5 < 77. i

Remark 2.~. In view of the last claim, one m a y ask, in the case p > pc, # = 1, what is the set of initial d a t a uo(x) E LI(1R ~) for which a solution to (2.4) does exist. Theorem 2.2 implies t h a t this set contains all u0 E L I(R n) N Lq(Nn), q >_ qc, and in particular, all uo(x) E LI(IR ~) N L~ However, Claim 2.3 says this set is not all of L 1(Rn). The situation is still not clear for # = - 1 . On the other hand, i f # = 1 a n d p _ > 2, Claim 2.3 can be strengthened as it is shown in Proposition 2.5 below. P r o p o s i t i o n 2.5 [11] - Let u(x, t) be a classical solution of (1.1), with # = 1, p > 2, in a strip IR~ x (0, T). Assume that lim u(., t) = uo

t---)O

in Lloc(Rn).

Then exp(u0) C L~oc(Nn). Finally, while (for p > Pc, # = 1) existence is not guaranteed for all u0 C Lq(Rn), 1 < q < qc, uniqueness can also fail, as the following theorem shows:

M. Ben Artzi

63

T h e o r e m 2.6 [11] - Assume 2 > p > pc and let 1 0 and # = - 1 . Then the solution u(x, t) is nonnegative and an integration of (1.1) shows that if, in addition, u0 E LI(]~ n) then u(., t) E L 1 for all t > 0 and the nonnegative function I(t) = f u(x, t)dx R~

is nonincreasing. Thus, the limit I ~ = lim I(t) >_ 0 always exists. It is t---. o o

interesting that the question whether or not I ~ = 0 is determined uniquely by p~ - (n + 2)/(n + 1), the same critical value as in the previous sections. We have the following theorem. T h e o r e m 4.1 [101 - Let 0 ~_ uo e C~(R n) N L l ( R n ) , u 0 r 0. Let u(x,t) be the solution to (1.1), with # = - 1 . Then Ioo > 0 r

n+2 = ~ . n+l

Remark ~.2. As was seen in Theorem 2.2, the well-posedness of (1.1) in L I(]~ n) was also linked to the same critical index p~. However, there is yet no direct argument connecting this well-posedness (essentially a short-time feature) with the long-time decay as expressed in Theorem 4.1. Remark ~4.3. In the case p < pc the equation is well-posed in LI(R~). Then, as in the discussion preceding Theorem 2.2, if 0 _< u0 c L 1(R ~) (and # = - 1 ) , it follows that u(., t) C C~(Rn)NLI(R n) for t > 0. Hence, Theorem 4.1 is applicable also, in the subcritical case, to all 0 _< u0 E L 1(Rn).

M. Ben Artzi

65

Remark ~.~ In the case p _< p~, the rate of decay of I(t) to zero becomes slower as p approaches p~. More precisely, let 1 0. On the other hand, if p = 1 and u0 is compactly supported then, for some A, 0 > 0 we have (see [3]), sup exp(At~ < oc. 0_ 0 and z < 0 respectively, see the following figure where each Fi is equal to 0fti \ F. Note also that, in the physical context, the heights of the domains are much smaller than their horizontal diameters.

F ga

Figure 1 The vector field ui stands for the velocity of a turbulent fluid in fti, pi represents its pressure and ki its turbulent kinetic energy (TKE in what follows). The quantity ai(ki) is the eddy viscosity, and we shall assume throughout this paper that the functions ai and "yi satisfy

{

a~CC~ 7~EC~

NL~(R) nL~

and and

VkclK, VkER,

a~(k)>_v,

1-II~llL(~,) +,~(4 - 2

~-2

" ~2

)11~11~L~(a~)'

and also 1

!

So choosing #i = ~C i leads to the desired result.

m

The next corollary is now a direct consequence of Lemma 3.1, see [11, Chap. I, Lemma 4.1]. C o r o l l a r y 3.2. - For i = 1 and 2 and for any data f i in L2(~i) d, (i) for any solution (ui, pi) of problem (3.3), the velocity ui is a solution of problem (3.5), (ii) for any solution ui of probIem (3.5), there exists a unique pressure pi in L2(~i) such that the pair (ui,p~) is a solution of problem (3.3). We now prove the existence of a solution of problem (3.5). We begin with an a priori estimate.

A model for two coupled turbulent fluids

76

L e m m a 3.3. - For every g~ in Ll(f~i) and f~ in L2(f~) d, 1 __0,which tends to g0 a.e. on ~-ti. Since the function &i is continuous and bounded, the subsequence (&i(g~ + pi)Hi)m>0 is dominated by c gi which belongs to L2(~ti), and tends to &i(g ~ + Pi)gi a.e. in ~ti, hence in L2(~ti). So, the function go is a solution of (4.8) and finally the function g~ = go + pi is a solution of (4.7). Since we have proved the existence of a solution of problem (4.7), we now pass to the case of real data. We approximate each ui in Xi, 1

2 ~ ( l e ? ! 2H , ( r ~ , ) i--1

-- (c'n

+

[Pi(Ul,

u~,)lH,(r,,))[g~


86

So using the bound (5.3) for Ilpi(ul, u2)[Ig,(a~) and taking each go on the sphere with radius #i = c'n + cn 89#, we deduce that the previous quantity is nonnegative. Thus, applying Brouwer's fixed point theorem yields the existence of a solution

(~?, ~ , ,,~, e0~) of ~m(~,T e~ == , ~ = ) = 0 ,

(5.4)

,

which satisfies

IlU~IIH'(~,>~ + Ilu~IIH*(~=)~ ~ #,

Ile~

_< #i,

i = 1 and 2.

Since the sequence (u~, t ~ u ~ , ~m)m is bounded, there exists a subsequence, still denoted by (u~, gore, u~, t ~ for simplicity, which converges to (Ul,g~, U 2 , ~ ) weakly in V1 x Hl(f~l) x V2 x Hl(f~2). Next, for a fixed (vl, gl, v2, g2), we pass to the limit in problem (5.4).

Proof (III). The limit on the equations for the velocities. We start from the equation ~ m ( u ~ , u ~ ) -- 0. For 1 < i < 2, there exists a subsequence still denoted by (g~ which converges to go strongly in L2(fli), hence a.e. in f~i. On the other hand, due to the continuity of Li, the sequence (gm = Li(u~ - U~))m converges to Li(ul - u2) weakly in H l(f~i). Due to the formula grad p? = grad

Tn(Gi(]gml2)) = 2Tnt (Gi(Ig?I2)) "yi(Igml2) grad g ? - gm,

there exists another subsequence (p,(u~, U~))m which converges to the function p/(ul,u2) weakly in H1(~2/) d and strongly in L2(fl/) d. So, the corresponding subsequence (t om +pi(u~, U~))m converges to t ~ +pi(ul, u2) a.e. in fli and, since 5i is continuous and bounded, for all fixed vi in Xi, the sequence ((~i(g~m + pi(ur~, urn)) VVi)m tends to &i(t ~ + pi(ul, u2)) Vvi a.e. in fl/ and is bounded in L2(fl/) d2' hence converges strongly in L2(fl/) d=. This yields the convergence of the first two integrals in the definition of ~m(', "): for i = 1 and 2,

lim s

&i(g~ + pi(u~, u'~)) Vu m . Vvi dx i

-- /a (~i(eOi Jr- pi('Ul, U2) ) VUi " VVi dx. i

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

87

The convergence of the third one follows from the compact imbedding of H 1 (F) into L 3 (F)"

liInor IF lU~n -- u~nl (U~n -- U ~ ) ( V 1 -- V2)dT -- ~F lul -- u21 ( u l -- u2) (vl -- V2)dT. Combining all this implies that the desired equation is satisfied by (Ul, u2).

Proof (IV). A stronger convergence result. For i = 1 and 2, we start from the formula J

Si(g~

u ~ ) ) V ( u ' ~ - ui) 9V ( u m - u i ) d x

4 --" / a

(~i(~Om + p i ( U ~ '

U~)) VU m

9V U m

dx

i

- Om + pi(u~, Um -- 2 / a ai(~i 2 )) V U ~ 9Vui dx i

+

v~

~ i

Om

+

m

)1

From equation (5.4), the first term in the right-hand side is equal to

-ff_r ]u~ - u~] (u~ - u ~ ) . u'~ d~" + /~ fi . u m, i

and using once more the compactness of H 89(F) into L3(F) implies its convergence. The convergence of the second and third term follows from the weak convergence of (VU~')m in L2(~i) a2 and the strong convergence of ((~i(~ Om + pi(u~, u~)) ~Tvi)m in L2(~ti). So, we obtain t h a t

lirn ff.q ~i(~ ~ + pi(uF, u~')) V ( u ~ - ui) - V(u~' - ui) dx = O, 4

which yields the strong convergence of (u~')m towards ui in Hl(~ti) d.

Proof (If). The limit on the equations for the TKE. Next, we consider each equation ~m(g0m) = 0. As previously, there exists a subsequence (c%i(t~~ + pi(u~', U~)))m which tends to &i(g~ + pi(ul, u2)) strongly in L2(~i). From part IV of the proof, there also exists a subsequence (i~Zu~'12)m which tends to IEZui]2 strongly in L l ( ~ i ) . Hence, the


88

sequence (&i(g ~ + pi(u~, u ~ ) ) I w T l ~ ) m converges a.e. in ~i and, since Tn is continuous and bounded, the sequence

+ p,(~7', ~))IV~?l~))~,

(Tn(~,(e ~

converges towards Tn(&i(g ~ + pi(ul, u 2 ) ) I W ~ l yields

~) strongly

in L2(f~i). This

- om + p,(~7, u~m ))IW?l ~) g, dx mlim ~ / ~ Tn (~,(e, i

-- /fl Tn (&i(go q_ p~(ul, u 2 ) ) I V u ,

I~) g~ dx.

i

Also, from the weak convergence of a subsequence ( p i ( u ~ , u ~ ) ) m Pi(Ul, U2) in H l ( ~ i ) d, we deduce

m

Vp~(u~, u 2 )- Vg, dx =

lim

/:

to

V p i ( u l , u2)- Vg, dx.

m - - - - ~ (:x3 i

i

So the desired equation is satisfied by t~i. Finally, the nonnegativity of the g~ follows from the standard maximum principle [7, Prop. IX.29]. We are now in a position to state the main result of this section. There also, we write the reduced variational formulation of system (2.5), where the equation on the t~i has now the same "transposed" form as in Section 4:

Find ui in Vi, 1 Jr- < f2, U2 > ,


90

so that passing to the limit yields

li~

1

J[]2 : < I'1, Ul > + < I'2, u2 > - J r ]ui - u2[ 3 dT.

We also derive from the first equation in (5.5) that

L ~i(~'l)]VUl'2dx+L 5i(~2)[Vu2[2dx 1

2

-- < f l , Ul > + < f2, U2 > -- IF In1 -- U2I3 dT, whence

lim f

n--+(X) y~-~ 1

&, (eT)IVuTI 2 d~ + f

J~"~ 2

&:,(G')IV"-'71~d~

#,

-/_

#-

1

3 and to (X3-2t) ' when t is < 3. It is also clear that the right-hand side in the fourth line of (7.8) belongs to Ht- 89(F) 2, hence to H2t-~ (F) 2. So using Corollary 7.3 (or Lemma 7.2) yields that (ui, qi) belongs to H t*(~i) 2 • Ht'-x(~ti), with t* = m i n { 2 t - 1, s0}, and applying Lemma 7.4 yields that gi belongs to H t* (~ti). Iterating n times this argument, where n is the smallest integer such that s* + n ( s * - 1 ) i s :> so, we obtain that the triple (ui,qi,gi), hence (ui,pi,gi), belongs to HS~ • H~~ • H~~ Then, ki also belongs to HS~ (~i).

Remark.

A more technical proof, relying on Meyers' argument [19] (see also [8]), allows for replacing, in the statement of Theorem 7.5, the assumption "the ui belong to H s~ (~i) 2'' by the modified one "the ki (or gi) belong to H s* (gti)", also for any s* > 1, however this new assumption seems stronger. Of course, these regularity properties can be extended to any convex polygons ~i, since the corresponding values of So can easily be computed from [22]. By a boot-strap argument, they also hold in the case where convection terms are added in the system. But, when the interface conditions are replaced by Manning's law, the regularity of the solution of the basic Stokes problem, a fortiori of the present system, seems unknown.

References [1]

[2]

C. Bernardi, T. Chacon, R. Lewandowski and F. Murat, Existence d'une solution pour un module de deux fluides turbulents couples, C. R. Acad. Sc. Paris 328 s~rie I (1999), 993-998. C. Bernardi, M. Dauge and Y. Maday, Polynomials in weighted Sobolev spaces: basics and trace liftings, Internal Report 92039, Laboratoire d'Analyse Num~rique, Universit~ Pierre et Marie Curie, Paris (1992).

C. Bernardi, T. Chacon Rebollo, R. Lewandowski and F. Murat

[3]

[4] [5] [6] [7] IS] [9] [10] [11] [12] [13]

[14] [15] [16] [17] [is] [19] [20]

101

P. Benilan, L. Boccardo, T. Gallou~t, R. Gariepy, M. Pierre and J. L. Vasquez, An L 1 theory of existence and uniqueness of nonlineai elliptic equations, Ann. Scuola Norm. Sup. Pisa C1. Sci. 22 (1995) 241-273. D. Blanchard and H. Redwane, Solutions renormalis~es d'~quatiom paraboliques ~ deux nonlin~arit~s, C. R. Acad. Sc. Paris 319 s~rie 1 (1994), 831-835. L. Boccardo and T. Gallou~t, Nonlinear elliptic and parabolic equatiom involving measure data, J. Funct. Anal. 87 (1989), 149-169. J. Boland and R. Nicolaides, Stability of finite elements under divergence constraints, SIAM J. Numer. Anal. 20 (1983), 722-731. H. Brezis, Analyse fonctionnelle, Collection "Math~matiques appliqu~es pour la ma~trise", Masson (1983). S. Clain and R. Touzani, Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients, Mod~l. Math. et Anal. Num~r. 31 (1997), 845-870. M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics 1341, Springer-Verlag (1988). T. Gallou~t and R. Herbin, Existence of a solution to a coupled elliptic system, Applied Maths Letters 2 (1994), 49-55. V. Girault and P.-A. Raviart, Finite Element Methods for the NavierStokes Equations, Theory and Algorithms, Springer-Verlag (1986). P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman (1985). R. Lewandowski, Analyse math~matique et oc~anographie, Collection "Recherches en Math~matiques Appliqu~es", Masson (1997). R. Lewandowski, The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity, Nonlinear Analysis TMA 28 (1997), 393-417. J.-L. Lions, Quelques m~thodes de r~solution des probl~mes aux limites non lin~aires, Dunod & Gauthier-Villars (1969). J.-L. Lions and E. Magenes, Probl~mes aux limites non homog~nes et applications, Vol. 1, Dunod (1968). J.-L. Lions, R. Temam and S. Wang, Models for the coupled atmosphere and ocean, Computational Mechanics Advances 1 (1993), 1-120. P.-L. Lions and F. Murat, Solutions renormalis~es d'~quations elliptique~ non lin~aires, to appear. N.G. Meyers, An LP-estimate for the gradient of solutions of second ordeI elliptic divergence equations, Ann. Sc. Norm. Sup. Pisa 17 (1963), 189206. F. Murat, Soluciones renormalizadas de EDP elipticas no lineales, Internal Report 93023, Laboratoire d'Analyse Num~rique, Universit~ Pierre et Marie Curie, Paris (1993).

102


[21] F. Murat, l~quations elliptiques non lin~aires avec second membre L 1 ou mesure, Actes du 26~me Congr~s National d'Analyse Num~rique, Les Karellis, France (1994), A12-A24. [22] M. Orlt and A.-M. Ss Regularity of viscous Navier-Stokes flows in nonsmooth domains, Proc. Conf. Boundary Value Problems and Integral Equations in Nonsmooth Domains, M. Costabel, M. Dauge et S. Nicaise eds., Lecture Notes in Pure and Applied Mathematics 167, Dekker (1995), 185-201. [23] G. Stampacchia, l~quations elliptiques du second ordre ~ coefficients discontinus, Presses de l'Universit~ de Montreal (1965). [24] L. Tartar, Interpolation non lin~aire et r~gularit~, J. Functional Analysis 9 (1972), 469-489. Christine Bernardi, Francois Murat Laboratoire Jacques-Louis Lions C.N.R.S. & Universit~ Pierre et Marie Curie Boke postale 187 4 place Jussieu 75252 Paris Cedex 05 France E-mail: [email protected], [email protected] Tomas Chac6n Rebollo Departamento de Ecuaciones Diferenciales y Ans Universidad de Sevilla Tarfia s/n 41012 Sevilla Spain E-mail: [email protected]

Numerico

Roger Lewandowski Equipe de M~canique, IRMAR Campus de Beaulieu Universit~ de Rennes 1 35042 Rennes Cedex 03 France E-mail: lewandow@maths, univ-rennes 1. fr Research partially supported by Spanish Government, MAR97-1055-C02-02 Grant.

Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 92002 Elsevier Science B.V. All rights reserved

Chap ter 6 D E T E R M I N A T I O N DE C O N D I T I O N S A U X LIMITES EN MER OUVERTE PAR UNE M E T H O D E DE CONTROLE OPTIMAL

F. BOSSEUR ET P. ORENGA

R@sum~. Nous pr@sentons dans ce travail le principe d'une m@thode num@rique adapt@e k la d@termination de conditions aux limites en mer ouverte d'un probl~me de shallow water. Cette m~thode repose essentiellement sur l'utilisation de la th@orie du contrSle optimal et de l'assimilation de donn@es. On se propose de reconstituer des conditions aux limites k partir d'un ensemble de mesures disponibles k l'int@rieur du domaine d'@tude. Le contrSle est effectu@ sur les conditions aux limites de la vitesse aux fronti~res ouvertes, en consid@rant des observations ponctuelles. Si au niveau th@orique, nous avons montr@ des r@sultats d'existence dans le cas lin@aire, nous n'avons p u l e faire dans le cas non lin@aire. Nous donnons des r@sultats num@riques dans le cas lin@aire et non lin@aire, tout d'abord sur une g@om@trie simplifi@e pour justifier la m@thode num@rique, puis sur le cas r~el d'une baie.

A b s t r a c t . We present here a numerical method adapted to the boundary conditions in open seas for a shallow water problem. Essentially, this method depends on the use of optimal control and data acquisition theory. We propose to reconstruct the boundary conditions from a set of available measurements taken in the interior of the region under study. The control is made on the velocity boundary condition on open boundaries, in considering observations at isolated points. If on the theoretical level we showed existence in the linear case, we could not achieve it in the nonlinear case. We present numerical results in the linear and nonlinear cases, first in a simplified geometry in order to justify the numerical method, then in the real case of a bay.

104

D d t e r m i n a t i o n de c o n d i t i o n s a u x f i m i t e s en m e r o u v e r t e . . .

1. Introduction Nous ~tudions dans ce travail un module de type sh a l l o w water qui, dans le cas d'une mer peu stratifi~e ou peu profonde, s'av~re suffisant pour representer la dynamique des fluides et peut constituer une ~tape pr~alable une ~tude plus approfondie des ph~nom~nes par un module tridimensionnel. Plusieurs ~tudes th~oriques et num~riques ont ~t~ faites sur ce module (Orenga, 1991; Chatelon-Orenga, 1997; Bisgambiglia, 1989). Pour notre part, nous nous proposons d'~tudier le cas de domaines avec des fronti~res ouvertes oh se pose souvent le probl~me de la connaissance des conditions aux limites sur la vitesse u. Celles-ci sont g~n~ralement ~valu~es grs ~ des donn~es exp~rimentales issues de campagnes de mesures, ou des donn~es calcul~es ~ l'aide d'un module de plus grande emprise. Dans le cas de la baie de Calvi, des mesures in situ ont pu ~tre effectu~es (Norro, 1995) grace ~ l'implantation voisine de la station de recherche oc~anographique STARESO. Toutefois, celles-ci sont en nombre insuffisant et ne nous donnent que peu de renseignements sur les conditions aux limites en mer ouverte ~ adjoindre au syst~me. De m~me, les r~sultats issus des programmes de recherche tels que Medalpex qui donnent des renseignements sur la circulation g~n~rale en M~diterran~e, ne peuvent prendre en compte les ph~nom~nes ~ l'~chelle de la baie. C'est dans le but de lever ces ind~terminations et d'utiliser au mieux les donn~es disponibles, que nous nous proposons d'utiliser des m~thodes d'assimi-lation de donn~es, d~j~ utilis~es avec succ~s dans de nombreux domaines, comme par exemple les pr~visions m~t~orologiques. Le probl~me est r~solu par la m~thode de Galerkin. Afin d'obtenir des conditions aux limites homogbnes, on effectue le changement de variable v - u - w, off u est la solution et w un rel~vement de la condition aux limites. Le contr61e est alors effectu~ sur w, en utilisant des donn~es ponctuelles issues de mesures effectu~es in situ ou de r~sultats de modules ~ plus grande ~chelle. Dans la section 2., nous rappelons bri~vement les ~quations du module utilis~. Dans la section 3., nous donnons les ~quations v~rifi~es par le contr61e optimal dont nous d~montrons, dans le cas lin~aire, l'existence et l'unicit~ dans la section 4. Les sections 5. et 6. sont, quant ~ elles, consacr~es ~ l'analyse num~rique des ~quations. Nous pr~sentons dans la section 7. un certain nombre de r~sultats obtenus sur une g~om~trie simplifi~e, constitute par un carr~ de cSt~ unitaire et qui nous a surtout permis de tester l'efficacit~ du code impl~ment~, puis sur le cas plus concret de la baie de Calvi (Corse), off l'on constate des ph~nom~nes d'~rosion des cStes dfis principalement ~ la modification des courants apr~s l'implantation de structures en mer (extension du p o r t , . . . ) .

F. Bosseur et P. Orenga

105

2. Equations du module Dans un module oc@anique complet, sont prises en consid@ration des variables biochimiques (concentrations chimiques, biomasses, ...) et hydrodynamiques (salinit@, temp@rature, vitesse et masse volumique). Dans notre cas, nous nous sommes limit,s g l'@tude des variables hydrodynamiques du syst~me. Pour d@crire l'@volution de ces variables, nous utilisons des modules de m@canique des fluides g~ophysiques qui se distinguent des probl~mes classiques d'@coulement, tels que ceux de Navier-Stokes, par les dimensions des domaines, les @chelles temporelles, la loi de conservation de la masse, la faible profondeur et surtout les conditions aux limites. A partir des @quations g@n@rales de conservation et des diff@rentes hypotheses simplificatrices li@es aux propri@t@s des fluides g@ophysiques (Nihoul, 1977), on @tablit les @quations du module tridimensionnel. Les ~quations du module que nous utilisons sont quant & elles obtenues en int@grant les @quations du module tridimensionnel non-stratifi~ sur la verticale.

x

u(t,x)

a

Fig. 2.1" Domaine d'dtude

9 ~'f repr@sente les c6tes ( u . n - 0), " 7e les fronti~res ouvertes ( u . n - G(x, t)). De plus, pour les fronti~res ouvertes, on distingue les parties de la fronti~re off le fluide est entrant (G(x,t) < 0) que l'on notera ~/-, des parties oh celui-ci est sortant (G(x,t) > 0), alors not~es ~/+. Cette diff@renciation est justifi@e par la n@cessit@ de fixer la hauteur d'eau sur la partie de la fronti~re o5 le fluide est entrant (Chatelon-Orenga, 1997). On d@signe par u(x, t), la

106

D d t e r m i n a t i o n de c o n d i t i o n s a u x limites en mer o u v e r t e . . .

fonction de f~x]0, T[ $ valeur dans 1t(2 reprdsentant la vitesse ~ l'intdrieur du fiuide et h ( x , t ) , la fonction de F~x]0, T[ ~ valeur dans ll{ ddsignant la hauteur de la colonne d'eau. On note 9 l'opdrateur c~, qui $ u = (ui,u2) E ]1{2 fait correspondre ( - u 2 , u l ) E I~ 2 ,

9 l'opdrateur R o t , qui ~ une fonction scalaire q(x, y) fait correspondre la fonction vectorielle Rot q - (Oq _ oO_~z) Oy ~ 9 l'opdrateur rot, qui ~ une fonction vectorielle u = (Ul, u2) fait correspondre la fonction scalaire rot u - ( ~Ox _ Oul ) Oy et on introduit Q

=

ax]0,T[,

r~

=

~•

r~-

=

~-•

2+

=

3, + x ] 0 , T [ .

Les 6quations du problbme de s h a l l o w water s'dcrivent ou _ A A u + ~1 V u 2 + rot u a ( u ) -5( +Du + wa(u) + gVh = f, u.n = a ( x , t),

rot u = O, ~(t = 0) = ~ 0 ( ~ ) , Oh ~-7 + div (uh) - 0, h = #(x, t),

dans Q, sur ~, sur ~, dans f~,

(2.1)

dans Q, sur E - , clans 9t.

h ( t = O) = h o ( x ) ,

Pour la rdsolution numdrique des dquations du module, et notamment pour pouvoir utiliser la base spdciale dans la mdthode de Galerkin, nous effectuons un changement de variable de mani~re ~ nous ramener ~ des conditions aux limites homogbnes. Ainsi, on pose ~ -- V'k-W~

off w = Vp,

(2.2)

avec p solution du probl~me -Ap--O Op

dans f~j2 (2.3)

-G

sur "7,

Gdg=O

107

F. B o s s e u r et P. O r e n g a

Les @quations du module se r@~crivent, en rempla~ant u par (v + w) + ~1 V v 2 + grad ( v . w ) + rot v a ( v ) + rot v a ( w ) + D v + w a ( v ) + g V h __ f Ow 1Vw 2 _ Dw - wa(w)

~t - A A v

-

ot

v . n -- O,

rot v - O, v(t

-

o) =

-

w(t

-

o),

Oh ~-? + div ( v h ) - - div (wh), h = #(x, t), h ( t = O) - ho(x),

dans Q, sur ~, sur E, dans f~,

(2.4)

dans Q, sur ~ - , dans ft.

Des r@sultats th@oriques d'existence et de r@gularit@ de solutions de ce probl~me sont donn@s dans Chatelon-Orenga (1997). Dans la suite, oh les r@sultats th@oriques sont d@montr@s dans le cas lin@aire, nous consid@rons le probl~me (7)) lin@aris@ not@ (P)l , -5~ ov _ A A v

+ Dv + wa(v) + gVh _ f

m

ow Ot

v . n - 0,

rot v -

0,

v ( t - O) - u o ( x ) - w ( t - 0), Oh -~ + h div v - O, h ( t - 0) - h0(x),

Dw-

wa(w)

dans Q, sur E, sur E, dans f~,

(2.5)

dans Q, dans f~.

off h repr@sente la hauteur moyenne sur le domaine, ne d@pendant pas du temps.

3. L e m o d 6 1 e

adjoint

Le principe de base du contr61e optimal est la minimisation d'une fonction cofit J mesurant les @carts entre la solution calcul@e et un ensemble d'observations disponibles. Ainsi, si l'on note X, l'espace des contr61es (g@n@ralement un espace de Hilbert), Xod, un convexe ferm@ de X et w E X, la variable de contr61e, le probl~me s'@crit t r o u v e r Wo E X~d r d a l i s a n t

inf

wE Xoa

J(w).

108

Ddtermination de conditions aux limites en mer ouverte...

En pratique, la minimisation de la fonctionnelle J(w) n~cessite la connaissance du gradient de J par rapport aux variables de contr61e. Parmi les diff~rentes m~thodes de d~termination de ce gradient, l'utilisation des ~quations adjointes du module lin~aire tangent semble ~tre la plus int~ressante, notamment au niveau num~rique (Lions, 1968; Talagrand-Courtier, 1987). Puisque l'on consid~re ici des observations ponctuelles, il est essentiel d'exiger comme condition pr~alable que les fonctions recherch~es soient au moins continues sur le domaine d'~tude. Dans notre cas, off la dimension du domaine est ~gale ~ deux, le th~or~me de plongement de Sobolev montre que ces fonctions doivent ~tre recherch~es naturellement dans H2(~). Compte tenu du type de rel~vement considerS, l'espace des contr61es est donc donn~ par L2(0, T; W ) , off W-

{ wEH2(~)2;w-Vp,

Ap-0,

0p

~n-0Sur3'I

}

9

Consid~rons X l , . . . X m , des points de ~, off m repr~sente le nombre d'observa-tions disponibles. On d~finit la fonctionnelle J(w) par

g(w) - I I c ( v + w ) - u~ll H = + ~IIwlIL~-(o,T;W) = ,

(3.1)

off C est l'op~rateur d'observabilit~ (d~fini ci-apr~s) et H - (L2(I~)) m l'espace des observations. Si les donn~es du probl~me sont suffisamment r~guli~res, il a ~t~ d~montr~ dans Chatelon-Orenga (1997) que la vitesse v appartient au moins

L~ (0, T; v), o~ V - {~p E H2(fl)2; ~.n - 0, rot ~ - 0 sur 3'}. L'op~rateur d'observabilit~ C est donc d~fini par

c

(L2(~)) m

L: (0, T; v u w) v

,

)

v(~m) Notons enfin la vitesse d~sir~e,

Ud

"-"

Ud E

H, sous la forme

lu,Xl,i idll

109


Alors, (3.1) se r66crit sous la forme

m

T

J(w) - ~-~ fo Iv(xj ' t;w) +w(xJ) -Udj(t)12dt j=l

+ llwll L2(O,T;W)"

(3.2)

Expression du probl~me adjoint Les 6quations adjointes sont obtenues formellement en multipliant les 6qua-tions du module lin6aire tangent I par les variables adjointes, not6es v* et h*, puis en int~grant sur Q = ~ • T[. On obtient alors le probl~me (P*) suivant: or* A A v * - v. div v* - w div v * + rot (va(v* ) ) ot + rot ( w a ( v * ) ) + v* a ( r o t v) - h grad h* - w a ( v * ) + Dv* --- Ejm=l (V(X j, t; W) + W(Xj) -- ~tdj (t)) @ (~(X -- X j) v*.rt = O, rot v* = O, = T) = 0 ,

dans Q, sur ~, sur E, dans ft.

Oh* ot v. grad h* - w. grad h* - g div v* - 0, h* = O, h* (t = T) = 0,

dans Q, sur E +, dans f~. (3.3) oh, (v(xJ, t; w) + w ( x j ) - Ud~ (t)) | 5(x -- xJ) est la distribution d6finie par

r --+

jr0 T

(v(x j, t; w) + w(xj) - Ud~(t))r

j, t)dt,

(

qui est une forme lin6aire continue sur L 2 0, T; (H2(f~) 2)

r e

).

Dans le cas off l'on consid~re le probl~me (P) lin6aris6, les 6quations adjointes s'6crivent

-Or* ~

A A v * - h grad h* - w a ( v * )

Ej=I (V( xj, (P*)t

+ Dv*

-

t; W) + W(Xj) -- Udj (t)) @ e l t a ( x -- x j)

V*.n = O, rot v* = O, v*(t= T)=0,

Oh* , ot g div v - 0 , h* (t = T) = 0,

dans Q, sur ~, sur ~, dans f~. dans Q, dans f~. (3.4)

1Obtenues en diff6renciant les 6quations (2.4) par rapport ~ la variable de contr61e.

110


Les propri6t6s de l'op6rateur adjoint nous permettent alors d'6tablir l'6quation v~rifi6e par le minimum local de la fonctionnelle J, qui s'e~crit 2, 'v'0 E L2(0, T; W)

I Ov*

+ v div v* + w div v* + rot va(v*) - Dv* + wa(v*) - h* grad h + C*(C(v + w) - Ud) -~- s

O~

/ L2(O,T;W ' ),L2(O,T;W)

"- O

(3.5) dans le cas non lin~aire, et

I Ov* --Oi

Dv* + wa(v*) \

+ C*(C(v + w) - Ud) + eAww, O) =0 / L2(O,T;W ' ),L2(O,T;W) (3.6) dans le cas lin~aire.

4. Conditions d'existence et d'unicit~ du contrSle optimal darts le cas lin~aire Avec les notations introduites au chapitre precedent, on a, T h ~ o r ~ m e 4.1 - On suppose que J(w) est donn~e par (3.1) et que G E H 1 (0, T; H~ ('y)). Le contrdle optimal w E L2(0, T; W) est caractdrisd de mani~re unique par les probl~mes (7~)~, (7)*)~ et l'dquation (3.6), avec

v E (L 2 (0, T; V) N H 1 (0, T; L2(~-~)2)), h E L (x)(0, T; H 1(~'~)), v* E L 2 (0, T; L2(fl)2), h* E L 2 (0, T; L2(gt)).

Ddmonstration. Les r~sultats d'existence et d'unicit~ du probl~me (P)t ont ~t~ d~montr~s dans Orenga (1991). Pour d~montrer que le probl~me adjoint admet une solution unique, on op~re par transposition (Lions-Magenes, 1972).

2Off AW repr~sente l'isomorphisme canonique de L2(0, T; W) sur L2(0, T; W').


111

Pour simplifier les notations, on considbre, dans les dquations (3.4), les constantes du probl~me dgales s un. Le syst~me se rddcrit alors

Ov* Ot

m

(v(x j, t; w) + w(x j) - Udj (t)) | 5(x -- xJ), (4.1)

Av* - Vh* - ~ j=l

Oh* div v* = 0, Ot v*.n = 0, rot v* = 0 sur ~/,

(4.3)

v* (t = T) = 0, h* (t = T) = 0.

(4.4)

(4.2)

Introduisons l'espace .= = {r C L2 (O, T; H2 (a) 2) AHI(O,T;L2(gt)2); r

= 0, rot r = 0 sur 7, r

= 0},

et le probl~me

or

A r + V ~ = F,

(4.5)

Ot O~ + div r = 0, Ot r

(4.6)

= 0, rot r = 0 sur 7, r

= o,

(4.7)

= 0.

(4.s)

On a le rdsultat suivant: Si F e L 2 (0, T; L2(a) 2) alors le probl~me (4.5)-(4.8) admet une solution unique dans =. En effet, l'application T ddfinie par

or off ~ vdrifie: 0~ 0--t- + d iv r - 0, est un isomorphisme de ~ dans n 2 (0, T; L2(ft)2). Par transposition, on en ddduit qu'il existe v* e L 2 (0, T; L2(Ft) 2) et h* e L2(O,T; L2(f~)) uniques, vdrifiant

/; ( V*

-~Or _ A r + V ~

)oh. dQ - M ( r

- ~ - + div v* - O,

112

D6termination

de c o n d i t i o n s a u x l i m i t e s en m e r o u v e r t e . . .

off: m

T

~..!

r... !

d@finit une forme lin@aire continue sur :. et off - est le dual de = Si e > 0, il y a existence et unicit~ du contr61e optimal w (Lions, 1969), caract@ris@ par

-~-Dv*+wa(v

*) +

--Ud

)+Aw ,8} W

L2(O,T;W'),L~(O,T; W)

=0,

m

for all 0 E L2(0, T; W).

5. Principe de la m@thode de r@solution 5.1. R~solution des @quations du module et des ~quations adjointes La m@thode utilis@e pour la r@solution des @quations du module et des @quations adjointes est bas@e sur l'utilisation de la m@thode de Galerkin. On transforme, par troncature d'une base de l'espace consid@r@, le syst~me d'@quations aux d@riv@es partielles par un syst~me d'@quations diff@rentielles ordinaires dont les inconnues sont les projections de la solution du probl~me approch@ sur la base. Dans le cadre de ce travail, nous avons utilis@ la base sp@ciale dont les propri@t@s ont @t@mises en @vidence dans Orenga (1992). En particulier, on a le r@sultat suivant: Thdor~me - S o i e n t V - {u C L2(f~) 2,div u E L 2 ( ~ ) , r o t u E L2(f~);u.n - 0 sur "7}, Ho(div 0, rot 0) - {u E L2(f~) 2, div u - 0, rot u - 0; u . n - 0 sur 7}, I'1, (') la n o r m e et le p r o d u i t scalaire d a n s O n c o n s i d b r e les p r o b l ~ m e s

-Au(Pl)

u . n -- 0

rot u - - 0

(7'2)

-ApAp grad p . n - 0

dans f~ sur -7

Au

L2(~)

ou

L2(~)2.

dans f~ sur .7 sur -),

(7'3)

q- A - q 0 - #q

dans f~ sur 7

113


et les p problbmes - - A r -- 0 (P4,)

/'--1

r--0

dans f~ sur 7i sur 7j

avec j ~ i avec i = 1 , . . . , p et j = 1 , . . . , p . On a alors 9 Si (A, p) est solution de (P2), a/ors (A, grad p) est solution de (Pl). 9 Si (#, q) est solution de (P3), alors (#,Rot q) est solution de (Pl). Cette propridtd montre en particulier 1'existence de solutions de (Pl ) divergence nulle. 9 Si ri est solution de (7~4~), alors (0, Rot ri) est solution de (Pl). 9 Si ~ est simplement connexe, alors 0 n'est pas valeur propre de (Pl), sinon l'espace propre associd ~ la valeur propre 0 est 1'espace

H0(div 0, rot 0), de dimension p, engendrd par les p solutions des problbmes (7)4~). 9 Soit {pi, i E N), un ensemble de solutions de (~2) formant une base orthogonale de L2(~), soit { q j , j E N}, un ensemble de solutions de

fo m..t

b. e o thogo..1

n (a)

soit

les

p solutions inddpendantes des prob1~mes (P4~). A10rs 1'ensemble des grad pi, Rot qj et Rot rk forme une base orthogonMe de L2(f~) 2 et

de V.

5.2. Approximation du gradient de la fonctionnelle Nous consid~rons ici le cas d'observations ponctuelles, off les observations sont connues en tout temps mais en un nombre restreint de points du domaine. Rappelons que l'espace des contr61es est donn~ par L2(0, T; W), la vitesse d~sir~e est telle que" Ud E (L 2 (I~)) m, l'op~rateur d'observabilit~ est not~ par: C" L 2 (0, T; Y U W) ~

(L 2 (I~)) m.

On introduit l'espace: (

Op )


114

Soit 7-/', le dual de 7/. On identifie 7-I et ~ ' , et on a

W ~__~7-l = 7-l' ,__+W '. On va d~terminer le minimum de J(w) par une m~thode it~rative. On ~crit l'~quation v~rifi~e par le minimum de la faqon suivante:

s (W' (9) L2(O,T;W ) -

;or. --Oi

)

(5.1)

Dr* + coa(v*), 0 L~(O,T;W'),L~-(O,T;W)

La m~thode consiste ~ consid~rer le second membre de l'~quation (5.1) fonction de w ~ l'it~ration pr~c~dente. Ainsi on va calculer w n ~ la n i~me iteration par

~(~. 0).~o T;w , + (~w~. ~0)(~.~.)~ : _(~o --

--~

1_ ~. ~0)(~(~./~ L2(O,T;W,),L2(O,T;W )

Dans le cas non lin~aire, on r~sout de la m~me mani~re

e(W,9)L2(O,T;W) + ( w d i v V*,91L2(O,T;W,),L2(O,T;W ) + (Cw, Cg) (L2(R))m -

(C v - u d ,

)

C9 (L2(R))m--

-~+vdivv*+rotva(v*)

- Dv* + wa(v*) - h* grad h, 0~

/ L2 (0,T; W' ),L2(O,T;W)

. (5.2)

Or, la r~solution de (5.1) ou de (5.2) n~cessite l'introduction de l'op~rateur C*, op~rateur adjoint de l'op~rateur C. En effet, on doit ~crire

et

(~v _ ~, ~ 0 ) ( ~ . ~ ~ =/~./~v - u~/. 0/~o ~;~ ~ ~o ~;~ , La difficult~ d'obtenir une caract~risation num~rique de l'op~rateur C*, nous a conduit ~ nous orienter vers une m~thode de r~solution plus appropri~e.

115


Nous proposons donc de r~soudre les ~quations (5.1) ou (5.2) par une m6thode d'approximation variationnelle, en utilisant une base de fonctions de l'espace W. C h o i x de la b a s e de W Nous devons tout d ' a b o r d d~finir un produit scalaire sur W; pour cela, on utilise l'6quivalence des normes suivantes (Dautray-Lions, 1988):

IlWllHk+l(a)2 et

IIIwl[]

off

jflwrfr-(llwll 2

L~(a) 2

+

[Idiv

~112Sk (,)~. + [[rot w II2

H

k (~)~. +

II~.n II 2. ~ §

1

(~)

Ce r~sultat nous permet de munir W du produit scalaire

puisque, d'apr~s les propri6t6s de l'espace W, div w = rot w = 0. En outre, si w = Vp est une fonction de W, alors p v~rifie le probl~me Ap-

0

p = #

dans sur %

Op

~-~n - 0

(5.4)

sur "Ys

oh # C H~ (%). Consid6rons h pr6sent les solutions (i des problbmes de N e u m a n n suivants: A~i - 0

dans

~i = ~bi

sur %

O~i = 0

(5.5)

sur 7f

oo

dans lesquels { r } i=1 est une base de H~ (%) et v~rifie:

~r

i-0,

'v'i E N.

116


On d~finit alors l'application

7~" H~ ('7) ~

Ha(f~)

qui est lin~aire et continue (Girault-Raviart, 1979). Avec ces notations, on a le L e m m e 5.1 - {V~i}i=l forme une base de W. D~monstration. Soit

0p r E ~(P E H3(Ft);Ap-- O, ~nn - 0 s u r '7.f~. ./ I1 existe # E H~ (%), tel que r {r

oo

~tant une base de H~ (%), il existe #n,v~rifiant #n -- E

air

Hi(~)

> ~,

i--1

et d'apr~s la continuit~ de l'application 7~: n

7~

H3

- r - y~ a~ i=1

(a) > O,

o~ fi = 7~r

et O vdrifie O-#

sur%

O0 ~ n --0

sur011

De plus 0 = ACn

> A @ = 0.

D'apr~s le r~sultat d'unicit~ de la solution du probl~me (5.5), on en oo d~duit que O - ~ et donc que {fi }i=1 est une base de

(pEH 3(~);Ap-0,

op

~ n - 0 S u r f f f ~ ")

(x)

En consequence, {V~ci}i=l est une base de W. D~montrons alors le L e m m e 5.2 - {V(i }i=1 forme une base de Ill.

II


117

Ddmonstration. Soit f - V~ E 7-/, telle que (f, V ~ i ) n - 0

Vie N

On a (f, V ~ i ) n -- s

VrI.V~ci

Or

soit (f, X;7~r / --

0~ ~~ r

ViEN

-- 0

(X)

{r }i:1 6tant une base de H~ (%) donc de L2(7), on en d~duit que o~ - 0 sur %. Finalement, ~1 v~rifie Arl - 0 et ~ = 0 sur 3', d'ofi on d~duit que r] - 0 et f - 0, ce qui d~montre que {V~i }oo /=1 est une base de 7-/. 1 On va maintenant approcher w par w~ E L2(0, T; W~), off Wr est le r sous-espace engend% par les r fonctions {0i V~i}i=l ; on raisonne dans le cas lin~aire, le cas non lin~aire ne posant pas plus de difficult6s. On note -

f (v*, h* ) -

Or*

(5.6)

Ot ~ Dv * + wa(v *),

Wr : E

#k Ok,

k--1

et avec i - 1 , . . . , r, l'~quation (5.1) s'~crit

+Z k=l

O (xj) O (xj)

-

j=l m

-- E (V -- Ud) (Xj).Oi(Xj) -- (f (v*, h*), Oi) L2(Q)2

.

(5.7)

j--1

6. Mise e n oeuvre num~rique 6.1. Notations On note 3d le maillage du domaine, Mi, i - 1 . . . N1, les points de l'int~rieur de f~ de coordonn6es (Xi, Yi) et Fj, j - 1 . . . N2, les points fronti~re, de coordonn6es (xj, yj).

--

i--1 ['j

} j--l"

118

D d t e r m i n a t i o n de conditions a u x limites en mer o u v e r t e . . .

6.2. Calcul des vecteurs de la base propre et de la base de W Nous avons vu ~ la section 5.1 que les ~l~ments de la base propre sont obtenus par r~solution des probl~mes (P2), (/)3), (P4,). Dans le cas d'un carr~ de c6t~ unitaire, les solutions de ces probl~mes sont connues de mani~re analytique. Dans le cas d'un domaine quelconque, les solutions sont calcul~es l'aide du logiciel Modulef, par la m~thode des ~l~ments finis. Les ~l~ments utilis~s sont de type hermite ~ trois degr~s de libertY, ce qui nous permet d'avoir acc~s, en t o u s l e s points du maillage, ~ la valeur nume~rique de la fonction solution ainsi que de ses d~riv~es premieres en x et en y (VidrascuGeorges, 1990). Pour construire num(~riquement la base de Wr, on r~sout les r probl~mes scalaires suivants: /X~ - 0 1 ~(xj,yj)

-

0

si j - a, off ( x j , y j ) E % si j ~ a,

La r(~solution num~rique de ces probl~mes est, comme pr~c~demment, effectu~e par la me~thode des ~l~ments finis ~ l'aide du logiciel Modulef, en utilisant les m~mes ~l~ments.

6.3. R~solution des ~quations du module et des ~quations adjointes Pour r~soudre les probl~mes (2.4) et (3.4), nous utilisons la m~thode de Galerkin, associ~e~ la base propre d~finie s la section 5.1. On transforme ainsi les syst~mes d'~quations aux d~riv~es partielles en des syst~mes d'~quations diff~rentielles ordinaires. Ces syst~mes sont alors r~solus par la m~thode d'Adams implicite, initialis~e par la m~thode d'Euler implicite.

