Patterns and Waves: Qualitative Analysis of Nonlinear Differential Equations

Patterns and Waves -Qualitative Analysis of Nonlinear Differential Equations- STUDIES IN MATHEMATICS AND ITS APPLICAT...

Author: Takaaki Nishida | etc. | M. Mimura | H. Fujii

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Patterns and Waves -Qualitative Analysis of Nonlinear Differential Equations-

STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 18

Editors : J. L. LIONS, Paris G. PAPANICOLAOU, New York H. FUJITA, Tokyo H. B. KELLER, Pasadena

KIN0KUNIYA:COMPANY LTD.-TOKYO JAPAN

NORTH-HOLLAND-AMSTERDAM

NEW YORK

OXFORD TOKYO

PATTERNS AND WAVES QUALITATIVE ANALYSIS OF NONLINEAR DIFFERENTIAL EQUATIONS

Edited by TAKAAKI NISHIDA MASAYASU MIMURA HIROSHI FUJI1

KINOKUNIYA COMPANY LTD.-TOKYO JAPAN

NORTH-HOLLAND-AMSTERDAM

-

-

NEW YORK OXFORD TOKYO

0Kinokuniya Company LTD., 1986

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN: 0 444 70144 3 Publishers KINOKUNIYA COMPANY LTD. TOKYO JAPAN

ELSEVIER SCIENCE PUBLISHERS B. V. P. 0. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U.S. A. and Canada ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK. N. Y. 10017 Distributed in Japan by KINOKUNIYA COMPANY LTD. Distributed outside Japan by ELSEVIER SCIENCE PUBLISHERS B. V. (NORTH-HOLLAND)

PRINTED IN JAPAN By The International Academic Printing Co., Ltd.

In recognition of the stimulating inspiration by his works, this book is dedicated to Professor Masaya Yamaguti.

Preface

This volume is dedicated to Professor Masaya Yamaguti in celebration of his 60th birthday. It is a compilation of contributions from mathematicians who have without a doubt been under the great influence of Professor Yamaguti, and from those people who have directly or indirectly during their scientific career been inspired by Professor Yamaguti. The title “Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations-” was chosen to symbolically reflect the subjects and topics contained within. Also, the title does not necessarily imply that the contents are restricted to nonlinear differential equations. These papers, in fact, deal with nonlinear problems related to the following such fields: General analysis, fluid dynamics, mathematical biology and computer sciences, and their underlying mathematical structures including nonlinear waves and propagations, bifurcation phenomena, chaotic phenomena, fractals, and so on. This volume is arranged into two parts: Part I contains papers which survey the developments in the analysis of nonlinear phenomena in our country in the past decade. Part I1 consists of up-to-date original papers concerning qualitative theories and their applications. Professor Yamaguti has left his great footprints on many fields of mathematics as well as mathematical sciences. Among other things, his contributions include his early works on the behavior of solutions of ODE’s (Mem. Col. Sci. Univ. Kyoto, 28 (1953) 87-96), characterization of strong and weak hyperbolicity of the Cauchy problem for linear PDE’s (Mem. Col. Sci. Univ. Kyoto, 32 (1959) 121-151, 33 (1960) 1-23), works on finite difference schemes for hyperbolic systems and pseudo-difference schemes (Math. Comp. AMS, 21 (1967) 611-619, Proc. Inter. Conf. Func. Anal. Related Topics (1969) 161170), mathematical treatment of pattern formation in biology (Adv. Biophy. 15 (1982) 19-65) as well as his recent studies in chaos related to the discretization of ODE’s and a new treatment of fractal theory which has a deep insight into the relationship between chaos, fractals and singular functions such as the Takagi function and the Weierstrass function (Physica 3D (1981) 618626, Japan J. Appl. Math., 1 (1984) 183-199).

vii

It is widely recognized that as a n “organizing center” of nonlinear sciences in Japan, he has inspired and given enormous impact upon many people through active and intensive discussions as well as his informal way of communicating and expressing his attitude, beliefs and approach to mathematical sciences. In this sense, Professor Yamaguti’s achievements and influence go far beyond the conventional notion of mathematics. He is known to have many friends in physics, biology, economy and philosophy, and a series of publications are planned to be released in the near future in collaboration with these people. It should be emphasized that this present volume reflects only a “mathematical projection” of his broad activities. The editors wish to express their gratitude to Professor Peter D . Lax, who presented us with a message of greetings. Professor Lax is a long-time friend of Professor Yamaguti since their meeting during Professor Yamaguti’s visit to Courant Institute of Mathematical Sciences in 1965-66. His personal relationship with Professor Lax has long been one of the incentives to deepen Professor Yamaguti’s research in nonlinear PDE’s and numerical analysis. Finally the editors wish to thank the authors to this volume for their cooperation in its preparation and production. Takaaki NISHIDA Masayasu MIMURA Hiroshi FUJII Editors

Message of Greetings

The Soul of Mathematics In the middle of our century, a time of resurgence of the human spirit following the conclusion of the second World War, a fierce fight broke out for the soul of mathematics. The purists, championed by Bourbaki, claimed that so many branches are budding and blooming today that mathematics will develop autonomously, with no need for input from the real world; applied mathematicians thought otherwise; their view was represented most forcefully by v. Neumann. He concluded his article “The Mathematician”*, with these words: “I think that it is a relatively good approximation to truth-which is much too complicated to allow anything but approximations-that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to a n empirical science. There is, however, a further point which, I believe, needs stressing. As a mathematical discipline travels far from its empirical source, or still more if it is a second and third generation only indirectly inspired by ideas coming from “reality”, it is beset with very grave dangers. It becomes more and more purely aestheticizing more and more purely I’art pour I’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections or if the discipline is under the influence of men with a n exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical. In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas. I a m convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.”

*

Reprinted in J. R. Neurnan’s “The World of Mathematics”.

ix

V. Neumann had very definite ideas where many of these empirical ideas might be coming from. I n a talk at Montreal delivered in 1946**, when fast electronic computers were merely figments of his imagination, he said: “We could, of course, continue to mention still other examples to justify our contention that many branches of both pure and applied mathematics are in great need of computing instruments to break the present stalemate created by the failure of the purely analytical approach to non-linear probelms. Instead we conclude by remarking that really efficient high-speed computing devices may, in the field of nonlinear partial differential equations as well as in many other fields which are now difficult or entirely denied of access, provide us with those heuristic hints which are needed in all parts of mathematics for genuine progress. In the specific case of fluid dynamics these hints have not been forthcoming for the last two generations from the pure intuition of mathematicians, although a great deal of first-class mathematical effort has been expended in attempts to break the deadlock in that field. To the extent to which such hints arose at all (and that was much less than one might d,esire), they originated in a type of physical experimentation which is really computing. We can now make computing so much more efficient,fast and flexible that it should be possible to use the new computers to supply the needed heuristic hints. This should ultimately lead to important analytical advances”. It was a great personal tragedy and a great loss for science, that v. Neumann did not live to see his prophecy fulfilled. But many scientists in many countries followed the road he marked out. One of the leaders in Japan is Masaya Yamaguti; this volume of articles, contributed by his former students and others influenced by him, are a n eloquent testimony to his scientific interests and style. On the occasion of his 60th birthday we express our admiration and affection for him, and transmit to him our best wishes for the future.

Peter D. Lax

** Contained in Vol. V of his Collected Works under the title “On the Principles of Large Scale Computing Machines”.

Contents

vi

Preface Message of Greetings The Soul of Mathematics Peter D. LAX

viii

Part I

Hyperbolic Partial Differential Equations On Weakly Hyperbolic Equations with Constant Multiplicities Sigeru MIZOHATA Strongly Hyperbolic Equations and their Applications Nobuhisa IWASAKI

1 11

Flow Problems Solutions of the Boltzmann Equation Seiji UKAI

37

Equations of Motion of Compressible Viscous Fluids Takaaki NISHIDA

97

Reaction-DiJFusionEquations Predation-Mediated Coexistence and Segregation Structures Masayasu MIMURA and Yukio KAN-ON

129

On the Structure of Multiple Existence of Stable Stationary Solutions in Systems of Reaction-Diffusion Equations Hiroshi FUJII,Yasumasa NISHIURA and Yuzo HOSONO

157

Chaos and Fractals Chaotic Phenomena and Fractal Objects in Numerical Analysis Shigehiro USHIKI

221

Fractals in Mathematics Masayoshi HATA

259

xi Numerical Analysis & Computations Parallel Computation Tatsuo NOGI

279

A Theoretical and Computational Study of Upwind-Type Finite Element Methods Masahisa TABATA

319

Part I1

Positive Solutions to Some Semilinear Elliptic Equations in L'(R") Masaharu ARAIand Akira NAKAOKA

357

On the Vlasov-Poisson Limit of the Vlasov-Maxwell Equation Kiyoshi ASANOand Seiji UKAI

369

A Discrete Model for Spatially Aggregating Phenomena Tsutomu IKEDA

385

A Note on the Blowing-up Problem of a Certain System of Nonlinear Parabolic Equations Nobutoshi ITAYA

419

L'Bquation de Kadomtsev-Petviashvili approchant les ondes longues d e surface de I'eau en Bcoulement trois-dimensionnel Tadayoshi KANO

431

Application of a n Iteration Scheme to the Analysis of Incompressible or Nearly Incompressible Media Fumio KIKUCHI

445

On a Local Existence Theorem for the Evolution Equation of Gaseous Stars Teru MAKINO

459

Fundamental Solution of the Linearized System for the Exterior Stationary Problem of Compressible Viscous Flow Akitaka MATSUMURA

481

Nonlocal Advection Effect on Bistable Reaction-Diffusion Equations Masayasu MIMURA, David TERMAN and Tohru TSUJIKAWA

507

On Small Data Scattering for Some Nonlinear Wave Equations Kiyoshi MOCHIZUKI and Takahiro MOTAI

543

xii

The Near-field Finite Difference Approximation for Wave Propagation Problems in Infinite Media Tomoyasu-Taguti NAKAGAWA

561

Energy Decay for Nonlinear Wave Equations with Degenerate Dissipative Terms Mitsuhiro NAKAO

583

On the Stochastic Integral Equation of Fredholm Type Shigeyoshi OGAWA An Approach to Constrained Equations and Strange Attractors Hiroe OKAand Hiroshi KOKUBO On the Existence of Progressive Waves in the Flow of Perfect Fluid axound a Circle Hisashi OKAMOTO and Mayumi S H ~ J I On Laminar Boundary Layers with Suction Taira SHIROTA Initial Value Problem for Kac’s Model of the Boltzmann Equation Yasushi SHIZUTA and Hideko NISHIYAMA The Initial Value Problem for the Equations of the Motion of Compressible Viscous Fluid with Some Slip Boundary Condition Atusi TANI

Index

597 607

631 645 663

675 685

This Page Intentionally Left Blank

Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equationspp. 1-10 (1986)

On Weakly Hyperbolic Equations with Constant Mu1tiplicities By Sigeru MIZOHATA Abstract. Let P(x, t , D,, D,) be a formally hyperbolic differential operator with constant multiplicity. This article gives a brief survey of the development of the research on the Cauchy problem for P. In the C"-case, the necessary and sufficient condition for the C"-wellposedness (Levi condition) is given in an explicit form by using the perfect factorization of P . Next, when the Levi condition is violated, the wellposedness in the spaces of Gevrey class is discussed. The best possible class of Gevrey is given also through the perfect factorization of P . Key words: Cauchy problem, hyperbolic equation, C" wellposed, Gevrey wellposed, constant multiplicity

S 1.

Introduction

We trace briefly the development of research on weekly hyperbolic equations, and next we summarize the results obtained hitherto from our viewpoint. Since a n article by N. Iwasaki appears in this issue, we restrict ourselves to equations of constant multiplicities. We are concerned with the equation of the form: P ( x , t ; D , D t ) = P m ( x ,t ; D ,D , ) + P , - , ( x , t ; D , D,)+

...

where P,(x, t ; E, r ) is a polynomial of homogeneous degree j in (5, r ) with and ( x , t )E R " x [ 0 , TI; D = -i(a/ax), D,=-i(a/at). coefficients in eZ,,, We assume always P,(x, t ; 0, T)=?', and PAX, t ; E, r ) =

ii (r-A,(x,

j=

1

t ; 5))"'

,

+m,=m, where , l j ( x , t ; E ) are real and distinct, and m, are constant. mi+ More precisely we assume l u x , t ; E)-J,(x, t ; 812 6 l E l ,

for all ( x , t ; E ) , where 6 ( > O ) is a constant. Received May 4, 1985.

S. MIZOHATA

2

In 1956 the work of A. Lax [9] appeared. Although it treats only the case n= 1 (one space dimension), this work elucidated the fundamental properties of this problem, and it has its own beauty. It presents two equivalent conditions. Condition (A).

Put d , = D t - I , ( x , t ) D z . P can be expressed in the form

...

a l ml a 2 m2

akmk+

2 a l ( m l - z ) +...

ak(llZk-')+

121

&f 1(x, I;D , Dt) 9

where M l are (arbitrary) differential operators satisfying order

(a,(ml-z)+ ... a k ( m b - z ) +1 M )<m-z.

(Let us note t h a t j + = m a x ( j , O).) The next equivalent condition, suggested by R. Courant, is more maniable. Let us consider the basis of the form almi-li

.. . a b m k - l a

---

where m,-Z,20, I,+ +Z,>l. Let S be the set of all differential operators which can be spanned, with coefficients in B ' functions, on the above basis. Then. Condition (C). P can be expressed as p=almt

.-.

a k m k + ~ ,

NES.

It is proved that (1) when P is of constant coefficient, (A) is necessary and sufficient for P-wellposedness, (2) (A) and (C) are equivalent, and (3) the Cauchy problem is C"-wellposed under the assumption (A) (or (C)). In 1957, A. P. Calderon-A. Zygmund [l], and in 1958 A. P. Calderon [2] appeared. The author began trying to retreat the works of I. G . Petrowsky [21], J. Leray [lo], L. Girding [6] from the view-point of singular integral operators. M. Yamaguti also showed keen interest in this subject, and we spent a good deal of time in enthusiastic discussions. Both M. Yamaguti and the author considered the elementary hyperbolic operator D , - I ( x , t ; D) b,(x, t ; D ) as a fundamental operator (see [14]). M. Yamaguti, however, aimed at a more delicate problem: the general extension of A. Lax's work. He succeeded with a publication in 1959 [22]. He replaces

in higher dimensional x , and extends the module S by all pseudodifferential operators of the form:

Weakly Hyperbolic Equation

3

where C n ( x ,t ; D)are pseudodifferential operators of the order 10. He showed that, under the assumption (C), the Cauchy problem for P is H"-wellposed. In 1968 S. Mizohata-Y. Ohya [18] appeared. This article takes a slightly different view-point from the previous one. It is a n extension of E. E. Levi [12], assuming the multiplicities at most double. It presents a necessary and sufficient condition for C"-wellposedness under the form of vanishing of the subprincipal symbol on the characteristic set of double multiplicity (see S 2). In 1971 H . Flaschke-G. Strang [ 5 ] appeared. It presents a necessary condition for C"-wellposedness in the case of general multiplicities under the form of (L') in S2, which they called the Lax-Levi condition. Finally J. Chazarain [30] showed that this condition is sufficient.

S 2.

C"-wellposedness in the Case of Constant Multiplicities We can summarize the above results.

We use perfect factorization. Let

P ( x , t ; D,D,)=Pm+P,,-l+

*..

,

where P , is the homogeneous part of degree j of P. Let k

P,(x, t ; E,

TI=

II (T--Aj(X,t ; El)"' ,

j=1

where Aj(x, t ; E) are real and distinct. We know that P can be factorized in the following way: P = P k * P k - ,- * - P , + R ,

-

where denotes the operator product and P , ( l < j < k ) is a pseudodifferential operator whose principal symbol is

(T-Ay(X, t ; 0 ) " j

.

To be precise,

P,=(D,-A,(x, t ; D))"'+al,,(x, t ; D ) ( D , - A ~ ( Xt ;, ~ ) ) " j - ' + +a,, ,(D,-A,(X, t ; D))"'-8+ * * fa,,,, ,

-

with order a,,,(x, t ; E)<s-l.

We remark that

.

(D,-A,(X, t ; D))k=(D,-A,(x, I ; D ) ) * ( D , - I , ( x ,t ; D)) * (D,-A,(X, t ; D)). k The operator R is a regularizing operator, which means the following:

R=

m

C r j ( x , t ; D)D,"-3 , j=1

S . MIZOHATA

4

where Y,(x, t ; E ) ( l < j < r n ) are null symbols. Then the Levi condition, namely the necessary and sufficient condition for C”-wellposedness, can be formulated as follows. (L)

order a,,,(x, t ; 0 1 0

for all s (l<J l . We assume, besides the assumptions made in § 1, all the coefficients a,,,(x, t ) of P belong to 7 2 ) . Then the Cauchy problem in 7(s) means:

Pu=f € C!([O, T I ; 72)) (Oijlrn-1)

a:ul,=,,=U,(X)€7:)

,

where t o € [ O , TI, and the solution u ( x , t ) € C y ( [ t oT, I ; 72)). However, if we assume a,,,(x, t ) € 72,)', then it is natural to seek a solution in 72,)'. Systematic treatment of the Cauchy problem in Gevrey class has been done by Y. Ohya 1191 and J . Leray-Y. Ohya [ l l ] . [19] proved its wellposedness for l<sp.

When l / s= p , only the local solution exists. In other words, when l/sO. We can conclude under the condition (4.1.2-5) that the linearizations Dd,of d, are effectively hyperbolic with respect to d x , at ( x , u, d,)=(O, u-, 0 ) in the sense of Definition 3.2 and Remark if we put R ( x , h, u)=hE(a) and Q ( x , u)=Dd,-d,E(a) with the principal part Qo(a). Therefore it is strongly well posed by Theorem 2.1. Theorem 4.1. Let u- be a formal solution at x,=O of the Monge-Ampire equation (4.1.1) d,(u)=O satisfying the conditions (4.1.2-3). If the conditions (4.1.4-5) hold at a neighborhood of ( x , u)=(O, u - ) , then there exists a unique solution of d,(u)=O on a neighborhood of the origin x=O such that u-u- is flat at x,=O (aa(u-u-)~,,=o=O f o r any a ) . Example. Let M be a two dimensional Riemannian manifold. If the curvature K of M is non positive at a neighborhood of a point x and if the Hessian V2Kof K does not absolutely vanish at points where K vanishes, then some neighborhood of x is isometrically embeddable into R3. (This is equivalent to finding a solution of the Darboux equation. See M . Spivak.) We need a simple lemma to prove the above theorem as a corollary of the unique extension theorem. Lemma. Let $ be an infinitely di'erentiable function being flat at xo=O. Then there exists a non negative infinitely di'erentiable $- in xo (independent of other variables) also being flat at xo=O such that $+$- is non negative. By using this lemma we can find a n infinitely differentiable Q(xo),flat at x,=O, such that d,(u-+$)gO

at x,IO

and also it is flat at x,=O. If we put P=@-g, where g=O at x o 2 0 and g= O(u-+$) at xoO. Such a solution u is able to extend again to x, t . And H f m = n s SH,8 that these scales are interpolation spaces and have the smoothing operators. It means the following.

1) There exists a sequence of norms the logarithmic convexity that ll4l2a+&2)t.

5~

II.IJS with

respect to HjS,which has

ll~ll~ll4l~-~

for O l ) , (1.1.5)

q(v, 8)=u’lcos Si-’‘qo(S)

,

r=l-- 4S ,

2 r’=l+-, S

where qo(S)>O is bounded and does not vanish near 8=lr/2. Extensive literature is available on the derivation of (1.1.1)-( 1.1.9, see e.g. [13, 14, 361, in which the following important properties of Q are also proved. Denote the bilinear symmetric operator induced from the quadratic operator Q by Q[., -1;

Further, define the inner product,

f(E)s(E)dS €= .

(1.1.7)

Theorem 1.1.1.

R”

(i)

Define thefunctions,

Then,for a n y f , g and O < j < n + l , (1.1.9)

~=O

*

( i i ) For a n y f 2 0 , (1.1.10) (iii) Q[fo]=O (1.1.11)

< k f , Q[fl>c 0 , v E R”,T>O independent of

6.

The functions h,’s are called collision (summational) invariants, because from (1.1.3) it holds that, for O l j l n f l , (1.1.12)

hj(E)+hj(E’)=hj(p)+hj(q’)

9

which are the conservation laws of the number (j=O), momentum ( l s j l n ) and energy ( j = n + l ) of particles during a collision. The function f o of (1.1.11) is called a Maxwellian and describes the equilibrium state of a gas having the mass density p, flow velocity v and tem-

40

S. UKAI

perature T . If p, v, T are constant in t , x, then f O is called a n absolute Maxwellian, and otherwise, a local Maxwellian. Besides its physical importance, f ois in the special situation that it happens to be a stationary solution to (1.1.1) (e.g. a n absolute Maxwellian for the case a ( x , E ) = O ) , thanks to Theorem 1.1.1 (iii). The statements in Theorem 1 . 1 . 1 are all true, of course, only when relevant integrals converge. Cutoff and non-cutoff potentials The simplest problem associated with (1.1.1) is the local (in t ) existence of solutions to the Cauchy problem. However, this is not a trivial problem because the nonlinear operator Q is unbounded, which comes from two different singular behaviors of q(v, 0). One is the strong singularity at 8=7r/2 in (1.1.5). Since f > l , the integral over 9-' in (1.1.2) does not converge i f f is only bounded, but it does i f f is smooth because p = E , pf=E' at 0=7r/2, so the quantity { - . .} in (1.1.2) then vanishes there, thereby cancelling the singularity of q(v, 8). Thus, Q is well-defined only for smooth f and behaves like a (pseudo-) differential operator, an unbounded operator. In $ 3 we present a method to control this unboundedness, which is analogous to, but much simpler than, that used for the abstract nonlinear Cauchy-Kowalewski theorem of Nishida-Nirenberg-Obsjannikov [30]. However, this method has a limited application, and at present we can only go farther in removing this singularity by cutoff approximations. The most successful cutoff is Grad's angular cutoff [18], with which the local and global existence has been established for a variety of the Cauchy and initial boundary value problems for (1.1.1). This will be seen in $$ 4-7. It is Grad's estimates for Q and related operators with this cutoff that largely promoted the study of (1.1.1). Grad's cutoff is equivalent to replacing q by its angular cutoff, 1.2.

(1.2.1)

q"v, 8 ) = x " M v , 0)

1

where xc(0)=l (l8-~/21 > E ) , = O ( ~ O - R / ~ / < E ) . Obviously, the ambiguity in choosing the value of e arises unless the convergence is established in some way. This will be discussed in S 3.2. Observe that the hard ball gas (1.1.4) has no singularities and is considered as a cutoff potential case. Even with cutoff, Q may be unbounded since q(v, 0)-w ( v - > m ) , as seen in (1.1.4) and (1.1.5) with s > 4 . This unboundedness is handled in $ 2 by a method similar to that of S 3.

1.3. Conservation laws Once local solutions are found, global solutions can be constructed if nice a priori estimates are available. The Boltzmann equation has L' estimates be the inner coming from physically natural conservation laws. Let

Solutions of the Boltzmann Equation

41

product (1.1.7) but over R Z n x R E n . Suppose a ( x , E ) = O and let f be a smooth solution to (1.1.1). Then, by integration by parts and by (1.1.9), we have, (1.3.1)

=, ~ E R ,O < i < n + l ,

which are the conservation laws of the total number (j=O), momentum ( l < j < n ) and energy ( j = n + l ) of particles. In $ 2 . 4 , we see that f ( t ) 2 O provided the initial datum f , = f ( 0 ) 2 0 (and q(v, 0)20), a physically natural conclusion because f denotes the density. Hence, (1.3.1) is L‘ a priori estimates. But it is not useful because it seems difficult to find L1-local solutions by the reason stated in s 2 . 2 (see also [33]), which presents a contrast to the spatially homogeneous case where (1.3.1) is essential in costructing global solutions [2, 31. In $ 4 , we propose a method to find global solutions, which takes advantage of nice asymptotic behaviors of solutions of the linearized problem at the cost of the restrictive condition that the initials are small in some sense.

1.4. Boundary conditions When the gas is contained in a vessel 52, the boundary condition which describes the reflection of gas particles by the wall 852 is imposed on f. Denote the outward normal to 852 at x€i?52 by n ( ~and ) set, (1.4.1)

s*= {(x, E) € a52 x W” I n(x).E2O} ,

(1.4.2)

r’f=fIs.

,

(same signs). are trace operators to S” whose existence will be established in S 5.1. Since particles at (x, E) E S+ (resp. S - ) are particles incident to (resp. reflected by) the wall as, r i f represent the corresponding densities. Therefore, the boundary condition must take the form, 7-f=M r +f

(1.4.3)

t

where M is the boundary operator determined by the reflection law. Classical examples are; (1.4.4) ( i ) (ii) (iii) (iv)

(perfect absorption) M=O, (specular reflection) Mr+f=f(t, x, E-2(n(x)-E)n(x)), (reverse reflection) My+f=f(t, x, -[), (diffuse reflection) n(x)*E’f(t, x, E’)dE’

Mr+f=g,(E)

9

n(z).e’>o

sw(E)=p,(2aTw)-“/2exp ( - lElz/(2Tw)), where T, is the temperature of the wall.

~ , = ( T ~ / ( 2 7 d ) ”, ~

M may be any (convex) linear com-

S. UKAI

42

bination of (i)-(iv). Also, (ii), (iii) may be generalized as ([24]), ( 1.4.5)

My+f=f(r, x , m ( x , E ) ) ,

m(x, E) E S-

,

and (iv) as

dux being the measure on 352, and so on. With these boundary conditions, we are given initial boundary value problems for ( l . l . l ) ,the subject of SS 5-6.

1.5. Large initial data In solving the Cauchy and initial boundary value problems globally in t , a n unpleasant restriction is required for initial data f , . Whereas the local existence can be established for arbitrarily large f , (SS 2,3,5), the global one requires the smallness condition on f,, as stated at the end of S 1.3. More precisely, f , should be near a n equilibrium described by a n absolute Maxwellian go (§§ 4-6). Of course, this is a technical restriction specific to our method of $ 4 . 1 , but at present, n o other methods are available. Thus, the global existence for f , far from go is a big open problem. For such f,, it is expected that the shock appears after a finite time, however smooth f , may be. Notice that all known solutions, local or global, are classical (smooth) ones, and so d o not present the shock. Therefore, the shock solution, if any, should be a weak one, but up to the present no definitions of weak solutions have been found which promise the global existence. Recently, Arkeryd [4] introduced a new concept to this problem, constructing global ‘non-standard’ solutions for arbitrary L1 initials f,. More precisely, his solutions solve (1.1.1) in the sense of the non-standard analysis that the variable t is considered in *R (the set of non-standard real numbers) and x , E in *Rn while the Lebesgue integral $ dE’do over R ” x Sn-’ is replaced by the Loeb integral $ L(*dE’ *do) over *Rnx*Sn-l where L(*d[’*do) is the Loeb measure induced from the Lebesgue measure dt’do. Although this result is remarkable, its conversion to the standard result seems difficult. For the spatially homogeneous case which will not be discussed in this note, global solutions exist for any f , , with and without cutoff, see [2, 31.

- -.

1.6.

.-

Fluid dynamical limits The motion of a gas can be described not only by the Boltzmann equation but also fluid dynamical equations such as the Euler and Navier-Stokes equations. The former gives a microscopic description in the sense that the gas


43

is looked at as a many-body system, while the latter gives a macroscopic one because the gas is regarded as a continuum like a fluid. Evidently, the former provides more precise information, but if the number N of gas particles is very large as is usually the case (N=1OZs,Avogadro's number), then the gas will behave much like a fluid, and at the limit N=oo both descriptions are expected to coincide. The Hilbert and Chapman-Enskog expansions reveal this. In 1912 the celebrated Hilbert, the first mathematician to attack the Boltzmann equation as well as many other big problems, gave a solution of (1.1.1) in a formal power series,

(1.6.1)

...

f=f'O'+Ef(1)+E2f'2)+

of € E N - ' (the mean free path). To determine the coefficients, assume a ( x , f ) = O and write ( 1 . 1 . 1 ) as 1

Df=, Q[fl ,

(1.6.2)

D=ac+E*V,,

which is obtained dividing (1.1.1) by N and writing f/N as f (the normalized probability density). Substituting (1.6.1) and equating the coefficients, he then obtains,

(1.6.3)

Q[f(o'l=o,

(1.6.4)

Df")=2 Q[f (O), f'"] ,

( 1.6.5)

D f C r= )

k+l

C Q [ f ( I ) f, ' k ' l - j ) ] ,

k> 1

J=O

.

By Theorem 1.1.1 (iii), f t o ) = f o , the Maxwellian ( l . l . l l ) ,with fluid dynamical quantities p, v, T still undetermined. Once they are known, (1.6.4) is a linear integral equation for f"). Applying his theory of integral equations, and by Theorem 1.1.1 (i), Hilbert has the solvability conditions,

(1.6.6)

E=O,

O<jE)E(x, E)b'2

(2.2.10)

.

Define the space (2.2.11)

X,.,,={f=f(x,

E ) I p a , p f E W R n xR")}

Ilfll..u.a=IIPu,pfll~~

9

.

In view of Theorem 2.2.1 and (2.2.7), we readily have the

Theorem 2.2.3.

For any p E [l, 001 and

IIU(t)fllp,u,p=

llfllP,u,p

7

(Y,

3/ E R , it holds that

t f sR

t

f E

x:.p .

Obviously U ( t ) has the group property U ( t + s ) = U ( t ) U ( s ) ,s, t E R . As for the continuity in t , we can prove the

S . UKAI

48

Theorem 2.2.4. Let p~ [l, 00) and a , P E R . X,”,a. Its generator, denoted by A , is given by

Then U(t) is a C,-group on

Af= -(E.Vz+4x, E ) . V e ) f , whose domain is maximal in X{,,+ sense of distribution.

Here the derivatives are to be taken in the

This theorem is not valid for p = m , but it is if U ( t ) is restricted in a suitable subset of X : , p . For example, we have, Theorem 2.2.5. For p = m , the previous theorem remains true i f X:,, is replaced by

z:,,={fE x,,n where x R = x R ( x , E ) is such that

x Rn)I l l ~ ~ f l l - , ~ , ~ - -( -R>-Om ) } ,

xR=

1 if 1x1

+ IEi > R , and =O

otherwise.

Another example will be obtained by @-operation of [21J, noting that the dual of X2a,-B is X ; , p and the dual of U ( - t ) is U(t).

2.3.

Construction of local solutions We are now in a position to solve the Cauchy problem for (1.1.1);

(2.3.1) Iff is a solution, it must solve the integral equation (2.3.2) which comes from (2.2.9). We shall solve this in the space X:,B of (2.2.10), but need to vary the index LY with time t to control the unboundedness of Q which is seen in the Lemma 2.3.1.

Theorem 2.1.1 remains true i f L:,p is replaced by X : , p

For the proof, it suffices to repeat the proof of Theorem 2.1.1 with the assumption (2.2.5) (ii) taken into account. In spite of Remark 2.1.2, this lemma fails in X,”,, if p < 00. Define the bilinear operator (2.3.3) and the nonlinear operator


49

Then (2.3.2) can be written as f = N [ f ] , so we shall find a fixed point of N . The simplest but powerful tool for this is the contraction mapping principle. However, if r > O , it does not work if a is kept fixed, because due to the above lemma, N loses the weight ErIz a t each step of iteration. O n the other hand, if a is varied with I , then it does work well since the integration in (2.3.3) with respect to t turns to a smoothing operator. T o be precise, introduce the space (2.3.5)

x=x,,.,p([-T, TI)=If(t, x, 5 ) I e - c l c l E f ( t ) ~ L " ( [ -TI; -,

II If1II = I IIf I II x =

ik!

Ile-"l"'"f(t)

ll-,a.j=

Ilf(t)Il-.a--.ltl ItlST SUP

X , j ) }

,

.j

Lemma 2.3.2. Under the assumption (2.1.1) and f o r any a>O and B 2 0 , there is a constant Co>O such that for any K > O and with T = a / ( 2 r ) , we have, for all f,9€X, (2.3.6)

IIINo[f, sllll

co.-'lllflll

lllglll

.

Proof. Put w(t, s)= IIU(t-s)Q[f(s), g(s)]ll,,a-.lsl,~-r.Using Theorem 2.2.3 and then Lemma 2.3.1, we get w(t, ~ ~ ~ ~ u - K ~ lllglll s ~ l~l Cl fal/ l~ll l l JIllslll, lll for Is1 I T . Hence,

where

The last integral is majorized by

K - ~ ,

so (2.3.5) follows.

The fact that the last integral is bounded although Er/2+cc (IEl+oo)

for

y>O implies that the integration in t plays a role of a smoothing operator.

Now N is a bounded operator.

Further, it is a contraction on the space

Va={f€

x I lllflllla)

if a>O is chosen appropriately. Actually, we first choose d= 1- 4 C 0 K - ~ l l f o l l ~ , ~ , > s 0,

and then set a= (1- 42)/(2C,,K-') ,

which is the smaller root of the quadratic equation C o ~ - ' a ~ - a +Ilfollm.a,e=O

K

so large that

S. UKAI

50

We easily see that az)and put u=p,f, w=p,g. From (1.1.3) we see that

v=e,

Qi

{ . a * }

[f,gl(E)=p,’(E)

Let llull=sup lu(E)I and let lula=sup Iu(p)-u(E)l/lv-Elg be the Holder quotient. We have


53

where

~=j

R n xS"-1

m, s)ip,-i(E')is-Eidd~'do .

From (1.1.3), 1~-El=IvcosOl, so we get by (3.1.1),

Z Ic

1

(1 +v-'+Z)r)Dap,-'(E')dE'

R"

1:

lcos 01-r'+dd0

The integral in 0 converges if --y'+6> - 1, whereas that in 5' can be evaluated using (2.1.9). The result is Zr+d. QZcan be estimated similarly. Thus Q,, and hence Q , are well-defined if 6 E (7'- 1, 11. For our purpose, however, this estimate is not convenient, and we shall deduce a n estimate in Gevrey classes re$;, defined as follows. Let pa be as above and define the norm

Definition 3.1.1. Let a , p, ,020 and f = f ( x , E ) E Cm(RnxR") satisfying

X,

v2l.

rg;:b is the set of functions

(3.1.4) where the sum is taken for all k, 1 E N". Remark 3.1.2. ( i ) Endowed with the norm (3.1.4), algebra. This is seen by Leibniz' rule and by

(3.1.5)

k!m!/(k+m)! 1, a contradiction.

but also nowhere

Convergence of cutoff approximation Suppose q(v, 8) satisfy (3.1.1) and consider its angular cutoff qe(v,8) given by (1.2.1). Obviously, qe also fulfill (3.1.1), so Theorem 3.2.3 applies. Furthermore, the constants T , /3, 1, a given there d o not change. Denote the solution thus obtained for qf by f ' . The original solution (i.e. for q) will always be denoted by f. 3.3.

Theorem 3.3.1. Under the situation of Theorem 3.2.3, let f and above, for the same initial f,. Then,

f E

be as

111 f-fq[I & due to (i) ([25]). Denote the domain of A by D ( A ) , and the resolvent set and spectrum of A by p ( A ) and u ( A ) , respectively. The set of discrete eigenvalues of A with finite multiplicity is denoted by ud(A) and the essential spectrum ([25, p. 2431) by a,(B). Finally, C,(p)= (A E C I Re ;I2/3}.


59

Define the operator B by D(B)=D(A).

B=A+K,

By (i) and (ii) in the above, B is a semigroup generator. Our aim is to deduce a n asymptotic behavior of e t B . To this end, we first discuss o(B).

Theorem 4.2.1. ( i ) a ( B ) c C - ( p +llKIl), p(B)3C+(p+IlKll). (ii) a,(~)=o,(~)co(~)cC-(p). (iii) a(B)n C + ( P + S ) is in o,(B) and is a finite set, where 6 is that of (4.2.1) (iii).

Proof. (i) and (ii) follow from [25, Theorems IX. 2.1 and IV. 5.351 respectively. Set G(R)=KR(R, A ) and write the second resolvent equation, R(R, B)=R(R, A ) + R ( I , B)G(I) .

(4.2.2) Solve this as

R ( I , B)=R(R, A)(Z-G(R))-' ,

(4.2.3) and conclude that

(4.2.4)

R € p(A) and I-G(R) is boundedly invertible

3

R E p(B) .

Now, by [25, Theorem VII. 1.91, a(B)n C+(p)is contained in a,(B), with possible accumulation points only on the boundary of C+(p). Put ao=p+6. In view of (4.2.1) (iii), one can find a r,>O such that IlG(R)]Iro}. Therefore, by the Neumann series, Z-G(R) can be boundedly invertible with (4.2.5)

ll(Z-G(I))-1/l p, we have,

'j* llR(r+ir,

~)ull2dr 1 / 2 . This will be done in the next section. In our situation, however, r is unbounded, so it violates (4.1.2) (ii). In § 4.5, we will give a modified version of Theorem 4.1.1 which ensures the global existence for (4.3.3). In the rest of this section, we state some fundamental properties of L and r. We assume the angular cutoff hard potential proposed by Grad in [18], which includes (1.1.4) and angular cutoff of (1.1.5) for s 2 4 . In [18], the followings are proved. First, (4.3.6)

L=--Y(E)x + K ,

where -Y(E)=Y[~~](E) (see (2.4.1)) and satisfies (4.3.7)

O < Y , I I J ( ~ ~ Y ~ < E ,> ~

with some positive numbers

yo,

IJ,and

<E>=(l+IEIz)'/*

~ 2 0 and ,

where K(E, E') is real measurable on R* x R" satisfying (4.3.9) ( i 1 W E , E')=K(E', E ) ,

,


63

with constants k,, k,>0 independent of E. Actually (iii) holds only for n=3, and should be modified for n f 3 . Here, we assume it for all n to simplify the argument. Denote by B(X, Y ) the set of all linear bounded operators from a Banach space X into another Banach space Y, and by C ( X , Y ) its subset consisting of compact operators. We write B ( X )=B(X, X ) and similarly for C ( X ) . Set, L2=L2(Rn),

L;={u(E) 1 <E>b(E)

€

L"(R")} .

Proposition 4.3.1. ( i ) K € B(L2)n B(L;, L;+J, p € R. ( i i ) K € B ( L 2 , L;). (iii) K C C ( L z ) . Proof. (i) and (ii) follows from (4.3.9) (ii) and (iii) respectively. Let 1 for IE[ < R and =O for 161 > R . In view of (4.3.9) (ii),

X,(E) be such that X,=

~ ~ ( z - ~ R ) K ~ ~ O .

H2={E€ R" I IEI < R)\Z,,

It is easy to see that mes El CeRn-' ,

mes E2I CR" ,

holds where mes means the measure in Rn and C 2 0 is independent of k, R, E , r. Then, for o 2 - u 0 + 6 ,

I=

L'

+

sz,

( 2 ~ ~ ) ~Thus, + ~ / (ii) ~ . follows. The above lemma assures (4.2.1)(iii) for our A ( k ) and K , uniformly for k € R n and with any d>O. Now that all conditions of (42.1) are confirmed, Theorems 4.2.1 and 4.2.3 apply to B(k) for all k € R n , with ,B=-u0 and for any d>O. R,, P,, Q, possibly depend on k , and will be denoted as R,(k), etc. The next step is to discuss these quantities. The following theorem is due to [16]. Set S 1 [ r ] = { k E R n I I k l S r } , SZ[r]=Rn\S1[r]. Theorem 4.4.2.

There exist positive numbers -,n+ 1, such that ( i ) f o r any k € Sl[~,l,

p j ( r ) € Cm([-ro, r , ] ) ,j = O ,

--

lie,

uo( < y o ) , and functions

u , ( ~ ( kn ) )C + ( ~ o ) = { ~ j ( k ),~ , " = R +, (dW. = P ~ ( I,~ I ) /f,(") = i&)K - p$%2+0

with py)€ R and py)> 0 , and

(I 4 3 )

( llil-0)

,

66

S . UKAI

for j = O ,

-,n+ 1,

where PSo)are orthogonal projections with

c P:"(k) ,

n+l

Po=

j=0

Po being that in Lemma 4.3.3, and P y ) ( k )e B(L2)with uniformly bounded norms, and ( i i ) for any k c S z [ l i o ]u(B(k))nC+(-a,)=Q,. , From this and Theorem 4.3.3, we conclude the

Theorem 4.4.3. There is a constant C>O such that, ( i 1 for any k E S1[liOI, n+l &B(k)

=

2 epj ( ' x' ) t P j ( k)U+( t ,k ) ,

j=O

[lU(t,k)IIICe-"ot,

r20,

( i i ) for any ~ € S , [ K - ,IlecB(k)/IICe-"oc, ], t20. In the above, the constant C can be taken independently of k because we have established (4.2.1) (iii) uniformly in k. Now we can deduce a decay estimate of ere. Let HL(R;) be the L2-Sobolev space, and define, Hl=L2(R;; H1(R;)) ,

Lg,2=L2(R;; Lq(R;))

.

Theorem 4.4.4. For any I E R and qe [1, 21, there is a constant C>_Osuch that, ( i ) IIecBull IC(1 t)-"ll u IIH l n Lq.2, (ii) IlecB(Z-Po)uII C(1+ t ) - a - 1 ' 2 1 [ u I I H ~ n ~ q 1 2 , with a=(n/2)(l/q- 1/2).

+

Proof. (4.4.5)

Referring to (4.4.3) and by Parseval's equality,

\

llefBuII~z= (ItIkl)2111etB(x)li(k, t ) l l h R ; ) d k. Rn

Split the integral over S , [ K o ] and S2[liO], and denote the respective integrals as ZI and Z,. By virtue of Theorem 4.4.3 (ii), I z ~ C e - z u ~ t I l u, l l ~ , whereas, by (i) and Theorem 4.4.2 (i) for P,(k), I,I

CCY I ~+ e, - ~2 ~ o ~ ~,~ u ~ ~ ~ c ) 1-0


61

so by the aid of Holder's inequality and then, thanks to the well-known interpolation inequality for the Fourier transformation,

where

which is majorized by C(l+r)-(nt*)'2if m 2 0 . These estimates prove (i) of the theorem. T o prove (ii), it suffices to take account of Theorem 4.4.2 (i) and put m=q' in the above. Remark 4.4.5. mum at q= 1.

Thus, a > 1/2 if q E [l, (l/n+l/2)-l) and a=n/4 is the maxi-

The nonlinear operator the space HL,a defined by (4.4.7)

K , p 3 24

-

r of (4.3.4)is not well-defined

L,ARf ;H1(R:)) II~II1,~--suP<EYllu(.,

in H,, but it is, in

7

E

E)llHhR;)

R , and =O otherwise, k being the dual variable where xR(k,E)= 1 for Ikl+ of x . It can be seen that etAof (4.4.9), and hence elB, are C,-semigroups on Hl,,9,although not in H l , p ,with the domains

D ( A ) = D ( B ) ~ t j , b l , p +, r , r ' = m a x ( r , 1)

(4.5.3) for any I ,

(4.5.4)

/?. Also, A-lrl-,

-1

maps H t , p x H l , pinto H l , p

if

I>n/2, 8 2 0 .

For the proof, see [42]. Therefore, it follows that N[u] is in Co([O,a);H1,& if so is u and if u , € & [ , ~ . Using (4.5.3) with 1, /3 replaced by 1-1, /3-f, we then conclude the Theorem 4.5.3. then,

Let u , uo be as in Theorem 4.4.2. Zf, in addition, U ~ E H ~ , ~ ,

UE

co([o, m ) ; H1,,dncl([o, m); Hl-l,,9-r,),

and u is a classical solution to (4.3.3), and hence so is f=gO+g;% to (4.3.1). Since H l , , 9 c H l - t , p -for e any E > O , the above two theorems assert the

Theorem 4.5.4. Let u , uo be as in Theorem 4.5.2. Then, u is a classical solution to (4.3.3) such that

u€mto, for any

E

m);

K , ~n )co([o,a);L , ~ - A

n cl(to, m ) ; H ~ + ~ , ~ - ~,, - J

> 0.

Remark 4.5.5. ( i ) Thus f(t)+go ( t - a ) at the rate t-", provided f o is close to go, a physically natural conclusion. ( i i ) If n 2 2 , one can take q=2. Then, a=O, but one can show that u(t)-FO strongly in H l , p . Remark 4.5.6. It is in the form of Theorem 4.5.4 that the results of [37,38] were stated. Theorem 4.5.2 has been proved in [28, 32, 381 independently. The periodic case The previous arguments also provide the global existence for the Cauchy problem (4.3.1) associated with the periodic (in x) boundary condition. This is not a trivial result because the special initial boundary value problem for (4.3.1), where the domain Q is a parallelepiped with the specular reflection boundary condition (1.4.4) (ii), is reduced to the periodic case, [19]. 4.6.

71


In this case it is natural to use the Fourier series, instead of the Fourier transform. Thus, for example, (4.4.5) is replaced by

+

jlecRuII,,l=kzn(l Ikl)zLllecB(k)fi(k, -1 lliz

,

where ii is the Fourier coefficient and HL=L2(R;;H L ( T ; ) ) ,T" being the ndimensional torus. In applying Theorem 4.4.3 to this, we may assume O < K,< 1, from which readily follows the Theorem 4.6.1. There is a constanr oo>O, and for any l € R , ( i 1 IleWI 5 Cllull, ( i i ) ~lecB(Z-Po)ull O . Then, IIINo[f,

sllllrclllflll lllslll *

Proof. Let Q,,j = 1 , 2, be given by (2.1.4) and N o , , defined by (2.3.3) with Q replaced by Q,. Noting the group property U(t+s)=U(t)U(s), and then using (4.9.2), we get IU(-t)NO,U,

s l k x, E l l =

11'

U(-s)QJf(s), g(s)ldsl

lJlllflll where

lllglll

Y

74

S . UKAI

(4.9.4)

q

e-alxta(€-€')12~S

-m

Since I X + ~ ( E - E ' ) ~ ~ ~ ~ ~ ( ~ + ( E - E ' ) . X v/ =D ~IE-E'I, ) ~ , we get W I C v - ' , and hence, J I C e x p (-a1x12-P1512) by means of (2.1.7). Now, the lemma follows for N,,,, and similarly for No,2more straightforward. Noting (2.1.5) then proves the lemma for No. Using this and noting the fact IllU(t)folll=Ilfollu,p by definition, we have, for the operator N of (2.3.4),

III"f1lll Ilfollu.p+clllfll12 III"f1 -"91 II I C(lIIf1I I + I I 191I I )I I If-91 I I * 7

As before, this implies that N is a contraction if f,,is small, and thereby, proves the Theorem 4.9.2. Let q(v, 0)be as in Lemma 4.9.1 and a, P > O . Then, there are positive constants a,, a, such that for each f, with IIfollu,8 l , a, E CL(V ) , ( - a , = v e . a l = 0 . According to Theorem 2.2.1, therefore, (2.2.2) has a unique solution S,(x, E)= ( X , 5 ) for any initial ( x , E ) € V as long as X stays in R. Denote the forward ( t > O ) stay time by t + ( x ,E ) and backward one by t - ( x , E). By definition, S,€V, -t-O.

f~ Lp(D) exists. Proof. Apply (5.1.11) to g E W z replacing p , R by q, - A Note that q € [ 1 , m) if p E ( 1 , a]. Then by (5.2.3), we have,

(5.2.5) where 11

llgllq

llP is the

9

respectively.

~ ~ ~ " ' ~ - ' ' ~ l l l~l ( A~ -~~ )~ ! JI l ll q; ~

norm of Lq(D) and

9

I.11;

that of Lq( V ) . Define,

Z,={(A-R)g I g € W $ } C L S ( D ).

(5.2.5) shows that for each h c Z , , there exists a unique g € W$ such that h = ( A - R ) g . Therefore, F(h)= -,- is a well-defined functional on Z,. Further, F is bounded because by (5.2.5),

Consequently, F has a bounded extension P to Lq(D) (the Hahn-Banach theorem [15]) and P ( h ) = ( f , h) with some f ELp(D) for any h € L q ( D ) (Riesz' representation theorem [15]). Restricting h in 2, and putting h=(A-R)g, we see that this f is a desired weak solution.


79

R,emark 5.2.2. For the case p = 1, the above proof works only if llMll< 1, and gives a weak solution in Lm(D)*=ba(D) (the set of bounded additive set functions which vanish on sets of measure 0 [15])2L1(D). When l[M[l1.

After a lengthy calculation using the asymptotic expansion of ,u,(K), we see that (D n and suppose h, be such that

(6.2.14)

hcEBO(Si[cil; YF*-)t

llhcll=O(l~l) (c+O)

.

Then, $c solves (6.1.7) in Lp-sense and, with r=2-1/p, $,EBD(SJc,l;L;;*")

(6.2.15)

9

ll$cll=~(IcI1-o~).

So far, we have not mentioned the conditions to be imposed on M . Here, we only point out that all the arguments from Proposition 6.2.3 on are valid for M of (1.4.4) (i)-(iii), and for M of (iv) if IT,--T,I

(6.2.16)

Salcl

holds with some a 2 0 , where T,=l is T of our Maxwellian ge. The last condition comes from the second requirement in (6.2.14). The proofs of the statements in this section are all long, and we refer the interested readers to [44]. 6.3. Existence and stability From Theorem 6.2.4 and Lemma 4.3.3, we can see the Proposition 6.3.1. Let n 2 3 , [0,1) and ,!3>n/2+1. P € [2,41 n ((n+0)l(n+0-2), n+O)

(6.3.1) and put

(6.3.2)

r= 1+2/p.

Suppose,

,

There is a constant C20 such that, for c Sl[cI], IIB;'r[u, v]llO, G(., c ) becomes contractive for small c, which proves the Theorem 6.3.2. Let n 2 3 , 8 € [0,2/7), p>n/2+1, and suppose (6.3.1) and (6.3.3). Then, there is a positive number c, ( I c l ) such that f o r any c€S1[co], (6.1.6) has a unique solution u, in X ; satisfying

(6.3.4)

IIu,IIX;0 .

Also, it can be shown that u, E Wp(V ) and satisfies (6.1.3) in Lp-sense. With this u,, we now solve (6.1.9). Since the second term on its right-hand side is linear, Theorem 4.1.1 must be looked at with y=O, and hence E,(t) must decay faster than t - l . Taking the inverse Laplace transform of (6.2.9) gives a n explicit formula of E , ( t ) ; (6.3.5)

E,(t)=rE~(t)e+(y-rE,(t)*e)*

T D,(t) 9 n;lE:(t)e ,

where E,(t)=exp (tB,"),9 means the convolution in I and D,(t) is the inverse Laplace transform of (I-Tc(,2))-1,see (4.2.7). Knowing Theorem 6.2.1 and following the line of Theorem 4.4.6, we have, Theorem 6.3.2.

(6.3.6)

Let 1 < 9 < 2 < p < o o and m=O, 1. Then,

IIE,(t)(Z--P,)mull.$.-I

C(l+t)-~-~'~lluII,$~""Zs ,

with ~=(n/2)(1/q-l/p)and C 2 0 independent of c , t , u.

89


By this and Theorem 6.2.2, etc., we have,

Proposition 6.3.3. Let n 2 3 and l e t c , be that of Proposition6.2.3. each 8 € [0, l), there is a constant CTO such thar, ll(~c(t)-Z)ully;~- i Clcl -V +t)-rIIuII,;~-

(6.3.7)

holds for all c€S,[cJ, with r=(n-1+8)/2 is even.

Then, for

,

if n is odd and =(n-1)/2 is n

Substituting these into (6.3.5) yields a desired estimate. Write the righthand side of (6.1.9) as N [ v ] ( t ) . In order to evaluate the second (linear) term of "v], it is necessary that r > l in (6.3.6) ( m = l ) and (6.3.7), while for the third, it suffices that 7>1/2, according to Theorem 4.1.1. For the former, therefore, we should take 8> 0 in (6.3.7) when n=3. Otherwise, we can choose 8=0. If 8>0, a divergent factor IcI-8 appears, but this can be cancelled by (6.3.4). In any case, a careful choice of parameters is required. Write p , 8 of (6.3.1) as p o , 8, and impose the additional condition p , n/2+1 , r=min ((n/2)(1/q-U~), (n/p,+l)/2, ( n / ~ + 1 ) / 2 ).

(6.3.8) Then,

r> 112.

Set

l l l ~ l l=y: l (1+ t)rllv ( t )11x5 . We have,

III"v1lll I I I"4

+

IC(Ilvollx$nzg ( I C I -@a+Ill~lll)llI~lll) 7 I I C(IC I +a+ II1411+ I IIWII l)llIy- WI II

- "wll

9

where a=lluell in X $ , p=p,. By (6.3.4), IcI-@a+O as c+O, so N is contractive if v, is small as well as c. Thus, we proved,

Theorem 6.3.4. Let n 2 3 and suppose (6.3.8). Then, there are positive numbers a,, a,, co such that f o r any c € S,[c,] and if IIvoll 0, and will be resolved by introducing a Banach scale again. We always assume a ( x , 5)=0 and Grad’s cutoff hard potential, and deal with the Cauchy problem only. Otherwise, all are open. In particular, the case SfR” where the boundary layer appears as well as the initial layer is a physically important open problem. Our result is local in t . A long time behavior which may involve the shock layer is also a n open problem. In this respect, however, [12] is suggestive, in which the Chapmann-Enskog approximation of the one-dimensional shock is discussed. 7.1.

The uniform existence of solutions We consider the Cauchy problem to (1.6.2) for all E > O , with a fixed initial data f,, and seek a limit of the solutions f’ as E+O. If such a limit exists and coincides with the first term f of (1.6.1), then the Hilbert expansion will have been justified to the 0-th order. For such a limit to exist, it is primarily necessary that f E exists on the time interval [0, T ] independent of E . According to Theorem 2.3.1, f’exists on [0, ~ € 1 but , T~ is easily checked to tend to 0 with E . O n the other hand, f’ exists globally in t if f, is near go, as shown in Theorem 4.5.2, but f, should approach g, as E + O , i.e., a,+O. The desired solutions exist i f f , is near go and analytic in x . To prove E go this, we shall make use of the result from S 4.4. Put f ’ = g o + g ~ ’ 2 ~where is as in (4.3.2). Then, (1.6.2) is reduced to (7.1.1)

U;=B%c+-

1 r [ u e ],

UfI,=o=u,

E

,

where

B.=-E.V,+L L

.

E

As before, we shall investigate the integral equation, (7.1.2)

uE(r)=etBEuo+€

Roughly speaking, (7.1.3)

lt

.

e(c-s)Bar[~E(~)]d~

0

is uniformly bounded for E > O , while

elBE

1 E

e t B ( Z - P , ) = I V , l a + ~e-‘Ot’€b , E

with uniformly bounded operators a, b and a constant o,>O.

Thus, the un-


91

bounded factor a - l is replaced by the unbounded operator lVsl, a pseudodifferential operator with the symbol Ikl, The last term of (7.1.3) contributes to the initial layer. Now, our situation is much like that in S 3 in which Q is a pseudo-differential operator in E, and (7.1.2) can be solved by introducing a Banach scale to control the unboundedness of IV,1. Since its order is 1, the scale should be that of analytic functions in x. To be more precise, recall E ( k ) of (4.4.1) and set E e ( k ) = -ik-E+a-'L, the Fourier transform of EL. Since Ez(k)=a-lE(ak), Theorems 4.4.2-3 apply to Bc(k)with k , t replaced by ek, t / e respectively. We write the result as follows.

where L 2 = L 2 ( R ; ) . Now (7.1.3) is visible since (Z-P,)P:o)(i)=O by Theorem 4.4.2 (i). By Proposition 4.3.2 and proceeding as in the proof of Theorem 4.4.6, we can infer the

Y V )3 u=u(t)

-

III~III~=III~IIId,~*l,a,l=SUP llmIla-Tt,L,p< t€I

O0

9

where =(l+ly/2)1'2,y € R n , i=F& and Z c R is a n interval. sequel, we fix a, I , p such that, (7.1.5)

a>O,

l>n,

In the

/3>n/2+1.

Then, X is a Banach algebra, and u E X is analytic in x in the strip R"+ , P by i{lyl O andput r=a/y.

Then, writing ~ ~ ~ * ~ ~ ~ = ~ ~ ~ * ~ ~ ~ ~ , , ~ ,

l l l ~ ~ l l l ~ ~ ~ ~ ,+ r ~E >l O~.l l l ~ l l l Proof. By (7.1.4) and Lemma 7.1.1, (i) is immediate, and (ii) also, using thrice the inequality,

with 0, u ( E ) / c and u,/e as 5. Write the right hand side of (7.1.2) as N'[u'](t). Then, NE[u]=etBEuO+ P A - l r [ u ] . Since X is a Banach algebra, and by Lemma 4.4.3, IIIN"~1lIls c{IIklII

+( 1+ $)lIluIli~)

IIIN"ul-~"~IlllsC

9

~ l l l ~ l l l + l l l ~ l l l ~ l l l ~* - ~ l l l

This indicates that if uo is small, then N' is contractive, uniformly for E > O . Thus, we proved, Theorem 7.1.3. Suppose (7.1.5). Let r>O andputr=a/r. Then, there are positive constants a,, a, such that if IIuollO, (7.1.2) has a unique solution uEE Y ([0, T I ) with

lllUflll s a ~ l l ~ o* l l Consequently, u'(t) is analytic in x in the strip R"+i{jy] 0. The limit uo((t)satisfies (7.2.1)

u0(t ) =E( t ) u 0 + F k 1 f[uo](t ) ,

on (0, r ] , but since this can be solved in Y by the contraction mapping principle, we can say that u o ( t ) E I ' and satisfies (7.2.1) on [0, 71. Recall that LPo=O, by which (Z-PO)P:2)(O,k)(Z-P,)=O follows. Hence, (7.2.1) gives (7.2.2) (7.2.3)

(Z-P,)uo(t)=-L-lf[uo(t)],

Luo(r)+f[uo(t)]=O ,

or

P,u"O) =Pouo.

Set f f = g o + g ~ ' 2 u Lfor € 2 0 . (7.2.2) is equivalent to Q [ f o ] = O , so f o is a local Maxwellian. In view of Theorem 1.1.1, (1.6.6) holds also for f',E > O . Going to the limit, we have, e

t=O

e= E

ds

9

here gives 7

which is just (7.2.3). Summarizing, we have,

Theorem 7.2.1. Let ue be that of Theorem 7.1.4. Then, uc(t)+uo(t) strongly in ~([6, 71) for any 6>0, with uo(t)EF. (i ( i i ) f O ( t ) = g o + g ~ / 2 u o ( tis) a local Maxwellian whose fluid dynamical quantities p(t, x ) , u(t, x), T(r, x ) solve the compressible Euler equation (1.6.7) with the initial condition (7.2.4).

94

S. UKAI

In (i), 6=0 is not permitted, i.e., the convergence is not uniform near t=O. In fact, f ' ( 0 ) = f o is not in general a local Maxwellian, but f o ( 0 ) is. Physically, this non-uniform convergence is called the initial layer. However, if the initial f, is itself a local Maxwellian then the convergence becomes uniform and the initial layer disappears;

Theorem 7.2.2. If,in addition, uo=Pouo,then (i) of Theorem 7.2.1 holds good with 6=0. The proof is simple but is referred to [43]. Note that we have, at the same time, constructed a solution to the compressible Euler equation, although within a class of analytic functions. An interesting converse is [lo]: Suppose (1.6.7) has a smooth (Sobolev) solution and construct the local Maxwellian f 0 ( t )by (1.1.11) from the solution. Then, we can construct a solution f S ( t ) to (1.6.2) which tends to f o ( t ) , on some time interval [0, r ] . In this case, initial layers are absent because f c ( 0 ) = f o ( O )is a Maxwellian. The method does not apply to non-Maxwellian initials. So far, we have justified the Hilbert expansion to the 0-th order. Justification to higher orders is not known, but a slightly different expansion is possible ([7]); f"(t)=f"(a,

t)+fO(E,

t/E)+Ef1,*(E,

t) ,

where ( i ) f , ( ~ t,) is sufficiently smooth in [0, 11x [0, T], with f o ( O , t ) = f o ( t )of Theorem 7.2.1 (ii), ( i i ) ~ O ( Ea), is sufficiently smooth in { ( E , a)/€€ [0, 11, EU E [0, r ] } ,and behaves like e-aa with a>O, (iii) f l r * is uniformly bounded and ~f'-* is sufficiently smooth, in [0, 1 ] x [O,r]\{(O,O)}. Further, f l , * has the form, f"*(E,

t)=f'(E,

f)+J'(E,

t/E)+ES2.*(E,

4,

and similarly for f 2 , * , fs2* and so on, each having a property like (i) (ii) (iii). As for the Chapman-Enskog expansion, we must mention [26] which shows that the solution of (1.6.2) with a fixed E > O approaches, as t + a , that of the compressible Navier-Stokes equation with the viscosity and heat diffusion coefficients proportional to E . This is not, however, sufficient for the justification of the expansion to the first order.

References [ 1 ] T. Arai, (in preparation). [ 2 ] L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal., 45 (1972), 1-34. [ 3 ] -, Intermolecular forces of infinite range and the Boltzmann equation, Arch.


95

Rational Mech. Anal., 77 (1981), 11-23. L. Arkeryd, Loeb solutions of the Boltzmann equation, Arch, Rational Mech. Anal., 86 (1984), 85-97. K. Asano, Local solutions to the initial and initial boundary value problems for the Boltzmann equation with an external force, I, J. Math. Kyoto Univ., 24 (1984), 225-238. -, On the initial boundary value problem of the nonlinear Boltzmann equation in an exterior domain, (in preparation). K. Asano and S. Ukai, On the fluid dynamical limit of the Boltzmann equation, Lecture Note in Numer. Appl. Anal., 6, North-Holland, 1983, 1-19. R. Beak and V. Protopopescu, Abstract time-dependent transport equations, (preprint). R. E. Caflisch, The Boltzmann equation with a soft potentials, Comm. Math. Phys., 74 (1980), 71-109. -, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33 (1980), 651-666. -, Fluid dynamics and the Boltzmann equation, Nonequilibrium phenomena I, The Boltzmann Equation, (Eds. J. L. Lebowitz and E. W. Montroll), NorthHolland, 1983. R. E. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194. T. Carleman, Probltme Mathtmatiques dans la Thtorie Cinttique des Gaz, Almquist et Wiksell, Uppsala, 1957. C. Cercignani, Theory and Application of the Boltzmann equation, Elsevier, Amsterdam, 1975. N. Dunford and J. Schwartz, Linear operators I, Interscience Publ., New York, 1957. R. S. Ellis and M. A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl., 54 (1972), 1825-1856. J. P. Giraud, An H-theorem for a gas of rigid spheres in a bounded domain, Colloq. Intern. CNRS, 1975, N236, 29-58. H. Grad, Asymptotic theory of the Boltzmann equation, Rarefied Gas Dynamics I, (Ed. J. A. Laurmann), Academic Press, New York, 1963. -, Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, Proc. Symp. Appl. Math., (Ed. R. Finn), AMS, Providence, 1965. K. Hamdache, Existence in the large and asymptotic behavior for the Boltzmann equation, to appear in Japan J. Appl. Math. E. Hille and R. S. Phillips, Functional Analysis and Semigroups, AMS, Providence, 1957. A. G. Heintz, Solution of the boundary value problem for the nonlinear Boltzmann equation in a bounded domain (in Russian), Aerodyn. Rarefied Gases, 10 (1980), 16-24. [231 R. Illner and M. Shimbrot, The Boltzmann equation; Global existence for a rare gas in an infinite vacuum, (preprint). 1241 S. Kaniel and M. Shimbrot, The Boltzmann equation, Comm. Math. Phys., 58 (1978). 65-84. T. Kato, Perturbation Theory of Linear Operators, Springer, New York, 1966.

96

S. UKAI

[26] S. Kawashima, A. Matsumura and T. Nishida, On the fluid dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Comm. Math. Phys., 70 (1979), 97-124. N. B. Maslova, Stationary boundary value problems for the nonlinear Boltzmann equation (in Russian), Aerodyn. Rarefied Gases, 10 (1980), 5-15. N. B. Maslova and A. N. Frisov, Solution of the Cauchy problem for the Boltzmann equation (in Russian), Vestnik Leningrad Univ., 19 (1975), 83-85. B. Nicolaenko, A general class of nonlinear bifurcation problems from a point in the essential spectrum, application to shock wave solutions of kinetic equations, in: Application of Bifurcation Theory, Academic Press, New York, 1977. T. Nishida, A note on a theorem of Nirenberg, J. Differential Geometry, 112 (1977), 629-633. -, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Commun. Math. Phys., 61 (1978), 119-148. T. Nishida and K. Imai, Global solutions to the initial value problem for the nonlinear Boltzmann equation, Publ. R.I.M.S., Kyoto Univ., 12 (1976), 229-239. A. Parczewsky, Local existence theorem for the Boltzmann equation in L’, Arch. Mech., 33 (1981), 971-981. Y. Shizuta, On the classical solution of the Boltzmann equation, Comm. Pure Appl. Math., 36 (1983), 705-754. Y. Shizuta and K. Asano, Global solutions of the Boltzmann equation in a bounded convex domain, Proc. Japan Acad., 53A (1977), 3-5. C. Trusdell and R. G. Muncuster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monoatomic Gas, Pure Appl. Math., Vol. 83, Academic Press, New York, 1980. [37] S. Ukai, On the existence of global solutions of a mixed problem for the nonlinear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. Les solutions globales de 1’8quation de Boltzmann dans I’escape tout entier 1381 -, et dans le demi-espace, C. R. Acad. Sci., Paris, 282A (1976), 317-320. The Transport Equation, (in Japanese), Sangyo Tosho Publ., Tokyo, 1976. [391 -, Local solutions in Gevrey classes to the nonlinear Boltzmann equation with[401 -, out cutoff. Japan J. Appl. Math., 1 (1984), 141-156. S. Ukai and K. Asano, On the initial boundary value problem of the linearized Boltzmann equation in an exterior domain, Proc. Japan Acad., 56 (1980), 12-17. -, On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. R.I.M.S., Kyoto Univ., 18 (1982), 477-519. -, The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J., 12 (1983), 303-324. -, Steady solutions of the Boltzmann equation for a gas flow past an obstacle. I, Existence, Arch. Rational Mech. Anal., 84 (1983),248-291,11, Stability (preprint). J. Voigt, Functional analytic treatment of the initial boundary value problem for collisionless Gases, Habilitationsschrift, Univ. Munchen, 1980. Department of Applied Physics Osaka City University Sugimoto 3, Sumiyoshi-ku Osaka 558, Japan

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 97-128 (19861

Equations of Motion of Compressible Viscous Fluids By T a k a a k i NISHIDA Abstract. We survey the global solutions of equations for one-dimensional motion of compressible, viscous and heat-conductive fluids. Initial value problems with fixed and free boundaries are treated about solutions global in time and about the asymptotic behaviors as time tends to infinity. Key words: equations of motion of compressible, viscous fluids, solutions global in time, asymptotic behaviors of solution, initial value problem with fixed boundary, free boundary problem

S 1.

Introduction

We consider the one-dimensional motion of viscous compressible and heat-conductive fluids:

in t 2 0 ,

-m<xo ,

apiap

, a p m , aeiao >o

and also

(1.3)

p>O,

r>O,

in p > O ,

8>0.

The initial and initial-boundary value problems for (1.1) are solved in a general situation under the natural assumptions (1.2), (1.3) and p>O, 8 > 0 locally in time. [24], [5], [28], [29]. Received October 15, 1985.

T.NISHIDA

98

Thus the main concerns with (1.1) are the global solvability and behavior of the solutions of the initial and initial-boundary value problems. The present theory for global solutions is far from complete. But there are two important special cases of equations of state of fluids. [l]. ( i ) polytropic gas:

where R and y > 1 are positive constants. ( i i ) barotropic gas:

(1.5)

p=ApT,

A=constant

.

Theories assuming the condition (1.4) or (1.5) and also the constancy of the coefficients p=constant>O ,

(1.6)

r=constant>O

,

have been fairly developed recently. [7], [8], [13], [15], [14], 1271 and [26] etc. In this paper we survey the two basic initial boundary value problems for (1.1) with (1.4) or (1.5) and (1.6). ( I ) Fixed boundary problem in Q - = { ( t , x); t 2 0 , O<x<X}:

t

(1.7)

u(t, O)=u(t, X ) = O

,

O J t , 0)=O,(r, X ) = O

.

(11) Free boundary problem in Q = { ( t , x ) ; O < t , O < x < X ( t ) } , where X ( f ) is a free boundary.

(1.8)

‘i

B,(t, O)=O , u(r, O)=O , p(t, X(t))=O , B(r, X(t))=O and

dX/dt=u(t, X ( t ) ) .

Problem I is a standard one for the system (1) and Problem I1 is a model for interface of a compressible fluid with the vacuum. We use the following notations: Q,={(t, x) E R 2 ;O i t < T , xEZ}, where Z is [0, 11 or R . .gU(Z)is the Holder space of Holder continuous functions with exponent u on I . .glta(Z) is the subspace of ~ 3whose ’ ~ 1-th derivative is also Holder continuous with exponent u. G+’/2+’(Q,) is the Holder space of functions on QT which is Holder continuous with exponent 012 and u with respect u €U~ , ~, E~ ~~ ~~ ~~ ~W1+“(Q,)= ; * ~ } . totandxrespectively. . ~ 3 ’ ~ + ~ ( Q ~ ) =u,,{ u,, { u € & + ’ / ~ , ~ ut, ; U , E ~ ~ H ~ 1~denotes ~ ~ }the. Sobolev space of L2 functions together with their x derivatives up to and including the Z-th order. B(0, T ; H 1 ) denotes the bounded continuous functions of t E [0, TI with vector values in H L . Lz(O,T ; H L ) denotes the square summable functions of t € [0, TI with vector values in H 1 . We also use the space LP of functions whose p-th power are summable, 1 i p < 00.

Motion of Compressible Viscous Fluids

8 2.

99

Initial Value Problem with Fixed Boundary

It is convenient to transform the system (1.1) with (1.4) or (1.5) to that in the Lagrangian mass coordinate. After a suitable scaling of the variables we have the following system:

where v =l/p is the specific volume. The boundary condition (1.7) is given by (2.2)

u(t, O)=u(t, 1 ) = 0 ,

B,(t, O)=B,(t,

1)=0,

rTO,

and the initial conditions are supplied: (2.3)

( v , U, O)(O, x ) = ( v ~ ,~ o ,B,)(x) ,

0 1 ~.~ 1

Here we assume the positivity (2.4)

vo(x), O o ( x ) > O ,

01XO V G W1+"(QT) ,

u, OE.G'ztu(Q,),

T.NISHIDA

100

{ II ""8;";;

(2.8)

M(T)- -

Yllzl,

,

~ll$,'"'

I W T )< 0jM(T) in Q , . 1

This is the first and a nice global existence theorem for the polytropic viscous gas and given by Kazhikhov-Shelukhin (1977). It is proved by a standard argument of continuation of the local in time solution by the a priori estimates in the form (2.8). But the proof for the estimate is technically involved (See [15].). This solution decays to the constant state ( 1 , 0 , 1) as time tends to infinity.

Theorem 2.2. The solution of the initial boundary value problem (2.1)-(2.3) decays to the constant state (1, 0 , 1) in H'(0,l) as ?--too. For some T>O the decay rate is exponential for t 2 T . This is proved by using several excellent ideas by Kazhikhov [14]. Since the Journal is difficult to obtain, here we trace his proof of this Theorem for the solution obtained in Theorem 2.1.

Proposition 2.3. density satisfies (2.9)

There exist constants m,,M , such that for any t2O the

O < m , < v( t , x) < M, < m ,

where

(t,x)€Qm,

em=[O, a)x [O, 11.

Before the proof of Proposition 2.3 we notice three identities for the sohtions of (2.1)-(2.3).

(2.10) (2.11)

\:u(t,x)dx=

s:

v,(x)dx=l

,

\'0(t, x)+ue(t, x ) / 2 d x = ~ 1 0 , ( x ) + u , l ( x ) / 2 d x =5 l 0

0

s:

(2.12)

V(t)+ V(t)dt=E,
O there exists x ( t )E [0, 11 such that

(2.15)

a Iu(t, x ( t ) )I B,

(2.16)

a l 0 ( t ,X ( t ) ) l / 5 ,


where

LY,j9

(2.17) Proof.

(2.18)

101

are two roots of the equation.

y-1-logy=

min (a, 1)

'

It follows from (2.12) that \~v-l--logv+0-l--log0dx~E1,

where E,=E,/min(a, 1). If v-l-logv+0-l-logO>E, for any x € [ O , 11, then it contradicts (2.18). Therefore at some point x(r) E [0, I], v-1-log v f 0-l-log0<E1, which proves (2.15) and (2.16). Lemma 2.5. For each t 2 0 there exists a t least one point xO=xO(t) € [0, 11 such that the following identity holds:

where

(2.20) Proof. Since by (2.1) we have ut=o,,

there exists a function +(t, x ) which satisfies u=(i),,

u=&.

Furthermore it follows from the definition (2.20) of u that this function the solution of the equation:

+ is

Multiplying this by v and using (2.1) this equation can be transformed to a divergence form:

( u + ) ~ - ( u + =p+,,--aO--U2 )~

.

Integrating it in Q, and using the boundary condition u=+,=O ary, we have the following equality:

on the bound-

102

T. NISHIDA

+

u2 a0 dxds = ?F( t ) ,

(2.21) where

Here we know that for each t > O there exists xo(t)€ [0, 11 such that

40, xo(t))=W

(2.22) In fact if not, there exists

*

such that

to>O

or

But multiplying this alternative by v(torx ) and integrating it on [0, 11 by using (2.10) we get an inequality for t = t o which contradicts the identity (2.21). Therefore since we can take

Lemma 2.5 follows from (2.21) and (2.22). Lemma 2.6.

The specific volume v(t,x ) has the representation:

v(t, x ) = D ( t , x ) exp

(2.23)

[

x 1 +-sf_

p

(-F1

so t

\ou2+aOdxds) 1

s‘ m, (+ \:1: 0

x)

exp

+

u2 a0 dydT) ds]

,

where

Proof. The first two equations of (2.1) give the following:

Integrating it with respect to t on [0, r] and integrating it on [ x o ( t ) ,x ] for t : fixed, we have


103

= p l o g v , , ( ~ ) - 0\ ~0~ ~ u ~ + a O d x d s + - \ x o u o ( ~ ) d ~ + ]u-u0dy, x 20

where thelequality (2.19);is'used.

Therefore we have

and so we have

aB exp (\t*ds)=exp PV

PV

0

(Ljt j1 u2+aOdxds P

0

0

Thus integration gives e x p (pl \8\1uz+aOdxdr)ds 0 0

e x p ( \ t *0 dPVs ) = l + \ t * D - lO( s P, x )

and (2.23) follows from this and (2.24).

Q.E.D.

Let us define (2.25)

M,(t)=max ~ ( tx),

,

M,(t)=max O ( t , x) O i X 5 l

O L X i l

.

Now we show the uniform bound for v from above. Since we have (2.11), D is uniformly bounded: (2.26)

O 0

(1'

w2+cu4d

1:

+)

~

~

K$+C----

u2u,2 dx U

0

(2.38)

IC,j ' s d x 0 2 , O. ( ii ) Moving boundary condition (Piston problem): (2.50) where X , ( t ) and X z ( t ) are given functions such that

5:

vo(x)dx+

s:

{X,(s)-Xx,(s)}ds>constant>O

.

Two global existence theorems (similar to Theorem 2.1) of the initial boundary value problem for (1.1) and (2.50) with Neumann zero boundary condition or Dirichlet boundary condition (2.49) for 0 are given in [14] by introducing auxiliary functions which transform the boundary data to zero. Some asymptotic behaviors of solutions for ( l . l ) ,(1.5), (2.50) are considered by Kawashima 1101.

S 3. Cauchy Problem Here we notice the initial value problem (2.1), (2.3), (2.4), x € R , which is solved globally in time by Kazhikhov [14]. Theorem 3.1.

(3.1)

Zf the initial data satisfy


109

then there exists a unique global solution which satisfies for any T > 0

Proof. In this case the equalities (2.10), (2.11) do not hold, but the equality (2.12) holds.

(3.3)

where U ( t )=

(3.4)

s'"_

+

u2/2+a( v - 1 -log v) 8- 1 -log 8 dx

Thus we have the following: Lemma 3.2. For each interval I , = [ n , n + l ) , there is a point xn(t)EZnsuch that

(3.5) (3.6)

a<j1,vdx,

1

SdxO

,

X

~

where we drop the overbars of all variables and parameters. and initial conditions are taken to be

,Q

The boundary

and (1.4)

u,(O, x)=uto(x) ( i = l , 2)

,

v(0, x)=vo(x),

x€S,

respectively, where a p n denotes the outerward normal derivative on First we state the global existence theorem on (1.2-4).

fi=QuaQ.

Theorem 1. Suppose that there is K , > 0 such that

aQ

and

Predator-Mediated Coexistence o s ~ , o ( x ,) uzo(x)

&(X)

9

131

SKl

, that for any x €a. Then (1.2-4) has a unique solution (ul(t,x ) , uz(t,x ) , ~ ( tx)) exists for all time. Moreover, there is K z > O such that

OSu,(t, X) , u ~ ( TX ,) , d t , x ) S K z for any

~ €and 0t>O.

The proof will be stated in Appendix. When the predator is absent (v=O), (1.2) is simplified to be - d,Au,

(1.5) -= at

+u,( 1 -u, -cu2) +

,

r>O,

x€Q.

dZAu2 u , ( u - ~ u ~ - u ~ )

For (1.5) with zero flux boundary conditions (1.3), de Mottoni [19] and HSU [9]proved the following: ( I ) If aO such that (3.2) has a unique one-parameter family of solutions ( ~ ( u )u(u)) , E R, x X bifurcating from ( e C , u*) for Iu] F. The solid curve is a stable branch and the broken curve is a n unstable branch. a=0.95, b=1.5, c=1.0, a1=a2=O.5, k=10.0, d = l , D=5.0 and r=3.1.

Figure 14. Schematic DI-branch sheet in ( r , €)-space. The upper sheet is stable and the lower one is unstable.

and there occurs no bifurcation from E , , , . Let r be further increasing beyond F ( r > F ) . In this situation, a coexisting equilibrium point E,,, no longer exists and either E,,, or E,,, is the stable equilibrium point. However D,-

Predator-Mediated Coexistence

149

branch still exists with a limit point S,. It is noted that there are two different types of solution branches to the limit €LO. When E is sufficiently small, the upper branch corresponds to the singularly perturbed solutions constructed in the preceding section, while the lower branch to that of new singularly perturbed solutions which we have never seen. Numerical results suggest that the upper branch is stable while the lower one is unstable (Figure 13). Moreover, we find that the limit point s, as a function of r is decreasing with r and the D,-branch disappears for large Y (Figure 14). In summary, we conjecture that there exist stable singularly perturbed solutions even for r @ (!, P), which exhibit sharp spatial segregation between u, and u2. 5.

Concluding Remarks

In the previous sections, we have shown mainly two different types of asymptotic states: one is the spatially homogeneous periodic or aperiodic solutions (Figures 1.2-4) and the other is the spatially inhomogeneous stationary solutions when E is sufficiently small (Figure 2). Both solutions exhibit coexistence mediated by a predator. We should remark that the argument in constructing the latter solutions is valid independently of the stability of E,,,. In this section, we are concerned with spatio-temporal segregated patterns as shown in Figure 3. Fix appropriately all of the parameters except for a parameter E to realize the situation where E , , , is unstable and there coexists a stable spatially homogeneous periodic solution branch when E is large (Figure 1.2) and D,-branch bifurcating from E,,,-branch which enters into the region where our singular perturbation approach is applicable when E is sufficiently small (Figure 12). It is noted that D,-branch in a neighborhood of ( E ~ u*) , is unstable, if it is super-critical, because E , , , is unstable. In this situation, it is observed by numerical simulations that when E is decreasing, D,-branch is still unstable but the spatially uniform periodic solution becomes unstable and a spatio-temporal periodic solution appears. This solution exhibits

u2

111

V

Figure 15.1. Spatially homogeneous periodic solutions where a =0.95, b=1.5, c=l.O, a1=a2=0.5, k=10.0, r=1.0, d = l , 0 = 5 . 0

and ~=0.04.

M. MIMURA and Y.KAN-ON

150

111

U

U?

Figure 15.2. Spatially inhomogeneous periodic solutions where a, b, c, al, at, k , r , d, D are the same as in Figure 15.1 except for ~=0.03.

Figure 15.3. Spatially inhomogeneous stationary solutions where a, b, c, a ~az, , k , r , d, D are the same as in Figure 15.1 except for ~=0.0025. S

spatially liorno~cneoos periodic solution

....-

3000000000000000000000~*********

.**

***

--. ..

spatially inliornogeneous periodic solution

'\,D,-branch \

,_ _ - -- - - a ___________ E+++-branch

\

-_--- ----

&

spatio-temporal segregation between 2-competing species. Furthermore, when E continues to decrease, the spatio-temporal periodic solution disappears and D,-branch becomes stable. That is, singularly perturbed solutions are stable (Figure 15). This behavior can be understood as a result of the inter-


151

action of Hopf and stationary bifurcations (see, for instance, Guckenheimer [7]). Thus, we confirm the existence of a secondary branch connecting the spatially uniform periodic solution branch with D,-branch (Figure 16). This suggests recovery of the stability of D,-branch when E is sufficiently small. That is, singularly perturbed solutions are stable. Moreover, a situation can be considered where there coexist spatially uniform periodic and aperiodic solutions and the globally existing D,-branch. For this situation, it is numerically confirmed that when E becomes sufficiently small, D,-branch recovers the stability after fairly complicated interactions of periodic and aperiodic solutions and D,-branch. Appendix

Proof of Theorem 1. Since a standard theory of semilinear parabolic equations yields the local existence and uniqueness of solutions of the problem (1.2-4), we may only show a n a priori estimate of L"-uniform boundedness on solutions. By using the assumption Osul0(x), u,,(x), vo(x)5K1,it is easy to see that

OSu,(t,x ) S M a x (1, K , ) and 0 5 u,( t, x) 5 Max (a,K,)

Let us show L"-uniform boundedness of v(t, x).

. From (1.2), we have

Applying the inequalities u ~ ( tx)dx ,

(i= 1,2)

to (A.l), we find that there is some K,>O such that (a,u,+azuz+v)(t, x)dxSK,

for any t > O . Thus, by using L"-uniform boundedness of u, and uz and L1uniform boundedness of v, Alikakos' theorem ([l, Theorem 3.11) shows L"uniform boundedness of v. Thus, we can secure the global existence of solutions of (1.2-4). Proof of Theorem 3. Since E , , , does not exist, there are equilibrium points on the boundary

152


of R:. By a simple calculation, one finds that some of them are locally asymptotically stable. Therefore we may prove that the solution of (2.1) approaches one of them as t tends to infinity. We may assume 1Al >O. The case of IA I < 0 can be treated similarly. Rewrite (2.1) as

(A.2)

d -log dt

u=e-Au

where e = t ( l , a, - r ) and log u=f(log u l , log uz, log v). Multiplying (A.2) by the cofactor matrix (Atj) of A , we have

('4.3)

Let u=zi be the solution of Au-e=O. Since E , , , does not exist, at least one of zi,, ii2 and V is negative. Suppose El be negative. Note that the first equation of (A.3) becomes

Then the right hand side of the above is negative, so that uf11u$2vA13 must be strictly monotone decreasing. By the boundedness and positivity of solutions, uf%f12vA13 tends to zero as t tends to infinity. Thus it turns out that the w-limit set of the orbit u belongs to the boundary of R3,. O n the other hand, if the initial values (ulo, uzo,u,) are on the boundary of R:, the solutions (ul, uz, v) are always on the boundary of R3, a n d approach one of the stable equilibrium points in the sense of a suitable two component system of (2.1). Moreover, by using phase space analysis, it is found that these equilibrium points are stable in a sense of the complete system (2.1). Thus , w e conclude that the w-limit set consists of a stable equilibrium point o n the boundary of R3,. Proof of Theorem 4. From IAI=0, (2.1) becomes

(A.4)

In Case (i), by assumption, one of the right hand side is not zero.

Therefore


153

the proof of this case can be reduced to that of Theorem 3. In Case (ii), the right hand side of (A.4) are zero. So the proof is obvious.

Proof of Theorem 6 . The eigenpolynomial of A4 is

By IAI>O, n o eigenvalues of (A.5) can be zero. When r is a bifurcation parameter, leaving the remaining parameters fixed, the critical value of r is obtained by R e 1(r)=O. The necessary and sufficient condition that Re 1(r)=O is f(r) =(uF+u$){(1 -b c ) u ~ u $ + ( a , k z u ~ + a z u f )v *ufu$v*lA } I=O

.

Sicne u* is the solution of Au+e=O, u:, uf and v* are all linear with respect to 1. Then f ( r ) is cubic with respect to r. By simple calculations, we know that limf(r) =c~,u$~v* r\c

and

limf(r)= a,k2ufZv* . r/?

Therefore we have two cases: (i) if f(r)+O on (c, P ) , E,,,-branch is stable and (ii) iff@) has two real roots r,, r*, E,,,-branch is unstable on (r*, r*) and Hopf bifurcations occur at r=r, and r*. Proof of Theorem 7. Let Ue='(Uel, uez,ve) be a periodic solution with period 0 of (2.1). stituting the transformation

By sub-

(A.6) into (2.1), we have the system for U = c ( U , ,U,, V ) of the form (A.7)

d U=A(t)U+ O(I UIZ) dt

where

i

-uei

-cue, A ( t ) = -hue, -ugt alkuo,

azuez

0

Let Y ( t )be the fundamental matrix of the linear part of (A.7). The stability of the periodic solution ue is determined by the eigenvalues of T(0).Denote the eigenvalues of V(0) by A,, A, and 1, which consist of lAllS lApl and A3= 1.