6.4. M~thode de calcul du minimum La principale difficult~ pour le calcul du minimum provient de la presence du terme (w, ~)L2(o,T;W) dans les (~quations (5.1) et (5.2). Or, nous avons vu en (5.3) que l'on pouvait munir W du produit scalaire

(e,,,

+

Pour calculer la valeur du produit scalaire de H~ (~,), on utilise alors le r(~sultat suivant (Dautray-Lions, 1988)"


119

Soit f, une fonction pdriodique ddfinie sur ]0, a[; on ddfinit la norme de l'espace de Sobolev H~ (0, a) 1

Ilsll~ -

~ (1 + j~)~lcJ~l ~

(6.1)

jeZ

o~ Cj -

f (x).e -i aI" J~ dx.

I

fO a

On associe alors ~ (6.1) le produit scalaire suivant: (f,g)-(E(I+J2)~CJ(f).-CJJ(g)), s

(6.2)

jeZ

off f et g sont deux fonctions p4riodiques d4finies sur ]0, a[. Dans notre cas, off la fronti~re % est constitu6e par la r4union de q segments de longueur ai, les fonctions (Ok.n) sont d6finies sur ]0, ai[ et sont nulles au bord. On a

cj(ok.n)

-~,q f a i

(O~.nl(x).~-~2,r jx ax

i--1 Jo

= - E i

(Ok.n)(x). sin U j x dx,

i---1

et -

a, -

.

o,

dx

i--1

=

i

(ai)

(Ol.n)(x). sin 27r j x dx.

i=1

Au total, le produit scalaire s'exprime donc sous la forme

(Ok.n, Ol.n) H~ ('),) = E

~(1+

j2)~[(~oa~(Ok.n)(x).sin(2----~jx)dx ) a i 9

(81.n)(x).sin - - j x dx

.

ai L'4quation (5.7) se pr6sente alors comme un systbme alg4brique de r 6quations, off les r inconnues sont les #k et m repr6sente le nombre de points d'observation, {xj, j = 1 , . . . , m}.


120

Ce syst~me matriciel, du type

A X = B, d'inconnues #1

#2 Z

__

9

#r est alors r~solu par la m~thode du pivot de Gauss. Une lois connues les valeurs des coefficients #k (k = 1 , . . . , r), on reconstruit la solution approch~e

Wr -- ~

#k Ok,

k--1

qui correspond ~ la valeur approch~e de w pour l'it~ration suivante de l'algorithme de contr61e. En effet, on doit donc, ~ ce niveau, calculer les nouvelles valeurs de v, h, v*, h* correspondant ~ cette nouvelle valeur de wr et ainsi de suite jusqu's convergence de la m~thode (Fig. 6.1).

6.5. Convergence de l'algorithme de calcul du m i n i m u m Au chapitre 3., nous avons donn~ l'expression de la diff~rentielle de la fonctionnelle, qui s'~crit dans le cas lin~aire 1 j , (wl , O) -

*,h* 1 + C* ( C v -

+ C* C w

+ eAw w, O)L2(O,T;W,),L2(O,T;W )

(6.3)

off A W est l'isomorphisme canonique de L:(0, T; W) sur L2(0, T; W ' ) et f ( v * , h * ) est d~finie comme en (5.6). Pour d~montrer, dans le cas lin~aire la convergence de l'algorithme de calcul du minimum, on raisonne par r~currence 3. 9 On se fixe w ~ quelconque et on calcule (v ~ h ~ et (v *~ h *~ en r~solvant les probl~mes (T')~ et (P*)~. 3Nous donnons un r~sultat de convergence uniquement dans le cas lin~aire, car dans le cas non lin~aire, nous n'avons pas de r~sultats concernant la convexit~ de la fonctionnelle.


121

Calcul par Modulef de w k ----X7pk, avec p~ solution de --A pk = 0 V pk.n

= G k

d a r t s ~2 sur y

~1~~1 ~

Calcul

wk+l

de

Calcu1 de

I

v , . J ( ~ 9)

N~gatif

Test ......

vwJ(.,9(~

y

Fig. 6.1: Schdma gdn6ral de la rdsolution

[

D6termination de conditions aux limites en mer ouverte...

122

1) Si w ~ v6rifie 1 j, , < (w~ O> - 0

, , V0 e L 2(0 T; W)

(6.4)

alors, d'apr~s le th~or~me 4.1, w ~ est le contr61e optimal. 2) Si w ~ ne v~rifie pas (6.4), on d~termine w I par

(C*C + ~ A ~ ) w I - - ( f ( v * ~

h "~ + C * ( C v ~ - u ~ ) ) .

L'op~rateur

T-

(C*C + eAw) " W---+ W',

~tant continu et elliptique, on peut ~crire

~, -_~-l(~(v,O,h,o)+c,(cvO_u~)). Utilisons (6.3) pour donner une expression de <J' (w~ w 1 - w~

il vient

L2<J' ( w~ ) , wl - w~ > -- L2(O,T,W,),L2(O,T;W )

- (s (v,O, h,o) + c* (CvO _ u~) + T~O,

_~-, (s(v.O ~.o)+ ~. (~o _~)) _ ~o>~o,~;~,,,,~o,~ ' = <S(v,O, h,o) + c* (CvO_ u~) + :r~o,

_,-l(S(~.o ~.o)+ ~.(~vO_ ~)+,~o)> '

oA~ - A ~ and, hence: lim P ( d ~ ) = P ( A ~

(8)

nJ'oo

It is also obvious that under our assumptions on the structure of U, the 1 2 xmi~) above events are independent for different pairs ( x rain, of neighbor minimum points. Consider standard bond percolation model using minimum points of U0 as sites, and let pc be the critical probability of the appearance of the infinite 1 We define the critical value r/c as follows: cluster" Pc - 5"

p(d~

1 - -~ ,

or, if such a r/c does not exist: r / c - inf{r/; P ( d ~ < 51 } -

sup{r/; P ( d ~ > 89

(9)

138

Effective diffusion in vanishing viscosity

This last equality is, in fact, an additional assumption which is supposed to be fulfilled later on. 1 Thus, using (8) , P(An~+.y) > For all ~/> 0 small enough, P ( A ~ > 7" n !2 for sufficiently large n. We fix such a n and denote it by n o ; we also denote p0 We say that a bond

-

no

P(A,c+~

1 Xmin) 2 (Xmin, is

).

open if the corresponding w belongs

1 X 2mi.)" A~:+~( X min,

to the set As proved in Kesten [13], for almost all realizations and for all sufficiently large N, the square [0, N] 2 contains at least c(p ~ N mutually non intersecting channels connecting left and right sides of the square. Finally, we arrive at the following conclusion: Conclusion. For sufficiently large N, [0, N] 2 contains at least c(p ~ N 1 -pipes connecting left and right sides mutually non intersecting smooth n-~ of the square such that along each of these pipes: U(y) < tic + '7.

(10)

Denote the above pipes by Q 1 , . . . , Q k ( N ) , k ( N ) > c(p ~ N. Without loss of generality we assume that for any function x ~ u(x) such that u(0, x2) - 0, u(N, x2) - 1 we have: Q m

Ou (y_____~d) y > 1 1 0~. - 2 no

(here I is a variable directed along the pipe after rescaling). Indeed, taking a smooth pipe included in Q m and choosing, if necessary, a larger value of no, one can achieve the above lower bound. After rescaling x = ~ y, ~ = l / N , we find: Q

On(x) 1 1 Og dx >_ -~ ~

withQ~-~Qm.

By the Shwartz inequality: e2

1 1 < (n~ 2 4 -

[/o

~

On(x) dx 0~

/o

< IQ~ml -

IVu(x)l 2 d x .

Thus, Q

IVu(x)l 2 dx > ~ 1 1 - 4 (n~ 2 cl(p~ "

F. Campillo and A. Piatnitski

139

Summing up over m leads to: k(N)

f IV(x)l

m--1

e

> c(p ~

1

1

c(p ~

4 (n~ 2 cl(p~

Q~m

1

Cl (pO) (2 Tt~ 2 "

From (10), we have:

s

IW(z)ldx

>_ k~) /s m--1

e-U(~)/t* lVu(x)l 2 dx

Q~ k(N)

->

e-(n~+'~)/~' E

/Q

m--1

> e_(n~+.y)ll, c(p~ -

cl(p~

IVu(x)12

dx

~m

1 (2 n~ 2"

Using Definition (7) of a(#), and taking into account the fact that -y is an arbitrary positive number, we obtain: lim inf # log a(#) > - ~ c . itS0

3.2. U p p e r b o u n d Let o1 and [:]2 be two neighbor cells, let say that [:]2 - [:]1 + el, and the corresponding maximum points of U0. We introduce the random set:

1 x E O1 ~ X 2max E [-12 X ma

and the events" 9 B ~ the set of w such that there is a path connecting x 1 and x 2 1 which belongs to A'( X =,~x, el) and which is included in G + (co). max

max

9 B~ the set of w such that there is a smooth curve of length not greater than n such that its ! - n e i g h b o r h o o d is included in G + (w) n

o

n

"

Comparing this setting with the one used for the proof of the lower bound, one can easily see that: ~7c - max{r/;

P(B ~

1 - minIr/; p ( B o ) > ~}. 1 < ~}


140

Thus, for any small positive 7 we have: 1

P(B~

> -~.

This implies the existence of no = no(7) > 0 such that: 1

nO

P(Bn~-'~) > 2" We use the notation pO - P(Bnc_~ no ). In the same way as above one can assert that for sufficiently large N, the square [0, g ] 2 contains at least c(p ~ g mutually non intersecting smooth 1n o pipes connecting bottom and top sides of the square such that along each of these pipes:

U(x) > nc - 7. We consider a specific test function ~ such that: (i) ~(0, x2) - 0 and ~(1, x2) - 1, ~ is continuous,

(ii) ~ is constant between any pair of channels (pipes), and also between {X;Xl = 0} and the first pipe from one side, and between the last pipe and {x; Xl = 1} from the other side,

(iii) crossing each channel (pipe), ~ makes a jump of amplitude 1/ (c(p ~ N) ; inside a channel ~ is linear in the direction orthogonal to the curve that forms the channels. Hence [V~[ < no/c(p~ inf v E HI([:])

1/N, we get:

and letting c -

[ eU(: )/" IVv(x)[2 dx

JD

0 such that, for all x. E Xs, the following decomposition is valid: OQ(x.) - Uj=IFj, 4 where Fj are connected components of OQ(x~) such that Uo(x) > Uo(x~) + ~ if x E F1 U F3, and min Uo(x) < Uo(x.) - fl,

xEF2

max

xEF2UF4

min Uo(x) < Uo(x.) - fl,

xEF4

Uo(x) ~_ Uo(x~) + ~ ;

(iv) for all xs E Xs" if x E OQ(x.) and Uo(xs) - ~ ~_ Uo(x) ~_ Uo(x~) + j3 then U1 (x) - O; (v) all the trajectories of the equation 2 - - V U 0 ( x ) starting at F2, are attracted with x~-~ - Xmi~ while the trajectories starting at F4 are i+ attracted with Xmin. Under the above assumptions on U0 and U1 the said neighborhoods do evidently exist if/~ is small enough. We are going to show now that for V0 < /~/2 the random variables a(Xmi., ei) are independent. To this end we consider arbitrary two neighbor minimum points Xm~. and xmi~ + ei and a minimizing sequence of curves { ~ ( . ) } such that 995(0) = Xmin, ~ ( 1 ) = Xmi~ + ei, ~ E X(Xmin,ei) and max U(~(t)) ~_ C~(Xmi~,e i ) + (f.

0~t~l

Due to the structure of Uo and the choice of Q(x~), the intersection of ~(.) with Q(x~) is nontrivial for all sufficiently small 5. It is also clear that ~5 only intersects OQ(x~) at the points located at F2 U F4. Denote T1 -- max{t; ~5(t) E F2},

T2 -- min{t > 71; ~ ( t ) E F4}.

Now one can replace the segments {~(t) ; 0 < t < T1} and {~p(t) ; T2 _< t _ 1} by the new ones in such a way that the curve ~5(.) obtained is continuous, still belongs to A'(Xmi~, ei) and satisfies the estimates:

U(~(t)) < a(Xmi.,ei),

for all t < T1 and t > T2.

Thus O~(Xmin,el) only depends on {Ul(x); x E Q(x~)}, and the statement of the lemma follows. D The proof of the next assertion is similar to that of the preceding lemma and will be omitted.

144


L e m m a 4 - Let Ul (x) be statistically homogeneous field (whose distributions are invariant w.r.t, any shifts) supported by Lipschitz functions, and suppose that

[Vl(x)[ ~ ~0, IV1(xl) -Vl(X2)[ ~ ")/l[X1 - x2[, x, x l , x 2 e 1~2, and that a{U1 (x); x e G 1} and a{U1 (x); x e G 2} are independent whenever dist(G 1, G 2) > p. Then for sufficiently small V0, "/1 and p the random variables a(Xmi., ei) are independent.

References [1] M. Avellaneda and A.J. Majda, Mathematical models with exact renormalization for turbulent transport. Communications in Mathematical Physics, 131 (1990), 381-429. [2] M. Avellaneda and A.J. Majda, Superdiffusion in nearly stratified flows. J. Stat. Phys., 69(3-4) (1992), 689-729. [3] M. Avellaneda and A.J. Majda, Application of an approximate R - N - G theory, to a model for turbulent transport, with exact renormalization. In Turbulence in fluid flows. A dynamical systems approach, The IMA volumes in Mathematics and its Applications, G.R. Sell et al., editors, 1-31, Springer Verlag New York, 1993. [4] M. Avellaneda and A.J. Majda, Simple examples with features of renormalization for turbulent transport. Phil. Trans. R. Soc. Lond. A, 346

(1994), 205-233. [5] J.J. Bear, Dynamicsof Fluids in Porous Media. Elsevier, New York, 1972. [6] A.Yu. Belyaev and Ya.R. Efendiev, Homogenization of the stokes equations with a random potential. Math. Notes, 59, 4 (1996), 361-372. [7] A. Bensoussan, J.L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, volume 5 of Studies in Mathematics and its Applications. North-Holland, 1978. [8] R.A. Carmona and L. Xu, Homogenization for time dependent 2-D incompressible Gaussian flows. Preprint. [9] A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows. SIAM Journal on Applied Mathematics, 54:333-408,. [I0] A. Fannjiang and G. Papanicolaou, Diffusion in turbulence. Probability Theory and Related Fields, 105 (1994), 279-334. [ii] J. Fried, Groundwater Pollution. Elsevier, Amsterdam, 1975.

F. Campillo and A. Piatnitski

[12]

145

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer Verlag, 1994. [13] H. Kesten, Percolation Theory for Mathematicians, volume 2 of Progress in Probability. Birkh/iuser, Boston, 1982. [14] S. Kozlov, Geometric aspects of homogenization, Russian Mathematical Surveys, 44, 2 (1989), 91-144. [15] S.M. Kozlov and A.L. Pyatnitskii, Averaging on a background of vanishing viscosity. Math. USSR Sbornik, 70, 1 (1991), 241-261. [16] A.L. Pyatnitski and S.M. Kozlov, Homogenization and vanishing viscosity. In B. Grigelionis, editor, Probability Theory and Mathematical Statistics. Proc. Fifth Vilnius Conference, 1989, 330-339. VSP/Mokslas, 1990.

Fabien Campillo SYSDYS, INRIA/LATP, CMI 39 rue F.Joliot-Curie 13453 Marseille Cedex France E-mail: [email protected] Andrey Piatnitski Narvik University College HiN Department of Mathematics P.O. Box 385 8505 Narvik Norway and Lebedev Physical Institute Russian Academy of Science Leninski Prospect 53 Moscow 117333 Russia E-mail: [email protected]




Chapter 8

VIBRATION OF A THIN PLATE WITH A "ROUGH" SURFACE

G. CHECHKIN AND D. CIORANESCU

1. Introduction Rough surface problems have attracted much attention in the context of wave propagation and scattering (see, for example, [1], [2], [5] and [19]). The frictional behavior of deformable bodies depends explicitly on the structure of contact surfaces. Micro-characteristics such as the roughness of the contact surface or material properties near the surface, influence the large scale behavior. The asymptotic analysis of these problems was treated for instance, in [4] and [31], while different homogenization problems in domains with rapidly oscillating boundary were considered in for instance in [6], [17], [18], [19], [29] and [30]. Based on classical Kirchhoff and Reissner plates theories, new models for bending of plates with rapidly varying thickness was proposed in [20]-[23] (see also [7], [16], [24], [25] and [33]). In these papers, motivated by the development of structural optimization (see for instance, [26]) the authors have studied symmetric, linearly elastic plates with thickness of order s varying on a length scale of order sv. They restricted their attention to locally periodic plates, and to loads transverse to their midplanes. The transverse loads and the symmetry allow, in particular, to reduce the limit problem to one fourth-order equation. There are three different regimes, depending on whether U < 1 (the case of relatively slow variation of the thickness), U = 1 (when the variation is on the same scale as the mean thickness), or U :> 1 (the case of relatively fast variation of the thickness). For each case, an effective rigidity tensor M ~ relating the bending moment to midplane curvature of the limit plate obtained as s --~ 0, was determined. In the limit problem, the vertical displacement of the midplane solves a

148

Vibration of a thin plate with a rough surface

fourth-order equation of the form

-

a,/3,-y,5= 1

The methods developed in [20]-[23] relied essentially on the symmetry of plates. In the present paper we drop out the symmetry assumption, and consider a nonsymmetric locally periodic plate with oscillating boundary. In contrast to the case of plates with symmetric geometry, the limit equations in the nonsymmetric case cannot be decomposed into equations describing the vertical displcement and equations describing the horizontal ones. Due to the absence of a symmetry in the initial problem, a nontrivial interrelation between the leading terms of the asymptotic expansion do appear. In particular, the limit problem is characterized by the coupling of vertical and horizontal displacements. In Section 1 we introduce necessary notation, define a class of thin domains and formulate the problem. In Section 2 we prove a priori estimates for the solution and investigate the asymptotic behavior of the moments and means of stresses. Then we introduce a family of auxiliary problems and obtain by a formal asymptotic expansion method the homogenized system. In the last section we prove a convergence theorem which justifies the results from Section 2. 1. S t a t e m e n t of the problem Let ft c ]Ra be a domain of thickness e, ~--{:;C I 0 No, then supp p(2 -j-) N supp p(2 - j ' . ) = 0. As we are interested in proving anisotropic estimates, we are also led to introduce an anisotropic Littlewood-Paley decomposition, in the spirit of [16]. To do so, let us define the operators A~ and S~ by

def VJ~E7"], A~--99(2-kD3)

and

S~=

A~,. k~Kk-1

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier

179

T h e o r e m 3 - Let ~ be the operator defined in Equation (2.3); for any p E [1, co] and for any c~ > 0,there exists a constant C such that, for any vector field f, we have < C2J(}-})(e22J)4,(~+ a) IIAyfIIL2(1R3),

(3.2)

_< c2J(1-} ) min {1, ( e 2 2 J ) + 2 + (y-k) }

• II/Xj/X~fllL~(~).

(3.3)

Proof. Like all Strichartz-type inequalities, (3.2) and (3.3) are consequences of the following dispersive estimates:

L e m m a 2 - For any function f, we have

IIAjA~O(~,0)SIIL~(~) < Cmin {2k-J,7- 89} 23Je-C~

IISIILI(~), (3.4)

and

v ~,2 _< Cmin {,1 T- 89 (j-k) } 22Je-C~ II/',j/XkG(~, O)f llL~h.~ where

,

~3 2f({) ) . g(r, O)f dej.)F_l {~eiT N-oI~I

First, let us remark that inequality (3.4) is nothing but Lemma 1 with a precise control of the constant. Proof of Lemma 2. We start by the proof of (3.4), and (3.5) will be obtained by very similar methods. We can write AjA~G(~-, O)f(x) = Kj,k(T, O) 9f ( x ) , with

Kj,k(T, O,X)

def j~a

eix'~+iT~T-Ol~le99(2-Jl~])~(2-k~3) d~,

(3.6)

hence all we have to prove is that

IIKj,k(',-, O,.)IIL~(~)_< ~C e_CO22J23j

T2

(3.7)

Anisotropy and dispersion in rotating fluids

180

Indeed, (3.7) implies that C _6022J ]]AjA~(r,O)f[IL~(IRS ) dt = ,I, et~sup (27r) - 3 / ~ + xfi s qp(2-Jlc~l)e-i}~-'tl~12]'(c~)~(t,~) dtd~ = sup (27r) -3 E

~B

k 1 and according to (4.8),

2-~IIAjAe ,m(D) T'o-i~~ b m l l L ~ ( L ~ ( ~ ) ) _x

(7)

[

ln ]

Now we define the 2 x 2 integral operator S by

s =

K-

lnA

T

(s)

'

To go further we introduce the decomposition in real and imaginary part of the Green function G(x, y)

C(x, y) = c~

- Yl) + isin(kl x - Yl)

4~1x - yE

(9)

4~j~ - yf

We obtain a similar decomposition for S in real and imaginary parts S= T+iR,

(10)

with T

--

i

Kr - ~nA

Tr

'

R

=

Ifi

Ti

.

(II)

From a mathematical point of view Ti, Tr, K~, Ki are symmetric, n(x)A is antisymmetric and Gi(x, y) is a regular kernel. So R appears as symmetric and regularizing while T* the adjoint of T satisfies [ T-

T* = I I =

0--n(x)A] --n(x)A 0

(12) "

Next we derive the following decomposition of R. If d is a direction of the unit sphere S 2 and (J(x),M(x)) are two fields on F we define the far field operator

a~J(a) = ~

a A (J(x) A a ) e - i k •

(13)

Integral equations and saddle point formulations for scattering...

196

We combine this into a new far field operator

M(x)

M

It may be proved t h a t

(a) - a ~ J ( a ) - ia A a ~ M ( a ) . (14)

R = (Am) * A ~

(15)

3. D e r i v a t i o n of the s y s t e m The idea is to consider a set of solutions of the Maxwell's system

VAE-ikZoH-O,

in D + (16)

V A H + i k Z o l E = O, in D +, with the following expansion at infinity _~

e iklxl _ o ~ ( ~ ) + e -iklxl ~.

Ixl ~

E(x)

eiklxl ZoH(~)

Ixl a~(~), Ixl

x

~,

(~ - VI

e_elxl

_out

Ixl (~A "~ (~))

-~

~

~ (Ixl ~A

) (17)

~n a~(~)), I~l-~ ~ ,

where u~-~ ~ , H) and a~in(E,H) are some tangential fields on the unit sphere, and the convergence e --~ e ~ holds in the sense lim 1 /R le(x) -- e~(x)l 2 dx = 0. R-,c~ -R

kllyllw~,

k > 0,

w i t h Wy =

v~

KerT*

,

(59)

just because standard functional spaces in which T is continuous are known to be based on H -1 (div, F) and H- 89(curl, F). Nevertheless it is at least possible to provide an abstract framework in which the inf-sup condition holds. Let us take Vx = X, Vy = Y, defined in (51). Definition 6 -

We define the quotient Hilbertian space W

___

Y KerT*

equipped with the norm inf

yo E K e r T *

JlY- YoJIY --IIT*yJIx +

inf

yo E KerT*

IIY- yolJx-

System (57) is well-posed as soon as the inf-sup condition sup ~x

(x, T * y ) x

> Cllyl]w

]l~ljx

(60)

-

holds for some strictly positive constant C > 0. This inequality can be derived as follows. Picking x - T * y in (60), we find sup

x~x

(x, T * y ) x

II~llx

> IIW*yllx.

-

(61)


206

Let y E Y. In appendix it is proved t h a t - I I T a projector (it is a Calderon Projector)"

with I I defined in (12), is

TIIT = -T.

(62)

Let yl = - Y I * T * y . We have t h a t T * ( y - Yl) = T * y + T*YI*T*y = 0. So yo = y - yl C K e r T * hence inf

Yo E K e r T *

IlY- Yollx < IlYllIX = IIII*T*yl]x = IIT*ylIx,

(63)

where we have used the isometric property YI*II - I. So we have sup ( x , T * y ) x Ilxllx

xeX

> 1 T* 1 inf IlY-Yollx = - ~11 YlIx -[- ~ yoEKerW*

1

IlYllw.

(64)

Thus it gives Lemma

C=!

2 -

The inf-sup condition (60) in spaces = X x W is true with

2"

Since the natural continuity T* 9Y -+ X holds and one has the bound (98) I I A ~ A ~ y l I x 0, it proves following [2] T h e o r e m 3 - The variational system (58) is well posed, that is, for every pair (g, gy) C ( V'x, Vy) - (Vx, Vy) there exists a unique (x, y) c Vx x W such that (x, 2) + ( A ~ x , A ~ ) - ( T ' y , ~) = (g, ~), (x, W*~) + ( A ~ y , A ~ ) = w ' (gy, Y ) w ,

V2 e Vx, V~ e Vy.

(65)

However when discretization is considered a difficulty arises with the use of (X, Y) = (Vx, V~) spaces. The reason is t h a t we want to avoid the construction of some discrete space compatible with the L 2 based space (X, Y). We would like to take a classical integral code based on the duality H- 89(div, F) and H- 89(curl, F), rearrange all the routines, and use the iterative algorithms described further in order to solve our new discrete integral system. Then the question of the convergence of the discrete solution to the exact solution arises. All our efforts in order to prove the convergence using this strategy failed. The reason seems to be t h a t the classical discretization of integral operator is based on H- 89(div, r) and H - 1 ( c u r l , r ) , and not on (Z, Y), [4]. Moreover some numerical results in 2D for the Helmholtz equation, [1], show t h a t this problem may be a real one. It seems that there are cases where the discrete solution obtained through the strategy described above does not converge to the exact solution, even in some very simple and regular cases. Of course this conclusion has to be re-evaluated if the

F. Collino and B. Despres

207

discretization of the integral operators are compatible with (X, Y). Note also that the kernel KerT* is a large space of infinite dimension. Some algorithms are very sensitive to the dimension of this kernel. It is our purpose now to modify the system and to present what we will call the penalized problem, with much stronger coercivity. Let/3 some positive penalisation parameter (for instance/3 = 1). Remembering that y = ix, we modify system (15) to obtain the penalized system (1 +/3)x + (Ar162* A ~ x -

T * y + i~y = g (66)

+Tx-

i3x + fly + (A r162 A ~ y = 0.

which appears to be a system of the form X

A simple calculation shows that X

X

[ y 1, [ y ])xxx -Ilxll~ + 91Ix + :> m i n ( ~ ,

iyli2x

2

+ tIA~yl]~ + [tA~x][z

(68)

1

and the system appears now as coercive in the x variable and also in the y variable. Another possibility is to modify system (15) according to (1 - 3)x + ( A ~ ) * A C ~ x - (T*y + if~y) = g (69) (Tx-

i/3x) + 3 y + (Ar162* ACCy = O,

where/3 is now some positive number less than 1 (let/3 = 89 The interest of (69) is that it corresponds to a saddle point for the Lagrangian 1 1

1

1

+xllTIl~ + xl]AC~xI[~ - Re ((x, T * y + ifly)x - i(v, ( A ~ ) y ) z ) . L

The problem appears as a penalized saddle point problem.

(70)

208


5. System for general impedance boundary conditions We turn now to the case of a general boundary condition. We assume that the electro-magnetic field is determined by some impedance boundary condition of the type

n(x) A (E~r(x) A n(x)) + ZoZ,.(H?r(X ) A n(x)) = F,

(71)

where Z~ is some impedance operator that we assume symmetric with a positive real part, i.e.

(NeZrJ, J) > 0,

VJ C D(Z~).

(72)

We associate to it its reflection coefficient operator

R = ( I d - Zr)(Id + Zr) -1,

(73)

which, thanks to (72), satisfies

ilRII_< 1 r IIRJ[I~(Tr) 0 a positive real number. Denote by D either ft or f~ x (0, T). For n E IN and 1 u

strongly in

L~(0, T; V ) n L2(O,T; H2(f~) n V),

w

strongly in

L ~ ( O , T ; H I ( ~ ) ) n L 2 ( O , T ; H 2 ( ~ ) n Hl(f~)) and

>p

strongly in

L~(O,T;L~(s

>

3. A priori estimates Throughout this paper the external fields f and g are assumed to be L2(QT) functions, small enough with respect to the viscosities coefficients of the model (a precise formulation of this hypothesis will be given later on; see Section 3.2) p, #~, Ca and Cd. Concerning the initial density p0, we assume that it is a continuously differentiable function (p0 E C1), and that there exist a,/3 such that 0 O, such

that ~k IIWF+I(~k)II 2 0, Au~ is uniformly bounded in L2(s,T; L2(s A similar conclusion hold for w n.

C. Conca, R. Gormaz, E. Ortega and M. R o j a s

229

Remark. Using classical compactness arguments, we conclude that, up to a subsequence, the approximate solutions (u ~, w '~, p") converges to a strong solution of problem (1.1)-(1.2). Alternatively, this can be proved using a different approach which we develop in section 4.

4. T h e c o n v e r g e n c e

of the sequence

This section is devoted to prove t h a t u ~, w n and p~ are Cauchy sequences. Let us introduce the following notation for the difference of two terms of a sequence. For n, s > 1, It n ' s ( t ) --- U n+s (t) -- It n ( t ) , W n's (t) -- W n + s ( t ) -- W n ( t ) ,

and

p',~(t)

=

p'+~ ( t )

-

p" (t).

It is clear t h a t u n,s, w n,s and pn,* satisfy the following equations p ( p n - l + s ~ a n,s t ) + ( # + # r ) A u n's -- 2 # r P ( r o t w n-1 ,s) + p ( p n - 1 _g(fln-l,sur~) _ g(pn-l+sun-l+s . ~ u n , s) -p(pn-l+sun-l's

p~-~+~'~

" V u n) - p ( p n - l ' s u n - 1 .

,s f )

~Tun)

(4.1)

+ (~a + ~ d ) B ~ ~'~ -- (~0 + ~d - ~a)V di~ ~ ' ~ + 4 p r w n'~ -- 2pr(rot u ~ - l ' s ) + p~-l,s g _ p n - l , s w ~ _pn--l+sun--l+s

. ~wn, s _ fln--l+sltn--l,s . VW n

(4.2)

_ p n - - 1,sun-- 1 . V W n

pt '~ + u "'~- Vp n+~ + u ~- Vp ~'s = 0.

(4.3)

The following lemma which can be easily proven, is fundamental in order to obtain error estimates. L e m m a 4.1 - Let 0 0 such that for all n > 2, we have 0 ~ ~ n ( t ) ~_ C

~ot ~ ) n _ l ( T ) d T .

Then (Ct) n-1

Cn(t) _< M ( n for all t E [0, T], and n >_ 2.

1)!

Existence and uniqueness of a strong solution ...

230

The next lemma, proven in W. Varhorn (1994, Lemma 3.10 p. 122) is a variant of Gronwall's lemma and it will be also needed. L e m m a 4.2 - Let "y c IR ('y > O) and let ~, f, g c C([0, T]) real functions (f _> 0, g >_ 0 on [0, T]) satisfying for all t C [0, T] the inequality

,~(t) +

j~Ot f ('r) dT 1, M :> 0 such that

faf(hx, o'Vuo,h)dx < M

(0.16)

for every h C IN.

Moreover, let us assume that there exists u0 c Wllo'~(JR n) such that

UO,h ~ Uo in LI(~t),

(0.17)

fhom(CrVUo) C LI(~~) for every cr E IR.

(0.18)

R. De Arcangelis

247

Then fhom is convex and finite in the whole IRn, r

_~ fhom(Z) for every z C IRn,

(0.19)

and for every A e [0, +oc[, p e [1, ~-1 [, fl e L ~ (f~) the values ih (f~, uO,h, fl) in (0.11) converge, as h tends to +oc, to the finite value

i~(~,uo, fl) =min

{J;fhom(Vu)dxq- ~,udx+)~~luiPdx 9

(0.20)

(cf. Theorem 5.1). We emphasize that in this case we do not need to assume

(0.8). When (0.13) is replaced by the linear coerciveness assumption (0.6), the homogenization process must be carried out in the context of B V spaces, but to do this some of the above conditions must be strengthened. Thus we assume that (0.6)+(0.8) and (0.16) are fulfilled, and tat, if f~' is an open set such that f~ C f~P, the following conditions hold 1

uo,h is constant in f~t Cl ~ S for every h E IN and S c C,

(0.21)

the integrals l S ( h x , VUO,h)dx are equi-absolutely

(0.22)

continuous in f~' when h C IN,

uo,h ~ Uo in LI(fY),

(0.23)

fhom(aVu0) C Ll(f~ ') for every cr E ]R.

(0.24)

Then, we prove that fhor, is convex and finite in the whole ]Rn, that (0.9) holds, and that for every )~ e ]0, +oc[, p e ]1, ~n--~_l[, fl e L~176 the values ih(f~, U0,h, fl) in (0.11) converge, as h tends to +cx~, to the finite value ioo(f~, u0, fl) = rain

fhom(Vu)dx +

/ho~m dlD ul

'~-LO fh~176176176 - ")n')d~n-l + Y[ofl'dx+ ~ /o i"Pdx "" ~ BV(')} (cf. Theorem 5.2).

Of course, problem in (0.25) reduces to the one in (0.20) when (0.13) is fulfilled. In particular, when uo,h --0 for every h E ~ , the above described convergence result for the problems in (0.10) follows as corollary.

248

Homogenization of Dirichlet minimum problems...

In both cases, convergence results in Ll(~t) for the minimizing sequences of the problems in (0.11) to minimizers of io~(~, u0, t9) are also proved. In particular, the above results continue to hold if a E L~oc(lRn) is Yperiodic, q E [1, +oc[, f satisfies

f(x, z) < a(x)+ Izlq for a.e. x e

]R n

and every z e IRn,

(0.26)

and if assumptions (0.15), (0.16), (0.18), respectively (0.22), (0.16), (0.24), are replaced by the integrals

/IVuo,hiqdx are equi-absolutely

(0.27)

continuous in ~t when h E IN, respectively by the integrals / I V u o , h

]qdx are equi-absolutely

(0.28)

continuous in f~'when h E IN (cf. corollaries 4.3 and 4.4). Our results are obtained by exploiting De Giorgi's F-convergence theory, together with some recent results and techniques introduced in [11], [9], and

[16]. 1. Notations and p r e l i m i n a r y

results

We first recall some properties of BV spaces. We refer to [21] and [31] for a complete exposition on the matter. Let ~ be an open set. For every u E BV(~) we denote by IDuI the total variation of the lRn-valued measure Du. Moreover, according to Lebesgue Decomposition Theorem, we have

Du(E) - / E Vudx + DSu(E) for every Borel subset E of where we have denoted by Vu the density of the absolutely continuous part of Du, and with DSu the singular part of Du, both with respect to Lebesgue measure. We recall that BV(~) is a Banach space with norm

[L"llBv(a)'ue B V ( 9 ) ~ f luLdx+ ]Dul(~). Jn

249

R. De Arcangelis

If f~ has Lipschitz boundary, B V ( F I ) continuously embeds in La-~-l(f~) and compactly in L p(f~) for every p e [1, ~ [. If f~ has Lipschitz boundary and u c BV(f~), then the null extension u0 of u to ]Rn is in BV(]Rn), and there exists a function in Ll(0f~) (endowed with the ?_/n-1 measure), called the trace of u and again denoted by u, such that D u o = - u n ~ "-1 in Oft. As consequence we have t h a t if f~' is an open set such t h a t -~ c f~', u C B Y ( f ~ ) , and v C B Y ( f ~ ' \ -~), then the function

W --

u v

inf~ ~t \f~ in

is in B V ( f ~ ' ) , v - u e Ll(0f~), and D w = (v - u ) n a ~ n-1 in 0f~. For every Lebesgue measurable set E we denote by IE! its measure. For every f" ]Rn ~ [0, Ac-CX:)[convex, we define the recession function fc~ o f f by .t

f cX~ " z

E ] R n )-->

lim

l=f (t z ) .

t -+ 4- c ~ t

It is well known t h a t f ~ is convex, lower semicontinuous, and positively l-homogeneous. We now introduce the r-convergence theory. We refer to [17] and [14] for a complete exposition on the subject. Let (U, T) be a topological space satisfying the first countability axiom. D e f i n i t i o n 1.1. - For every h C IN let Fh" U --, [-oo, +ec]. We define the r - ( T ) - l o w e r limit and the F - ( T ) - u p p e r limit of {Fh} as F - ( T ) l i m i n f Fh: u C V ~ inf r~ liminf Fh(Uh) " Uh ~ U in T~ h--* + o o

~ h--+ + o o

)

and

F-(T) limsup F h ' u C U ~-~ inf { lim sup Fh(Uh) " Uh --* U in T}. h-++oo

h--++cx~

I f in u one has

F - (T) lim inf Fh (u) = F - (T) lim sup Fh (u), h--,+c~

h--~+c~

we say t h a t in u there exists the F - ( T ) - l i m i t of { F h } , and we define it as

F - (T) lim Fh (u) -- F - (T) lim inf Fh (u) = F - (T) lim sup Fh (u). h--*+c k}l < h---,+cx~

sup / ~ luh -- UO,h[dX = ~1 / ~ l~- ~01d~, -< kl lim h-~+~ by (0.15) we deduce the existence of kE C ]hi such that

f ( h x , Vuo,h)dx < ~ for every k _> ks. (2.2) lira sup f{ h--,+c~ xEf~:Iuh(x)--UO,h(x)l>k} Moreover, again by (0.15), let A be an open set such that A _C f~, s p t ( u u0) C A, and limsup]

f ( h x , Vuo,h)dx _ h~,A. By the convexity properties of f we get

/ f(hx, Vwt,k,h)dx _ kE, t C ]0, 1[, and observe that tl~tk,h-- Uo,hliL~(a) _ k~, t E ]0, 1[ such that 1 + t - t 2 < (7.

(2.9)

R. De Arcangelis

257

We now remark that for every t C ]0, 1[, wt,k ~ wt in L I ( ~ ) as k tends to +c~, and that wt ~ u in L l ( ~ ) as t increases to 1. Therefore by (2.9), (1.2), and (0.4) the lemma follows letting first k go to +oo, then t go to 1, and finally e decrease to 0. I By using Lemma 2.1 we can prove the estimate from above for the F - ( L 1 ) - u p p e r limit of {F0,h}. We start with the case dealing with Sobolev functions. P r o p o s i t i o n 2.2. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as ii1 (0.4), fhom be given by (0.5), {U0,h} c_ Wllo'cC~(]Rn), s u0 C Wllo'I(IR~). For every h E IN let Fo,h be defined by (1.4). Let f~ be a bounded open set with Lipschitz boundary, and assume that (0.7), and (0.14)+ (0.18) hold. Then

F - ( f l ( a ) ) l i m s u p F o , h ( a , u)

>

--

d D S U ) d l D S u l n t - f o a fhC~m((Uo--u)nf~)d~-~n-1 f h o m ( V u ) d x + ~ f h ~ m ( dlD~u]

for every u e BV(f~). Proof. Let u c BV(f~), and assume that F-(LI(f~))liminfh--,+oo FO,h(a, u) < +oc, then there exist {Uh} C_ Wllo'~(IR~), and {hk} C_ IN strictly increasing such that Uh E uO,h + wl'~176 for every h C IN, Uh --* u in Ll(f~), Uhk is constant in f~ O ~ S for every S c (J and k E IN, and

r-(L~(a)) lim inf F0 h(a, h--,+oe '

u) =

lim ~ f ( hkx, Vuhk )dx.

k ~+oc

(3.4)

Let e > 0, and let, by (3.2), A c_ f~' be a bounded open set with Lipschitz boundary such that f~ C_ A, and f lim sup I

h--.+oo J A\-~

f ( hx, Vuo,h )dx < ~.

(3.5)

Then, by (3.3), (3.1), and (1.1), it is clear that Uh---*W--

u u0

in f~ inAkf~

L1 in

(A),

and that Uhk is constant in A O h@S for every S r C and k E IN. In fact, given S E C and k E IN, this comes trivially if ~ S C f~, whilst, if h-LkSN (]Rn \ f~) =/= 0, it follows once we observe that (0.8) implies the coincidence of the constant value taken by Uhk in f~ C3 h![S with the one taken by UO,hk in 1 S. Moreover, f

lim inf I

k --, + oo J A

f (h k x, V u hk )dx > F- (L 1(A)) lim inf Fhk (A w) > --

k ---++ oo

>_F-(LI(A))lira

inf Fh(A, w).

h--,+oe

'

--

(3.6)

R. De Arcangelis

261

By combining (3.4) with (3.6), (3.5), and Lemma 3.1, it results

F-(LI(~)) lim inf F0 h(ft, u) = h--*+c~ '

(3.7)

f f = lim infk_~+~ L f(hkx, VUhk)dx - l isuPk__,+~ m ]A\a f(hkx, VUo,hk)dx _> _>

i.

fhom(Vw)dx +

S.

fho~m d[D,wl dID'w I - e >

> dD'u - ~ fhom(VU)dx_V ~ fh~om( d]Dsul)dID~ui+ +

~o fhom ~ (\diD~w[/dl dD'w D~w] a

e.

By (3.7) the proposition follows as e tends to 0, once we recall that m

Dw = (u0 - U)~-~n--1 in Oft.

4. Representation results for homogenized functionals In this section we collect the previously obtained estimates to get some integral representation results, in Sobolev and B V spaces, for the F-(L1)limit of the functionals in (1.4). T h e o r e m 4.1. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be given by (0.5), {U0,h} C Wllo'~(]Rn), and uo E Wllo'~(]Rn). For every h e IN let Fo,h be defined by (1.4). Let ft be a bounded open set with Lipschitz boundary, and assume that (0.7), and (0.14)+(0.18) hold. Then fhom is convex and finite in IRn, for every u e uo + w l ' l ( f t ) the limit F-(Ll(ft))limh_~+~Fo,h(ft, u) exists and

F-(L~(~)) h-~+~limF0,h(a, u) =

~ fhom(Vu)dx.

Proof. The properties of fhom follow from Theorem 1.5, whilst the remaining part of the theorem by Proposition 2.2, and Lemma 3.1. m T h e o r e m 4.2. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in (0.4), fhom be given by (0.5), {U0,h} C Wllo'F(IRn), and U0 E w~l'l(]Rn).loc For every h E IN let Fo,h be defined by (1 .4). Let ft be a bounded open set with Lipschitz boundary, fY be an open set such that ft C ft', and assume that (0.7), (0.8), (0.16), and (0.21)+(0.24) hold.

Homogenization of Dirichlet minimum problems...

262

Then fhom is convex and finite in ]R~, for every u C BV(~t) the limit F - ( L I ( f t ) ) limh~+cr Fo,h(~t, u) exists and

r-(Ll(f~)) h-*+c~ lim Fo,h(f~, u) =

=

/f~ f h o m ( V u ) d x A - /Ftfh~m ( dlD~ul dDSu )diDSul~-~oa f~c~~

"

Proof. The properties of fhom follow from Theorem 1.5, whilst the remaining part of the theorem from Proposition 2.3, and Proposition 3.2. I In particular, by the above results we deduce the following corollaries. C o r o l l a r y 4.3. - Let Cy be a collection of subsets of Y, C be given by (0.3), f be as in the first two lines of (0.4), fhom be given by (0.5), {uo,h } C_

Wllo'cC~(]Rn), u0 E Wllo':(]Rn), a e L~oc(]R n) be Y - p e r i o d i c , and q e [1, +c~[. For every h E IN let Fo,h be defined by (1.4). Let ~t be a bounded open set with Lipschitz boundary, and assume that (0.7), (0.14), (0.17), (0.26)1, and

(0.27) hold.

e 0+W0 ,l(a)

/horn

the limit F-(LI(ft))limh_~+~ Fo,h(gt, u) exists and

r-(Ll(n))

lim Fo h(~) u) = ] fhom(Vu)dx. ' ' Ja

h---,+cx~

Proof. We prove that the assumptions of Theorem 4.1 are fulfilled. It is clear that (0.15) and (0.16) follow trivially from (0.26) and (0.27). Let us observe now that (0.17), (0.27), and the Ll(~t)-lower semicontinuity of the functional v E Wl'q(~t) ~-~ fa ]Vviqdx imply that u0 E wl'q(ft). By virtue of this, since by (0.26) and Jensen's inequality it results fhom(Z) _

fogt fh~ +oo

+ fa fho~m dDSu dlD~ul + u0 -- u)na) d~'~n-1 + f~ /~udx + )~f~ lulPdx if u C BV(a) if u e n 1(ft) \ BV(f~).