154


It follows from Abel's theorem that I,R,=det V(O)=exp {$," tr A(t)dt}F(u, v), and the monotonicity of u requires that the equality holds only at x=O and 1. Let the definition domain of E=E(E, v ) be denoted by T . Then,

where Eb =ma x {E;(hs), E?(bh,)}and E"(y)=min {E:((y), EZ((y)}i

On Multiple Existence of Stable Solutions

187

Then,

(3.4) and

(3.5) where ii(E, Y) and _u(E,Y ) are two consecutive zeros of

F(u, Y ) = E with h-(v)O small enough. In the next section, we discuss also construction of large amplitude solutions using the singular perturbation technique. An important problem which is not included in these arguments is the stability of D"(E,a) solutions for n > l . In this section, we first show the stability of D ~ ( E a), for small enough E > O and a20, under Assumps. and . As a remark, a violation of leads to a n instability theorem. We next show that for n22, D*(E,0) (large amplitude solutions with mode n 2 2 ) on the shadow ceiling u=O, are all unstable. We note here that the "recovery of stability" of Dn's when o > O is the main subject in our future discussions. The results in this section are due to [34] and we ask the readers to refer to it for details. (See also [33].) To show the essence of our argument, we first consider the case o=O. So, we denote Sf as D:(E,0) in this section. The linearized stability problem of D:(E,0) is reduced to the study of the eigenvalue problem : iL'W'+

,

f:7j+=REWL

x EI , x€dZ,

J%=O

where 77' is a constant function, and

(4.1) Here,f,"=f,(uE,

vf),

and so on.

{c,

Lemma 4.1. Let Sturm-Liouville problem :

fn}

(n=O, 1,

-

- ) be the orthonormal system of the

(4.2)

( 1 ) Then, the principal pair of eigenvalue and eigenfunction (C;, &)

H. FUJI].Y.NISHIURA and Y.HOSONO

192

satisfies:(*)

< C , exp ( - r / E )

0
0

small)

and

oi>o

(4.4)

( X E U

&dx = O(dT),

7

where r and C,are positive constants independent of E > 0. ( 2 ) Moreover, there is a positive constant p > 0 , such that for n= 1 2 , (4.5)

- p ,

namely, (Ls-R)t is a compact operator of Lz(Z)into L Z ( I ) n{&}', for &>O. The next lemma We let Q; be the projection of Lz(Z)onto Lz(Z)n {#,}l. is the first key to our arguments.

Lemma 4.2. More precisely,

(P-R)t becomes a nzultiplication operator in the limit

for any bounded u E L 2 ( Z ) ,and Re R >

-,u,

where fZ=f,(u*(x),

E

LO.

y*);

The following prepares the second key.

Lemma 4.3. 1 ( 1 ) lim -O.

f,", &>=

> 0 is a positive constant.

Now, before stating our main Theorem, we need to divide the eigenvalues into two classes. Let A , ( i = O , 1) be defined as:

Non-critical eigenvalues (4.8) (*)

A,"'{;jEl

IR'126>0, & > O } ;

To extend the argument to ~ 2 0 we , need to show: O 0, and hence, is a non-critical one. To see this, it is enough to observe that such 2''s satisfy the Sturm-Liouville problem, and hence is equal to some 6 . Thus, if k 2 1 , then, Re R"=P=G< - p , by Lemma 4.1. Next, we point out that (C:, (&, 0)) can not be a n eigenpair to (E)f, since from (1) of Lemma 4.3, the left side of the second equation of (E)€can not be zero. Accordingly, it suffices to consider only those eigenfunctions which satisfy V'fO,

E>O.

Using the Sturm-Liouville eigenfunctions, I is now written as:

(assuming that Re I > - p without loss of generality!). From Lemma 4.2, it follows that

where f z = f u ( u * ( x ) ,Y*)

;:, I:zf'"'

, etc.,

and A*=-?-

-

Lemma 4.4 (Non-critical eigenvalues). Let I' be a non-critical eigenvalue which stays in A , for small E > O . Then, there exist positive constants 6 , and €dl independent of E such that (4.12)

ReP0) can be similarly shown. Concerning the critical eigenvalues A,,, we have the following Proposition 4.1 (Critical eigenvalues). A , includes onZy one eigenvalue A:, which is real and simple. When E 1. 0 , 2: behaves a s : (4.1411

2;--Aloa

,

where

See, Fig. 4.1 (a).

Corollary 4.1.

Under , it holds that

Remark 4.1. Suppose that J J ( v * ) > O bility occurs, i.e., &>O.

1, < 0 .

(see fig. 4.1 (b)), the crinical insfa-

Proof. First, we rule out the two cases: ;i&>O, P E A , , . Next, from Eq. (4.10), R=R' is a solution of (4.15)

F(I, &)=R2--R

RE>O

&>+C

\,

for any

[*Idx] = O ,

where the integral $ I [*]dx is the first term of the right side of Eq. (4.10). In view of exponential decay of C: (Lemma 4.1), Lemmas 4.2 and 4.3, and using the implicit function argument, we can show our proposition.


Fig. 4.1. (a) Spectra of D:(E,0).

195

Fig. 4.1. (b) Functional form of f ( u , v) when .Ti("*) >O.

Stability o f D:(E, a) for small a>O. An extension of the preceding arguments to D:(E, a), ( E , a) E Ql, namely, D'-singularly perturbed solutions of class I (see, Sec. 5 ) is given in [34]. Here, we show the outline of the discussion. Proposition 4.2.

There is a small rectangle Q , E { ( E , ~ ) I O < EOO for v>u*, and vice versa. Under (global Sg), G*(v) is monotone decreasing in each of the intervals v < v < u * and u*O. Note also that this is a jump along R, connnecting the two bi-stable states (v*, h-(v*)) and ( v * , h + ( v ) ) . The “depth” of the jump is determined by the condition: (5.1)

J(h-(v*), h,(v*); v*)=

s

h+(u’)

f(s, v“)ds=O

.

k-(v*)

The reduced sets of Neumann layered class, i.e., class 11, R(o)’s are obtained by attaching Neumann slits to one and/or both ends of the interval Z. Let us define two functions k + = k * ( v ) ,defined on Y*, respectively, by the relations : J(k,(v), h , ( v ) ; v ) = O ,

and

U E Yi.


205

The reduced solutions (U#n(x),V;(x)), corresponding to R,(u) are plotted in Fig. 5.5. II=

h,

(c'

( c))

I

U 1

!

"t

I 1

We note that

and

By definition, the "heights" af(a) (See, Fig. 5 . 5 above) of Neumann slits are determined by

af(d=k*(v34)

,

and hence, satisfy the generalized Fife condition: (5.2)

J(k*(vZ(o)),M v 3 u ) ) ; vZ(a))=O

.

0 '

Fig. 5.6. Reduced set R#(o)of (the bold solid line).

H. FUJII,Y. NISHIURA and Y. HOSONO

206

Proposition 5.3 ([27]). Assume . Then,for each n 2 1, it holds that There exists a rectangle 0 ) [ O < E < E , , O O , one of the eigenvalues, R-, is smaller than -1 and the other, A,, is between 0 and 1. The eigenvector of linear map DF, are given respectively by

(3.6)

tE,

and

tE-,

where Et=(l+, 1) and (-=(A-,

1)

for r € R . Let E s = { r f + l t € R } and P = { t f - I t € R } denote the eigenspaces. Note that if w e Es, then (DF,)"w tends to the origin as n tends to infinity, and that if w e E" then (DF,&)-"wconverges to the origin when n tends to infinity since R- < - 1 and 0 < A, < 1. Linear space Es is called the stable eigenspace and Eu is called the unstable eigenspace at ( p , p ) . Linear space E" is called the unstable eigenspace. If h is very small, then A, is near 1 and I - is near -1. Eigenspace Es is near the "diagonal line" A = { ( u , v)€R21u=u} and E" is almost orthogonal to A . Now, let us look at the mapping Fh near the fixed point ( p , p ) . By the theorem of Hartman-Grobman [8], we know that F, is topologically conjugate to the linearized map DF,. More precisely, there exist a neighborhood V of ( p , p ) , a neighborhood U of the origin ( 0 , 0 ) € R 2 , and a homeomorphism q5: U- V such that for any ( u , v) € U ,

holds.

If w € $ - ' ( E S ) ) ,then F,l(w) converges to the fixed point ( p , p ) . If converges to the fixed point as n tends to the infinity. The set fioc=q5-1(E*)is called the local stable manifold and W;.,,=q5-'(Eu) is called the local unstable manifold. The stable manifold w" and the unstable manifold Wu of the fixed point are defined as

w € q5-'(Eu) then F;"(w)

They are smoothly immersed curves i f f is smooth. In our case, the fixed point ( p , p ) is said to be a saddle point since both w" and W' are non-trivial.

Chaotic Phenomena and Fractal Objects

229

Next, suppose that the ordinary differential equation has a n unstable equilibrium point, say x=q, withf(q)=O andf’(q) >O. By a similar argument, fixed point ( q . q) of Fh has its own stable manifold and unstable manifold, becuase, in this case, the eigenvalues satisfy - 1 < L ( q )< 0 and R+(q)> 1. The stable eigenspace Es is defined a s the linear subspace spanned by eigenvectors belonging to the eigenvalue which is smaller than 1 in norm. The unstable eigenspace Eu is spanned by eigenvectors with eigenvalues greater than the unity in norm. Fixed point (4, q) is a saddle point, too. Consider the following situation. A point, say w E R 2 , is said to be doubly asymptotic if F;(w) tends to a saddle point, say P, as n tends to infinity and if F i n ( u )also tends to a saddle point. If there exists a doubly asymptotic point w such that

(3.9)

lim Ph(w)=lim F;”(w)=P n-oa

n-sm

then o is called a homoclinic point. The existence of homoclinic points was first discovered by H . PoincarC [22]. A homoclinic point belongs to the stable manifold W s and the unstable manifold Wu of the saddle point at the same time. S. Smale [25] proved that if these two curves, w” and W u , intersect at w and are not tangent to each other at 0, then there exists a n uncountable closed set S, which is invariant by the mapping Fhr such that the dynamical system Fh restricted to S (or its some iterated composition) is equivalent to “coin tossing”. The existence of a homoclinic point with transversal intersection of invariant manifolds implies the existence of infinitely many homoclinic points and periodic points (and a n uncountable number of aperiodic points). By numerical experiments, M . Yamaguti and S. Ushiki [35] observed the existence of homoclinic points in several cases. They considered it as a n origin of the strange phenomena of numerical solutions of (3.1). S. Ushiki [29] gave a rigorous proof for the existence of Smale’s horse-shoe in Fh for the case of f(x) =x( 1 -x). According to this proof, Fh has horse-shoes for any non-zero time step h. The proof is based on a transcendental property of complex analytic functions. This method can also be applied to some holomorphic dynamical systems. The outline of the proof is as follows. Suppose f(x)=x(l-x). Then ordinary differential equation (3.2) has two equilibrium points. The point x=O is a n asymptotically unstable equilibrium point and x = 1 is a n asymptotically stable equilibrium point. Corresponding to these two equilibrium points, there are two fixed points, P=(O, 0) and Q= (1, 1). Both of these are saddle points. Stable nanifolds and unstable manifolds associated with these saddle points are denoted as W ( P ) ,Wu(P),W ( Q ) ,and W’(Q). They are depicted in Figure 3.1. Since we consider only the case f ( x ) = x ( l - x ) , so that f(1-x)=f(x), we see that Ws(P)and W u ( Q )are symmetric with respect to the line L = { ( u , v ) l u + v = l } c R 2 . Similarly, Wu(P)and

230

S. USHIKI

S

P

Fig. 3.1.

W”(Q) are symmetric with respect to L . By an elementary argument, we obtain that W ( P ) n L f Q ) and W u ( P ) nL # @ . Using the symmetry of invariant manifolds, these imply that W ( P )n W s ( Q ) #0 and W’(Q)n W ( P ) f 0 . As non-empty intersection does not imply a transversal intersection, we must study more precisely how they intersect each other. In general, unstable manifolds and stable manifolds can intersect with tangency. Especially, if we treat the system as a family of dynamical systems parametrized by h, then such “degenerate” situation is inevitable. In fact, non-transversal intersention of invariant manifolds can occur and it can be observed numerically. Even though the transversal intersection is not always expected, “transversal intersection” in the topological sense can be proven. And “topologically traversing” intersection of unstable and stable manifolds of a saddle point can give rise to a “horse-shoe’’ dynamical system in the topological sense, which was studied by K . Yano [37]. Finally, to prove the topological transversality of the intersection of invariant manifolds, we must appeal to the analytic property of invariant curves. In order to prove this fact, a detailed analysis of invariant manifolds is necessary. It is found in S. Ushiki [29], which will be reviewed in the next section.

S 4.

Saddle Connection Curves for 2-Dimensional Analytic Dynamical Systems

As we have considered dynamical systems on the plane, the dimension of the unstable manifold and the stable manifold associated with a saddle point is always one. These invariant curves are always smoothly embedded curves if the dynamical system is smooth. They are analytic if the system is analytic. Note that F is not supposed to be diffeomorphic. In this case the invariant “curves” W s and Wu are not necessarily manifolds any more. Let F : R2-+R2be a real analytic mapping. Assume that the origin is a fixed point, i.e., F ( O ) = O . Let a and p be the two eigenvalues of a differential map of F at the origin. Assume that 0 is a saddle point, i.e.,


23 1

The saddle point is possibly critical when P=O. In this section, we treat only the analyticity of unstable “manifold”. When F is a diffeomorphism, then by considering the inverse map F-I: R2+R2 in place of F , a n analytic formula for stable “manifold” can be obtained. If F is not invertible, then unstable “manifold” will be the image of a n analytic map and the stable “manifold” (stable set) will be a complicated entangled set with many branching points. An analytic parametrization of unstable “manifold” is given as follows. By applying a linear change of coordinates if necessary, we can assume that the Jacobian matrix DF at the origin is diagonal. The mapping F : R2+R2 can be expanded into Taylor series around the origin as,

(4.2) A mapping $: R+R2 which parametrizes the unstable “manifold” W” of the origin is constructed by setting its Taylor coefficients as described in the following. Supposing $(E)=( f ( E ) , g(E))to be analytic in a neighborhood of the origin, expand the components f ( E ) and g(E) into the power series:

Observe that if $: R-R2 parametrizes the unstable “manifold”, it could satisfy the following function equation

(4.4)

F $(El =$((YE) , 0

which was called the fundamental equation by H . PoincarC [21]. Moreover, the linear part of $ should be tangent to the unstable local manifold at the saddle point. Hence, we can assume f l = l and gl=O. Now compute both sides of (4.4) as a formal power series and compare the coefficients with respect to p. The formal power series to be computed are given by

and

All the coefficients f n and g, can be determined inductively, starting from

S . USHIKI

232

f,=1 and gl=O.

For non-negative integers p and q with p f q 2 2 , let

where the summation is done over all combinations of positive integers i,, ..., i,,jl, . . - , j , with i,+ +i,+j,+ ..- +jq=n. Since p + q > 2 , all the indexes i,, . ., i,, j , , . , j q are smaller than n. gn-lare known, then @:.q can be determined Hence, if f,,..-,f n - , , g,, by (4.8). Coefficients f , and g , are given by

..-

-

--

and (4.10)

for n 2 2 . The formal power series computed by this procedure converges in a neighborhood of the origin. Theorem 4.1. If 101 > 1> then formal power- series f(E) and g(E) defined above converges in a neighborhood of the origin. The proof needs a delicate estimation for the coefficients using a majorant series. See S. Ushiki [28] for details. The theorem assures the analyticity of the mapping q5: R-R2 near the origin. We constructed a n analytic mapping 0: R-R2 in the neighborhood of the origin of R . It satisfies the fundamental equation (4.4). Theorem 4.2. If F : R2-R2 is analytic on the entire plane, R 2 , and if lal>l>[email protected] h e n $ : R - R 2 d e f i n e d a b o v e i s a n a l y t i c o n R . Proof. An analytic continuation of Q can be dfined by using the fundamental equation (4.4). Suppose that power series Q(E) is convergent for ]El < r . For any E € R , find a positive integer k such that < r . Define $ ( E ) by (4.11)

$(E) =Fh(Q(a-XE)) .

The value $(E) does not depend on the choice of Ic, since (4.4) holds in the neighborhood of the origin. In such a way, analytic function Q is defined on the entire space R .


233

Now, let us consider holomorphic dynamical systems. Suppose F : C2+C2 is a complex analytic mapping defined on C2. Assume the origin, O=(O, 0), is a fixed point a n d let a and /3 be the eigenvalues of a Jacobian matrix at the fixed point. By the same argument, we have the following theorem. Theorem 4.3. Zf la1 > 1> 1/31 then there exists a holomorphic parametrization $: C-C2 of unstable “manifold”, satisfying the following conditions: ( i ) q5(0)=0, ( i i ) Image ( d $ , ) = P , (iii) Image (+)= W , and (iv) F o $(O=+(at). The Taylor coefficients of q5 are computed as in the case of the real analytic case. This theorem, for the case of a rational map F : C2->C2,was proven by H . PoincarC [21]. Note that the obtained mapping F : C-,C2 is defined on the entire complex plane. It is a n entire mapping. See S. Ushiki [27] for the proof. Let us now make use of this fact to proved the non-existence of “saddleconnection curves”. In the preceding section, we proved the existence of intersection points of unstable manifold Wu(P)and stable manifold W*(Q). These invariant curves are analytic. If they intersect, the intersection takes place at discrete points locally, or two curves coincide entirely. Let h : Z-R2 be a n embedding (one-to-one non-singular differentiable mapping) of the unit interval into R2. Suppose F : R2->R2 has two saddle points, P and Q (P and Q may be identical). We call h a saddle-connection curve if ( i ) h(0)= P , h( 1)= Q, ( i i ) h(10, 1[) does not contain fixed point of F, (iii) h(Z)is invariant under F. Theorem 4.4. Zf real analytic diffeomorphism F : R2-bR2 can be extended to a complex analytic autornorphism F : C2-C2 of two dimensional complex Euclidean space, then there exists no saddle-connection curve. This theorem shows that in our dynamical system, W u ( P )and W s ( Q ) cannot contain a common arc in their intersection. For the proof of this theorem, see S. Ushiki [27].

S 5.

Analytic Formula for Invariant Manifolds

When we look at global bifurcation phenomena as the apparition of homoclinic points, saddle connection curves, or degenerate tangency of invariant manifolds, we need some numerical approach to study the behavior of invariant manifolds near the bifurcation point. In the case of analytic

S. USHIKI

234

dynamical systems on the plane, invariant manifolds associated to saddle points are calculated by the formula given in the preceding section. For higher dimensional dynamical systems, there exists analytic formula for unstable manifolds and stable manifolds associated to saddle points. Let f:Rn-R" be a real analytic map defined in a neighborhood of the origin. Suppose the origin, O=(O, - ,0 ) E R", is a fixed point of f, i.e., f ( O ) = O . We assume that the Jacobian matrix D F , at 0 is diagonal. Let al, -,a , be the eigenvalues of DF,. We assume

--

--

(5.1)

so that 0 is a saddle point. Let azL=(al, ax)and a , = ( a k C I , an). Let 6=(6,, -,6,) denote a multi-index with k components. All components of 6 are non-negative integers. We denote by 161 the length 6,+ +6,. If x = ( x I , * . -,x,) is a vector with m components and if p=(pl, p,) is a multiindex with the same number of components, we adopt the abbreviation - . a ,

--

...,

...

xP=~fl

...

X Pmm .

Using this convention, we assume

(5.2)

asfat

for any multi-index 6 with 161 2 2 and i = l , k. We call a point P € R* a n unstable point of 0 if there is a sequence of points P , € R " , i = O , - 1 , -2, - . - ,such that P,=f(P,-,) for i=O, - 1 , -2, ..-, P=P,, and that P, tends to the origin as i tends to -m. We denote the set of unstable points of 0 by W". Wu is called the unstable set of 0. I f f is a diffeomorphism, then W uis nothing but the unstable manifold of 0. Local unstable manifold W: is defined for small E > O as follows. Let BE denote the ball of radius E centered at 0. A point P E B , belongs to W: if there is a sequence of points P, Be, i=O, - 1 , -2, -,such that P,=f(P,-,) for i=O, - 1 , -2, P=P,, and P, tends to the origin as i tends to -a. Note that m a - ,

-

..-,

m

(5.3)

W"=

u f"W:)

k=n

.

Let E" denote the linear subspace spanned by the eigenvectors of DF, associated with eigenvectors a,, -,a,. Let 8 : E'-Eu be the linear map DF, restricted to the invariant subspace E".

--

Theorem 5.1. There exists a real analytic mapping q5: U-R" defined in a neighborhood of the origin of E". such that ( i 1 q5(0)=0, ( i i ) D@ is non-singular,


235

(iii) $( U ) = WE, (iv) f o q5=# 0. 0

Taylor coefficients of $ are given below. Note that iff: R"-Rn is defined globally on R", so that the iterated compositions f k = f o f are always defined, then using the fundamental equation (iv) in the theorem, q5 is defined on the entire space Eu and q5(Eu)= W" holds. Moreover, i f f is a n analytic diffeomorphism, then the same argument holds for W . When f is a n analytic mapping of a real analytic manifold into itself, a similar theorem holds. So it is true in the case of complex analytic mapping. In order to give the explicit formula for Taylor coefficients of q5, we x , ) € R n , and d=(dl, d,) be a introduce several notations. Let x = ( x l , multi-index with n components. Let f ( x ) = ( f l ( x ) , . , f , , ( x ) )and the Taylor expansion as 0

.-.

. ..,

. . a ,

where the summation is done over all multi-indexes d with length [dl 2 2 . For the sake of consistency of notation, let d(j)=(O,

. . a ,

0, 1,0,

. . a ,

0)

3

denote the multi-index with n components of length 1, which has only one positive component. Similarly, let a(j)=(O,

. - - , O ,l,O, ..-,O) 3

denote the multi-index with k components of length 1. Let f i , d ( j ) = aifY ti=j and f,,,cn =O if ifj. We have D F o = ( f i , d ( j ) ) .Using this notation, (5.4) can be rewritten as fi(x)=

(5.5)

C

ft,dxd

for

i = l , ..-,n.

Id121

..

Let (=(El, -,5,) denote a vector in E u = R k . Suppose q5: E'-Rn be a real analytic map with q5(0)=0. Let $(f)=(Q,(E), -..,q5,(()) and their Taylor expansion be

q5Yt(E)=

(5.6)

c $,.SP,

i=l, ...,n.

161>1

We define the Taylor coefficients $ i , S as in the following. For indexes k , of length 1, and for i= 1, -,n, set

S ( j ) , j = 1,

(5.7)

--

a ,

-

S . USHIKI

236

so that the Jacobian matrix, D$o, at the origin is the inclusion map E u c R " . Hence it is of maximal rank. For the sake of consistency, we set $ L , c o , . . . , o , =O. Next, let us formally calculate the coefficients of products of (5.6). Let d= ( d l , * -,d,) be a multi-index. Consider d,-th power of $((E). Just set ( e = d , for simplicity of printing)

-

c $t,sS")"= c $:,sEs

(5.8)

16121

161>e

Then the Taylor coefficients $:,s are given by

$h=c$d* *

(5.9)

$z.P

Y

where the summation is done over all the combinations of multi-indexes r', - . . , y e such that yl+ +ye==6. For e=O, we If e=l then $:,a=$t,s. set

--.

(5.10)

$P,co,

. o ) = l,

if

$P,s=O

161 21

.

Observe that if e 2 2 , then all f , - - . ) r ewhich appear in (5.9) have lengths strictly smaller than 161, since they must have positive lengths. Next, to compute the product

we set the Taylor coefficients (5.11) as

(5.12)

(Q(E)Id=

c $;Es

.

Then, for Id1 2 1, we have

(5.13)

$:=

c

* *

-

$:yrn

,

where the summation is done over all the combinations of multi-indexes - ,y" with k components, such that r'+ + y n = 6 . When Idl=l then

f ,

---

-.

(5.14)

'$~'"'$z.d

The Taylor coefficients of f,($(E))

.

are computed by setting

as (5.16)

ft,s=cft,d$:

9

where the summation is done over all multi-index d satisfying 1612ldl. (5.16) can be rewritten as


237

(5.17)

Now put all these into the fundamental equation (5.18)

fo$=$oB.

Then we obtain the equation of formal power series (5.19)

That is to say (5.20)

for 161 2 2 . For 161 =1, the equalities of coefficients hold automatically. As we noted before, (5.20) gives all values of $i,8 since the right hand side contains only $ t , r ’ ~ with lyI < 161, and the divisors do not vanish by our assumptions (5.1) and (5.2). Theorem 5.2. (5.20).

The function $: E”-R” in Theorem 5.1 isgiven by ( 5 . 6 ) , (5.7),

The proof of the convergence of this formal power series is found in S. Ushiki [28]. 6.

Ghost Dynamics in Numerical Studies near the Hopf Bifurcation Point

F. Brezzi, S. Ushiki and H . Fujii [3] studied the relevance of numerical methods in view of the bifurcation theory. Much work had been done for numerical methods used in the numerical study of stationary states. In their study, they regarded the relevance of Euler’s scheme for a numerical method employed in the Hopf bifurcation problem. When we treat a n ordinary differential equation or a partial differential equation which has a Hopf bifurcation, some “dynamic” aspects intervenes in the bifurcation, since a generation of oscillatory solutions is concerned. In their work, they concluded that Euler’s finite difference scheme can reproduce sufficiently well the phenomenon of Hopf bifurcation in a neighborhood of the bifurcation point. At the same time, they showed that the relevance of the numerical approximation holds within a certain limited extent. Global bifurcation behavior can be quite different from the original differential equation even though the time step is taken very small. And if the time step is not “sufficiently” small, then the numerical dynamics produce a strange behavior similar to the so-called “chaos”. We consider a family of ordinary differential equations

238

(6.1)

S . USHIKI

du/dt=f ( u , s ) ,

u € R"

, s€ R .

Suppose (6.1) contains a Hopf singularity, say at (uo,so)=(O, 0). Let us consider Euler's finite difference scheme

As a classical result, it is well known that if we fix the time step r > O , then (6.5) possesses a n s-family of "invariant circles" whenever F, satisfies the Hopf condition in the mapping sense. It seems quite natural to expect that, as r tends to zero, there exist invariant circles, uniformly in r , and this r-family of circles converges to the limit cycle of the original system. The uniform convergence of this family of invariant circles to a family of limit cycles is not guaranteed by the classical Hopf bifurcation theory for discrete mappings. It is because the eigenvalue of discrete dynamical system (6.3) approaches 1 as r tends to zero. The Hopf bifurcation point, say (u,,s,) depends on the time step r. Hence a delicate analysis and a more detailed error estimate are necessary to bring a uniform convergence of invariant circles. Proposition 6.1. There exist positive constants ro, so and c,, and a smooth function 6=6(r), r € N,'=]O, r o ] ,such that for each r E N : , F, has a Hops bifurcation point ( 0 , 6 ( r ) )€ R" x R in the sense of mappings. Here, 6 ( r ) satisfies that 16(r)l O , k-m

(iii) an uncountable subset Soof S such that for every x , y € So, lim inf IIFL(x)-Fk(y)jl=O . k-m

The proof is found in S. Marotto [17]. A generalization of this theorem in the case of a “hyperbolic snap-back point” was obtained by K . Shiraiwa and M. Kurata [23]. M. Hata [9] studied the existence of a scrambled set in expansive maps on Rn.

24 1


Part 11. Normal Forms Theory and Strange Attractors

S 8. Formal Normal Forms for Degenerate Singular Points of Vector Fields Motivated by the existence of “strange” attractors in nonlinear systems of ordinary differential equations, the author a n d his coworkers were interested in the “normal forms” of degenerate singular points of systems of ordinary differential equations. Usually, the interest of mathematicians is directed toward the normal forms of simplest singularities (least degenerate singularities). If we want to understand global bifurcation of strange attractors, however, we must look at a very degenerate system, since a degenerate system has quite a complicated structure and can give rise to strange attractors in its unfoldings. Let us begin with the definition of jets of vector fields. Let &?denote the Lie algebra of the (germs of) smooth vector fields defined in a neighborhood of the origin, 0, of R*. For a positive integer, k, let -.’/x denote the subset of Z consisting of those (germs of) vector fields whose Taylor coefficients at 0 of orders strictly smaller than k, vanish. Subspaces -Hk, k = l , 2 , are Lie ideals of &Y.We have the following sequence of linear subspaces: . . a ,

Define quotient Lie albebras, H A ,for k = l , 2 , ..., by H k = = / - / . f l k + l . If we fix a system of coordinates around the origin, then there is a one to one correspondence between H k and the polynomial vector fields of degree k which vanish at 0. The correspondence is given by the Taylor expansion and the truncation at degree k. Two elements, say X and Y , of 2’are said to have the same k-jet at 0 if their Taylor coefficients at the origin of orders smaller than or equal to k coincide. The Lie algebra H k can be regarded as the space of k-jets at the origin. Define linear spaces, H k , for k = 1, 2 , . -,by Hk=A‘,/J//X+,. Note that the linear space, H,, is isomorphic to the linear space of homogeneous vector fields of degree k. This isomorphism depends on the choice of the system of coordinates. We use the standard coordinates of R n in the following. We denote the canonical projections as j k : P + H k and j k , p : Hp+Hk, for p > k . Since we fix the standard coordinates and identify the elements of H k with polynomial vector fields of degree k, we have natural inclusions H-Hk and H,+Hk for 1I i l k. Lie algebra Hk has a decomposition as a linear space:

Decomposition H k = H k - ’ @ H, is often used. For X E 2 , X k will denote the k-jet in Hk represented by X , i.e., X k = j k ( X ) . We denote the Hk-component of X k by X,. X , is the degree k homogeneous part of the vector field, X . Similarly as decomposition (8.2), X k can be decomposed as

S. USHIKI

242

(8.3)

Xk=X,+...+X,

and

Xk=Xk-l+Xk.

In order to compute the normal forms of singular points of a system of ordinary differential equations, one must look at how a coordinate change works on the vector fields. When we compute the effect of infinitesimal changes of coordinates, we must use the Lie algebra calculus. Let us introduce some notations for "graded Lie algebra". For X and Y in Z, [ X . Y j = X Y - YX denotes the Lie bracket and [ X , Y ] , denotes the k-jet of [ X , Y ] . After our systematic use of superfix and suffix, [ X , Y ] , denotes the H,component of [ X , Y ] , . If X , € H , and X , € H , then [ X , , X,] E H k + , - , . Lie bracket [ X , , X,] defines a bilinear map H k xH,+Hk+,-,. If Xx€ H k and Yk€ H k , then [ X k , YkIkis a well defined element in FP. It does not depend on the choice of the representatives for X k and Y k . The adjoint operator, adk(Xk):Hk-.Hk, of Xk€ H k is defined by adk(XA)(Y k )= [ X k , YkIk

(8.4)

for

Y kE H k ,

Its i-th component with respect to the decomposition (8.2) is denoted as ad:( X k ): Hk+H,

(8.5)

(1Ii I k )

and is defined by

(8.6)

adl(Xk)(Y k ) = [ X k Y , k ] t = [ X k Y, k ] % for

- + X , and Y * = Y , + - - - + Y,, then adk(Xk)(Y k ) = [ X , + .- - + X k , Y,+ - - - + Y,jk

In other words, if X x = X , + . (8.7)

Y k EH x .

a

=[XI, Y , l + [ X , , Y , I + . * * + [ X , ,Ykl + [ X , , Y , l + . . . + [ X , , Yk-I1

+...

+IX,, Yll and (8.8)

a d : ( X k ) ( Y k ) = [ XY, ,J + [ X 2 ,Y , - , l + - - . + [ X , , Yll

.

Now let us look at how a coordinate change transforms a vector field. Let X 6 2" be a vector field defined on a neighborhood of the origin, O€R", and let q5 be a local diffeomorphism around 0. The ordinary differential equation defined by X : (8.9)

dx/dt= X ( X ),

x

R"

is transformed by the coordinate change, y = $ ( x ) into (8.10)

d y / d t = ( d ~ / d x ) . X ( ~ - ' .( v ) )


243

Hence the coordinate change y = $ ( x ) transforms X into (d$/dx).Xo $-I, where Jacobian matrix d$/dx is evaluated at $ - ' ( y ) . The transformation formula for jets of a vector field and diffeomorphism is given as follows. Let Sk denote the group of k-jets of local diffeomorphisms a t 0 € R". Dkis a Lie group. Let Xk€ H k and $k € 9fh. Although k-jet does not define a vector field nor diffeomorphism, the k-jet of transformed vector field (8.11)

-

(d$'/dx)

Xk

0

(p) -1

does not depend on the choice of representatives for X k and p . We denote the k-jet of (8.11) by Adk(qV)(Xk). We say two k-jets Xk,Y k€ Hkare k-equivalent as truncated vector fields if there exists a k-jet of diffeomorphism, @k, such that (8.12)

Yh=Adk(qP)(Xk).

In other words, two vector fields X and Yare k-equivalent as truncated vector fields if there exists a coordinate change such that the transformed vector field (8.10) and Y has the same k-th order Taylor expansion at 0. The truncated k-equivalence is a n equivalence relation. Let - T k ( X k denote ) the set of all kjets YkEHkwhich are k-equivalent to X k as truncated vector fields. Since -Qk is a Lie group and its operation (8.11) on H kis a Lie group action, the set Sk( X k ) is a manifold.

Definition. The k-th order normal form of vector field singularity on R n is the set of representatives of the orbit space Hk/GJh. The choice of representatives for each equivalence class is not unique. The choice must be strategically done in view of the use of normal forms in various problems. In order to select the representatives, we must compute explicitly the orbits S h ( X k ) . This computation can be done using the calculus of the Lie algebra Hkof the Lie group S k . Infinitesimal transformation of coordinate change and the infinitesimal deformation of vector fields can be calculated by looking at the tangent spaces of the orbits 9 k ( X k )which is obtained by the action of Lie algebra. Detailed computation is found in S. Ushiki [30]. The normal forms of first order are given by the theory of Jordan's normal forms of matrices, since H 1 is isomorphic to the linear space of n x n matrices, S ' = G L ( n , R), and Adl($l)(X1)=$l.X1.($l)-l. Hence higher order normal forms, which we are going to compute, can be considered as the generalization of Jordan's normal forms. If none of the eigenvalues of the linear part of a vector field has a zero real part, then the vector field in a neighborhood of the origin is topologically conjugate to its linearized mapping. Hence, we consider only the cases where the linear part has multiple zero eigenvalues or purely imaginary eigenvalues.

S . USHIKI

244

In the following statements of normal forms, we adopt notations for vector fields used in geometric theories. For the sake of simplicity, we employ notations as a, in place of atax, 8, for a/ay, etc. By identifying these differential operators with a basis of the tangent space, the system of ordinary differential equations, dx,/dt=f,(x,, * * * , x J , i=17 . - . , n ,

(8.13)

can be rewritten as a vector field in the form: (8.14)

X=f,(x,,

*

-

*

,X n ) a s l + * * - +fn(xl,* * * x,1azn 7

The k-jet, Xk,of (8.14) is represented by a polynomial vector field of degree k obtained by truncating its Taylor expansion at the origin. Theorem 8.1. Assume vector field, X , on R2 (or C z )has ya, as 1-jet at the origin. Then X can be transformed by a change of coordinates into a vector field whose 4-jet is one of the following form: (a) ya, + ( i - x 2 f u x y wx3+qx3~)a,, x y v,x3 v,x2y+ qx3y+sx4)a,, (b) ya, (i or (c) ya, ( w,x3 w,x2y+ qx3y+sx4)a,, with w: w;= 1. Parameters u, v,, v,, w, q, s, w,, w, are uniquely determined fr om the 4-jet at the origin of the given vector field, X .

+ +

+ +

+

+

+

If we use the terminologies and notations defined above, this theorem can be re-stated as follows. If X,=yd, then X4 is 4-equivalent to (a), (b), or (c) above in the truncated sense. In other words, (a), (b), and (c) are normal forms of fourth order for vector fields with X,=ya,. In the above theorem, cases (b) and (c) are special cases. Almost all vector fields have their normal forms belonging to (a). I n the following theorems, we use the term “generically” to mean that the transformation can be executed for all vector fields satisfying the assumption in the theorem, except those vector fields which satisfy a n additional algebraic condition on Taylor coefficients. Let X be a vector field around the origin, O E R 3 (or C3). Theorem 8.2.

+

If X I = -ya,+xa,,

+

then X 6 is generically 5-equivalent to

+

+

+

+

( 1 bz+ ez2+gr4)i3, (azr dzZr frK)i3, ( ir 2 k zz czs)i3, , where a,= -yd,+xa,, ra,=xa,+ya,, determined f r o m the 5-jet X5.

andparameters a , 6 , c , d , e ,f , g are uniquely


If X,=yd, then its third order normal f o r m is generically

Theorem 8.3. yd,

245

+( t x 2+axy-t z2+cyz+ e x s+hxyz+ iyz2)d,+(bxz+dz2+fx3+gz3)d, ,

where parameters a , 6 , c , d , e , f,g , h , i are uniquely determined f r o m X s .

Theorem 8.4. yd,

If X,=yd,+zd,

then the third order normal f o r m isgenerically

+zag+ ( -tx2+axy+ bxz+cx2y+dxz2 +ex3)&,

where paramters a , b, c, d , e are uniquely determined f r o m X 3 .

We refer to S. Ushiki [30] for the proof.

S 9. Truncated Versal Family Suppose X k € Hk is a n element in normal form of degree k. A family of k-jets of singular vector fields, A : Rm-Hk, with A (0) = X k , is called a truncated deformation of X k . Definition. A truncated deformation A : RT'-Hk, A ( 0 ) = X k , is a k-th order versal f a m i l y if, for any deformation A : Rm-Hk, there exists a continuous mapping I : Rm-RT with 1(0)=0 such that A ( a ) and A(1(a))are k-equivalent as truncated vector fields for all a in a neighborhood of the origin 0 c Rm. Since k-equivalence relation in H X is defined by a Lie group action Adk: Hk-Hk, versal families can be obtained as transversal sections to the orbit d k ( X k ) c H k . Especially, families of normal forms given in the preceding section give transversal sections of the orbit 9 * ( X k )restricted to affine sub@ H k . Hence, by adding a versal family of its linear part space X,+Hz @ X I , in the linear space H,, a versal family of k-th order is obtained. Versal family for linear part X , is given by V. I. Arnold [l]. As a n example, let us consider a 4-jet X4, at OER2, in normal form: 9

k

x

--.

(9.1)

X4=Ydz+(X2+U0xY

+w,x3+qox~Y)d,.

The versal deformation of the linear part X,=yd, is a two parameter family (9.2)

X1+(P,x+PzY)d,

*

Hence five parameter family (9.3)

+

+ +UXY +wx3+9X3Y)a,

X4 ( P1x PZY

with parameter ( p l ,p z , u, w,q) E R6 is a fourth order truncated versal family. Versal families for other normal forms can be obtained similarly. Versal families of linear parts are given as follows (see V. I. Arnold [l]). (9.4)

X,+plrd,+p2as+p3zd,

for X I = &

at O c R 3 ,

S. USHIKI

246

(9.6)

X,+(P,X+P,Y+P,Z)~,

for X,=yi?,+za,

a t O E R8 .

When we consider vector fields which have some symmetry, similar arguments can be made and analogous computations respecting the symmetry give normal forms under the presence of symmetry. S. Ushiki, H . Oka and H . Kokubu [31] calculated normal forms of vector fields around 0 E RS with X,=yd, under the symmetry ( x , y , z)+(-x, - y , z). Its third order normal form is generically given by

(9.7)

X 3 = X 1 + ( ~ x z + a y z + c x ~ + d x ~ y + e y z ~ ) t ?i ~x 2++( b z z + f z 3 ) d z.

H . Oka and H. Kokubu [I91 gave the third order normal form for XI= yl,+za, with symmetry ( x , y , z)->(-x, - y , - z ) as (9.8)

X 3= X I +(_ ~ x ~ + ~ x ~ ~ + / ~ x ~ z + .~ x ~ z + ~ x z ~ ) ~ ~

Versal families for symmetry cases can also be given. families for the linear parts are given by

(9.9) (9.10)

~ 1 + ( P , x + P z ~ ) ~ , + P 3 z & for (9.7)

Symmetric versal

7

X l + ( ~ l x + ~ z ~ + ~ 3 z for ) & (9.8) .

Normal form theory for parametrized families of vector fields was developed by H . Kokubu [13]. An application to the bifurcation problem of some reaction-diffusion equations is studied there. Normal form theory for constrained differential equations are developed by H . Oka and H . Kokubu. See their article in this volume for details.

S 10.

Application of Versal Family to “Renormalization” of Strange Attractors

The study of normal forms and versal families of degenerate singularities was motivated by the existence of “strange attractors” such as the Lorenz attractor and Rossler’s attractors. The existence of strange attractors and their structures have global features. When we want to study them in a n analytic manner, we need to localize them in some sense. We supposed that the global aspect of strange attractors and their bifurcations can be treated by looking at a very degenerate singularity if it contains such strange behavior in its unfoldings or versal families. In fact, S. Ushiki, H. Oka and H. Kokubu [31] found that the total family of the Lorenz system [I51

(10.1)

+

a(y- x)a, ( - X Z Y X -y)a,

+( x y -bz)&

is equivalent to a subfamily of the versal family of (9.7). Just set X = d T x ,


241

Y = d T ( y - z ) , and Z=(1--b/20)(20z-x2), then (10.1) becomes (10.2)

Ya,+ ( AX+B Y f 0 X . Z - X 3 ) a ,+ ( C X + X z ) a , ,

where A = u ( r - 1 ) , B=-0-1, C=--6, a=2o--6. (10.2) is a subfamily of (9.7)+(9.9). Family (10.2) includes four parameters whereas (10.1) contains only three parameters. In other words, Lorenz family (10.1) is embedded into (10.2). In this large family, we can construct a “homotopy” which connects the Lorenz system to a n integrable system, by smoothly changing the coordinates and the time scale. The global property is not affected except the limit system. Change the time variable t into t’/p and set the “renormalization” change of coordinates as x = p X , y = p 2 Y , z=pZ. Then the “renormalized system” is

Similar results for Rossler’s family is found in S. Ushiki, H . Oka and H. Kokubu [31].

Part 111. Fractal Objects in Holomorphic Dynamical Systems

S 11.

Chaotic Dynamics on a Complex Domain

Modern research in holomorphic dynamical systems began with Mandelbrot’s experiments on Julia sets and the so-called Mandelbrot sets (see B. Mandelbrot [16]. A . Douady and J . H. Hubbard [ 4 ] , D . Sullivan [29] and M. Herman [12] revealed the strikingly rich and beautiful dynamical behavior of holomorphic dynamical systems. Since real analytic dynamical systems include holomorphic dynamics, a study of holoinorphic dynamical systems is necessary to understand, at least, the apparently mysterious property of real analytic dynamical systems. Let us begin with one dimensional holomorphic dynamical systems. Let c = C U {a} denote the Riemann sphere, and let f: be a complex analytic map. Holomorphic mapping is a rational map and can be written as a quotient of mutually prime polynomials, say f ( z ) = P ( z ) / Q ( z ) . We call d=max (deg (P), deg (Q)) the degree off. Holomorphic dynamical systems are found in numerical analysis, where iterations of complex analytic functions are concerned. As an example, consider a polynomial equation

c->C

(11.1) in complex variable z.