By (5.6) and (5.8), equality (5.5) follows. Finally, since A > 0 and p > 1, again by (0.6) it soon follows that the functionals u e L1 (ft) H F0,h(~, u ) + fa Zudx+ A f~ lulPdx are equicoercive in L 1(ft). In conclusion, the theorem follows from (5.5), and Theorem 1.2, once we observe that, by (0.22), (0.23), (0.6), and Sobolev Imbedding Theorem lim

h--~ + oo

_ \10,1/2[ 2

41/

> -

3

(z + Vv)dy

\]o,1/212

I 41/o -- ~

2

(Y\10,1/212

(Uz q- v)nY\lO,1/2[2d'][ n - 1

--

4 -- 3 [z[2 for every z E IRn, from which the left-hand side of (6.1) follows. In order to prove the remaining inequality, we take z c IRn and define v (z) as the Y-periodic extension of --XlZ

v (z)" (xl,x2) c Y ~-~

1 --

X2Z 2

- ( 1 - Xl)Zl - ( 1 -- Xl)Z 1

--

X2Z2

if (Xl, x2) e ]0, 1/2[• if (xl,x2) E ]1/2, 1[•

1/2[ 1/2[

(1 - x2)z2 if (Xl,X2) e ]1/2, 1[• -XlZl - (1 - x2)z2 if (Xl,X2) e ]0, 1/2[• --

1[ 1[,

then v (z) C VVllo'c~(IRn), Uz + v (z) is constant in S for every S c C, and

fhom(z) ~ / y [z + Vv(Z)]2dy = 2]zl 2 Let us now take ~ = Y, and u0: x E ] R n ~ X l , then it is clear that all the assumptions of Theorem 5.1, except (0.14), are fulfilled with u0,h = u0 for every h E ]N, and that for every h E ]hi there cannot be functions in u0 + W0~' (Y) t h a t are constant in each set of the type (~, i or (0, ~)+]~h,0[ 2 for some i E { 0 , . . . , h 1}. Consequently, for every t3 E L ~ ( Y ) , and A C IR, it results

inf { / y f (hx, Vu)dx + / y /3udx + A / y ]u[ dx :

R. De Arcangelis u E uo + W0t'~ (s

1

269

}

u constant in f~ N ~ S for every S E C -- +cx~

whilst, by (6.1), min

{ j f fhom(Vu)dx+jf~ ~udx + A 9fn luldx . u c uo + w l ' l ( ~ ) }
0} are the two hemispheres. Expression (2.3) models an elastic bounce against the solid wall, with a random deflection of the velocity direction. K is an integral kernel which describes the reflection law of the particles. Indeed, K ( x , Iv]2/2, w ' ~ w)[wzldw is the probability for a particle hitting the wall at point x with velocity v' - [vlw' to be reflected with the same Iv[ and velocity direction w in the solid angle dw. We shall comment on the physical relevence of this model in Section3. Note t h a t B = B(x, [v[2/2) operates on the angular variable w only while x and Iv[ are mere parameters. We are interested in the situation where the potential r confines the particles close to the boundary F. Therefore, we assume H y p o t h e s i s 2.1 - For any given fixed x E F, the function z E ( - o o , 0] -~ r z) is decreasing and lim r

Z--~ --(:X3

z) - +oc.

This hypothesis could somehow be relaxed but we shall leave this point to future work. We define: -

r

z -

0),

z)

-

r

z) -

9

(2.4)

We note that r 0 ) - 0. Under hypothesis 2.1, the motion of the particles can easily be pictured. After some excursion in the domain of negative z, the particles are attracted back to the wall by the potential field. As they hit the wall, they are reflected elastically into the domain and a new ballistic loop begins. As a result of this succession of bounces, the large scale dynamics in the direction parallel

P. Degond

277

to the boundary should resemble a diffusion process. Therefore, we are led to rescale the longitudinal space coordinate x and the time t according to x' = c~x_,t' - a2t, where a O: H y p o t h e s i s 3.2 - (i) flux conservation: f~ es 2 K ( x , z ; w ' - - + w ) l w z l d w = l ,

V w ' C S +2l

(3.1)

W c S 2 ~' e S+2

(3.2)

0i) Reciprocity:

K(z_,s; w ' - + w) = K(x, ~;-w -+ - w ' ) ,

~

(iii) Positivity: K(x, e; w' --+ w) > 0, for all w c S 2 w' c $2+ _y

(iv) B(~,~) i~ ~ compact o p ~ t o ~ e~om L~(S~) onto L~(S~). Relation (3.1) expresses the conservation of the normal flux of particles of given energy e at the point x of the boundary. Indeed, the magnitude of the incoming normal flux J~- is "

J z (x_, ~, t) = ]~

ES 2

f(x, z = 0, t~!~)I~ll~zl d~.

Transport of trapped particles in a surface potential

280

Using boundary conditions (2.6), we have:

--

[ f(Ivlw')lv]lw'~ldw' - J+, dw'ES~_

(3.3)

where the integrations with respect to w and w t have been exchanged, hypothesis (3.1) has been used and d + denotes the outgoing normal flux. This hypothesis is crucial for the validity of theorem 2.1. In practice, this assumption is not rigorously satisfied because surface interactions, like secondary emission [25] can induce jumps of the electron energy. However, over a wide energy range, these jumps are small and the interaction process can be approximated by the elastic process (2.3) with good accuracy. Even if the discrepancy between the two processes is large, the limit SHE model (2.9)-(2.10) can still give a good qualitative picture of the phenomena (see an example in a slightly different context in [15]). The reciprocity relation (3.2) is a macroscopic effect of the time reversibility of elementary particle-surface interactions. It may not always be true (see [11] for references about the validity of the reciprocity relation), but it considerably simplifies the analysis and we shall take it for granted. By the change a; ito - w in (3.1) and the use of (3.2), we easily deduce the following 'normalization' relation:

K(x_,r

V~E$2

(3.4)

The normalization relation, together with the positivity of K has important consequences. First, from (2.3), 139~(w) for w c S 2_ appears as a convex 2 From Jensen's inequality, we deduce the folmean value of 9~(cv~) over S+. lowing inequality, which bears some similarities with the Darroz6s-Guiraud inequality in gas dynamics [11]:

/s

I zld 0, we define the average density F ~ and current J~ by F~(x, ~, t)

=

1 f f~ hE(x, v)dvdz, N(x, e)

J~(x,e,t)

=

la / vf~ 5E(x,v) dvdz.

(4.5)

We prove: L e m m a 4.4 - F ~ and J~ satisfy the continuity equation (2.9).

Proof. We multply (2.5) by 5~(x, v) and integrate it with respect to v and z. First, using Green's formula, we have, recalling (2.8) and omitting the arguments in the delta measures"

/(Vz

Oz

~ Iv12 + ~(x, z) - ~

Oz Ov=

= =

f /

Vzf~Sdv[z=O 1

~z/~l~=o~(~l~l

dzdv -

f Vzf~5'-5"~zdVdz o~/ + / Vzf~5' ~dvdz 2 - e)d~ = 0,

(4.6)

P. Degond

because

0~

0r

Oz

Oz

283

and by virtue of the flux conservation relation (3.3). 5 t denotes the derivative of the delta measure. Then, we compute 9

:

= ave-Y~

+ V~_r

= c~ V ~ . J~ - V~_r

,

(4.7)

because f ~f~'e~ez = -(o/o~)(f j . ~ e v e z ) . Finally, collecting (4.6), (4.7) and (4.5) leads to the continuity equation (2.9) for F ~ and Y~. I Obviously, F ~ --~ F as c~ ~ 0. Thus, the continuity equation (2.9) for F and Y will be proved as soon as we know t h a t g~ ~ Y. This is the aim of the next two sections. We start with proving t h a t the auxiliary equation has a non e m p t y set of solutions.

4.3. The auxiliary equation In this section, we consider the most general problem of which (2.14) is a particular case. Indeed, let g(z, v) be given and let us consider the problem of finding X such that:

-~zN

+ O z 0%-7 ~ - g'

(~) =

(~-(~1).

(4s)

Again, we omit the dependence upon the x variable in the forthcoming discussion. We prove: L e m m a 4.5 - Problem (4.8) has a solution if and only if g satisfies

gSE(x, v)dzdv = O. Furthermore, if this condition is satisfied, the solution X is unique under the condition f XS~(x, v)dzdv - 0 and the set of solutions is the one-dimensional linear manifold {X + F(Iv[2/2 + ~) with F(e) arbitrary.


284

Proof. We use the change of coordinates (4.3). With the same notations as in section 4.1 the problem is then written: 0

-Vz(Z, Uz)-~z ~ = O,

-),+(~) = B*(-y-(~)).

(4.9)

Let u be fixed. We integrate the first equation (4.9) with respect to z 2 between the turning point Z(uz/2 ) and 0 and obtain for u~ > O:

~(0,V,~z)

~(0,~,-~z)

-

~(z,_~,~)=-

0(z,~,~z)l~(~,~)l-~e~,

(4.10)

-

)?(z,_~,-~)=

/; o(z,~,-~)l~z(~, ~)l-~ez,

(4.~1)

Since ~(Z, v, Uz) = 2 ( Z , v , - u ~ ) = x ( Z , v , 0), we deduce that 2(O,v_,-Uz)2(O, v, Uz) - G(v, u~) with

G(v_, uz) =

/;

(O(z, v_, uz) + O ( z , v , - u ~ ) ) IVz(Z, U~)l-ldz .

(4.12)

This is equivalently written ~'-(2) = J ~ ' + (~) + a .

(4.13)

We insert this relation into the last equation (4.9) and obtain

( I - B*J)~/+ (2) - B*G.

(4.14)

By lemma 3.2 (ii), a solution 3`+ ()~) exists if and only if s (13*a)(lulw)lwzldw - O,

which, in view of the flux conservation relation for B* (which results from (3.4)), is equivalent to s~ a ( l ~ l ~ ) l ~ z l d ~

- o.

This relation can equivalently be written, for all s -

0

~1~1~ - ~ =

(~2/2)

) I~zld~

O(z,V, Uz)lVz(Z, U z ) l - l d z

O(z,V, Uz)5 z

lul2/2:

-~lul2 - ~

)(1 ~

-~1~12 -

IVz(Z, Uz)l-lluzldu

~) [uzldu dz. (4.15)

P. Degond

285

Using the change of variables (4.3) to change to the variable v in the integral (4.15), we find" o =

/~J(/R O0

g(z,

1'2 + r ~,~z)5 ( ~[~

- ~) dv

)

(4.16)

dz,

3

which is the condition of lemma 4.5. Therefore, under this condition, there exists a solution 7+()~) of (4.14), which determines the outgoing trace of )~. Then, the incoming trace 7-()~) is determined by (4.13). Once the traces are known, 2(z, u) for all possible values of z are obtained by integrating (4.9) between z and 0. Two solutions of (4.14) for 7 + (2) differ by a constant function of w, i.e. a function of lu] 2 only. Then, the associated 7 - ( 2 ) differ by the same function of I~12 and so do the associated ~. Back to the v variable, we deduce that two solutions of (4.8) differ by a function of Ivl 2 + @(z). A unique solution can clearly be singled out by imposing that

/ xSe(x, v)dzdv = O. This ends the proof. 1 The function g(z, v) = v obviously satisfies the solvability condition of lemma 4.5 by oddness. Therefore, the auxiliary function X defined by (2.14) exists and is unique under the constraint f )iS~(x, v)dzdv = 0. Vie are now ready to derive the current equation. 4.4. T h e c u r r e n t e q u a t i o n L e m m a 4.6 - The current equation (2.10) is satisfied.

Proof.

We multiply equation (2.5) by XSe(x, v). Using Green's formula and the fact that

/("'Oz

or

0~

Oz

Oz

(1..

we compute, omitting the arguments in the delta measures:

K-

-=

=

.

0.0..

( /v,f'x__ddv )

~~

+

) ("

Ivl + ~ ( ~ _ , z ) - .

' 0 0)

)

d.dz

Vz-~z + 0--~cgv-~ X__5 dvdz.(4.17)

z---O

Now, we have, using (2.6) and the second equation (2.14)"

286

T r a n s p o r t o f t r a p p e d particles in a surface p o t e n t i a l

Ivl 2 [ ~_ 7+(f) Js

=

(7+(x) - u* (7- (x)) I~: ld,.,.,= o, -

-

(4.18)

_

and with the first equation (2.14)"

S ( ' '+0) f"

- V z - ~ z 4- 0--~ Ov---~ X_ 5 d v d z =

J,-

v 5 dvdz = aJ" .

(4.19)

Therefore, inserting (4.18) and (4.19) into (4.17), we deduce that K " = a g ~. This a posteriori justifies the definition of XLFrom (2.5) and (4.17), we deduce:

"': - ~

i

"s"-"("v) 'v'z-i (v

~ _ - ~_+ . . ~ ) s. ,_,.(.,v) , v , z .

Taking the limit a --~ 0 and using (4.2), we obtain: J = - S ( v . V~_ - V~_r V,__) F ~ 5~(x, v) d v d z .

But

(.. v . - v.r v~) Y - . .

(V . - V.r ") Y.

So, we get J - -

(i

X_ | v 5,(x, v) d v d z

)(

V a - V~_r

')

which leads to equ. (2.10) and (2.13). This ends the proof.

F, m

5. E x a m p l e s In this section, we consider particular examples of b o u n d a r y collision operators B. A rotationally invariant B means that K only depends on Wz, / ~z' and the angle between _~ and _w', i.e. K ( ~ ' ~ w) - K ( ~ z , ~ z , u~ . u~,), where u~ - w/iw_ I C S 1 and S1 is the unit circle. If in addition, K does not depend on u~ .u~,, B is said to be isotropic. We show t h a t in the case of an

P. Degond

287

isotropic B, the density-of-states and diffusivity can be expressed in terms of the bounce period of a particle, given by

T(x, U2z/2) =

IVz(Z, Uz)l-ldz,

2

(5.1)

(_~,~,~/2)

where u~ is the transverse velocity at the origin z - 0. L e m m a 5.7 - ff B is isotropic, we have

27r~ jfo 1 T(x, ep)dp,

N(z_.,s)

=

D(z__,e)

-- "fie2

/01

T 2 (z_, ep)(1 - p)dp I ,

(5.2)

where I is the 2 x 2 identity matrix. Proof. We first begin with N. Using the change of variables (4.3), and the coarea formula, we have, ignoring the _z-dependence: N(s

--

]r

(~

3

( 1

)

~]ul2-5

"

]Vz(Z, Uz)l-lluzld~tdz

(u~/2)

1]'n ~ T ( u ~ / 2 ) 5 ( 1 -~[~1 ~ - ~ ) I ~ z l d ~ = ~ = -~

~ T(~l~z[~)t~ld~

Using w E B(0, 1) as parametrization of 5;~, where B(0, 1) is the unit ball in N2 and noting that Iwz [dw = &o, we obtain: N(s) = 2~ ]B(o,n T(s(1 -Iw__}2))dw. Changing to cylindrical coordinates in B(0, 1),-we finally get:

/01

N(e) = 47r~

f01

T(e(1 - p2))pdp = 27re

T(sp)dp,

which is the first formula (5.2). Now, we turn to D. We use the notations of section 4.3. First, we note that with g - vi, (i = 1,2), equ. (4.12) gives G(v, uz) = viT(u~/2). Therefore, G is an odd function of vi and since B is isotropic, BG = 0. It


288

follows from (4.14) and lemma 3.2 (i) that 7+(X,) does not depend on the angular variable w. In fact, we can take "y+ (Xi) = 0 since any other choice will simply add a function of 1vl2/2 + ~ to x~, which will not modify the value of D. From (4.13), it follows that ~/-(Xi) = G. Then, integrating (4.9) between z and 0, we deduce, for uz > 0: x~(z, v, ~z)

=

2 v~ ~-(z, ~z/2),

Y(i(z, v, -Uz)

=

2 v, (T(u2z/2) - T(Z, Uz/2)) ,

where

T( Z, ?.t2/2) z

(5.3)

Z0

[Vz(~,Uz)[-ld~.

Now, it is clear that the vector X is proportional to v, so that the diffusivity D - d I is a scalar with / / 1 ) d XlVl~z(X, v)dzdv - )(lVl(~( ~lu[ 2 -- ~ [Vz(Z, Uz)[-l[uzldudz. Additionally, the term proportional to 7 in X1 is odd with respect to Uz and has a vanishing contribution to d. Thus, using the coarea formula and with the same computations as for N, we get:

-

j/

z 0 arbitrarily fixed, Q :> 0, S E L~(A4) and p > 2. The function G is increasing and /~ represents a bounded maximal monotone graph in IR 2 (of Heaviside type). We also consider the associate stationary problem

(PQ,~)

- d ~ ( l ~ l ' - 2 X T ~ ) + G(~) e Q s ( x ) Z ( ~ ) + f ~ ( ~ )

on M .

Through the paper we shall use the notation div(]Vu]p-2Vu) = Apu. Problem (P) arises in the modeling of some problems in Climatology: the so-called Energy Balance Models introduced independently, in 1969 by M.I. Budyko [15] and W.D. Sellers [64]. The models have a diagnostic character and intended to understand the evolution of the global climate on a long time scale. Their main characteristic is the high sensitivity to the variation ofsolar and terrestrial parameters. This kind of models has been used inthe study of the Milankovitch theory of the ice-ages (see, e.g. North, Mengel and Short [60]). The model is obtained from the thermodynamics equation of the atmosphere primitive equations via averaging process (see, e.g. Lions, Temam and Wang [53] for a mathematical study of those equations, Kiehl [50] for the application of averaging processes and Remark 1 for some nonlocal variants

298

Diffusive energy balance models in climatology

of (P)). More simply, the model can be formulated by using the energy balance on the Earth's surface: internal energy flux variation = Ra - Re + D, were Ra and Re represent the absorbed solar and the emitted terrestrial energy flux, respectively and D is the horizontal heat diffusion. Let us express the components of the above balance in mathematical terms. The distribution of temperature u(x, t) is expressed pointwise after standard average process, where the spatial variable x is in the Earth's surface which may be identified with a compact Riemannian manifold without boundary A/l (for instance, the two-sphere $2), and t is the time variable. The time scale is considered relatively long. Nevertheless, in the so called seasonal models a smaller scale of time is introduced in order to analyze the effect of the seasonal cycles in the climate and in particular in the ice caps formation (see Remark 2 for the connection with the associate time periodical problem). To simplify the presentation we assume t h a t the internal energy flux variation is simply given as the product of the heat capacity c (a given constant which can be assumed equal to one by rescaling) and the partial derivative of the temperature u with respect to the time. For a more general modeling see Remark 1. The absorbed energy R~ depends on the planetary coalbedo ~. The coalbedo function represents the fraction of the incoming radiation flux which is absorbed by the surface. In ice-covered zones, reflection is greater than over oceans, therefore, the coalbedo is smaller. One observes t h a t there is a sharp transition between zones of high and low coalbedo. In the energy balance climate models, a main change of the coalbedo occurs in a neighborhood of a critical temperature for which ice becomes white, usually taken as u = - 1 0 ~ The different coalbedo is modelled as a discontinuous function of the t e m p e r a t u r e in the Budyko model. Here it will be treated as a maximal monotone graph in IR 2

m ~(u)=

[m,M] M

u -10,

(1)

where m =/3i and M =/3w represent the coalbedo in the ice-covered zone and the free-ice zone, respectively and 0 < ~i < ~w < 1 (the value of these constants has been estimated by observation from satellites). In the Sellers model, t3 is assumed to be a more regular function (at least Lipschitz

J.I. Diaz

299

continuous), as for instance m

?.t ~ u i ,

m +( M

) ( M - m)

< uw,

where ui and uw are fixed temperatures closed to - 1 0 ~ both models, the absorbed energy is given by Ra = QS(x)~(u)where S(x) is the insolation function and Q is the so-called solar constant. The Earth's surface and atmosphere, warmed by the Sun, reemit part of the absorbed solar flux as an infrared long-wave radiation. This energy Re is represented, in the Budyko model, according to the Newton cooling law, that is, R~ = B u + C. (2) Here, B and C are positive parameters, which are obtained by observation, and can depend on the greenhouse effect. However, in the Sellers model, Re is expressed according to the Stefan - Boltzman law

Re

--

(TU 4 ,

(3)

where cr is called emissivity constant and now u is in Kelvin degrees. The heat diffusion D is given by the divergence of the conduction heat flux Fc and the advection heat flux Fa. Fourier's law expresses Fc = k~Vu where k~ is the conduction coefficient. The advection heat flux is given by Fa = v.~Tu and it is known (see e.g. Ghil and Childress [35]) that, to the level of the planetary scale, it can be modeled in terms of ka~TUfor a suitable diffusion coefficient ]Ca. So, D = + ( k V u ) with k = kc-+-ka. In the pioneering models, the diffusion coefficient k was considered as a positive constant. Nevertheless, in 1972, P.H. Stone [68] proposed a coefficient k = i~Tul, in order to consider negative feedback in the eddy fluxes. So, in that case the heat diffusion is represented by the quasilinear operator D = div(IVul~Tu). Our formulation (P) takes into account such a case which corresponds to the speciM choice p = 3 (notice that the case p = 2 leads to the linear diffusion). These physical laws lead to problem (P) with Re(u) = G ( u ) - f. In Section 2 we start by presenting some results on the existence and uniqueness of solutions which generalize some previous results in the literature for a one-dimensional simplified formulation. Such simplification considers the averaged temperature over each parallel as the unknown. So, the two-dimensional model (P) is reduced in a one-dimensional model when A4 is the two dimensional sphere and considering the spherical coordinates. Therefore, the model becomes

(p1)

=

0(x)

in ( - 1 , 1) x (0, T), in ( - 1 , 1),


300

with p(x) = (1 - x 2 ) ~ where x = sin0 and 0 is the latitude. Notice t h a t again there is no b o u n d a r y condition since the meridional heat flux ( 1 x 2) ~ luxlp-2ux vanishes at the poles x = J:l. We also include in this section some comments on the free boundaries associated to the Budyko type model (the curves separating the regions { x " u(x,t) < - 1 0 } and {x " u ( x , t ) > - 1 0 } ) . We end the section with a result on the stabilization of solutions as t -+ c~. Some references on the question of the approximate controllability for the transient model are given in Remark 4. Section 3 is devoted to the study of the number of stationary solutions according to the parameter Q, when 13 is not necessarily Lipschitz continuous and p >_ 2. We start by estimating an interval of values for Q where there exist at least three stationary solutions and other complementary intervals for Q where the stationary solution is unique. A more precise study of the bifurcation diagram of solutions for different positive values of Q is available once we specialize foo(x) - C with C a prescribed constant. Then problem (PQ,f ) becomes

(PQ,c)

- div(IVu]p-2Vu) + G(u) + C C QS(x)/3(u) on A/I.

We denote by E the set of pairs (Q, u) c IR + x V, where u satisfies the equation (PQ,c). We show that, under suitable conditions, E contains an unbounded connected component which is S - s h a p e d containing (0, ~ - 1 ( _ C ) ) with at least one turning point to the right (and so at least another one to the left). We end Section 3 with a remark on a simplified version of problem (PQ,c) for which it is still possible to find more precise answers: if Ol < Q < 02, for some suitable positive constants Q1 < 02, then we have infinitely many solutions. More precisely, there exists k0 c IN such that for every k c IN, k _> k0 E ]N there exists at least a solution uk which crosses the level uk = - 1 0 , exactly k times.

2. The transient model 2.1. O n the existence of solutions Motivated by the model background described in the Introduction, we introduce the following structure hypotheses: p _> 2, Q > 0, - (HM) A/I is a C ~ two-dimensional compact connected oriented Riemannian manifold of IR 3 without boundary, - (Hz)/3 is a bounded maximal monotone graph in IR 2 i.e m < z < M ,

~

- -

w

Vz e/3(s), Vs c IR, - (HG) (~ 9IR -~ IR is a continuous strictly increasing function such t h a t (}(0) - 0, and [(j(a)] _> Cla]" for some r _> 1,

,

301

d.I. Diaz

- (Hs) S : M ~ IR, S c L ~ 1 7 6 Sl ~ S(X) ~ SO ~>0 a.e.x C A/I, - (Hr f e L ~ 1 7 6 x (0, T)), ( r e s p . - (H~~ f e L ~ 1 7 6 x (0, oo))), -

(H0)

u0 e L ~ 1 7 6

The possible discontinuity in the coalbedo function causes that (P) does not have classical solutions in general, even if the data uo and f are smooth. Therefore, we must introduce the notion of weak solution. The natural "energy space" associated to (P) is the one given by V := {u: M ~ R, u c L 2 ( M ) , ~7~4u E L P ( T M ) } , which is a reflexive Banach space if 1 < p < oo. Here T M denotes the tangent bundle and any differential operator must be understood in terms of the R i e m a n n i a n metric g given on A4 (see, e.g. Aubin [8] and D~az and Wello [26]).

D e f i n i t i o n 1 - We say that u : j~4 --, IR is a bounded weak solution of (P) if i) u e C ( [ O , T ] ; L 2 ( A 4 ) ) r q L P ( O , T ; V ) A L ~ 1 7 6 x (0,T)) and ii) there exists z e L~(A// x (0, T)) with z ( x , t ) e f l ( u ( x , t ) ) a.e. ( x , t ) e A4x(0, T) such that

L

u(x,T)v(x,T)dA

+

=

-

1o

V, xV dt+

< [~Tu]p-2Vu, V'v > dAdt +

Q S ( x ) z ( x , t) v d A d t +

Vv e LP(0, T; V) N L ~

f v dAdt +

~(u)v dAdt =

u o ( x ) v ( x , O) d A

such that vt e L p' (0,T; Y'),

where V, x y denotes the duality product in V ~ x V.

We have T h e o r e m 1 - There exists at least a bounded weak solution of (P). Moreover, if T -- +co and f verifies (H}~), the solution u of (P) can be extended to [0, co) x A//in such a way that u C C([0, co), L2(]vl)) N L ~ ( A / / x (0, co)) r L~oc(O, oo; V). The above result can be proved in different ways. As in the case of the one-dimensional model (Diaz [19]) we can apply the techniques of Diaz and Vrabie [30] based on fixed point arguments which are useful for multivalued non monotone equations. We start by defining the operator A : D ( A ) C

302


L2(M) ~ L 2 ( M ) , A ( u ) = - A p t + G ( u ) i f u e D(A) = {u c L2(M) : - A p t + G(u) C L2(AA)}. The Cauchy associated problem (Ph)

du --~(t) + A t ( t ) ~ h(t)

t e (0, T), in X = L2(AJ)

u(o) = to,

u0 c L2(A/I),

is well posed (it has a unique mild solution in C([O,T];L2(M)) for every h c L2(0, T; L2(A/t)) by the abstract results of Brezis [14]) since we have Proposition

1

-

Let

u e D(r

(4)

u r D(r

where G(u) =

jr0u G(a)da

with D ( r

{u C L2(M), Vu e L P ( T M ) and

f ~ G(u)dA < +c~}. Then i) r is proper, convex and lower semicontinuous in L2(AA). ii)A = 0r and D(A) = L2(M), and iii) A generates a compact semigroup of contractions S(t) on L2(A/~). Besides, from Brezis [14] we know that u, solution of(Ph), verifies that u E LP(O,T;V), v/tut C L2(O,T;L2(A/~)), u c WI'2(5, T;L2(AJ)), 0 < 5 < T. Let us prove the existence of solutions for the problem (P) via a fixed point for a certain operator s Let Y = LP(O,T; L2(A/~)) and define ~_." K ---, 2 Lp(~ by the following process: Let us define

K = {z e LP(O,T; L ~ ( M ) )

: [Iz(t)[In~(M) 2 and 1 d 2 dt 1] u

u II~<M)

--

(CzQ It s

}2~4]~ C1,2,c r

IIL~<M)

~

- ~ ,

(11)

for the case p = 2 where e and a = a(E) .Now, we distinguish two cases:

Co

Cl,p,~ -< 0 and p > 2, then

CASEI: ifCtQ]lS]]~ ld

and the result holds by Gronwall's Lemma. 1

If p = 2 and CzQ [] S []~

0 on the manifold (A/I, g), D " A/t C IR 3 -+ IR a, D ( x ) = Yc = 6x. So, if u is any function defined on +9l, its local representation in the new coordinates is 5 " A//a -+ IR, u(~) = u(~) and we have 05 ~ (- 5 : )

1 0 u 5: = 5 Oxi (~)

i = 1,2, 3.

So problem (P) in the new coordinates becomes

(Pa)

-6PdivM~ ( ] V M ~ S I P - 2 V M ~ 5 ) - ~ ( 5 ) e Q S f l ( 5 ) + f (0, ~) - u0(-~ )

{Su

in (0, T ) • on Aria.


308

Clearly, if 5 is a solution of (P~) then ~t(Sx, t) is a solution of (P). Moreover, the uniqueness of (P~) implies the uniqueness of (P), and conversely. Let us see that there exists 5 > 0 such that the solution of (P~) is unique. Let u~ and 5~ be two solutions of (P~) with u~ verifying the strong nondegeneracy property. Arguing as before we arrive at

1d f lu~(t)_fi~(t)12dA~ 2 dt J~ 5 < IVu~Ip-2Vu~ -IVfi~IP-2Vfi~, Vue - Vt~ > dA~

+5 p f J.~4 5

+[

-

-

=

Q [

-

-

J,.~[ 5

JiM 5

for some z~ E/~(u~) and ~ E / ~ ( ~ ) . Here, S~ is defined by S~ :Ad~ -~ ]R, S~(2) - S ( ; ) . (10) and (11) allow us toestimate u~ - ~ for u~ and ~ solutions of (P~) ld 2 dt

~__(C1,5 Q I] S5 + II ~ -

IIL~(.Ad~) --

C1,2,a,5

C~ ( M ~ )

(12)

~;~ II2

L2(JM~) nc C1,2,a,5"

A careful study of the dependence on 5 of the involved constants (see Diaz and Tello [26]) allows us to see that if we define the constant

K.,~ = G,~Q

II s~ IIL~<M~>- c1,2,o., 5

we have that lim Kp,~ = 0. This reduces the proof to Case 1 and the proof 5--.0

of (i) follows. For the proof of (ii) we use the second part of Lemma 1 and SO

ld

2 dt II u - ~ II~ 0 such that the set {x E ( - 1 , 1) : luo(x)+ 101 _< e0} has a finite number of connected components Ij with j = 1, .., N and for any j there exists xj c Ij such that uo(xj) = - 1 0 , a n d luox(x)l >>50 for some 50 > 0 and any x C /j close to xj, then there exists a solution u(x, t) satisfying the strong nondegeneracy property on (0, T*) for some T* (see Diaz and Tello [26]). Some results on solutions with IVul ~- 0 on the level where /3 becomes multivalued for a similar bidimensional problem are given in Gianni [36]. 2.3. O n t h e free b o u n d a r y for B u d y k o t y p e models The discontinuity of the albedo function assumed in the Budyko model (/3 multivalued) generates a natural free boundary or interface ~(t) between the ice-covered area ({x e A/I : u(x,t) < - 1 0 } ) and the ice-free area ({x c A/l : u(x,t) > - 1 0 } ) . The free boundary is then given as r = {x e A// : u(x, t) = -10}. In Xu [72], the Budyko model for p = 2 is considered in the one-dimensional case. He shows that if the initial d a t u m u0 satisfies

t o ( x ) - - U o ( - x ) , uo e 6 3 ( [ - 1 , 1]), U'o(X) < 0 for any x e (0, 1) and there exists r e (0, 1) such that (to(x) + 1 0 ) ( x - r < 0 for any x e [0, r

(((0), 1],

then there exists a bounded weak solution u of (P) for which one has r = {r U {r with x = r a smooth curve, r = r and r e C ~ ( [ 0 , T * ) ) , where T* is characterized as the first time t for which 4+(t) = 1. He also gives an expression for the derivative r (t) (some related results for a model corresponding to p(x) = 1 can be found in Feireisl and Norbury [33], Gianni and Hulshof [37] and Stakgold [67]). We point out that the uniqueness result can be applied for such an initial datum. For the study of the free boundary in the bidimensional case see Diaz [22] and Gianni [36]. The interpretation of the size of the separating zone r for other models is in fact a controversial question. So, some satellite pictures (Image of the Weddell sea taken by the satellite Spot on December 10, 1987) show that the separating region between the ice-free and the ice-covered zones is not a simple line on the Earth but a narrow zone where ice and water are mixed. Mathematically it could correspond to say that the set

M(t) = {x E All: u(x, t ) = - 1 0 } is a positively measured set. In the following we shall denote this set as the mushy region (since it plays the same role than in changing phase problems, see e.g. D i a z , Fasano and Meirmanov [23]). Using the strong maximum principle, it is possible to show that if p = 2 the interior set of the mushy region M(t) is empty even if the interior of


310

M(0) is a n o n e m p t y open set (see Gianni and Hulshof [37]). As we shall see, this is not the case when p > 2 (recall t h a t p - 3 in Stone [68]). A o

necessary condition for the Budyko model (with R~ - B u + C) for M(t)7~ is t h a t C - 10B E [~iQS(x), [3~QS(x)] for a.e. z e Ad. (13) It is possible to show t h a t if p > 2, this condition is also sufficient. Here we merely present a result for the one-dimensional case (see Diaz [22] for the bidimensional case)" T h e o r e m 3 - Let p > 2. Assume (13) and uo E L ~ ( I ) such that there exist xo E I and Ro > 0 satisfying

M(O) = {x e I ' u o ( x ) = - 1 0 } D B(xo, R 0 ) ( = {x e

I'lx

-

x01
0, (16) where C is a positive constant and O(t) > 0 when t is large enough with O(t) = O(1) when t --~ c~. Then, thanks to a technical lemma due to Nakao [56], we conclude that ~(t) = O(1) which is equivalent to u E L~(0, oo; V). The following theorem proves the stabilization of the solutions u satislying (15). As usual, given u bounded weak solution of(P), we define the ca-limit set of u by ca(u)= { u ~ E V n L ~

3tn ~ +oo such t h a t u(tn, .) -~ u ~ in L2(A//)}.

T h e o r e m 4 - Let uo E L ~ ( M ) A V and let u be any bounded weak solution satisfying (15). Then, i) ca(u) ~ 0 and if u ~ E ca(u), 3tn ~ +(x~ such that u(.,t,~ + s) ---, u ~ in L 2 ( - 1 , 1; L2(Ad)) and uo~ E V is a weak solution of the stationary problem associated to fo~ ; ii) in fact, if u ~ E ca(u), then 3{t~ } --, +oo such that u(-, tn) ~ uo~ strongly in V.

J.l. Diaz

313

Proof. Let u ~ be an element of w(u). Then,

ll~(tn + s)- ~(tn)ll~.(~) < 2 Ilu~ll~,=( ( t ~ - l , t ~ -

+ l);L2(Ad))"

-

Since ut C L2(0, oc; L2(jt4)) , [lut]12L2((t_l,t_t_l);L2(./M)) --~ 0 when t~ -~ oc and by the Lebesgue convergence theorem we conclude that u(., tn + ") U~ in L 2 ( ( - 1 , 1); L2(Ad)). To prove that u ~ is a solution of (PQ), we consider the test functions v ( x , t ) = ( ( x ) g ) ( t - t~) with ( E V O L ~ ( A d ) and ~ c Z)(-1,1), ~__ 0, fll ~ --- 1. Then

fti~+l /Ad ut~q~(t -- tn) + fftlt~+l/Ad I V u F - 2 V u

]i i

J M O(u)(7)(t - t~)

f t +l

9 1

-- tn)

f

+ =

. V~(t

f(x, t)~(t

- t~)

z ~ ~ ( u ( x , t)).

Jtn-1

Changing variables, namely s = t - t~ and defining U~ (x, s) = u(x, t~ + s), we obtain that Un

__.x u o o

weakly in weakly in

'

IVUnlp-2VUn ~ Y

L ~ ( ( - 1 , 1); V) L ~ ( ( - 1 , 1);LP(TA/I))

Va > 1 Ycr > 1.

Applying Aubin's compactness result (see e.g. Simon [66]), a well known property of the maximal monotone graphs (see Brezis [14]) and Lebesgue's theorem, w e g e t t h a t z, - - zoo C/3(uoo) weakly in L~ (Ad x ( - 1 , 1 ) ) Vcr > 1 and ~(U~) --+ ~(uoo) in L I ( A d x ( - 1 , 1)). Letting n --, oc, we arrive to Y-V~p+

6(uoo)~=

QSzoo~+

foo~

V~ r V r q L ~ 1 7 6

1

Now, the main difficulty is to prove that ~ Y(s,-)~(s)

1

[ W ~ IP-2Wo~.

Due to the coercivity of the p-Laplacian operator we obtain the following inequality: lim n--+oo 1

(IVUnIP-2VUn-IVXI~-=VX)

9( W o o

-

VX)~(~)

gAds >

O,

(17)

314


which holds for X E V. We arrive to the desired convergence by taking = u ~ + A~ and applying a Minty type argument to(17) as in Diaz and de Th~lin [28]. The proof of (ii) uses the coerciveness of the operator and the fact t h a t ( ] V U n l p - 2 ~ U ~ - [ V u o o [ P - 2 V U o o ) 9(VUn - Vuoo)~p(s) dAds --, O.

The inequality I ~ - r to obtain

-< (]~[p-2r

[VU~ - Vuoo[P~(s) d A d s = 0, Vp. n---,oo

1

This implies t h a t there exists a subsequence { S n } n ~ , where $n E (--1, 1) such t h a t lim ] [Vu(t~ + s~, .) - Vuoo[PdA - 0 J~

rt---, oo

and so we prove the assertion,

m

R e m a r k 6. If uoo is an isolated point of w(u), it is easy to see that in fact the above convergences hold as t --, oo (and not merely for a sequence tn -~ oc ). The proof of this convergence is an open problem in the remaining cases. In fact, in some cases the set of stationary points is a continuum (see Remark 11) and the convergence when t ~ oe is far from trivial (for some results in this direction see Feireisl and Simondon [34]). R e m a r k 7. A result on the convergence (in a suitable sense) of the free boundaries to the free boundary of the solution of the stationary problem is given in Diaz [22] (see also Gianni [36]). R e m a r k 8. The question of the approximate controllability was considered in Diaz [?] and [21]. To avoid technical difficulties, in these articles the manifold A//is replaced by an open regular bounded set ~ of IR 2 (here IR 2 can be also substituted by IR N with N C IN ) and p is taken as p = 2. As a boundary condition on (0, T) x/)f~, it is chosen the one of Neumann type since it leads to a set of test functions for the weak formulation very similar to the one corresponding to the case of a Riemannian manifold without boundary. The case of an internal control is considered by taking f ( x , t ) = v ( x , t ) x ~ with v the control to be searched, and X,~, the characteristic function of w, a given open bounded subset of ~. Thus, the new formulation is now the following: given Yo, Yd : ~ - - * IR and ~ :> 0, find v~ : w x (0, T) ~ IR such t h a t d ( y ( T : v~), Yd) ~ E where, in general, y ( T :v) represents the solution

d.I. Diaz

315

of problem

(P~)

l Yt -- Ay -t- ~(y) E QS(x)/~(y) ~- vxco Oy

in ft x (0, T )

~nn - 0

on Oft x (0, T)

y(0,-) = y0(')

on ft,

where n is the outer unit vector to 0f~. It is shown t h a t the answer to the approximate controllability question depends on the asymptotic behaviour of the nonlinearity G(y) of the equation. If, for instance ~(y) = lylP-2y, then the approximate controllability property holds when p E (0, 1] but if p > 1, an obstruction phenomenon appears, implying the impossibility of the controllability for general data. Some results concerning a special class of data for the superlinear case p > 1 are presented in Diaz [21]. We point out that in 1955, John von Neumann [57] proposed to control the climate by acting on the albedo and that this still remains a mathematical open question. Finally, we mention the "rain making" (see Dennis [17]) as a practical example of the application of control problems in environment.

3. O n t h e s t a t i o n a r y

problem

We consider the problem (PQ,f) obtained in the last subsection. Following Diaz, Herns and Tello [24], we made in this section the additional assumptions - ( ~H* ) ~ satisfies (g~) and liml~l_~~ I~(s)[ = +oc - ( H f ~ ) fc~ E L ~ ( A d ) and there exists Cf > 0 such that --[If~IIL~(.M) 0 and 6 ( - 1 0 + e) + [Ifo~llg~(~) < S0M

9

6(--10

-- C) -Jr- Cf

-- Sl?7"t"

A function u E V N L ~ ( A 4 ) is called a bounded weak solution of (PQ,I) if there exists z E L~ z(x) E/3(u(x)) a.e. x E $r such that

Vv dA + f .~ 6(u)v

dA-/

QS(x)zv dA + J " f~v dA,

for any v E V. 3.1. E x i s t e n c e of at least t h r e e s o l u t i o n s for s u i t a b l e Q V~re start with a multiplicity result given in Diaz, Herns

and Tello [24]

316


Theorem

5 - Let urn,

UM

be the (unique) solutions of the problems

(Pro)

- Apu 4- ~(u) -- Q S ( x ) m 4- foo(x) on .M,

(PM)

-- Apu + 6(u) = Q S ( x ) M + fo~(x) on .M,

respectively. Then: i) for any Q > O, there is a minimal solution u_u_(rasp. a maximal solution ~) of problem(PQ,f). Moreover, any other solution u must satisfy Um - 1 0 + eo, which is a contradiction ( u3 necessarily crosses the level-10), il C o r o l l a r y 1 - Let Re(u) = B u + C with ~ given by (1), - 1 0 B + C > 0 and s_x < M __ Then we have i) if O < Q < - ISO1BM- t - C ~ then (PQ , I) has a So -m" unique solution, ii) if -XOB+C SoM < Q < -10B+C S~m ' then (PQ,I) has at least

three solutions, iii) i f - x oSom B + c < Q, then (PQ,I) has a unique solution. R e m a r k 9. As pointed out in Hetzer [44], the uniqueness of solutions for Q small and Q large still holds if conditions (H~) and (Hg) are replaced by G c

C1 (JR),/~ C C I ( I R - { - 1 0 } ) , rn 0. Indeed, if Q is small enough, we can construct a supersolution showing that any possible solution u satisfies that U < u _< - 1 0 - e on AA. Then, any solution u must satisfy - A p t + JZ'(x,u) - f ~ ( x ) with ~'(x, u) "- G(u) - QS(x)~(u). Since ~-(x, u)is a strictly increasing function on [ U , - 1 0 - el, for a.e. x c Ad we have the uniqueness of solutions. The assumption on ~ leads to a similar conclusion when Q is large enough. 3.2. S - s h a p e d bifurcation d i a g r a m

As a continuation of the previous results we can improve the answer for the special formulation

(PQ,c)

- div(IVulP-2Vu) + ~(u) + c

c QS(x)/3(u) on M .

Following Arcoya, Diaz and Tello [6], we shall describe more precisely the bifurcation diagram and in particular, we shall prove that the principal branch (emanating from (0, G - I ( - C ) ) c IR + x L~ is S-shaped, i.e. it contains at least one turning point to the left and another one to the right. By a turning point to the left (respectively, to the right), we understand a point (Q*,u*) in the principal branch such that in a neighborhood in IR + x L~ of it, the principal branch is contained in {(Q,u) c IR + x L ~ ( A J ) / Q _ Q*}). A previous result is due to Hetzer [43], for the special case of p = 2 and /3 a C 1 function. He proves that the principal branch of the bifurcation diagram has an even number(including zero) of turning points. Our main

J.I. Diaz

319

result already improves this information showing that indeed, this number of turning points is greater than or equal to two. Semilinear problems with discontinuous forcing terms on an open bounded set and with Dirichlet boundary conditions have been considered in Ambrosetti [2], Ambrosetti, Calahorrano and Dobarro [3], Arcoya and Calahorrano [5] (see also Drazin and Griffel [31], North [59] and Schmidt [65] in the context of energy balance models). We make the additional assumption G ( - 1 0 + e) + C < S2M -(He) G(-10-e)+C>0 and { 7 ( - 1 0 - e ) + 6 Slm" We start by considering the problem with/3 a Lipschitz function (as in the Sellers model). T h e o r e m 6 - Let/3 be a Lipschitz continuous function verifying (H~).