(11.2)

P(z)=O The Newton’s method applied to this equation:

S . USHIKI

248

C-tc

gives a holomorphic dynamical system f: with f(z)=z-P(z)/P’(z) a rational function. In terminologies defined below, a solution of (11.1) corresponds to a super attractive fixed point of (11.2). The problem is how the basin of attraction of these fixed points are composed, what their boundary is, etc. A point z G C is said to be normal if there exists a neighborhood, U , of z such that the sequence of iterated composition { f k l u } k = l , . .restricted ., to U is equicontinuous. The set of normal points (11.3)

F ( f ) = { zC ~ I z is normal for

is a n open set and is called the Fatou set o f f . (11.4)

f)

Its complement set

J ( F )=C\F(f)

is called the Julia set o f f . The Julia set is a closed set. The Fatou and Julia sets are both totally invariant under f, i.e.,

Here are some fundamental properties of the Julia set. Theorem 11.1. Z f d 2 2 then J ( f ) # @ . Theorem 11.2. J ( f ) is perfect and uncountable.

In most cases, the Julia set is a “fractal” set. Although J ( f ) is never empty, the Fatou set can be empty. A point p € c is said to be a fixed point if f ( p ) = p . The value I=f’(p) does not depend on the choice of coordinates and is called the eigenvalue of the fixed point. Fixed point p is classified by its eigenvalue as follows. It is super attractive if I=O, attractive if I R I < l , repulsive if 111 >1, and neutral if 1R1=1. When p is neutral, let R=ezrrie. Neutral fixed point p is rationally neutral if 0 is rational, irrationally neutral if 0 is irrational. A n irrationally neutral fixed point, p , is called diophantine if there exist positive constants C and E , such that (11.6)

IB-qlrl > C / r Z t f

holds for all integers q and r with r>O. For fixed point p , let A ( p ) denote the set of points z E C such that fn(z)--tp as n + a . A ( p ) is called the basin of attraction of p . If p is a n attractive fixed point, then A ( p ) is a n open set and p E A @ ) . I n this case A ( p ) is called a n attractive basin. If d 2 2 and p is rationally neutral, then p € a A ( p ) . In fact the “Flower Theorem” by Fatou and Julia shows that at least some “petal” region is included in A ( p ) .


249

Theorem 11.3. I f d 2 2 , P = l , and I m f l f o r l < m < n , then there are an integer k and nk real analytic curves which are pairwise tangent at p and which bound petals. The union of the petals is forward invariant, and any orbit in a petal is asymptotic to p . In this case, A ( p ) is called a parabolic basin. When f is topologically conjugate to a n irrational rotation in a neighborhood of the fixed point p , then there exists a n invariant disk around p . Such a disk is called the Siegel disk. If p is a n irrationally neutral fixed point and its rotation number 8 is diophantine, then p has a Siegel disk. An attractive basin, the interior of a parabolic basin and the interior of a Siegel disk are included in the Fatou set F ( f ) . If f “p)=p and f “ p ) f p for 1 Ii< k , then p is said to be a periodic point of period k . The eigenvalue of a periodic point is given by ;i=(fk)’(p). According to its eigenvalue, periodic points are classified similarly as the fixed points. Classification of the basin of attraction etc. are given similarly. Besides these components of F ( f ) , the Fatou set can have a (periodic) invariant annulus called the Herman ring, on which f works as a n irrational rotation. D. Sullivan gave a classification for connected components of F ( f ) . Each connected component of F(f) is called a stable region.

Theorem 11.4 (Sullivan). All stable regions are eventually periodic. Periodic stable regions are either attractive basins, parabolic basins, Siegel disks, or Herman rings. For the proof, see D. Sullivan [29]. For a n introductory survey of holomorphic dynamical systems, we refer to P. Blanchard [ 2 ] . M . Shishikura [24] developed a surgery theory of complex analytic dynamica1 systems and perturbation of systems. He proved the following theorems.

Theorem 11.5. For all integers p 2 1 , there exists a rational function of degree 3, which has Herman rings of order p . Theorem 11.6. For an irrational number 8 , the existence of a rational function with a Siegel disk of rotation number 8 implies the existence of a rational function with a Herman ring of the same rotation number, and vise versa. Theorem 11.7. A rational function of degree d has at most 2(d-1) cycles of stable regions, with cycles of Herman rings counted twice. Moreover, there exist at most ( d - 2 ) cycles of Herman rings. Conversely, f o r any prescribed combination of numbers of cycles for each type of stable region, satisfying the restrictions above, one can find a rational function of degree d , which just has the prescribed number of cycles. S. Ushiki, H . - 0 . Peitgen and F. v. Haeseler [32] studied the structurally

S. USHIKI

250

stable regions in the parameter space parametrizing a family of rational functions of degree 2.

s 12.

Julia Sets for Quadratic Maps and Rational Maps

Numerical experiments with computer graphic techniques are said to have inspired mathematicians working in holomorphic dynamical systems. Here, we show some pictures of Julia sets. As the simplest case, consider a family of quadratic maps

(12.1)

f(z)=z2+c.

The infinity z=oo is always a super attractive fixed point. We have the following cases. (a) J ( f ) is toally disconnected, (b) there is a (cycle of) super attractive basin, (c) there is a (cycle of) attractive basin, (d) there is a (cycle of) parabolic basin, (e) there is a (cycle of) Siegel disk, (f) none of the above. The coloration principle of the pictures is as follows. Take a small disk around the (super) attractive (periodic) point. For each point z € C , compute f " ( z ) , n = 1, 2, Select a color for the pixel corresponding to z as a function of the smallest n such that f " ( z ) falls in the small disk. Pictures 1 and 2 represent case (a). If c=O, then (12.1) has a super attractive fixed point at z=O. Pictures 3 and 4 represent cycles of super attractive basins of period 2 and 3, respectively. Picture 5 is for case (c). Pictures 6, 7 and 8 are for case (d) with A = l , - 1 , and i respectively. A quadratic map with a Siegel disk is represented by picture 9. Values of parameter c are as follows.

-

- a .

picture

C

1

1

2

-0.75+0.23

3

-1

4

- 1.22561 +0.7448621'

5

-0.15

6

0.25

7

-0.75

8

0.25+0.5i

9

0.37418+0.1934111'


25 1

As the second example, consider the two parameter family of rational functions of degree two: f(z) =z(z+l)/(l+ p z )

(12.2)

.

Picture 10 represents the case with two attractive fixed points. Picture 11 represents the case with one attractive basin and a Siegel disk. In picture 12, f(z) has a Siegel disk and a parabolic basin. Picture 13 represents the case where there are two distinct parabolic basins. In picture 14, there is only one cycle of attractive basin. Both of the critical points are attracted to a n attractive cycle of period three. Parameters for these pictures are as follows. picture 10

R

P

-0.5+0.71'

,u=R

11 12

eo.sr

13

ezni/s

i

14

-0.692683+ 1.1997621'

p=R

el

O.5etIb errt/z

-

Rational mappings of a degree smaller than 3 do not have Herman rings. Picture 15 is a n example of a Herman ring discovered by M. Herman [12] for (12.3)

f(z)=(et/z)(z-a)z/(l-2)2

with a=0.25, t = 1.9. The green annulus represents the Herman ring and the other annuli are its preimages. Picture 16 is a cycle of Herman rings of period 2 found by M. Shishikura [24]. His mapping is given by (12.4)

f(Z ) = z'( Z- b)/(z- C) +a

for a =0.864375+0.2103381', b= 0.076868 -0.2503721' and c= - 0.080948 0.249204i. The brown annulus appears to be a cycle of Herman rings. Other rings are their preimages.

S 13.

Mandelbrot Sets for Rational Functions

B. Mandelbrot [ 161studied the bifurcation diagram of holomorphic dynamical systems. He executed numerical experiments for the family of quadratic maps (12.1), and found surprisingly beautiful and complicated objects. The Mandelbrot set M c C is defined by

M = {c f?C I {f"(O)},=,,,, ... is bounded}

.

S. USHIKI

252

Pictures of the Mandelbrot set are found in B. Mandelbrot [16]. Many beautiful full color pictures are found in H. 0. Peitgen and P. Richter [20]. Here are just a few of pictures of the Mandelbrot set. Picture 17 is the global shape of the Mandelbrot set. Pictures 18 to 20 are its enlargements. The regions of parameter c represented in these pictures are as follows. picture 17

region -2.11Re (c)O. Proof.

It suffices to show the non-differentiability at x=po.

We note that

I p 2 1, since P-n-1-Po

P-n-Po

P-n-1-Po

-

~

W(P-,-l)--PO

1

as n - w .

W'(P0)

Suppose, on the contrary, that there exists a finite derivative A=(i?/i?x)F(a,p o ) . Then, the equality F(a, p-,-,)-aF(a, p-,)=P-,-, implies A=(l+aA)-I. On the other hand, we have A =F(a,p-zM-l)-FF(a,po) M-

=l+a

P - 2 M - I -PO P-2M-pO

+

...

+a2X

P-ZM-I-PO

First consider the case a1

since

>O .

~ - 2 M t 2 , - 2 - ~ O + ~ ~ ~ - 2 M + 2 j - 1 - ~ o ~

Hence A 2 1, contrary to the assumption A-'=l-Alal< l-!rllal 10. Next, consider the case a>O. Then we also have A M > l since a2p2P-2Mt2j-2-P0

PO - P - Z M Hence A 2 1 , contrary to A - ' = l + a I > l .

t 2 j-

1

This completes the proof.

0

In general, it is a difficult problem to study the differentiability of (2.3) at repulsive periodic points. It is a n open problem whether there exists a nowhere differentiable generating function for which the dynamical system w is not onto.

266

M. HATA

S 3.

Substitution Operator S, The Weierstrass function (1.1) for b=2 can also be represented in the form

c an cos

7l2O

(27TX) =

c a" cos (qP(x)).

7L20

Thus the Weierstrass function and the series (1.3) corresponding to b=2 are particular cases of the following series: (3.1) where F(0, x)=g(x) is a smooth function on Z. It is easily seen that the series (3.1) is a unique continuous solution of the functional equation

To deal with the series (3.1), it will be convenient to introduce a substitution operator. Let E be a complex Banach space of all complex-valued continuous functions on Z with uniform norm. For a given continuous dynamical system o:ZkZ, we will define the substitution operator S, by

(3.3)

S,(f)(x)=f(w(x))

for x € Z .

As is easily shown, S, is a bounded linear operator of E and its spectrum o(S,) is contained in the unit disk. It is known that the substitution operator (3.3) is one of the Bourlet operators satisfying a multiplication formula (Targonski [411). Moreover it is a linear ring endomorphism of our Banach algebra. Note that the eigenvalue ) problem for the substitution operator leads to the Schroder equation f ( w ( x ) = Af(x). Using the operator S,, the series (3.1) can be written as

where the operator (Id-uS,)-' is known as the resolvent operator of S,. Therefore (Id-aS&' maps g,(x)=cos zx to the Weierstrass function and gl(x)=x to the series (1.3) for b=2; that is, it maps some snooth functions to nowhere differentiable functions. If the operator S, is completely continuous, then a family on functions { w , 02, ...} must be a compact subset of E . In this respect, we have the following :

Theorem 3.1 ([44]). Suppose that there exists a sequence { p - , } , , , such that w(p,,)=p,fp-, and o(p-,,)=p-,+,for n > l . Then we have u(S,)={z; lzl 0. Then the cone map S, has the desired properties. Note that its partial spectral radius is given by C such that Sau=nl(0)u. d ( 0 )and there exists a n eigenvector More generally, we will consider the operator

(3.4)

T,(g)(x)=

c amSWn(g)= c a,g(w"(x))

9

b>O

Tl>O

where C a,, is a n absolutely convergent series. Plainly T , is a bounded linear operator with llToll < Cnzo lanl. By the well known representation theorem, there exists a function T ( X , y ) , defined on ZxZ, satisfying T,(g)(y)=

1:

g(x)dT(x,Y )

where ~ ( xy ,) is of bounded variation with respect to x for each y and is continuous with respect to y as x = l . Actually, we can obtain the concrete expression for ~ ( xy ), as follows:

On the operator (3.4), we have the following:

Theorem 3.2. Suppose that a power series C n t O a,,zn has a radius of convergence > 1 and has no roots in the unit disk. Then the operator T , is a homeomorphism of E. Proof. It is clear that the series C,,,b,z"=(C,,,~,z")-~has a radius of convergence > 1 and therefore C b, is absolutely convergent. Then, for any f E E , define

Hence,

This implies T,(E) =E.

Next we assume Cnro a,S;(g)=O.

Then

This implies that T, is one to one. Thus, T , is a homeomorphism of E.

0

M. HATA

268

c,"==o

Corollary 3.3. Suppose that a polynomial c,zn has no roots in the unit c,S; is a homeomorphism of E. disk. Then the operator C,"=o Note that the conclusion of the above corollary is equivalent to the fact that the linear functional equation

+ - +C,S(W(X))+Codx)= f ( x )

CNS(W"(X))

* *

has a unique solution g € E for any f € E. It is also interesting to consider the higher dimensional substitution operator in the form

swl,...,"n(f)(xl,*

e . 7

Xn)=f(W1(x1)+

---

+W,(X,))

,

which maps E into the space E , of all continuous functions defined on the n-dimensional unit cube. In this respect, there is a remarkable result:

Theorem 3.4 (Kolmogorov [22]). There exists a family of continuous monotone increasing functions w P g ,defined on I , l < p < n , l < q < 2 n + l , such that the substitution opeartor S* on E Z n + l defined by

S*(fl,

-

2nt 1 *,f2,+1)=

c s,l,.....on, (f,,

q= 1

is onto; that is, S*(EZnf1) =En. This is known as the representation theorem of continuous functions of n variables by superposition of continuous functions of one variable and addition.

S 4.

Difference equations

In this section, we will discuss various properties of the function in the form

Obviously the Takagi function (1.2) and the series (2.2) are particular cases of (4.1). First of all, Hata and Yamaguti [14] proved the following theorem using particular orbits of the dynamical system $.

Theorem 4.1. Suppose that the series (4.1)converges everywhere. Then the series C c, is absolutely convergent. Moreover, they showed that the operator L defined by

Fractals in Mathematics

269

is a linear homeomorphism from the space of absolutely convergent series onto its image. They also generalized this result to the series (3.4) for o=$. Faber [7]showed that the series (4.1) has no finite derivative at any point if lim sup,,, 2"lc,l >O. This result was accomplished by K6no as follows: Theorem 4.2 (KGno [23]). The series (4.1)has nofinite derivative at any point Moreover, if lim sup,,, 2nc,=0, it is difer-

if and only i f lim sup,,, 2"lc,l > O . entiable on a set of continuum.

He also studied further properties on the series (4.1). In particular, he showed that the family { p ( x ) - 1/2}n20is a concrete example of a multiplicative system but not strongly multiplicative. In [14],we showed that a continuously twice-differentiable function in the form (4.1) must be a quadratic function. This result was also strengthened by KBno so that it holds true even in the class of smooth functions in the sense of Zygmund. Although there are n o simple functional equations the series (4.1) must fulfill in general, we can obtain a family of difference equations whose unique continuous solution is the series (4.1). It is convenient to denote the set of lattice points {(n, rn); Oess supa(x). Integration of (5.6) shows jlaull=O, which with (5.5) shows Au=O and so u=O. Thus -A+a is injective so that it is not surjective, which completes the proof. Lemma 5.2.

For any z € R n and R > 0, we have

Z(z, R ) = \

k,(x-y)dya(x)(1- e - c u ) / C

.

Replace @(x,u ) in (E)' with the right hand side of (6.5) and then replace Cu and Cf with u and f, respectively, to obtain (Eo). Hence, by Lemma 2.4, as for the existence of the solution, it suffices to consider (Eo).

Examples. Let a ( x ) and b ( x ) be positive bounded functions. ( i ) @ ( x ,u ) = a ( x ) ( l - e e - b ( z ) usatisfies ) the conditions (6.1)-(6.4) with C= sup b ( x ) . ( i i ) @ ( x ,u)=a(x)u/(l+b(x)u)satisfies (6.1)-(6.4)with C = 2 .

Semilinear Elliptic Equation in L'(R")

367

References [ 1 ] P. BCnilan, H. Brezis and M. G. Crandall, A semilinear equation in LL(RN), Ann. Scoula Norm. Sup. Pisa, 2 (1975), 523-555. [ 2 ] T. Gallouet and J.-M. Morel, Une equation semi-IinCaire elliptique dans L1(RN), C.R. Acad. Sci. Paris, 296 (1983), 493-496. [ 3 ] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. [ 4 ] M. Mimura and A. Nakaoka, On some degenerate diffusion system related with a certain reaction system, J. Math. Kyoto Univ., 12 (1972), 95-121.

Masaharu ARAI Faculty of Economics Ritumeikan University Kyoto 603, Japan Akira NAKAOKA Department of Mathematics Kyoto Institute of Technology Kyoto 606, Japan


Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 369-383 (1986)

On the Vlasov-Poisson Limit of the Vlasov-Maxwell Equation By Kiyoshi ASANOand Seiji UKAI Abstract. The Vlasov-Maxwell equation is shown to converge to the Vlasov-Poisson equation at the limit of the infinite light velocity. The initial layer and the asymptotic expansion of solutions as the light velocity tends to infinity are discussed in detail. Key words: Vlasov-Maxwell equation, Vlasov-Poisson equation, VlasovPoisson limit, initial layer, asymptotic expansion

1. Main Results

The change of the density distribution of charged gas particles is described by two types of equations: The Vlasov-Maxwell equation and the VlasovPoisson equation. The latter describes the motion of plasma when the magnetic field generated by the plasma is small. The purpose of this paper is to study the relation between the two equations. Roughly speaking, the solution of the Vlasov-Maxwell equation converges to the solution of the VlasovPoisson equation when the light velocity tends to infinity. Letf,=fi(f, x, u ) be the density distribution of charged gas particles of the type i ( i = l , 2 , -.-,N ) at time t 2 0 and position x E R S with velocity uERS. Let E = E ( t , x) and B = B ( t , x) denote the electric and magnetic fields generated by the charged gas particles. The Vlasov-Maxwell equation is,

(1.1)

supplemented by Received May 29, 1985. Revised July 20, 1985.

K. ASANOand S . UKAI

370

(1.2)

VZ-E=4nKf,

V,-B=O.

and x are the scalar and vector products in R3,V, is the gradient in v, while c is the light velocity and ai=qilm,, qi and mi being the electric charge and mass of a single particle of i-species. Further, Jf and K f are the current and charge densities generated by f=(fi, f 2 , fN); Here

x and V , in

..-,

(1.3)

Notice that (1.2) can be deduced from (1.1) for t > O if it is satisfied at t = O , i.e., if the initials f o = ( fi,o,f2,0, -,f,,,) and (Eo,B,) satisfy

--

(1.4)

V,.Eo=4nKfo,

VZ.Bo=O.

This is a physically reasonable compatibility condition and will be assumed throughout the paper. On the other hand, the Vlasov-Poisson equation is,

(1.5)

This equation is formally obtained from (1.1) and (1.2) by setting B=O or c = w . With the notations and function spaces defined by (1.19)-(1.23) below, our main results are stated in the following three theorems.

Theorem 1.1. (1.6)

Let 123, a 2 0 , p 2 0 , P E R , and suppose ft,o€fC,p,p

(1CiSN)

9

(Eo,Bo)CH'

.

Suppose further that (1.4) is fulfilled. Then, there exist positive constants C, T , r and the following holds. ( i ) (Uniform existence) For each C E [ l , cu), (1.1) has a unique classical solution f=(fl,f 2 , f ) and u=(E, B ) on the time interval [0,TI such that

-..,

Vlasov-Poisson Limit of the Vlasov-MaxwellEquation

IUI 1 , T I I uo I L + Clf o 1 1 . 0 , p. B.

(1.10)

371

*

( i i ) (Continuity in c ) A s functions of c, f and u satisfy 1

[A.llF,b

f~

nM w , m); c m ,TI; w , ~ - J )

j=O

Thus, the solutions exist on the interval [0, T ] independent of c € [ l , m) and are continuous in c. Further, they have limits at c=oo which we call the Vlasov-Poisson limits. More precisely, we show Theorem 1.2 (Existence of limit). Let f,u be as above. ( i ) They can be extended to [ 1, 001 as functions of c so that [~.31:,~ [A.4IL

f € B O ( [ l001; , CNO, TI; f C : ; , p - J )

7

v E >o

9

u € B O ( [ la, 1 x [O, Tl\I(w, 0)l; BL-2(R3)) ,

holds good, and in particular, they converge as c--rm in the topology indicated here. W e write the limits as f"=(fT,fy,

- - - , f F ),

u"=(E", B")

.

( i i ) Bm=O while (f",E") is a unique solution to the Vlasov-Poisson equation (1.5) on [0, TI satisfying (1.11)

(1.12)

(f"7

Ern)€Co([O,TI; f f b , p , p X A L t ' )

3

I f m l l . o l p , B I T ~ ~ l f o l L , o . ~ 7, B

IIE"llo,T+

lV.J"lz-i,T
0 but not uniformly on (0, TI. Physically, this implies the development of initial layer. Also, comparing (1.7) and ( l . l l ) , we see that the limit f" which is a solution to (1.5) belongs to a better function class thanf, the solution to ( l . l ) , and similarly for E" and E. Finally, we shall discuss a n asymptotic expansion which is somewhat complicated due to the presence of initial layer. To simplify the notation, we introduce the operators L , A , A defined by

(1.13)

L ( f , E ; f ',E') = (ad,+ v.V,f,+a,E'.V,f,+a,E.V,f,'),N_, Af=(-4~Jf, 0) , Au=(V,XB, -V,XE).

,

K. ASANOand S . UKAI

372

The first of these is a n N-dimensional vector while the remaining two are 6dimensional. We seek the expansion of the form k

f=f"+ (1.14)

cc-Y

j=O

,

k

u=u-+C c-'uj , j=O

where f", u" are as in Theorem 1.2. The coefficients f j = ( f : ' , f ; , -..,fi,), u j = ( E f ,Bj) still depend on c. We wish to determine them as solutions of the following equations whose derivation will be described in S 5 . Note that the 0-th term of the expansion (1.14) is ( f " + f o , u"+uO). The equation for (fo, uo) is the nonlinear equation,

L(f0,EO;f",E"+EO)=O , acUo-CAUo=PIAf "+ / I f o , f"c=O=o, u"c=o=P,uo ,

(1.15)O

where P I is the projection defined in S 3 (see the remark below (3.3)), and u, is the same initial as in (1.1). The equation for ( f j . u j ) , l < j < k - 1 , is the linear inhomogeneous equation,

j-1

+~1 x W

Fj = - (LY, C (I? * 0,fi -' 7=1

*

V&-')

+L Y ~ Ux Bo- 0,fi

-I):=

,

and the equation for (fk, uk)is the nonlinear one just obtained by substituting (1.14) into (1.1) and taking account of ( 1 . 5 ) and ( 1 . 1 5 ) ~ ,OCjLk-1:

where F k is a given function of (fj , uj), 0 1j < k- 1 and their derivatives, and V , c. The expansion (1.14) is verified by the Theorem 1.3 (Asymptotic expansion). and 1.2, let OO such that

If

(5.2)

nll,o,p,B,pT+

IUn12*TS

c

holds for all n. Then by Lemmas 2.1, 2.2, 3.2, it follows that (5.3)

f n € [A.l]f.an[A.3]f,B,

u n € [ A . 2 I i n[A.4I1,

n20

.

Using these, we repeat the argument of [ l ]to see that

f,-f un+u

in B O ( [ l00); , CNO, T I ; HZp-A) , in B O ( [ lm); , Co([O,T I ; H1-l)),

strongly as n+m, with some limit (f,u). By (5.2) a n d the interpolation theorem, this convergence is also true if 1-1, 19-1 are replaced by I - E , B-E, for any E > O . This and (5.3) then imply that for any 6>0,

un+u strongly in B O ( [ lm , ] x[a, T I ; B2-2(RS)). Now the first half of Theorem 1.2 follows since the limit (f,u ) obviously coincides with the solution of Theorem 1.1, and the latter half comes directly

382

K. ASANO and S. UKAI

from Lemmas 2.2, 3.2 and Theorem 4.1. The asymptotic expansion in Theorem 1.3 is obtained as follows. let f" be that of Theorem 1.2 and assume the expansion,

First,

k

f=f-+Xc-fjj.

(5.4)

3=0

Substitute this into (3.8) which holds for our (f,u) by going to the limit in (5.1). Then we have,

where

riO=ecLAuo+jt ec(L-s)AA( f "(s)+f O(s))ds , 0

e c ( t - s ) A A ~ ( s ) d s ,l < j i k

uj=\'

.

0

Using (3.41, we decompose ao further as l i o =u"+ uo

,

s: 1'

u"(t)=Pouo+

PoAfm(s)ds,

uO( t ) =ectAf',UO+

ec(c-s)A (P,/2frn(s)+/2f0(s))ds .

0

By Lemma 3.2 and since our f " is that of Theorem 1.2, u" defined above is just that of Theorem 1.2. Also, recalling that (3.8) is a unique solution to (3.1), we see that uj solves (formally) the Maxwell equation in (1.15)1, O i j < k . Substitute (5.4) and (5.5) into (1.1) to deduce the equation for f j in (1.15)j. Now the proof of Theorem 1.3 can be completed by the help of Lemmas 2.2, 3.2 and 4.1, and by proceeding as in the proof of Theorems 1.1, 1.2 and 4.1. The detail is omitted.

References [ 1]

K . Asano, On local solutions of the initial value problem for the Vlasov-Maxwell

equation, 1984, Preprint. On the incompressible limit of the compressible Euler equation, 1985, Preprint. [ 3 ] K. Asano and S. Ukai, On the fluid dynamical limit of the Boltzmann equation, Lecture Notes in Numer. Appl. Math., 6, Kinokuniya/North-Holland,1985,l-19. [ 4 ] C . Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Internal report no 101, Centre de Math. Appl. E.P.P., 1983. [ 2]

-,

Vlasov-Poisson Limit of the Vlasov-Maxwell Equation

383

[ 5 ] P. Degond, Local existence of solutions of the Vlasov-Maxwell equations and

convergence to the Vlasov-Poisson equations for infinite light velocity, 1984, Preprint. [ 61 S. Mizohata, The theory of partial differential equation, Cambridge Univ. Press, 1973. [ 7 1 S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, 1984, Preprint. Kiyoshi Asano Institute of Mathematics Yoshida College Kyoto University Kyoto 606, Japan Seiji Ukai Department of Applied Physics Osaka city University Sugimoto, 3, Sumiyoshi-ku Osaka 558, Japan



A Discrete Model for Spatially Aggregating Phenomena By Tsutomu IKEDA Abstract. The objective of the present paper is to nuinerically study the behavior of a solution of a mathematical model for spatially aggregating phenomena of population. For this purpose, we propose a discrete model, which preserves important properties of the continuous model with the aid of its nonlinear artificial viscosity term. Using this discrete model, we study the behavior of a solution and the stability of pulse-like stationary solutions. Key words: spatially aggregating population model, standing pulse-like solution, monotone finite difference method, nonlinear artificial viscosity, asymptotic behavior

1. Introduction

In the present paper, we study a finite difference approximation for the following nonlinear degenerate diffusion equation involving a nonlocal convection term P(m,r)

(

ULx, d = ( U 7 n ) z z ( xf)+ , U(x, t )

{ s'

U(Y, W Y -

2-T

SXtr

U(Y, W Y } )

X

2:

in R X (0,

m)

subject to the non-negative initial condition (1.1)

U(x,0)=Uo(x)20

for x € R .

Here, m > 1 and O S r s m are parameters, and U(x, r ) 2 O denotes the population density at position X E R and time t > O . We assume that Uo€C(R) and the support of Uo, which is denoted by supp [UO], is compact. The equation P(m,O) agrees with the porous medium equation, which appears in the theory of fluid flow through a porous medium (Bear [4] and Scheidegger [29]). In the case of r > O , P(m, r ) represents a mathematical model for spatially aggregating phenomena of population, proposed by Mimura and Yamaguti Received July 2, 1985.

386

T. IKEDA

[22]. The second term on the right-hand side of P(m, r ) ecologically shows a n aggregating mechanism of individuals, which is motivated by the notion of “centripetal instincts” (Hamilton [14]). In fact, the term provides a mechanism that moves individuals at position x to the right (resp. left) direction when

JZ

Jz-r

The first term ( corresponds to the transport of population through a nonlinear diffusion process called density-dependent dispersal (Gurney and Nisbet [12] and Gurtin and MacCamy [13]). The diffusion speed mUm-’ decreases with the density U and vanishes at position x where U(x)=O. Consequently, P(m, r ) is provided with a homogenizing process and a dehomogenizing process. We may expect that a delicate balance between these two processes gives rise to a spatial pattern, which shows a n aggregation of individuals. We first review mathematical works related to the Cauchy problem P(m, r ) subject to (1.1) and the stationary solution of P(m, r ) . The distinctive feature of the Cauchy problem P(m, r ) , which is caused by the degeneracy of diffusion at U=O, is that a n initial distribution with compact support spreads out at a finite speed and loses its initial smoothness (Aronson [2], Kalashnikov [18] and Oleinik et al. [27] for P(m,O)). A solution of P(m, r ) is therefore defined in a generalized sense (Aronson [l], Caffarelli and Friedman [5] and Gilding and Peletier [lo] for P(m,O)). For a class of Cauchy problems including P ( m , r ) , Nagai [23] has shown the unique existence of a generalized solution, and he has proved the finite propagation property. A stationary solution W of P(m, r ) , which ecologically exhibits a spatially aggregating pattern of individuals, is defined to be a non-negative valued function belonging to L1(R)n L-(R) that satisfies

in the distribution sense. The trivial function W=O is always a stationary solution of P(m, r ) , and we naturally are concerned with the non-trivial stationary solutions. In the present paper, a non-trivial stationary solution W of P(m, r ) is called a standing solitary pulse (abbr. a n ss-pulse) if supp [W] is connected and W>O on the interior of supp [W]. The porous medium equation has n o non-trivial stationary solution; while P(m, a)has ss-pulses (Mimura and Yamaguti [22]):

Theorem 1.1. For each q > O , P(m, a)has an ss-pulse W such that

A Discrete Model for Spatially Aggregating Phenomena which is unique up to coordinate translation Moreover,

(11 - 11

387

denotes the supremum norm).

-

where 11 denotes the usual norm of L1(R), and diam (W) denotes the length of SUPP[WI. 17 For the general case of O O . Using J"[. ; m , r ] , we rewrite (2.3) as

Qh r)

uc(x, t)=J"[u;m , r],(x, t ) in R x ( 0 , a).

3. A Discrete Model for Spatially Aggregating Phenomena In this section, we propose a discrete model Q , ( m , r ) for spatially aggregating phenomena of population, which is obtained as a finite difference

A Discrete Model f o r Spatially Aggregating Phenomena

391

approximation for Q ( m , r ) . The population density is obtained by differentiating the solution of Q,(m, r ) . Let h be a positive number, which denotes the spatial mesh size. We use the following notation ( a : a non-negative number):

(3.1) X ” ( a ) = { v , € C ( R ) ;v, is linear on each interval (ih,i h f h ) ( i E Z ) , lim v,(x)=-a and lim v,(x)=a} , z+-m

z-m

(3.2) X h = { C h € L m ( R )C; h is constant on each interval (ih,ih+h) ( i E Z ) } ,

v*=v,(ih)

(3.3)

(3.4) (3.5)

for

v , € X h = U Xh(a) and i E Z , a20

Vv,=(the derivative of v , X~h ) € x h , {VIvh=(the value of Vv, on (ih,ih+h))

for v , € X h and i € Z ,

Xkono(a)={vhE X h ( a ) ;Vv,zO o n R} and X2,,,,= U Xkono(a). a20

In general, it is not easy to derive a discrete model that fulfills the requirements (Pl)-(P3) described in Section 1. To see this, let us consider a simple case r = m . In this case, Q(m,a)has no nonlocal interaction:

Qh, a)

u,=

(u;)~+(u~)~ in R x ( 0 , a) I

It is natural to use the central finite difference approximation 1

t(V‘~h)m-(V*-luh)m~ (u, E X&m0)

for the first term (u;)~. However, by the same reason as that in the case of the Burgers equation ut=(u2),, the central finite difference approximation does not apply to the second term (u2), if u, is approximated by the forward difference. We consider the application of schemes for the Burgers equation to Q(m, 00). The requirement ( P l ) is not fulfilled by a scheme derived from a nonmonotone scheme (the Lax-Wendroff scheme ([ZO])for instance). On the other hand, a scheme derived from a monotone scheme may fulfill (Pl). In fact, a n application of the Lax-Friedrichs scheme ([7]):

392

T. IKEDA

fulfills ( P l ) under a condition on the time increment r,,. the Enguist-Osher scheme ([28]): For u:

e X;on,,(T 1

And application of

, find { u ; } ; = , c X ~ , . , ( ~ llVu:lll)

/lVu:lll)

such that

1

(u:+'- ~ , " ) = - { ( V ' U ~ ) ~ - ( V I - ~ U ~ ) ~ } h

+ 1 If-(u:+

1)

-f- (u,"1+f+(u,"1-.f+(U?-

111

9

where f ( u ) = u 2 , f+(u)=f(min {u, 0}) and f-(u)=f(max { u , 0}), also fulfills (Pl) under a condition on rn. However, neither scheme has a stationary solution w such that supp[Vw] is compact. This fact is easily shown by expressing (L-F) and (E-0) in the divergence form:

For (L-F) the flux is given by F:+112=G:,llz+h2Vtujt/4r,, and F:,,,,=G:+l12+ {f-(u:+l)-f-(u:)-f+(u:l 1)+f(uzI))/2 for (E-0) where G:+1,2=

(ViUhn)llL+

1

-{(u1)2+ 2

We return to the general case of 0 5 r 5 00. by

(u:.kl)z}

.

We define b,[ * ; r] : X~,,,,,+X"


393

where ai[v,; m, r ] denotes the value of a nonlinear artificial viscosity

-

on the interval (ih,ih+h). Xh(0) defined by

-

Using Jh[ ;m, r ] , we introduce L,[ ;m, r] : Xiono-+

Now, our discrete spatially aggregating population model is:

t

1

-(u;tk1-u;)=L,,[u;t; m , r ]

for n=O, 1, 2,

-

a ,

rn

where the time increment rn is determined so that

two propositions where r,=h2/max {2m( ~ ~ V U ~ /hllVu:lll}. ~ J ~ ) ~The - ~following , assure that Q,(m, r ) has a solution for any u: E Xi",,,, that is,

(3.14) if u;E X&,no(c), then u:+'E X&,.Jc) Proposition 3.1.

If

0,

E X ~ o n o ( cthen ) , Gh=vh+rLh[uh;m, r ] E Xh(c),and

(3.15) Proof.

{ 4 , ( x )- v , ( x ) } d x=0

.

Since L,[v,; m, r ] € P ( O ) , 8,€ Xk(c). By integration by parts, a,

-m

under the condition (3.13).

i=-m

T.IKEDA

394

Proposition 3.2. Let V,E X:o,,o(c). Then, 9,=v,+rL,[v,; X&"Jc) under the condition

t

2r max llV~,]\:-~,

(3.16) Proof.

(3.17)

h 2

- Ilb,[v,; rlllm} S h 2

m, r ] belongs to

.

We put

h p,[w,] = (V,w,Jrn-l +a,[w,; m, r ]=max ( V i ~ h ) m - l , 16" w,; rl I

t

for W, E Xk0,,,, and i E Z. Then, L,[w,; m, r] is rewritten as

With the aid of the nonlinear artificial viscosity (3.11),

h 2

for i E Z .

pt[w,]--lbb,[w,;r]I~O

(3.19)

The expression (3.18) of L , [ - ;m, r] yields

ai+ 6=(vi, -

= :

+

{pi+Jv,l

't

+-h

Jv, ; m , rl -L J v , ; m. rll h 1 2 bttl[vh;r l } (Vr+lv,)f y W - ~ ~ P J ~ , I N V A J

- v,) rWt

+-

h p i - I [ v h ] - - b t - l [ v h ; r ] (Vb,-lu,)

2

1

for i E Z .

The coefficients of Vitlvh and V,-,v, are non-negative by (3.19), and that of Viv, is non-negative under the condition (3.16). We thus complete the proof. 0 A stationary solution w h of Q,(m, r ) is defined to be a function belonging to Xkon0such that (3.20)

Lh[wh;m,r]=O

on R

.

Since Q,(m, r ) is of conservation form, if w,(x) is a stationary solution, then w,(x-ih) is a stationary solution for i E Z . The function w,=O always is a stationary solution of Q,(m, r ) . A spatially aggregating pattern is obtained by differentiating a non-trivial stationary solution of Q J m , r ) . For the continuous model, the standing solitary pulse means the stationary distribution of the population density. However, for brevity, a non-trivial

395


stationary solution w h of Q,(m, r ) is also called a standing solitary pulse (abbr. an ss-pulse) if supp [Vw,] is connected. The key of Q,(m, r ) is the nonlinear artificial viscosity a,[-; m, r ] . We have shown that a h [ - ;m, r ] permits Q,(m, r ) to fulfill the requirement ( P l ) (Proposition 3.2). We shall see in the forthcoming sections that a,[.; m, r ] permits Q,(rn, r ) to fulfill the other requirements. 4.

Spatially Aggregating Population Model Q,(m, a)

In this section, we study the discrete spatially aggregating population model Q,(rn, a). After observing how a,[ ;m, a]works and proving a comparison theorem, we show that Q,(m, co) satisfies discrete analogies of Theorems 1.1 and 1.5. We introduce two functions U ( K , v) and J(u, v), defined on the half plane D = { ( u , v ) E R 2 ;u ~ u } ,

-

(4.1) (4.2) Then, bt[vh;a], a,[v,; m, a]and JJv,; m, a]( v , Xkon0) ~ are reduced to

1

m, w]=a(v,, v , , ~ ), b2[vh; w l = ~ ~ + v, ~ + a,[v,; ~

(4.3)

(i

>’

JI[vh;m, 031=J(vt, vt+J- - IIVvhlll ,

respectively, and Q,(m, a)is rewritten as (4.4)

~;+~=u;+p,J(up,~p+~)-p~J(u,”-,, up)

where p,=rJh.

for

i € Z and n20

,

We divide the half plane D into the following three regions:

(4.5)

(Figure 4.1). By (4.1), (4.2) and (4.5), a(u, v ) is continuous in D , and is of class C1 on DoU D , U D a(u, v)=O on Do and a(u, v) > O on D\d, ,

,

396

/!q/ T. IKEDA

0

D

0

DD-

D-

(a)

1<m Z

(b) m - 2

Figure 4.1. Regions DO,D+ and D - . (4.7) J(u, v) is continuous in D , and is of class C’ on DoU D , U D -

,

where the symbol - denotes the closure. We easily see that J(u, v) and the partial derivatives Ju(u,v) and Ju(u,v ) are expressed in the following form:

(4.8) ( J ( u ,v)=v2 on

(4.9)

IJ,(~,

6 , and

m v-u

*--l

v)= - h ( k )

‘J,(u, v)=O on D ,

J(u, v)=uz on

6- ,

on D , ,

+u

and Ju(u, v)=2u

on D- ,

(4.10)

‘J,(u, v)=2v

on D ,

and J J u , v)=O

on D-

.

Observing ( 4 . 3 , (4.9) and (4.10), we see (4.11)

J,sO on D and J,O on D,UD+ ,

h

-la+Pi,

2

h yIp+rI}

for

aspsr,

and obtain the following comparison theorem:

Theorem 4.1. Let v: € X;o,o(c) and then, under the condition

vie X ~ , , , ( c ) . If v:(x) 5 v:(x)f o r x € R,

397


both O;=v;+rL,[v,W; m, 001 and v^i=uk+rL,[vi; m, w] belong to A'&,no(c), and

(4.15)

$ ~ ( x ) S O ~ ( x ) for

xGR.

and put 8t=v;+

Proof. For 0 0, we associate the number

E ( c , h ) = m i n { q ~ R ; ( v , ~ ) ~ .6 + }

(4.18)

(See Figure 4.1.) By (4.9,

(4.19) (4.20) (4.21)

[ ( c , h) > - c

and [ ( c , h)= -max

{v E R ; ( - c ,

E(c, h)+c as h+O

7) E B-},

,

( v , c ) E D + if E ( c , h ) < v < c , if - c < 7 j l < - e ( c , h ) .

{(- c , ? ) E D -

Moreover, there exists a number a,=a,(c, h) ([(c, h)O for l i l < N

(Zh,n$d(WS -r$i

I$,, r} of a

and $,(x)ZO for Iil < N .

405

function #,E X h ( 0 )

for ~ E R ,

We define a sequence { $ ; t } ~ - ~of functions $;E X h ( 0 ) by (4.55)

# ; = ( l + ~ ) - ~ $ , for

n20.

BY (4.49) and (4.54), K$,Z/Z,l on R for some large number K>O. K$;I-Z;t. Then, (4.53) and (4.54) yield @;

(4.56)

Let @=

+' I@;+ ( L n 0 3 ( i h ) +

Ki#;+'-$a --(L,,$Xih)} 20: +(Zh,n@Xih) = (1-pna," -pn/3:)@: +p,a:@:-l +p,,/3:~@,"~~

-

for lil$itl

for O s i < N ,

(4.68)

Q,S(1+q)w,-ls(l+q)2t-1

for O < i d N ,

]ilSN,

for O < i z N .

Let us show that the pair {$h,r=apq2/2} satisfy the third condition of (4.54). By (4.63), (4.61) and (4.64), (lh,,$,)(0)=p,(a,"+p,")($l-$O)=-E(n(aOn = - p,(a,"

+p,")PqQl$l

+Po")pq2/(1+pq2) 5 - -21 apq2= -740 .

Let O < i < N . We rewrite (Zh,n$h)(ih) as

A Discrete Model for Spatially Aggregating Phenomena

407

For - N < i < O , rewriting (l,,,,$,)(ih) as

we can similarly show (lh,n$h)(ih)S -r$$. 5.

0

Spatially Aggregating Population Model Q,(m, r ) (0 < r < 00)

We study in this section the stationary solution of Q,(m, r ) for O < r < By integration by parts, J,[v,; m, r] is rewritten as

A stationary solution

W,

of Q,(m, r ) satisfies Jh[wh;m, r]=O on R

(5.2) 5.1.

00.

.

Decomposition of a stationary solution We show in this subsection a discrete analogy of Theorem 1.2. We let s(r, h)=min { i h z r ; i E Z} .

(5.3)

Proposition 5.1. Assume W , to be a stationary solution of Q,(m, r ) . If Vw,=O on an interval [nh, nh+h] ( n E Z), then there exists an integer i such that

(5.4)

, Vw,=O

n h + h s i h + s ( r , h) , on the interval [ih, ih+s(r,h)]

Proof. Let i=min { k E Z ;n h f h - s ( r , h ) s k h , w,=w,} and j =m ax { k E Z ; k h s n h + s ( r , h), wr=w,}. I f j hzi h+ s ( r , h), then we obtain (5.4). Suppose that j h < i h + s ( r , h). Then,

T. IKEDA

408

which contradicts (5.2). [7

Theorem 5.2. A function w, € X&,nobecomes a non-trivial stationary solution of Q,(m, r ) if and only if W , is expressed in the f o r m w,(x)=

(5.5)

where

c wP)(x)

for

k ~ , 1

x€R,

{w?)},~,,is a set of ss-pulses wi*) of Q,(m, r ) such that

(5.7)

dis (supp [ V w i L ) ]supp , [VwLL’)l)2s(r, h)

f o r k € A , k’ E A , k f k ’

.