Then E contains an unbounded connected component which is S-shaped containing (0, G-~(-C)) with at least one turning point to the right contained in the region (Q1,Q2) x LC~(2vl), and another one to the left in (Q3, Q4) x L ~ ( A J ) . Proof. Step 1. E has an unbounded component containing (0, G - I ( - C ) ) 9 We claim that the following result, due to Rabinowitz [61], can be applied to our case: "Let E a Banach space. If F 9IR x E -~ E is compact and F(0, u) - 0 , then E contains a pair of unbounded components C + and C in IR + x E, IR- x E, respectively and C + N C - - {(0, 0)}". To do so, we consider the translation of u given by v := u - G -1 ( - C ) . Obviously, v is a solution of -Apv + r - QS(x)~(v) on AA (24) where G(a) = ~(G-[-~-I(-C))-t-C and r = / ~ ( G - t - ~ - I ( - C ) ) . We define in an analogous way to E. Let E = LC~(2vt) and define F(Q, v) = ( - A p + G)-l(QS(x)~(v)). Then F is the composition of a continuous operator and a compact one (recall that p >_ 2), so F is also compact. On the other hand, if Q = 0 problem (24) has a unique solution v = 0 , so F ( ~ 0) = 0. In conclusion, E contains two unbounded components C + a n d C - on IR + x L~176 and IR- x L~ respectively and C+ N C - = {(0, 0)}. Since E is a translation of E, E contains two unbounded components C + and C - on IR + x L~(Ad) and IR- x L~ respectively, and that C + n C - = { (0, g - 1 ( - C ) ) } . Since Q _> 0 in the studied model, we are interested inC +. In order to establish the behaviour of C +, we also recall that for every q > 0, there exists a constant L = L(q) such that, if 0 < Q _< q, then every solution UQ of (PQ,c) verifies IluQllL~(~) n0},

limsup C~ "=

{p C

X : for any neighbourhood U(p) of p in X

n---~(x)

U(p) N Cn r 0 for infinitely many n}. A lemma due to W h y b u r n [70] shows that if i) lim~_.~ i n f Cn r 0 and ii) t2~__1C n is precompact, then limn_.~ sup Cn is a nonempty, precompact, closed and connected set. Proof of Theorem 7. The method of super and sub solutions proves that if Q > Q2, then there exists a solution of (PQ,c) greater than - 1 0 + e. Analogously, we know that if 0 _< Q < Q3, then (Po) has a solution smaller than - 1 0 - e. It is clear that these functions are not the unique solutions of (PQ,c) in those intervals and that the uniqueness holds at least in the Q-intervals [0, Q1) and (Q4, oc). Since we can not apply directly Rabinowitz theorem to our problem, we consider the family ]~n - - n ( I 1 --1 ), n C ]hi to approximate/3 in the sense of maximal - (I - n/3) monotone graphs when n --~ oc. Notice that since/3verifies (H}), then/?n is a Lipschitz bounded nondecreasing function (see Brezis [14]) and that fin(S) = fl(S) for any s r [--10 -- e,--10 + e + ~M] , Vn. Let u~ be the solutions of the approximated problem (P~))

- ApU~ + G ( u ~ ) + C = QS(x)/3~(un) on .Ad

and let En the bifurcation diagrams for (P~)). Let us denote by Sn the component of En containing (0, G - I ( - C ) ) . By Theorem 6, every Sn is an unbounded, connected and S-shaped set. First of all, we are going to prove that lira sup Sn is a connected and closed set of solutions to problem (PQ). In order to apply W h y b u r n ' result, we consider the sets C j (j > Q4) defined as S~ A ([0, j] x L~(f~)), Vn C IN containing (0,~-l(-C)). It is

322


easy to see that these sets are connected and that i) is verified. Let us check (ii),Un~__lC~ is precompact. Since X is a Banach space, it suffices to prove that every sequence {(Ql, ut)}l~IN C Un~__IC~ contains a subsequence {(Qza,uta)} converging in X. From Qt c [o, j], there exists Q E [O, j] and a subsequence of {Qt}which we still call {Ql}, such that Qz --+ Q. On the other hand,ul is a solution of the problem

Taking ul as a test function in this equation, we obtain the estimate

I V u , ] ' d A -4-

-~lm[2dA 2, there exist u e L~(;~4) and asubsequence {ula} of {ul} such that uzk --~ t i n L~(jM). If p = 2, then {u~} is a bounded sequence in the Sobolev space H2(.M). From the compact embedding H2(Ad) C C(Ad), we deduce the existence of a subsequence {ulk} and u C C ( M ) , such that uzk ~ u in L~(Ad). Thus t2~__lC j is precompact. Then by Whyburn's result CJ ~_ lim~__.~ sup C~ is a connected and compact set in X. Moreover, since every S~ is unbounded and fixed Q, the solutions UQ are uniformly bounded in L ~ (AX), for Q < Q, we have that C~ N ( { j } • L ~ (AJ)) r 0, for all j c IN. Now, we prove t h a t the set C y is contained in E. Let us see that for every Q c [Q1, Q4], we have that every (Q, u) E C j verifies that u is a solution of (PQ) (notice that it is true for every Q e (0, Q1] u [Q4, +cr from C~ - C j in these intervals). Let (Q, u) c C j - lim~__.~ sup C3n, that is, there exists a subsequence of (Q~, u~) e C~ such that (Qua, Una) ~ (Q, u) in IR• From estimate (25) and the compact embedding H2(Ad) c L~(AA) (for p = 2 ) a n d V c L ~ ( , M ) (for p > 2), we deduce the existence o f t c L ~ ( A J ) and a subsequence of { (Qua, u~a )} which we call { (Qna, Una ) }, such that

(Q~k,Unk) ~

(Q,u)

in IR • L ~ ( A J ) ,

Since/3~ ~ / 3 in the sense of maximal monotone graphs of IR 2, we have that ~na (Una) ---" Z C /3(U) weakly in L2(3A). Using a Minty's type argument we deduce that u is a solution of the problem (PQ,c). Thus (Q, u) E E and CJ c E. Since for all n and j, C ~ N ( { j } x L~(AA)) # O, there exists {(j, u~)}~e~ such t h a t (j, Un) C C j , that is,

- A p U n + ~;(Un) -- j S ( x ) ~ ( u n )

- C

in Ad.

Using that the operator (Ap + 6)-~ is compact in L ~ ( M ) , there exists a subsequence una --* U in L~(A/I). Thus (j,u) E C j and C j A ({j} x

J.I. Diaz

323

L~(A/I)) r 0. Since j > Q4, Uj is the unique solution of (PQ,c). On the other hand, we know that EN(j, ec) x L~(A//) = EMA(j, cxD)x L~ So, we have obtained a connected unbounded set which starts in (0, G - I ( - C ) ) . The proof ends with the argument used in the proof of Theorem 6 for

Q2 < Q3. m Remark 10. We point out that our results remain true for the more general equation

-div(k(x)lV~l~-2V~) + ~(~) + C e QS(x)~(~) on M, where k(x) is a given bounded function with k(x) _> k0 > 0 a.e.x c A/l, representing the eddy diffusion coefficient. When Ad = 81, it is usually assumed that S(x) = S().) and k(x) = k(A, r with A the latitude and r the longitude. So, in that case, the corresponding solutions are not r Remark 11. By using a shooting method, it is possible to show (see Diaz and Tello [27] that there exist infinitely many equilibrium solutions for some values of Q when we study the one-dimensional problem

-(lu'lp-2u')'+Bu+C c Q~(u) x E (0,1), (P1,Q,C)

u'(0) = u ' ( 1 ) = 0 .

If Q1 < Q < Q2 then (P1,Q,C) has infinitely many solutions. Moreover, there exists K0 C IN such that for every K E IN, K >_ K0 C IN there exists at least a solution which crosses the level U K = - - 1 0 , exactly K times. Remark 12. After my lecture at the Coll~ge de France, Professor J.L. Lions pointed out to me the reference Rahmstorf [63] where a S-shaped diagram bifurcation curve arises in the context of the Atlantic Thermohaline Circulation in reponse to changes in the hydrological cycle.

References [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18, 4 (1976), 620-709. [2] A. Ambrosetti, Critical points and non linear variational problems. Supplement au Bulletin de la Societ~ Math~matique de France. M~moire 49 (1992). [3] A. Ambrosseti, M. Calahorrano and F.Dobarro, Global branching for discontinuous problems, Comment. Math. Univ. Carolinae, 31 (1990), 213-222. [4] S.N. Antontsev and J.I. Diaz, New results on localization of solutions of nonlinear elliptic and parabolic equations obtained by energy methods, Soviet Math. Dokl., 38 (1989), 535-539.

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327

J.-L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237288. J.G. Mengel, D.A. Short and G.R. North, Seasonal snowline instability in an energy balance model, Climate Dynamics, 2 (1988), 127-131. A.M. Meirmanov, The Stefan problem, Walter de Gruyter, Berlin-New York, 1992. M. Nakao, A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan, 30 (1978), 747-762. J. von Neumann, Can we survive technology?, Nature, 1955. (Also in Collected works. Vol VI, Pergamon, 1966.) G.R. North, Multiple solutions in energy balance climate models, in Paleogeography, paleoclimatology, paleoecology, 82, Elsevier Science Publishers, B.V. Amsterdam (1990), 225-235. G.R. North, Introduction to simple climate model, in Mathematics, climate and environment, (J.I. Diaz and J.-L.Lions eds.), Masson, Paris (1993), 139-159. G.R. North, J.G. Mengel and D.A. Short, Simple energy balance model resolving the season and continents: Applications to astronomical theory of ice ages. J. Geophys. Res., 88 (1983), 6576-6586. P.H. Rabinowitz, A global theorem for nonlinear eigenvalue problems and applications, in Contributions to nonlinear functional analysis, (E.H. Zarantonello ed.), Academic Press, New York (1971), 11-36. J.M. Rakotoson and B. Simon, Relative rearrangement on a measure space. Application to the regularity of weighted monotone rearrangement. Part. II. Appl. Math. Lett., 6 (1993), 79-82. S. Rahmstorf, Bifurcations of the atlantic thermohaline circulation in reponse to changes in the hydrological cycle, Nature, 378 (1995), 145149. W.D. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, J. Appl. Meteorol., 8 (1969), 392-400. B.E. Schmidt, Bifurcation of stationary solutions for Legendre-type boundary value problems arising from climate modeling. Ph.D. Thesis. Auburn Univ. 1994. J. Simon, Compact sets in the space LP(0, T; B), Annali Mat. Pura et Appl., CXLVI (1987), 65-96. I. Stakgold, Free boundary problems in climate modeling, in Mathematics, climate and environment, (J.I.Diaz and J.-L.Lions eds.), Masson (1993), 177-188. P.H. Stone, A simplified radiative-dynamicM model for the static stability of rotating atmospheres, J. Atmos. Sci., 29, 3 (1972), 405-418.

328


[69] L. Tello, Tratamiento matems de algunos modelos no lineales que aparecen en climatologia. Ph. D. Thesis. Univ. Complutense de Madrid, 1996. [70] G.T. Whyburn, Topological analysis, Princeton Univ. Press, Princeton, 1955. [71] I.I. Vrabie, Compactness methods for non linear evolutions, Pitman, London, 1987. [72] X. Xu, Existence and regularity theorems for a free boundary problem governing a simple climate model, Aplicable Anal., 42 (1991), 33-59. J. Ildefonso Diaz Departamento de Matems Aplicada Facultad de Matems Universidad Complutense de Madrid 28040 Madrid Spain E-mail: [email protected]

Studies in M a t h e m a t i c s a n d its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 15 UNIQUENESS AND STABILITY IN THE C A U C H Y PROBLEM FOR MAXWELL A N D ELASTICITY SYSTEMS

M. ELLER, V. ISAKOV, G. N AKAMURA AND D. TATARU

1. Preliminaries By x = (XO, X l , . . . , X n ) we denote t h e coordinates in R • R n. For t h e differentiation o p e r a t o r s we set Oj - O / O x j and D j = 7l O / O x j S o m e t i m e s we shall call x0 the time coordinate, a n d use t h e alternate n o t a t i o n t for it. Given a positive scalar function a in R n we define the scalar associate wave o p e r a t o r in R • R n by ['-1a - - aO2t - A .

By H k we denote the classical Sobolev spaces, with the norm d e n o t e d II-llkOften we shall use on these spaces t h e weighted norms,

I]u]]2;~--/ ~

7"2(k-]a])lSaaul2dx

(1.1)

]~t 0 V (x,~) e Fs, ~ r O, p(x,~) = {p,r

=0

(2.2)

D e f i n i t i o n 2.2 - a) A smooth function r is strongly pseudo-convex with

respect to P on F if it is pseudoconvex and 1

i--~{I~r162

> 0 V ( x , ~ ) E F, ~ ~ 0, T > 0, p r

0.

(2.3)

b) A smooth oriented surface S is strongly pseudo-convex with respect to P on F if it is pseudoconvex and 1 ~=0, T>0, i-~ {p-c, p,}(x, ~) :> 0 V (x, ~) E Fs, pc(x ' ~) = {pc, r

~) = 0.

(2.4)

M. Eller, V. Isakov, C. Nakamura and D.Tataru

331

The relation between pseudoconvex functions and surfaces is obvious, Remark 2.3. a) An oriented surface S is (strongly) pseudoconvex with respect to P on F iff it is a level set of a function r which is (strongly) pseudoconvex with respect to P on F, so that V r ~ > 0 on F. b) Moreover, if the oriented surface S = {r = 0} is (strongly) pseudoconvex with respect to P on F then e ~r is (strongly) pseudoconvex with respect to P on F for large enough A. c) The (strong) pseudoconvexity condition for both functions and surfaces is stable with respect to small C 2 perturbations. For second order operators the condition (2.4) is void for noncharacteristic surfaces, therefore P r o p o s i t i o n 2.4 - Let P be a second order operator with real principal symbol. If a noncharacteristic oriented surface S is pseudoconvex with respect to P on F then it is strongly pseudoconvex with respect to P on ['. Now we discuss the corresponding Carleman estimates. Since this is all we use later on, in the sequel we only refer to second order partial differential operators P(x, D) with real principal symbol. The classical result (see [4]) is the following one: T h e o r e m 2.5 - Let f~ be a compact subset of R n+l. pseudoconvez with respect to P in f~ then rile'Cull 21;7" < Clle~r

If ~b is strongly

2

(2.5)

whenever u E H 1 is supported in t2 so that the R H S is finite. The substitution v = e~r

reduces the estimate to

2 which is essentially a subelliptic estimate for PC. The strong pseudoconvexity condition therefore expresses the subellipticity of PC in terms of its symbol. If instead D is a bounded domain with smooth boundary and u is a function in Q, then the appropriate estimate should include the Cauchy data of u on the boundary, see Tataru [ll]. T h e o r e m 2.6 - Let f~ be a compact subset of R n. If r is strongly pseudoconvex with respect to P in f~ then

lle' ul121;T,fl --< C(lleTCPull~0)

-1- T(lte~-r

Jr-Ile~'~b~-~t1

whenever u E H 1 is supported in ft so that the R H S is finite.

(2.6)

332

Uniqueness and stability in the Cauchy problem ...

In effect such an estimate might hold even with only one boundary data on the RHS, provided that the corresponding boundary operator satisfies a strong Lopatinskii boundary condition with respect to de, see [11]. If the coefficients of P are analytic with respect to some variables then the unique continuation results can be improved. We state here the Carleman estimate corresponding to the simplest result of this type, namely for the wave equation with time independent coefficients. In this case, we set F = {~0 - 0}, where ~0 is the time Fourier variable and consider functions and operators in R n+l. 2.7 - Let P be a second order hyperbolic operator with time independent coefficients. Furthermore, let r be a quadratic function in which is strongly pseudo-convex with respect to P on F. Then there exist d > O and C > O such that

Theorem

T]le-~D2eTdpul121;T--~ C ( l l e - ~ D ] e r C P ( x , D ) u l l ~ + ]leT(dp--de)?-t[121;7")(2.7) for all u E H 1, supported in ~, provided 7 is large enough and e is su]ficiently small. This result was proved in [10]. In effect it holds even if we only assume that the coefficients of P are analytic in time and that r is analytic. However, the proof is substantially simpler in the special case considered above. The critical point is the conjugation argument, which needs to be carried through also with respect to the Gaussian. With the substitution w - e~+u the estimate (2.7) reduces to

7-][e-~D~w[I 21;r --< C(]]e-~D2op(x,D + iT-Vr

2+[le-derw]] 21;T )

(2.S)

Furthermore

e ~V2ot -- (t + i e - D o ) e - ~ D~ T Since ~zr is linear, this implies that

e ;-~D~op(x,D+iTVr

P(x,D+iTVr162

(2.9)

Hence if we set v = e ~ V2ow then (2.8) reduces to T[[V][ 2I;T --< C[[Pe,r

2 -~-[[e--deTw[]

2I;T)

with P~,r - P(x, V + i ~ - V r

e0t(Vr

If e is small enough then P~,~ is a small perturbation of PC. Then the pseudoconvexity condition for r implies some subellipticity for P~,r in the region [~0[ _< cT. Outside this region, the Gaussian yields exponential decay in z and leads to the last RHS term.

M. Eller, V. Isakov, G. N a k a m u r a and D.Tataru

333

3. Uniqueness and stability for principally scalar systems Here we apply the results in the previous section to the systems of the type

Pjuj + bj(z, t; Vu) + cj(x, t; u) = f j, j = 1, ..., m

(3.1)

Here Pj are second order operators with real principal part and C 1 coeffiON3 cients and other coefficients in Lzo~,bj, cj are linear functions of Vu, u with L~o~-Coefficients , u = (Ul, ..., urn). First we obtain the Carleman estimates for such systems. T h e o r e m 3.1 - Let r be a smooth function which is strongly pseudoconvex with respect to all Pj in a given compact set K . Then

7-11e"%112 < clle~-~fll 2 1,7"

--

,

r > r0

(3.2)

whenever u is supported in K and solves (3.1). Proof. First we apply the Carleman estimates in Theorem 2.5 to uj with respect to Pj. This yields

~tIe~%jll~,~ ~

clle~Pj~jll 2,

and further

~-II~-~ujll~ < cllC~jll 2

2

Summing with respect to j, the first terms on the right are absorbed on the left for sufficiently large ~- and we obtain (3.2). m Next we consider the unique continuation problem for the corresponding homogeneous problem

Pjvj + bj(x, t; Vv) + cj(x, t; v) = O, j = 1, ..., m

(3.3)

T h e o r e m 3.2 - Let S = {r = 0} be an oriented surface which is strongly pseudoconvex with respect to Pj for all j. Then we have unique continuation across S for Hloc solutions u to (3.3). More precisely, given any xo E S and v E Hloc solving (3.3) near xo which is identically zero in {r > 0} near S, this implies that v = 0 near xo.

Proof. W i t h o u t any restriction in generality we assume t h a t x ~ = 0. Since S is strongly pseudo-convex, by R e m a r k 2.3 we can also assume t h a t r is strongly pseudoconvex with respect to Pj. Consider the modified function ~ ( x , t) = r t)+e(r2-1xl2). For small e, r is still strongly pseudoconvex with respect to all Pj near x0. Furthermore,

{r > o} o {r < o} c B(o, ~)

334

Uniqueness and stability in the Cauchy problem ...

Hence, by choosing r sufficiently small we insure that (3.3) holds in {r > 0}. Now let X be a cutoff function which is 1 in ~ > 0 and 0 in ~ < - e r 2. Then the function u = Xv solves (3.1) with f E L 2, supported in supp VX C {r < 0}. Apply the Carleman estimates (3.2) to u. We obtain rlle'r

.,- _
0}. Since r > 0 this concludes the proof, m The next step is to establish the corresponding stability estimate. For this it is more interesting to work with boundary value problems. Thus, let ft be a bounded domain in R ~+1 with C 1 boundary Oft. Let S be a part of 0ft where one prescribes the Cauchy data for u

u = go, O~u = gl

onS

(3.4)

Given a smooth function r let

~E = a n { c

< r

O~E -- Of] n {E < r

T h e o r e m 3.3 - Let r be strongly pseudo-convex with respect to all operators Pj on-~ and 0~o C S. Let u C H I ( ~ ) solve the Cauchy problem (3.1),(3.~). Then with some positive constants C, A = A(e) C (0, 1) we have [[Ulll,a~ _< C(e)(F + F ~ M 1-~)

where F = Ilg0l[1,s + Ilglll0,s + []f[[0,ao,

M = II~PIl,ao-

Proof. We want to use the analogue of the Carleman estimate (3.2) but for boundary value problems,

Tlre~uf[21;-r,a

-< c ( [ l e ~ r

0,a -I-

2 .,oa T(llerCU[ll;

N O,Of~)) (3.5)

+ II~ ' ~ 0u II2

for u solving (3.1) in ft. This follows in the same way as (3.2) if we use (2.6) instead of (2.5).

M. Eller, V. Isakov, G. Nakamura and D.Tataru

335

To proceed we need to localize to f~0. Hence, introduce a cut-off function X E C ~ ( R n + I ) , X = 1 on f~/2 and X - 0 outside f~0. T h e n work with Xu. After commuting we get

P j ( x u j ) -- x f j + A1;ju where A1;j is a linear differential operator of first order with measurable bounded coefficients depending on c, supported in ft0. Hence, applying (3.5) to Xu we obtain the inequality 1;"r

_ ~-0(e). Increasing C(e) we can take T0(e) = 0. Hence we can minimize the right hand side with respect to T. If M < F , then Theorem 3.3 follows if we simply set T -- 0. Otherwise, we choose T so t h a t

Fer(e~-E/2) = Me-r~/2 and then the right hand side equals

C(e)M1-;~F ~,

E

A = 2---~

and we obtain the conclusion of the theorem with A = e/(2(I)).

ll

If a portion of the b o u n d a r y is strongly pseudoconvex, then the above result can be used locally near S. The corresponding uniqueness result follows easily from Theorem 3.2. C o r o l l a r y 3.4 - Assume that the surface S C Oft is strongly pseudo-convex with respect to P j , j = 1,...,m. If u E HI(f~) solves the system (3.3) with

336

Uniqueness and stability in the C a u c h y p r o b l e m ...

f -- 0 and has zero Cauchy data (3.~) on S then u - 0 in some neighborhood ofS in~.

Another corollary gives an explicit description of an uniqueness domain when S = ~ • ( - T , T) E C 2 in the two following cases: 1) ~/ = 0fY, the space origin is i n ~ ' a n d 2 ) ~'C {-h<xn < 0 , x 2 + . . . + x n2 _ l < r 2 } , S = O~ ~ N {Xn < 0}. Here we let x ~ = x r R n. To formulate the result we need the weight function r t) x 2 + ... + x n2 _ 1 + (Xn --/3) 2 -- 02t2 -- S to define n

>

C o r o l l a r y 3.5 - A s s u m e that the coefficients a(j), j = 1, ..., m of the principal part Pj - [:]a(j) of the system (3.1) satisfy the following conditions 02a(j)(a(j)+Ota(j)/2+a-1/2(j)ltVa(j)l

02a(j) _< 1

) < a(j)+l/2x.Va(j)-l/2/30~a(j)

on ~ , a ( j ) E C 1 ( ~ )

(3.6)

and ft' c B(O; OT), ~ - s = O in case 1) and h(h + 2~) < 0 2 T 2, ~2 + r 2 = s in case 2). A s s u m e that u C H I ( ~ ) solves the system (3.1). Then Ilul[1,a~ 0) associated with equation (4.8) for profiles U(y) that contain at least one inflection point. Meshalkin and Sinai [18], followed by Yudovich [19] investigated the instability of a viscous shear flow U(y) = sin m y using techniques of continued fractions. More recently Friedlander et al [13], [20], [21] showed that these techniques could be used for the invisid equation (4.8) with U(y) - s i n my. Eigenfunctions are constructed in terms of Fourier series that converge to C~-smooth functions for eigenvalues a that satisfy the characteristic equation. We write OO

(~(Y) =

E

an

e iny

.

(4.10)

n=--oo

The recurrence relation equivalent to (4.8) yields the following tridiagonal infinite algebraic system" dn+m = 13r~(a) dn + d n - m , n C Z

where /3n(O') :

2a

(k 2 + n 2)

T"

k2 + n 2 - m 2

(4.11)

(4.12)

and dn - an(k 2 + n 2 - m2).

(4.13)

The system (4.11) is treated using continued fractions to yield the characteristic equation for cr namely (

1 )( 1 ) flJ + [flj+m,~j+2m,-.-] flj-m + [/~j--2m,/~j--am,..-] + 1 = 0 (4.14)

for each integer j - 0, 1 , . . . [m/2]. Here [ , . . . ] denotes an infinite continued fraction, i.e. =

~jTm + /3j+2m + f~j+ 1 a~,, +...

(4.15)

In [20], [21] an analysis is given of all the roots a with Re a > 0, of equation (4.14) for each fixed integer m and 0 > 1, was demonstrated in [21] using homogeneization techniques to compute the spectral asymptotics. It is proved that one root of the asymptotic characteristic equation is given by a / k = + v / ( U 2)

as rn --, c~

(4.16)

where ( 9 } denotes the 27r-average with respect to the fast variable m y . Numerical analysis is used to describe the qualative behaviour of the distribution of roots of the asymptotic characteristic equation in the parameter space of the wave numbers n and k. The following is an example of a stream function 9 satisfying an equation of the form (4.3) which exhibits both the features of exponential stretching at a hyperbolic stagnation point and oscillatory shear flow behaviour. We consider ~= l (cos(x+my)+acos(x-my)) (4.17) m where a is a constant such t h a t 0 < a < 1. This flow has hyperbolic points at x + m y = 2n~r , x -

m y = (2j - 1 )

~r.

Hence by the results of Section 3 there is a non empty unstable essential spectrum associated with this flow. The fluid Lyapunov exponent # can be calculated explicitly in this example to give # -- 2 a 1/2 .

(4.17a)

Again homogenisation techniques can be used to demonstrate the existence of unstable eigenvalues for equation (4.7) with @ given by (4.17) with m >> 1 [22]. We introduce a change of variables ~ = m y and write (4.7) in the form 0

+ x

02 )

[

o]

(sin (x + r/) - a sin (x - r/)) ~0 _ (sin (x + r/) + a sin (x - r/)) N

[0 2

02

~-Sx2+m2~2+(1+m

]

2) r (4.18)

We seek Block eigenfunctions of the form r = e ipn/m G ( x , rl)

(4.19)

On the unstable spectrum of the Euler equation

360

where p is an integer such that p 0 there exists a 5 > 0 such that Ilw(O)llx < 5 implies (i) there exists a unique solution w(t) e L~((0, oc); X) N C([0, oo), Z) and

On the unstable spectrum of the Euler equation

362

(ii) IIw(t)lIx < : for a.e. t e [0, cxD). The trivial solution w = 0 is called nonlinearly unstable if it is not stable.

Note: by this definition the "blowing up" of a solution is a particular case of instability. T h e o r e m 5.1 - Let (5.1) admit a local existence theorem in X . and L satisfy the following conditions. (1)

Let N

IIN(w)liz ~ Collwllx Ilwllz fo~ w e X with [Iwllx o.

(5.2) (2) A spectral "gap" condition, i. e. a(e Lt) = a+ U a _ with a+ ~ r where ~+ c (z e C l ~ M~ < Izl 0, donc y 6 ftn, et donc f t ~ = IR 3. Le m~me type de raisonnement p e r m e t de montrer que si a = - o o , alors flo~ = O. Reste donc ~ 4tudier le cas off a c IR. Si K est un compact de H ~'~, alors par d4finition de H ~'~ en (10), on a VyeK,

V~(x~).y=-a+m(y),

avec

re(y) >0,

donc pour n assez grand, uniform4ment en y 6 K , on a ~(Xn + h~y) > O. Inversement, si K est un compact de c H ~'~, alors pour tout y E K , pour n assez grand uniform4ment en y, on a ~(xn + hny) < 0, et le r4sultat s'en d4duit imm4diatement" f t ~ -- H ~'~. Pour conclure la d4monstration de la proposition, consid4rons une donn4e concentrante (p, r hn, xn, tn), d'onde de concentration lin4aire associ4e Pn, et soit p,~'~ l'onde de concentration lin4aire associ4e ~ ( ~ ' ~ , ! ~ ' ~ , h~,

Xn,tn), avec (9~a'w, ~)a,~o) d4=f(~H~,~qp, 1H~,~ r Par conservation de l'4nergie, on peut 4crire

2

Xn,,)) _ 7)a(99( X ~hnX n )) I EO(Pn - p~'W) : hn3 /]R IV (~)f~(q~pc~'w(x - xn 3 hn .... V(lfl(x)r

+h~3/~3

dx

x-xn X--Xn )!2 An ) - lft(x)~)(- hn ......) dx

_

dy

I1 suffit donc ~ pr4sent de d4montrer la convergence de Pan f v e r s / ) a ~ f dans/:/1(IR3), Mnsi que celle de l a n g vers l a ~ g darts L2(IR3), pour t o u t couple (f, g) c/2/1 x L2(IR3). 2.7 - Soit fin un domaine convergeant vers f l ~ , dans le sens de Ia ddfinition 1.3, off f~oo est un demi-espace, @ ou IR 3. Alors on ales propridtds suiyantes:

Lemme

(ii)

Vf E/2/1 (IR3),

(iii)

Vg e L2(IR3),

Ddmonstration du lemrne.

lirn 79anf = T)a~ f lim l a . . g = l a ~ g

n---,cx~

dans dans

IJt1(IR 3) ; L2(IR3).

Les cas off f t ~ = IR 3 ou 0 sont 4vidents. Dans le cas off fto~ est un demi-espace, il est facile de d4montrer le point (i)" on

376

Ddcomposition en profils des solutions de l'dquation des ondes...

suppose par exemple que f t ~ est d~fini par f t ~ = {x E IR 3 ix3 > 0}, et l'on consid~re une fonction p C C ~ ( f ~ ) . Alors pour toute fonction u c t:tl (IR3) support~e dans ~oo, la fonction u ~ d~f = u . p ~, oh pe d~f = 1/E3p(./E), tend vers u dans/:/1(IR3), et est support~e dans f t ~ , donc (i) est d~montr~. Le point (ii) se d~montre de la mani~re suivante: on constate facilement d~f

que pour tout domaine M, et pour toute fonction f , la fonction v = T ' M f est d~finie comme la solution de Av--Af

dans

M

vlo M ~- O. d~f

Alors il suffit de d~finir v~ = P ~ n f , et soir v une de ses limites au sens des distributions; on a pour toute fonction r C C ~ ( f 2 ~ )

IR3AV n-~ dx =/~ts Aye dx, puisque pour n assez grand, on a r ~ C ~ (gt~). P a r passage ~ la limite faible dans la condition aux limites de Dirichlet, on en d~duit que v = 7 ) a ~ f . Finalement la convergence forte est due au fait que

IVv.I 2 dx

=

/IR ~ v n A ~ n dx

=

-

-

Vvn" Vfdx, 3

f

f

et donc comme ] V v . V f dx - - ] [Vv[ 2 dx, le r~sultat suit. JIR 3 J]R3 Le point (iii) s'obtient de mani~re similaire, nous laissons les d~tails au lecteur. [] Ce lemme termine la d~monstration de la proposition 2.6. [] Le r~sultat suivant sera utilis~ fr~quemment dans la suite; sa d~monstration est du m(~me type que les calculs ci-dessus conduisant au point (ii) du lemme, nous ne le d~montrerons donc pas ici. 2.8 - Soit Mn un ouvert de ]R 3, de limite M. Soit (fo, fn~) une suite bornde de E ( M n ) , convergeant faiblement (resp. fortement) vers un couple (fO, f l ) dans/-:/I(IR 3) x L2(IR3), avec (fO, f l ) C /_:/1 X L 2 ( M ) . Alors la solution de Proposition

{ Dfn--O

dans

IRt x Mn ,

f nlIR,, • OMn = 0

O fn)t =o = (fo, fx)

I. Gallagher et P. Gdrard

377

converge faiblement (resp. fortement) vers f duns Llo~(IR; /[/l(]n3)), e t 0tfn converge faiblement (resp. fortement) vers Otf dans L~oc(IR; L2(]R3)), off dans I R t • ff~txOM = 0 (f, Otf)lt=O -- (fo, . D'autre part, si M~ est l'extdrieur d'un domaine strictement convexe, alors la convergence a lieu aussi dans L~o~(~;

fl~

2.2. Propagation de la (h~)-oscillation Commenqons par rappeler la d6finition d'une fonction strictement (hn)oscillante. Dans la suite nous noterons A l'op6rateur auto-adjoint non born6 suivant: 7)(A) = { (u,/t) c/:/1 • L2(gt) !/t E H01(gt), An c L2(fl)}, Ad~f ( 0 = A

1) 0 "

D ~ f i n i t i o n 2.9 - Soit (hn) une suite de rdels strictement positifs, tendant vers O, et soit (fn,gn) une suite bornde dans /2/1 X L2(]R3). La suite (f~, gn)est dite (h~)-oscillante si lira lim ,,ll[[llAl>hn---z-(fn'gn)[[ftX• -,

R---~cx) h---~0

La suite (fn, g~) est dite strictement (h~)-oscillante si elle vdrifie d'autre part s--~0 h - - * 0

-- ~'s

'

L 2 ( I R a)

Remarque. Nous dirons qu'une suite (v~) bernie dans C~ T],/:/l(a)) telle que Otv~ est born6e dans C~ ~st (strictement) (h~)oscillante si (Vvn, OtVn) l'est au sens de la d~finition 2.9. Nous ne rappellerons pas ici la notion de composante strictement (h~)oscillante d'une suite bernie de L2(IR3), et renvoyons ~ [1], Lemme 3.2 (iii), pour une d~finition. Enon~ons maintenant le %sultat suivant, de propagation de la (h~)-oscillation stricte. Nous laissons sa d~monstration au lecteur (voir [4]). P r o p o s i t i o n 2.10 - Soft (Pn,~n) une suite bornde de E(ft), telle que la suite ( ~ 7 ~ n , ~ n ) est (strictement) (h~)-oscillante. Soit v~ la solution de (5) associde; aIors (Vvn, OtVn)(t) est (strictement) (hn)-Oscillante, uniform~ment en t pour tout t E IR.


378

D'autre part, soient V n et vn les solutions de l'dquation des ondes lindaire, 1 assocides a des donndes (VOn,Vn) et (v~'~ ~1) respectirement, off (v~On,~ln) est o v~). Alors pour toute la composante strictement (hn)-oscillante de (%, suite (tn), ( V ~ (t~), Ot~n (tn)) est la composante strictement (h~)-oscillante de (VVn(tn), OtVn(tn)). 2.3. Un r6sultat de non concentration Nous allons dans cette section donner quelques idles de la d~monstration du r~sultat suivant.

P r o p o s i t i o n 2.11 - Soit pn une onde de concentration lJndaire associde une donnde concentrante (~, ~, hn,xn, 0). Alors pour tout intervalle de temps I c IR borng, pour route suite (sn) telle que lim E~/hn = +oc, on a n---*c~

lim [[PnlIL~(I\[-~,~]),L~(a) = O.

n-,c~

Ddmonstration. Nous allons en donner le principe g~n~ral, mais n'entrerons pas dans les d~tails des calculs (on renvoie g [4] pour les arguments precis). Commenqons par remarquer que la proposition 2.11 est d~montr~e si l'on montre que (i) pour tout temps T 7~ 0, et pour toute suite T~ ~ T, on a lim [Ipn(Tn)llL6(~) = 0;

n - - - - ~ (:X)

(ii)

pour toute suite r lim

--~ 0, avec r flP (

~ +ec, on a

)ltLO( ) - 0 .

n---~ oo

Le cas (ii) peut ~tre consid~r~ comme une version d~g~n~r~e du cas (i), et se d~montre de faqon analogue, mais plus simple (par changement d'~chelle); nous n'y reviendrons donc pas. La m~thode de d~monstration de (i) s'appuie sur le principe de concentration-compacit~ de P.-L. Lions (voir [11]-[12])" introduisons la densit~ d'~nergie

en(t,x) d~f (latpn(t,x)12 + ]V~p~(t,x)12) dx, qui v&ifie

Oren

--

div~ (0tp~ V ~p~ ).

Alors si eo~ est un point d'adh~rence de e~, alors eo~ est continue en temps, valeurs mesure, et par le principe de concentration-compacit~, il suffit de d~montrer que

eoo(t)({xo}) -- O,

Vxo e ~,

Vt # O.

(11)

I. Gallagher et P. G~rard

379

Le calcul de e ~ se fait par utilisation des mesures semi-classiques de Wigner (voir [7], [8], [13])" soit # la mesure semi-classique associ~e ~ (Vt,xpn). I1 est bien connu que

L'r X ]P,.~

#(t,x, dT, d~) R/h~J'~-J)[[L2(aa) = 0, R---*c~ n---,c~

I. Gallagher et P. Gdrard et

383

~-~olimlimsuplln_~r ll~l~/h(~)~)llL~(~) ~ =

et de m~me en remplafant j par k. Alors lim Ill2 )f(k) IIL~(a) = 0.

n ---+(2~

Ce lemme conduit ~videmment directement s la proposition 3.12, puisque les suites Vv(J)(t (j), .)et Vv(k)(t (k), .), pour des suites quelconques (t~)) et (t (k)) de r6els, v~rifient les hypotheses du lemme 3.13 et la proposition 3.12 est d6montr6e, m

Ddmonstration du lemme 3.13. Supposons par exemple que lim h(nk)/h(~j) = n --+cx)

O. Commen~ons par supposer que les fonctions f(j) et f(k) ont un spectre tel que respectivement h(nJ)i~i C C(oj) et h(~k)]~I ~ C(ok), o~ C(Oj) et C(k) sont des couronnes fixes de IR. On a alors

Ilf~(J)IIL~oR~) _ et

Ch~)liVf(~J)llg~(~)

ilf~(k) IIL~(~) --< Ch(~k)IIVL(k) IIL~(~)

et donc

IIL(k)I1L~(~) _< Ch(~k).

IIL(5)llL~(~) ~ Ch(nj) et

Alors par l'in~galit~ de Hausdorff-Young, on a

Donc finalement 1/2

1

h~ ) ( h ( ~ )

0. On choisit alors la donn6e concentrante (99(1), ~p(1)x(1),t(1) ) tel que 1 -~(~(Vh)

IIV99(1) I]~2(IRa) -~-I]~Z;(1)l122(IR3) > et, quitte A extraire une sous-suite,

D(hl)Vh __~ (99(1)r Alors Ph(1) est l'onde de concentration lin~aire associ6e ~ la donn~e concentrante (99(1), ~2(1), h,x(1), t(1)). Le lemme suivant se d~montre par un simple calcul, utilisant la proposition 2.6 et le lernme 2.7. L e m m e 3.16 -

Soit w (h1) d=e f y

EO(Vh)

h - - ~ h,(1)"

Alors

-- E0(v(h 1)) -}- E0(W(h 1)) nu o(1),

h --, 0.

On poursuit alors le d6veloppement par rdcurrence: on suppose que

K-1 Vh(t,x) = ~ p(k)(t,x) -t- W(K-1)(t,x), k=l avec orthogonalit~ en ~nergie, off chacun des

p(hk)

est une onde de con-

centration lin~aire avec orthogonalit6 des temps t(hk) et des coeurs x (k) de concentration. Alors on exhibe par la m~me technique que ci-dessus une nouvelle onde de concentration lin~aire p(hK), de la mani~re suivante: on peut, d'apr~s le lemme 3.15, supposer que 5(w (K-l)) > 0, et l'on d~finit la donn~e concentrante (~o(K), ~ ( g ) ,

x (K), t (g))

I[V99(K) II~2(IR3)-Jr-[Ir

de telle sorte que 1

3) :> ~(~(W (K-l))

(16)

et, quitte 5~ extraire une sous-suite,

D(h/{) Wh(K-l) __x (99(K) ~)(/6)). Alors p(/{) est l'onde de concentration lin~aire associ~e ~ la donn~e concentrante (99(K), g2(K),h,x (K),

t (K)),

et l'on a, comme dans le lemme 3.16

386


ci-dessus, K

Eo(vh) = E E~

+ E~

+ o(1),

h --~ 0,

k--1

K -- Vh-- E k=l

vec

L'orthogonalit~ des temps t (k) de concentration est due ~ la proposition 2.11. Celle des coeurs est laiss~e au lecteur: elle se montre par un argument classique de changement d'~chelle en espace, en utilisant la proposition 2.8. Reste alors ~ d~montrer que VT > 0,

lim lim ,,IIw[K) ,, ,~ IIL~([-T, TI,L6(a)) - - O .

K-.cx~ h-+0

Ce r6sultat resulte du fait que la s6rie de terme g6n6ral IIVp(k)ll 2L2(IR3) -~2 I1~(k) [IL2(IR 2 3) est convergente, et done IIV~ (k) IIL2(IR3) -]- I]~/)(k) 1122(IR3) tend vers z6ro quand k tend vers l'infini. en (16), on a

Mais par d6finition de (~(K) ~(K))

(~(W(hK - l ) ) ~ 2 (]Iv~(K) II~2(IR3) -~-]'~(K)]]~2(IR3)) , donc lim

h--~0

5(w (K-l)) tend vers zdro quand K tend vers l'infini. Le lemme 3.15

donne alors le r~sultat. La proposition 3.14 est d~montr~e,

4. L e p r o b l ~ m e

non lindaire: dc!monstration

m

du th~or~me

2

Nous allons proc~der en plusieurs ~tapes, qui nous p e r m e t t r o n t d'obtenir un r~sultat plus precis que celui ~nonc~ dans le th~or~me, avec notamment une ~tude autour du temps de concentration tn; comme dans la section pr~c~dente, nous allons pour simplifier les notations consid~rer une famille (Ph) d'ondes de concentration lin~aires. D'autre part, nous ne nous int~resserons qu'au seul cas off lim th h-~0 -h- = + ~ ' et laissons les deux autres cas ( - c ~ ou 0) au lecteur. Les lemmes 4.17, 4.18 et 4.19 suivants conduisent directement au th~or~me.

I. Gallagher et P. G&ard

387

4.1. Avant le t e m p s de c o n c e n t r a t i o n L e m m e 4.17 - Sous les hypothbses du thdorbme 2, on a

/ lim lim {

sup

A--,c~ h--,O \--T O, on peut h ~crire

L5 ([-~,~],L 1o) __ < C~(h) avec lim r

( IIQ~II2~([-x,xl,/,o) + IIQhllx

(

205 [-A,A] L

,i IO)

)

= 0 pour tout A > O. On conclut alors par bootstrap

h--+O

superlin6aire (voir par exemple [I], lemme 2.2) ~ partir de l'estimation

IIQ~ - Q~ll~([-~,~l,/,o) -< ce~(h)llQ~'ll~~ +cr

-

Q~IIu~ Ls([-A,AI,L~~

"

Pour terminer la d6monstration du lemme, on introduit la fonction Rhx d~f Qh _Q,Xh' avec

Qh(s, y) d~f hl/2qh(t h + hs, xh nt- hy). Alors R~ v~rifie l'dquation suivante:

{en2

hi ~ h + Q2) -IQ21 ~Qh~ -- 0 dans + In2 + Q~'~(n~

IRs x Dh,

t ~ h,x l I R s x Of~h ---- 0

(~, o~)1~=-~

= ( Q ~ , O~Q~)I~=-~ - (P~, o ~ P ~ ) I , = _ ~ .

D'apr~s le lemme 4.17, on a lim lim Eo(R~h,-A) - 0,

X--+o~ h---, 0

I. Gallagher et P. Gbrard

389

donc en notant de mani~re g~n~rique s(A, h) une fonction v6rifiant lim lim r

A--*c~ h---~0

h) - 0,

l'estimation de Strichartz (2) fournit, pour tout To > -A,

IIR~IILS([_A,Tol,LlO(IR3)) < Cr

h)

5

j=l < C

5-j ilR~hllJLs([_A,To],LIO(IR3))IIQ~]]L~([_A,To],L~O(IR3))

~(A,h)-t-__

)

. (lS)

I1 est facile de voir, par un changement d'~chelle, que la constante C ne d~pend ni de h, ni de A. Mais d'apr~s (17), on peut remplacer dans (18) la fonction Qh~ par Q~. En outre, la fonction Q~ converge, quand A tend vers l'infini, vers Q - dans Ls(]R, LI~ avec v1Q-

-t- I Q - 1 4 Q

-

dans IR~ x f t ~ , Q~R~xOa~ -- 0 lim E o ( Q - - P, s) = O.

= 0 8----> ~

(20

Cette convergence a lieu pour tout temps, car le domaine limite ~ est soit l'espace IR3 entier, soit un demi-espace, pour lesquels on sait que les solutions de l'~quation des ondes non lin~aire critique sont dans L5 (IR, Ll~ (voir [3], le cas du demi-espace d~coulant du c a s ] R 3 par r~flexion antisym~trique). On a donc en fait

liR llL0([- ,ToJ,L o( )) --< Cs( , h) 5

5--j

j--1 et l'on va conclure par d~formation en temps (voir aussi [2]): si le temps To est assez petit, uniform6ment en h et en A, clots l'estimation ci-dessus implique directement, par bootstrap superlin6aire, que lim lim ,,,o,,IIR~IIL~([-A,Tol,LI~

A-+oo h---,0

-- O.

D6finissons ~ pr6sent = sup { T E IR ] A--.cx~ lim h~O lim ]]R~[ILS([_A,T],LlO(IR3))--0}. rmax d6f

390


On a Tm~x >_ To; soit T1 < Tm~x, alors on peut 6crire

]lR~hllL~([_~,Tm~l,L~O(n~)) ~ C~(A, h) + e l i R h~ il 5 5 ([T1 ,Tmax ] , i 10 (in3))II Q - I I t 5 ([T1 ,Tmax ] , i 10 (iRa)).

I1 suffit alors de choisir 7'1 assez proche de Tmax, uniform~ment en h et en )~, pour conclure que lim l i m A--.c~ h - . 0

]]R~IILS([_A,Tma~],LlO(IR3)) : O,

et par l'in~galit~ d'~nergie, lim lim

sup

Eo(R~h, s) = O.

A- .c 0 est assez petit, alors lim lim IIR~llLS([Tmax Tmax+n] LlO(IR")) = 0, A---,c~ h--,0 ' ' ce qui contredit la maximalit~ de Tmax; enfin on montre de la m(~me mani~re que lim lim ,,,~,,IIR~AIILS([-A,+oc[,L~~ - - 0, A--~co h---~0

et avec l'in6galit6 d'e~nergie, le lemme est d~montr6. D

4.3. Apr~s le temps de concentration L e m m e 4.19 - Sous les hypothSses du thdorSme 2, on a

/ lim lim |

sup A---*oo h---+O \ t h T A h < t < T

Eo(p + - qh, t) + lip + -- qhllL~([th+Ah,T],ilo(a))) = O.

D~monstration. Elle est tr~s simple, au vu des deux lemmes p%c~!dents: il suffit d'appliquer les %sultats de scattering obtenus dans [1]-[2], qui s'~tendent s notre cadre par %flexion antisym~trique si f ~ est un demiespace, ll

R~f~rences [1] H. Bahouri et P. G~rard, High frequency approximation of solutions to critical nonlinear wave equations, American Journal of Mathematics, 121 (1999), 131-175.

I. Gallagher et P. G@rard

[2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [121 [131 [14] [15] [16] [17]

391

H. Bahouri et P. G@rard, Concentration effects in critical nonlinear wave equations, Geometrical Optics and Related Topics (F. Colombini and N. Lerner eds.), Progress in Nonlinear Differential Equations and Applications, 32 (1997), Birkhs Boston, 17-30. H. Bahouri et J. Shatah, Global estimate for the critical semilinear wave equation, Annales de l'Institut Henri Poincar~, Analyse non lin@aire, 15 (1998), 6, 783-789. I. Gallagher et P. G~rard, Profile decomposition for the wave equation outside a convex obstacle, Journal de Math~matiques Pures et Appliqu~es, 80, 1, pages 1-49, 2001. P. G~rard, Oscillations and concentration effects in semilinear dispersire wave equations, Journal of Functional Analysis, 141 (1996), 60-98. P. G~rard, Description du d~faut de compacit@ de l'injection de Sobolev, ESAIM ContrSle Optimal et Calcul des Variations, 3 (1998), 213-233, (version @lectronique: http://www.emath.fr/cocv/). P. G~rard et E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Mathematical Journal, 71 (1993), 559-607,. P. G~rard, P. Markowitch, N. Mauser et F. Poupaud: Homogenization limits and Wigner transforms, Communications on Pure and Applied Mathematics, 50 (1997), 323-379,. P. G@rard, Y. Meyer et F. Oru, In@galit~s de Sobolev pr~cis~es, S~minaire l~quations aux D~riv~es Partielles, l~cole Polytechnique, d~cembre 1996. J. Ginibre et G. Velo, Generalized Strichartz inequalities for the wave equation, Journal of Functional Analysis, 133 (1995), 1,50-68. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Annales de l'Institut Henri Poincar@, 1 (1984), 109-145. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, II, Revista Matematica Iberoamericana, I (1985), 145-201. P.-L. Lions et T. Paul, Sur les mesures de Wigner, Revista Matematica Iberoamericana, 9 (1993),553-618. G. M6tivier et S. Schochet, Trilinear resonant interactions of semilinear hyperbolic waves, Duke Mathematical Journal, 95 (1998), 2, 241-304. J. Shatah et M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, International Mathematics Research Notices, 7 (1994), 303-309. H. Smith et C. Sogge, On the critical semilinear wave equation outside convex obstacles, Journal of the American Mathematical Society, 8 (1995), 4, 879-916. H. Smith et C. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, preprint.

392


Isabelle Gallagher D~partement de Math~matiques Universit~ de Paris-Sud 91405 Orsay Cedex and Centre de Math~matiques t~cole Polytechnique 91128 Palaiseau Cedex, France E-mail: Isabelle. Gallagher@math. polytechnique, fr Patrick G6rard D~partement de Math~matiques Universit~ de Paris-Sud 91405 Orsay Cedex, France E-mail: Patrick. [email protected]. fr

Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 18

U P W I N D D I S C R E T I Z A T I O N S OF A S T E A D Y GRADE-TWO FLUID MODEL IN TWO DIMENSIONS

V. GIRAULT AND L. R. SCOTT

1. I n t r o d u c t i o n A fluid of grade two is a particular non-Newtonian Rivlin-Ericksen fluid (cf. [28]) whose equation of motion is 0 Ot

u)

u + curl(u

+

u) • u (1.1)

- (a~ + a 2 ) A ( u . V u) + 2(a~ + a 2 ) u . V ( A u ) + V i5 = f . This system of equations is completed by the condition of incompressibility: div u = 0,

(1.2)

and suitable initial and b o u n d a r y conditions. Here f is an an external force (usually gravity), u is the velocity,/5 is the pressure, ~ is the viscosity and a l and a2 are material stress moduli, the three parameters being constant and divided by the density. It is considered an appropriate model for the motion of a water solution of polymers. Dunn and Fosdick prove in [13] that, to be consistent with t h e r m o d y namics, the viscosity and normal stress moduli must satisfy // > 0

,

O~1

> 0

and a l + a2 = 0.

The reader can refer to [14] for a thorough discussion on the sign of a l . W i t h these assumptions, setting a = a l , (1.1) simplifies and leads to the equation of motion 0 0-~(u - a A u) - tJA u + c u r l ( u - a A u) x u + V p = f , where the modified pressure p is related to i5 by 1 1 p -- i5 -4- ~ u . u -- c~(u- ~ u -4- ~ t r A2),

394

Upwind discretizations of a steady grade-two fluid

where A1 = V u + (V u) t is the symmetric gradient tensor. Interestingly, in [21, 22], Holm, Marsden and Ratiu derive these equations with v = 0 and c~1 + a2 = 0, as a model of turbulence. They are called averaged-Euler equations, and c~1 is an averaged length scale. It can also be interpreted as a measure of dispersion and in this respect, these equations describe a dispersive fluid model (cf. [18, 21]). The equations of a grade-two fluid model have been studied by many .authors (Videman gives in [31] a very extensive list of references), but the best construction of solutions for the problem, with homogeneous Dirichlet boundary conditions and mildly smooth data, is given by Ouazar in [26] and by Cioranescu and Ouazar in [7, 8]. These authors prove existence of solutions, with H 3 regularity in space, by looking for a velocity u such that z = curl(u - a A u), has L 2 regularity in space, introducing z as an auxiliary variable and discretizing the equations of motion (in variational form) by Galerkin's method in the basis of the eigenfunctions of the operator c u r l c u r l ( u - c~ A u). This excellent choice of basis allows one to recover estimates from the transport equation O~Z a - ~ + vz + c~(u. Vz - z . Vu) = v c u r l u + a c u r l f.

(1.4)

Whenever c u r l f belongs to L2(~) 3, this construction is optimal because, in contrast to fixed-point arguments, it uses all the information conveyed by (1.1)-(1.4). Thus, it allows one to derive global existence of solutions with minimal restrictions on the size of the data, cf. [3] and [9]. A fixed-point argument cannot use all four equations because they are redundant. It is particularly important to preserve (1.1) since it implies that the energy is bounded without restrictions on the data. This point will be crucial for the numerical analysis of schemes discretizing (1.1). The transport equation (1.4) substantially simplifies in two dimensions since the second nonlinear term z . V u vanishes. In this case, z = (0, 0, z) with z = curl(u - ~ A u), where curl is the operator curlv =

OVl

OV2

Ox2

Oxl

Hence z is necessarily orthogonal to u = (Ul, u2, 0). This vanishing term has a very important consequence: all the analysis can be performed without having to derive an a priori estimate for u in W 1,~ (f~)2. The same property will hold in the discrete case, provided the discrete scheme is suitably chosen.

V. Girault and L. R. Scott

395

In this article, we propose finite-element schemes for solving numerically the equations of a steady two dimensional grade-two fluid model, with a non-homogeneous tangential boundary condition. Defining z as above, the equation of motion becomes -uAu+zxu+Vp=f

inft,

(1.5)

the incompressibility condition is unchanged: divu=0

inf',

(1.6)

the boundary condition is u=g

on Oft w i t h g . n = 0 ,

(1.7)

where n denotes the unit exterior normal to Oft, and the transport equation becomes uz + a u . Vz = ucurl u + acurl f.

(1.8)

Girault and Scott in [16] prove that (1.5)-(1.8) always has a solution u in HI(f~) 2 and p in L2(ft), on a Lipschitz-continuous domain, without restriction on the size of the data, provided curl f belongs to L2 (ft), thus extending to rough data a result of Ouazar [26]. This unconditional existence result relies entirely on the fact that u does not need to be bounded in W 1'~ (f~)2. Similarly, our finite-element schemes are chosen so the numerical analysis can be performed without having to derive a uniform W 1,~ estimate for the discrete velocity. As expected, the difficulties arise from the transport equation (1.8). As is observed in [17] and [10], a straightforward argument shows that either the discrete velocity must have exactly zero divergence, or its non-zero divergence must be compensated by an extra stabilizing term in the transport equation or by a compatibility condition between the spaces of discrete pressure and discrete auxiliary variable z. Roughly speaking, let Xh, Mh and Zh be discrete spaces for the velocity, pressure and variable z and, as usual, let us discretize (1.6) by

Vqh E Mh, (qh, div U h ) = O.

(1.9)

Clearly, if we want to derive an unconditional a priori estimate from the discrete analogue of (1.8), we must be able to eliminate the nonlinear term. But even in the simplest case, Green's formula gives

/a (Uh" V Zh)Zh dx = --~1/ (diVUh)(Zh)2dx.

396


Hence, we can eliminate this right-hand side either by adding to the lefthand side of (1.8) a stabilizing, consistent term, so that it becomes uz + a u . Vz + 1 (div u)z = vcurl u + acurl f,

(1.10)

or by asking that (Zh) 2 e Mh,

(1.11)

and applying (1.9). Keeping this in mind, we propose to discretize (1.10) or (1.8) by an upwind scheme based on the discontinuous Galerkin method of degree one introduced by Lesaint and Raviart in [23]. This means that in each element of the triangulation, Zh is a polynomial of degree one, without continuity requirement on interelement boundaries. On one hand, if the form (1.10) is used for discretizing the transport equation, then we can approximate the velocity and pressure by the standard 1 P 2 - / P 1 Hood-Taylor scheme, where f~k denotes the space of polynomials of degree k in two variables (cf. for example [15]). On the other hand, if we discretize the transport equation in the form (1.8), then (1.11) implies that Ph must be a polynomial of degree two, discontinuous across elements. In addition, the fact that the pressure and velocity spaces must satisfy a uniform discrete inf-sup condition implies that each component of Uh can be a polynomial of degree three plus two bubble functions of degree four, with continuity requirement on interelement boundaries (cf. [15]). Thus, denoting by C(Uh; zh, Oh) the discrete nonlinear part of (1.10) in variational form (el. (3.5)), our scheme is: Find Uh in Xh + gh, Ph in Mh and Zh in Zh such that Y v h C X h , ~ ' ( V u h , V V h ) + ( Z h XUh,Vh)--(ph, d i v V h ) = ( f , Vh), Yqh E Mh , ( qh, div Uh) = 0, VOh ~ Zh , v (Zh, Oh) + C(Uh; Zh, Oh) -- ~ (curl Uh, Oh) + C~(curl f, 0h).

(1.12) (1.13)

(1.14)

Here ga is a suitable approximation of g and the functions of Xa vanish on 0Ft. Without restriction on the size of the data, we establish that this scheme always has a discrete solution in a Lipschitz polygonal domain and that this solution converges strongly to a solution of the exact problem. Furthermore, if the domain is convex and the data small, this solution can be computed by a converging successive approximation algorithm, with arbitrary starting guess. In addition, we prove an error inequality that leads to


397

error estimates when the solution is sufficiently smooth. For both velocitypressure discretizations, the error is of the order of h 3/2, a result that remains valid as a tends to zero. With the lP2 - lPl Hood-Taylor scheme, this is the best that can be achieved, considering that the discretization of the transport equation loses inevitably a factor h 1/2. For the iP3 - iP2 scheme with discontinuous pressure, whose transport equation is simpler, this result is disappointing considering that the interpolation error for the 'velocity and pressure is of order h 3. These two results complete those of Girault and Scott in [17].

Remark 1.1 Another possibility is that Zh be constant in each element of the triangulation. Then (1.11) implies that Ph must also be a piecewise ,constant and we can associate with it the incomplete lP2 finite-element ,of Bernardi and Raugel [15] for the velocity, or even the non-conforminig element of Crouzeix and Raviart [12]. Otherwise, if we use the stabilizing term of (1.10), we can discretize the velocity and pressure with the "minielement" of Arnold, Brezzi and Fortin [15]. The analysis below extends to these examples and it can be shown that their error is the order of h 1/2. Remark 1.2 The results presented here are much more valuable than what Baia and Sequeira derive in [2]. Their analysis is of very limited use because, in order to guarantee the convergence of their algorithm (or even any algorithm), they must start with a first guess that has an error of order h 3/2. And since they prove no a priori estimate, they cannot construct this first guess, which in fact amounts to solving their problem directly. The remainder of this paper is divided into three sections. Section 1 briefly recalls the analysis of the exact problem (1.5)-(1.8) and compares it with the formulation proposed by [2]. The finite-element schemes are described in Section 2 and their error is estimated in Section 3. We end this introduction by recalling some notation and basic functional results. For any non-negative integer rn and number r >_ 1, recall the classical Sobolev space (cf. Adams [1] or Ne6as [25]) W'~'r(gt) = {v e Lr(s

; Okv e L~(f~) Vlkl _< m } ,

equipped with the seminorm

IVlw'~'~(~) -- [ E fg21Okvlrdx] 1/r Ikl-m and norm (for which it is a Banach space)

O~k~rn


398

with the usual extension when r - c~. The reader can refer to [20] and [24] for extensions of this definition to non-integral values of m. When r = 2, this space is the Hilbert space Hm(f~). In particular, the scalar product of L2(~) is denoted by (., .). The definitions of these spaces are extended straightforwardly to vectors, with the same notation, but with the following modification for the norms in the non-Hilbert case. Let u = (Ul,U2); then we set

IlullL~(a) = where

II. II

Ilu(x) I1" d x

,

denotes the Euclidean vector norm.

For vanishing boundary values, we define

H1 ( ~ ) - { v e i l 1(~); vlo~ =0}. We shall often use Sobolev's imbeddings: for any real number p > 1, there exists a constant Sp such that

Vv c Hg(~), IIvlIL~(~) 0 be a discretization parameter and let 7-h be a regular family of triangulations of ft, consisting of triangles K with maximum mesh size h (cf. [6], [5])" there exists a constant a0, independent of h, such that h~VK c :Yh, - ~ _ a0,

(3.1)

flK

where hK is the diameter and PK is the diameter of the ball inscribed in K. We first recall how upwinding can be achieved by the discontinuous Galerkin approximation introduced in [23]. Consider the discontinuous finite-element space

Zh = {Oh E L2(~) ; VK E "Yh, OhlK E J~D1}.

(3.2)

As interpolant, we shall mostly use an approximation operator (cf. [11], [29], [4]) Rh C s Zh N C~ for any number p __ 1, such that, for m - 0 , 1 and 0 _< 1 _< 1 VZ C W I + I ' P ( ~ )

, [Rh(Z) - Z]w,~,p(f~) ~___C h l + l - m l z I w z + l , p ( f t )

.

(3.3)

Let Uh be a discrete velocity in H~(ft), and for each triangle K, let

OK_ = {x c OK; C~Uh. n < 0).

(3.4)

Note that, when running over all triangles K of Th, OK_ only involves interior segments of Th, because Uh" n -- 0 on 0~. Then we approximate


402

the nonlinear terms a [ ( u . V z,O)+ l ( ( d i v u ) z , 0)] by a / ~ (div Uh ) ZhOh dx

+ Z

(f~ ~(u~. v z~)0~ ax

(3.5)

K ETh

-~- ~fOK_ IOLUh"nl(Zhnt -- z~xt)oihntd8)' where the superscript int (resp. ext) refers to the trace on the segment OK of the function taken inside (resp. outside) K. Note also that when summing over all triangles, OK_ is not counted twice because Uh. n changes sign across adjacent elements. Rather, in the above sum, the boundary integrations are taken over complete interior segments.

3.1. The Hood-Taylor scheme Let us first recall the standard Hood-Taylor discretization of the velocity and pressure. The discrete pressure space is

Mh -- {qh e Hl(~t)M L02(a) ; VK e Th, qhlg e ~~

(3.6)

and we interpolate the functions of L02(~t) by a regularization operator analogous to Rh, rh E s Mh), such that for 0 _< 1 _< 2, Vq c Hl(~)) n L0e(gt), Ilrh(q) - qll/~(~)

~- C hl[qIu~(~).

(3.7)

The discrete velocity space is X~,T

-

{v e c~

; VK e ~ , vlK e ~ , v. nlo~ - o } ,

X h : X h , T C1 g 1 (gt) 2 ,

f

W h --

{V E X h , T ; Vq C Mh , .In q div v dx

= 0},

(3.8)

(3.9)

y. - w~ n H](n) ~ . If all triangles K of Th have at most one edge on the boundary 0~2, the pair of spaces (Xh, Mh) satisfies a uniform discrete inf-sup condition (el. [15]). But as in [17] and [19], we can obtain better results from a local infsup condition that yields an approximation operator satisfying sharp local estimates. Indeed, we can prove that there exists an operator Ph C


403

s Ns such that the support of Ph(v) is contained in a neighborhood of the support of v, and

Vv e H~(a), IIv- Ph (v) llL, (a) _ 2 and all real numbers s with 1 < s < 3, m - 0, 1, and VK e Th, Vv e H~(gt), Vqh e M h , / g qh d i v ( P h ( v ) - v ) d x = 0.

(3.12)

Let gh -- Ph(r) where r is any lifting of g in W. This lifting is only a theoretical convenience because on one hand, gh can be constructed directly by interpolating g appropriately on Oft and on the other hand, gh does not depend on the particular lifting chosen; in addition, gh satisfies

IlghllH1/:(oa) = IIPh(wg)llH1/:(Oa) < IlPh(wg)llH,(a) _< Co IWg[H,(a) < Co T IlgllH1/~(oa) ,

(3.13)

where T is the constant of (1.19) and Co and all subsequent constants Ci are independent of h. Then, as written in the introduction, our discrete scheme is" find Uh in X h + gh, Ph in Mh and Zh in Zh solution of (1.12)-(1.14):

V'Vh E X h , l](V Uh, V Vh) Jr- (Z h X Uh, Vh) -- (Ph, d i v v h ) = (f, Vh) , Vqh C Mh , ( qh , div Uh) -- O, VOh C Z h , v (Zh, Oh) + C(Uh; Zh, Oh) -- V (curl Uh, Oh) + a (curl f, Oh), where c is defined by (3.5). As in [17], the trace preserving properties of Ph and its sharp local estimates allow one to construct a Leray-Hopf's lifting of gh satisfying: L e m m a 3.4 - For any g c H l / 2 ( 0 ~ ) 2 such that g . n = 0 and for any real number ~ > O, there exists a lifting Uh,g of g such that

lUh,glHl(a) _~ c~-l/2-1/tl]gllH1/:(Oa)

, 2 ~ t < c~,

(3.14)

and if hb < s, where hb denotes the m a x i m u m mesh length of triangles in a tubular neighborhood of the boundary, then for all v C Ho~(~) 2 and for t < s < c~, (recall that I]" I] denotes the Euclidean vector norm),

IIIlu..~ll Ilvllll,_..(~) < cell2-11~llgllH~,':~(oa)lVlH~(a),

(3.15)

where the constants C depend on t or s, but are independent of h, ~ and g.

404


In order to prove existence of solutions of (1.12)-(1.14), let us recall the :following identity established by Lesaint and Raviart in [23]" L e m m a 3.5 - For all vh in Xh, for all Zh and Oh in Zh, we have

~(v~; z~, 0~) = ~

(- f~ ~(v~. v O~)z~ ~x

K ~Th

Io~vh. nlz~Xt(0~xt-

+Z

0hnt)ds)

(3.16)

(div Uh)OhZh dx

(3.17)

K_

a f (div Uh)0h Zh dx. 2 J~ Note that when Oh is in Ht(~t), (3.16) reduces to

a/a C(Vh; Zh, Oh) -- -- / ~ a(Vh" V Oh)Zh dx - -~ Note also that, when

0 h -- Z h C Z h ,

c(vh; zh, zh) - ~1 g ~ ~

then

f o ~ _ [avh " nl(z~Xt -- zihnt)2ds"

(3.18)

Therefore, choosing Oh = Zh in (1.14) and applying (3.18), we obtain

I~uh. nl(z~ x t - z~nt)2ds

Ilzhll~(~) + 5 ~ KeT-h

K_

(3.19)

= ~ (curl Uh, Zh) + ~ (curl f, Zh). Equation (3.19) allows us to prove the following existence theorem. T h e o r e m 3.6 - There exists a constant C1 ~ O, independent of h, such that for all v > 0 and a C Kl, for all f in H(curl, ~) and all g in H1/2(0~) 2 satisfying g . n - O, if hb < Clv2+tllg[[-2-t H1/2(0~)

,

for some t > 0

(3.20)

then the discrete problem (1.12)-(1.1~) has at least one solution (Uh,Ph, Zh) in (Xh + gh) • Mh • Zh and every solution satisfies the following a priori estimates:

lUhIH~ ~ ~5'2Ilf]lL~(~) + K1 (h)Co Tllgl]H~/~(o~), v

(3.21)


IlPhlIL~(~) ~ ~-* (S211flIL~(~) + v'CoTllg[[Hl/~(0~)

405

(3.22)

+ S4S41U~IH~(~)Ilzhll~(~)),

PlzhllL~(~) ~ V~[UhlHl(~) ~- I-~!llcurlfpJL~(~), 1

~K~~fO,C [auh'n[

(z~X t

--

z~nt )

2ds

(3.23)

(3.24)

(V/-2/]lUh[HI(Ft) d-[a[[lcurlf[[L2(a))[[Zh[lL2(gt), where Co is the constant of (3.13), ~* is the constant of the discrete inf-sup condition, and

Kl(h)-

1+

s~4 [[Zh[[L2(~). V

In addition, we have for any real number s > -~

~

+ 2v/2S2 [[fllL2(a) /2

(3.25)

+ 2 [am[]]curl fllL2(~),

/2

where the constant C2 depends on s and t, but not on h or v.

Extracting subsequences (that we still denote by the index h), the uniform a priori estimate (3.25) combined with (3.21), (3.22), (3.24) show that, on one hand, (uh,Ph, Zh) converge weakly to functions (u,p, z) in H~(gt) x L2(~t) x L2(~t), and on the other hand, the quantity ~'~Ke~-h fOK_ Ia u h " nl(z~ xt -- zihnt)2ds converges to a non-negative number, say S. Passing to the limit in (1.12), we see that (u,p,z) satisfies (1.5). Next the strong convergence of Uh is easily established, as in [17]. Owing to this strong convergence, using Rh(O) with smooth 9 for test function in (1.14), and applying (3.17), we can readily prove that (u,p, z) is a solution of Problem P. Using again the strong convergence of Uh and the weak convergence of Zh, and comparing with (1.8), we see that the right-hand side of (3.19) converges to v [[zl122(a). Here, we need the following consequence of Green's formula established in [16]:

/(

u- V z ) z d x = O,


406

for z a solution of (1.8). Therefore h--,olimu Ilzhll2 _ "Yh~ , with a constant ~/ > 0 independent of h. independent of h, such that

-}- ~

1(

Then there exist constants Ci,

C3lPh(U) -- utile(a) + x/2 llrh(p) -PIlL'(a)

/Jmin

(4.3)

/2

+ liz~ - zliL~(a)(c~ w h e r e Pmin i s

)

the minimum of

+ C~hl/~),

PK for

all K in Th.

Remark ~.9 The bound (4.3) is one in a family of estimates established by [17] in Wl'r(gt) 2 for all r in (2, o0). The assumptions on the boundary d a t a g, the domain and the triangulation depend upon the value of r. The choice r = 2 + 1/4 is convenient here because on one hand, it is sufficient for deriving the error inequalities below and on the other hand, the condition (4.2) allows for a wide range of refinements of the triangulation. Remark ~.10 Since, W1'2+1/4(~) in continuously imbedded into L~176 an easy variant of Theorem 4.8 gives with possibly different constants but under the same assumptions:

Ilu~ - uffL~(a) < IfPh(u) - utiLe(a) + , / 91

(c~ip~(u) - uf.,(a)

/3min

+ v~ll~(p) /y

- pfr~(~)) + ffz~ - z l l ~ ( a ) (c~ +

(4.4)

c~h'/~).

Now, in order to prove an error inequality for Z h -- Z, we assume t h a t z belongs to HI(Ft). It is shown in [17] t h a t this assumption holds on a convex domain, when g and f are sufficiently smooth and small. It is restrictive, but so far, relaxing this assumption appears difficult. The following arguments are written when c is in the form (3.5); they simplify when the stabilizing


409

term is missing. Taking the difference between (1.8) and (1.14), inserting any element Ch of Zh, choosing 8h = Zh--~h, and applying (3.16) and (3.18), we obtain

1 fo

IOLUh" n l ( ( Z h

-- ~h)ext

--

(zh -- ~h)int)2d8

K ETh + E (--OZ/K U h ' ~ ( Z h , - - ~ h ) ( ~ h - - z ) d x K ~q-u + ~ K _ [(~Uh " n'((Zh -- ~h)ext -- (Zh -- ~h)int)(~h -- Z)extds) (4.5)

62/a div(uh -- U)(~h

+-~ a ]~

--

Z)(Zh -- ~h)dX

div(uh -- U)Z(Zh -- ~h)dx

+ a ~ (Uh -- u). V Z(Zh -- ~h)dx = ~,(~ - Ch, zh - Ch) + ~ , ( c u r l ( u h

- u), z~ -

r

Clearly, the most troublesome terms in the left-hand side of (4.5) are the third and fourth terms. In each element, the third term can be split into: KUh

V(Zh -- ~h)(~h -- z ) d x = f K ( U h -- u ) . V(Zh -- ~h)(~h -- z ) d x + / K u . V(Zh -- ~h)(~h -- z ) d x .

Now, instead of choosing ~h -- R h ( z ) , that clearly cannot give anything better in the second term than an error of the order of h, let us take advantage of the discontinuity of the space Zh and choose ~h -- ~h(Z), the L 2 projection of z on iPl in each triangle K: ~h(Z) C iPl, such that

Vq E ~ 1 , /K(~Oh(Z) -- z)q d x -- O. This operator has locally the same accuracy as Rh. Moreover, we have for any constant vector c:

]~

u. V(z~-

Qh(z))(~ (z)- z)dx = f ( u - c ) . V(zh - ~ (z))(~h (z)- z)dx,

410

Upwind discretizations of a s t e a d y grade-two fluid

because V(Zh -- ~h(Z)) is also a constant vector. Thus a local inverse inequality and (3.1) yield: f ]a ./ ( u - c). V(Zh -- Qh(Z))(~h(Z) - z ) d x I Cl

-- ZIIL2(K)]IZ h -- ~h(Z) I]L2(K)

]OL]I[ u - - C I I L ~ ( K ) I I ~ O h ( Z )

PK < C2~ol~llUlw,,~(g)ll~(z)-

zllL~.(g)llzh

--

(4.6)

~(z)llL~(g),

where ci denote various constants independent of h. Similarly, applying the same local inverse inequality and the approximation property of Qh, we obtain f ]~ ./ (Uh -- u ) . V ( z h -- Qh(Z))(~h(Z) - z ) d x I

(4.7)

The fourth term can be split into" f

/

]~uh. nl((z~ - Q~(z)) ext- (z~ - e~(z))int)(e~(z)- z)eXtd~

KeTh JDK_
O, and for all (Xr, v) 6 Fx]0, 1[, 0tEk(xr, v) • n --

Cf

(~f#fOtt + rotr rotr ) n,.H~-l(x~, ~) d~,

(25)

(E~ (x~, ~) • n ) ( t - 0) - 0.

0t H k (Xr, v) x n - 0t H k (xr) x n

_1

#3

(e~#fOtt + rotr rotr ) II,iEk-1 (xr, ~) d~,

(26)

(H~ (Xr, ~) • n)(t = 0) - 0.

Proof. Let us prove (25). Applying II H to the first equation of (22) shows that: 0~(E k x n) - #f 0 t ( H , n k-l) + rotr (E k - l " n). (27) We take now the scalar product of the first equation of (23) by n. n-ro~tr - 0 we obtain: efOt (E k - l " n) -- rotr H k-1.

As

(2s)

Combining (27) and (28) yields Or(OrE k x n) -

1 ) k-1 #~Ott + -- rotr rotr Hijn f ~f

We get identity (25) by integrating the previous equation between ~ and 1 and by using the derivative in time of the boundary condition given in (22). Identity (26) is obtained in a similar way. m Remark 2.1. From equation (28) one can easily deduce that the normal components (E k. n) and (H k- n), k >_ 0, are determined by ef0t (E k. n) - rotr HL.Hk,

#f Ot (H k- n) - - r o t r II,,E~,

(E~..)

(H k- n) (t - 0) - 0.

(t - 0) - 0.

(29)

These equations will be useful later in order to prove the error estimates. Construction principle of the effective boundary conditions. In order to obtain an approximation of order k of the scattered field (EW, HW) a good candidate would be the truncated expansion --k uk

k-1

( E , , H v) - ~ i--0

r]i (E~, H~)

422

Stability of thin layer approximation of electromagnetic w a v e s . . .

(cf. theorem 0.2) . According to (20) , (Ev, --k --k H~) satisfies Maxwell's equations on fl~ coupled with the boundary condition on F --k

k-1

Hll E . (Xr) - ~-'~i=0

~i

i

xr e F.

IIttE,(xr, 0),

(30)

On the other hand, we notice that, having Ht.E ~ and HttH ~ by (24), equations (25) and (26) are nothing but first order differential equations with respect to v that permit to compute step by step IIttE k and H,tH k for each order k as functions of the boundary values of IItt H k on F. So one can express the right hand side of (30) as a function of the boundary values of HHH~1 0 < i < k - 1. Then the question is: can we express it as a function --k

of (H v x n)? The response is no, in general. Meanwhile, we may establish formally the existence of a linear tangential operator B~ such that -Ek v x n - B

k, ( i i -H- k~ ) + O ( r l k)

on

F.

To obtain our effective boundary condition of order k, we chose to omit 0(~? k) in the previous identity. That is why the approximate solution, de"~

""

--k

mk

noted by (E~, H~), will be different from (E,, H , ) . We shall discuss the validity of this choice via the stability study. We are going to give the details of this construction for k - 213, 4. Remark 2.2. As we assumed that the thin layer characteristics are independent of the the thickness 77, the limit problem, when ~ goes to 01 is obtained by omitting the thin layer and applying the Dirichlet boundary condition (4) on F. This coincides with the effective boundary condition of order 1" combine the first equation of (24) with the system (20) written for k-0. 2.2.2. E f f e c t i v e b o u n d a r y

condition of order 2 and 3

According to condition (30) we can derive the effective boundary conditions of order 2 by computing E ~ (Xr, 0) x n and E 1 (Xr, 0) x n. E ~(Xr, 0) x n - 0 by (24). From (25), written for k = 1, and from the second equation of (24), we deduce that 1 0tEl (xr, v ) x n - - - - ( 1 - v ) ( c f # ~ O t t + r o t r r o t r ) I I j t H ~

(31)

Cf

We have then: --2

T]

OtEv(xr ) x n - - - -

--2

(r #fOrt + rotr rotr ) IIttHv(xr ) + O(r/2)

~f

(where O(r/2) - ~Cf (~f #f(gtt + rotr rotr

1

) II,,Hl(x~)).

H. Haddar and P.

Joly

423

If we denote by (Ev~,H~) the approximate scattered field, the effective boundary condition of order 2 is: atEv~

x

n

-

-

~

6f

(~f #, art -I- rotr rotr ) rill Hvn on F x R +.

(32)

In this particular case of a straight boundary, condition (32) is in fact of order 3 (but will be false with curved boundary or non linear materials (see section or section 0.0.1)). Indeed, in order to compute (E2(xr,0) x n) using equation (25), we need to compute (H,H~(xr,y)). Since II,E ~ - 0, equation (26) shows that (after time integration)

II,,H~(x~,v)- II..Hl(x~),

(x~,v) e Fx]0, 1[.

(33)

So we have from (25) written for k = 2, 1

cOtE2(xr, L,) x n -

(1 - ~,) (~f #~Ott + rgt~ rot~ ) II,,H~(x~).

(34)

~f

We deduce from (24), (31) and (34) that --3

?7

OtEv(xr) x n - - - -

Cf

--3

(r #fOrt + rotr rotr ) H,Hv(xr ) + O(~3),

which gives, when omitting O(~3), the same expression as (32). The fundamental point is that the coupling between this condition and Maxwell's equation leads to a stable problem. T h e o r e m 0.1 - Sufficiently regular solutions (EW,HW) of {(6), (32)} satisfy the a priori energy estimate:

s { dt

(t) fi , (t)) + ,

(t) } - 0,

(35)

N

where, if we set, ~o" (Xr, t) - H, H~ (Xr, t), (Xr, t) E F x R +, #f

2

1

E~'(t) = ~ II~'(t)llL. + ~-~

fo

Li2

rOtr ~v(.i-) dT L2

Identity (10) is then well satisfied. Proof. In the classical energy identity for Maxwell's equation,

d---tg~ (t) -

(E' x n ) . ~v dxr

(36)

424

Stabifity of thin layer approximation of electromagnetic waves...

we explicit the right hand side using (32). We integrate in time this condition, take the scalar product with ~o~ and integrate over F. We use the duality between the operators rotr and rotr to deduce that: (En x n). ~on dxr - -r/~-~Ern(t).

(37)

Identity (35)is yielded by (36) and (37).

m

2.2.3. Effective b o u n d a r y c o n d i t i o n of o r d e r 4 We need to compute (E3(xr, 0) • n) using equation (25), written for k - 3, which requires (H..H2(xr, u)). Equation (26), implies for k - 2 0tH2(xr, u) x n - 0 t H 2 ( x r ) x n - -

1/o?

ef ][/f 0tt + rotr rotr)H..Efl (Xr, ~) d~.

Pf

(38) From (31) we have, 0tH,Ef~(xr, u ) _ 1 (1 - u)(ef#f0u - Vr divr ) (H~

• n).

et

(39)

We apply the operator (of #f0u + rotr rotr ) to (39) (using rotr Vr - 0) and integrate in time. Hence (ef #f 0tt + rotr rotr ) II,,Ef1(xr, u)

(40)

= (1 - u)#f 0t (el #f 0u - Vr divr + rotr rotr ) ( n ~ (xr) • n). Combining this identity with (38) yields Hf2(Xr,U) •

H~(xr) •

( e . # . O t t - ~ ' r ) (H~

•

(41)

where Ar -- Vr divr - r o t r rotr, is the Laplace-Beltrami operator. From (41) and (25) we finally deduce Ot Ef3 (Xr, u) • n -

(

n,,H~(x~)-

1

(1 - u) (el #f 0u + rotr rotr ) sf ~1 (1 + u - v ) 8f ~fd~tt - /~r II,,H~

)

)9

(42)

Combining (42) with (24), (31) and (34), we obtain the formal identity: --4

OtEv(xr ) x n =

r] (ef#fcgt t + rotr rotr ) ( 1 -

ef

m4

~ ( e l # f O r t - /~r ) ) I I , , H v ( x r ) + O(r] 4) 3

(43)

H. Haddar and P. Joly

425

which gives the following effective boundary condition of order 4 on F x R +

OtEnv x n - -~-- (e~ #fOrt + rotr rotr ) (1 - ~ (el # f O r t - A r ) ) II,,~I~v ~.f

3

"

(44) A n i n s t a b i l i t y r e s u l t . Contrary to the case of the second order condition, The condition (44) coupled with Maxwell's equations does not lead to a stable problem. In particular, one cannot obtain a priori estimate like in

(10).

Let us study for that the case of the 2D problem. We denote by (x, y, z) the space coordinates system, and assume that all the fields do not depend on the z variable. For sake of simplicity, we make e%= ef - 1 and #0 - #f 1 and assume that f~v - y < 0. If we consider (E~, H~) satisfying Maxwell's system (6) coupled with the boundary conditions (44), the we observe that the z component of Ev~ (resp. of H~) is a solution of the initial boundary value problem 7)~ (resp. 7~nn).

O t t u - (Oxx + Oyy)u - O, V~

u + 7 7 ( 1 - ~ - 3( O t t - O x x ) ) O n u - O , ('U,, Or?2) -- (UO, ~t 1),

t>O

y--O,

t > O, y < O,

OnU + rl(Ott - 0 ~ ) ( 1 - ~-(Ott - O~))u - 0 3 o

u) -

(45)

t -- O, y < O.

O t t u - (O~ + Oyy)u - O, V~n

t > O, y < O,

t -

t > O, y - 0[46)

o, y < o.

These problems correspond to the classical decomposition of the electromagnetic waves into two independent polarizations: T.E. (system 7)nn) and T.M. (system 7)~). T h e o r e m 0.2 - For fixed rl, the problems 7)~ and T)n~ are well posed in the sense of Kreiss [5]. However both of them are strongly unstable in the following sense: there exists (u~(y), u~(y)) a sequence of initial data and un(y, t), the corresponding solution, such that:

where C E R + and is independent of 7, limn~0 JJunJJL2(]_oo,O[x]O,T[) = +oo, VT > 0.

Proof. We seek a solution of P~ or P~ of the form: y, t) -

),

426

Stability of thin layer approximation of electromagnetic waves...

where, k E It, s = a + i w , (a,w) E R 2, and fi E L 2 ( - c ~ , 0 ) . equation implies that our particular solution is of the form ~(y)

Ae

,

The wave

AER,

with Re((s 2 + k2) 89 > 0. Let us set a - ~(s 2 + k 2) 89 One checks that the effective boundary condition of either Pd or ?n leads to the characteristic equation: l + a ( 1 - ~ 1 ~2 ) - 0 . (47) This equation admits one real solution n0 > 0. The two other ones have negative real part, so they are not admissible. Hence acceptable waves t~ 2 satisfy: s 2 + k 2 - ~ , which is equivalent to (w-0anda2+k2-a2)

( ~ - or~

a-0andk

2 - w 2-a~

.

(48)

Let ~ be fixed. Relation (48) shows that plane wave solutions of P~ or Pn~ are such that Re(s) < ~o. 1/ This means that the boundary value problems ?~ and P~ are well posed in the sense of greiss (see [5]). -

-

However, one notices that when 7/ goes to 0, the higher bound of the

Re(s) goes to +oc. This means the existence of plane wave solutions that blow up for t > 0, when rl --+ 0. For instance, let us consider the worst case, corresponding to k - 0 (which is nothing but the 1D approximation) and a - a~ = ~o. v We fix m E IN, and set f ' ( y ) = e~'~ y ' y < 0. The sequence of initial data

:"(~) /"(U)

-

IIs'(y)ll...

, u ~ ( y ) - II:'ll,m

satisfy {IJu0iiHm + IiuoiiH m } < 1 + ~ < c , when 77 --+ 0. They correspond to the plane wave solutions: -

(~--~o)m+l

IIf~JIL2(O'T)

Therefore

=

.f,

whereas

~ 12-~o exp( no~7T

IIu~?IIL2(]_cx),O[x]O,TD"+ "1"-00 when rI --+ 0.

permits to

(t)

Ilu~l[L~(]_~,0[)IIf'llL~(o,T).

We have: IJu'TlIi2(]_~,o[• When rl-+ 0 : I I u ~ l l L 2

s"i i s(y+t) .ll.m

I


427

S t a b l e c o n d i t i o n of o r d e r 4 We are going to built, from the unstable fourth order condition (44), another condition that gives the same order of approximation, but leads to stable problems. This is reminiscent of the techniques used for absorbing boundary conditions [3]. Let us consider first the simple case of the 1D approximation: i.e. we omit the tangential derivatives in the effective boundary conditions. We take the problem (45) as example. The boundary condition becomes when we apply Fourier transform with respect to t: 1 2 032 )On~t - + r/(1 + jr/

(49)

0

where w is the dual variable of t. Comparing this condition with the stable condition of order 2: fi + rl0n~ = 0, we observe that the instability comes from the substitution of (1 + ~1 ?]2 032 ) for 1. A natural idea is to replace (1 + 1 ?72032 j~ by another function g(r/w) such that" g(~/03) - 1 + y1 T]2032 + O((~w) 3 ). As (49) is obtained in a formal way by neglecting the terms of order greater than 4 (with respect to 7), we keep unchanged the order of approximation. We chose g(r/03) - 1/(1 - ~1 T]2032 ). Hence, instead of using (49), we suggest to use the condition (1 _ 1 2~n 2 ^ w )u + n O ~ - O,

that corresponds to (1 + ln20~t)u + nO~u - o.

This new condition leads to stable initial boundary value problems. This result will be shown later in the general case via energy estimates, but we can already see that the previous construction of unstable plane wave solutions fails: the characteristic equation (47) becomes I + ~ 1 ~2 + ~ - - 0 ,

~ -- v/s,

and has no roots with positive real part (which can easily checked). For the 3D case, we would like to apply the same ideas to (44). Unfortunately, working directly on the expression of (44) do not lead to the appropriate condition with regards to energy estimates. That is why we begin by writing differently the differential operator in the right hand side of (44). As, rotr Vr = 0, we have (el #f 0tt + rgtr rotr ) (1 - ~ (~f #f 0u - / ~ r ) ) -

e~#~Ott ( 1 - ~ (efptfOtt-Nr + r o t r r o t r ) ) +ro-*trrotr ( 1 -

~ro-*trrotr)

(50)

428


Let P be one of the two differential operators (El pf O f t - / ~ r + rotr rotr ) and (rotr rotr ). Like previously, we apply to (50) the formal approximation

(1-~P)- (I+~P)-1 -~- O(~4), and obtain the condition (51), where we need to introduce two new variables r and r on F, and where ~ ( x r , t ) - I I , H ' ( x r , t ) , (xr,t) E F x R +.

OtE~v x n - - rl (~ #fOtt~2 ~ 4- rotr rotr Cv) ~f

onF•

( 1 + ~3r o t r roUt) Cn - ~ n (1+

~2 - (/ ef~#f r(~t 3

+ r~

r~

+.

(51)

Cn - ~ n,

To the third equation of (51) we associate the initial conditions r

= o) = o,

= o) = 0

(52)

Such doing, we have constructed a boundary condition, that when coupled with Maxwell's equations, leads to a stable problem: T h e o r e m 0.3 - Sufficiently regular solutions (E~, Hv~) of { (6), (51), (52)} satisfy the a priori energy estimate:

d d--t { s (E~v(t), I--I~(t)) + rl S~r (t) } - O, where, S~r (t) - ~2 IIr r~32

and

(~2 gl(r

111 f0 (rotr Cge) (T) d7 iiL22 + + ~1 g2(r )~

2+ ~

(53)

81 (Ca (t)) -- r #f IlOtr ~ (t)ll L2 2 + Ildivr Cn (t)IlL2 2 + 2 IIrotr Cn (t) II2L2~ t 2 g2(r -- ~o (rotr rotr Ca) (T)dT . L2

Proof. We first write the L2(F) scalar product of the first equation of (51) by ~ fr ( E ' x n ) . ~ dxr = - ~ (#f fr ~n . OtCndxr + ~1 fr ~" . fot-rOtr rOtr r

dT dxr) .

(54)


429

When we take L2(F) scalar product of the second equation of (51) by /0 t (rotr rotr r

dT

and

by part, we get

fr ~P~ " ( fo r~

r~ r ~ dT) dxr

=

Cn dT

2 dt

fO rotr

ir

L 2 + ~3

ir

fot rotr -~ rotr

Cn

ff')

(55)

dT- L2

and when we take L2(F) scalar product of the third equation of (51) by Ore ~ and integrate by part, we get

/ ~''Otr

1d dxr - ~ dt

( II

r

,

2 (t)llL2 + -~s162

)

.

(56)

The energy identity of theorem 0.3 is obtained by using (56) and (55) in (54) and substituting (54)in (36). m 2.3. E r r o r estimates Let k 6 {1, 2, 3, 4}. We recall that the effective condition of order k on the boundary F is the Dirichlet boundary condition when k - 1, the condition (32) when k - 2,3 and the condition (51)-(52) when k - 4. We shall determine, using the stability results (theorems 0.1 and 0.3), the order of approximation between the exact solution (E~, H~) and the approximate one (E~v,H~). Our main result is summarized by theorem 0.4 below. N

In fact, rather than working directly on the difference (E~ - Ev~, H~ - H~), we shall insert the asymptotic expansion (19). So we consider (E k, H k, E k, nk)k>0 the sequence (of sufficiently regular fields) satisfying (20), (24), (25), (26) and (29). Remark 2.3. At least, when (Eo, Ho) is regular (let say in ~P(~tv)) the existence of the expansion (E k, H k, E k, Hk)k>0 can be easily shown by a recurrence on k, as explained in the construction of the equations (20), (24), (25), (26) and (29). Moreover, one can check that (E k, H k) are polynomial functions with respect to v, of degree lesser than k. T h e o r e m 0.4 - Let k 6 {0, 1,2,3}. If (Ev~,H~) is a sufficiently regular solution of (6) coupled with the effective condition of order (k + 1) and (E~, Hv~) is a sufficiently regular solution of the exact transmission problem, then, there exists for all 0 < T < +co a constant Ck(T) independent of ~, but depending on {(E~176 ~ ~ 9 9 9 (E k , H k,E k , n k)} and T, such that: sup O 0 of Xr ~ min(Icl (Xr)l, Ic2(Xr)l), on F. The mapping (73) is then an isomorphism for r / < c -1. The covariant derivative operator Vr : Let v be a scalar function defined on F. We define ~ on ~ by: ~(x) = V(Xr), x and Xr related by (73). The operator Vr is defined by

(vr ,)(x~) = (w)(x~, 0),

Xr E F

(76)


438

It is a tangential operator that can be expressed in the contravariant basis of Txr, by V~ v - (0~o v)~. (77) To prove (77), we use (73) and differentiate ~ with respect to (~1, ~2, s). We get

(0~o ~)(x~, ~) - (W)(x~, s). ( ~ + ~ c ~ ) , (0~)(x~, ~) - (W)(x~,

~). ~.

Making s - 0 and using (asO) - 0 (see the definition of ~) show that (Vr v).z'~ - Or and (Vr v ) . n - 0. We deduce (77) since (T 1, T2,II) is the dual base of ( r l , r2, n).

Consider now a tangential vector field v defined on F and a field of symmetric matrices 74 defined on F such that T~n - 0. We define (divr v), ((T~Vr). v) and (divr T~) by the following expressions, where, denotes the scalar product in R 3 ((79) and (80) are original definitions), divr v - (Vr ( v . ei)) .ei - (Or v ) . r " ,

(78)

(nVr).v

(79)

- ( ( n V r ) ( v . ei)) .ei - ( 0 f . v ) - ( 7 ~ T"),

divr 7~ - (Vr 9( n e ' ) ) 9ei - (0~. 7~)T".

(80)

Remark that the definition of divr v coincides with the definition of Vr 9v (n

-

n,).

As in the case of straight boundary, the operators rotr and ro-*tr are defined, from divr and V r , by relation (13), that we recall here rotr v -

divr (v x n),

ro~tr u -

(Vr u) x n,

where v (resp. u) is a vectorial (resp. scalar) field defined on F. We still have (we omit the proof)" rotr Vr - divr rotr - 0. Also divr (resp. rotr ) is the adjoint o f - V r (resp. rotr ) for the L2(F) scalar product. We prove now the basic result used in the formal derivation of the effective boundary conditions (it differs from the classical representation of the curl operator (see for example [14]) by making explicit the dependence on s). L e m m a 0.6 - Let v be a vector field defined on f~7. Let (Xr, s) E Fx]0, r/[ be the parametric representation o/ ~ . Then we have rotv-Tr sv-

Os(vxn),

~h~r~, %'v - [(n, v~ ). (v • n)] n + [n~ V~ (v. n)] • n and Tie defined by:

7~s (l-I, + s C) - Hii, 7~s n -- 0.