Proof. Assume w h to be a non-trivial stationary solution of Q,(m, r). be connected components of supp [Vw,] (A: a n index set). For each Let k € A, define W p ) by (5.8) Then, c,=

W i k ) ( x ) = ( V w h ) ( x )for x € S ,

(U2)ll W?)lll< (U2)

IIvwhll19

w:~)(x)=\’

(5.9)

and

Wik)(x)=O for x4: S,

.

and

Wi*)(y)dy-c,€ X&o,,(c,) .

-m

Each supp[Vwp)] ( G S , ) is connected, and {WL*)}~~,, satisfies (5.5) and (5.6). By Proposition 5.1, {wp)},..,, also satisfies (5.7). Let us show that each wp) satisfies (5.2), by noting that (5.10)

rn, r] is determined by Vi+yh’S (ljlhSs(r, h ) ) .

Jt[vh;

(See (3.6), (3.7), (3.11) and (5.1).) Let x € ( t h e interior of S,). VwLk)(x+y)=Vwh(x+y) for 1yI 5 s ( r , h) by (5.7), and (5.11) J , [ w p ) ; m, r](x)=Jh[wh;m, r](x)=O

Then,

for x € (the interior of S,)

by (5.10). Let x @ S,. Then, Vwp)(x)=O, and by (3.10) or (5.1),

A Discrete Model for Sparially Aggregating Phenomena

Jh[wLk); m, r](x)=O

(5.12)

for

x

409

S, .

Now, each wr) is a n ss-pulse of Q(m, r). Assume {w?)}~.,,to be a set of ss-pulses of Q,(m, r) satisfying (5.6) and (5.7). Then, the function w, given by (5.5) belongs to X:,,,, and satisfies Jh[wh;m, r]=

(5.13) by (5.7) and (5.10). 5.2.

C Jh[wiX); m, r]=O

ksA

Hence, w, is a stationary solution of Q,(m, r).

0

Standing solitary pulses We now proceed to the discussion on a discrete analogy of Theorem 1.3.

Proposition 5.3. Assume w, € Xkon0(c)to be a stationary solution of Q,(m, r). Let u be an arbitrary positive number, and put q=hum-2. Then, z,(x)=umw,(xu2-m)

(5.14)

€ X&ono(Cum)

is a stationary solution of Q,(m, rum-2). Proof. Put

It follows from the definition of z, that and ( V Z , ) ~ = U ~ ~ ( V ,W , ) ~

Vz,=u2Vw,

(5.15)

J iq

(5.16) ih+r

(Vw,)(x){w,(x-r)-w,(ih)}dx

for i E Z

,

ih

Siha

ihafha

(5.17)

=-urn - -u2m-2

{w,(~+r)+wh($-r)-2w,($)}dy ih+h

(d7w,)(x)dx=u2m-zhb,[w,; r]

for i E Z ,

\ih

a,[z,; m, ra](Vz,)(iq)=max (5.18)

ra]l -(Vzq)m-l}(Vzg)(iq) Ib,[w,; ~ ] I - ( V W , ) ~ - ~

=uZrnah[w,; m,r](Vw,)(ih)

for i E Z

By (5.15)-(5.18), z, is a stationary solution of Q,(m, rum-'):

.

T. IKEDA

410

. (5.24) M ( m , h)=max {diam (Vz,); IIVzhllrn=l,zh is a n ss-pulse of Q,(m, a)} Theorems 4.7 and 4.8 imply (5.25)

M ( m , h)+F(m) as L O . We put

Let W, be a n ss-pulse of Q,(m,

m).

(5.26)

q = h IIVw,II,'-m'2 ,

z , is a n and define a function z , ( ~ ) = ~ ~ V w , ~ ~ ~ ~ ' ~ w ~It ( is ( hshown / q ) x ) that . ss-pulse of Qh(rn, m) by the same method as that of Proposition 5.3. More over,


411

Now, a discrete analogy of Theorem 1.3 is stated as Theorem 5.5.

(i)

Q,(m, r) has no non-trivial stationary solution w, such

that

( i i ) An ss-pulse W, of Q,(m, a)is a n ss-pulse of Q,(m, r) if

Proof. ( i ) Assume W, to be a non-trivial stationary solution of Q,(m, r). Assume that Vtwh= IIVw,[lm and Vt--Iwh< I[Vw,[[.. for some ic Z. Then, we obtain

In the case of Viwh< llVwhllrn for all i E Z , we also obtain the same estimate as (5.31). Hence, Q,(m, r) has no non-trivial stationary solution satisfying (5.29). ( i i ) Let w, be a n ss-pulse of Q,(m, m). Then, by (5.28) and Proposition 5.4, W, is a n ss-pulse of Q,(m, r) under the condition (5.30). 0

T. IKEDA

412

Examination of the conjecture We demonstrate in this subsection the results of numerical examinations of the conjecture given in Section 1. We discuss the existence of a n ss-pulse W and the length diam ( W ) of supp [ W ] solely. By the same method as that of Proposition 5.3, it is shown that if W is a n ss-pulse of P(m, r ) , then, for each o>O, Z ( x ) = ~ ~ W ( x ois ~a -n ~ss-pulse ) of P(m, rum-e) and (ra"-z)zIIZll-2-"=r2~~W11~-". Values of 1.5 and 3.0 were chosen for the parameter m. We fixed h=0.1, and prepared various ss-pulse w h of Q,(m, m), by using (4.25). Each of these ss-pulse was used as a n initial value of Q,(m, r ) . As stated in Proposition 5.4, w, is a n ss-pulse of Qh(m,r ) if r 2 d i a m (Vw,).

5.3.

( a ) m = 1.5

Figure 5.1.

(F(1.5)=4.206...)

(b) m = 3.0

(F(3.0)=2.587...)

Examination of the conjecture for the existence of an ss-pulse.

A

3-

3-

2-

I

( a ) m = 1.5

Figure 5.2.

2.

(F(1.5)=4.206-..)

(b) m = 3.0

(F(3.0)-2.587..')

The length diam ( W ) of the support of an ss-pulse W.


413

For some r and w, such that r < d i a m (Vw,), supp [VK;] (u;: the solution of Q,(m, r ) with ut=w,) tended to some finite interval, and Vu; tended to some non-negative function W,sO as n 00. In this case, we judged that Vu; and diam (Vu;) (n: sufficiently large) approximate a n ss-pulse W of P(m,r ) and diam (W), respectivly; and we made a dot at position ([[Vu;[lm,r ) in Figure 5.1. Figure 5.2 shows diam (W)/r ( W : a n approximate ss-pulse) plotted against r [IWII_l-m'z. Figures 5.1 and 5.2 indicate that the conjecture is correct for the existence of a n ss-pulse W and the length of supp [ W ] ,respectively.

6. Behavior of the Solution of P ( m , r ) In this section, we express our views, which are supported by various numerical studies using Q,(m, r), on the stability of standing pulse-like solu-

Figure 6.1. An approximate solution of P(2.0,3.9) (h=0.3, U " ( x ) = l + cos ( 4 1 8 ) for 1x1< 18, Uo(x)=O for 1x1 > 18).

Figure 6.2. Approximate solutions of P(2.0, 3.9) (h=0.3, U"(x)= max {O, cos (bxx)) for 1x1 18).

T. IKEDA

414

(1)

X

a = 12.0

tu _g

(2)

a = 15.0

(3)

a = 21.0

X

Figure 6.3. Approximate solutions of P(2.0, 3.9) (h=0.3,1 UO is given .by (6.1)).

(1)

a = 10.8

(2)

a = 13.5

.. ..-.

Figure 6.4. Approximate solutions of P(2.0, 1.8) (h=0.3, Uo is given by (6.1)).


415

tions of P(m, r ) and the behavior of the solution U(x,t ; U a ) of the Cauchy problem P(m, r ) with the initial condition U(x,0; Vo)=Uo(x) ( O < r < a).We denote by #( V) the number of connected components of supp [ V ] .

Stability of standing pulse-like solutions. Our view on this subject is that each standing pulse-like solution of P ( m , r ) is stable in a sense. Figures 6.1 to 6.6

rl

(1)

a = 15.0

(2)

a = 21.0

-

U

I

.

_-

X

Figure 6.5. Approximate solutions of P(1.5, 3.9) (h=0.3, U o is given by (6.1)).

(1)

a = 12.0

(2)

a = 14.7

(3)

a = 16.0

X

. -

Figure 6.6. Approximate solutions of P(3.0, 3.9) (h=0.3, U ois given by (6.1)).

416

T. IKEDA

( 1 ) Uo(x)

i s g i v e n by ( 6 . 1 ) w i t h a = 33.0

( 2 ) U0(x)=2/9

(-13<x O ,

8

41)

9

.

q-$g(p,q)>O

If E is not empty and, further, p , q, uo, and v o satisfy

(2.11)

then, defining p o and go anew by

(2.12)

we have the following a priori estimates from above of the solution (u, v) for (1.1)-(1.2), 12

05U(X,

(2.13)

t)51uol'o'+-po=P,

8

12

O ~ V ( X ~, ) j l v o l ( o ) + s q o = q ,

Thus we have:

( O l r 5 T ).

Blowing-up Problem of a Parabolic System

423

Theorem 2. If the set E (see (2.10)) is not empty and, further, p , q, uo, and

v0 satisfy (2.11), then there exists a unique temporally global solution (u, v) for (1.1)-(1.2) which, as restricted to Zx[O, TI, belongs to H;++"xH~T++" for an arbitrary T x ( 0 , m) (also, see [3], [4], [ 5 ] ) . Therefore, the solution (u. v) does not blow up. Moreover, it holds that (2.13)'

0 5 u ( x ,t ) S p ,

O S v ( x ,t ) i q ,

(OZto, p---->OY l2 P4 8 l+q

p q >O} 8 l+p

q - - -l2

We divide our problem into 3 cases. 1) In the case of 8 > 12,

(2.20)

Therefore, if uo and v, satisfy

then there exist p and q such that

(2.21)’

Thus, the solution (u, v) for (1.1)-(1.2) does not blow up. Moreover, it holds that (2.21)”


425

2) In the case of P = 8 ,

(2.22)

Now, we remark the inequality

which implies that, if u, and v, satisfy (2.24)

IuoI (0)- /ool(0) < 1 ,

then the solution (u, v) does not blow up, since there exist p and 4 such that

3) In the case of P > 8 , we can treat the problem in the same way as in Example 1.

S 3. Blowing-up In the preceding section we have endeavored to obtain a sufficient condition under which the solution (u, v) for (1.1)-(1.2) does not blow up. However, it is very difficult to obtain a necessary and sufficient condition on u, and v, for the blowing-up of the solution (u, v). In this section we shall show that there are some cases in which the solution (u, v) blows up. Now, let $(u, v) and &u, v) satisfy

-(->=”(’>, a 1

av

$5

au

6

and (u, v) be a global solution for (1.1)-(1.2). Dividing both sides of the upper and lower equations of (1.1) by $ and 6,resp., we have

(3.2)

By the equality (3.1), there exists a function @(u, v) (u, u 2 O ) such that

N. ITAYA

426

(3.3) For example, @(u, v) defined by (3.4) (3.2), we derive from

satisfies (3.3). Hereafter, we adopt this function as @. (3.2)

a

(3.5)

v)=(u+v),,+{B(u, v)+B(u, 241 .

-@(u, at

Here, we note that @ has the following properties,

d dw

I:;

-@(w,

(3.6)

-@(w,

w)=-+-

1

$(W,

w)

w ) = -~ $(w.

>o, #(W, W )

{ $ L ( w ,w)+$,(w,

41

W)z

and that, therefore, @ ( w , W ) is monotonically increasing and concave. NOW, multiplying both sides of (3.5) by K

X

s(x)=-sin-x 21 1

(3.7)

I!(

s(x)dx= 1

and integrating them from 0 to t in r and from 0 to I in x , we have a n equality

(3.8)

1'

@(u(x,t ) , v ( x , t))s(x)dx

For the left-hand side of (3.8), firstly, by the definition (3.4) of @(u,v), we easily have a n inequality

21' @(u(x, t ) , v ( x , r))s(x)dx, 0

(OSt< 03)


427

Secondly, by the inequality

(3.10)

@(u+v, u+v)LO(u, v)

(N.B.: Ou, @,>O)

,

the concavity of @, and Jensen's inequality, we have

Hence, if there is a suitable function F(w) such that

B(u, v)+B(u, v ) L F ( u + v )

(3.12)

,

then all that remains for us is to make similar arguments to those in [4] and [ 5 ] , and to show that in some cases there arise contradictions to the assumption that (u, v) is a global solution for (1.1)-(1.2). Thus, we have:

Theorem 3. Let $, $, 4, and q be the same as in the preceding sections, $ and $ satisfying (3.1). (i) Zf there exists a function F ( w ) which satisfies (3.12) and has a form F(w)=Clw~-Ccz(where C,, C,, and /3 are constants such that C , >0, C z 2 0 , and /3> l), then, under a certain condition on uo and vo,the solution (u, v) for (1.1)-(1.2) blows up. (There are also cases to which Theorem 2 is applicable.) (ii) Zf B(u, v) and B(u, u ) satisfy C,(u+u)+C,zB(u, v)+B(u, v ) ~ C 3 ( u + v ) - C c , (where C3, C,, and C, are constants such that C,>O, C,LO, and C,LO), then, under a certain condition on uo, vor C,, and @, the solution ( u , v) blows up. (There are also cases to which Theorem 2 is applicable.)

As for the proof of the above theorem, refer to [4], [ 5 ] , and the examples to be given below. The inequality (3.11) is necessary to demonstrate (ii) of the above theorem. Remark. (i) For $(u, v)=$(u, v), the equality (3.1)implies $ u = $ u = $ z I , which shows that $(u, v) has a form $(u, v)=$,,(u+v). (ii) For $ ( u , U ) = $ ~ ( U ) Therefore, (3.1)is satisfied. and $(u, v)=$,(v), q5u=$,=0. Example 1. In the case of $(u, v ) = p (const. >O), $(u, v ) ~ (const. p >O), and $(u, v)=q(u, v)=u2+v2, it is obvious that

(3.13)

@(u, v)=p-'U+p-'v,

{ B(u,

u)=,i-1(u2+v2)

B(u, v)=p-'(uZ+v2)

.

By (3.8)and (3.9), we have, (a0=p-l+p-l)

(3.14)

aoJ(t)=ao

,

N.ITAYA

428

2

5'

(p-luo+,E-lvo)(x)s(x)dx-k2

0

+a,

5: 1' dt

5'

J(r)dr

0

( u 2 + v 2 ) ( xr)s(x)dx ,

0

Thus, if it holds that (3.15) then the solution (u, v) blows up. Example 2 . In the case of #(u7 v ) = l + u , J(u, u ) = l + v , +(u, v)=a(l+u)v, and $=b(l+v)u ( a , 6, const. >O), we have

(3.16)

@(u, v)=log(l+u)(l+v)

,

B(u,v ) = v ,

B(u, v)=u

.

By (3.8) and (3.11)7it holds that (3.17)

@ ( J ( t )J(t))=2 , log ( l + J ( t ) )

1:

21' {log ( l + u , ( x ) ) ( l + v , ( x ) ) } s ( x ) d x + (c-k2)J(r)dr, 0

).

( k = F , c-min { a , b}

Hence, if c is larger than k 2 = r 2 / l 2and uo+vo#O, then the solution (u, v) blows UP. Example 3.

The system of equations

(3.18)

[ u ( x , O)=uo, v ( x , O)=vo (uo, v o 2 0 , and € H Z t u ) ,

u(0, t)=u(Z, t)=v(O, t)=v(Z, t) = O,

etc.]

is Petrowsky-parabolic as seen from the view point of classification. However, w - u + z , and z-u-v satisfy


429

(3.19)

[ w ( x ,O)=u,+u,

(>=O), z ( x , O)=uo-uo, w(0, t)=w(l, t)=z(O, t)=z(l, t ) = O , etc.]

.

Therefore, w e can treat t h e system of equations (3.18) in almost the sa m e way as i n the preceding cases.

References A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964. H. Fujita, On the blowing up of the solutions of the Cauchy problem for ut= Au+ulta, J. Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109-124. N. Itaya, On the counter-blowup effect of the viscosity coefficient in nonlinear parabolic equations, Collected Papers in Commemoration of the 50th Anniversary of the Foundation of K6be University of Commerce. (In Japanese). -, A note on the blowup-nonblowup problems in nonlinear parabolic equations, Proc. Japan Acad., 55, Ser. A. (1979), 241-244. -, On some subjects related to the blowing-up problem in nonlinear parabolic equations, Lecture Notes in Numer. Appl. Anal., Vol. 2, Mathematical Analysis on Structures in Nonlinear Phenomena, Kinokuniya, 1980, 27-38. -, Re-discussion on the blowing-up problem in nonlinear parabolic equations, Jimmon-ronshO of KBbe Univ. Comm., 16, No. 4 (1981), 150-157. (In Japanese). 0. A. Ladyzhenskaya, et al., Linear and Quasi-linear Equations of Parabolic Type, Nauka, 1967. (In Russian). J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, 1983. A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, RIMS, Kyoto Univ., 13, (1977), 193-253.

Kbbe University of Commerce 4-3-3, Seirybdai, Tarumiku Kbbe 655, Japan



L’iquation de Kadomtsev-Petviashvili approchant les ondes longues de surface de l’eau en ecoulement trois-dimensionnel Par Tadayoshi KANO Abstract. A mathematical justification for Kadomtsev-Petviashviliequation as an approximate equation for long waves of water surface of three dimensional flow. Introduction On Ctudie dans cet article les ondes longues d’ampleur finie de surface de l’eau e n Bcoulement trois-dimensionnel qui se distinguent d’ondes de surface en eau peu profonde. On donnera une justification mathkmatique pour l’kquation de KadomtsevPetviashvili [ 7 ] , [14]comme une Bquation approchCe des ondes longues, ce qui correspond, pour ainsi dire, ti une justification mathkmatique pour 1’Bquation de Korteweg-de Vries [12] et celle de Boussinesq [2] dans le cas de 1’6coulement deux-dimensionnel [lo].

S 1.

Equations non-dimensionnelles des ondes longues de surface de I’eau

Les ondes de surface de I’eau en Ccoulement trois-dimensionnel sont rCgies par les Cquations aux dCrivCes partielles suivantes par rapport a @=O(t, x, y , z ) , le potentiel de vitesses, et a r = T ( t , x, y ) qui dCfinie la surface libre de l’eau par z - f ( t , x, y)=O:

(1.1)

.a

@zl+@y,+@zr=O

dans Q ( t ) ,

(1

@‘,=O,

(1.3)

Ot+-

(1.4)

rc+ (wz +rpy)O~=o

1 2

z=o, (Dz2

Received February 1, 1985. Revised May 21, 1985.

+@,2+ D S Z )+gz=O -

T. KANO

432

oh Q(r)={(x,y , z ) : ( x , y)GR2, O < z < r ( t , x , y ) } , t > O , est le domaine rempli de

I’eau. Dans 1111, on a montr6 I’existence locale par rapport a u temps d’une solution unique pour (1.1)-(1.4) dans une Cchelle d’espaces de Banach S = Up,,, B, de fonctions analytiques, en connaissant le potentiel initial @(O, x, y , z ) et la forme initiale r(0,x , y ) d e la surface de I’eau. On l’a dkmontrk dans une forme non-dimensionnelle OG intervient un parametre non-dimensionnel d=h/I: le rapport de la profondeur moyenne de I’eau h a la longueur I des ondes de surface. On a demontrk, de plus, que notre solution Btait indkfinirnent differentiable par rapport a ce parametre 8 , obtenant ainsi une justification mathematique du developpement de Friedrichs en cas de l’bcoulement trois-dimensionnel, voir aussi [5], 161, 191 et [191. En s’appuyant sur ces resultats, on va expliciter maintenant les equations non-dimensionnelles adequates et le developpement de Friedrichs correspondant pour 1’Ctude des ondes longues de surface de I’eau. On e n deduira par la suite, comme Bquations approchees pour les ondes longues de surface de I’eau, I’Bquation deux-dimensionnelle de Boussinesq et, en particulier, 1’6quation de Kadomtsev-Petviashvili [7], [14]. Pour ce faire, introduisons tout d’abord l’ampleur non-dimensionnelle des ondes de surface de l’eau: soit zl la difference de la profondeur z de I’eau en mouvernent de la profondeur h de I’eau en repos:

.

z=h+z,

(1.5)

Soit, d’autre part, a le dbplacement moyen de la surface z = r ( t ) de la surface de I’eau en repos. Alors, en posant z,=az,’, on dBfinit I’ampleur non-dimensionnelle yj par r=h+av.

(1.6)

Appliquons a (1.1)-(1.4) le changement d e variables: ( t , x , y ,

Z)H

(tl’, xl’, y,’, z,’) defini par

(1.7)

Posons

-=a, h I

a h

--=€

.

Alors que I’on a btudik, dans [ll], le probleme non-dimensionnel tel que I’on

L’kquation de Kadomtsev-Petviashvili des ondes superficielles

433

ait 6-0 lorsque R+w tout en laissant a pouvoir 2tre finie, on va ktudier dans cet article des ondes de surface de I’eau telles que 6z=(h/R)z et E=a/h soient de mbme ordre comme infinitksimaux lorsque R+w, a+O, e n suivant les suggkstions d’Ursell [21] et de Stokes [20]. Pour la simplicit6 de calcul, on pose carr6ment (1.9) dans ce qui suit. I1 faut noter que l’on Btudie e n fait le probleme pour 6 E [0, 11, et non pas seulement pour 6>1, pouvant rester finie, voir [lo].

O . Moreover, he discussed the penalty finite element method as the discretization of (2). It becomes clear in his analysis that the scheme must be based on appropriate mixed finite element models to have nice approximation properties, although p is a subsidiary quantity in (2). Otherwise, the finite element solutions by (2) behave quite badly. In actual implementation, the explicit use of the mixed method may sometimes be avoided by the use of the so-called selective reduced integration, but the implicit role of the mixed method is in a sense essential. Arnold [ l ] considered a model parameter dependent equation, which has essentially the same properties as (1). In this case, E is a physical parameter, and it is practically meaningful to obtain solutions for E other than zero. He stressed the importance of the robustness of finite element approximations with respect to the parameter, which he showed may be realized by appropriate mixed finite element models. Kikuchi [7] also considered a similar equation and its approximation, and generalized the results of Arnold by means of the asymptotic expansion. In this paper, we present a n iteration scheme to solve the above type of problems. First, we consider a parameter dependent problem in a product Hilbert space as a generalization of problem (1). Then we introduce the iteration scheme with two theorems on its convergence. The basis of the present method is the usual penalty method, but, by using a n iteration technique, we can now obtain solutions for a fairly wide range of E . Although detailed results will be reported in [8], we will show the outline of the convergence proofs as well as some basic properties for completeness. In particular, the iteration process is convergent for E = O . Its relation to the augmented Lagrangian method is also noted: see Fortin and Glowinski [5]. Then the mixed finite element method is introduced and it is shown that the iteration process is also applicable to the arising approximate equations under certain hypotheses. Finally, we give some numerical examples for (1)to see the validity

447

Iteration Scheme to Nearly Incompressible Media

of the present approach. In particular, we observe the dependence of the iteration process on the parameter, and we may see that our approach has nice convergence characters when the above-mentioned hypotheses hold. It is to be noted that our approach is also effectively applicable to numerical analysis of the incompressible Navier-Stokes equations, where the use of some iteration processes is inevitable due to the nonlinearity of the problem, see Mizukami v11. This work is in part supported by the Grant-in-Aid from the Ministry of Education.

2. Preliminaries The norm of a Banach space X is denoted by 11. l.y, and if X is a Hilbert For two Banach spaces X space, its inner product is designated by and Y , L ( X , Y ) implies the Banach space of all linear bounded operators from X into Y. The norm of L ( X , Y ) is conventionally written by I I . I I L ( X , Y ) but The null space and the range of a n operator is often abbreviated as /I.II. T € L ( X , Y ) are denoted by N ( T ) and R ( T ) , respectively. For T € L ( X , Y ) and a subset Z of X , TIZ implies the restriction of T to 2. Let V and W be real Hilbert spaces. We consider two bounded bilinear forms: (a,

,

a ( * , a ) : V x V-R'

(3)

b(*,

,

We assume that a(

where R' is the set of all real numbers. tive in the sense that

(4)

V X W-R'

a):

a(u, u ) 2 0 ;

VU€

a,

0 )

is non-nega-

V .

We can define A E L( V , V ) ,B E L( V , W ) ,and B* € L( W, V )by (5)

(Au, u),=a(u,

V)

;

(Bu, ;Ow=(u, B*A),=b(u, 2 ) ;

(6)

vu, V

€

V ,

VuG V , I € W ,

We will only consider the case where

(7)

BfO

Let

If,

E~

f

N(B)#{O)

*

be a (small) positive constant, and consider the problem: given E E [ O , E ~ ] find , {ut, &} E V x W such that

g } E V x W and

(8) or, equivalently,

44,V ) + b ( V , &)=(f,V ) , ; b(u,, PI- 44,P)w =(9,P)w ;

VV€

VP €

V, w,

F. KIKUCHI

448

(9)

Au,+B*A,=f,

Bu,-d,=g.

We will regard the above as a parameter dependent problem with the parameter E chosen from [0, 4. From the second relation in (9), g must belong to R(B) for E=O. Then R, is necessarily in R ( B ) for E # O . For E = O , A, is indefinite in its component in N(B*). If R ( B ) is closed, then R ( B ) is the orthogonal complement of N(B*) in W , and hence it is sufficient to look for 1, in R(B) even for E=O. Hereafter, we will consider the above problem as a generalization of problem (1). As for its solvability conditions, the following two are wellknown: see e.g. [ll, [2], [3], and [ti].

[HI] There exists a positive constant k , such that

[H2] There exists

(11)

Q

positive constant k , such that

IIB*~II"Bk,ll~I[,;

VA€NB)

I

It is clear that R(B*) is closed when [H2] holds. Then, by the closed range theorem [14], R ( B ) is also closed, and (11) is equivalent to (12)

IIBullwLkzllullv ;

vu.s R(B*) .

Moreover, [ H l ] is equivalent to the following when [H2] and (4) hold, as is proven in [8]: there exists a positive constant k , such that

(13)

Q(U,

u)+llBul12,Lk311ullT. ;

VuE V .

Note that k , is expressed by a continuous function of k,, k,, and [IA[I. Sometimes, (13) is easier to deal with than [Hl]. Under the two hypotheses above, we have the following results as are proven in [l], [2], and [6]. Lemma 1. Assume that [Hl] and [H2] hold, and consider problem ( 9 ) with { J ,g } € Vx R ( B ) and 8 € [o, E ~ ]given. For E = O , there exists a solution { u ~A€} , which is unique in V X R(B) (but WIQYnot be so when it is considered in Y X W ) . Similarly, for E E 10, E,,], there exists a unique solution in V X W , again denoted by {us,As}, where A6 belongs to R(B). In both cases. {us,R6} satisfies

(14) where C is Q positive number dependent continuously on k,, k2,E I[B*II only, and hence is independent of E € [ O , E ~ ] .

~ I[AII ,

and 11B11=

For E ~ O the , second relation in (9) gives 1,=~-~(Bu,-g). Substituting this into the first equation in (9), we have a single equation in u, only:

449


+

Au,+ E-'B*Bu,= f E-'B*g ,

(15) or, equivalently,

a(u,, v)+~-'(Bu,,Bv),=( f, U ) , + E - % ( U ,

(16)

g) ;

V U €V .

These expressions correspond to (2), and often appear in the description of actual physical problems; see Lions [9], Arnold [l], and Kikuchi [7]. They are also commonly used in the penalization of (8) or (9) for E=O. One of the merits of such a n approach is that RE can be dealt with as a subsidiary quantity. Furthermore, R , obtained by the penalty method automatically belongs to R(B) since small positive values of E are used, and hence we need not handle the indefiniteness of RE for E = O . 3.

An Iteration Scheme Fix

E*

(17)

€

[0,

and rewrite (9) as Au,+B*R,=f,

Bu,-E*R,=(E-E*)R,+g,

for which we can consider the following simple iteration scheme: (18) ( i )

=O

( i i ) for i = O , 1,2, (19)

E R(B) ,

.-., decide {u:),

+B * p =f ,

€ V XR(B) recursively

Bu!t)--*J!t)

= ( € - € * ) l i i - ' ) +g

by

.

The existence and the uniqueness of each {@, A:)} in V x R ( B ) follow from Lemma 1. Clearly, {u!O), R , ' O ) } is the approximate solution obtained by equating E to E* in (17). If E*>O, (19) may be rewritten by

The left-hand side of the first equation of (20) is exactly of the same form as that of (15) employed in the penalty approach. Thus we can deal with {R:i)}:=o as subsidiary quantities, and hence we can take full advantage of such a n approach. It is easy to check that {@, A:')} is rewritten by i

(21)

= C (E-c*)!u(f) j=O

,

=

i (E-c*)jJ(j)

,

3 =O

where the coefficients { u ( j ) ,A")} E V X R ( B ) for j=O, 1,2,

--

belong to V XR(B)

F. KIKUCHI

450

and are uniquely defined by (22)

Au(O)+B*J(O)=f

Bu(0)-EE*;j(O)-

A u ( j )+B*J(j)= O

Bu(j)-€*A(/)

--9

7

021).

=A(j-1)

Note that the coefficients do not depend on i. Similar type of expansions at E*=O are analyzed by Lions [ 9 ] , Temam [13], and Kikuchi [7].

We have the following results for the above expansion.

Theorem 1. Assume that [Hl] and [H2] hold. Then, for any {f,g } € V X R(B) and any E * E [0, E ~ ] , each of the coeficients { u ( j ) ,R ( j ) } for j 2 0 in (21) is determined from (22) uniquely in V x R ( B ) with the estimation (23)

llu(j)II"+ IIJ'."

I l f ll"+ Ilsllw)

llw 5 Cj+Y

9

where C is the positive number appearing in (14). Moreover, {uLt), Q)} for each E [0,4 satisfies (24)

Il@) - 4 I l v + I I P - A c l l w 5

Cit21E- €*I

ll"

""llf

+ llsllw )

for i 2 0 , and it converges to rhe solution {uE,A C } € V x R ( B )of (9) as i-+m if E C [ O , E ~ ]is subject to (25)

IE-E*I

< l/C .

Remark. From (25), {&), converges for E = O if E * < l/C, which is one of the most important cases in applications. In particular, E* need not be too close to zero even when we want to obtain solutions with E almost equal to zero, since the convergence is assured so long as (25) holds. This fact is practically important since we are likely to have numerical instability if we use too small a n E * . Proof. Applying Lemma 1 to (22) with E equated to E * , we can assure the existence and the uniqueness of each { u ( j ) ,A(')} ( j 2 0 ) in V x R ( B ) with the estimations

II u(O)ll"

+ llA(o)llw i C(llfll"+ 11-911w)

IlU(j)ll"+

9

~ l ~ ~ ~ ~ l l w i c l l A ~ (j21) ~ - l ~ 1~ l w

from which (23) follows. From (22), we find that (21) satisfies for i 2 0

Au!l)+B*l,")=f,

BULL)-E *1,( ~ ) = ~ + ( E - - * ) ; I ! I ) - ( ( E - - * ) ~ + ~ R ( I )

which, together with (9), gives A(uii' -U,)+B*(A!"-I.~)=O, B ( u , ' i ) - ~ e ) - E ( J ! t ) - -) =; (- ( € -

€* 1t + l J ( t )

,


451

Applying Lemma 1 to the above with (23) taken into account, we can conclude (24). It is now clear that {u:), A:)} converges to {u,, At} as i+m if (25) holds, and the proof is complete. / I / / Notice here that the above iteration scheme reduces to a special case of the augmented Lagrangian method when a ( . , - ) is symmetric and e = O : see Fortin and Glowinski [ 5 ] . Suggested by the convergence proof of such a method, we can obtain another theorem on the convergence of the present iteration scheme.

Theorem 2. Assume that [Hl] and [ H 2 ] hold. Then the sequence {{uLi), {ue,As} of (9) in v x R(B) as i+m i f

ALi)}};=o based on (18) and (19) converges to the unique solution (26)

OI€52€*

(l€--E*I

S € * ).

Remark. Theorem 2 is in a sense complementary to Theorem 1. In fact, the constant C in Theorem 1 is difficult to evaluate in actual problems, while such a quantity does not appear in Theorem 2 : we can a priori make the scheme convergent for the considered E by choosing E* to satisfy (26). Especially, the iteration is convergent at e=O for any ~ ~ 2 0On. the contrary, when E* is smaller than l/C, Theorem 2 gives less information than Theorem 1: particularly, it yields nothing on convergence rate. Note also that the starting approximation & - I ) to A, need not be zero for the convergence of the iteration, provided that 1L-l) lies in R ( B ) . Furthermore, if O < E < ~ E * , A:-') need not be in R ( B ) , as may be seen by carefully checking the proof.

Proof. Define {u('), p")} E V x R(B) by u(')=u,'t)-uu, and p(')=A:')--AC i 2 0 . Moreover, put p ( - l ) = -A, E R ( B ) . Then we find for i20 that A ~ ( i ) f B * ~ ( i ) = o , Bv(L)-

E

for

*p ( t ) = (&-€*)p(t-l).

For E*=O, the conclusion of the theorem is trivial.

For E * > O , we have

and hence

), gives Furthermore, we find that Au(f)+(l/€*)B*Bu(i)=-( 1- ~ / e * ) B * p ( ~ - lwhich

F. KIKUCHI

452

From the above two relations, we obtain

l (26) and ( A u ( ~v)(, ~ ) ) , Z O from (4). That is, {IIp(')Ilw};==o since ( l - ~ / e * ) ~ S from is a non-increasing non-negative sequence and hence has a limit. Then the ~)), left-hand side of (a) converges to zero, and hence both ( A U ( ~ ) , U ( and [lBu(i)llwconverge to zero as i k c o . By (13), this fact implies that u ( t )- 4(t)-u,+O

in

V

(i-tco)

.

Now the relation A U ( * ) + B * ~ ( ~ )gives = O that B*p(*)+O in V ( i - a ) , which assures by [H2] that p(i)=,?~i)-,+O

in

R(B)

(i-co)

,

since p ( i ) belongs to R ( B ) for each i20 (for O < E < ~ E * , (a) directly assures that p*'L)+Oin W ) . Thus {uJi), ,?J$)}+{uc,&} in V x R ( B ) (ik-..), and the proof is complete. ////

4. Application to FEM The techniques presented in the preceding section are easy to apply to the finite element method. In the standard finite element method, we prepare a family of spaces { V h xW h j k e nwhere r the index set A is contained in [0, h,] for a positive constant h, and has zero as a n accumulation point, and, for each h E A , V h and W h are respectively finite-dimensional subspaces of V and W. Usually, h implies the representative element size, and we are interested in the asymptotic behaviors of the numerical solutions as h 10. Let P , € L( V, V h )and QhE L( W , W h )be the orthogonal projection operators, and define A , E L( V f i ,V h ) ,B, E L( V h , W h ) ,and BZ € L( W h ,V h )by

It is clear that A,=P,AI V h , B,=Q,BI V h ,and Bf=P,B*I W h . If we use the standard Galerkin method, the finite element approximation {uhL,,?fie} € V " x W h to the solution {uc,,?€} of (8) or (9) is determined from the condition


453

or, equivalently,

where E , f, and g are the same as those appearing in (8) or (9). Clearly, Q,g must belong to R(B,) for the above to be meaningful for E=O. Therefore, for the validity of the present problem for all g c R ( B ) , it is necessary that (31)

IHo1h

Q , R ( B ) c R ( B h );

VhE A

Note that the above is equivalent to: N ( B ? ) c N ( B * ) for all h E A . The present finite element approximation is of mixed type since it employs two unknowns uhe and However, as in the continuous case, we can eliminate A,, from (30) for E > O : by using the relation Ah,=(l/E)(B,u,,-Qhg), we have

Therefore, we can also utilize the penalty approach for solving the discrete problem (30) with E = O . As is pointed out in [l], [2], [lo], [12] and in many others, such process is in fact effective if Q , has certain special structures. In particular, it can be easily implemented when suitable selective reduced integration formulas are found. It is also possible to apply the iteration scheme considered in the preceding section to solve the discrete problem above. In order to assure the validity of the discrete problem, it is quite natural to assume the discrete analogs of [Hl] and [H2]. [Hl], ( 34)

uh)zk?llUhll?J;

[H2],

(35)

There exists a positive constant kT independent of h E A such that VuhEN(Bh)

*

There exists a positive constant k$ independent of hE A such that

IlB?Ahllvzk$llkllw ;

VAhE R(B,)

.

As in the continuous case, (35) is equivalent to (36)

IIBh4hllwzk$ll4llv ;

v U , E R(B?) *

Furthermore, we can show that [Hl], is equivalent to the following when [H2], and (4) hold: there exists a positive constant k? independent of h E A such that

F. KIKUCHI

454

We can express k r by a continuous function of k r , k$, and IlAll. Under hypotheses [HO],, [Hl],, and [H2],, we can obtain essentially the same results as in the continuous case with respect to the uniqueness and existence of the approximate solution as well as the convergence of the iteration scheme. Especially, in the statement of the results, there appears a positive constant corresponding to C of Lemma 1 , which can be taken to be independent of h E A .

5. Numerical Results We apply our method to finite element analysis of the following problem related to the two-dimensional Stokes equation:

(38)

+

-AZ+gradp=O

,

-ddivZ--p=O

in f2 ;

z=zo on

af2,

where 9 is the square domain shown in Fig. 1 , G={ul, us} is the velocity, p is the pressure, and Zo denotes the prescribed value of ii on the boundary all. We consider the so-called cavity flow problem, where the boundary condition is inhomogeneous and is shown in Fig. 1. Note that Z0is discontinuous at the upper left and right corners of f2. We test two types of triangular finite element discussed by Crouzeix and Raviart [ 4 ] ,in which the approximate velocity ii,={uRI, u,~}and the approximate pressure p h are given as follows.

Element-1. Over 9, uhl and un2are continuous piecewise quadratic functions, while p, is a piecewise constant function. That is, in each triangle, uh1 and u,, are quadratic polynomials, while p h is constant. Element-2.

Over f2, u , ~and u,, are non-conforming piecewise linear func-

x2= 1

u1'UZ'O

r I

~

u1= 1 , Uz= 0 --

R

C

XI= 1

Fig. 1. Q and boundary conditions.


455

tions: they are linear polynomials in each triangle, and are generally discontinuous along the sides of the triangles except at the midpoints. The functional form of p h is the same as that of Element-1. Note that p h is discontinuous, and hence the operation for Q h can be performed elementwise. The selective reduced integration technique is available for both elements, see Bercovier [2]. As for the stability and error analysis of these two elements, see Crouzeix and Raviart [4]. It is especially to be noted that Element-2 has nice properties corresponding to [HO],, [Hl],, [H2], etc., although it is non-conforming. We also consider two types of uniform meshes (Mesh-1 and Mesh-2) shown in Fig. 2. For Element-1, u , ~ ,is set equal to 1 at the upper left and right corners of 9, where K , is discontinuous. Numerical solutions are obtained for various combinations of E and E* by the use of the double precision arithmetic, and the numbers of iterations for convergence are counted. The employed stopping condition is

(39)

That is, the iteration is stopped when (39) is first satisfied, and the corresponding i’s are given in Table 1 for Element-1 and Table 2 for Element-2, respectively. Note in the second relation of (39) that p i i ) and are constant in each element. Convergence is generally rapid when E is close to E * . As we have discussed in the analysis, convergence is always attained for E = O , although it becomes slower as E* becomes larger. We can see that even the choice of is available in practice to analyze the case of E = O , and we need not use too small E* which may bring us numerical instability. Moreover, it appears that the numbers of iterations are in general insensitive to the choice of elements or meshes. To see the dependence of the numerical solutions on E , we give computed velocity vectors in Figures 3 through 6 for ~ = l 0.1, , 0.01, and 0: they are all

MESH-I

MESH - 2

Fig. 2. Triangulations employed for computations.

F. KIKUCHI

456

Table 1. Number of iterations for convergence: Element-1.

X

h

1 1

1 10-1 10-2 10-8 10-4 10-5

50

6 7 69 6 9 6 9 69

0 (-:

10-1

1

E*

2 1 53 7 3 76 7 6 7 6 76

1

10-3

10-2 2

-

-

1 1 3 1 4 1 4 1 4 14

1 1 4 1 5 1 5 1 5 15

1

-

2

_

1

-

10-4 2

-

-

1

-

-

-

-

-

1 6 6 6 6

1 6 6 6 6

-

-

1 4 4

1 4 4 4

4

10-5 2

- -

1

2

-

-

-

-

3

3

1 2

1 2

-

4

4 1 3 3

1 3 3

divergent)

Table 2. Number of iteration for convergence: Element-2. 1

€*

X

I 38 48 4 9 49 49 49

10-1 10-2 10-5 10-4 10-5

0 (-:

J2

10-1

10-2

10-3

10-4

10-5

2

1

2

1

2

1

2

1

45 59 6 0 61 61 61

1 1 1 1 1 1 1 1 1 11

1 1 2 1 3 1 3 1 3 13

_

-

_

_

_ -

5

6 1 4 4 4

5

1

1 6

5 5 5

6 6 6

1 3 4 4

3 1 3 3

2

-

4 1 3 3

1

2

-

-

3 1 2

3 1 2

divergent)

u,=1

E =I

-

& 0.1

Fig. 4. Computed velocity vectors (~=0.1,Mesh-2).


E = 0.01

x2

457 E=O

U,=l

xL U1'1

...*

.

C

C

"

.

.

. . . * *. . . . . , , . ' " ' .

Fig. 5 . Computed velocity vectors (e=O.Ol, Mesh-2).

Fig. 6. Computed velocity vectors (e=O, Mesh-2).

based on t h e use of Element-1 a n d Mesh-2, a n d the vectors a r e shown only a t vertex nodes. N o t e here that the choice of E* has no influence o n the results so long as the iteration is convergent. W e c a n see that a vortex appears when E becomes closer t o 0, a n d qualitative difference m a y b e found between solutions f o r smaller E ' S a n d those for larger ones. Moreover, the numerical solution f o r ~ = 0 . 0 1is quite close t o that for E = O .

References D. N. Arnold, Discretization by finite elements of a model parameter dependent problem, Numer. Math., 37 (1981), 405-421. M. Bercovier, Perturbation of mixed variational problems. Application to mixed finite element methods, RAIRO, Anal. Numtr., 12 (1978), 211-236. F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers, RAIRO, 8 (1974), 129-151. M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, RAIRO, 7 (1973), 33-76. M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1983. V. Girault and P.-A. Raviart, Finite Element Approximation of the NavierStokes Equations, Lecture Notes in Math., No. 749, Springer Verlag, BerlinHeidelberg-New York, 1979. [ 7 1 F. Kikuchi, Accuracy of some finite element models for arch problems, Comput. Methods Appl. Mech. Engrg., 35 (1982), 315-345. An abstract analysis of parameter dependent problems and its applications 181 -, to mixed finite element method, J. Fac. Sci., Univ. Tokyo, Sec. IA, 32 (1985), 499-538.