( n , C v) • n,

439

H. H a d d a r and P. J o l y

Proof. For a scalar function u defined on f~fn, one gets (using (73)),

0~o u - ( V u ) . (n,, + ~ c ) r . ,

G u - (Vu). n. As TEe(H, + s C ) - H,, T E s n - (H, + s C ) n - 0, TEs and (H, + s C ) are symmetric, we deduce that (TCs~'1,7~sr2,n) is the dual basis of ((H, + s C)vl, (H, + s C)v2, n). Hence, (Sl)

V u - (O{~ u) T G r " + ( O s u ) n .

From rot v - e i x V ( v . ei), we obtain using (81), rot v -

(nsV a) x ( O ~ v ) - (OsV) x n.

Let us set B~ v - (TCs~-~) x ( 0 r (B~ ~) x ~ - - ((o~o

~). ~)

n~

(82)

and prove that Brs - Trs. ~

=

-(o~o

=

-(o~o

(~. n) - c ~ . (~. ~) n ~ ~

~)n~

~

+ (c~.

~) n ~ " .

As the tangential and symmetric tensors TCs and d commute, the previous equality yields, n., (B~ ~) - ~ x ((B~ ~) x ~) - I n , v ~ ( v . ~)] x ~ - ( n ~ c v).

(83)

On the other hand, (Bs v ) . n - ((0f~v) x n).7~sT a - (Of~ (v x n)).TEsTa + v-(TEST a x Cra). By applying the lemma 0.7 below, to .4 - d and B - 7~s, we deduce that the last term in the previous equality is 0. Consequently, this equality becomes, using (79), (B~ v ) . n - (ns Vr )" (v x n).

(84)

Identities (83) and (84) show that B s - "Frs. Hence, lemma 0.6 is proved by (82).

l

L e m m a 0.7 - Let A and 13 be two s y m m e t r i c and real matrixes in s such that A n -

13 n -

0 and having the s a m e e i g e n w c t o r s .

3, R 3)

We have:

~4~'~ x 13r ~ - O.

Let (r162 be the eigenvectors of ,4 and B and (a~,a2) (resp. (b~, b2)) the corresponding eigenvalues of ,4 (resp. /3). Let us set: ~-~ = Ttr r -[- Ttr r and r ~ - T~'1 r + T~'2 r Then Proof.

Ar~ x Br ~

+ a2 T~,2r

X (bl T ~ ' 1 r

-

(al T~,I r

=

(al b2 T~,IT ~'2 -- a2 bl T~,2T ~'1) r Xr

~'2r


440

By definition of the dual base ( r ~, r 2 ) , we have, setting 5 = 1/(T~,~ 72,2 -7"1,2 T2,1 ), T 1,1 _. (~7.2,2 '

T 1,2 __ __(~T2,1 '

T2,1 __ _(~7.1,2,

T 2 ' 2 --" (~7"1,1.

T h e n , one easily checks t h a t Tn,1 T n'2 - - O a n d Tn,2 T n ' l = O. We prove finally the following geometrical identity t h a t enables us later to write in s y m m e t r i c way the effective b o u n d a r y condition of order three. Lemma

0.8 -

We have divr ( 2 H H., - d) = - 2 G n,

where G = det C, is the Gaussian curvature. Proof. Following the definition (80), we have, divr ( 2 H II..-C) = ( 0 ~ ( 2 H H i , -

C))r ~. So divr (2HH.. - d ) . n -

[(0e~(2HH,. - d ) ) v s ]

9n - [ ( 0 e ~ ( 2 H I I H - d ) ) n ] . l

= [(C - 2 H H,.)0~ n ]. v s - [ (C - 2 H H..) d vs ]- v s - t r ( d 2) - t r ( d ) 2 = -~ To conclude, we are going to show t h a t divr ( 2 H H,, - C ) . v ~ E {1, 2} (which means: I I , d i v r ( 2 H II, - d) - 0).

- 0, where

divr ( 2 H II,, - C ) - r Z - [ ( 0 ~ ( 2 H II,, - d ) ) ~ . s ]. ~.Z _ [ ( 0 ~ ( 2 H II,, - C))~-Z] = 0 0 ( e l l ) - [(0e~ C ) r ~ ] . 7-s We have (using O 0 0 ~ n [(o~o c ) ~ ] .

~-

0~.Oon )

-

[o~o (c ~ ) ] .

~

-

[coco r,]. ~-~

=

[ o o (c ~ ) ] .

~

-

[coco r,]. ~-,

=

o~, (c

whence divr ( 2 H I I , - d ) . r Z Let us set: a n _ ( c ~ ) .

~. ~)

- (c ~ o ~

-0OTs

+ c ~-O~o

~,),

- - (d v s 0 O r s + d 7-s0~ ~'Z) - - B . 7"n. Since C 7-s - anS 7.n and C r s _ a s 7-n,

B - a s Tn 0 0 r s + a ns r ~ 0 ~ vZ - - a s~ o e ~ + (=

~

a~" ~ o e o

v~

+ 0 ~ TZ) a~ 7"n

B u t we have 0 0 7"s - 0 ~ ( 0 ~ x r ) divr ( 2 H H,. - C) .7-~ - 0.

-

0 ~ 7"Z. Consequently, B -

(Sh) 0 and m

H. H a d d a r and P. J o l y

441

2.4.2. Scaling and asymptotic expansion Scaling. As in the case of the straight boundary, the scaling corresponds to the change of variables (12). According to lemma 0.6, the differential operator rot becomes in the new coordinates systems (Xr, u) E F x]0, 1[: 7~rv + s X 0~. Notice that the operator 7~r~ is not singular with respect to rl. Equation (17) becomes 1 0~(Hfn x n ) er Ot Enf - 7~r ~ Hnf + -~

0,

#~OtHn~ + 7~r~'En~ - 1 0v(E~ x n ) -

0.

(86)

Our problem is constituted now by equations (1), (16), (86) and (18).

Asymptotic expansion. We use the ansatz (19). Moreover, we need an analogous expansion of the operator 7~rv. We use, for instance, a Taylor expansion of 7~nv with respect to the parameter (~u g)" ']~-,v -- nil -t- ~i~176(--,/] e) i, so we have T~rv -- ~ 0 ( - r w ) i ~v

Tr/, where,

(87)

- [(e ~ v , ) . (v x ~)] ~ + [e ~ v ~ ( v . ~)] x ~ - (e ~+~ ~) x n.

The formal identification lets unchanged equations (20) and (21). However, equations (22) and (23) change to the following ones, where k > 0, 69u(E k+l x n ) -

, f 0 t n k + ~ i =k 0 ( - l ] ) i .-[-.ii~k-i ,r--'f ,

Ek+l(Xr,1) x n - - 0 ,

{

(88)

for Xr e F.

0v(H k+l x n) - - e , cgtEk + E i = 0 ( - v)' '-Fr'H~k-i, H k + l ( x r , 0 ) x n - - n k + l ( x r ) X n,

(89)

for Xr e F.

The zero order terms are determined by (24). To determine the higher order terms, we use the lemma 0.9. We introduce the notations, where i is an integer, rotr('~ u - (Ci Vr u) x n,

rotr('~v - (C i Vr )" (V X n).

442


L e m m a 0.9 - For all k > O, we have for all (Xr, u) E Fx]0, 1[, 0t

E k+l

(xr, u) x n -

-fl

#, Ouli,Hk(xr, ~) d~

Y~i=o Y ~ j = o ( -~)

-

H j (Xr, ~) d~.

rgtr(k-') r~

Y~i:o(-~) k-i (C k-i+1 0tE~(xr, ~)) x n d~.

(90) 0tHk+l

(

Xr u) x n ,

-

v 1

fo

- fo

--

k

(9-t H v k+ 1

Z,=o

(Xr) x n -

f0 ef0ttH"Ek(xr,~)d~ v

i

)-~j=o (_ ~ ) k - j rgtr(~-,)rOtr(,-j)E j (xr ~) d~ (C k-i+l 0tH~(xr, ~)) x n d~

k

(91) Proof. We are going to prove (90) only. One deduces (91) by analogous arguments. By applying H.. to the first equation of (88) we obtain

0v(E k+l x n) -

a,0t(rI,,H~)

ki=o( _u)k_i (C

k + Y~i=o(--u)k-irot(r k-') (E~. n)

i+1 0tEf(xr,~)) i

x

n.

(92)

On the other hand, applying II. to the first equation of (89) shows that for i>0, ~f0,(E~ 9n ) - ~ j = ~ o ( - U ) ~-J rotr('-~) n fj . (93) Equation (90) is obtained from f~ 0t(92) d~, by using the expression of Ot (E~. n) in (93), and by using the boundary condition at u - 1, given in (8s).

m

0.0.1

2.4.3. E f f e c t i v e C o n s t r u c t i o n

Of course the principle is the same as the case of the straight boundary. C o n d i t i o n o f o r d e r 2. The terms H,E ~ and H , H ~ are given by (24). Making k = 0 in (90) and using (24) shows that

OtE~(xr,u) x n -

- ( 1 - u)

( #fOrt + ~ r o t r rotr ) II, Hv(xr 0 ),

(94)

which is exactly the same expression as (31). Hence, the condition of order 2 in the case of curved boundary is given by (32). Indeed, the stability theorem 0.1 still applies. However, the condition of order 3 has no longer the same expression.


443

C o n d i t i o n of o r d e r 3. Make k - 0 in (91). Using (24) we get,

II.,H~(x~)- vcn~

IIHH~(xr,v ) -

(x~,v) e r•

1[.

(95)

Consider (90) when k = 1, OtE2(xr,v) x n

-

- f l #fOttlittHl(xr,~ ) d~ -4- f~ (COtE~(xr,~)) x n d~ _ fl

1 rotr rotr H I (xr ~) d~

1 (r~t(1) rotr + rotr rotr(x))H ~ (Xr, ~) d~, + f l ~ ~ff

(96) I I , H ~ is given by (24), IIttH~ is given by (95) and we have COtE~ - COtIIttE 1, where OtHttEI~ is given by (39). We obtain after simplifications and using

(75), Ot E 2 (Xr, V) X n -- # f O t t ((1 - v)II,,H~(x~) - (1 - v2)(c - H)II..H~ (xr)) 1 rotr ~,

rotr ((1 - v) II,,H~(x~)1 ~1(1

+7,1 ~1(1__

(97) 0 H~(xr))

V2 ) ~gt~ ~otp) H~(x~). 0

The ultimate simplification is due to lemma 0.8 and is given by the following lemma: L e m m a 0.10 - Let v be a tangential vector field defined on F. We have the identity: rotr(1) v + rotr (C v) - 2H rotr v Proof. By definition, rotr(1) v rotr(~) v + rotr (C v)

(C Vr )" (v x n). Hence, using (75),

=

( C V r ) " (v x n) + Vr 9( ( 2 H n l l -

C)(v x n ) )

=

2H divr (v x n) + (divr (2H Hi, - e))" (v x n),

and according to lemma 0.8, (divr (2H IIll - C))" (v x n) - 0.

m

Applying this lemma to relation (97) yields 0tEfe(xr, v) x n = -#~ 0tt ((1 - v)II,~H~(x~)-(1 - v2)(8 - H)II..H~ 1 ((1 - v ) r o t r rotr H,.Hlv(xr) - (1 - v 2) rotr g r o t r Hi.H~

~f

(98)

444

Stabifity of thin layer approximation of electromagnetic w a v e s . . .

Combining (98) with (94) and (24) leads finally to the following third order condition

OtE~ • n - -~-- (~r #f [1 - rl(C - H)]Ott + rotr (1 - r/H)rotr ) II,.H:~, Cf

(99) on F • 1~+. Notice that this condition differs from the second order one by corrector terms involving the geometry characteristics C and H, only. In the case of straight boundary, C = 0 and H = 0, the two conditions are of course identical. We give now the stability result for this condition. N

N

Theorem 0.5 - Sufficiently regular solutions (E~,Hv~ of { (6), (99)} satisfy the a priori energy estimate: d d--t {$~(F,~(t), H~(t)) + ~$r~(t)} - 0, N

where, if we set, qo~ (xr, t) - II.. H~ (xr, t), Xr e F, g~ (t) - -~

-~7

-

.OtcP'7

+~

(1-~H)

Iforotr qon dT I dxr

We have stability as defined by (10) when, r/ i~f (max (1Cl + c21, Ic1

and where

Cl

-

-

C2[)) < 2,

and c2 are the eigenvalues of C.

Proof. The proof of the a priori estimate is rather straightforward. It follows the same steps as the proof of theorem 0.1 and is based on the symmetry of [1 - r/(C - H)]. The stability (defined by (10)) is obtained when the eigenvalues of 1 r](C-H) and the real ( 1 - 0 H ) are positive; i.e. when ~ m a x (ICl + c21, Ic1 - 521 2 a.e. on F. In this case we have $r~ (t) _> 0. II

3. Thin layer approximation: the non linear case We generalize the construction of thin layer approximations to the case of non linear materials of ferromagnetic type. We consider here directly the case of a curved boundary.


445

3.1. Description of the model We keep the same notation as in section. However the material of the thin layer ~t~ is no longer linear and obeys to the following equations"

{

r OtE~ - rot H~ - 0,

#f Ot(H~ + MT) + rot E~ - 0, on f~,

(ET, H~)(t - 0) - (0, 0),

(100) where M~ is called the magnetization field and is linked to H~ through the ferromagnetic law s as follows:

0tM~ - s

H,no,; x),

M~ (x, 0) - M~(x),

x E a~,

(101)

H~o, - H~ - VM~I,n(M~; x), where, for a.e. x 6 ~ , (m,h) 6 l~ 3 X l~ 3 ~ s h; x) E l~ 3 is a C c~ function and m , ~ On(m; x) 6 R is C cr and positive function, s satisfying:

(i) (ii) (iii)

h:

; s (m, h; x) is linear, m E l ~ 3 X E ~'~f~

s

re, h E R 3 x E f ~

(102)

s (m, h; x) . h >_ 0, m E R 3, xEf~fv.

The diffraction problem is constituted by the equations (1), (2), (100), (101) and (4). E x a m p l e . The Landau-Lifshitz law of ferromagnetic materials, without exchange field (see [13] or [16]), is a particular case of this general framework. It corresponds to: s

a:(x) I m x ( h x m ) , h; x) - 7 h x m + iMP(x)

(I)n(m; x) -- g1 i n ~ ( x ) _

ml +1g K 2 (x)

Im - (pn (x). m) pn (x)[ 2 ,

where, 7 is the absolute value of the gyromagnetic factor, that is a universal constant, K~ and a n are positive scalar functions, M~ is the initial magnetization, H~ is a given static magnetic field, and finally, pn is a unit vector called easy axis of the magnetization. One easily checks that this law satisfies the required properties. We have for example: s

x). h-

~',(x) I Ih x ml iMp(x)

> _ o.

446


Apriori estimates. The stability of the coupling between Maxwell equations and the ferromagnetic law relies on the following fundamental a priori estimates. The first one is a consequence of the nonlinear law itself. According to the property (102) (ii), if we take the scalar product of (101) by M~, we see that OtlMT(x,t)[ 2-0

~" IMT(x,t) l - I M ~ ( x ) l

Vt_>0anda.e.x

E ~7. (103) The second one is the equivalent to the classic energy estimate for Maxwell's system:

~

s (E~(t),I-I~(t)) + g~ (E~(t) H~(t)) + p~

(M~(t)) dx

< 0. (104)

Proof. Using the continuity relations (2), we have: d~gV (E~(t), I-I~(t)) + g n (E~(t),I-I~(t)) - -#~

O t M ~ . I-I~ dx.

(105)

7 On the other hand, property (102) (iii) yields, for a.e. x ~_ f ~ ,

0~MT. H7 - 0~MT. (n~o, + VM~'(MT)) _> 0~'(M,'). Consequently - #f

7 Ot MT

. H 7 dx _ O, we have for (Xr, v) E Fx]O, 1[, cOtEk+ 1

Xr, v) x n - - f l Pf Oft (H,H~ +

(

k Y~-j=o(-~) i k - j r~ - f : ~1 Y~-i=o

+fl

n,M, k) (x~, {) d{ r~

Hj (xr, ~) d~

i=o(-{) k-i (C k-i+1 0tE~(xr, {)) x n de

(113) c0tn k+l

(x~.~,)

• n - a,H~+l(x~) v 1

k

• n-

f [ ~.a.II..E.~(x~.,q

d,'

i

- fo 7, E~=o Y~-j=o (_~)k -J rotr(~-), rotr(~-j) E~ (xr, ~c) d~

- f 0 ~ i =k o ( - ~ ) k-i (C k-i+l 0tH~(xr, ~)) x n d~ -- fo Y~i=0 k ( __{)k-i rotr(k-') 0, ( M ,i. ll)(Xr , {) d{ .

(114) 3.3. C o n d i t i o n of o r d e r 2

We need to find the analogous to (94). Making k - 0 in (113) and using (24) yields Ot

E 1 (Xr, u) x n

= - (1 - v) -

(

1 -, ) o # f O t t + ~ r o t r rotr H , H v ( x r )

L 1 #,OttII, g ~

(115)

d~

Let M~ be an approximation of order one, that will be specified later, of the magnetization Mfv. We can write in a formal way: M ~ - M ~ + 0(7/). If we combine this with (115) and (24), we can write an effective boundary condition of order 2, like: .... c0tEv~(Xr) x n -

...

r1

.--...

B~(H,Hv~)(Xr)- r/#f J0 cgttIIliM~c(Xr,V ) dr.

(116)

Now, the question is how to determine M~, using the ferromagnetic law s Let us give the general principle of the answer (i.e. how to determine M~,


449

an approximation of order k of M~)" suppose that H~ is an approximation of order k of H~" H~ - I-I~ + rlkH~, where H~ is uniformly bounded with respect to rl. Thus

OtM~ - s

H~ - VM(I)(M?))+ r/ks

H~).

As M~ has a uniform bound with respect to ~ (by (103)), we conclude (at least formally) that s H~) is also bounded uniformly with respect to Vl. So s VM(~(M~)) constitute an approximation of order k of cOrMS. A natural way to define M~ is then impose . v

cgtg ~ - s

H7 - ~TM(I)(M~)), M7(t -- 0) -- M0.

(117)

Doing so, we see that the definition of M~ amounts the one of HT. This can be done using the asymptotic expansion k-1

H7 - E

~li H~ + O(rl k)

(118)

i--0

To determine H k, we split it into two parts" IIliH k that can be computed by (114) and H~H k that can be computed by applying H~ to (112), that yields 1

Ot (H k- n) -- -Or (M k" n) _ __ E i k= 0 ( - u ) i rotr(') E k - i

(119)

#f

We are going to see that applying expression (118) is sufficient to derive a stable effective boundary condition of order 2, while the condition of order 3 requires slight modifications. We come back now to the second order condition, where, according to the previous considerations, we need an approximation of order 1 of H~. Combining (24) and (119) applied to k = 0, we see that

HT(xr, u)

-

n,H~

+ n~ (M0 - M ~

=

H.. n ~ (xr) + II• (M0 - M~)(xr, v) + O(~/)

u) + O(~/)

(120)

450

Stability of thin layer approximation of electromagnetic w a v e s . . . N

N

So, setting HT(xr,v) - n,~H~,(x~) + H• (116) and (117) leads to following condition

- M~)(Xr,V) and applying

0tEv~(Xr) x n - B2V(H..Hv~)(xr)- ~/#f ]0 IIitM~(xr, v ) dr, where Mf~ satisfies for a.e. (Xr, v) E F x]0, 1[, OtM~ = s

H7o~), Mf~ = O) = M0,

fiTo,(X~, v) = n..H~,(x~)+ n~(Mo - MT)(x~, v ) - V~,(~(MT)(x~, v). (121) According to the following theorem, this condition is stable. Theorem 0.6-

Regular solutions ( E ~ , H ~ , M ~ ) o f {(6), (121)} satisfy,

for all t >_ O,

Ig ~ l - I M o l

a.e. F•

and

_d {E~(~,(t) fi~,(t))+ ~ E~,(t)+ ~ E,,(t)} < 0 dt ' - " where, if we set, cp'7(Xr, t) - IIHfi~(Xr, t), (Xr, t) e ~f

2

$~r (t) - -2 ]]r

"~" ~

l l]ft

F x

It +,

2

]]JO (rOtr r

, L2

C ( t ) - 7#* II H~ (Mo - ~ 7 ) ( t ) I 1 ~ + , , fr •

ll O(MT(t)) dxr dr.

Proof. As (121) has the same structure than (101), property (103) still applies on the scaled domain. From Maxwell's equations and (121), we get (see the proof of theorem 35) 1

d{s dt

'

0tH, MT(xr v)'~n(xr)d~'dxr.

ffI~(t))+~s

(122) For a.e. (xr, v) C F• OtH,.~I~. r

1[,

_ Ot~'I~. r

_ Ot~.inf. (fi~o~ + VM(I)(M:7) - H.(M0 - ~fn))

So, using property (102)(ii), Otn.,M'~. ~"

1 >_Or{ O(M~) + ~[ 111(Mo - M~)] 2 )

Integrating over Fx]0, 1[, shows -.,

JfF

~01 O~IIj.M~(x~, ~~ ~,). ~,'(x~) d~, ax~ ___ - ~ d C(t)

(123)


451

The energy estimate of theorem (0.6) is deduced by identifying the right hand side of (122) with the left hand side of (123). m 3.4. C o n d i t i o n of o r d e r 3 Let us begin by the "Maxwell" part (like for the condition of order two). We point out the differences with the linear case. Equation (95) changes to II,Hf~ (xr, v) - I I , H ~ ( x r ) - vC HO(xr) -

fo

Vr ((Mr~ - Mo)" n ) ( x r , ~) d~, (124)

and relation (98) becomes

OtE~ (x~, ~) • ~ -

-#fOrt

((1 - v)II,Hlv(Xr) - (1 - v2)(C - H ) H . , H ~

- • g f ( (1 - v ) r o t r rotr IIi, H l ( x r ) - (1 - v 2) rotr H r o t r H,,H~

-#,Ott

f l ( n . . M f l ( x r , ~) + f l (2H _ C)ii.,MO(xr, T)

+#fOrt fl f:

Vr ( M ~ n ) ( x r ,

dT) d(

T)dT d~. (125)

If we regroup (125), (115) and (24), written for v - 0, and if we set Mf~ = M ~ + ~M~ + 0(72), an approximation of order 2 of the magnetization M~ (to be specified later), we can write the condition of order three in the following way. 0tE~(xr) • n -

B~(H,H~)(Xr)

-~

fo1[1 + r~u ( 2 / - / -

~

Oft

+ ~ .~ o~

~01 (1 -

r

-

(x~, u) du

(126)

~

~,) v~ ( M ~ . n)(x~, v) a~,

We used the fact that fo f (~) (f2 g(v) dr) d~ - fo (fo~ .f (v) dr) g(~) d~. To obtain an approximation of order 2 of Mf~ we need to compute 111Hf1 . Equation (119), written for k - 1 and combined with (115), leads to (we use rotr Vr - 0) Hfl(xr, v) 9n - - ( M 1. n ) ( x r , v) + (1 - v ) d i v r

~

(n,,H~

-

(127) ldivr (II, (Mo - M~

{) d{.


452

We obtain the following approximation of Hf~ by regrouping (120), (124) and (127):

Hr~(xr, ~') = [1 - ~ uC] II,,H~n(Xr) + rl

Vr ((M0 - M ~ ) . n)(x~, ~) d~,

+ H i (M0 - Mf~)(xr, u) + ~ (1 - u) { divr (IIliH~)(xr) } n

(128) Contrary to the case of the second order condition, when we use (128) (of course without O(~2)) to determine Mf~ via (117) and we couple this with (126) in order to built a third order condition, we are unable to prove stability via energy estimate like in theorem 0.6 (this doesn't mean the instability of the obtained condition). However, we are able derive another expression of the third order condition, that differs from the initial one by terms of order ~3, but that enables us to get energy estimates. The next modifications are suggested by difficulties encountered when trying to prove energy estimates. In (126), we replace the operator [1 +fl u ( 2 H - C ) ] by (1 + ~ u 2 H ) ( 1 - ~ uC). As the two expressions differs by a O(rl 2) term of order 7/2, the new condition differs from the initial one by O(~/3) terms, which is consistent with the order of (126). Remark that T ] r rl d e f

-

n (1 + r/u 2H)

(129)

is the development of order 3 of the volume element df~fn. Meanwhile unstead of using (128) we set, using the usual notation qo~ (xr, t) - II HHn (xr, t), Hfn(xr,u)

-

[1 - ~ u C ] ~ o n ( x r ) +

VrCX(xr,~)d~ (130)

+ (r where r r

,

u ) + ~ ( 1~-o nu()Xd ir v) )r r n

n,

is a scalar field defined on F x]0, 1[ by" - (Mo - Mf ) 9n

rrl

1 divr (IIjl(Mo - M~))(Xr f) df ~

9

(131)

Comparing the right hand side of (128) to the right hand side (130), we see that Hf~ _ ~fn + O(y2),


453

which satisfies the requirement of the construction of the third order condition. In conclusion, if we set r

u) - rn(1 - r/uC)II,M~(x~,

u) -

rl (1 - u) Vr ( M ~ . n ) ( x r , u), (132)

the new third order condition can be rewritten as

N

- Bg(q0n)(Xr)

x n

where r

is related to M~n by (132), and M~ satisfies a.e. Fx]0, 1[,

OtMnf - s

n, n,not),

-

r/#,

/ol

0tEvn(Xr)

OttCn(Xr,U) du,

Mtn(t - 0) - M0,

H,not - H~ - ~TM~(Mfr/).

The field H~ is given by (130). (133) The main justification of this condition is the following a priori estimate.

1 theorem 0.6 remains valid when we replace T h e o r e m 0.7 - For 71 < if-re' the effective boundary condition (121), by condition (133), and the expressions o[ Cnr and E~ by Ern(t) - m2 fr[1 - ~(C - H)]OtqO n . Otqondxr

1

+5-~et f r ( 1 - ~ g ) _

f

.fat

(rotr ~O~)(T)dT

2

(134)

dxr

1

x]0,1[ Proof. Like in the case of the second order condition, we have the classical identity

d .trig2() + ~grn(t)} - - # f dt

; / o 10tCn(Xr

We are going to compute the right hand side of of Cn in (132), one has O r e '7 . ~o '7 -

r '7 0 t M ~

.

[1 -

~ u C ] ~ o '7

u)-q0n(Xr) du dxr.

(135). Using

- ~(1 - u ) V ~

(OtM~

.

(135)

the expression

n). ~.

(136)

[ 1 - 7 Y C]q0n can be expressed using (130), where H~ - H~o, + VMO(M~) as shown by (133). Sue5 doing, we get r n OtMnt 9[1 - r/uC]~o ~ - OtM~. {r n (H,no,

-r

n + VMq,(M~)) - r / ( f o Vr r

d~) - 7(1 - u)(divr ~o") n}.

(137)

454


1 r~? For ~ < WH-T' > 0. Thus, property (102)(iii) yields rV 0tM~. H~ot _> 0. Then using (137) in (136), yields, after integrating over F • [0, 1] and multiplying by -#f,

-#f

/~/o I Ore "1. ~'~ du dxr _< -#f ~-~ ~/~/o ~(I)(M~(t)) r v dxr du

+#f

0t(M7" n) r" du dxr

r

(/o~Vr r d~) 9Otl-I,~I~ du dxr +rl #f /01(1 - u) {/~ Vr (0M~. n). ~v + (0tM7 9n) divr (~a') dxr ) du +7/#f fr fo1

(138) The last term in (138) is zero (integrate by parts). Let A be the value of the the second line of (138). Using the expression (131) of r one has A = #f (Ix + 7//2) where,

/~fo 1r -

n) r ~ du dxr

~/o 11~ 0t

i~

Hz (Mo - MT) r ' du dxr

+~? J(r J~010 t ( ( M o - MT). n ) ( j f 1 divr (II,, ( M o - M~))d~)du dxr and,

12-- ~r ~O1 (~o" Vr r

d~) " OtII,, ~IT du dxr

=

~r~olCvn'Ot(~x

-

/~/ol ((Mo - MT). ~ n) Ot (/1 divr (II,(Mo - ~I7) ) d~) du dxr - 7/

divr(IIjl(Mo-~I~))d~)

/~/ol,r-~ 5,o/~ 1 divr (Hjl(Mo t

dudxr

~I~)) d~

i~ du dxr.

This gives

'/~/o 11~ Ir

A - -#f ~-~

2 r '7 du dxr

which, combined with (138), shows that

-#f

/~/o 1Ore ~ . ~a'~ du dxr

< -~g'f(t)

(139)


455

We obtain the desired energy estimate by combining (139) and (135).

References [1] A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math., 58 (1996), 1664-1693. [2] F. Collino, Conditions absorbantes d'ordre ~lev~ pour des modules de propagation d'ondes dans des domaines rectangulaires, I.N.R.I.A, no. 1794 (1992). [3] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651. [4] B. Engquist and J. C. Nedelec, Effective boundary conditions for acoustic and electromagnetic scattering in thin layers,Ecole PolytechniqueCMAP (France),278 (1993). [5] H. Kreiss, Initial Boundary Value Problems for Hyperbolic Systems, Comm. on Pure and Appl. Math.,13 (1970),277-298. [6] B. Gustafson, H. Kreiss, and A. Sundstrom, Stability theory of difference approximations for mixed initial boundary value problems, Math. Comp.,26 (1972), 649-686. [7] A. Haraux, Nonlinear evolution equations-global behavior of solutions, Springer-Verlag, 1981. [8] J.L. Joly, G. Metivier, and J. Rauch, Global solutions to Maxwell equations in a ferromagnetic medium, S~minaire EDP, Ecole polytechnique (France) ,1996-1997, no. 11. [9] K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations, IEEE Trans. Antennas and Propagat., AP-14 (1966), 302-307. [10] A. Visintin, On Landau-Lifchitz equations for ferromagnetism,Japan J. Appl. Math.,2 (1985), 69-84. [11] L. N. Trefethen, Group velocity interpretation of the stability theory of Gustafsson, Kreiss and SundstrSm, J. Comp. Phys., 49 (1983), 199217. [12] P. Joly and O. Vacus, Mathematical and numerical studies of ld non linear ferromagnetic materials, In Numerical Methods in Engineering 96, ECCOMAS, 1996. [13] P. Joly and O. Vacus, Mathematical and numerical studies of non-linear ferromagnetic materials,M2AN, (1997). [14] I. Terasse, R~solution math~matique des ~quations de Maxwell instationnaires par une m~thode de potentiels retard~s, Ecole polytechnique (France), 1993. [15] Y. Choquet-Bruhat, G~om~trie diff~rentielle et syst~me ext~rieur, DUNOD, Paris, 1968.

456


[16] O. Vacus, Mod@lisation de la propagation d'ondes en milieu ferromagn@tique, Ecole Centrale de Paris, 1997. [17] H. Haddar and P. Joly, An Asymptotic Approach of the Scattering of Electromagnetic Waves by Thin Ferromagnetic Coatings,Mathematical and Numerical Aspects of Wave Propagation,Siam, (1998) ,June. [18] H. Haddar and P. Joly,Conditions @quivalentes pour des couches minces ferromagn@tiques, @tude du probl~me monodimensionnel,I.N.R.I.A.,3431, (1998),May, Th~me 4. [19] H. Haddar and P. Joly, Effective Boundary Conditions For Thin Ferromagnetic Layers; the 1D Model. , (99) ,Submitted to Siam J. Appl. Math.. [20] H. Haddar and P. Joly, Stability of thin layer approximation of electromagnetic waves scattering by linear and non linear coatings, (2000) ,preprint. [21] H. Haddar and P. Joly, Electromagnetic waves in laminar ferromagnetic medium. The Homogenized Problem., Mathematical and Numerical Aspects of Wave Propagation, Siam, (2000) ,July. Houssem Haddar and Patrick Joly INRIA, Domaine de Voluceau-Rocquencourt BP 105 78153 Le Chesnay C@dex France E-mail: houssem.haddar@inria, fr, [email protected]

Studies in Mathematics a n d its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter

20

R E M A R Q U E S SUR LA LIMITE c~ ~ 0 P O U R LES FLUIDES DE G R A D E 2

D. IFTIMIE

R 6 s u m 6 . On consid~re la limite a ~ 0 dans l'~quation des fluides de grade 2. On montre la convergence faible des solutions vers une solution faible de l'6quation de Navier-Stokes, en supposant que les donn~es initiales convergent faiblement dans L 2. A b s t r a c t . We consider the limit a --+ 0 for the equation of the second grade fluids. We prove that weak convergence of the solutions to a weak solution of the Navier-Stokes equation holds by assuming t h a t the initial d a t a weakly converges in L 2.

1. Introduction I1 existe dans la nature des fluides qui n'ob~issent pas aux classiques ~quations de Navier-Stokes. Des modules plus compliqu~s ont dfi ~tre d~velopp~s pour les ~tudier. Ainsi, Rivlin et Ericksen [10] introduisent les fluides de type diff~rentiel. Un cas particulier de ces fluides est constitu~ par les fluides de grade 2. L'analyse de Dunn et Fosdick [6] montre que l'~quation d'un tel fluide est donn~e par Ot(u - a A u ) - v A u + ~-~.(u - a A u ) j V u j J +u. V(u-

a/ku) - -Vp

+ f,

(1)

div u - 0, oh a > 0 est une constante mat~rielle, v > 0 est la viscosit~ du fluide, u le champ de vitesses et p la pression. Pour a - 0, on obtient les ~quations classiques de Navier-Stokes Otu- vAu + u. Vu - -Vp

+ f,

divu-0,

(2)

458

Remarques sur la limite a ~ 0 pour les fluides de grade 2

de sorte que l'~quation du fluide de grade 2 est une g~n~ralisation simple des ~quations de Navier-Stokes. Les premiers r~sultats math~matiques pour les fluides de grade 2 ont ~t~ obtenus par Cioranescu et Ouazar [4]. Ces auteurs montrent l'existence et l'unicit~ des solutions, globale en dimension 2 et locale en dimension 3, pour des donn~es initiales appartenant ~ H 3. L'existence et l'unicit~ globale des solutions tridimensionnelles ont ~t~ obtenues par Cioranescu et Girault [3] pour des donn~es initiales petites dans H 3. La m~thode de d~monstration repose sur des estimations d'~nergie. Un autre point de vue est adopt~ par Galdi, Grobbelaar van Dalsen, Sauer [7] et Galdi, Sequeira [8]. Ces travaux utilisent une m~thode de point fixe pour obtenir des r~sultats similaires. Tous ces r~sultats sont ~nonc~s dans des domaines born~s mais l'extension ~" ne semble pas poser de difficultY. Une question qui se pose naturellement est de savoir si les solutions des ~quations du fluide de grade 2 convergent vers une solution des ~quations de Navier-Stokes lorsque a --+ 0. La r~ponse n'est pas ~vidente et ne d~coule pas des travaux precedents car toutes les estimations pr~c~dentes "explosent" lorsque a -+ 0. Le but de ce travail est de montrer que la convergence vers une solution des ~quations de Navier-Stokes a bien lieu, et cela sous des hypotheses tr~s g~n~rales. La seule hypoth~se "artificielle" sera la borne Ca -1/2 pour la norme H 1 de la donn~e initiale. Avant d'~noncer nos r~sultats, rappelons un r~sultat classique d'existence des solutions faibles pour l'~quation de Navier-Stokes qui est du ~ Leray [9], voir aussi [5], [12]. On appelle solution faible des ~quations de Navier-Stokes sur [0, T) un champ de vecteurs de divergence nulle

u e C~([O,T);L2)nL~oc([O,T);H 1) qui v~rifie l'~quation (2) au sens des distributions. Le th~or~me classique de Leray affirme l'existence d'une telle solution, unique en dimension 2, d~s lors que u0 e n 2, divu0 = 0 et f e n~oc([O,T);g-1); de plus, on peut supposer que cette solution v~rifie l'in~galit~ d'~nergie suivante: [lu(t)ll~ + 2 ,

IIVu(T)II~ dT 0 on (0, c~). Classically a(x) = x 4 and that is the case we will treat here. Physically f must satisfy the boundary conditions : lim a(x)(Oxf + f + f2)(x) = lim a(x)(Oxf + f + f2)(x) = 0, X---~O

X--+OO

(1.3)

470

Remarks on the Kompaneets equation

which express the conservation of photon number density. These nonlinear boundary conditions can be satisfied by seeking a solution in an appropriate functional space. We point out here the fact that equation (1.2), written formally in an expanded form, reads:

o~f = x 2 0 ~ f + (~20~f + 4xo~f) + 4~f + ~20~(f2) + 4~2f 2.

(1.4)

As far as the order of the principal part of the differential operator and the degree and the type of nonlinearity are considered, equation (1.4) reminds of the Burger's equation

0~ = O~u + 0~(~). However it is clear that the singular term X2Oxxf, as x ---, 0 +, makes equation (1.4) more delicate to handle. But we attract the reader's attention to the fact that this is not the only difficulty with equation (1.4). Indeed, the terms x20~f and x f add some degree of difficulty to the appropriate study of that equation, especially if one wishes to prove a local or global existence theorem in a Banach space of the type LP(d#) for some p E [1, +c~] and some positive measure d#. In an attempt to convince the reader, we may give the following example of a linear equation, which contains terms of the above mentioned nature (however slightly different because we want to show an explicite example). For x E ]R consider the linear equation 1

{ o~(t, x) = o ~ - xO~ + 4~2~

(1.5)

~(o, ~) = ~o(x). Now the principal part of the differential operator is Ox~u, which is not singular, but if one tries to prove a global existence result for a given u0 C LP(]R) for 1 < p < +c~, one sees that the usual techniques do not apply. For example when uo(x) := e x p ( - x 2 / 4 ) , then one may show that the unique solution of (1.5) in LP(]R) is given by

u(t,x) "= (2t + 1)-1/2et/2 exp/'[ ( 2 t - 1)x2}\ \ 4(2t + 1) / " Clearly this solution blows up in LP(IR), and this for

any p, at time Tmax -

1/2. This is to emphasize that, in some cases, it is not a good idea to consider such terms as xOxu or x2u/4 as perturbations of Oxxu, the principal part of the differential operator. It is also clear that if one considers the solution

O. Kavian

471

of (1.5) in the weighted Lebesgue space LP(d#), where d# "= e x p ( - x 2 / 2 ) , with the linear operator u ~-~ Ox~u-xOxu, such difficulties are easily avoided. Regarding the Kompaneets equation (1.2)-(1.3), we are going to construct a solution in the Hilbert space

{ /0

L2 "= u ;

lu(x)12xe~dx < c~

}

(1.6)

endowed with its natural norm and scalar product. To do so we consider the unbounded linear operator ( B , D ( B ) ) defined by

Bu "= x G u + ( x - 1)u = e-Xx20~(x-le~u),

(1.7)

D(B) "= {u E L2 ; Bu E L2}.

We will show that B is closed and that D(B) = D(B*). Then, as one can be convinced after some simple calculations using the fact that

B* u = - x O x u - 3 u ,

and

B* Bu = - x 2 0 ~ x u - ( x 2+3x)O~u-(4x-3)u,

we write equation (1.2) in the following form:

I Off + B * B f = - B * ( f + x f 2) f(t,.)eD(B*B)

on (0, T) x (0, c~) for t e (0, T)

(1.8)

f(0, . ) = knit(')In this setting the boundary conditions (1.3) are derived from the fact that we seek solutions satisfying f(t) E D(B*B), when fInit is an appropriate given initial data. Our main result is the following. T h e o r e m 1.1. - For any given fInit E D(B*B)), there exists T > 0 such

that qu tio (1.8)

u.iq,,

olutio. f e C([O, TI,D(B*B)). D

oti.g

by Tmax "-- T m a x ( f I n i t ) the supremum of all such T's, then either Tm~x = +ce or Tmax < (x~ and lim [[xu(t)lTo~- +oo. t--~Tmax

Also a comparison principle holds for solutions of (1.8), i.e. if two intial data fInit, fInit E D(B*B) satisfy fInit ~ fInit then the corresponding solutions satisfy f (t) _ 0,

which are the classical Bose-Einstein equilibria distributions. Now it is clear t h a t by the comparison principle stated in theorem 1.1, if 0 0 then Tmax = +oo. However we point out the fact that our functional space setting is not optimal, as the Bose-Einstein equilibrium ~0 does not belong to L2. Before going into the details of the proof of these results, we would like to mention t h a t the main ingredient in the proof of T h e o r e m 1.1 is the following Nash type lemma, which, we believe, is interesting in its own right. L e m m a 1.3. - Let H be a Hilbert space, ( A , D ( A ) ) a densely defined, dosed linear operator acting on H and a Banach space X such that D ( A ) N X is dense in H. Assume that A*A generates the continuous semigroup S(t) := e x p ( - t A * A ) which satisfies

V~o e x,

IIS(t)~ollx _< co II~ollx.

(1.9)

Then the following properties are equivalent: (i) There exist Cl > 0 and fl > 0 such that for all u c D(A) r3 X one has

[llt[[~/+/3 _~ C1 [IAu]]~ IIu[l~x.

(1.10)

(ii) There exist c2 > 0 and fl > 0 such that for all uo C X one has Vt > o,

]lS(t)uollH 0 and fl > 0 such that for all ~ c

L l(d#) and all t > 0 one has

Ils(t)~lloo _< c2 t-~/2ll~lll. Moreover if in either of above properties one has /3 > 2 , then they are equivalent to the following Sobolev embedding: 3). there exist two constants c3 > 0 and t~ > 2 such that for all ~ E D(L) one has ]]PI[2~/(Z-2) _< c3 (Lp]p). (The n u m b e r / 3 may be thought of as a geometrical dimension for the underlying measure space). The remainder of this note is organized as follows. In Section 2 we establish some of the properties of the operator B defined in (1.7) and we prove lemma 1.3. In Section 3 we prove local existence in time and uniqueness in an appropriate space using semigroups and fixed point theorem. In Section 4 we study the qualitative properties of the solutions and we discuss global existence for some solutions, as well as demonstrating finite time blow-up for some initial data.


474 2. P r e l i m i n a r y

results

In order to solve the Kompaneets equation (1.2), we introduce the Hilbert space L2 :=

{ fo lu(x)12xe~dx } u ;

< oo

(2.1)

with its natural inner product ( u [ v ) : = f o u(x)v(x) xe~dx and associated norm. On this space we consider the differential operator (B, D(B)) acting on L2 and defined by

B u "- e-~x20~ ( ~ u )

= xO~u + (x - 1 ) u

(2.2)

with domain

D(B) :=

{

u e L2, 9

/o

xhe -

~

I(

O~ e~u(x) X

)l

2d x

< oe

}

.

(2.3)

In order to determine the adjoint of ( B , D ( B ) ) we introduce an operator (B1,D(B1)) (which actually is the formal adjoint of B) by setting

B l u := -(xOxu + 3u) = -x-2Oz(xau), D ( B I ) := {u e L2 ; B l u r L2}.

(2.4)

As usual for a closed operator (B, D(B)) we define its graph norm by

II~llD(m := (11~112+ IlBull 2) 1 / 2

for

u C D(B).

L e m m a 2.1. - (B, D(B)) and ( B 1 , D ( B , ) ) a r e densely clel~ned closed operators in L2 and C~(O, oo) is dense in D(B) and in D(B1) endowed with

their respective graph norms. Moreover for any u c D(B) U D(B1) one has the boundary limits lira xu(x) = 0 = lira e x/2 xa/2u(x). x--+0

(2.5)

x--* Cx:~

In particular there exists a constant C > 0 such that for u E D(B) (resp. u e D(B1)) one has II(x + xa/2e~/2)u[Io~ < CIlullD(m (resp.

II(z

+

xa/2e~/~)ull~ < CIi~IID(B,)).

Proof. The density of C ~ ( 0 , oe) in D ( B ) , or D(B1), equipped with the graph norm is straightforward. Let u be in D(B); for any 0 < xa < x2 < oe we have the basic estimate : X2

eX'uXx'l=Xx

(eX,X,x )dxl = J

3xox xl

1

(2.6)

O. Kavian

475

Using the rough estimate obtained from

( f ~ 2 --~dx) e~ 1/2 <eX2 /2 ( fz ~ d x l / 2 1

--

1 "-~)

-

ex2/2

--

2X21'

(2.7)

we begin with the small x limit. From the above rough estimate and (2.6) we get XllU(Xl)I

Letting

--- x 2

X l --~

oxl l UXl( X 1)1

I

~ X2 eX2u(X2)

9

1

x2/2

0 in the right hand side, followed by lim

x---+O

x2

12xeZdx)

X2 X2 ~

0,

yields the result

xu(x) = O.

The large x limit is obtained in two steps. First, (2.7) and (2.6) give

eZ2/2]u(x2)l

--e

_x2/2e~2u(x2) x2

X2

0. Using this property one can develop existence and uniqueness theories for semilinear perturbations of the

478

Remarks on the K o m p a n e e t s equation

heat equation. This so-called ultracontractivity property may be proved by knowing the heat kernel and using Young's inequality for convolutions. However (see E.B. Davies [2] for a general discussion) for general semigroups one may use the equivalence of properties 1), 2) and 3) given at the end of section w 1. Here we establish a Nash inequality related to B* in order to prove the ultracontactivity of the semigroup e -tBB* . We begin with the proof of the general lemma 1.3. P r o o f of Lemma 1.3. Assume that property (i) of lemma 1.3 is satisfied. Let uo E X A D ( A ) be given and set u(t) : - e-tA*Auo be the solution of Otu - - A * A s ,

u(O) - so.

Then if ~(t) := Ilu(t)ll~, one has ~'(t) = - 2 1 1 A u ( t ) l l fact that Ilu(t)llx 2c~-lcoZlluoll X- z from which we infer that

~(t)-z/2 > ~ci-~coZll~ollx~t + ~(o) -~/2 >_/3~i-lCoZll~ollx~t. Clearly this yields (1.11), that is Ilu(t)llH (C1/~)I/J3Co

0, i.e. setting

( c,~lluoll~-) ~/(~+2) t.--

DIIAuolI2H

one obtains inequality (1.10) of the lemma, m In order to get the regularizing property of the semigroup e - t u B * , we apply lemma 1.3 to A := B* by proving inequality (1.10) when x

{

.-

~ ; II,~llx " -

/o

lu(x)leX/2dx < c~

}

.

(2.13)

We begin by seeking a pointwise estimate for u E D ( B * ) . L e m m a 2.6. - For any u C D ( B * ) = D ( B ) we .have

e-X~2 l u ( ~ ) i _< v " i ....... x

IIB*uJl 1/2 I1~il 1/2

Proof. For u E C ~ (0, c~) we have y3u(y)Ogy(y3u(y))dy = 2

(X3U(X)) 2 -- --2

yhu(y)B*u(y)dy

y 4 e - y , lu(y)B*u(y)]yeYdy.

~ 2 X

Now for 4 _ 4 we get lu(x)[ 2 < 2x-2e-~llullilB*uil , which is the desired result for x >_ 4. beginning with the identity (X3U(X)) 2 - - 2

'0 x

When x < 4 we argue similarly

y3u(y)Oy(y3u(y))dy.

m Next we prove t h a t the semigroup e -tBB* acts on X. L e m m a 2.7. - The semigroup e -tBB* induces a contractive, p o s i t i y i t y _preserving semigroup on X ; more precisely we have

lle-*BB*uoIIx

< e-Tt/4lluollx.


480

Proof. It is sufficient to prove the lemma for uo E X such that Uo _> 0. Now for uo E C ~ ( 0 , cx~), and uo > 0 set u(t) := e-tBS*uo, that is Otu = - B B * u and u(0) = uo. One has u(t) > 0 and u(t) E D(BB*) while d

dt

f0

u(t)eX/2dx

BB* u eX/2dx oo

e x

= fO x2e-X/2Ox("~Ox(x3u))dx" But integrating by parts twice yields _d_ / u(t,x)eX/2dx = fo ~176 u(t,x)x30z ( 1~-~eX/2(2- gx ) dx

dt

= - fo ~176 u(t,x) IX-~ + (x -4 4)2]eX/2d x < - 7 fo ~ u(t x)eX/2dx, 4 and the lemma follows for uo _> 0. 1 Finally in the following lemma we state the desired regularizing property of the semigroup e - t B B * . L e m m a 2.8. - For any u E X N D(B*) one has

II~[I6 < 4[[B*~l1211ull 4 X

~

and for ali t > 0 and uo E X one has Ile-'BB'Uoll _< t-1/4 Iluollx.

(2.14)

Proof. Using the uniform estimate given in lemma 2.6 we have I1~112 =

/0

[u(x)12xeXdx =

/0

x}u(x)l lu(x)leXdx

< v~fIB*ulll/2llul]~/2

/o

[u(x)leX/2dx,

~nd hence 1[~113/2 _ v~llB*~lll/211~lIx. Next, using lemm~ 2.7, ~s ~ corollary of lemma 1.3 we obtain (2.14). 1 Combining this result and lemma 2.5 (ii) we have the following regularization result: C o r o l l a r y 2.9 . - For u0 E X and all t > 0 one has

IIB*e-tBB*uoII 0, we denote by C([O,T],L2) the space of continuous functions from [0, T] into L2, equipped with the norm

II~IIc([0,T1) :-- sup II~(t)ll. 0 0 such that equation (2.12) has a unique solution f e C([O,T],L2)n CI((O,T),L2). Moreover f e C ( [ O , T ] , D ( B ) ) a n d in particular f is a strong solution to

Otf -- - B * ( B f + f -4- x f 2) f ( o , ~) - fi.it

(~).

on (0, T) x (0, oo)

on (0, oo)

Moreover if Tmax is the maximal time of existence, then either Tmax = +oo or else one has Tmax < +cxD and limtTT,,ax IlB f(t)ll = +oo. Proof. We begin by checking that q : D(A) ~ L2 is locally Lipschitz. Using the pointwise estimate of lemma 2.6 and the fact that B*(xu 2) = 2xuB*u + u, we have easily I1~(~) - ~(v)ll ~ c (1 + IfB*ull + liB*vii)[IB*u - B'vii and also

II,I'(U-l~) - r

~ C (1 + II~ll + Ilvll)11~ - vii

(3.3)

O. Kavian

483

for some constant C > 0 (we use also the fact that [lull _< IIB*ull by lemma 2.4). Now for u c C([0, T], L2) we define: t

(F(u))(t) := A e - t B B * f I n i t -

Ae-(t-s)B*St~(A-lu(s))ds.

It is straightforward to show that

tlF(u)(t) - F(v)(t)] I 0.

Proposition

4.1. - Assume that fl,Init _~ f2,Init are two initial data in D(B) and denote by fl and f2 the corresponding solutions in Cl((o, T), L2) N C ( [ O , T ] , D ( B ) ) for s o m e T > O. T h e n one has fx ~ f2 on

[0, T] x (0, cx3).

In particular if fInit ~ O, then f >_ O. Proof. The argument is a classical one: set g := fl - f2 and q(t,x) := x ( f i -F f2); recall t h a t due to the above remarks q E LCc((O,T) x (0, oc)). As g satisfies the equation

Otg + B*Bg = - B * (g + qg), upon multplying this by g+ in the sense of the scalar product of L2 we get

d~

Ig+(t'x)[2xeXdx + -

/o /o

_< e

/0

IB(g+)(t'xl2xeXdx-

g+B(g+)(t,x)xe~dx -

IB(g +)(t,

/o

q(t,x)g+B(g+)(t,x)xe~dx

zl2ze~dz

+ (c(~)+ I]q(t,-)ll~)

Ig+(t,x)[2xe~dx

where we have used Young's inequality a/3 _< ~a 2 + Ce/3 2. Choosing ~ = 1/2 for instance, Gronwall lemma and the fact t h a t g+(O,x) - 0 imlpy g+ (t, x) -- 0, t h a t is fl _< f2II The same observation leads to the following: Lemma

4.2. - ff f is a solution such that Tmax < cx~, then l i m s u p I[zf(t, ")[1~ = +oc. t TTm~x

Proof. Indeed, assuming that Tm~x < oc and Ilxfll~ ~ c for some constant C > 0 and all t < Tma• multiplying equation (2.11) by f, using the approach developed above we get d~

[f(t,z)

,2xeXdx + -~1/o

]B f(t,x)[2xeXdx 2

there exists T, (ao) < oo such that T ( T, (ao). Proof. Let h(t) "- f o f (t, x)~(x)x2dx. One checks t h a t h'(t) -- = _

x4(Ozf + f -t- f2)Ox~(x)dx

~00~176

/o

+

/o

( f + f2)x4

(z)d


486

where we have performed another integration by parts on the first term involving O~f. As O~(x4~(x)) = (4x 3 - x4)~(x), we get finally

h'(t) =

/0

f(t,x)(2x 2 - 4x)x2~(x)dx +

Upon using the fact that 2x 2 inequality

/0

f(t,x)2x4~(x)dx.

4x > - 2 in the first term and Jensen's

J~o f(t'x)2x4~(x)dx >-

(/0

f(t'x)x2~(x)dx

in the second term, we end up with the differential inequality ht(t) >_ - 2 h ( t ) + h(t) 2, or equivalently

(e2th(t))' >_e -2t (e2th(t)) 2 " This implies that e -2t

1

0 < h--~ < h(0)

1 2 ~

e -2t 2 '

and therefore T < T.(ao)"= -log((h(0) - 2)/h(0))/2. One may check also that the test function ~(x) " - e -x may be replaced by a function of the type ~(x) "- e - ~ for some )~ > 0. In this case one may see that if the initial data satisfies:

j~

oo fInit (x)e- ~x x 2dx >

then the corresponding solution blows up in a finite time.

B

References [1]. R.E. Caflisch and C.D. Levermore, Equilibrium for radiation in a homgeneous plasma Phys. Fluids, 29 (1986), 748-752 [2]. E.B. Davies, Heat Kernels and Spectral Theory Cambridge Tracts in Mathematics 92, Cambridge University Press, Cambridge, UK, 1989 [3]. E.B. Fabes and D.W. Strook, A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash, Arch. Rat. Mech. Analysis, 96 (1986), 327-338

487

O. Kavian

[4]. [5]. [6]. [7]. Is]. [9]. [10].

M. Fukushima, On an Lp estimate of resolvents of Markov processes~ Research Inst. Math. Science, Kyoto Univ., 13 (1977), 277-284 O. Kavian and C.D. Levermore, On the Kompaneets equation, a simplified model of the Fokker-Planck equationIn preparation A.S. Kompaneets, The establishment of thermal equilibrium between quanta and electrons, Soviet Physics JETP, 4 (1957), 730-737 (translated from J. Exptl. Theoret. Phys. (USSR), 31 (1956), 876-885) J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954 A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Math. Science 44, Springer, New-York, 1983 M. Reed and B. Simon, Methods of Modern Mathematical Physics, (volume IV, Analysis of Operators)Academic Press, New York 1978 N.Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Func. Analysis, 63 (1985), 240-260 Otared Kavian Laboratoire de Math~matiques Universit~ de Versailles 45, avenue des Etats Unis 78035 Versailles cedex France E-mail:

Appliqu~es (UMR

7641)


Studies in Mathematics and its Applications, Vol. 31 D. Cioranescu and J.L. Lions (Editors) 9 2002 Elsevier Science B.V. All rights reserved

Chapter 22

S I N G U L A R P E R T U R B A T I O N S W I T H O U T LIMIT IN THE E N E R G Y SPACE. CONVERGENCE AND COMPUTATION OF THE ASSOCIATED LAYERS

D.

LEGUILLON,E.

SANCHEZ-PALENCIA and C. DE SOUZA

1. Introduction We consider a class of elliptic singular p e r t u r b a t i o n problems depending on a small parameter ~. The energy space V for z > 0 is strictly contained in the energy space Va for the limit problem (E = 0). Obviously, the dual spaces are such t h a t V~ is strictly contained in V'. Classical singular p e r t u r b a t i o n theory is concerned with the case when the loading f is contained in Va~, SO t h a t the problem for ~ > 0, as well as the limit problem ~ - 0 make sense in the variational formulation. We consider here the case when f E V ' but f r t, where the variational formulation makes sense for ~ > 0 but does not for ~ - 0. The energy of the solution tends to infinity as r tends to 0. Two examples in dimension one, borrowed from [9] are considered in sections 3 and 5, using the formal asymptotic m e t h o d of " matched asymptotic expansions" [14]. The solutions exhibit an i m p o r t a n t layer phenomenon, and the energy of the solution concentrates in such a layer as ~ - 0. Sections 4 and 6 are devoted to a rigorous justification of the above formal results. To this end, the exact problem for u s in the variable x is written in terms of a new variable y - x/G, the " inner variable" for the description of the layer. The p e r t u r b a t i o n problem changes drastically, becoming a sequence of problems in domains depending on r which tend to an u n b o u n d e d limit domain. Moreover, the different terms of the expression of the energy are changed in different ways, so t h a t a new concept of energy appears in the variable y, and this energy remains bounded (after an appropriate scaling of the unknown u e, in the example of sections 5 and 6). Then the convergence is proved using an elementary estimate of the new energy. The problems examined here may be considered as a first a t t e m p t to u n d e r s t a n d asymptotic properties of thin elastic shells, where the structure

490

Singular perturbations without limit in the energy space...

of the limit space V~ depends on the shape and the fixation conditions of the shell. In some cases, the space V~ is very large, going even out of the space of distribution and accordingly V~ is very small, so that loadings f do not belong to V~ (see [6,11,12], and [7] for a model problem). The corresponding difficulties of the numerical computation are mentioned in [4,13]. The special structure of the solutions u ~ with small ~ needs a mesh refinement in the region of the layers, as explained in [9] for the one-dimensional examples of sections 3 and 5. A first attempt to elaborate strategies of numerical computation for the problems of shells is done in section 8 for a problem analogous to that of section 3 in dimension 2. The loading f is a ~I distribution localized along a curve g. One may think to use refined meshes in the vicinity of C, in particular anisotropic meshes involving flatened triangles in the direction normal to the layer as used in other problems involving layers [1,2,3]. Instead of this, we used an iterative adaptive mesh procedure. Starting from a conventional F.E. mesh, new refined meshes are sucessively derived using numerical estimates of the computed solution on the previous mesh [5,8]. Such a procedure has been successfully used for computing fluid mechanics problems involving shock waves, boundary layers and wakes. Traditionally, automatic mesh generator produce "isotropic" meshes where triangles are as close as possible to equilateral ones. In the present case it will lead to meshes containing a drastic large number of elements. Thus, we select a mesh generator (BL2D [8]) able to constructanisotropic meshes, i.e. triangles having a large aspect ratio.

2. Singular perturbations Let V be a real Hilbert space, a(u, v) and b(u, v) two continuous and symmetric bilinear forms oil V. In addition, the form b is coercive, so that it may be taken as scalar product in V. Let the form a satisfy

a(v, v) > O, a(v, v) = 0 ~

(2.1) v = O.

In other words, a 1/2 is a norm on V. Let Va be the completion of V with this norm. Obviously, V, Va and their duals V', Va~ satisfy Y C V~, V~ C V' (2.2) with dense and continuous embedings. We consider the following family of problems with parameter s c (0, 1] : Problem P~ : Let f c V/, find u ~ c V satisfying

a(u E, v) + s 2 b(u ~, v) = < f, v >,

Vv e V.

(2.3)

D. Leguillon, E. Sanchez-Palencia and C. de Souza

491

Obviously, u E exists and is unique. Its energy is defined by 1 [a(u~ , u ~) + s 2b(u e, ue)] . E(u E) = -~

(2.4)

In usual examples, this energy is an integral on a certain domain, and the energy in a part of this domain makes sense. Problem P0 : let f E V~. Find u EVa satisfying

a(u, v ) = < f, v >,

Vv c V,.

(2.5)

with the energy 1

E(u) = -~a(u, u).

(2.6)

T h e o r e m 1.1 - Let f E V~ be fixed. The solutions u ~ and u of P~ and P0 are then well defined and satisfy u e --~ u

in Va strongly

E(~ ~)

, E(~)

(2.7)

(2.8)

We shall not give here the proof of this classical result which may be found, for instance in [13] sect. VI.1.4. In particular we note that (2.8) is a corollary of (2.7), taking v = u s or u in (2.3) and (2.5) respectively. Moreover, ,are have (see [6] or[7]) T h e o r e m 1.2 - Let f C V' and u s be the solution of P~, then a) E(u ~) is bounded iff f c V~', b) If f r V'~, then E(u ~) -----,oc. Remark 1.3. When f ~ Va~, the limit problem does not make sense as a variational problem, but it may happen in elliptic problems that it does in the Lions and Magenes sense [10]. 3. F i r s t e x a m p l e i n d i m e n s i o n

one

Let us consider V =/-/2(0, 1) and the forms 1

a(u, v) =

jr0 ~0

u'(x)v'(x)dx,

(3.1)

~"(x)v"(:~)dx.

(3.2)

1

b(~, v) =

492


The completion of V is Va = H0~(0, 1). We shall identify H = L2(0, 1) to its dual, so that V' = H - 2 ( 0 , 1), V" = H - l ( 0 , 1). (3.3) Let us take f = 5~/2, where 51/2 denotes the Dirac distribution at x = 1/2. We note that

e v',

r v'.

(3.4)

and we are in the situation of Theorem 1.2.b. The problem P~ is: d2

d4 ) +

--

u

~X 4

= 51/2 '

du ~ du ~ ue(0) = uE(1) = ~-x (0) -- -~x (1) = 0

(3.5)

(3.6)

The limit problem P0 does not make sense as a variational problem because of (3.4). Nevertheless, as 6~/2 is smooth near the boundary, it belongs to the space ~ - 2 of [10] and P0 is a "Lions-Magenes problem" (see also Remark 1.3) : d2 dx 2 u = 6~/2 (3.7) u(0) = u(1) = 0

(3.8)

whose solution is U(X)

/ __ ~

X

t x--1

for 0 < x < 1/2 for 1/2 < x < 1

(3.9)

which presents a " Heaviside step" at x = 1/2. The formalasymptotic expansion of u ~ is easily obtained by the method of the matched asymptotic expansions [14]. It appears t h a t u ~ exhibits an internal layer in the vicinity of x = 1//2 which is described by an inner asymptotic expansion in terms of the "inner variable" (3.10) y ~ ~x - ~ 1/2 the leading term of this asymptotic expansion is ley

vOtyjr, =

1

1

..~e-Y m _

2 which is the " smoothed" step.

for y < 0 (3.11)

1 for y > 0


493

There are also " small boundary layers" in the vicinity of x = 0 and x = 1 described in terms of x/e and (1 - x)/e respectively, accounting for the lost boundary conditions on the derivative (see (3.6) and (3.8)). Out of these three layers, the convergence of u e to u is uniform. It is then seen that the energies of u e in the layer (3.11), in the small layers and out of the layers are of orders O(e-1), O(e) and O(1) respectively. We then see that the total energy tends to infinity, according to Theorem 1.2.b ; moreover it " concentrates asymptotically" in the layer (3.11). 4. C o n v e r g e n c e

to the layer

The layer (an internal layer) (3.11) is a well defined function of the variable y. We show that, writting down the problemP~ in the variable y, it is possible to prove that the solutions converge to (3.11); this convergence holds in the topology of some energy space of functions of y, which does not coincide (at least concerning limits as e --. 0) with the energy spaces for the variable x. Let us write explicitly the variational formulation of P~ in the case of Section 3" Find u ~ e H02(0, 1) such that, Yv e H02(0, 1)

fo l(u~'v' + e2uE'v")dx _ --~-~x dv (1/2)

(4.1)

Then we define

v (y) =

+ 1/2).

(4.2)

The problem for U ~ is obtained from (4.1) using (4.2) and an analogous formula for the test function. This gives, after multiplying by e Findu ~ c V~ such that, VW c V~

l/2s [(OyUE)(OyW) + (02yUE)(O2yW)] dy = -OyW(O)

(4.3)

i/2e

where VE - H02(-1/(2e), 1/(2e)). Here, it is useful to consider the functions of this space continued with value zero for l Y i> 1/(2e). In order to define an appropriate " limit space" we note that, as e tends to zero, the " boundary conditions" are sent to infinity, which is not very easy to handle. We also note that the right hand side of (4.3) only contains the trace of the first derivative, so that it is not modified by adding a constant to the test functions. Consequently, we are passing to the limit" up to additive constants". To this end, for each V~ we also consider the space Ve of the functions of H02(-1/2e, 1/2e) defined up to

494


an additive constant. The spaces are ordered by embeding as ~ decreases. Then we define the " limit space" 12 as the completion for the norm

II w 112-

F

~

[(0~w) 2 +

(0uW) 2] dy,

(4.4)

(X)

of the space (4.5) S

The " limit problem" writes- Find U E 12 such t h a t VW E 12

/_§

[(OyU)(OyW) + (02U)(O2W)] dy = - 0 y W ( 0 ) .

(4.6)

We note t h a t it is a classical variational problem in the Hilbert space 12 as the right hand side is obviously a continuous functional on it. Then the solution U is well defined (we shall see later t h a t it is the layer (3.11) up to an additive constant). Then we have 4.1 - Let U s c Vr be the solution of (4.3) defined up to an additive constant, and U the solution of (4.6). Then :

Theorem

rs ----, U

in 12 strongly

(4.7)

Proof. Take W = U s in (4_3). Then, considering for each U s the corresponding equivalence class U s defined up to an additive constant, we have

Ff u~ IIv~ c

(4.8)

and after extracting a subsequence 9 U c-

, U*

in 12 weakly

(4.9)

which implies t h a t the first and second derivatives converge in L2(IN) weakly. Let us fix W belonging to a certain V~ (and then to the Vs with smaller ~) in (4.3). We may write ~ s and 17d instead of U s and W. Then, passing to the limit (4.9) we obtain (4.6) with U* instead of U.As the considered W are in a space dense in 12, we see t h a t U* = U, i.e. the subsequence (and then the whole sequence) tends to U. It only remains to prove that the


495

convergence in (4.9) is strong. Let us denote by B the bilinear form in the left hand side of either (4.3)or (4.6). We have: II 8 ~ - u I1~= B ( 8 ~ - U, 8 ~ - u )

= B ( U ~, Ue) -

2B(U, (7E) + B(U, U)

(4.10)

= - o ~ : ~(o) + 2 o ~ : ~(o) - o ~ u ( o )

where we used (4.3) with W = U ~ and (4.6) with W - U ~ and W = U. But the right hand side of (4.10) tends to zero by virtue of (4.9) as it involves a continuous functional on 1/. m It is not hard to check t h a t (3.11) (up to an aditive constant) is the solution of (4.6). Indeed, the equation associated with (4.6) is

( - 0 ~ + 0 4) u = ~'(y)

(4.11)

and we note that (3.11) is a solution of (4.11) for y ~: 0. Moreover, at y = 0,, has a discontinuity of the second derivative, which implies a 5' term for the fourth derivative, so t h a t v~ solves (4.11). Finally, we must check that v ~ in (3.11) (up to an additive constant) isan element of V. Indeed, the completion process passing from (4.5) to V allows functions tending to two different constants at +c~ and - o o , whereas (4.5) only contains functions vanishing (i.e. equal to a single constant) for large I Y I- Let U tend to the different constants at +oo and - o o . It may be approximated by functions UL tending to the same constant using a matching on a large interval of length L --+ oo ; the first and second derivatives of UL are of order O(L -1) and O ( L - 2 ) , respectively. Then

v~

[I U - UL i1~= O [L(L -2 + L-4)] --+ 0.

(4.12)

As a result, the boundary layer (3.11) is the limit of the functions U~ in the topology of 1~, i.e. in the energy of the " inner problem" for the variable y (which is not the energy for the variable x). 5. S e c o n d e x a m p l e

in dimension

one

We are now considering a second example with (3.1)-(3.3), when the right hand side is given by

f(x) = x - p - 2 + (1 - x) -p-2.

(5.1)

with some p E (0,1/2). We note that in this case f is a second derivative of a function of L 2, so t h a t it belongs to V' = H -2. Nevertheless, it is singular

496

Singular p e r t u r b a t i o n s w i t h o u t limit in the energy s p a c e . . .

at the boundary of the domain, and then it does not belong to the =-2 space and the Lions-Magenens theory does not apply. Moreover, we shall see that the limit of u ~ does not exist in a usualsense. We must perform a re-scaling to prove the convergence to the corresponding boundary layer. The equation is always (3.5) with the right hand side (5.1). The (formal) asymptotic expansion of the solution u e takes the form : ~tCout - -

s

r = s - P v ~ (y) + uin

(5.2)

"~ ...

...

,

Y=

X/E

(5.3)

where "out" and "in" denote the outer and inner matched expansions. Of course there is an analogous inner expansion in the vicinity of x -- 1. Note that the first one is such that its leading term is constant with respect to x; ~y is a constant coming from the study of the boundary layer(5.3). As for the inner expansion, v ~ is the unique solution of to d2

d4 )

+

v~

=

y e (0,

(5.a)

dv o

v~

=

(o) = o

v ~ is bounded on (0, c~).

(5.5)

(5.6)

It appears that (5.4)-(5.6) has a unique solution which tends to a certain constant 3' as y ~ +r This is the constant which appears in the outer expansion (5.2). It is apparent that the boundary layer is somehow "autonomous", as (5.4)-(5.6) is a well-posed problem with a given right hand side. The outer expansion (5.2) is in some sort a "sequel" of the boundary layers: in fact its leading term is nothing but the horizontal asymptotic of the function v ~(y). It is noticeable that an accurate finite element computation of u z needs a very small mesh step h in the layers, whereas a coarse mesh out of the layers works very well. Moreover, if the mesh step in the layers is not sufficiently fine, the computation of the layer is obviously poor, but, in addition, the region out of the layers (which depends on them, as we just pointed out) is also inaccurately computed. Concerning the energy, a simple computation of (5.3) shows that the energy in the layers is of order O(~--2p--1), whereas out of the layers,as the leading term in (5.2) has a vanishing energy, it is of order o(E-2P). We observe again that the total energy tends to infinity and it concentrates in the layers.


497

6. C o n v e r g e n c e in t h e s e c o n d e x a m p l e First we give a variational formulation of the problem (5.4)-(5.6). Let V be the completion of the space of function of H02(0, oc) which vanish for sufficiently large y, with the norm [(0yw) 2 + (0~w) 2] dy.

II w I [ ~ =

(6.1)

It is easily checked as at the end of Section 4, that V contains functions tending to a constant different from zero at infinity. Then, the variational formulation of (5.4)-(5.6) is: Findv ~

VwCV (6.2)

(v ~ ~)v =

y-~-2~(y) ay

We must check the following result: L e m m a 6.1 - The right hand side of (6.2) is a continuous functional on V.

Proof.

As the function y-p-2 is locMy of class H -2, its behaviour at infinity must only be checked. We may consider

fro+~ p(y)w(y) dy where p is a smooth function, equal t o may choose p such that

y-p-2 for

Jo

+ ~ ~(y)dy

Let us construct

~(y) =

~

(6.3) sufficiently large y. We

0.

(6.4)

~(~)d~.

(6.5)

Y

which is smooth and satisfies ~(0) = 0

(6.6)

y-p-2 9 (y) =

p+l

for large y.

(6.7)

498

Singular p e r t u r b a t i o n s w i t h o u t limit in the energy space...

Let us take w C Ho2 (0, c~) vanishing for large y. We have ~(y)w(y)dy

=

~' ( y ) w ( y ) d y

=

~ ( y ) w ' (y)dy

(6.8)

< II 9 IIL~II ~ ' I1~< c

Ii ~ Ilv

which proves the l e m m a as w is any function in a dense set of )2. I In order to prove the convergence to the layer at x = 0, we write the problem PE ( ( 3 . 1 ) - ( 3 . 3 ) ) with the right hand side (5.1) after the c h a n g e : u ~(x) = e - P U E(y),

y = x/e

(6.9)

namely: Find U e e/-/02(0, l / z ) s u c h t h a t for all W e/-/02(1, l / z )

fo 1/~ [(o~u~)(o~w)+ (o~u~)(o~w)] dy

[,,. + (1

.]

(6.10)

The function U E m a y be continued with value zero for y > l / e , so t h a t it is element of ~. T h e proof of the convergence is then analogous to t h a t of Section 4, and even simpler, as we do not deal with equivalence classes. T h e o r e m 6.1 - L e t U E and v ~ be the solutions to (6.10) and (6.2) respectively. Then, U ~ ~ v ~ in V weakly (6.1 1) R e m a r k 6.2. In Theorem 6.1 the convergence is only weak, and the reasoning (4.10) for the strong convergence does not work because of the presence of the term f near y = 1/z. Nevertheless, by the linearity of the problem, we m a y decompose it into two problems for the two terms of f in (5.1). For each one (the solution of which is somehow analogous to that of Section 5), we m a y prove strong convergence in P to the corresponding layer. Oppositely, when we consider simultaneously the two terms in (5.1), strong convergence does not hold, as the limit is the layer in the neighbourhood of x = 1, whereas U ~ bears the energy of both layers. 7. A n e x a m p l e

in dimension

two

In this section we consider a problem analogous to t h a t of Section 3 but in Ft - (0, 1) x (0, 1) of the (Xl,X2) plane. The equation is (~2A2 - A)u ~ -- f.

(7.1)


499

The chosen boundary conditions are not the Dirichlet ones, but special ones allowing on one hand anti-symmetry continuation for x E (1,2) and on the other hand, such t h a t the solutions for f independent of y is it self independent of y. The sake of such a choice is obviously to compare twodimensional solutions with the previously obtained one-dimensional ones. This conditions are u=0 for x = 0, (7.2)

s(Ou/On) = 0 u=0

(7.3)

forx=l

s2Au=0

Ou/Ou=O

for x = 0

(7.4)

forx=l fory=0,

E2(OAu/On) = 0

(7.5) y=l

(7.6)

for y = 0, y = 1

(7.7)

Clearly conditions with the factor ~ only are concerned with s > 0. The spaces V and V~ a r e : V=

{v e g 2 ( ~ ) , u satisfies (7.2), (7.3), (7.4), (7.6)} Y~ = {v 6 g l ( f t ) , u satisfies (7.2), (7.4)}

(7.8) (7.9)

Let us define the bilinear form a e (u, v ) = s 2 / a A u . A v dx + J~ V u . V v dx.

(7.10)

The space Va is classical. As for V, we have L e m m a 7.1 - There exist two constants cl and c2 such that for any u, v in V: l aE(u,v) I0 converges strongly in W~'2(gt) to ~. Moreover, the same estimate as given in (34) holds for the sequence { R ( T n ) H ( O n ) R T (7 n) }n>o

523

R. Lipton and A.P. Welo

and we can proceed along the same lines as in the proof of Theorem 3 to show that lim RF(~ ~, .7~ , e(0 ~, 7~), V@) - RF(O, 7, ~(0, 7), V@).

(52)

to--+0

We now identify minimizing sequences of designs for the R P problem. We consider any nested family of partitions denoted by {T~}~>0. For each value of n we consider the optimal design for the discrete problem R P ~ denoted by (0~, ~ , e(0 ~, ~ ) ) . T h e o r e m 7. - T h e sequence {(0~,~,~(0~, ~))}~>o, is a minimizing se-

quence for the R P problem and satisfies the monotonicity condition: for n < n', R P ~ = RF(-~ ~, ~ , ~(~, zy,~), V~)

< _ R p ~' _ RF(~ 'r

,

,

),

and lim RF(-~ '~, ~'r 6(~ ~, z/,~), V@) - RP.

~---~0

Proof. The monotonicity follows immediately from the fact that n < n' implies that D O C D~). We note that the monotonicity property implies the existence of the limit lim RF(O '~, -~, ~(~'~, ~'~), V@).

t~--+0

Since D~ C Do we have:

R P < RF(~ ~, ~/'r ~(0~, ~/'~), V~),

(53)

for every n > 0. On the other hand, for a nested family of partitions {T~}~>0 and for any given (0, 7, r 7)) in Do, it follows from Theorem 6 that there exists a sequence {(0 ~, 7 ~, r ~, 7~))}~>0 for which:

RF(~ ~, ~ , ~(~, zy,~), V~) _ RE(O h, 7 ~, ~(0 ~, 7~), V~),

(54)

and lim ~--+0+

RF (-0~,-~, ~(0~, ~ ) , V~)

_ ~o > O, V# e D, then the discrete problems (18) and (19) are well-posed. I f furthermore V g = W N, the minimum-residual statement is equivalent to standard Ga/erkin approximation: u n ( p ) = uN'GaI(#), where uN'Vat(p) e W N satisfies

RP~(v;uN'G"~(~);~) = O, Vv e W N.

(27)

A blackbox reduced-basis output bound method for noncoercive...

542

An analogous result applies/'or the dual. Proof. We consider the primal problem (18); analysis of the dual problem (19) is similar. To begin, we recall t h a t Pty N 6. V/V satisfies

(p[~, v)r = t(v), Vv e v N. It thus follows that, for any w N 6- W N,

P ~ wiv = Ptv ~ _ TNwN; from our minimum-residual statement (23) we then know t h a t u N 6- W N satisfies

( T ~ u N, T y v ) v = t(T~v), Vv e W ~.

(28)

We now choose v = u N in (28) and note that, since T ~ u N is the supremizer over V N associated with u ~r,

( T ~ u N , TNu~r)y = a(u N, T f uN; #) _> and thus

1

BNII~IIYIITy~NIIY,

1

Ilu~rllY _< ~--~-Ilellv, < ~lltllv,. We have thus proven stability; uniqueness follows in the usual way by considering two candidate solutions. Finally, we consider V Iv - w N : from standard arguments we know that, for/~N (/t) > /~o > 0, the Galerkin approximation (27) admits a unique solution u g,vat. But since [[pNuN'aal[[y -- O, UN'Gal must be the (unique) residual minimizer, and hence u N = u N'~al. m We can then prove that u/v, C g are optimal. Indeed, we have Lemma

2 - IT ~lv (t~) >_ ~o > O, V# 6 T), then

min Ilu(~) - w NIIv, Ilu(~) - u N(#)IIY -< 1 + ~2 7 ) w"~w" with an analogous result for the dual. Proof. Since for any w N E W Iv, w N - u N is an element of W Iv, we have from (25) t h a t

~NIIwN -- UNIIYIITN(w N -- UN)IIY

Y. Maday, A. T. Patera and D. V. Rovas

u ~, T ~ ( w N

uN); #)

=

a(w ~

-

inf liT.wilY = Z(~) > Z0 > 0, ~v I1~11~

IIw~ll.

as desired.

I

Thus, for V N = Y, the hypothesis of Theorem 1 is satisfied with/~0 =/~0; we are guaranteed stability. To ensure accuracy of the inf-sup parameter - - and hence asymptotic error bounds from Theorem 2 - - we shall first need

L e m m a 4 - If ~M iS C]lOSell such t h a t sup inf [[#_ #m[[ _.+ 0 ~,~z~ ,,,e{1 ..... M}

547


as M --+ oo, and if X(#) is sufficiently smooth in the sense that

II sup IIV~xll IIv < ~ , DE/)

then

inf

w~r E W ~

(35)

IIx(~) - w NllY -~ o, v~ e ~ ,

as M (and hence N)--r c~. Note [l" 11refers to the usual Euclidean norm. Proof. Recalling that X(#), the infimizer, is defined in (7), we next introduce

~N(~) e w y as ~N(#) = X(#m" 0,)), m* (#) = arg

min

me{1 ..... M }

l# - #m I.

Thus

IIx(~,) - ~NCi,)llr

< (

inf

me(1 ..... M }

< (sup --"

I1~- ~11)II sup IiV~xll IIY

inf

DEW me{1.....M}

#E'D

I1~ - ~mll)ii sup II%xll IIv, v~ e z~, #ET)

and therefore for all # E :D, inf

wNEW N

IIx(#)--wNIIv

I(~ N - B)I2B since ~N >_ ~ from Lemma 3.

II

The hypothesis of Theorem 2 is thus verified for the case V N = Y, W N W N. The quadratic convergence of ~N is very important" it suggests an accurate prediction for /~.w and hence bounds - - even if W N is rather marginal. 4.3.2. T h e C h o i c e V N = Y, W N = W~ v

Method 2

In this case the X ( # m ) , m = 1 , . . . , M , are no longer members of W N. We see that Lemma 3 still obtains, and thus the method is stable - - in


549

fact, always at least as stable as W N = W N . Furthermore, since WoN still contains W M and W~ r, we expect i l u - u N IIY and 0]r c g ilY to be small, and hence from Theorem 1 I ( s - sN)(#)I should also be small. There is no difficulty at the level of stability or accuracy of our output. However, Lemma 4 can no longer be proven. Thus not only is (36) of Theorem 3 obviously not applicable, but - - and even more importantly u (38) no longer obtains: we can not expect Bg (#) to tend to B(~u) as N --+ oo. In short, the scheme may be too stable, fiN may be too large, and hence for any fixed a < 1 we may not obtain bounds even as N -+ oo. In short, in contrast to the choice W N = W N , the choice W N = WoN no longer ensures that ~N(#) is sufficiently accurate. In practice, however, BN (#) may be sufficiently close to fl(#) that a ~ g (#) < ~(#) for some suitably small a. To understand why, we observe that, in terms of our eigenpairs (Ti, wi) of Section 2.3, ~(Ti) wi(# m) - 1 Ti.

u(#rn) =

(39)

o___

For "generic" ~, u(# m) will thus contain a significant component of T i-(~) and hence x ( # m ) . It is possible to construct ~ such that s = 0, and hence we cannot in general count on X(# m) being predominantly present in WoN; however, in practice, ~ will typically be broadband, and thus W - WoN may sometimes be sufficient. Obviously, for greater certainty that our error bound is, indeed, a bound, W = W N is unambiguously preferred over W = WoN. 4.3.3. T h e C h o i c e

V N = W N,

W N = W N ~

Method 3

We know from Lemma 1 that this case corresponds to Galerkin approximation, but with W M present in our spaces. We first note that not only does Lemma 3 not apply, but unfortunately we can prove that ~N(#m) < ~(#m), m = 1 , . . . , M : -> -

),

sup inf

sup

>

a ( w , v; flwilYfl

sup _

) (40)

ilr

since X(# m) E W N c Y. Stability and accuracy of the output could thus be an issue, though not necessarily so if fin (#) is close to ~(/~). As regards the accuracy of ~N(#), Lemma 4 still applies, however (36), (37) (and hence (38)) of Theorem 3 can no longer be readily proven.

550


Nevertheless, in practice, ~N may be quite close to/~. To understand why, we recall from Section 2.3 that X(# m) is not only our infimizer, but also proportional to T~-X(#m). It follows that if X(# ~) is the most dangerous mode in the sense that sup

~ ( x ( , ~ ) , ~ ; , ~)

< sup

~(w, v; ~ ) ,

~ew~ IIx(#")llrllvllv - ~ew." Ilwllrllvllr

w e w~,

(41)

then

BN(#') = sup .(x(t.'). ,,; #")

-(x(#"). T..~ x(#"); #m)

since both X(# m) and Turn(X(#m)) are in WN; note that (41) is a conjecture, since the supremizing space here is W[ v, not Y as in Section 2.3. Under our assumption (41), we thus conclude from (40) that ~lv(#m) = ~(#m).

(42)

By similar arguments we might expect/~N (#) to be quite accurate even for general # E D, as both X(#) and TuX are well represented in the basis. (From this discussion we infer that for nonsymmetric problems a PetrovGalerkin formulation is desirable.) The above arguments are clearly speculative. In order to more rigorously guide our choice of V ;v, we can prove an illustrative relationship between the Galerkin V iv = W N, W ;v = W ~ (superscript "Gal")and minimum residual V/v = Y, W N = W ~ (superscript "MR") approximations: T h e o r e m 4 - For 831 # E D,

~XN,Ma(~) < A~,G~,(~),

(43)

where A N,MR and A N,Gal refer to (26) for the minimum-residuM and Galerkin cases, respectively. Proof. We first note that

B N'Gal =

inf sup a(w,v;#) < inf sup a(w,v;#) ,o~w~ ~ewr Ilwllvllvllv -weW,,, ~ev Ilwllvllvllv

__ ~ N , M R

(44) for all # E 2). We then note from (23) that A N , MR

~_

1

~N.MR IIR'" (" uN'M~; ~')IIY'I1Rd"(', cN, MR;u) IIY' 1 or~N, MR

1

IIP~uN, MRIIv lID.N r ~,MR IIv

,r~N.MRIIP~w~IIvlIDN.~'NIIv. VW~v e Wx~, V~oN e W~r,

551

Y . M a d a y , A . T. P a t e r a a n d D. V. R o v a s

where P ~ - W N --~ Y and D N" WIN --+ Y are here defined for V N = Y. Thus A N , MR

1) problems). Our truth space X~f is a linear finite element approximation with 200 elements. We consider the two-parameter Helmholtz equation defined in Section 6.1, with n l - ]0, 0.5[ and fl2 = ]0.5, 1.0[. For simplicity, we present a "compliance" case in which

t(v) = t~

= fo.55

V,

,]0.45

corresponding to an imposed (oscillatory) distributed force for the input and an associated average displacement amplitude measurement for the output. In the below we shall consider the four methods associated with the four choices of spaces of Sections 4.3.1, 4.3.2, 4.3.3, and 4.3.4: Method 1 refers to the choice of Section 4.3.1, V N = Y, W N = W N, and will be denoted by the line pattern (--) in all plots; Method 2 refers to the choice of Section 4.3.2, V N = Y, W N = WoN, and will be denoted by the line pattern (---); Method 3 refers to the choice of Section 4.3.3, V N = W N , W N = W N , and will be denoted by the line pattern ( - - ) ; and Method 4 refers to the choice of Section 4.3.4, V N = WoN, W N = WoN, and will be denoted by the line pattern (..-). In general, subscripts, unless otherwise indicated, refer to the Method index above. Throughout this section we take a - (1.1)-1: it follows from Theorem 2 that a sufficient (though not necessary) condition for bounds is that /~N be within 10% of/~. For most of the results of this section, we choose an effectively one-dimensional parameter space D which is the subset of Z)' = ]11, 11[ x ]1,20[ in which neighborhoods of the two resonance points p - (kl, k2) = (11, 7.5) and p - (kl, k2) = (11,14.4) have been excised such that fl0 = 0.02. (Of course, in practice, we would not know the location of these resonance points, and we would thus consider 7) - 7:)' - - which would only satisfy our inf-sup stability condition, "in practice," as discussed in the previous section.) To begin, we fix M - 3, and hence N - 2M - 6 since we are in "compliance," with #1 = (11,2), #2 = (11,8) and thus #3 = (11,14), and thus 8 M = {#1,#2,#3}; we shall denote this the "M = 3" case. We first investigate the behavior of the discrete inf-sup parameter, the accuracy of which is critical for both the accuracy and bounding properties of our output prediction. In Figures 1 and 2 we plot the discrete inf-sup parameter ~N, i = 1 , . . . , 4, and the ratio ~ N / ~ , i = 1 , . . . , 4, respectively, as a function of k2 for fixed kl (see Section 6.1); recall that the index i refers to the method under consideration. We first confirm those aspects of the behavior t h a t we have previously demonstrated. First, f/N and BN axe never less than t~N, as shown in Lemma 3 and Section 4.3.2, respectively; and ~N > ~N, as must be the case since the inf space is smaller. The choice V N - Y ensures stability. Second, we see t h a t fin _> ]~3N and ~2N _ ~N, as demonstrated in

Y. Maday, A. T. P a t e r a a n d D. V. R o v a s

563

(44) and Section 4.3.4 respectively; the methods with smaller supremizing spaces are perforce less stable. Third, we see (by closer inspection of the numerical values) that BN is never greater than B at the sample points, consistent with (40); in fact, we observe that equality obtains at the sample points, (42), and hence at least in this particular case the conjecture (41) appears valid. Fourth, we notice that ~ can be either below or above ~, and is clearly the least "controlled" of the four approximations. (Indeed, for other parameter values we observe near zero values of BN at points quite far away from the true resonances of the system.)

Figure 1: The discrete inf-sup parameter for Methods 1, 2, 3, and 4 as a function of k2 (see text for legend). The symbol x denotes the exact value of ~.

Figure 2: The ratio of the discrete inf-sup p a r a m e t e r to the exact inf-sup parameter for Methods 1, 2, 3, and 4, as a function of k2 (see text for legend). The thick line denotes the "sufficient" limit: if f~N < 1.1~, bounds are guaranteed.

It is clear from Figures 1 and 2 t h a t B/v is indeed a very accurate predictor of f~ over most of D for Methods 1 and 3; we anticipated this result in Theorem 3 and the discussion of Section 4.3.3. We now study the convergence of BN to B as N increases. For this test we consider a sample s M = {#m, m = 1 , . . . , M}, with the #m randomly drawn from l:); the particular parameter points selected are given in the second column of Table 1. (Note t h a t for a given M, indicated in the first column of Table 1, S M con= sists of all #m, m = 1 , . . . , M.) We present in Table 1 the vaJues of f~N _ fl for Methods i =1, 2, 3, and 4 for k2 = 11 (and hence D = (11, 11)). We note that, indeed, fin converges very rapidly for i = 1 and i - 3 - - the two methods in which we include the infimizers in V n - - whereas for i - 2 and i = 4 we do not obtain c o n v e r g e n c e - not surprising given the discussion

564

A blackbox reduced-basis output bound m e t h o d for n o n c o e r d v e . . .