458

F. KIKUCHI

[ 9 ] J. L. Lions, Perturbations Singulieres dans les Problemes aux Lirnites et en ContrBle Optimal, Lecture Notes in Math., No. 323, Springer Verlag, BerlinHeidelberg-New York, 1973. [lo] D. S. Malkus and T. J. R. Hughes, Mixed finite element methods-reduced and selective integration techniques: a unification of concepts, Comput. Methods Appl. Mech. Engrg., 15 (1978), 63-81. [ll] A. Mizukami, Finite element analysis of the steady Navier-Stokes equations by the multiplier method, to appear. [12] J. T. Oden and N. Kikuchi, Finite element methods for constrained problems in elasticity, Intern. J. Numer. Methods Engrg., 18 (1982), 701-725. [13] R. Temam, Navier-Stokes Equations, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1977. [14] K . Yosida, Functional Analysis, Springer Verlag, Berlin-Heidelberg-New York, 1965.

Department of Mathematics College of Arts and Sciences University of Tokyo 3-8-1, Komaba, Meguro-ku Tokyo 153, Japan


On a Local Existence Theorem for the Evolution Equation of Gaseous Stars By Tetu MAKINO Abstract. The equation of the hydrodynamical evolution of an adiabatic gaseous star, which consists of the compressible Euler equation coupled with Poisson's equation, is waiting for mathematical treatises. Although many interesting studies from the numerical point of view supported by the development of big computers can be found among astrophysicists, rigorous mathematics have not caught up with them. The author has heard nothing about even the proof ofthe existence of solutions for the associated Cauchy problem. Now this article is devoted to establishing the existence of local solutions under a suitable condition on the initial data. The discussion is along a standard line except for the crucial difficulty of integrating the Euler equation for solutions with compact support, which does not appear in the hydrodynamics on the Earth. However the result is too restrictive so that the author hopes that he has presented a tentative treatise initiating the further development of mathematical theory on this interesting equation. Key words: Cauchy problems, quasi-linear hyperbolic systems, hydrodynamics, self-gravitating systems, astrophysics

1. Introduction

W shall investigate t h e C au ch y problem

(1-i)

(3) (4-0) Received April 5 , 1985.

p Ir=o=pO9

460

T. MAKINO

v, It=o=vot

(4-i) Here K , 7 ,

K

are positive constants.

(i=l, 2, 3)

.

The unknowns are the functions p=

p(t, x), ~ = ~ (v2, v u,)=v(t, ~ , x), p = p ( t , x ) = K p ( t , x ) ~ $=$(t, , x) of t>=O and x= c ( ~ lx2, , x 3 )€ R S , while po=po(x) and vO=vo(x)are initial data. We wish to establish the existence of a solution p(t, x), v(t, x), p ( t , x), # ( t , x) on a domain [0, T ] x R 3 = { ( f , x ) / O 5 t 5 Tx, € R 3 } for given po and vo.

Equations (l), (2), and (3) describe the evolution of a star regarded as an isentropic ideal gas with self-gravitation. The variable p means the density of the gas, p the pressure, @ the Newtonian gravitational potential and u the velocity. Equation (1-0) is the equation of continuity, and (1-l), -2), -3) express the conservation of momentum. In this paper the system (1) consisting of (1-O), -l), -2), -3) will be called the Euler equation. Equation (2) is the equation of state, 7 being the adiabatic exponent. We keep in mind that r = 5 / 3 if the stellar material is treated as a monoatomic gas, 7=4/3 if the radiation pressure is taken into account and supposed to be excellent, and other values of r have significances of their own (see [l],Chap. 11, Section 12 or [ 6 ] , Section 53). Equation (3) is Poisson's equation, tx standing for the constant of gravitation. The solution of (3) of physical interest is that given by Newtonian potential:

(3)" For a detailed discussion in astrophysical contexts of the Cauchy problem (1)(4) we refer to P. Ledoux and Th. Walraven [6]. An existence theorem of the problem will be established along the following line: First, taking a n arbitrary p(r), not necessarily a solution, we integrate Poisson's equation (3) by (3)" to get the potential $ ( t ) (Section 4); next we integrate the Euler equation (1) together with (2) with respect to the now given potential $ ( t ) under the initial condition (4) to obtain a new density distribution p ( t ) (Section 3); if p(r) coincides with the first p ( t ) then it should be a solution of the problem (1)-(4). The final work will be done by applying the fixed point theorem (Section 5 ) . A model of this procedure can be found in the study of Vlasov's equation by S . Ukai and T. Okabe, [81. When we progress in this way, we meet with difficulty at the integration of the Euler equation. The standard mathematical treatment of the Euler equation is to transform it to a symmetric hyperbolic system to which Friedrich's theory is applicable. We are acquainted with such a study by S . Klainerman and A. Majda [ 5 ] . In their study, and in most other studies by mathematicians establishing rigorous theories on fluid dynamical equations, the density p of the fluid remains to majorize a positive constant uniformly throughout the whole space during the motion. However, in our astrophysical context the density is expected to have compact support or, at least, to

Evolution of Gaseous Stars

46 1

vanish at infinity, for, otherwise, the Newtonian potential (3)" would be divergent (Olbers' paradox). Unfortunately, if we trace [5] in order to obtain a symmetric system, we find that vanishing of ,o causes the coefficients to be degenerated or unbounded (see [ 5 ] , ( 2 . 2 ) ) . This is the difficulty of the Euler equation in astrophysical contexts. This paper is devoted to proposing a solution of this difficulty. It is a change of variables from p to a suitable variable w=w(p), by which one can reduce the Euler equation to a symmetric system whose coefficients behave gently even if p vanishes. In conclusion we can say that the change of variables

will lead us to the following result.

Theorem. Assume that the initial data po(x)and v o ( x )are continuously differentiable and that p o ( x ) is nonnegative everywhere and has compact support. If (1)

1 )- A k (

u(0))l

dOO, x’=O}, that is, E € Cm(R3\ Rzl). ( ii ) E-E, E L:,,(R3). ( i i i ) E has a f o r m , E=E,+E,+E,+E,+E,+E,, which satisfies the following (iv)-(viii). ( iv ) E, satisfies Eo€C”(R3\{O)) and I E , ( ~ ) I S C , ( ~ + I X ~ ) - ~where ~XI-~ we represent rapidly decreasing function by C,( 1+ Ixl)-”. ( v ) E, satisfies E,€C”(R3\Rzl) and C,(l+lxl)-Nlxl-l for lE,(X)lS{CN(l+lxl)-Nlx’~-l for ( vi )

x,> 1, x12u+lxl

vu>O,

O>l, if M 1 (resp. M = l ) ,

+ I E , ( ~ ) l = O ( l x l - ~,/ ~(resp. )

B,=O, O 1 , vu>O,

x12u+lxl

O < ~ E ~ T , ’C, s.t.

-

B,LE+

IE,(x)/=0(1~1-~’~), ~E3(~)lSC8(l+l~ . l)-z

A. MATSUMURA

486

Remark. The essential differences from the incompressible part are represented by E,' and E," in (ix). For more precise property as to each component of E , refer to the lemmas in the following sections. 3.

Preliminaries

We define the Fourier transform and inverse transform for x € R3 and its dual variable E € R 3 by

s

a ( i ) = F [ ~ ] ( E ) = ( 2 ; r ) -e~ /az ~ ' b (x)dx, (3.1)

s

t ( x ) =F-l[u](x) = ( 2 ; r ) - 3 / 2 etz*Eu(E)dc.

The Fourier transform of the equation (2.6) takes the form (3.2)

A(E)s(e)=Z,

E € R3 .

So our purpose is to investigate F - ' [ A - ' ( e ) ] . Denote the eigenvalues of A(E) (resp. B(E)) and the corresponding projection matrices by I,([) and p,(E) (resp. A,(E) and P,(E)). Then, by (2.3), it obviously holds that

R",(E)=

(3.3)

--icEI+A,(E) ,

P,(E)=P,(E) ,

and 1, is one of the roots of the characteristic equation det IRI-23 =(A+v1[l2))"f(1)=0 ,

(3.4) where f ( A ) =R3

+ (i+ 7 )I El"+

+ +

(s71El4 ( a 2 jZ) IEl"A+ a"lEl

.

We define 1, by A , , ( E ) = - Y I E ~ ~ and {1,};=, by the three roots of f ( A ) = O . us list up the basic properties of 1, which are well investigated in [ 3 ] .

Let

Lemma 3.1. ( i ) Re A, O and 0 5 j 5 3 . ( i i ) 1, is semisimple for I E l > O , that is, rank (1,1--B)=3 ,

for

IEI > O .

(iii) There exists a positive constant r , such that for O < 1615 r l , { I , } ; = , are distinct roots and can be expanded by the Taylor expansion as I,(E) = C,"=, a,tk)/ E l k . (iv) There exists a positive constant r2 ( > r J such that for 151 2 r 2 , {A,};=, are distinct roots and can be expanded by the Laurent expansion as i , ( E ) = CiZ2b,tk)IEIk. By the property (i), we have det A(E)fO for IEI > O which implies that the equation A(D)W=O in 9 ' ( R 3 ) admits only polynomials as a solution. This

Fundamental Solution for a Viscous Flow

487

fact proves the uniqueness of the solution in 9' of (2.6). The property (ii) enables us to define the partial inverse A-'(E)P,(E) for V I E 1 > O by A-'(E)P0(E)=

(&J(E) - icEJ-IP,(E). Lemma 3.2.

F-l[(A,-icEl)-lP,]=

Proof. The direct calculation gives the relations

(3.5)

(3.6)

=U,, ,

(lzi,j 5 3 )

where w=E//1El. This completes the proof. This lemma shows F-l[?,-lP,] exactly corresponds to the incompressible part of the velocity profile. Let {xt(t)}:?=l be C"(R;) functions which satisfies

~ ~ - 1 ,x 2 - x 3 = O ,y1=x3-0, xz-l xI=xz=O, xs=l

(3.7)

xl+xZ+x3=

for

tSr12/4,

for r 1 2 5 t 5 r z 2 , for t 2 4 r Z 2 , for t E R 1 .

Then A-l is divided as A-'(E)=C;=, xt(lEIZ)A-'(E). Since the property (i) implies that x 2 ( l f l * ) A - 1E( ~C;(R3), ) we have Lemma 3.3.

F-l[xz(IEIz)A-'(E)]is in Cm(R3)and rapidly decreasing.

By the properties (iii), (iv), xrA-' ( k = l , 3) can be represented as 3

(3.8)

xk(lEIZ)A-YE)=C (~,(E)-icE,)-'xk(IE12)PI(E). 2=0

] k = l , 3 and l s i ( 3 . Thus it is sufficient to estimate F - 1 [ 2 , - 1 x 9 , for In the following, we shall analyze the case \El >> 1, \El> 1, where a,, A , and R , are in Cm(R3)

and

It is easy to see that this lemma corresponds to the properties of E, and

Esin the Theorem. Let us turn to the estimates for X~(R”,-~P,+R”~-~P,). Since (5.5)

F-’[X~(R”,-’P,+R”,-’P,)] =2 Re F-1[~IR”2-1P2] ,

it suffices to estimate Re F-’[X~R”,-~P,] in what follows. The expansion

shows that the leading term is ic,lEl(l-Mo,) for M< 1 and therefore the case M < l is easier to treat than M z l .

Lemma 5.2. If M< 1, then F-1[~lR”2-1P2] € C”(RS)and IRe F - ’ [ X ~ R ” ~ - ~ P5C(l+lxl)-2 ,](~)I .

Proof. First, P2(5)is expanded as

A. MATSUMURA

492

Fl(E)

f FZ(E)

and estimate each term as follows.

+

+

F3(E)

9

For F,, since a f F , E L' for la1 2 3 , it holds

IF-'[F,](x)l ~ C , I X I - (~N 2 3 )

(5.9)

for

1x1 2 1

.

For F, and F4, since At(F3+F4)€ L1,we have (5.10)

IF-1[F,+F4](x)15Clxl-z

for

1x121

.

For F,, since Fl has a form F,= IEl-'(B+K(w)) ,

(5.11) where

represents a constant matrix and K ( w ) represents a matrix satisfying

(5.12)

K ( w )E C"(Sltl=,)

s

and

K(w)du=O ,

it follows from the arguments on singular integrals that (5.13)

+

I x I 2F-l [ F ,I =F-' [ A @ IE I - 1 K ( 0 )IE I -1) 1 , =F-"B8t+v.p. K(o)IEl-31 , =B+K(x/lxl)

.

Hence it holds (5.14)

jF-'[F,](x)i ~ C I X ] - for ~

1x121

.

Combining (5.8)-(5.14), we have F-l[xI~z-lP,] E C" and IF-'[&lP,](x)l 5 C ~ X Ifor - ~ 1x1 2 1 . This completes the proof of Lemma 5.2. This lemma corresponds to the property of E, for M>l, 8,=8,

If M > 1 (resp. M = 11, then F - ' [ X ~ ~ , - ~€PCm(R3) ,] and it holds +

\ R e F-l[~IR",-lP,l(x)l = O ( l ~ l - ~ ,/ ~ (resp. )

IXI-~'~)

and or,, (6.13)

Re

sirn

X1K2e*Z'Edr=Re

+R

XIKzetx'Edr

-R

--m

=Re

+R

K,eiz.Edr+ Re

-R

Ks(0)+K&O)

\

+R

(xl-

l)K,etx'~dr

-R

1

For K6(w),using (6.9) and integration by parts again, it is easy to see

s:

IK,(w)[S CR-'lxl - 2 + C l ~ l - ~( 1 +r2)-'dr .

(6.14)

For K 6 ( w ) ,we complexify r, say z, and apply the Cauchy's integral formula to K 5 ;

= K h ) +K,(w)

9

where C = { z [ Im z=O, IRe z [SR}u { z I IzI = R , Im Z ~ O } . It is not difficult to see the integral K 8 on Iz[= R has the estimate IK8(w)I5 C R - ' l x [ - 2 .

(6.16)

For the crucial term K,(o), it follows from residue calculus that for a suitably large R, (6.17)

K8(w)= -(27cix4/Ix 10. w x ) ( z e i ~ x ~z )2z l0r =~i e0( i - , v c o s a ) - - (27ci~Jlx 10. wz)e-clsl( l - N c o s a ) w . w1 s-( c( 1-M cos 8)I xlw-w,) , 9

which leads to the estimate, using (6.8), (6.18)

[ K8(w)1~ ( 2 ~ ~ ~ / ~ ~ [ x ~ ) e - 1~+~c lox l~( l "- ~M( cos ' - "0)) ~ ." ~ ~ ) (

Therefore, introducing the new variable t= 1-M cos 8, (6.19)

a,sasa,+ao

I KAw)Idm

SClx[-l 5Clxl-2

\:

e - e 8 ~ l s l t ( l + t l x ~ ) d (t f, o = l - M c o s (8,+B0))

.

Thus, combining (6.13)-(6.19) and taking R++M, we have (6.20)

l z 5 ~ x ~ l ~ c l x.l - z

A. MATSUMURA

496

By the same arguments as for Z5, I,, and Z3 have the same estimates l~E(x)l? 113(x)1 s c l x l - 2 .

(6.21)

Hence by (6.11), (6.20) and (6.21), we obtain

Im)Isc(19,)lxl-2

(6.22) where C(O,)-+m

as 0,-0,

or T-0,.

1

Thus the proof for Case 1 is completed.

Case 2. 6,o,

l-Mcos00}

Then, similar to having the form of I , in (6.34), the essential integral to be estimated is given by (6.52)

I,= -2~lxl-'

\

KI,($)@+~K O=O,

\

K18(m)da-2r D2

\

K18(o)do D1

Since the integrands in (6.52) are even functions of $, we treat only the part of D, U D, with $ 2 0 and furthermore divide the part into four parts such as DXresp. D:)= {o€ D,(resp. 0,)I $ , ( E ) 5$}, D:(resp. D ; ) = { o € D,(resp. D,)1 0 O in each I- and I,. The boundary conditions are assumed to be, respectively,

+

(11)

u,(O)=O,

u(O)=O

u(P)=cU,

and (12)

u(P)=a,

u(+m)=0

9

v(P)=r

t

where a € (0, 1) and y > O are arbitrarily fixed constants. Since n,(y) does not satisfy the condition @)=a, we need to construct another approximate function in a neighborhood of y = p . Using the usual stretched variable f = ( y - p ) / ~ in (9), we obtain

Here, the subintervals I- and I, are transformed into [-PIE, 01 and R,. Consider the limiting case when e = O in (13). Then the second equation in (13) becomes vc=O. Hence, we take v(E)-fi,,(/3)=/3. Then (13) simplifies to a scalar equation of u only

Nonlocal Advection Efect on Reaction-Diyusion Equations ( 14)

O = ~ ~ ~ + ' p ( 2 P ) u e + f ( u ) for CER,

513

.

The boundary conditions (11) and (12) are assumed to be

(15)

u(-m)=l,

u(O)=a

respectively. Solutions of (14), ( 1 5 ) and (16) in R-=(-m,O] and R+=[O, cm) are obtained in the following.

Lemma 1. Consider the two boundary value problems with a parameter I (17,)

I

R

O= W&UW:+f(W')

for E E R , , w-(-m)=l , W-(O)=a, W + ( O ) = a , W+(+ca)=O .

Then there exists 6 > 0 such that for any fixed I € A 8 = { I I II-I*(f)IO and 6,>0 such that for any u f ( 0 , u s )and If A,,, there exists t(u, A) E X , satisfying ( i ) P(t(u, A ) ; u, 1 ) = 0 , ( i i ) lim IIt(u, A)llxo=O uniformly in and

010

(iii) t(u, 1) is uniformly continuous with respect to u and A in the X,-topology. Consequently, Lemma 3 yields a solution ( u - ( y ; E , p), v - ( y ; E , p)) of (9) and ( l l ) , which takes the form u-(Y; € 9 P)=fi-(y/P;

(25)

E/P,

P(2P)/2)

for

1-

.

Remark 2. (19) and (ii) of Lemma 3 lead to dulim ~ - - ( p ; CIO

dy

lim v-(p; E ,

E,

d WP)=--(O; dt

B)=p

(p(2P)/2)

uniformly in /3

.

Nonlocal Advection Effect on Reaction-Diyusion Equations

517

We next consider (9) and (12) in I + = [ @ ,+a). Using the transformation z = ( y - p ) / ~ we , rewrite (9) and (12) as

and u(O)=a ,

(27)

u(+Oo)=O

,

v(O)=y=v-(P;

E,

p) ,

respectively. Again we use a parameter I (=y(2/3)/2) instead of p. Let us seek a solution ( i P ( z ; E, I ) , V + ( z ;E , I)) of (26) and (27) which takes the form

t

(28)

u+(z;E , I ) = W + ( z I; ) + r ( z ; E , 2 ) P ( z ; E , I ) = v - ( p ( I ) ;E , / 3 ( I N + E V + k I)+s(z; E , 2 ) ,

s:

We note that the boundary conditions of where F + ( z ;,I)= W + ( eI)de. ; t = t ( r , s) become r ( 0 )= r (

(29)

+

00)

=s(O)

=O

.

Substituting (28) into (26), we have

where

+ +

+

+ +

Q (‘)( t ; E , 2 ) =Y,, p(2(V - cPt s))( W: r,) Wt, +2p’(2(v-+cVt + s ) ) ( E W+s,)(Wt + + r ) + f ( W ++ r )

and Q ( s ’ ( tE;, I)=s,--EY

.

Let Q ( t ;E , I ) be a mapping from X p = X;,,(R,)x k;,,,(R+) into Y,= X;(R,) x Xj(R+) for any fixed p ( O < p < r + ) . Lemma 4. There are E~ > 0 and 6, > 0 such that for any E E ( 0 , el) and R E A,,, there exist K7and K8 independent of E and I such that ( i ) lim IIQ(0;E , R)Ilu,=O uniformly in A € Aa3, €10

( i i ) llQt(tl;E , ;O-Q,(t,;

E,

;Oll~,-Yp~K~lltl-tzll~,for any t l , f,€ X ,

and

(iii) QJO; E , I ) has an inverse satisfying IIQ;’(O;

E,

R)IlYp-t,SK8.

Proof. Note that Q = t ( Q ( l ) ,Q ( * ) )is described by Q(.)(O;E , 4=p(2(v-f E ~ + ) ) W : + ~ E ( ~ ’ ( ~ ( ~ _ + E ~ and ~ ) ) Q(l)(O; ( W + )E ,~A)=O. - ~ I W(i) ~ directly follows from Remark 2 . (ii) can be easily obtained. We will prove (iii) in a way similar to (iii) of Lemma 2. QJO;E , 2 ) is given by

M. MIMURA,D. TERMAN and T. TSUJIKAWA

518

where Q:r)=&+p(2(v-+~7+))-+2~p'(2(u-+&+))W++f'( d dz

W+)

and d err)=240'(2(~-+E V + ) ) W : + ~ E ~ "+EV+))( ( ~ ( V _W + ) ' + 2 9 ' ( 2 ( ~+- EP+))W+dz

i

It is sufficient to show that for any G=t(G(r), G ( * ) E ) Y p , there uniquely exists t = t ( r , s) E X, satisfying and

for K B independent of E , R and G. We note that dldz has an inverse with [l(d/dz)-'11,$~~,,0 such that (61)

f(U)>--L(U-l) 2

for

I-~ 0 such that f o r any [p*-r, ,B*+r], Y,(y; p) belongs to s' f o r some y . Moreover, after entering Ss,Y,(y;P*+r) leaves s' through E; and Y,(y;p*-r) leaves .Ye through E;.

E€

[O, 4 and

Proof. This follows from Lemma 16 and the fact that SE is open and [,B*-r, p*+r] is compact.

Throughout the remainder of this subsection, we fix E E ( O , E J . [p*-r, p*+rl, let

For /3E

M. MIMURA, D. TERMAN and T. TSUJIKAWA

534

and If Y,(y;@)belongs to pp= YE@&8).

s" for all ~ > 3we~ let ,

Ya=+m.

If Jp is finite, we let

Lemma 22. For each @ E[B*-r, @*+r], J p isfinite. That is, after entering SKar y=pg, Yc(y;@) must leave 9 . Proof.

Note that in 9,

p= - &UO.

Since SKis bounded, each Y,(y;p) must either leave Sr for some y > j j ~ ,or approach a critical point which lies in the closure of 9. Because there are no such critical points, the result follows. Lemma 23. For each ,!? E [@*-r, b*+r], P, belongs to E c . Proof. We prove that a trajectory can only exit s' through E". A trajectory cannot exit SI through &, because on Ej, P = - E U < O . T o prove that a trajectory cannot exit SI through E;, we let n=(O,4B*, -1) be a vector normal to E; pointing into 9 ,and X = ( - E U , W , ~ V W - ~ E U ~ - ~the ( Uvector )) V2&U2+f(U)>O, which implies that trajectories cannot leave S through &. It remains to consider the edges of 3 through E;. Along Zl={( V, U , W ) ; O< V 0. We then set V(x)=

s:

U(y)dy

0

The phase plane corresponding to (66) is shown in Figure 5. The boldly drawn curve corresponds to the solution of (66)satisfying (67). We remark that V* is finite, since U approaches zero in the exponential order. This is the solution we wish to perturb to the case s>O. Before doing so we discuss further properties of solutions of (66), for which it is necessary to introduce the following notation. For v > O , let ( U ( x ;v), W ( y ;7)) be the solution of (66) which satisfies (U(O;v),W(O;v))=(v, 0). Let a=-f'(O), and choose 6>0 so that

536

M. MIMURA,D. TERMAN and T. TSUJ~KAWA W

\

Figure 5. Trajectories (U,W) of (66) and (67). Define u, el and epby u = { ( U , W ) I W 6 } , e , = { ( U , W ) I O< U 0. (iii) After enrering u, ( U ( x ;v*-r), W ( x ;v*-r)) leaves u through e , . (iv) After entering u, ( V ( x ;v*+r), W ( x ;T*+Y)) leaves u through e,. We do not give a poor of this lemma since it follows from straightforward phase plane analysis. The relevant trajectories are shown in Figure 5. The following notation is needed to discuss the three dimensional phase ; U,(X;7). WE(x;7)) space determined by (54). For 7>0, let Z E ( x ;~ ) = ( V * ( Xv), be the solution of (54) satisfying ZJO; v)=(O, 7,O). Define S,El, Epand E , by

Nonlocal Advection Effect on Reaction-Diffusion Equations

537

Since ( U ( x ;v), W ( x ;v)), u, e, and e, are, respectively, the projections onto the

(U,W ) plane of Zo(x;q), S,, El and E,, the following lemma is an immediate consequence of Lemma 26. r, Lemma 27. Let r be sufficiently small. Then, for any ~ € I o = [ ~ * - v*+r], Z,(x; q) belongs to Sfor some x>O. Moreover, after entering S for thefirst time, Z,,(x;v*-r) leaves S through El,and Z , ( x ; ~ * + r )leaves S through Ez. Now, S , El and E, are open sets. Therefore, by continuous dependence of solutions on a parameter, Lemma 27 remains true if Z, is replaced by 2, for small E . That is,

Lemma 28. Choose r as in Lemma 27. There exists E,, > 0 such that for any and 7 E Z,, then Z,(x; v ) belongs to S for some x >0. Moreover, after entering S for the first time, Z , ( x ; v*-r) leaves S through E l , and Z,(x; v*+r) leaves S through E2. a E [ 0, E,]

We assume throughout the remainder of this subsection that E € (0, aO) is fixed with lY

(69)

EOO lZ,(x;q)ES}

and y,=inf { x > x , 1 Z,(x; ?) sf S} . Define I , and Ze by 11={77€Zo l Z e ( Y 7 l ; q ) ~ ~ 1 l

and

I , = {v

zo

I ZAY, ; 77) c Ez}

*

Lemma 29. I , and Z, are nonempty. relatively open subsets of Zo. Proof. From Lemma 28, we know that v*-r GZ, and ;1*+rEZ,. W'=

-2E u2 -f( U )

2-

3 1

2&U--

U T - 2&S--

";Iu>o ,

On E,,

M. MIMURA, D. TERMAN and T. TSUJIKAWA

538

because of (68) and (69). Hence, whenever a trajectory leaves S through El, it must do so transversally, not tangentially. This, together with the fact that E, is open and the continuous dependence of a solution on initial data, implies that I , is a relatively open subset of I,,,on E,, U’=

wO

.

X-+m

The proof is broken up into a few steps.

Lemma 30. y , is infinite. That is, Z,(x) belongs to S f o r any x > x , . Proof. Suppose that y , is finite. Clearly Z,(x) cannot leave S through the V-axis since each point on the V-axis is a critical point of (54). By the assumption, Z,(y,) cannot belong to E, U E,. The only remaining possibility is that Z,(y,) belongs to E,, which we show, is impossible. Let n=(O, -1, 1) and

(70)

X = ( V ’ , U’,W ’ ) = ( U , w, -2E(VWS.U2)-f(U))

.

One finds that n is a vector normal to E8 pointing into S , while X is the vector field defined by the right hand side of (54). Recall that on E,, W-u--ou2+2u=u 8E a because of (69). Therefore, no trajectory may leave S- through So.

H

The following result completes the construction of the small pulse.

Lemma 32.

There is a positive constant V, such that

.

lim Z,(x)=(V,,0,O) 2-f-

Proof. Inside S , V'=U>O

,

U'=WO. Once the solution of (1.3) is shown to exist, we compare this w(t) with the free solution w-(t) in the energy space SYe with norm (1.6)

1

IIU(t)l l e =-J$IIA""(t)

1;

+ IIa,U(t)1I;Y

*

We denote by @ the following square in R2:

and by 9the part in

@ of

the closed quadrangle with vertices

Let d = d ( l / p , l/q) be a piecewise linear function of (l/p, l/q)€ 9 defined by (1.%

d = -I+---

n P

n 4

(in case m=O) ,

545

Small Data Scattering

Fig. 1. The case m=O ( n = 3 , p=4).

n---P -n+2+-+-

I=

4 n-2 P

Fig. 2. The case m>O ( n = 3 , p=4).

in P,P,P, n 4

in P,P,P,

in P,P,P,

n

in P,P,P,

where P,=(1/2+ l/(n+2), 1/2- l/(n+2)) and each P,PjP, means the part in CT of the closed triangle with vertices P,, P, and P,. For each p > l , the subdomain 9, of, 9is defined by

Cf. Fig. 1 and Fig. 2, where gP is shaded. We know from works of Strichartz [7] and Marshall-Straws-Wainger [ l ] that the free solution S(t)$ satisfies the following Lp-Lq estimates. Proposition 0. Let ( l / p , l / q )€ 9and let d = d ( l / p , l / q ) be defined by (1.7) (ix.,(1.7)0or (1.7)J. Then w e have

(1.9)

llS(t)$llq~Cltl-dll~llpfor any

R-W) ,

where C is a positive constant independent of t and $.

As regards the nonlinear term f(w)we require: Assumption. There exist a positive integer k, constants l < p l < and a belt domain g S c 9such that for any ( l / p , l / q )€ak,

. - -< p L

546

K. MOCHIZUKI and T. MOTAI

(1.12) Moreover,

(1.13)

2

c

l l f ( u ) - f ( d l l p ~c u=1 IIlu116y,'+

Il~ll2i'HlU- 41,

if two points (l/p, l/q), (l/p, 1 / ~E@* ) satisfy

(1.14) Under these conditions on f ( w ) we have the following

Theorem 1. ( i ) (Existence of the scattering operator) Suppose that .B'@n 9pl# 0 . For (Up, l/q) E.B'~ n eP, let d = d ( l / p , llq) and V= V(k,q, d) be defined by (1.7) and (1.5), respectively. Then there exists a 6 > 0 with the following properties: If w - ( t ) € Z e nV and Ilw-llv<S, then there exists a unique solution w ( t ) of(1.3) such that w ( t ) € X e nV, llw[lvs(4/3)11w-llv and (1.15)

~ ~ w ( t ) - w - ( t ) / ~ e - + Oas

t+-m

.

Furthermore, there exists a unique solution w + ( t )E 3Fen V of (1.2) such that (1.16)

Ilw(t)-w+(t)Il,-+O

as

t++m

.

The correspondence S : w - ( t ) - w + ( t ) defines the scattering operator. ( i i ) (Uniqueness of solutions) Let B be a convex domain in a k n g p , . Suppose that w - ( t ) € Z e nV(k,q, d) for each (l/p, l/q) € B and IIw-[IV(k.g,d) is bounded in B by a suBciently small constant. Then for any pair of points ( l / p , l/q), (l/fi,I/@) € B , the two solutions w(t) E V= V(k,q, d ) and G ( t )E P= V(k,4, d ) , where d=d(l/fi, l/@),of(1.3) coincide with each other. The proof of this theorem will be given in the next S 2. In $5 3-5 we give applications of this theorem to several concrete problems. The power nonlinearity f ( w ) = , I l ~ l p - ~isw considered in S 3. In this case we obtain the following results: Let r(n) be defined by

+

( n z 3n- 2

+ d ( n 2+3n-

2)2- 8n(n- 1)

if

m=O

if

m>O.

(1.17)

Assume that

n=1-4


Then choosing k = l , pl=

547

.-.=pl=p and S k = ~ where p,l,

(1.19) we cam show that f(w) satisfies the above Assumption and @ p , l n 9 p # 0 . Moreover, if we assume

1()$Ll-ko, 1 4

P

4

P

n

which gives the condition that (l/p, l/q)€.GPpp,k. (3.15) and (3.16) imply the Sobolev embedding Hk,r

CL(PV-l)r'P

and

Hk.; G L ( P v - l ) r ' P

556

K. MOCHIZUKI and T. MOTAI

Q.E.D.

and we obtain (1.13).

S 4. The Sine-Gordon Equation In this section we consider the sine-Gordon equation dtw(t) - Aw(t)

(4.1)

+sin w(t)=0

in ( x , t ) € R"x R, where n 2 2 . We put

(4.2)

f(w)=sinw-w

and write (4.1)in the following form:

+

a;w(t)- Aw(t) w(t)+ f(w(t)) = O

(4.3)

.

Lemma 4.1. For any multi-index a we have

( Q r ( u ) = O if /a1< 3), where ap,a#(u)and ar(u) are smooth functions with bounded derivatives. Proof.

Note that

f(u)=sin u-u= -

\:?

~ ~ ( u ) = ( cu-l)a,u=os (1-0)

f(u) = -

s:

cos (0u)d0u3,

s:

Proof.

s:

,

C O S ( ~ U ) ~ ~ U ~ ~ cos , U -(Ou)dO~a,~aju

(1- 0) cos (0u)dOU ~ ~ : , , U

where a:,=a,a, and a;,,=a,ajak. equality give (4.4).

Lemma 4.2. gknp3+0.

(1-0) cos (8u)d0U2a,U,

Let k, p l ,

These and repeated differentiation of the last Q.E.D.

.. .,pL and B kbe as given in (1.20). Then we have

In case n=2, we have k=l, 1=2 and


557

Thus, the point P,=(3/4, 114) is on the lower boundary {0=3/q-l/p}. In case n 2 3, we have

Obviously, PI=(1/2+l/(n+ l ) , 1/2- l/(n+ 1)) satisfies the first condition 0 1 314- l/p. The second condition is also satisfied since we have {n2-(2k+ l)n-2k}(l+ 2) ---,

and

k n

B-2r-3r-g

Q.E.D.

and (1.13) is proved.

S 5.

Wave Equations with Cubic Convolution

In this section we consider the wave equation (1.1) with the following nonlinearity :

(5.1)

f(w)=(V*Iw12)w

9

where V= V ( x ) is a real function belonging to L’(Rn), z > 1 and n 2 3 , and means the convolution of V a n d [ w I 2 . Lemma 5.1. For z satisfying (1.21), let have

akbe defined by

(1.22).

*

Then w e

@,nP3+0.

Proof. The second inequality of (1.21) is obtained as the condition that 3/q- l/p > 1- l / z on the vertex 6n(n- 1)

6n ’

’ 6(n-

6n

of .9’8. Thus, if (n+1)/4O , and @(x,u)u>O, it is already known that the energy

Received February 15, 1985.

5 84

M. NAKAO

for a solution u decays at the rate ( l + r ) - z / r (exponentially if r=O) as t - + q provided thatf(r) tends to 0 rapidly (cf. [ 5 ] . For more general or related results see Nakao [7, 81, Yamada [13] and Haraux [2].) Quite recently, in [9] we discussed the decay rate of solutions of linear wave equations (possibly higher dimensional) with u ( x , u,)=a(x)u, and P ( x , u)= 0 under the weaker assumption that a ( x ) 2 0 and l/a E L p ( Z ) for some 0 < p < 00. There we proved that E(u(t))5 C,( 1 t ) - ' ~with some constant C,depending on the initial data and k , where k denotes a certain index of the regularity of the solution u. Related results can be found in Russel [ l l ] where the assumption on a ( x ) is implicit and different from ours. See also Iwasaki [3] and Dafermos [l], where it was proved by dynamical method that lim,,,E(u(t))=O if ( u o , ul)€fZ1(Z) x L 2 ( Z )and a ( x ) > O on some open set in Z. The object of this paper is to extend the result of [9] to nonlinear equaUnfortions of the form (0.1) with u(x, u,) like u(x)Iu,jru,, a ( x ) > O , r > - l . tunately our technique is applicable only to equations of spatial dimension 1. For higher dimensional equations further devices and additional assumptions on u ( x , u,) and p ( x , u ) will be required. Here we state the precise hypotheses on u and 1.

+

A,. u ( x , v) is measurable in x E Z for each v E R, continuously differentiable in u € R-{O} for a.e. X E Z and satisfies the conditions: (0.3)

u ( x , ~ ) v ~ a ( x ) j v 1 ~with +~

r > -1 ,

where a ( x ) is a nonnegative measurable function on Z such that (0.4)

a E L"(Z)

and

a-l( .) E Lp(Z)

l/a)

where C( .) is a continuous function on (0, m). A,. !(x, u ) is measurable in x E Z for each u E R, continuously differential in u E R for a.e. x E Z and satisfies the conditions: ( i ) P ( x , u)u>k, $ , " / ( X , ?)d?>O for some k,>O, and ( i i ) for any M>O there exists C ( M )> O such that

A Nonlinear Wave Equation

5 85

Concerning the data (uo,ul, f )we assume A,.

(uo,u , ) € f i 1 n n H , xand ~ , f f W:;:(R+;L2(Z)).

Hereafter we denote by C, and C, various positive constants depending + ] ~ u l ~ ~ ~Also, l , by C we denote on ~ ~ ~ o l l ~llulllL2 l + and ~ ~ u o ~ ~ H ~respectively. generic positive constants. We set

Il~ll,=(\~

l ~ l ~ d x ) ~ ’O ~< , q<w,

and

llullm=esssup ZEI lu(x)l

for a measurable function u on I . For simplicity we write /I I] for 11 /I2. For a nonnegative integer k and l l q < 00 we denote by W:;:(R+;X ) the set of X-valued measurable functions u on R+ satisfying ~ , ( \ ~ ~ ! ~ u ( t ) ~ ~ ~ dfor t ) any ” * O ~

(with usual modification if q=

a). For

convenience we set

and

The following existence theorem is standard and the proof is omitted (cf. Lions & Strauss [4],Nakao [6] etc.).

Theorem 0. Under the hypotheses A,, A, and A, the problem (O.l), (0.2) has a unique solution u such that UE

w : ~ ( R +L2)n ; w : ~ ( R +~ ;, ) n L : , , ( ~ ~ ; ~ , n.i i , )

In what follows we are concerned with the decay property of the solutions in Theorem 0. Now. we shall state our results.

Theorem 1. Assume that r>O and p>2/r. (0.8)

Then, we have

E(u(t))lC0(1+t)-2/‘

provided that

so(?) a , , 6,
O. Then, $(t) has the following decay property: ( i ) i f a > O , 0 = 1 and Iimt-.- (logt)lt'/ag(t)=O, theiz$(t) O , 19< 1 and Iimt-- t(1--8)(1t1'U) g(t)=O, then $(t)< Co'(l+t)-(l-e)/U and (iii) i f a=O, 0 < 1 and g(t)O, where C,' denotes positive constants depending on C,, $(O) and other known constants. 2.

Proofs of Theorems 2 and 3

Starting from the inequality (1.1) in Proposition 1 we shall prove Theorems 2 and 3. First we note that the inequality (1.2) and the assumption on s,(t) imply

(2.1)

E(u(t))+lt O

1

36 0 ( i ) ( 7 f 2 ) ' ( 7 i 1 )

a ( x , u,)u,dxds<E(u(O))+C

I

Next, we consider the differentiated equation

i=l

591

A Nonlinear Wave Equation

where we set U=u, and @(x, u)=(a/au)B(x, u). When - 1< r < 0, d ( x , u,) may have a singularity at u,=O and the argument below is somewhat formal. However it can be made rigorous through appropriate approximate solutions (cf. [lo]). We utilize (2.2) to estimate ~ ~ u t z ( r ) ~ ~ . Proof of Theorem 2 . Assume that p(x, u)-0. from (2.2)

Since d ( x , u , ) 2 0 we have

which implies

Hence we have l ~ u t z ( r ) ~ ~03 ~ Cand 1 < consequently, by (1.1) and (2.1),

+

+

sup E(u(s))IC1{D(r)4p(r+2)’(4*+~~t2) D(j)2(1t1) D ( p + ~ 3 ( t ) ~ ) t_<S_ct+l

I Cl{D(t)v+6(t)2)

or

where we set p=min ( 4 p ( r + 2 ) / ( 4 p + p r + 2 ) , 2 ( r + l ) , r + 2 )

.

It is easily checked that 4p(r+2)/(4p+prf2) ’I= {2(r+ 1)

if ~ > - 2 ( r + l ) / r ( r + 3 ) if rO) to (2.3) we obtain (0.9). Proof of Theorem 3 . At this time we assume that r>O and 2 / r > p 2 2 / ( r + 2 ) (we set 2/r=03 if r=O). Recall that E(u(t))is bounded ( ( 2 . 1 ) ) . Then, by (2.2) we have

M. NAKAO

592

(2.4)

E(u,(r))IE(uc(O))+

s:

<E(ut(O))+C,

u)l Iu,I lu,,Idxds+

IP'(X,

s:

1:

IStu,tldxds

(l+llfcll)./Eo)ds

which together with the boundedness of

s:"

Ilf,(s)llds implies

E(u,(t))lC,(1+t)2.

(2.5)

Therefore, noting that 4 p ( r + 2 ) / ( 4 p + p r + 2 ) < r + 2 < 2 ( r + l ) , (1.1) the difference inequality for E ( u ( t ) ) : (2.6)

sup E(u(s))'+( 2

we can derive from

P7)l4p

t<Slt+l

p>2/(r+2), r 2 0 , and let us derive a sharper estimate than (0.11) under the more restrictive condition A,' on (a/au)/?(x,u). Due to the condition A,' we have, instead of (2.4),

1; 1

E(uc(t))<E(uc(O))+C IE(u,(O))

+c

st

lUla IUCI

Irrttldxds+]t

(E(u(s))"t a ) / 2

Ilfcll

IlUCtlldS

+ llS~ll~ll~c,lld~

and, using the estimate ( O . l l ) , (3.1)

+

E(u,(t))<E(ut(0)) Cl

with ro=(pr+2p-2)l(pr+2).

S:

+

{(I+s)-'o('cn) IlftW ll}ll4t(4lids

Since

$0" Ilf,(s)llds
O). Also we note that the conditions (3.16) are satisfied for all k under our assumptions on &,(() and s(()in Theorem 4 . The proof of Theorem 4 is now complete. 4.