Table 1:

M

1 2 3 4 5 6

The error B~v - B for Methods i =1, 2, 3, and 4, for k2 = 11, as a function of M. #M

(11, 4.7351) (11,19.0928) (11,11.4848) (11,13.6038) (11, 1.4975) (11, 2.6998)

i=1 1.81e 1.70e 3.52e 9.25e 6.57e 1.90e -

01 01 04 06 09 11

i-2 2.92e + 4.28e 2.76e 5.76e 4.91e 4.81e -

00 01 01 02 02 02

i=3 -1.88e -2.03e -1.24e 3.99e 2.43e 4.49e

-

01 01 04 06 09 08

i=4 1.08e + 4.14e -9.85e 5.31e 4.13e 3.83e -

00 01 02 02 02 02

of Section 4 . 3 . Note also t h a t whereas the convergence of M e t h o d 1 is (and m u s t be) monotonic, this is not necessarily the case for M e t h o d 3. We conclude t h a t Method 2 a n d in particular M e t h o d 4 are not very reliable: we can certainly not guarantee asymptotic bounds for any given fixed a < 1; for this reason we do not recommend these techniques, and we focus primarily on Methods 1 and 3 in the remainder of this section. However, in practice, all four m e t h o d s may perform reasonably well for some smaller a, in particular since the accuracy of the inf-sup p a r a m e t e r is only a sufficient a n d not a necessary condition for bounds. Indeed, for our M = 3 case of Figures 1 and 2, Methods 1, 2, and 4 produce bounds for all k2 less t h a n approximately 18 and Method 3 in fact produces bounds for all k2 in D; consistent with T h e o r e m 2, bounds are always obtained for all m e t h o d s so long as aB N B) is due to the poor infimizer a p p r o x i m a t i o n properties of W N for larger k2; if we include an additional sample point, ~u4 = (11, 20), we recover bounds for all D. (In fact, even for lower k2 the infimizer approximation is not overly good; but thanks to the quadratic convergence proven in T h e o r e m 3 the inf-sup p a r a m e t e r remains quite accurate.) T h e fact t h a t M e t h o d 3 produces bounds over the entire range is consistent with the "less stable" a r g u m e n t s of Section 4.3.3. However, by these same arguments, in particular T h e o r e m 4, we expect t h a t the bound gap the controllable error in the o u t p u t prediction - - will be larger for M e t h o d 3 t h a n for M e t h o d 1. To d e m o n s t r a t e this empirically, we plot in Figure 3 A N / I s l , i = 1 and i = 3, as a function of k2, for the M = 3 case of Figures 1 and 2. We observe that, indeed, the bound gap is significantly smaller for M e t h o d 1 t h a n for Method 3. Note also t h a t the normalized bound gap is quite large for the k2 at which we no longer obtain bounds for M e t h o d 1; no doubt these predictions would be rejected as overly inaccurate and

Y. Maday, A. T. Patera and D. V. Rowas

565

Figure 3: The normalized bound gap A~/Is I for Methods i =1 and i =3 as a function of k2.

requiring further expansion of the reduced-basis space (thus also recovering the inf-sup parameter accuracy and hence bounds). As regards the convergence of the bound gap, we present in Table 2 convergence results for A ~ , i = 1 and i = 3, for k2 = 11 (and hence # = (11,11)), as a function of M (analogous to Table 1 for the inf-sup parameter). (Note for Methods 2 and 4 the convergence is slower, with the bound gap typically an order of magnitude larger than for Methods 1 and 3; this suggests that the inclusion of the infimizers can, in fact, reduce the approximation e r r o r - as might be anticipated from (39).) We observe that the differences between Methods 1 and 3 become smaller as M increases; however it is precisely for smaller M that reduced-basis methods are most interesting. "We conclude - - given that the two methods are of comparable c o s t - that Method 1 is perhaps preferred, in particular because we can also better guarantee the behavior of the inf-sup parameter. Note that the difference in the true error for Methods 1 and 3 is much smaller than the difference in the error bound for the two methods; this is expected, since the inf-sup parameters do not differ appreciably. It follows that the effectivity (defined in (34)) of Method 1 is lower (and hence better) than the effectivity of Method 3; this is not surprising, since for Method 1 the approximation is designed to minimize the bound gap. We close by considering a second set of numerical results included to demonstrate the rapid convergence of the reduced-basis prediction as N increases even in higher dimensional parameter spaces: we now consider 7:) = ]1, 20[ x ]1, 20[ (without excision of resonances, and hence satisfying our infsup condition only "in practice"). In particular we repeat, the convergence


566

Table 2:

Table 3:

M 1 2 3 4 5

6 7

The bound gap for Methods i =1 and i = 3, for k2 = 11, as a function of M.

M

#M

I 2 3 4 5 6

(Ii, 4.7351) (II, 19.0928) (II, 11.4848) (11, 13.6038) (11, 1.4975) (11, 2.6998)

i=1 2.23e 2.13e 5.37e 4.01e 6.43e 1.65e-

04 04 06 08 11 14

i=3 1.73e 6.21e2.68e 5.80e 6.50e1.62e-

01 01 05 08 11 14

The b o u n d gap and effectivity at # = (11,17), as a function of M , for Methods i = 1 and i - 3, for the two-dimensional p a r a m e t e r space Z~ - ] 1 , 20[ x ]1, 20[. #M (7.5388, 14.2564) (2.9571, 7.1526) (4.2387, 9.0533) (17.7486, 15.9503) (9.7456, 14.0523)

(3.8279,16.3388) (11.2113,

8.0970)

1.30e 1.22e 5.82e 1.10e 2.99e 2.81e 5.40e-

04 04 05 05 08 08 12

24.20 3.51 1.08 2.56 3.47 3.71 3.78

6.09e 3.05e 8.77e 1.24e 3.07e2.88e 5.44e -

04 04 05 05 08 08 12

12.18 11.88 1.49 3.42 3.71 3.97 3.85

scenario of Table 2, but now choose our r a n d o m sample from the t w o dimensional space l) = ]1,20[ x ]1, 20[; we present, in Table 3, the bound gap and effectivity (defined in (34)) for Methods 1 and 3 for a particular "new" p a r a m e t e r point # -- (11, 17). We observe, first, t h a t we obtain bounds in all cases (~/N > 1) m indicative of an accurate inf-sup parameter prediction; second, t h a t the error (true and estimated) tends to zero very rapidly with increasing M , even in this two-dimensional p a r a m e t e r space; and third, t h a t Method 1 again provides smaller bound gaps (and lower effectivities) t h a n Method 3, consistent with Theorem 4 - - though the difference is only significant for very small M. Note t h a t u and the o u t p u t s are order 10 -3, so the relative errors are roughly 1000 times larger than the absolute errors in the table. Results similar to those reported in Table 3 are also obtained if we consider the error over a r a n d o m ensemble of test points # rather t h a n a single test point.

Acknowledgements.

We would like to t h a n k Dr.

Luc Machiels of


567

Lawrence Livermore National Laboratories and Prof. Jaume Peraire of M.I.T. for numerous helpful discussions, and for suggestions leading to simpler and more illustrative proofs. This work was supported by the Singapore-MIT Alliance, by AFOSR Grant F49620-97-1-0052, and by NASA Grant NAGl-1978.

References [1] M. Ainsworth and J. T. Oden. A posteriori error estimation in finite element analysis. Comp. Meth. Appl. Mech. Engrg., 142:1-88, 1997. [2] B. O. Almroth, P. Stern, and F. A. Brogan. Automatic choice of global shape functions in structural analysis. AIAA Journal, 16:525-528, May 1978. [3] I. Babuska and J. Osborn. Eigenvalue problems. In Handbook of numerical analysis, volume II, pages 641-787. Elsevier, 1991. [4] It. E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial differential equations. Math. Comput., 44(170):283-301, 1985. [5] A. Barret and G. Reddien. On the reduced basis method. Z. Angew. Math. Mech., 75:543-549, 1995. [6] R. Becket and R. Rannacher. A feedback approach to error control in finite element method: Basic analysis and examples. East - West J. Numer. Math., 4:237-264, 1996. [7] J. P. Fink and W. C. Rheinboldt. On the error behaviour of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech., 63:21-28, 1983. [8] M. B. Giles and N. A. Pierce. Superconvergent lift estimates through adjoint error analysis. Technical report, Oxford University Computing Laboratory, 1998. [9] M. D. Gunzburger. Finite element methods for viscous incompressible flows. Academic Press, 1989. [10] P. Ladeveze and D. Leguillon. Error estimation procedures in the finite element method and applications. SIAM J. Numer. Anal., 20:485-509, 1983. [11] L. Machiels. Output bounds for iterative solutions of linear partial differential equations. Preprint. [12] L. Machiels, Y. Maday, I. B. Oliveira, A. T. Patera, and D. V. Rovas. Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris, Sdrie I, to appear. [13] L. Machiels, Y. Maday, and A. T. Patera. A "flux-free" nodal Neumann

568

[14] [15]

[16]

[ls]

[19] [2o] [21] [22]

[23] [24] [25]

A blackbox reduced-basis output bound method for noncoercive... subproblem approach to output bounds for partial differential equations. C. R. Acad. Sci. Paris, Sdrie I, 330(3):249-254, Feb 1 2000. L. Machiels, Y. Maday, and A. T. Patera. Output bounds for reducedorder approximations of elliptic partial differential equations. Comp. Meth. Appl. Mech. Engrg., to appear. L. Machiels, A. T. Patera, J. Peraire, and Y. Maday. A general framework for finite element a posteriori error control: Application to linear and nonlinear convection-dominated problems. In ICFD Conference on numerical methods for fluid dynamics, Oxford, England, 1998. L. Machiels, J. Peraire, and A. T. Patera. A posteriori finite element output bounds for the incompressible navier-stokes equations; application to a natural convection problem. Journal o.f Computational Physics, submitted. Y. Maday, L. Machiels, A. T. Patera, and D. V. Rovas. Blackbox reduced-basis output bound methods for shape optimization. In Proceedings 12th International Domain Decomposition Conference, Japan, 2000. To appear. Y. Maday, A. T. Patera, and J. Peraire. A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem. C. R. Acad. Sci. Paris, Sdrie I, 328:823-828, 1999. D. A. Nagy. Modal representation of geometrically nonlinear behaviour by the finite element method. Computers and Structures, 10:683-688, 1979. A. K. Noor and J. M. Peters. Reduced basis technique for nonlinear analysis of structures. AIAA Journal, 18(4):455-462, April 1980. M. Paraschivoiu and A. T. Patera. A hierarchical duality approach to bounds for the outputs of partial differential equations. Comp. Meth. Appl. Mech. Engrg., 158(3-4):389-407, June 1998. M. Paraschivoiu, J. Peraire, Y. Maday, and A. T. Patera. Fast bounds for outputs of partial differential equations. In J. Borgaard, J. Burns, E. Cliff, and S. Schreck, editors, Computational methods for optimal design and control, pages 323-360. Birkh~iuser, 1998. A. T. Patera and E. M. Ronquist. A general output bound result: application to discretization and iteration error estimation and control. Math. Models Methods Appl. Sci., 2000. To appear. J. S. Peterson. The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Star. Comput., 10(4):777-786, July 1989. T. A. Porsching. Estimation of the error in the reduced basis method solution of nonlinear equations. Mathematics of Computation, 45(172):487-496, October 1985.


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[26] R. von Kaenel. Reduced basis methods and output bounds for partial differential equations. Dipl. thesis, MIT, EPFL, 2000.

Yvon Maday Laboratoire Jacques-Louis Lions Universit~ Pierre et Marie Curie Bo~te courrier 187 75252 Paris Cedex 05 France E-mail: [email protected] Anthony T. Patera Massachusetts Institute of Technology Department of Mechanical Engineering Room: 3-266 Cambridge, MA 02139-4307 U.S.A E-mail: [email protected] Dimitrios V. Rovas Massachusetts Institute of Technology Department of Mechanical Engineering Room: 3-264 Cambridge, MA 02139-4307 U.S.A E-mail: rovas~mit.edu




Chapter 25 SIMULATION OF FLOW IN A GLASS T A N K

V. NEFEDOV AND R. M. M. MATTHEIJ

I. Introduction The manufacturing of glass is a complicated and expensive process. The glass is produced in a so-called glass oven or tank. The raw materials such as soda and sand are dumped on one side of the oven, which is heated from above by gas burners. Ovens are constructed in such a way that the glass stays inside for some time, and after about 20 hours flows via feeders to the production lines on the other side. There are, therefore, several processes involved, such as flow and heat transfer and various chemical reactions. In order to study how different oven configurations affect the production, numerical simulation is required, as full scale experimental studies are too expensive to carry out. Since the complexity of the problem is quite appalling, existing codes for tackling this problem are in for improvement and further sophistication, like adaptivity in gridding, to make the simulations more feasible (with respect to memory and computing time). In [6] a local refinement procedure was combined with a glass oven model. The method proposed was based on staggered grid finite volumes [7]. In this paper we study a different way to simulate the glas flow, based on locally uniform grids. The next section describes the mathematical model, namely the equations we are dealing with, the boundary conditions and the specific physical parameters. In the third section we discuss the discretisation procedure. In section 4 the solution method is described, including some implementation details. Section 5 deals with a local refinement technique called local defect correction (LDC). Its application to the glass flow is discussed in the two last sections, viz. a stirred flow in section 6 and bubbling in section 7.

2. Mathematical model The oven in fact consists of an oven proper (melting tank) and a second part, connected by a small channel, from which the glass is going to a feeder

Simulation of flow in a glass tank

572

channel for further processing, leading to the actual product eventually, see Fig.1. The mathematical model consists of equations, describing physical processes occuring in the glass flow, complemented by suitable boundary conditions. In order to restrict this still general model to our particular case of interest we perform a dimension analysis based on glass glass flow properties.

2.1. Main equations Since melted glass can be considered a viscous Newtonian fluid, we can model it by the (incompressible) Navier-Stokes equations

(p(x)u, Vu) = F V - ( p ( x ) u ) = 0.

-

Vp + V - ( # ( x ) V u ) ,

(1)

Here p, p are the viscosity and the density of the glass respectively, and F = (0, O, _pg)T is a gravitational force. Since the glass flow exibits steady state behaviour for large time scales, these equations are written in a time independent form. The motion of the glass is caused by temperature differences; thus the set of equations is incomplete without the energy equation. For a steady lowvelocity flow with negligible dissipation we can derive the energy equation in the following form

V . (Cpp(x)uT) = V - ( k ( x ) V T ) ,

(2)

where k and Cp are the conductivity and the heat capacity, respectively.

2.2. Boundary conditions A typical glass oven configuration is depicted in Figure 1. We can define the boundary values for this geometry as follows: the velocity has Dirichlet values on the inflow, outflow and, assuming no-slip, on the walls too. The top layer is modelled as a symmetry plane, that is the normal component is zero; for the rest we prescribe homogeneous Neumann boundary conditions. For the temperature the situation is different, since the heat loss from the walls and heat influx from the top layer are not known a priori, but can be expressed via a Robin-type boundary condition. In particular we can write -(u,n)lr,~1,~

= uo,

-(u,n)lro~,1,~

= ul,

V. Nefedov and R. M. M. Mattheij

573

Figure 1: Sketch of the glass tank (horizontal and vertical cross sections)

(u, T)lr,~,,o~ -- (u, T)lro~,,o~ -- O, ulr~,~ = 0, (u, n)lr,o~ = 0,

0(u,r) On

Ir,o~ = 0,

OT TIr,~,,o~ = To, ~-~nlro~,,o~ = 0,

0TIr~o = H ( T ) ' OT On ~--~nlr~o,, = Q(T).

2.3. Physical parameters The above described equations, together with the boundary conditions, are still too general, since they might as well represent any viscous flow in a tank. In order to resctrict the model to the particular case of the glass flow we are interested in, we need to specify the parameters of the equation, in particualr the viscosity, the density, the heat conductivity and the heat capacity. The most important of these coefficients are the viscosity and the density. The viscosity does significantly affect the flow pattern; it decays exponentially as the temperature grows (Vogel Fucher Tamman law)

574

S i m u l a t i o n o f flow in a glass t a n k

5

Viscosityof glass

x 104

1 9

|

10(~ 1200 1400 1600_ 1800 2{~ tem-/Yeratuf~K

Figure 2: Viscosity of tv-glass

# g l a , s ( X ) -- # 9 1 a s , ( T ( x )

-- a ~ e b~'/(T-c~').

(3)

The coefficients a t , bz, c~, in (3) are specific for the glass type (tv-glass, window-glass etc). The most significant factor in a glass flow computation is the density. It may be modelled as a linear function of the temperature, more precisely Pgla,,(T) = ap(1 - bp(T - Cp)),

(4)

and changes by only about 10%; nevertheless it drives the flow via the convective term and the gravitation. 2.4. D i m e n s i o n a l

analysis

In order to examine the behaviour of the momentum and the energy equations, in particular to determine whether diffusion or convection prevails, we make the equations dimensionless. Let us rewrite them as new variables 1

1

"2.=

T m Tmin

AT

Here X, U are characteristic length and velocity respectively and AT -T,~ax - Train is a maximum possible temperature difference. The gradient

and the divergence operators in the old and the new variables are related a~

-

v-=

1

:=

1

V. Nefedov and R. M. M. Mattheij

2550

575

Densityofglass

.~2500 E

~.2450 (n q)

2400

'%

23~0~ 12'00 14()0 16'00 18'00 2000 temperature. K

Figure 3: Density of tv-glass

A tilde over the gradient and the divergence thus indicates that it is taken with respect to the new variables. After subsitution the Navier-Stokes and the energy equations will look like U2 -~-(Off, Vfi) = F -

1 U x g p + ~--~9-(#9fl),

(5)

v . (p~) = 0, ATU ~~ AT X Cp(pfi, VT) = ~ V - ( k ~ T ) .

(6)

Since the density p changes by only about 10% we assume p to be constant. After dividing both parts of the first equation of (5) by p U 2 / X we obtain an equation in dimensionless variables

( ~ , 9 ~ ) = P --9 ~ + 9 . ( ~ Here

y=X

1 Vfl) .

1 P=~P"

pU ~F,

By analogy we divide both parts of (6) by A T U C p / X to obtain - ~ (~, VT) = 9 " (R~.1 Pr

9T).

Here the Reynolds (Re) and Prandtl (Pe) numbers are defined by Re :=

pUX #

,

Pr :=

~c~ k

Simulation of flow in a glass tank

576

IN

IN

gAN

gAN

gN

A

gN

^

A

gN

A

gN

g"

9

9

~

'

9

gN

IN

IN

gAN

~N

~N

gN

A

A

A

A

gN

A

gN

~

9

~

A

9

9

9

IN

A

IN

A

IN

A

IN

A

IN

A

A

A

A

A

q

A

9

ab

ab

ab

~k

4

Figure 4: Staggered (left) and collocated (right) grids

The Reynolds number expresses the ratio between convection and diffusion, while the product of Reynolds and Prandtl numbers indicates correspondence between radiative and convective heat transfer. In the case of the glass tank (ommiting physical units) X-20,

U-0.01,

p.~2490, 2 5 _ p < 2 5 0 ,

1.2_2) with a regular b o u n d a r y F, Q - ~tx]0, T[, Z - F x]0, T[, ~/is a regular manifold in F of dimension D o in Le(~t) [2]. For 1 < t~ < oc the domain of A~ is Oy T)(Ae) = {y E W2'~(gt)] 0--~A = 0 on F}.

For g = 1, T)(A1) is the set of functions y in Ll(f~) such that there exists z e LI(Ft) satisfying fn z ( x ) v ( x ) d x = fn y ( x ) A v ( x ) d x for all v e D(A). For any 1 < g < c~, 0 belongs to the resolvent of - A e and there exists 5 > 0 such Re a(Ae) _> 5 (it is a consequence of (A1) and of the fact that a(Ae) is independent of g). Therefore, for a > 0, there exists a constant K = K(g, a) such that

JlA~Se(t)~JJL~(~) ~_ Kt-~ll~[[Lt(a), Henry, [23], A~ is the a-power of Ae). Thanks to this result the following lemma can be established. L e m m a 2.1 [2], [26] - For every 1 < ~ < A < oc with ~ < oc, there exists a constant K1 = K1 (A, g) such that

(2.3)

[LS~(t)~IIL~(a) O. For every I k>0,

i>l,

j>d',

k

N

1

N

a+~+x-~"_o m9 L d' (f~), we have: w(t) = Sd,(t)A~,r = A~,Sd,(t)r Observe that if/3 obeys k < 2/~, then D(A~) r wk'J(f~). Moreover, if d' < j < oc, with (2.3) and (2.4) in Lemma 2.1, we have:

IIw(t)llw~,j(a) = IIA~,Sd,(t)r -
N~D to have g2D + 2~ N 1 1 From estimates for convolution ~-j~+ 89< N ( ; 1- - ~ 1 ) + 1 + / ~ < 1 + (~1 q" we deduce

NN- D- D- 1 may be deduced from the previous one. 2 - Let us prove t h a t the mapping t h a t associates y~, the solution to equation (2.12), with u is continuous from Lq(O,T;L~(~/))into Le(O,T;L~(~)). Let w be the solution of the Cauchy problem (2.6) with a - 0, d - r. If D O" < N-D-2N-D, due to (2.13), we can choose fl > N~D to h a v e N 2 D 2t-~--~--N < N(~21 _ 7)1 + / ~ < 1 ~ ~1 ql. We can conclude with the same arguments as in Step 1. The continuity of the mapping u H Tr.(y~) m a y be obtained in the same manner. 1 -

-

C o r o l l a r y 2.1 - Let y be the solution to equation (2.12).

The mapping that associates y with u, is continuous from Lq(O,T;L~(7)) into L~(Q) for every r satisfying 1 _< r < inf { ( N

N+2 q,

N

N-D+2 (N-D-~)

~,) ( N - 2 - ~ )

N-D N-D-2

I

(2.16)

The mapping that associates yl~ with u, is continuous from Lq(o, T; L a ('7)) into Ls(E) for every s satisfying N-1

N+I

_

l<s N - D then dl < N - D 52 --51 --q and d2 - d l . With assumption (4.2), we have "

N-D #("

2

D

--

1

+~aa + q

N - D - 2 ~

N-D 1 )

g

2a I

1)(N 2

< 2a'

D

2)

§

(4.16)

1

(4.17)

then N(a-1)(N-D-1)

2 ( N - D) 1

/z-

q'

1 -~

q

g

(/~- 1)g(g-

D-

2)

2 ( g - D)

g-D

N ( a - 1 ) ( N - D - 1) N 2 ( N - D) < N- D

1

(4.18)

+2 '

(/z- 1 ) N ( N - D - 2) 1 2 ( N - D) + ~ .

(4.19)

L e m m a 4.2 - Let f~ be in L0(0, T; (L'(f2))g), where ~ > 1 and ~ > 1. We consider the equation: _ 019+Ap=_ cot

divfzinQ,

Op = f ~ . ~ o n E , On A

p(T)=O inf'.

(4.20)

The solution p to equation (t.20) belongs to L I ( O , T ; W I ' I ( D ) ) A Le(0,T; Lr(~)) for all (~,r) satisfying:

7)_ I and I + / ~ + N ( ~ _1 ;1) < 7"1 Thus the mapping

t ~-, t - 8 9 -.!) belongs to Li(O,T). convolution, we have

(

(

Therefore, using estimates on

(T + t - T)- 89189 ~_ C[IhIIL~(O,T;(L,(~))N),

and the proof is complete. 4 - We can proceed in the same manner to prove t h a t the trace of p on ~• belongs to L~(O,T;L~(~/)). m L e m m a 4.3 - Let g be in L$(O,T; L~(F)), where ~ > 1 and ~ > 1. We

consider the equation:

_

Op-~- Ap = 0 in Q,

Ot

Op = g on E,

OnA

p(T) = 0 in ~.

(4.27)

The solution p of the equation (4.27) belongs to Le(0, T; L r ( ~ ) ) for all (~,r) satisfying: N-1 1 N 1 1

~ N - D - l , for all (~, T) satisfying:

N-1 1 N-1 1 1 2---Y- + -5 < 2---7- + -~ + ~

(4.30)

the trace of p on-rx]O, T[ belongs to L~(0, T; L~('r))

~

N-1

1

~ -

-~
1 and • > 1.

Let b be a nonnegative function belonging to L~(O, T; L~(r)) for some (~, ~) satisfying (2.23) and (~.1). Consider the equation --

_

-~

Op

OPot ~-Ap - - div h in Q, ~nA +bp - h. ~ on E, p(T) - 0 in f~, (4.32)

The solution p to equation (~.32) belongs to LI(o, T; WI'I(f~))NLe(O,T; Lr(f~)) for every (~, r) satisfying (~.21). Moreover, the trace of p on E belongs to L~(0,T; n'(r)) for every (~,s) satisfying ~

~ > 1. -- 1 ' -~From Lemma 4.5, and under condition (4.1), it follows that p2 belongs to

L%=-i(T- t,T; wl'r~-; (D)) n L e ( T - f, T; L~(f~)). Then pe e LI(T - t,T; WI'I(g/)) n Le(T - t-, T; Lr(12)). Let ~1 and ~2 belong to L I ( T - t,T; W I ' I ( f ~ ) ) n L e ( T - E, T; Lr(f~)), ar, d let pc, and PC2 be the solutions of equation (4.40) corresponding to ~1 and ~2. Still from Lemma 4.5 and under condition (4.1), we have: [[Pr -P~2 I[L 1 (0,3; W 1,1 (f~))NL e (T-t,T;L" (f't) ) __~ C [ l a [ I L ~ ( T _ ~ , T ; L k ( n ) ) 1 1 ~ I - - ~ 2 [ [ L I ( T _ ~ , T ; W I , , ( 1 2 ) ) n L e ' ( T _ ~ , T ; L , ' ( ~ ) ) ,

where C depends on T, but independent of t. The mapping t C~ fT-t T Ila(X'T)llki~(a) dT is absolutely continuous,

t then 3t- > 0 such that C (fmax{t-~,0} [la(., T)llki~(f~)dT)~

q a-l'

~>~',

s_>

d

1

_g,

a-l' 1

1

=+- 0 b e c a u s e u < q . Now we set ~1 7}' 1 - 2}" 1 Let us prove t h a t the pair (~, s) obeys and 71 = i n f { ( ~ - l ) N( N- D- D - 1 ) , ~-7 - 1 ~ N(,~- l ) ( N - D - 1 ) q 2 ( N - D) If~l ._ q'l

~1 a n d

tion (4.10).

I f ~1

71 __ .!.n--1)(N--D--!)N_D , c o n d i t i o n q,1

=

71 a n d

71

=

~,1

1

g-1

s

2s

< - +

1

+ ~.

(4.46)

(4.46) f o l l o w s f r o m a s s u m p (4.46) f o l l o w s f r o m

1 condition e,

(4.44). If ~1 __-- ,~--lq and ;1 __ (n-1)(N-D-1)N_D , condition (4.46) follows from l condition (4.46) follows from a s s u m p t i o n (4.6). I f ~1 -_- n-1 q and s1 -__- a1' e, (4.45). - D- 1, big enough, so t h a t (4.41) and Due to (4.46), we can choose d < NN- D (4.42) be satisfied. due Second case: a _> N-D-2'N-D-1 choose ~~ = v-l"q Since /2 < NN -- D D -- 12 ' to (4.3), (4.13) and (4.14), we can choose g satisfying (4.43), such t h a t g-

D-q 1 t

D-1

1 < g < (v-i~-g-D-2), N(~-I)(N-D-1) 2 ( N - D)

1 g-1 < ~+2(ND-

1)

u-1 q

g-1 2s

1 (4.47) t-~,

1

(4.48)

and N(~-

1)(N-

D-

2(N-D)

1)

N-

1

< 2(N-D-l)

N-

2g

1

+2"

1 Die Observe t h a t ~1 - ~1 > 0 because u < q, and set ~i _ i n f { ~ -q1 , q,I 7}. distinguish two subcases. When (,~-I)(N-D-1)N_D < ~ N - D - 1 1 -- ~'1 we set -~1 __ (~-I)(N-D-1)N_D . Let us prove

t h a t the pair (~, s) obeys (4.46). If ]1 -__ q,1

1

condition (4.46) follows from

a s s u m p t i o n (4.10). If ~l = trq .... , condition (4.46) follows from a s s u m p t i o n

(4.6). W h e n ( ~ - I )N( -ND- D ' I )

> --

1

N-D-I

_ !~ due to (4.47) and (4.48) (depending on

1 we can choose s such that 0 < 7 1 < N-D i -1 the value of ]),

1 a n d such

t h a t (~, s) satisfy (4.46). big enough, so t h a t (4.41) and Due to (4.46), we can choose d < N g- D- D- I ' (4.42) be satisfied. 2 - In the same way, using a s s u m p t i o n s (4.2), (A10) and ( A l l b ) , we can

626

Control localized on thin structures for semilinear parabolic...

prove t h a t there exists (fr k) obeying (2.22) and (4.1), such t h a t the set of pairs (~, r) satisfying ~> q -a-l'

~ > k', r > d -a-l' 1 + =" 1 < 1 -z

r

1 N (a-1)(q+~-~)
k', -1+

k--~'

1 < - -1 -k-a"

r

-1+

r

1 N 1 (4.49) ~+~rr+~,

1 < 2 -k N - D '

big enough. is non e m p t y for all d < N N- D- D- I ' 3 - From L e m m a 4.6, it follows t h a t the trace of the solution p to equation (4.37) belongs to L~(O,T;L~(F)), where (g,s) is the pair defined in Step 1. 1 1 We set ~1 -- ~1 + 7' ~ -- ~1 § ~' then bp E L~(O, T; L~(r)). Let 7r be the solution to the equation 07r 07r . Ot. F ATr . . 0 in. Q,. Onm

bp on ~

7r(T)-0

in ~

(4.50)

N-DD -- 21) ' then/~ _> q~, a n d / 3 > a~ > N - D - 1 If a < N - D - 21 ( r e s p . a > N N -- D (resp. ~ > N - D - 1 _ a~). Using Lemma 4.3, we deduce t h a t the trace of r on 7x]0, T[ exists, and belongs to Lq' (0, T; L ~' (7)). 4 - From L e m m a 4.6, it follows t h a t the solution p to equation (4.37) belongs to Le(0, T; Lr(f~)), where (~, r) is defined in Step 2. Let ~r be the solution to the equation -

0# - O---t-F A ~ r -

-

- a p in Q,

0# OnA - - 0

on ~,

7 7 ( T ) - 0 in f~,

(4.51)

we can prove t h a t the trace of # on 7x]0, T[ belongs to L q' (0, T; L ~' (7)). 5 - Let # be the solution to the equation - --+ at

A# - - div h in Q,

0# OnA

=h.gonE,

#(T)=0inf~.

N-D

(4.52)

> NSince ~ < 1 + N - D1 - I ' for d < N - D - 1 big enough, we have From L e m m a 4.2, setting ~ = ~ = ~ in (4.23), it follows t h a t #]~• belongs to L ~--br-(0, T; L~--~-(7)). From (4.6), we have ~

D.

> q'. Since ~-1 >

N - D > 2 > a', the trace of # on 7x]0, T[ belongs to L g'(0,T; L ~'(7)). Notice that p -- zr + # + #. The proof is complete, m ( A 1 2 ) - Assumptions needed in the proof of Theorem 4.3. ( A 1 2 a ) - Assumptions needed to estimate the term bp. Jr 2

+

~r 2p

0 because u < q. Now we set ]1 = inf{ tr 2 , q, 1 Observe t h a t ~v 1 ____ i n f { ~ i 1 1 and 7 p , ~, e }. Let us prove t h a t the pair (~, s) obeys

- 1

N(a-

2 If ~q , _-1 1 If ~q , 1-_

~1 1

t

1)

2p

1

0 because u < q. We set ]1 = i n f { ~ - I 2 , q'1 consider two subcases.

(4.69) 1 ~}, and

P. A. Nguyen and J. -P. Raymond

629

When < N - D 1- 1 t1' we set s1 = ~ ~ X . Let us prove t h a t (~,s) obeys x - 7, x condition (4.67) follows from assumption (4.53). If (4.67). If ~x = ~7 = ~-x ~ condition (4.67) follows from the fact t h a t ( ~ - 1 ) < 1 < N - D N - D - 1 1 e'X due to (4.68) and (4.69) (depending on the value 1 x and such t h a t (g, s) of ~), we can choose s such t h a t 0 < ;x < N - Dx- X ~' satisfy (4.67). Due to (4.67), it follows that we can choose 1 < 54 < 2, close enough to 2, so that (4.63) and (4.64) are satisfied. 2 - In the same way, using assumptions (4.2), (A10) and (A12b), we can prove t h a t there exists (k, k) obeying (2.22) and (4.1), such t h a t the set of pairs (~, r) satisfying

r>k'

, r >- ~ - 1 ' - 1' N < ~1+ ~ +N 1 ( n - l ) ( ~4 + zp~-'7-)

-

:1+

r

1 < 1

~-~'

-1+

r

1 < ~1

-1+

-k-a"

r

(4.70) 1
13 > a' > N - D - 1 (resp. /3 > N - D - 1 > a'). Using L e m m a 4.3, we deduce t h a t the trace of 7r on "yx ]0, T[ exists, and belongs to L q' (0, T; L ~' (7)). 4 - From L e m m a 4.6, it follows t h a t the solution p to equation (4.37) belongs to Le(O,T;Lr(gt)), where (~,r) is chosen in Step 2. Let r be the solution to equation (4.51), we can prove t h a t the trace of r on 3,x]0, T[ belongs to

Lq' (O, T; La' (,./)). 5 - Let # be the solution to equation (4.52). Since ~

> N-

D, from

L e m m a 4.2, it follows t h a t #i~• belongs to L--~ (0, T;L~-~-r-~(7)). We have a' < 2 < p < ~-~-l" Moreover, we can choose 54 < 2 big enough, to obtain q' < 2 < ~-~_~. Thus #l~xl0,X[ belongs to L q' (0, T ; L ~' (7)). Notice t h a t p = 7r + 7? + ~. The proof is complete, m -D In the sequel, we set # = rain (a, N -ND -~

).

( A 1 3 ) - Assumptions needed in the proof of Theorem 4.4. N 0 < inf{1 + ( N -

D + 2 ~"

q.~2p

D) (N

a,D

.-7 0

q,)2 ' 1 + (N

~,D

~)2 }.

(4.71)

630

Control localized on thin structures for semifinear parabolic...

( A 1 3 a ) - A s s u m p t i o n s needed to estimate the t e r m bp.

( 0 - 1)(N

D

a'

2N If a
k', -

r>e, -

r>k' -

'

-1+1 _ r

~ < 1,

is non e m p t y for all e < (N-~-r-q - - rN) (2O - - 1 ) D

1

-r +

1 -k

Y - D - 1. Let

P. A. Nguyen and J. -P. Raymond

633

7r be the solution to e q u a t i o n (4.50). Using L e m m a 4.3, we deduce t h a t t h e trace of 7r on ~x]0, T[ exists in LI(O,T;LI(9/)). 2 d - F r o m L e m m a 4.8, it follows t h a t the solution p to e q u a t i o n (4.37) belongs to Le(O,T;L"(~)), where (~,r) is chosen in Step 2. Let # be the solution to equation (4.51), we can prove t h a t the t r a c e of # on -yx]0, T[

belongs to LI(0, T; Ll(q,)). 3 - From Steps 1, 2b and 2c, it follows t h a t the trace P[~x]0,T[ of t h e solution to e q u a t i o n (4.78) belongs to LI(0, T; L1 (7)). By a c o m p a r i s o n principle and using Step 1, we can prove t h a t it belongs to L q' (0, T; L ~' (9/')). E1 E x a m p l e s . Let us give e x a m p l e s in the three dimensional case, for which (A5), ( A 7 ) - ( A 1 3 ) a r e satisfied. 9 Suppose t h a t N = 3, D = 1, q -

a = 2. We have

N - D N - D - I

---

2 < --

N - D

N m - 2 = C~ a n d 2 - 2, 0 < 2. If p > 2, (A9) is satisfied. Thus, if # < 2, v < ~, all t h e conditions are satisfied.

(7
3, (A9) is satisfied. Thus, if # < 2, v < 3, a < 3, 0 < 2 a n d p > 3, all the conditions are satisfied.

9 Suppose t h a t N = 3, D = 0, and t h a t K u is b o u n d e d in L ~ (7 x ]0, T D. Therefore we can take a and q as big as we want. We have a ->- N N -- D D - 2 - - - 3 and a ->- NN -- DD -- 12 __ 2. Conditions (2.1) and (4.2) are satisfied if # < 3. Conditions (2.2) and (4.3) are satisfied if v < 2. Conditions (4.6), (4.14) and (4.19) on a correspond to a < 3. Conditions (4.71), (4.74) a n d (4.76) on 0 correspond to 0 < 3. If p = oc, (A9) is satisfied. T h u s , if # < 3, v < 2, a < 3, 0 < 3 and p - co, all the conditions are satisfied.

634

Control localized on thin structures for semifinear parabofic...

4.1. O p t i m a l i t y conditions for (P1) L e m m a 4.9 - Let (an)n be a sequence of nonnegative functions converging to a in Lk(0, T; Lk(~t)) for every (k,k) satisfying (2.22). Let (b,~)n be a sequence of nonnegative functions converging to b in Lt(0, T; Lt(r)) for every (~,~) satisfying (2.23). Let zn be the solution of Oz

+ A z + anz = 0 in Q,

Oz OnA F bnz

u ~ on E,

z(O) = 0 in ~,

and let z be the solution of Oz O---t+ A z + az = O in Q,

Oz OnA + bz = us

onE,

z(O) = O i n n .

(4.85)

Then (Zn)n converges to z in Le(0, T; Lr(12)) for all (~, r) satisfying (2.13), and in Lq(O,T; wl'd(fl)) for all d < N - D - 1 to z(T) for the weak topology of Lr(12) for all 1 N, satisfying sup { N ( N - 3 ) ( 0 - 1) N ( N - 3 ) ( # - 1) N 2 ( N - 1) ' 2 ( N - 1) } < 1 -~ 2jl

~1 < 1 2 2"

N-1 and k < (N-3)(ttN-1 1), such t h a t Next, we can choose e < (N-3)(o-1) N

N

0 < 2ee < 1 - ~ 2jl

~1

2'

(5.5)

Control localized on thin structures for semilinear parabolic...

640

sup { N ( N - 3 ) ( # - 1) N 2(N-l) '2jl

N N N 2e } < ~ 0, ce qui montre que le spectre de L e s t positif, strictement k l'exception de la valeur propre 0. En ce plaqant dans un espace de type L 2 ~ poids, on peut obtenir que le spectre de L soit discret. Enfin, en se restreignant au sous-espace invariant d~fini par

/

(z - r

-- f

~dx,

le spectre de L devient minor~ par un nombre strictement positif. Le second r~sultat important est celui de Osher et Ralston [2], dans lequel le flux f e s t quelconque. La m~thode employee est la bonne, qui utilise la propri~t~ de contraction dans L 1 du semi-groupe associ~ ~ la r~solution du probl~me de Cauchy pour (6). Mais, curieusement, trop de d~tails techniques en cachent la g6n~ralit~ et les auteurs n'obtiennent la convergence que pour un choc v~rifiant la condition de Lax f'(Ud)

< S < f'(ug),

(S)

650

Stabilitd des ondes de choc de viscositd ...

au lieu d'in~galit4s larges. Ici, une restriction importante a lieu, qui est peut~tre n4c4ssaire : la condition initiale r + c~ est ~ valeurs dans l'intervalle I d'extr4mit~s u 9 et Ud. Enfin, un article r4cent de Matsumura et Nishihara montre la stabilit~ du profil d'un choc caract4ristique (c'est-g-dire qui v4rifie f ' ( u d ) -- s o u f ' ( u g ) -- s). Cependant, la perturbation initiale est suppos4e petite. De plus, la m4thode est bas4e sur des estimations d'~nergie et la perturbation doit donc appartenir g u n espace L 2 g poids, qui n'a pas de sens physique. Nous pr4sentons donc ici un r4sultat plus complet, sans hypoth~se de petitesse sur c~, dans l'espace naturel LI(IR) et sans autre restriction que r + ~(x) E I pour tout x E IR. T h ~ o r ~ m e 1.1 - Soit r un profil de viscositd entre Ug et Ud. Soit a E L I(IR) une perturbation initiale telle que +

e I,

pour presque tout x E IR. Soit xo -- (Ud -- Ug) - 1 f i R (~(x)dx et soit enfin V(x,t) = r - st - xo). Alors lim /iR iu(x , t) - U(x, t)ldx - O.

t--*+c~

2. R a p p e l s ; r ~ d u c t i o n h un cas particulier La th~orie des op~rateurs monotones permet de construire un semigroupe (S(t))t>0 qui r~sout le probl~me de Cauchy pour l'~quation (6) lorsque la donn~e initiale u0 est dans L~(IR). Comme l'~quation (6) satisfait le principe du maximum, ce semi-groupe jouit des propri~t~s suivantes. S G 1 (r~gularit~) Pour tout a E L~176 et born~ sur IR pour tout t > 0.

S ( t ) a est ind~finiment diff~rentiable

S G 2 (principe du maximum) Si a 0. S G 3 (conservation de la masse) Si a L 1(IR) et on a pour tout t > 0

/iR(

b E LI(IR), alors S ( t ) a -

S ( t ) a - S ( t ) b ) d x = / i R ( a - b)dx.

S(t)b E

D. Serre

651

S G 4 (contraction) Sous les m(~mes hypotheses qu'en (SG3), l'application

t ~/~t

I S ( t ) a - S(t)bldx

est d4croissante. I1 n'est pas n4cessaire ~ la compr4hension de ce rapport de d4montrer les assertions ci-dessus. Elles sont classiques. Le th4or~me se d4duit ais4ment du lemme suivant 9 L e m m e 2.1 - Soit r un profil de viseositg qui joint ug ~ ua. Soit a une perturbation, fonction mesurable sur IR, telle que r + a soit compris entre deux translatds de r (il existe ~/, ~ E ]It tels que r + "y) 0, une fonction an E L 1 et des nombres r~els/3,-y tels que +

_o est donc relativement compacte dans LI(IR). L'ensemble w-limite A : " r oh Bs est l'adh6rence dans L~(]R) de {v(t); t > s}, est donc non vide puisque A - r est l'intersection d6croissante de compacts non vides. Cet ensemble est celui de toutes les valeurs d'adh6rence, pour la distance d(z, w) = Ilz - will, des sous-suites (u(t~))ne~ off t~ ~ +c~. L'ensemble w-limite est invariant par le semi-groupe S puisque si a E A, oh a = l i m n _ ~ u(tn), alors S(t)a = lim~-~oo u(t + t~). Pour la m~me raison, S(t) est surjectif sur A car on a aussi a = S(t)b oh b est une valeur d'adh6rence de la suite ( u ( t n - t))ne~. La propri6t6 de r~gularit~ [SG1] implique donc que A est inclus dans C ~176 Soit m a i n t e n a n t k E IR. La fonction t --, I l u ( t ) - r k)lll , d6croissante, a d m e t une limite not6e c(k) quand t --, c~. Si a E A, on en d6duit que lid - r k)lll = c(k). Cependant, S(t)a appartient encore h A. I1 s'ensuit que t ~-, I i S ( t ) a - r k)lI1 est constant. Notons provisoirement w(t) S(t)a et z(t) = S ( t ) a - r k). On a :

0 = ~-~[[z(t)l[1 --

ztsgn zdx.

Or zt + ( f ( w ) - f ( r k)))x = zx~, oh on a utilis~ l'~quation de profil pour r Multipliant ceci par sgn z, on en d~duit

Izlt + ((f(w) - f ( r

k)))sgn z)x = zxzsgn z,

ce qui donne apr~s integration sur IR"

d / dt

]zidx = /

zxxsgn z dx.

Finalement,

0 = / ~ zxxsgn z dx

D. Serre

et donc

I

653

'

0 = ./,~ ax~sgn a dx.

(9)

Cependant les estimations a priori faites lors de la construction du semigroupe S m o n t r e n t que wxx est int~grable sur ]R et donc aussi axx. On a donc, d'apr~s le th~or~me de convergence domin~e, 0 = ~-,01im/~a~xj~(a)dx Oh j e ( T ) - - V / e 2 -[- T 2. Int~grant par parties, il vient f

a~3~ 2.,, (a) ax.

O=lim[ ~---*0 Jla

Soit Y0 un point oh a s'annule et soit 5 > 0 tel que 51a~(yo)l < 1. Pour e > 0 assez petit, on a lal < e sur ] y o - 5e, yo + 5el puisque a est continument 1 j ( ~~) avec J(T) = (1 + T2) -3/2 Ainsi diff~rentiable. Or j~'(T) : -~

IR a,3e 2 .,, (a) dx > -1 [yo+~e J(1)a2dx, C. J yo--Se

dont le second m e m b r e tend vers 25J(1)ax(yo) 2 quand e tend vers z~ro. On en d~duit que ax(yo) = O. Finalement nous avons prouv~ que

'Ca e A, Vk e IR, Vx e ]It,

(a(x) = r

---> (a'(x) = r

(10)

Pour conclure, on note d ' a b o r d que a est compris entre r et r comme limite de telles fonctions, donc a prend ses valeurs strictement entre ug et Ud. Aussi la fonction x H k(x) - : x - r o a(x) est-elle bien d~finie et r~guli~re (rappelons que r est strictement monotone). P a r construction, a(x) = r ce qui donne en d~rivant a'(x) = r k'(x)). Vtilisant (10) avec k - k(x), il reste r

- k(x))k'(x) = 0

et donc M(x) - 0 puisque r ne s'annule pas. Finalement, k est une constante et a - r k). Cependant, les ~l~ments de A satisfont

/

(a - r - a)dx = O,

ce qui fixe, on l'a vu, la valeur de k : on a k - - y o . On a donc prouv~ que l'ensemble w-limite est r~duit t~ un seul ~l~ment (au moins un puisqu'il n'est pas vide et seulement celui-l~) : r yo). Puisque

654

S t a b i l i t d des o n d e s de c h o c d e viscositd . . .

la famille (v(t))t>_o est relativement compacte dans LI(IR) et comme elle n'a qu'une seule valeur d'adh~rence quand t --, +co, elle est convergente, c'est-~-dire que lim Ilu(t) - r Y0)lli -- 0. t--,~c~

Ceci ach~ve la preuve du lemme et donc celle du th~or~me.

Remerciements Je remercie Heinrich Freistiihler pour l'int~r~t qu'il a port~ ~ ce travail, les discussions fructueuses et pour avoir attir~ mon attention sur des travaux ant~rieurs. Je remercie aussi J.-L. Lions et ses collaborateurs pour leurs constants encouragements. References [1] A. M. II'in, O. A. Oleinik, Behaviour of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of time. AMS Translations, 42 (1964), 19-23. [2] S. Osher, J. Ralston, L 1 stability of travelling waves with application to convective porous media flow. Comm. Pure and Applied Maths., 35 (1982), 737-751. [3] A. Matsumura, K. Nishihara, Asymptotic stability of travelling waves for scalar viscous conservation laws with non-convex nonlinearity. Comm. Math. Physics, 165 (1994), 83-96.

Denis Serre Unit~ de Math~matiques Pures et Appliqu~es CNRS UMR 128 ENS Lyon 46, All~e d'Italie 69364 Lyon France E-mail: [email protected], fr

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