Proof of Theorem 5

Here we assume A," and consider the case: 0 1 r 1 2 and 2 r / ( r 2 + 2 r + 4 ) < p . The proof of Theorem 5 is done in a parallel way to that of Theorem 4 . First we observe (cf. ( 2 . 4 ) , ( 3 . 1 ) ) that

(4.1)

Now, we see

M. NAKAO

594

and moreover

(4.3)

where we have used the assumptions {(r+2)&2}rT -p and u E L", and the fact:

From (4.1)-(4.3) and the boundedness of E(u(t))it follows that

Substituting (4.4) into (1.1) we obtain as is usual

(4.5)

sup E(u(s))'+ (2 tN,(o)]= 1, G , is the random operator corresponding to the kernel (Z,,g)(r, s, 0).(For example, it may suffice to put;

where [ ] A [means ] the operator norm of A as a n operator from L2 into itself.) Therefore, following a n argument similar to that in the proof of Proposition 3, we confirm that for a sufficiently large n ( > max ( N , , N , ) ) the equation (1)' can be rewritten into the following form; x ( t ) = ( C e , f ) ( t ) + a ( B , x ) ( r ) (n>max W1,Nz)) ,

(15) where

).K(f, (16) ( i 1 ( ~ ~ f ) ( r ) = f ( r ) + { ( Z - G ~ ) - ' ( z ~ ( ~ f ) ) } ( l1) - {K*(Z-G=)-l(z~(~f))}(r) 9

1: d=W,

( i i ) (B,x)(t)= B J t , s,

Bn(t,s, o)x(s)ds

s, w ) + K ( t , U U - G J 1 ( Z n ( W - , s, 4 ) ) W -{K,(Z-GG,)-'(Z,(21L(., s, o)))}(t) f

It is important to notice that the kernel B , is of Hilbert-Schmidt type for almost all o and that, limn-IB,-Blzdsdf=O (P-a.s.). Now let a , be such that (a,)-l g Z(B),then the operator (I-a,B) is invertible with probability one and for this value of a the equation (1) has a unique S-solution which we denote by x"(.). This given, we set

$:$:

1:

N,(o)=max i n ; l a o l { \ ~ l ~ , - ~ l ~ s 1/2 d tl~(z-aoB)-l/l>l} }

then as in the case of the operator G,, we see that;

Stochastic Integral Equation of Fredholrn Type

P[(a,)-' E SJw ;B A

Vn

605

> N,(o)l= 1 ,

hence for a sufficiently large n ( > m a x ( N l , N , , N 8 ) ) the equation (1)' can be solved uniquely for each o and we denote the solution by x,. If we put, X J t , ~ ) = l ~ ~ , ~ ~ , ( ~o) ) -(N,=max x,(t, ( N l ,N , , N , ) ) then it is not difficult to see that, s-limndSX,(*)=x"(.) P-as. Thus we have obtained the following: Proposition 4. Under the assumptions ( H , l)', ( H , 2)' and ( H , 3), the following statement holds; For every constant a , such that ( a o ) - Isf ,Z(B), the modified equation (15)'

has a unique solution X,, which has the following properties; ( i ) s-limfi+mX,(.)=x"(.) (P-a.s.) ( ii ) For a sufficiently large n the function X , solves the equation (1)'

References [ 1 ] Ogawa, S., Quelques propri6tts de I'inttgrale stochastique du type noncausal, Japan J. Appl. Math., 1 (1984),405-416.

Kyoto Institute of Technology Matsugasaki, Sakyoku Kyoto 606, Japan


Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 607630 (1986)

An Approach to Constrained Equations and Strange Attractors By H i r o e OKA and Hiroshi KOKUBU Abstract. Because of the very thin nature in their shapes of several strange attractors of ordinary differential equations, it is likely that they are expressed in terms of constrained equations. In order to clarify such a situation, the notion of generalized vector fields is introduced, and their normal forms as well as unfoldings are discussed. Constrained systems are characterized in the class of generalized vector fields. An application is proposed for the case of the Lorenz attractor by a computer simulation. Key words: constrained system, strange attractor, normal form, generalized vector field, infinitesimal deformation, unfolding

Introduction Several strange attractors i n three-dimensional systems of ordinary differential equations, such as t h e Lo r en z attractor (Fig. l), the Rossler attractor,

40

i

20 -

10 t . . . . , , . . . , -10 0

.

.

.

r

10

.

.

. i

20

x-

Figure 1. The Lorenz attractor. (reprinted from [S]) Received July 6, 1985.

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H. OKAand H. KOKUBU

etc., have nearly two-dimensional very thin natures in their shape. Taking notice of this character, some authors regarded them as the ones constrained on adequate surfaces and tried to find out some properties of these strange attractors from this point of view ([Rl], [Tl], [Loz]). In order to make such studies systematically, we extend our object to a class of ordinary differential equations involving the one not solved by its derivatives. Such a differential equation is called a n implicit differential equation or a generalized vector field. In this paper, we begin with the local study of implicit differential equations around a point in the phase space and, especially, we prepare a theory of normal forms and versal unfoldings for them. Our next purpose is to characterize constrained differential equations in the space of implicit differential equations. We give a definition of the constrained system in § 4 and discuss the singular perturbation of constrained systems employing their normal forms and unfoldings. Finally we propose a n example of singularly perturbed equations which exhibits a Lorenz-like strange attractor by numerical experiments. The connection between this system and the Lorenz equation can be explained in terms of degenerate singularities of a (generalized) vector field. This paper is a survey article for our recent works ([OKl], [OK2], [O]). The organization is as follows: S 1. Historical survey and motivations § 2. Definition of generalized vector fields S 3. Normal forms and versal unfoldings for generalized vector fields S 4. Constrained systems and their perturbations S 5. Constrained Lorenz-like attractor § 1.

Historical Survey and Motivations

Investigations of strange attractors by constrained systems seem to have been begun by Takens and Rossler independently. In this section we obscurel y mean a system constrained on a surface by ‘constrained system’ and later we give a definition of it. Takens [Tl] gave a n example of a constrained system as shown in Fig. 2. Though he only illustrated a picture, his constrained system seems to model the Lorenz’ strange attractor [Lor]. In fact, he alluded to the resemblance of the graph of its first return map to that of the Lorenz attractor. Rossler [ R l ] also considered such systems on a double fold type surface (Fig. 3), and in [R2], he proposed four typical constrained systems as prototypes of continuous chaos. One of these systems called ‘the spiral type chaos’ is shown in Fig. 3 (a). Among these prototypes, a constrained system modelling the Lorenz attractor is contained, which is essentially the same as the Takens’ model (Fig. 2), though thz latter is constrained on the cusp surface

Constrained Equations and Strange Attractors

609

, Figure 2. Takens’ constrained model [Tl].

Figure 3. (a) The spiral type chaos proposed by Rossler [R2]. (b) A strange attractor observed in the equation (1.1).

(the Whitney’s pleat). Moreover, Rossler [R2] gave a n explicit form of differential equation as follows:

1

x=-y+ax-bz

p=x+1.1

E i = (1- z”(x+ z ) - EZ

(.=$)

which realizes a spiral type strange attractor by a numerical experiment when ~=0.03,a=0.1 and b= 1.

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H. OKAand H. KOKUBU

We generalize this equation into the following form :

where E is a small parameter (or a small multi-parameter). When E=O, the equation reduces to,

t

X=fo(x, Y ) 0= g o b , Y ) .

For the surface defined by the equation g o ( x , y ) = O , we use the term characteristic surface. The characteristic surface of the equation (1.1) consists of three planes z = i1 and z= - x , and the observed attractor is along the stable part of this surface. (Fig. 3 (b)) As for the constrained model of the Lorenz attractor, neither model indicated above reflects the symmetry of the original Lorenz equation, which is invariant under the transformation, ( x ,Y , 4

-

( - x , -Y,

4.

Recently, Lozi [Loz] proposed another constrained model respecting this symmetry. His model, shown in Fig. 4, has also the cusp type constraint surface.

Figure 4. Lozi’s constrained model of the Lorenz attractor [Loz]. Since constrained models for the Lorenz attractor (or other strange attractors appearing in O.D.E. systems) can be considered as a sort of the idealization of the original attractor, one may expect to use such constrained models for the study of strange attractors. All these constrained models mentioned above, however, illustrate only naive pictures and do not have any analytic expression such as differential equations. The only exception is the Rossler’s equation (1.1). This equation


611

seems suggestive, for it is a n exmaple which can be considered as a constrained model of the spiral type chaos with analytic expression. Though Rossler does not clarify the method of its derivation, and though the Rossler's equation has too special a form to generalize, this example shows that the equation of the form (1.2) may play the role of a constrained model with analytic expression. Our interest is to find out some general method to obtain a n equation of the form (1.2) from a given strange attractor. We note that, as long as the small parameter E is non zero, the equation (1.2) is a system of ordinary differential equations solved by their derivatives, but when E becomes zero, some of the variables are n o longer independent and the equation is not, in general, within the class of ordinary differential equations solved explicitly by the derivatives. Therefore we enlarge the class of differential equations containing both equations of the form (1.2) (especially, the equation with E =0) and autonomous ordinary differential equations solved by the derivatives. In this framework, we try to investigate a general correspondence between constrained models and strange attractors. We consider the differential equations of the following type: (1.3)

A(x)2= V ( X )

x E R"

where A is a matrix-valued function of X, and z, is a vector-valued function. We call it a n implicit differential equation or a generalized vector field on R". The class of implicit differential equations contains both equations of the form (1.2) and ordinary differnetial equations solved by the derivatives. Moreover, by a coordinate change x = $ ( y ) , the equation (1.3) is tranformed to, A ( $ ( Y ) ) .W v ) 3 =V ( $ ( Y ) )

.

Therefore, this class is invariant under the action of coordinate transformations. In the subsequent sections, we concentrate on the local classifications of implicit differential equations and their perturbations. At the end of this section, we remark on the recent work of Benoit [B]. For the equations on R S with the small parameter multiplying one of the derivatives, he gave a definition of trajectories of reduced system with E = O , and obtained a condition for the convergence of trajectories of a singularly perturbed system to those of the reduced one as E tends to 0. Moreover he investigated some topological and analytical properties for trajectories of such equations. In his work, he mainly employed methods of the non-standard analysis. It seems for us that his results may give a useful tool for our study: once we have obtained a constrained model of the form (1.2) of some strange attractor, we could analyse such equations using Benoit's results.

S 2.

Definition of Generalized Vector Fields In this section we give a general coordinate-free description of generalized

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H . OKAand H . KOKUBU

vector fields on a manifold. Throughout this paper, M is an n-dimensional C"-manifold and T M denotes the tangent bundle of M with the bundle projection IT. By X ( M ) , we denote the set of all vector fields on M , that is, the smooth sections of the vector bundle T M . A bundle homomorphism A of the vector bundle T M is the C"-mapping from T M to itself whose restriction on each fiber T,M ( x E M ) is the linear endomorphism A ( x ) of T , M . The set of all bundle homomorphisms of TM is denoted by H O M ( T M ) . Definition 2.1. A generalized vector field on M is the pair ( A , v ) of a bundle homomorphism A of T M and a vector field v on M . Therefore the set of all generalized vector fields on M , which is denoted by @ X ( M ) ,is nothing but the product of HOM ( T M ) and X(M). On a n arbitrarily chosen local chart, we can identify a generalized vector field ( A , v ) with the following local expression:

(2.1)

4'5)* i= v(E)

which is a n implicit differential equation on Rn introduced in S 1. Thus the generalized vector field on M can be considered as a globalization of the implicit differential equation on R". From this fact we define the solution of a generalized vector field as follows: Definition 2.2. Let y be a smooth map from a (possibly, infinite) interval Z to M . We say the map y is a solution of the generalized vector field ( A , v) if y satisfies,

for all t € Z . The above expression makes sense, for dy(t)/dt is a tangent vector of M at y ( t ) . Note that the existence and the uniqueness for solutions of generalized vector fields are not guaranteed in general. We give two examples illustrating each case. Example 2 . 3 . x x = - 1 By putting z = x 2 , this equation reduces to z= -2, so we can easily obtain the explicit solution: z =x2 =x,2 - 21

where x , is the initial condition. From this expression, it is clear that all solutions arrive at the origin, x=O, in finite time, and cannot be extended beyond this time. Especially there is no solution starting the origin at t = O . Fig. 5 (a) exhibits such a configuration of solutions.


613

(a) (b) Figure 5. Configuration spaces of solutions. (b) x i = - x (a) xx=-1 Example 2.4. x k = - x We can also solve this equation explicitly and obtain the configuration of solutions as in Fig. 5 (b). In this case, for each initial condition, there exists a solution which can be extended at infinity. But this equation does not have the uniqueness for solutions at x=O as indicated in Fig. 5 (b).

In the above two examples, the existence and the uniqueness are violated only at x=O. This holds in general; for any generalized vector field ( A , v), the existence and the uniqueness of solutions break only where A is degenerate. Next we consider the notion of equivalence and transformation for generalized vector fields. Let ( A , v) be a generalized vector field on a manifold M . As stated above, we identify the generalized vector field with the differential equation (2.1). Thus it is natural to consider that the transformed equation of (2.1) by a coordinate change C=$(E) is equivalent to (2.1) itself. Moreover multiplying the non-singular matrix valued function P(E) to both sides of (2.1) does not change the equation essentially. Strictly speaking, P is the bundle isomorphism of T M , that is, a n invertible bundle homomorphism. (The set of all bundle isomorphisms on M is denoted by ISOM ( T M ) . ) Consequently we arrive at the following definition of equivalence:

Definition 2.5. Let ( A , V ) and (A’, v’) be generalized vector fields on M . We say these generalized vector fields are equivalent if there exist a bundle isomorphism P of T M and a diffeomorphism $ of M such that,

-

( A’, v’) = (P* T$ * A * ( T$)-’, P T# * u*

holds. The pair (P, (p) is called a fransforrnation of the generalized vector field. In the sequel, we focus our attention on the study of the local structure of generalized vector fields around a n arbitrarily chosen point x, in M . Expanding a generalized vector field into the Taylor series at xo, we change it, up to order k, by transformations of the generalized vector field as simply as possible. Such simplified generalized vector field is called a k-th order normal form for generalized vector field. T o obtain all k-th order normal forms for generalized

614

H. OKAand H. KOKLJBLJ

vector fields can be interpreted as the local classification of them up to order k .

S 3.

Normal Forms and Versa1 Unfoldings for Generalized Vector Fields

This section is devoted to proposing a general framework of the normal form theory for generalized vector fields and to give several examples. For the results obtained in this section, we refer to [OK2]. First we briefly review the method to obtain normal forms for (ordinary) vector fields. The normal form theory for vector fields has been developed by PoincarB, Siegel, Sternberg, Arnold [A2], Takens [T2], Ushiki [U], and others, We mainly follow Ushiki’s method. Let v be a vector field on M . Our interest is to know how the vector field v changes by a given diffeomorphism q5. For this purpose, taking a one-parameter family q5t of diffeomorphisms connecting the identity at t=O and $ a t t = l , we investigate the way of deformation &v of v in terms of a differential equation on the space Z ( M ) of vector fields. Recall that every vector field Y o n M generates a diffeomorphism as the time-one-mapping of the flow defined by Y, that is, q5=exp Y . We call Y the infinitesimal generator of q5. In this situation we consider the following one-parameter family of diffeomorphisms, @=exp r Y , and deform a given vector field v by @. Then a formula in differential geometry gives,

where [ , ] denotes the Lie bracket for vector fields. The left hand side of (3.1) is called the infinitesimal deformation of v by Y. Since {V}forms a oneparameter group of diffeomorphisms, (3.1) defines a differential equation on Z ( M ) , that is, (3.2) which describes the way of deformation of v. Integrating this equation Conversely, in from t = O to t= 1, we obtain the transformed vector field &v. order to simplify some terms of v, we have only to find appropriate infinitesimal generators and to solve the equation (3.2). This is the idea of the normal form theory for vector fields. Especially the Jordan normal form theory for

Constrained Equations and Strange Atrractors

615

matrices is nothing but the first order normal form theory for vector fields. For details and the practical method of computation, see Ushiki [U]. The normal form theory for generalized vector fields is essentially the same but slightly more complicated. As the first stage, we consider the theory of the leading order. Let ( A , v) be a generalized vector field on M and let x , be a point in M . We consider the following two cases: ( i 1 v(xo)+O, ( i i ) v(xo)=O. 3-1. When v(x,)#O, by fixing a n arbitrary local chart, we can consider the pair, ( A , ; ~,)‘(A(Xo), v(x0))

of a matrix and a vector where A ( x , ) is the matrix obtained by the restriction of A on the tangent space of M at x , . Such a pair ( A , ; v,) is identified with a n n x (n+ 1)-matrix and we call it a n extended matrix. The equivalence for generalized vector fields induces a n equivalence for extended matrices. Definition 3.1. Extended matrices ( A , ; v,) and (A,’; v,’) are equivalent if there exist non-singular matrices P, Q € GL(n,R) of order n such that, (A,’; v,’)=P.(A,; v,).Q

where Q denotes the non-singular matrix,

of order n+ 1. The classification of extended matrices with respect to the equivalence is very simple: we obtain the following theorem by a n easy linear algebra argument. Theorem 3.2.

Every extended matrix ( A , , v,) € M ( n , n+ 1; R) (where v,#O)

is equivalent to one of the following two forms:

where r is the rank of A,,

ek=Yl,O, - . . , O ) E R k

( k = n - r or r )

H. OKAand H. KOKUBU

616

and Z, is the unit matrix of order r. 3-2. We consider the case v(xo)=O: the point x, is called a n equilibrium point of the generalized vector field ( A , v). In this case, the linearization of ( A , v) at x, is naturally defined as,

Mxo), W

X O ) )

where Dv(x,) is the Jacobian matrix of v at xo. This yields the following definition.

Definition 3.3. An (n-dimensional) linear generalized vector field is a pair of two real square matrices of order n. It is convenient to identify a linear singular vector field ( A , B ) with the expression AA+B by introducing a parameter 1. We call such a n expression a matrix pencil of order n. ?en (n, R) denotes the set of all matrix pencils of order n. Clearly, Pen (n, R ) = M ( n , R) x M(n, R) where M(n, R) is the set of all real matrices. We define a n equivalence relation among matrix pencils similarly to the previous sub-section 3-1.

Definition 3.4. We say two matrix pencils AA+B and A’A+B’ are equivalent if and only if there exist non-singular matrices P and Q of order n such that the relation,

A’A+B’=P* (AA+B)* Q-l , holds as a polynomial of A. We divide matrix pencils into the following types:

Definition 3.5. ( i ) A matrix pencil AA+B is non-singular if A is nonsingular, that is, det A f O . Otherwise we say A1+B is singular. ( i i ) A singular matrix pencil AR+B is non-degenerate if det ( A J + B ) f O as a polynomial of 1. Otherwise we say AA+B is degenerate. Note that a non-singular matrix pencil A + B is equivalent to, Z,J+

Q(A-’B)Q-l

by choosing P=QA-’ in the above definition 3.4, where I,, is the unit matrix of order n. Moreover, by a suitable choice of Q , it reduces to,

where J is the Jordan normal form of A-lB.

Therefore the equivalence


617

classes of non-singular matrix pencils have one-to-one correspondence with the Jordan normal forms of matrices. So the classification of matrix pencils is a generalization of the Jordan normal form theory for matrices. The classification of matrix pencils in the above sense was completely done by Weierstrass and Kronecker with the help of elementary divisors. We only state the result in a rough manner. For the proof and details, see Gantmacher [GI. Theorem 3.6 ([GI). Every matrix pencil is equivalent to a block-diagonal matrix pencil, each block of which has one of the following forms: ( i ) Z,A+J,(c) where J J c ) is a Jordan cell of order m of Jordan normal form with the eigenvalue c. W e denote ImA+Jm(c)by the symbol c".

m

We assign it the symbol pm.

The symbol is

(iv) The transpose of (v)

-

(O)]g

E

~

.

E~

.

The symbol is 7".

The symbol is (9,h ) .

h

W e call this block-diagonal matrix pencil the Weierstrass-Kronecker normal form (abbrev. W-K normal f o r m ) of a matrix pencil. Moreover the W-K normal form is uniquely determined up to the order of disposition of blocks. In order to represent a W-K normal form, we use the symbols indicated above; e.g. the symbol (0, 1).7,P-O signifies the matrix pencil,

We can easily count up all equivalence classes of matrix pencils of fixed order n by combining these symbols. As a n example we show, in Table 6, the classification of matrix pencils of order 2 except for non-singular ones.

H. OKAand H. KOKUBU

618

Table 6. Classification of the singular matrix pencils of order 2. Degenerate

Non-degenerate Norma' form

Symbol

Codim.

Normal form

Symbol

Codim.

2

2 4

3-3. In the above two sub-sections, we considered the normal form theory for generalized vector fields of the leading order both at a n equilibrium point and at a non-equilibrium point. Here we proceed to the higher order normal form problem, for which we adopt the same strategy as that for (ordinary) vector fields: we take a one-parameter group of transformations for a generalized vector field generated by a n infinitesimal generator, and compute the infinitesimal deformation. Let ( A , u ) be a generalized vector field on M , that is, A E HOM(TM) and u E Z ( M ) . Let (P, $) be a transformation of the generalized vector field, where P EISOM(TM), the set of all bundle isomorphisms of T M and $ E Diff ( M ) , the set of all diffeomorphisms of M . (P,$)#(A,u ) denotes the transformed generalized vector field of ( A , u ) by (P,$), that is,

(3.3)

(P,$ ) $ ( A ,u)=(PoT$oAoT$-l, PoT$ouo$-')

.

This means that the product group of ISOM(TM) and Diff(M) acts on HOM(TM) x Z ( M ) ,where (3.3) induces the group structure as follows:

(P.$ ) - ( Q +)=(PoT$oQoT$-', , $04), and yields the semi-direct product group ISOM(TM) >a Diff ( M ) . Thus the following proposition is obtained:

Proposition 3.7. The semi-direct product group ISOM( T M )>a Diff ( M ) acts on 8 X ( M ) in the manner as ( 3 . 3 ) . We define the exponential mapping from @)3E(M)to ISOM(TM)>aDiff ( M ) . To begin with, we prepare several notations. Let ISOM(TM, T M ) be the set


619

of all pairs ( F , f) where F is a smooth invertible bundle mapping of TM and f is a diffeomorphism of M satisfying, mF=fon,

(n is the bundle projection of T M )

.

ISOM(TM, T M ) forms a group under composition, and there is a group isomorphism, u:

ISOM(TM)>aDiff( M )+ ISOM(TM, T M ) ,

defined by,

(P,4)

+

(POT474 ) .

On the other hand, we can identify any bundle homomorphism R E HOM(TM) with a vector field on TM in terms of the following local coordinate representation : (x, E ) 3 (x,0,E, R(x1.E) .

(Here the bundle projection of T ( T M ) is Tn.) We define the mapping, K:

HOM(TM) + 5 ( T M )

by this identification and denote the image K ( R )of R by YE 5 ( M ) , T Y is a vector field on T M , that is, TnoTY=T(no Y)=T(id,)=Id,,

K,.

Since, for any

,

we can consider the sum K,+TY. For any t c R , the exponential mapping of this vector field on TM defines a n element of ISOM(TM, T M ) , which is denoted by exp t(r,+TY). Using above notations, we define the exponential mapping for generalized vector fields. Definition 3.8.

The exponential mapping of ( R , Y) E @ 5 ( M )is defined by, exp t ( R , Y)=u-loexp t ( r , + T Y ) ,

for sufficiently small t . The infinitesimal deformation of the generalized vector field ( A , U) is, thus, given by,

d ( exp t ( ~ Y, W , dt

u) .

t=D

To compute the infinitesimal deformation, we identify the bundle homom orphism A with a (1, 1)-type tensor field through the natural vector bundle isomorphism,

620

H. OKAand H. KOKUBU

Horn ( T M )N T*M@ TM

,

where Hom ( T M ) is the vector bundle over M whose fiber at x E M is the vector space Horn ( T , M , T , M ) , the set of all linear maps from T,M into itself. The next lemma is a fundamental result in differential geometry.

Lemma 3.9.

where @=exp t Y and - P Y A denotes the Lie derivative of the tensor field A with respect to the vector field Y . From this lemma and a n easy calculation, the infinitesimal deformation is obtained as follows:

Theorem 3.10. exp t(R, Y ) # ( A v, ) = ( R o A - P Y A , Rev-[ Y , v]) . With a local coordinate representation,

a

where R = C Ri,-@dxj,

axi

A=

a C Aij--@dxj, ax,

Y=

C

Yi-

d

8%

and

v=C vt-. d

ax,

Using this theorem, we can calculate normal forms for generalized vector fields after Ushiki’s method to obtain those for vector fields. Since computations for individual cases are complicated, and various types of degeneracy occur for non-linear terms, we cannot exhaust all possible normal forms with the specified leading terms. We give below several examples, each of which has some type of the W-K normal form as its leading terms. These examples deal with most nondegenerate cases in their non-linear terms, called generic normal forms.

Proposition 3.11.

where a is a constant.

The generic normal form of order 2 for

,uzis given

by,


Proposition 3.12. symmetry,

62 1

The generic normal form of order 3 for p 2 - 0 with the

( x ,Y , z )

-

(-x, -y, z)

is given by, (az+ O ( ? ) ) x = y i x z + p x 3 j=X

-txZ+bz2+qz3

Z=

where O(2) denotes terms of order 2 and a, b, p , q are constants.

Proposition 3.13. The generic normal forms of order 3 for (0, 1).$.0 with the same symmetry as that of Prop. 3.12 is given by, (az+ 0 ( 2 ) ) x = y i x z + p x 3 j=bxz+qx3

(3.4)

z = ~ x ~ ~ z ~. + s z ~ 3-4. The final subject in this section is the perturbation of generalized vector fields. For any given generalized vector field ( A , u), we consider a smooth family ( A 2 ,vl) parametrized by 1~ R*. We call the germ of the family ( A 2 ,vl)at 0 E RXa k-parameter unfolding of ( A , v) if ( A o ,v o ) = ( A ,v). Our interest is to obtain a n unfolding of ( A , u ) which contains all possible types of perturbations of ( A , v), which is called a versal unfolding. Definition 3.14. A n unfolding ( A 2 ,v2) (1 G Rk)of ( A , v) is called versal if, for any unfolding (Ap’,up’) ( p E RL)of ( A , v), there exist a mapping 1=B(p) and a family of transformations ( P 2 ,$ 2 ) parametrized by 1such that the followings hold:

‘W)=O ( 4 ’ 7

Up’)

(Po,$o)=(IdTM, id,) , and , for any p E R’ $scp, )#(&p) vs(p)1

= (Peep, 9

9

9

.

In other words, any unfolding of ( A , u ) is derived from the versal unfolding of ( A , u).

Of course, the versal unfolding for a given ( A , u ) is not unique. We say a versal unfolding of ( A , v) is miniversal if the number of parameters is minimal among all versal unfoldings of ( A . u ) and we call the number the codimension of ( A , u). The notion of versal unfolding for matrices was first introduced by Arnold [ A l l . We can construct a miniversal unfolding of a matrix A in the following way. The set of all matrices conjugate to A is called the orbit of A , which is a submanifold of M(n, R). A versal unfolding of A is given as the germ of a family transversal to the orbit at A . We introduce a natural inner product into M ( n , R) given by,

622

H. OKAand H. KOKUBU

=tr (AsCB).

Then a miniversal unfolding is obtained by taking one complementary (say, for example, perpendicular with respect to the inner product defined above) to the tangent space of the orbit at A . All such considerations work for extended matrices (3-l), matrix pencils (3-2) and even for generalized vector fields (3-3) as they are. Introducing appropriate inner products, we can show the following: Theorem 3.15. A miniversal unfolding of an extended matrix ( A o ;v,) is given by ( A o ;v , ) + ( G ; u ) where G and u satisfy, A , . t G + v o ~ t u = O , and c G . A , = O .

Corollary 3.16. Miniversal unfoldings of normal forms of extended matrices given in Theorem 3.2 are,

0 0 . 0 0 1,' eT where A=(Atj) and

Theorem 3.17.

Y=(Y*)

sre unfolding parameters.

A miniversal unfolding of a matrix pencil AA+ B is given by,

where G and H satisfy, G . t A + H - t B = O , and t A . G + t B - H = O .

Corollary 3.18. Miniversal unfoldings for ,urnand ( 0 , ~)-Y,I"-' are given by the following ones, respectively:


623

where a , and ,E, are parameters. Especially the codimension of these matrix pencils is m and m+ 1, respectively.

From the corollary 3.18, we can easily obtain the bifurcation diagrams of matrix pencils. As a n example, we show the bifurcation aspects of (0, l).vl. The codimension of the matrix pencil (0, 1)-$ is 3. Hence all possible perturbations of (0, l).yl are contained in a 3-dimensional space indicated in Fig. 7.

Figure 7. Bifurcation diagram of versal families for the matrix pencil (0, I).$.

We can obtain versal unfoldings for generalized vector fields in a similar fashion. Here we do not go into details but give only one example. We consider the unfoldings ( A 2 ,u2) of the normal form (3.4) given in Proposition 3.13 satisfying the following conditions: ( 1 ) v,(O)=O for any 1, ( 2 ) ( A 2 ,v,) has the symmetry of Proposition 3.13, ( 3 ) A , [resp. v,] is a t most of order 2 [resp. 31.

Proposition 3.19. for (3.4) is given by,

In the above category of unfoldings, the versal unfolding

i

(E

(3.5) where

E,

a , /3 and

+az+ O(2))i=ax +y kx z + p x S 9 =,EX +bxz+ qxS

r are parameters.

i =y z i xz iz2+

SZS

H. OKAand H. KOKUBU

624

S 4.

Constrained Systems and Their Perturbations Let us recall the equation (1.2) of

S 1,

(4.1)

Within the framework of generalized vector fields, we shall completely characterize such equations including the case of E = O . A bundle homomorphism A of TM is called of constant rank if, for any x E M , the rank of A(x) is independent of x. In this case, we say A is of corank r if the rank of A equals n - r where n is the dimension of M and r is a non-negative integer.

Definition 4.1. A constrained system of corank r on M is the pair ( A , v) of a bundle homomorphism A of T M of corank r and a vector field v on M . A constrained system on M is a constrained system of corank r for a n integer r . We denote the set of all constrained systems [resp. of corank r ] on M by B E ( M ) [resp.&X('j(M)]. The set & f ( M ) is a subset of @ f ( M ) . It is easy to see that, for any constrained system ( A , u ) of corank r and any transformation (P,$), the transformed system (P, $ ) # ( A ,v) is again a constrained system of corank r . Thus the group ISOM(TM)>aDiff ( M ) acts on the space Q5(rj(M) (and, as the result, & f ( M ) ) . Especially, applying the map, GE'O'(M)+ f ( M ) ,

( A , v)

-

A-'v

,

we can show that the category of & f ( O ) ( M and ) the action of ISOM(TM)>a Diff ( M ) to it is equivalent to the category of the space Z ( M ) and the action of Diff ( M ) to it by the coordinate transformation. In other words, the constrained system is a natural extension of the vector field. In this framework, the equation (4.1) can be identified with a family ( A s ,u,) of constrained systems parametrized by E satisfying ( A o ,u,) G QX("(M), that is, a n unfolding of a constrained system of corank r . This fact leads us to the following idea: a singularly perturbed family of vector fields is nothing but a n unfolding of a constrained system. At the end of this section, we analyse several examples of perturbation problems from this point of view. We can also define the characteristic surface for our constrained systems.

Definition 4.2. characteristic surface

Let ( A , u ) be a constrained system of corank r on M . The 9of ( A , u ) is given by, 9= {xE M I u(x) E Im A(x)} ,

where ImA(x) is a linear subspace of T,M consisting of all images of linear endomorphism A(x) of T,M.

A standard transversality argument shows the next proposition:


625

Proposition 4.3. For any generic ( A , v) in G X ( 7 ) ( M ) ,the characteristic surface of ( A , v) is a smooth submanifold of M of codimension r. The normal form problem for constrained systems is solved by reducing it to that for generalized vector fields. Let ( A , u ) be a constrained system of corank r. Similarly to S 3, we put,

('40;vo)=(A(x,), U ( X 0 ) )

7

for a n arbitrarily chosen point x, in M , and we call ( A o ;v,) the leading part of ( A , v) a t x,. Since ( A , u ) is o f corank r , A , is a linear map of rank n-r. If u,fO, the leading part of ( A , v) is classified as in Theorem 3.2. For simplicity we give results only for the two dimensional case.

Theorem 4.4. Suppose that ( A , v) is a constrained system on a two dimensional manifold M and that its leading part (A,; u,) at x, 6 M is equivalent to

(:

:;3.

Then, the infinite order normal form of ( A , v) is given by

(4.2) In other words, every finite order parts of ( A , u ) at xo except for the leading part is eliminated by suitable transformations. This result corresponds to a formal version of the rectification theorem for vector fields. For the case (ii) of Theorem 3.2, there exist various kinds of normal forms according to the degeneracy of higher order terms. We give below a partial result.

Theorem 4.5. Let ( A , v) be a two-dimensional constrained system of corank 1 whose leading part ( A o ,v,) is equivalent to

(:

;;9 .

Then the first order normal form of ( A , v) is one of the following:

H. OKAand H. KOKUBU

626

Moreover the case (i) is also the infinite order normal form. For the case (ii), its generic second order normal form is given by, iy+ax2

(4.4)

where a is a constant. Its characteristic surface is the parabola given by y= *ax2. In contrast with the case of generalized vector fields, we cannot obtain, a t present, versal unfoldings for constrained systems in a general manner. The main difficulty is that the space & X ( M )is neither a vector space nor a manifold. We give below a n example of versal unfolding of a constrained system, which is based on a cumbersome computation for this special case.

Proposition 4.6.

The versal unoldifng of (4.2) is given by,

3 (3

(;I

(4.5)

9

where E is an unfolding parameter. This versal unfolding can be expressed in terms of differential equations as follows:

t

(4.5)

ER= 1

9=0.

As well as the above system, we consider the following, which are unfoldings of (4.3) and (4.4), respectively:

(4.7)

t t

€X=a-rX j=1

+

ER=a +y+ /3x ax2

Y=l-rx

where E , a and /3 are parameters. Each of them describes a typical local orbit structure of the equation of the form (4.1). For instance, the phase portrait of the well-known Van der Pol equation, Ejt-(l-x2).R+X=O

,

is given in Figure 8, in which the local orbit structures around the points A , B and C correspond to the phase portraits of the equations (4.9, (4.6) and (4.7), respectively.


627

Ty

&-

I

Figure 8. An illustration of the phase portrait of the Van der Pol equation for small positive E .

S 5.

Constrained Lorenz-Like Attractor

The last section of this paper is a n application of our theory of normal forms and their unfoldings for constrained systems. As we have mentioned in S 1, the equations [Lor], (5.1)

i=-ux+uy,

P=rx-y-xz,

i=-bz+xy,

have a nearly two-dimensional strange attractor, called the Lorenz attractor (Fig. 1) for some values of parameters u, r and b. We have proposed, in [OKl], the following differential equation: E X =y

+a x z -

xs

3 =A x + B y + pxz

(5.2)

i =cZ+w ,

as a constrained model of the Lorenz attractor with analytic expression. In fact, the equations (5.2) exhibit a Lorenz-like strange attractor in a computer simulation; when parameter values are, E:

0.03, A : 0.7, B: 0.7, C: -1.0, a: 0 . 7 , p: - 1 . 0 , 7: 1.0,

628

H. OKAand H. KOKUBU

Figure 9. A strange attractor observed in the equation (5.2). These two figures exhibit the same attractor from different directions.

Figure 10. The graph of the Lorenz plot of the attractor of Fig. 9. the strange attractor indicated in Fig. 9 is observed. We have also taken the Lorenz plot, that is, the successive local maxima in the z-coordinates of the numerical data. The graph of the Lorenz plot is shown in Fig. 10. See [OK11 for more precise data of numerical experiments. We have derived the equation (5.2) somewhat heuristically in [OKl]. Here we shall give a n explanation for the reason why the equation (5.2) and the Lorenz equation seem to have similar attractors in a computer simulation.


629

The equation (5.2) can be transformed into, E X =BEX+ Y + a X Z -

[

x3

P =A X + ( p - B a ) X Z + i=cz+y x 2 ,

BXS

by the linear change of variables, X=x,

Y=~-BEx,

Z=z.

We can show that this system is a n unfolding of the generic third order normal form of the constrained system with symmetry whose leading part is (0, 1).$.0. On the other hand, Ushiki and we [UOK] have shown that the Lorenz equation can be regarded as a subfamily of the following system of ordinary differential equations,

1

x=y 9 =A’x+ B ’ y i x z + a’yz+px3 2 = C’Zl x2+ p’z2+sz3 .

+qx2y+ ryz2

This is the versa1 unfolding of the generic third order normal form [U] of the ordinary vector field whose linear part is conjugate to the Jordan normal form,

iH B 9. The linear vector field of the above type corresponds to the matrix pencil 02-0,that is,

k : $+k : :I. 1 0 0

0 1 0

As is illustrated in Fig. 7, and easily checked by simple calculation, the matrix pencil (0, 1 ) - $ - 0 is contained in the closure of the orbit of the matrix pencil 02.0. In other words, the linear singular vector field of type (0, l ) . v l - O is a (singular) limit of the linear vector field of type 02.0, which gives a relationship between the equation (5.2) and the Lorenz equation. We have shown in [UOK] that some degenerate singularities of vector fields can be considered as ‘organizing centers’ of strange attractors: we have proved that a strange attractor which is qualitatively the same as the Lorenz attractor is observed in an arbitrarily small perturbation of a degenerate singularity of the vector field of type 02.0. It seems, therefore, for us that the above mentioned relation of the equation (5.2) and the Lorenz equation

H. OKAand H. KOKUBU

630

m a y provide a clue t o finding o u t a general correspondence between strange attractors i n systems of ordinary differential equations a n d their constrained models.

Acknowledgement. W e express our sincere gratitude t o Professors M a s a y a Yamaguti a n d Shigehiro Ushiki for their valuable advice a n d encouragement. References [All

V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys,

[A21

26 (1971), 29-43. __ , Lectures on bifurcation in versa1 families, Russian Math. Surveys, 27

(1972), 54-123. [B] [GI [Lor] [Loz] [O] [OK11

E. Benoit, Thesis, UniversitC de Nice, 1984.

F. R. Gantmacher, The Theory of Matrices, Vol. 2, New York, (1959). E. N. Lorenz, Deterministic nonperiodic flow, J. Atom. Sci., 20 (1963), 130-141. R. Lozi, Thesis, Universith de Nice, 1983. H. Oka, in preparation. H. Oka and H. Kokubu, Constrained Lorenz-like attractors, Japan J. Appl. Math., 2 (1985), 495-500. in preparation. [OK21 -, [Rl] 0. E. Rossler, Chaotic behavior in simple reaction systems, Z. Naturforsch., 31a (1976), 259-264. , Continuous chaos, New York Acad. Sci., 316 (1976), 376-392. [R21 [S] C. Sparrow, The Lorenz Equation: Bifurcations, Chaos and Strange Attractors, Appl. Math. Sci., Vol. 41, Springer-Verlag, 1982. [Tl] F. Takens, Implicit differential equation: some open problems, Lecture Notes in Math., 535, Springer-Verlag, (1976), 237-253. [T21 , Singularities of vector fields, Publ. Math. I.H.E.S., 43 (1973), 47-100. S. Ushiki, Normal forms for singularities of vector fields, Japan J. Appl. Math., [U] 1 (1984), 1-37. [UOK] S. Ushiki, H. Oka and H. Kokubu, Existence d’attracteurs &ranges d a m le dtploiement d’une singularitt dtgCnCrte d’un champ de vecteurs invariant par translation, C. R. Acad. Sci. Paris, 298, SCr. I, (1984), 39-42. ~

~

Department of Mathematics Kyoto University Kyoto 606, Japan


On the Existence of Progressive Waves in the Flow of Perfect Fluid around a Circle By H i s a s h i OKAMOTO~ and M a y u m i S H ~ J I Abstract. We consider a free boundary problem for incompressible perfect fluid with surface tension. The problem to be considered is as follows: A perfect fluid is circulating around a circle r (see Fig. 1). The outward curve r is a free boundary to be sought. We assume that the flow, which is confined between r and 7, is irrotational. On the free boundary, surface tension works and makes the free boundary circular. On the other hand, the centrifugal force caused by the circulation of the flow makes the fluid go outward. Hence the balance of these two kinds of forces determines the geometrical properties of the free boundary. We show that there exist progressive waves, which are periodic motions of the fluid and are the exact solutions corresponding to the solitary waves.

Fig. 1. Key words:

S 1.

free boundary, bifurcation, progressive wave, surface tension

Introduction

W e consider a nonstationary flow of perfect fluid with a free boundary a r o u n d a circle. T h e problem t o b e considered here can be regarded a s a model f o r a flow ar o u n d a celestial body. W e consider a plane through an equator of a celestial body a n d a two-dimensional flow i n this plane. W e assume that f o r a fixed time t t h e flow region is enclosed by tw o closed Jorda n curves and r(t). is an equator of a celestial body a n d ~ ( t is) a free

r

r

Received April 2, 1985. Revised July 20, 1985. + . . . Partially supported by the FGjukai.

H. OKAMOTO and M. SHOJI

632

boundary which is outside r. These two curves enclose a doubly connected domain, which is denoted by Qr(,,. Then the fluid lies in Q,(,, (see Fig. 1). For simplicity we assume that the inward curve r is the unit circle in the plane. We also assume that r ( t ) is represented as r ( t ) = { ( r , 0); r = r ( t , 0 ) } in terms of a function r = y ( t , 0 ) and the polar coordinates ( r , 0). Then the problem is formulated as follows. Problem. Find a time-dependent closed Jordan curve r ( t ) ,a stream function V = V(t,r , 0 ) and the pressure P(t, r , 0 ) satisfying the conditions (1.1-8)

-0

in

QT,,

,

--

Here u, g and w o are prescribed positive constants. A is the Laplace operator with respect to ( r , 0 ) . Krc,) is the curvature of r ( t ) , the sign of which is chosen to be positive if it is convex. For the physical meaning of this problem and derivation of these equations, see [7] or [8]. We only note that V is a stream function for the flow, i.e., the velocity vector is given by (aV/ay, -aV/ax) and that (1.4, 5) is the Euler equation written in terms of V. Our purpose in this paper is to show the existence of progressive wave solutions, i.e., solutions of the following form 7=7(0-~ct)

,

V= F(r, O-cr)

,

where c is a constant. When c=O, this solution reduces to a stationary solution, which is studied in [7]. We can easily find a simple stationary solution given by

Progressive Waves in the Flow of Perfect Fluid

633

where r , > 1 and a > O are constants (see S 2 below). This is a radially symmetric solution and we call it a trivial solution. We will prove that there exist progressive wave solutions in a neighborhood of the trivial solution. The propagation speed c is determined by the wave number and the magnitude of circulation. The important consequence of this result is that the trivial solution is not asymptotically stable, since the progressive wave solution is a time periodic solution. This fact constitutes a striking contrast to the case of viscous fluid (see Beale 111). Indeed, in 111, Beale considers viscous incompressible fluid flow above a plane-like bottom with finite depth (a model of flow in the ocean). He proves that the rest state (motionless fluid with a horizontal plane as a free surface) is asymptotically stable by virtue of the viscous and capillary forces. Furthermore the rate of convergence to the motionless state is proved to be O(t-''2) (see, Beale and Nishida [13]). We also show that if the circulation of the flow is zero, there is no stationary solution other than one in which the free boundary is a circle. This fact is worthy of notice because even if the circulation is zero there are progressive wave solutions which are not circles. Finally we remark that we show the existence of the progressivz waves by the bifurcation theory. In utilizing the theory, we take the propagation speed as a bifurcation parameter. This fact provides for distinction between our problem and the problem for flows in a n infinite domain over the straight line or plane (i.e., the ocean problem). Indeed, the wave number ranges over all the real numbers in the case of the ocean problem. On the contrary, in our problem, the wave number can take only the integer multiples of 2n. This makes the spectrum discrete, whence we can use the bifurcation theory. Our problem differs from the dissipative system of evolution equations like reaction-diffusion equations defined on R. In some dissipative system, the existence of the progressive wave solution is known but its propagation speed is uniquely determined by the system. Therefore such progressive waves do not fall into the framework of the bifurcation theory (see 14, 61). This paper is composed of four sections. In section 2 we reformulate the problem and give a precise version of the theorem. Section 3 is devoted to the proof of the theorem concerning the existence of progressive waves. In section 4 we show the uniqueness theorem for the stationary flow when the fluid does not move but only the surface tension works on the free surface. Finally we give a derivation of the formula used in S 3 concerning variation of domain. Acknowledgment. The authors wish to express their hearty thanks to Prof. H. Fujii who kindly read the original version of the manuscript and gave us much useful comment and encouragement.

634

S 2.

H. OKAMOTO and M. S H ~ J I

Existence of Progressive Wave

We begin with the fact that there is a trivial stationary solution in which the free boundary is a circle. The stationary problem is to find a closed Jordan curve and a function V such that the conditions (2.1-5) below are satisfied: (2.1)

AV=O

in

Q,,

(2.2)

V=O

on

r,

(2.3)

V=a

on

r,

(2.4)

1

2 /VV12-x+cKr=constant r

,

on

r

Here Qr is a doubly connected domain bounded by and 7. Note that in the stationary problem the condition (1.3) reads that V is constant on r. So we denote the constant by a. The constant a represents the magnitude of the circulation. The case of a=O corresponds to the case where the circulation vanishes. If a=O, then V=O by virtue of (2.1-3). Hence, in this case, the fluid does not move anywhere. The conditions (1.4-6) reduce to (2.4), which is known as the Bernoulli equality. We now define r , > l by l r ( r o 2 - l ) = u 0 . Let To be a circle of radius ro with the origin as its center. We put V,(= V,(r))= (a/log r,,) log r for 1< r < r,. Then {rn,V,} is a stationary solution, i.e., satisfies (2.1-5). Our aim is to prove the existence of progressive wave solutions. We exclusively consider the solutions near the trivial solution. Hence what we really do is to show the existence of functions U E C3+"(S1)and V € C3+"(Dr) such that r is given by y ( t , O)=r,+-u(O-ct). Here the symbol C3+"implies the Holder space. These functions are governed by (1.1-8). If we introduce a new variable $=O-ct and if we note that a/& is replaced by -c(a/&b), then these governing equations are, in the present case, expressed as follows: r-

(2.8) (2.9)

:r

(r-Vr: )+-=O :T

in

a,,


635

(2.10)

P=aKr

(2.11)

on

r,

(2.12)

The equation (2.10) is equivalent to saying that cr(aV/ar)+(l/2)IVV I2+P--g/r does not depend on q5. On the other hand, by (2.6), the equation (2.9) is rewritten as follows: c-

:(

T)+-

a 1 -lVVlz+P-ar(2

r-

r

>-

-0

in Qr

.

Hence we see that ~ r ( a V / a r ) + ( l / 2 ) 1 V V ~ ~ + P -does g / r not depend on r . fore the condition (2.9-11) is reduced to the equation below: (2.13)

av

1

cr-+-lVV12+aK,-x=constant dr 2 r

on

There-

r.

Next we consider the equation (2.8), which is rewritten as

Hence we have V(7(q5),$)= -(~/2)r($)~+constant. Now we can reformulate the problem. We first define symbols:

ru: a

closed Jordan curve represented by (r,+u($), $)

Qu={(r,$)€R2; lO. (2.1-5) other than y = ~ , , ,

Our

Then there is no solution to

V=O.

Remark. Here we do not assume that y lies near r0. We only assume that y is a C*-curve. The assumption that g > O is indispensable. In fact, if


641

g < 0 , existence of nontrivial solutions can be proved by the bifurcation theory.

Proof of Theorem 2. Let {r, V } be a solution. Then, by (2.1-3), we see that V vanishes identically in 9,. Therefore oKr=g+E r

(4.1)

on

r,

where 6 is a constant. We have only to show that r is a circle. We first prove the case where o < g . Let A (resp. B ) be a point on which has the largest (resp. smallest) distance from the origin. Then we have

9 - = a -~ 9, ( and ~ )O A Z O B . OA

OB

By the definition of A and B, it holds that K , ( A ) Z 1/OA and that K , ( B ) S l / O B . Therefore we obtain a/OA 5 aK,(A)=g/OA-gglOB+aK,(B) Z g / O A - g / O B + a/OB. By this inequality and the assumption a < g , we obtain O A S O B , which implies that r is a circle. We now prove the case where a z g . We assume for simplicity that the curve r is represented by the equation r = r ( O ) . Then by (2.5) we have

It is known that total (oriented) curvature of a closed Jordan curve is equal to 2n, i.e.,

\r

Kr(r(O)'+ ~'(B)2)1'2dB= 2~ .

As for the proof, see, e.g., Hopf [ 5 ] . By (4.1) and this fact (4.3)

2na=g

S:

we have

( r ( ~ ) 2 + r ' ( e ) z ) " z / r ( ~ ) d ~,+ E L ,

where L, is the length of the curve it on [ 0 , 2 n ) . Then we obtain

r.

We multiply (4.1) by

(4.4)

The second equality is proved by the formula

r2 and

integrate

642


Eliminating E from (4.2-4) we obtain

By the isoperimetric inequality the left-hand side is nonpositive and it vanishes if and only if 7 is a circle. Then, by o Z g > O , we have

On -

the other

:5

r(B)dBIL7 and that This is i.e., 7 is a circle, which completes the proof. Q.E.D.

hand, it obviously holds that

:$ (y(B)2++r’(B)z)1/z/r(B)dB~ - 2 ~ . Hence the equality must hold.

possible if and only if f = O ,

Appendix. Here we show how the formulas (3.1-4) are derived. The procedure here is the same as that in [7]. In fact we borrow a lemma which is proved in [7]. We first prepare notation. Let f be a C1-mapping from Cmt1+ftU(S1) into Cm+l+a(S1). Here m>O a n d j 2 0 are integers and O m + 2 + j + a , then the derivative of T is represented by (A.4)

D,T( u)v = auu

I

rl

+I(W,)v -

(r,+u)v’-u’v afo {(ro+u )+~( K ’ ) ~ } ~ / dB ~

Here V is the solution of

(A.5)

AV=O

in Q,,

Progressive Waves in rhe Flow of Perfect Fluid

643

(A.6) Here D , f is the Frichet derivative.

The function 2’( W,) is given by

The derivative of T , is represented by

If we a d m i t this lemma, t h e proof of (3.1-4) is easy. In fact we put f ( u ) = - ( ~ / 2 ) ( r , + u ) ~ + a + ( c / 2 ) r , ~ . Then t h e l e m m a shows that (‘4.7)

D,T(O)w= D,T,(O)w

dU azv, w , =-+dr ar2

w h e r e U is given by (3.5). Observing that iVVl=aVu/i3v, w e obtain (3.1) by (A.7). O t h e r formulas are easy to see. Q.E.D.

References J. T. Beale, Large-time regularity of viscous surface wave, Arch. Rational Mech. Anal., 84 (1984), 307-352. M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalue, J. Funct. Anal., 8 (1971), 321-340. H. Fujii, M. Mimura and Y.Nishiura, A picture of the global bifurcation diagram in ecological interaction and diffusion systems, Phisica D, 5 (1982), 1-42. R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982). 343364. H. Hopf, Differentical Geometry in the Large, Lecture Notes in Math., No. 1000, Springer, Berlin, 1983. H. Hosono and M. Mimura, Singular perturbation approach to travelling waves in competing and diffusing species models, J. Math. Kyoto Univ., 22 (1982), 435-461. t 7 1 H. Okamoto, Bifurcation phenomena in a free boundary problem for a circulating Bow with surface tension, Math. Methods Appl. Sci., 6 (1984) 215-233. On a nonstationary free boundary problem for perfect fluid with surface 1 8 1 -, tension (to appear in J. Math. SOC.Japan). [ 9 1 -, Stationary free boundary problems for circular flows with or without surface tension, in Proc. U.S.-Japan Seminar on Nonlinear PDE in Appl. Sci., eds. H. Fujita, P. D. Lax and G. Strang, Lecture Notes in Numer. Appl. Anal., Kinokuniya/North-Holland, Tokyo/Amsterdam, 1983, 233-251. -, On the 4-dimensional O(2)-equivariant bifurcation equation arising in a stationary free boundary problem for a perfect fluid (preprint).

644


[ll] H. Fujita, H. Okamoto and M. Shaji, A numerical approach to a free boundary problem of a circulating perfect fluid, Japan J. Appl. Math., 2 (1985), 197-210. 1121 M. Shaji, An application of the charge simulation method to a free boundary problem, to appear in J. Fac. Sci. Univ. Tokyo. [13] J. T. Beale and T. Nishida, Large-time behavior of viscous surface waves, to appear in Lecture Notes in Numer. Appl. Anal. Vol. 8, North Holland-Kinokuniya. [14] H. Okamoto and M. Shbji, Dynamical system arising in nonstationary motion of a free boundary of a perfect fluid, Kbkytiroku of RIMS No. 559, 19-41. Department of Mathematics Faculty of Science University of Tokyo Tokyo 113, Japan


On Laminar Boundary Layers with Suction By Taira SHIROTA Abstract. The basic mathematical questions of the two-dimensional, incompressible and laminar boundary layer theory are considered, where prescribed pressure gradients are not always negative. From certain a priori estimates, the existence of separation points of solutions to Prandtl equations with suction and Prandtl’s approximation to Navier-Stokes equations along Fife’s consideration are obtained by means of continuations of Oleinik’s local solutions. In this regard it continues the work of Matsui and myself. Key words: laminar boundary layer, separation points, Prandtl equations, suction

1. Introduction Let x , y be coordinates parallel and normal to the wall and u, v the corresponding components of velocity. Then the Prandtl equations of motion considered here are

u -au + v - = -au -+v-, dp (1.1)

ax

8%

dx

ay

ayz

au av -+-=o

aY

in

aY

[0,Z)X [0,a)

with boundary conditions

(1.2)

u=O

and

v=vo(x)

on y=O

and

(1.3)

u(x,y)+U(x)

as y+oo

uniformly in x on any compact subset of [0, I ) . Here ( d p / d x ) ( x ) = p , ( x )is the prescribed, non-negative pressure gradient, v , ( x ) the prescribed suction and U ( x ) the main-stream velocity related by

(1.4) Received April 1, 1985.

2 p ( x ) + U2(x)=constant .

T. SHIROTA

646

Since the governing equations are parabolic if x is taken as the evolution variable and u20, it is also necessary to specify a n initial profile for u at some station of x : (1.5)

u(x,y)=uo(y)

for x=O

and

CJ(O)>O

.

From a n engineering point of view ([lo], [l l ]), the station of zero skin friction of solutions to the problem is calculated approximately or numerically, and the value of the suction velocity, which is sufficient to prevent separation, is predicted approximately for flows with the downstream stagnation point. But from the degenerate parabolicity of (1.1) in y=O together with boundary conditions (1.3) and (1.4), even the existence of the exact, classical solution of the boundary layer equations up to and including the point of zero skin friction has not yet been obtained. Thus, from a mathematical point of view the behaviors of the exact solution and its derivatives at such a point cannot be used directly in discussion as in a n engineering approach. On the other hand, for the Prandtl approximation to Navier-Stokes equations Fife has defined a certain laminar class of the solutions of Navier-Stokes equations and succeeded in obtaining a n approximation theory in the case where the pressure gradients are strictly negative ([l] and its references). But it is very difficult to obtain a proper description of the high Reynolds number, steady, laminar Navier-Stokes’ flows, even if only the flow field upstream of the separation is considered. The purpose of this paper is to provide a mathematical basis for the theory of Prandtl equations mentioned above. In Section 4 we shall show the existence of the separation point, assuming behaviors of the main stream and the suction velocity near the rear stagnation point, whose definition is given in Section 2. In Section 5 we shall try to develop a connection between the preseparation solutions to Navier-Stokes equations and the Prandtl equations along the Fife’s considerations. Our results will give no ultimate solution to the problem, but they are hoped to be useful for further mathematical investigation of the problem ([2]).

2. Notations and Definitions For the sake of simplicity we are concerned with flows past a flat section of boundary. Then the stationary Navier-Stokes equations are uU,+UU?l=y(C1,,+Uyy)-Px

(2.1)

C1Z),+UUy=Y(Z),,+21,,)--Py

U,+Uy=O

, , in

[0, I] x [0,

.

a)

(The subscripts in (2.1)denote the partial differentiation with respect to the corresponding variable.)

On Laminar Boundary Layers with Suction

647

The Prandtl equations (1.1) are formally derived from (2.1) by the following manner: By the transform x=x,

and

qeu-1’2y

72x9 v ) = ~ - ” z 4 x Y, ) ,

a(x,v)=u(x, Y ) 9 P(x, v)=p(x, Y ) 9

we have ii,,-liii,-cii,-p,=

(2.1)’

-j, = -lJ(d,,

-Mi,,

,

+uscx+ui)c,

fue,)

,

ii,+s,=o . Neglecting the terms of the right-hand sides in the above, we obtain the dimensionless Prandtl equations ii,,

- vii, = p,( x )

-Eli,

u,+v,

,

=0

where p , ( x ) = p , ( x , 0). Then returning to the original coordinates ( x , y ) and setting also

4%Y)=u1’2v(x,77) ,

,

u ( x , y ) = U ( x , ‘7) PAX) = P , ( x )

9

we obtain (1.1). Now the substitution of the independent variables in (1.1) of the form

(2.3)

x=x

9

$=#(x,y)

3

where

reduces formally (1.1), (1.2), (1.3) and (1.5) to Mises’ form: putting w ( x , #)= u2(x,y ) , we have

with the boundary conditions (2.5)

w ( x , Y ) =o

on y = O ,

(2.6)

w(x,#)-U2(x)

as $-roo

T. SHIROTA

648

uniformly in x € [0, I ] ,

(2.7)

407

=.10(9)

9

where

Next in order to indicate sets of data or solutions to the above equations, for an interval [0, I ] let Bo([O,I] x [O,m)) be the Banach space of uniformly bounded continuous functions in [O,I ] x [0, m) with supremum norm. For any a ( O < a < 2 / 3 ) and any y,>O let C"([O,I ] x [ y o , m)) be the Banach space of Holder continuous functions, i.e.,

IM{IXl--Xz1"2+ IY, -YzlY for ( x i , Y , ) E 10, I1 x [yo7 a) ( i = 1 , 2 ) and Iu(x1, Y I ) -u(xz, Yz)l

M=M(yo, u )

.

nyo>o

Let C"([O,I ] x (0, m)) be C"([O,I ] x [ y o , m)). Furthermore let B"([O,I ] x (0, 00)) be Bo([O,I ] x [0,m))n C"([O,I ] x (0, m)). We also define spaces Bo([O,a)), C"((0,m)) and B z t U ( ( 0m)) , by the same way. Then we shall take a function u € B z t u ( ( 0 ,a)) as an initial datum in (1.5) if uy(0)>O,

u(O)=O,

(2.8)

u-U(0) VU,,(Y)

as y+m -P,(O)

w($)

-u o ( 0 ) u , ( ~ =)0 ( y 2 )

as Y->O ,

Let us denote the set consisting of

€ I % a if and only if

" , "+ w(O)=O

(2.9)

,

and

which is a strong compatibility condition. such data by I z t a = Z Z t a ( ~U, ( 0 ) )and

by Pi". Obviously

for y 2 0

u,(y)>O

, 4 Gw$h$h B"((0, m))

,

w+(O)>O,

9

ws($)2O

49)+ U2(0) lJ4WwILIL -2p,(O) - V O ( 0 ) O=0(9) ~

for $ 2 0 , a s $-+m and as 9 - 0 .

Let F& "([O, I ] ) be the space consisting of functions w ( x , $) such that

On Laminar Boundary Layers with Suction 0

, 0, , 04 , v'Gw,+€B"([O,I1 X ( 0 , m)) for (x, $1 € I0,ZI x (0, I k$l-B , w ( x , 9)> 0

649

9

Iw,I

q d x , 0) > 0

for

XE

00)

,

[O,I]

where kdepends only on wand ,8€ (0, 1/2). Let F2,'"([0,Z))=nosl~ O for x E [0, I ] ([6]). Here we say that (u, v) is a classical solution to (1.1)-(1.5) if

Let the space consisting of such functions described above be represented by F2([0,ZJ) and put

Now we define the separation points of solutions to (1.1)-(1.5) (2.4)-(2.7) respectively.

and to

Definition 2.1. A point (s, 0) is a separation point of a solution (u, v) to (1.1)-(1.5) if ( u , v ) E F Z t a ( [ 0s),) and

(2.11)

lim inf

K&X,

y)=O

,

x<s, (x,~)-+(s,O)

or equivalently a point (s, 0) is a separation point of a solution (2.7) if w E F2,'"([0, s)) and

(2.12)

lim inf

w+(x, + ) = O

w

to ( 2 . 4 ) ~

.

xI,>infvl,>O. For the interpretation of this definition see Fife [ l ] and its references and Section 4. Here and hereafter all functions described above are assumed to be continuous.

3.

Lemmas on Local Solutions

Here we shall consider the local solution o ( x , 9)to (2.4)-(2.7) with initial data wo($)E rye constructed by Oleinik [ 6 ] . First we remark that the solution 0 ( x , ~ ) ~ F ~ ~ “ ( 0for , 1 some , ] ) Z,>O is monotone non-decreasing in $ 2 0 for all x € [0, Z,], provided wo(9)€ IFa ([4]).

Lemma 3.1. Let w ( x , $ ) be the locaI solution constructed by Oleinik with initial data € I F : “ . Then w ( x , 9)E F2,’“([0,I,]) for some lo, and for some $o > O and k>O

Corollary 3.2. Under the same assumption in Lemma 3.1, the section o ( x , - ) belongs to ITafor any x E [0, Z,], i.e., the section w ( x , .) satisfies the strong compatibility condition (2.9) replacing 0 by x . The above assertions are very useful to continue the local solution obtained in [6] and is first announced in [4]for a special case, but the proof in our case is accomplished by the same way as in [4] and hence omitted here. In the following, in particular in Section 5 we need to obtain uniform

On Laminar Boundary Layers with Suction

65 1

estimates of solutions with respect to the data. T o obtain such estimates, we set k , and k , be mino,,,lo W(x) and max,,,,,, j2p,(x)I for some positive 1,O

Then from Lemma 3.5, it implies that

for e 1 .

--E

See Kac [3] or Griinbaum [2] for the proof of these facts.

Let

.=\= Z(O)d0. -77

Then we have the expression L = - v + K , where K is a compact symmetric operator. It is easily seen from (2.2) that I,=Rz=O and that 1, -p*/2.

We substitute Pf=Coeo+C,e,

into (2.5) and then form the inner-product of (2.5) with eo,e,. Co=Co(G(E, 4eo, eo)+Cz(G(E, 9 e 2 ,e,)

(2.6)

We obtain

,

C,=C,(G(E, 4 e 0 , e,)+C,(G(E, Alez, ed'

Here

We set

A nontrivial solution exists if and only if

W E , R)=det (DjAE,4 ) = 0

(2.8)

.

We substitute i = e z C into (2.8) and compute each entry as follows.

The last term equals (G(E,E2C)ive,-G(0, O)ive,, e,)

= -E(G(E, E'C)(iv+EC)G(O, O)ive,, e,)

.

Y.SHIZUTA and H. NISHIYAMA

668

Here we used v e , = l / ~ e , - , + l / n f e , + , .

It follows that

Djk(E,E2C)= -E2C(G(E, E2C)ej,e,)

.

+Ez(G(5,S'C)(iv+EC)G(O,O)iue,, e,)

Let us set DjAE, E") =E"jAE,

(2.9)

6) .

Then the equation (2.8) is equivalent to the equation

W E , 1:)= det (MJE, 5 ) )= O .

(2.10)

When E=O, this reduces to the equation (2.11)

(tl-+- 3c+-=o

3

M ( 0 , C)=C'-3

.

2123

The two roots are

We apply the real implicit function theorem It is easily seen that z, 0 such that

(i)

For each a>O small enough, there exists a constant

IINE, IIl<Ml holds for any (5, I ) satisfying IEla, Re 12 - p / 2 . Here M I is a constant not depending on E, I . ( i i ) For each 6 > 0 , there exists a constant j3>0 such that

II W E , 4ll I M, holds for any ( E , I ) satisfying l E l 2 6 , R e , ? - j 3 . pendent of 5, 1.

Here M , is a constant inde-

Proof. See, for example, [6] for the proof. Note that the corresponding fact is well-known for the Boltzmann equation with cut-off hard potentials. We use now a result from the semigroup theory. The semigroup etB^(E)

Kac's Model of the Boltzrnann Equation

671

is represented as the inverse Laplace transform of the resolvent R(E, A). Although the integral is not absolutely convergent in general, we may shift the path to the left by virtue of Proposition 3.1. Let 6 be sufficiently small. Then, for IElO ( T > O ) ; (iv) f E 'G::;/z(eT-i2x[O, TI); m ) x (0, > 0, 2pf( v ) (,% p', K , P, S)(p, 0) E V'+"((o, P , r , So ( aS/a@) 3 ~ ~ 2 0 : (vi) the compatibility conditions between the system of (1.1) and the initialboundary conditions (1.2)-( 1.4), which we omit to write down because of clearness, are valid. Then there exists a unique solution ( p , v, 0 ) of (1.1)-(1.4), which belongs to [B'+"(&J)n { p " Z p ( x , t ) >O)l

x w:f:.l+a/y(QTf)x [ ~~?:f~~~+~/~((e,~) n { O * Z O ( X , t)>o}] (p* and 0" are positive constants) f o r some T' E (0, T I .

Remark. The initial-boundary value problem for (l.l),(1.2), the other slip condition (cf. [l]) (1.5)

v.n=O,

v-r=KPn-r,

K>O

,

and the general singular boundary condition for the temperature (1.6)

r,(O-O,)-(l-r,)VO-n=g

,

Osr,Sl

Slip Boundary Problem for a General Fluid Flow

671

will be discussed in the forthcoming paper [7]. (1.3) is a singular case of (1.5) in the sense that (1.3) corresponds to the case K = m in (1.5). In the present paper we use the characteristic transformation h':,$,,: 0, onto QTby virtue of (1.3)l and is given by the formula

(x, t)w(xo(x, t), to=& which is one to one from

x=xo+

(1.7)

5:"

C(X0,

r)dr ,

O(Xo,

t0)=17:,~,,v(x, t ) ,

for only the first equation in ( l . l ) , differing from the previous ones 13, 4, 51. Then the equation ( 1 . l ) I is uniquely solved by p ( x , t)=n::;"P(xo,

(1.8)

to)

p(x0,to)=po(xo)e w [

,

- ~ i o v r ~ ( Txwo] , ,

where 17Ev;t0 is the inverse mapping of n:;It,,, V:= C V , V=(V,, V,, V 3 ) , V,= a/axo,,( j = l , 2, 3), 9=(g,,)=(ax/i3x0)-l. Hence the problem (1.1)-(1.4) can be reduced to the following initial-boundary value problem with respect to w= q ~e ),- w o , wo=t(uo,oo): D

t , w; V ) w + Q ( x ,

(1.9) I

1

or

in

QT

w/,=o=o

on 9 ,

B ( x , t , w ; V ) W = $ ( X ,t , w )

on

n,V,f6,,n,V,-2n,n,n,V, Bik=d

r, w)

O

m,V,

( j = 1 , 2 , k= 1 , 2 , 3 ) ,

(j=k=4) ,

O

,

rT,

D

0

0

A. TANI

678

with p and (v,O)replaced by (1.6) and w+w, respectively. Therefore it suffices to solve the initial-boundary value problem (1.9). And we only give the proof of the main theorem concerning the boundary condition (1.4)l, since we can prove such quite to the other one (1.4)2.

S 2.

Linearized Problem In this section we consider the following linearized problem of (1.9):

l"j = *( x,

I,

w ; V)$+Q(x, t , w )

in

QT

,

on 9 , on r T .

*d=0

b ( x , t , w ; V)@=$(x,t , w ) Here w is a given function belonging to

9T {w -

,O

the row vectors of matrix B ( x , t, w; i f ) g ( x ,t , w ; i5, v ) ( ( x , t ) E rT, fixed) are linearly independent modulo M = n : = , (E3--5:(?)), where M ~ tX , w; , ie, v ) is an adjugate matrix of M ( x , t , w ;ic)-u14, and E;(‘)’s are the roots in f 3 of det [ M ( x , t , w; it)- vZ4] =0 with positive imaginary parts. Proof. The roots

tT(r)(5’,v ) of det [ M ( x , t , w; iE)-v14]=0 are given by

In the present case, the boundary operator B ( x , t, w; i f ) i s

0

O1

If we denote by C:=, a ( s ) f i -the l remainder term when we divide B(x, t , w ; i f ) . M ( x , t , w; i f ,v ) by M , then a ( 4 ) = ( a j i ) ) 1 s jis , kgiven s, by the following formulae: h

A. TAN]

680

(2.4)

a::)=

If v = O , then we define a(4)as the limit value of the where u,=p/(p+p’). above formulae as v+O. After considerably lengthy calculations, we obtain (2.5)

det a ( 4 )

= p Z ( ~ , E ~ a , 3 ~ 4 3 ) - l ~ ~ ~ 1 ~ ~ ~ ~ 3 ~ ( ~ ~ ~ l ~ + ~ ~ ~ 3 ~ ) 3 (

x (E; (1) -E; x (EJ ( 4 ) -E,

(4))2((E:

(3)

(l))z(,e;

(4)

-E; -E,

(1))(6; ( 3 )

(3))

-E;

(4))

.

Since B,>O ( r = l , 2, 3,4) follows from Lemma 1 and the condition (2.3), it is obvious that det > 0. Hence it is sufficient to take 6’ as a n arbitrary positive constant smaller than 6, but for later purposes we take 6’ in such a way that

a’=-

1 min {us,u4,6) 4

.

Q.E.D.

Poisson kernel and Green matrix of R3,. Poisson kernel HI and Green matrix H , of R3, are defined in the same manner as those in [3, 4, 51: 2.2.

x B b , t , w ; V)Z,(Y,-E, ro’-r0; x,

,

I;w)IYo,3=od~o’

where r+ is a contour enclosing all El(‘)( r = 1 , 2 , 3 , 4 ) , a 4 = ( a i j * r )is) the inverse matrix of a(’) and 2, is the fundamental solution of the system of equations

aw

--=.P/(x,

ar

t, w;V J W

Slip Boundary Problem for a General Fluid Flow

681

By Lemma 2.5 in [ 5 ] , we have the following estimates of a4 for any q)€{Re q 2 -p,IIm 41, 177’1 5p8(lE’14+lqlz)/1’4,15’14+lq1z>O) (for B, and &, see Lemma 2.5 [ 5 ] ) , (C’=E’+iv’,

(otherwise) . Tracing the proof of Lemma 3.14 in [3], we obtain

Lemma 3.

(otherwise) ,

I Dr7D,’H0I(y,r ; E, r,,)5 C 3 ( ~ - ~ 0 ) - ( 2+ r3 )t/ 2’ exp s ’ ~ - ~ l ~ - E l ~ ./ ~ ~ - ~ ~ ~ l 2.3. Solution of (2.1) Similarly to [ 3 . 4 ] , we can obtain the following lemmas concerning the regularizer R of the problem.

A. TANI

682

where

1

’Rp$ = -

1: SKI dr,

Hick”)(a-jj’,t-ro)c(k”)(jj’)$(jj’,ro)dy’ .

where C, ( 2 1) and C, increase monotonically in T and M , . C+O as T+O, and N = N ( T ,M , , M , ) increase monotonically in T , M I and M,. Returning to the problem (2.1), we need to evaluate .@ and q5. seen that E W3+, implies the estimates

r

$411F,fa)

11$i,

9

1 \ 6 9

$311?;a)

sc8

.

From (1.7), (1.8) and (2.2) it follows that (l-ta)

+

IIPllQ,

5 CAT7 Ml) CIdT, MJM2

1 1 ;9 : 11

5 C,(T, Ml)+ ClO(T9 MAM2

7

hence 9

It is easily

Slip Boundary Problem f o r a General Fluid Flow

683

S 3. Nonlinear Problem (1.9) We construct the sequence {w,(x, t ) } of the successive approximate solutions as follows :

t

w,(x, t)-O€G, , w,(x, t ) is defined as a solution

* of (2.1) assuming W = W , , - ~ E ~ ~ .

Then the result in § 2 implies that w,(x, t ) uniquely exists and belongs to G T ,n=O, 1,2, -. Applying the estimates in $ 2 to the equation concerning W " - W ~ - ~we , obtain

.

(3.1) where C,,-O as T-0. Therefore the sequence {w,(x, t ) } converges to some function w(x, t ) uniformly if we choose T'E (0, TI so as to satisfy C,,(T', M,, M , ) < 1. The uniqueness of the solution of (1.9) is proved by the fact that the difference of two solutions supposed to exist satisfy the inequality analogous to (3.1). The positivity and boundedness of p and 6 are obvious from our construction method. Thus the proof of our main theorem is completed.

A. TANI

684

References [ 1 ] J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik, 8, Springer, 1959. [2]

B. A. COJIOHHWKOB, 0 KpaeBbIX

a a ~ a q a xAJIR JIWHeMHLIX

[3]

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

o6qero

naph6onusec~uxCWCTeM

MaT., 83 (1965), 3-162. A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. RIMS, Kyoto Univ., 13 (1977), 193-253. -, On the free boundary value problem for compressible viscous fluid motion, J. Math. Kyoto Univ., 21 (1981), 839-859. -, Two-phase free boundary problem for compressible viscous fluid motion, Ibid., 24 (1984), 243-267. -, The initial value problem for the equation of motion of general fluid with some slip boundary conditions, Keio Math. Semi. Rep., 8 (1983), 77-83. -, The initial value problem for the equations of the motion of compressible viscous fluid with general slip boundary condition, in preparation. AW@@epeHqWaJIbHbIXYpaBHeHWZi

Department of Mathematics Keio University Yokohama 223, Japan

BWJIB, TpYAbI

Index A

a priori estimate, 40, 100, 118 absolute Maxwellian, 40 activator-inhibitor, 158 ADE (Alternating Direction Edition), 296 ADENA-I, 299 ADENA-11, 303 ADENA computer, 299 AD1 method, 294 adiabatic exponent, 460 adjoint, 61, 78 advection, 507 aggregating phenomena of population, 385 Airy, 439 alternative theorem, 601 approximate solutions, 591 artificial viscosity, 393 astrophysical contexts, 461 asymptotic behavior, 58, 111, 112, 387, 403, 481 asymptotic expansion, 371 asymptotically stable, 82, 83 attractive, 248 attractive basin, 248 augmented Lagrangian method, 45 1 B

Banach scale, 90, 91 barotropic gas, 98 barycentric domain, 347 bifurcation analysis, 145 bifurcation point, 636 block Jacobi method, 294 blowing-up problem, 419 Boltzmann equation, 38, 663 boundary condition, 41, 645 boundary layer, 90

boundary operator, 41 boundary value problem for stochastic differential equation, 597 Brownian motion, 599 B(u, u), 420 B(u, u), 420 E(u, u), 420 B(u,u), 420 buffer memory unit, 299 Burgers equation, 11 1

C canonical equation, 27 canonical one form, 26 Cantor’s ternary set, 259 Cauchy problem, 40, 57 Cauchy-Kowalewski theorem, 40, 50 cavity flow problem, 454 cell Peclet number, 323 central difference scheme, 227, 269 C,,-group, 48 chaos, 222, 227, 239 Chapman-Enskog expansion, 43, 94 characteristic equation, 46 characteristic (integral) curve, 75 circumcentric domain, 349 class I, 168 199 class 11, 169 200 classical solution, 50, 70, 92 codimension, 621 collision cross section, 38 collision integral, 663 collision (summational) invariants, 39 compact, 601 competition system, 240 compressible Euler equation, 30, 44, 93 compressible Navier-Stokes equation, 44, 94

686

compressible viscous flow, 481 compressible viscous fluid, 675 compressible viscous and heatconductive fluid, 481 condensation of singularities, 26 1 conservation law, 39, 40 constructed by Oleinik, 650 constrained Lorenz-like attractor, 627 constrained system, 608, 624, 629 contraction mapping, 549 contraction mapping principle, 49, 381, 548 convergence of cutoff approximation, 56 critical eigenvalues, 193 critical line, 2 15 crosswind diffusion, 332, 339, 353 C,-semigroup, 57 cubic convolution nonlinearity, 543 curvature, 632 cutoff assumption, 44 cyclic reduction method, 288 D

D”,174 D: ( E , a), 168 de Rham’s functional equations, 269 decay estimate, 64, 72, 670 decay property, 583, 590 decay rate, 112, 584 decomposition of a stationary solution, 407 degenerate singular point, 241 densely defined scattering operator, 544 difference inequality, 590, 592 diffusion-induced instability, 147, 159 diffuse reflection, 41, 81 dimensionless Prandtl equations, 647 discrete maximum principle, 326 dissipativity conditions, 583 doubly asymptotic, 229

Index

E

effective hyperbolicity, 13 effectively hyperbolic, 1 1, 15, 17 eigennilpotent, 60 eigenprojection, 60 eigenvalue, 58 energy, 583 energy estimate, 119 energy identity, 589 energy inequality, 587 energy method, 113, 587 energy space, 543, 544 essential spectrum, 58 Eulerian coordinate, 1 16 Euler equation, 460, 632 Euler’s finite difference scheme, 222 evolution of a star, 460 existence theorem, 585 explicit scheme, 282 exterior stationary problem, 48 1,482 F

fast Poisson solver, 296 Fatou set, 248 Fife-Mimura class, 199 finite difference scheme, 123 finite propagation speed of exponential decay, 1 15 fixed boundary problem, 98 fixed point theorem, 465 Flower theorem, 248 fluid dynamical limit, 42 fold-up principle, 174 formal normal form, 241 Fourier series, 71 Fourier transform, 60, 64 fractal, 259 fractal object, 247 free boundary, 63 I free boundary problem, 98, 116 Friedrichs’ scheme, 566 frozen branch, 2 1 1 fundamental equation, 231 fundamental existence theorem for

Index

687

ordinary differential equations, 467 fundamental matrix, 14 fundamental solution, 481, 482, 484

hyperbolic conservation laws, 112 hyperbolic snap-back point, 240 hyperbolic-parabolic type, 112

G

I

Galerkin method, 452 rn,172 Gauss curvature, 22 generalized vector field, 61 1, 612, 613, 616, 619, 623 Gevrey class, 7, 52, 53 ghost solution, 225 Gierer-Meinhardt model, 166 globally in time, 99, 122 global shadow assumptions, 162 global singular assumptions, 163 global solution, 57, 58,69,71,74,81, 89, 98, 672 global solutions near zero, 72 Grad’s angular cutoff, 40 gravitation, 122 Green’s formula, 77, 78 Green matrix, 680 Gronwall inequality, 106

ILLIAC-IV, 291 implicit differential equation, 61 1 implicit function theorem, 668 implicit scheme, 283 incompressible flow, 48 1, 482 incomplete HV-decomposition, 314 incompressible or nearly incompressible media, 445 index, 584 infinitesimal deformation, 614, 619 initial boundary value problem, 42, 74, 80, 98 initial-boundary conditions, 583 initial layer, 90, 91, 94, 371 initial value problem, 108 integro-differential equation, 663 interior transition layers, 200 invariant manifold, 230, 233 inverse power law potential, 38, 52 iteration scheme, 445 J

Hamiltonian, 25 hard ball gas, 38, 40 hard potential, 62 Hausdorff dimension, 274 Henon’s mapping, 225 Herman ring, 249 hidden symmetry, 173 high Reynolds number, 646 Hilbert expansion, 43, 89 Holder quotient, 52 Holder space, 98 holomorphic dynamical system, 233, 247 homoclinic point, 229 Hopf bifurcations, I37 horse-shoe, 229 “hump” effect, 159

J(t), 427 Julia set, 247, 264

K Kac’s model, 663 k-jet, 241 Koch’s curve, 259, 271

L lacunary series, 262 Lagrangian mass coordinate, 116 laminar and pre-separation, 650 laminar, preseparation class, 657 Lane-Emden equation, 478

688

Lane-Emden function, 478 Laplace transform, 60 Lax-Wendroff scheme, 566 le developpement de Friedrichs, 432 le nombre d’Ursell, 443 le theorkme abstrait non-lineaire de Cauchy-Kowalevski, 442 Lebesgue’s singular functions, 270 I’equation de KadomtsevPetviashvili, 431 I’equation de Korteweg-de Vries, 43 1 I’equation deux-dimensionnelle de Boussinesq, 432, 437 les ondes longues d’ampleur h i e de surface de l‘eau, 431 Levi condition, 4, 15 Levi-Lax condition, 4 limiting absorption principle, 77, 84 line method, 118 linearized Boltzmann operator, 62, 81, 83 linear Cauchy problem, 46 linearized equation, 664 linear hyperbolic system of first order, 562 linear initial boundary value problem, 77 linearized system, 48 1 linear Vlasov equation, 374 linear wave equations, 584 Li-Yorke, 222 Lp-Lq estimates, 544, 545 local bifurcation, 169 local bifurcation theory, 167 local Maxwellian, 40, 93 local singular-shadow assumptions, 161 local solution, 44,48, 52, 650 local stability assumptions, 164 local unstable manifold, 228 locally in time, 97 Lorenz attractor, 246, 608, 627 loss of smoothness, 52, 54 L2(R,”)-valued difference equation, 566 L2-stable, 569

Index

M Mach angle, 485 Mach cone, 481, 483 Mach number, 483, 484 main-stream, 645 Mandelbrot set, 247, 251 marginally stable, 639 matrix pencil, 616, 622, 629 maximal covariance, 173 maximum principle, 115, 326, 421 Maxwell distribution, 664 Maxwell equation, 376 Maxwellian, 39 Maxwellian gas, 664 memory bank, 297 memory conflict, 298 MIMD, 284 miniversal, 62 1 Mises’ form, 647, 658 mixed finite element/finite difference scheme, 576 mixed finite element method, 445 Monge-Ampkre equation, 22, 25 monotone matrix, 335 monotone scheme, 391 monotonicity, 77, 326 moving boundary condition, 108 multiple coexistence, 165 multiplicity, 58, 60

N Nash-Moser implicit function theorem, 19, 32 nested disection ordering, 309 Neumann layered class, 200 Neumann slit, 200 neutral, 248 Newton’s method, 247 Newtonian potential, 460 non characteristic Cauchy problem, 13 noncritical eigenvalues, 192 noncutoff potential, 40, 52 nonlinear effect of slip/separation,

Index

58 1 nonlinear hyperbolic conservation laws, 43 nonlinear systems, 19 non-linear wave equation, 25, 543, 583 non-negative, 358 non-negative solution, 357, 358 nonnegativity, 57, 80, 326 nonnegativity of solution, 50 non-standard solution, 42 normal, 248 normal form, 241, 243 normal forms for generalized vector field, 613, 620 normal forms for (ordinary) vector field, 614 normal form problem for constrained systems, 625 number density of gas particles, 38 numerical examinations, 4 12

0 odd-even reduction method, 288 one-dimensionalization, 292 one-dimensional shock profile, 82 orthonormal basis, 597 Oseen’s hydrodynamical potentials, 483 overshoot, 332, 353

P parabolic basin, 249 parallel computation, 279 parallel simulation, 282 parallelization by dimension, 289 parameter dependent problems, 445 Parseval’s equality, 60, 66 partial upwind, 333, 340 Peclet number, 321 penalty approach, 446 perfect absorption, 41 perfect factorization, 3

689

perfect fluid, 631 period doublings, 138 periodic boundary condition, 70 periodic point, 249 Petrov-Galerkin approximation, 342 Petrowsky-parabolic, 428 phase space methods, 5 19 @(u, v), 426 piston problem, 108 Poincare’s inequality, 589 point Jacobi method, 291 point SOR method, 292 Poisson equation, 290, 379, 460 Poisson kernel, 680 Polya’s space-filling curve, 271 polytropic gas, 98 population model, 239 positive solution, 358 positive trace, 14 positive-type matrices, 335 power nonlinearities, 543 Prandtl approximation, 657 predation-mediated coexistence, 129 predatory-prey models of LotkaVolterra type, 129 pressure, 445 pressure gradient, 645 prey-predator, 159, 165 prey-predator system, 240 principal symbol, 18, 20 probability density, 43 probability density of gas particles, 38 progressive wave, 631 propagation speed, 112, 633 (pseudo-) differential operator, 40 pseudodifferential operators, 3, 54, 91 pseudoeigenvalue, 72 pulse, 507

Q quadratic maps, 250 quadrature point, 345

690

R random integral equation, 600 random operator, 601 rational map, 250 reaction-diffusion equation, 246, 507 recovery line, 2 15 recovery of stability, 168 reduced set, 201 reduced solutions, 202 reductive perturbation method, 111 regularity, 70, 584 regularizer, 681 regularizing operator, 3 repulsive, 248 residue calculus, 495 resolvent, 58 resolvent set, 58 resolvent spectrum, 58 reverse reflection, 41 Riemann function, 262 Riemann invariant, 113 Riemann problem, 116 Riesz’ representation theorem, 77,78 Ritz-Galerkin finite element approximation, 323 Rossler’s attractor, 246 R-S scheme, 302 R(a), 201

S S. Russell, 439 S scheme, 302 saddle connection curve, 230, 233 scale of Banach space, 44, 54 scattering operator, 543, 546 Schauder base, 270 scrambled set, 224, 240 secondary bifurcation line, 213 Seelig’s model, 166 selective reduced integration, 453 self-gravitation, 460 self-similarity, 271 semigroup generator, 59

Index

semilinear equation, 357 separation point, 649, 653 shadow method, 168 shadow system, 185 shallow water waves, 443 shock layer, 90 shock wave solution, 116 Siege1 disk, 249 SIMD, 284 sine-Gordon nonlinearity, 543 singularity, 591 singular-shadow edge, 169, 208 singular integrals, 492 singular integral operators, 2 singular integral operators of Calderon-Zygmunt type, 371, 379 singular limit point, 207 singular perturbation technique, 134, 140 singular perturbation method, 168, 51 1 singular wall, 207 slip boundary condition, 675 slip/separation condition, 579 small data scattering, 543 smoothing operator, 49, 55, 68 snap-back repeller, 240 Sobolev embedding, 548, 550 Sobolev space, 98 Sobolev’s theorem on smoothness of composed functions, 469 soft potential, 71 soil-structure interaction, 576 solitary wave, 631 solutions, 507, 583 sound speed, 484 spatially homogeneous case, 42 spectrum, 601 specular reflection, 41, 70 splitting-up method, 31 1 S-solution, 598 stability of the discrete model, 389 stable manifold, 228 stable region, 249 stationary flow, 82, 481

Index

stationary Navier-Stokes equations, 646 stationary patterns, 397 stationary problem, 82 stationary solution, 40, 82, 83 stochastic integral equation of Fredholm type, 597 stochastic integral of noncausal type, 597 Stokes equation, 445 strange attractor, 241, 246 stream function, 632 strictly hyperbolic, 15 strong compatibility condition, 648 strong hyperbolicity, 13 strong solution, 78 strongly hyperbolic, 11, 12, 15 strongly well posed, 20 structure matrix, 274 subprincipal symbol, 14 subsonic, 481 substitution operator, 266 successive approximations, 380, 38 I suction, 645, 653 super attractive, 248 supersonic, 48 1, 483 surface tension, 631 s(x), 426 symmetric hyperbolic, 563 symmetric hyperbolic system, 460 symmetry-breaking bifurcations, 159 symmetry-breaking destabilization, 168

T Takagi function, 261 test function, 78 the

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