Nonlinear Partial Differential Equations in Applied Science: Seminar Proceedings (Mathematics Studies)

NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN APPLIED SCIENCE PROCEEDINGS OF THE U.S. - JAPAN SEMINAR, TOKYO, 1982

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NORTH-HOLLAND

MATHEMATICS STUDIES

Lecture Notes in Numerical and Applied Analysis Vol. 5 General Editors: H. Fujita (University of Tokyo) and M. Yamaguti (Kyoto University)

Nonlinear Partial Differential Equations in Applied Science; Proceedings of The U.S. -Japan Seminar, Tokyo, 1982 Edited by

HlROSHl FUJITA (University of Tokyo) PETER D. LAX (New York University) GILBERT STRANG (Massachusetts Institute of Technology)

1983

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK. OXFORD

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Lecture Notes in Numerical and Applied Analysis Vol. 5 General Editors H. Fujita University of Tokyo

M. Yamaguti Kyoto Universtiy

Editional Board H. Fujii, Kyoto Sangyo Universtiy M . Mimura, Hiroshima University T. Miyoshi, Kumamoto University M. Mori, The University of Tsukuba T. Nishida, Kyoto Universtiy T. Nishida. Kyoto University T . Taguti, Konan Universtiy S . Ukai, Osaka City Universtiy T . Ushijima, The Universtiy of Electro-Communications PRINTED IN JAPAN

PREFACE Nonlinear equations come to us in tremendous variety, each with its own questions and its own difficulties. At one extreme are the completely integrable equations, with constants of the motion and a rich algebraic structure. At the other extreme is chaos, with turbulent solutions and statistical averages. Between these two possibilities, algebraic and ergodic, lies the full range of nonlinear phenomena. There are smooth solutions which develop shocks, or bifurcate, or maintain slow and nearly periodic variations that imitate the linear theory. Each of these questions requires a separate treatment, and the subject would be simpler if we know for every equation which behavior to expect. Nevertheless these equations, the nonlinear partial differential equations which arise in applications, share one crucial property. They are all vulnerable when the right pattern in found. It is a slow process, to uncover and reveal their structure, but it is moving forward. The papers in this volume reflect a part of that progress. They were presented at the U.S.-Japan Seminar in Tokyo in July 1982. One goal of the seminar was to establish personal contact among those mathematicians who are actively working for these difficult but fascinating equations in the U.S. and in Japan. The other goal was a wider one, that is, to invoke most advanced scientific talks and discussions on major topics in this developing field of applied analysis. Thanks to the cooperation of all participants from the U.S., Japan, and some third countries including China, the seminar was successful in both sense mentioned above and we believe that these proceedings of the seminar which contain all papers delivered there will contribute much to the progress of the study of nonlinear problems. Finally, we, who served also as the coordinators of the seminar, wish to express our gratitude to the governmental agencies, i.e., National Science Foundation and Japan Society for the Promotion of Science, for their support and to industrial companies in Japan for practical assistances which they gave as institutional participants. Last but not least, our gratitudes go to all of our committee members and staff members of the secretariat of the seminar for their enthusiasm and devotion. September 15, 1983

H. FUJITA P. D. LAX G. STRANG

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CONTENTS

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PROGRAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ronald J. DIPERNA: Conservation Laws and the Weak Topology.. . . .

v ix 1

Hiroshi FUJI1 and Yasumasa NISHIURA: Global Bifurcation Diagram in Nonlinear Diffusion Systems ...............................

Yoshikazu GIGA: The Navier-Stokes Initial Value Problem In Lp . . . . . 37 Ei-Ichi HANZAWA: Nash’s Implicit Function Theorem and the Stefan 55 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tosio KATO: Quasi-linear Equations of Evolution in Nonreflexive Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hideo KAWARADA and Takao HANADA: Asymptotic Behaviors of the Solution of an Elliptic Equation with Penalty Terms . . . . . . . . . . . 77 Robert V. KOHN:

Partial Regularity and the Navier-Stokes Equations

........................................................

101

Kyiya MASUDA: Blow-up of Solutions of Some Nonlinear Diffusion 119 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hiroshi MATANO: Asymptotic Behavior of the Free Boundaries Arising in One Phase Stefan Problems in Multi-Dimensional Spaces . . . . 133 Akitaka MATSUMURA and Takaaki NISHIDA: Initial Boundary Value Problems for the Equations of Compressible Viscous and HeatConductive Fluid ......................................... 153 Sadao MIYATAKE: Integral Representation of Solutions for Equations of Mixed Type in a Half Space ............................. 17 1 Tetsuhiko MIYOSHI: Yielding and Unloading in Semidiscrete Problem 189 of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alan C. NEWELL: Two Dimensional Convection Patterns in Large Aspect Ratio Systems ................ ..................... 205 Hisashi OKAMOTO: Stationary Free Boundary Problems for Circular Flows with or without Surface Tension ...................... 233 G. PAPANICOLAOU, D. MCLAUGHLIN and M. WEINSTEIN: Focusing Singurarity for the Nonlinear Schroedinger Equation ....... 253 Mikio SATO and Yasuko SATO:

Soliton Equations as Dynamical Sys-

viii

Contents

tems on Infinite Dimensional Grassmann Manifold ............ 259 Gilbert STRANG: L’ and L” Approximation of Vector Fields in the Plane ........................................................ 273 Takashi SUZUKI: Deformation Formulas and their Applications to Spec289 tral and Evolutional Inverse Problems ....................... Seiji UKAI and Kiyoshi ASANO: Stationary Solutions of the Boltzmann 313 Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Teruo USHIJIMA: On the Linear Stability Analysis of Magnetohydrodynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Hans F. WEJNBERGER: A Simple System with a Continuum of Stable 345 Inhomogeneous Steady States .............................. Masaya YAMAGUTI and Masayoshi HATA: Chaos Arising from the MoDiscretization of O.D.E. and an Age Dependent Population del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 John G. HEYWOOD: Stability, Regularity and Numerical Analysis of the Nonstationary Navier-Stokes Problem .................... 377 LIN Q u n and JIANG Lishang: The Existence and the Finite Element Apau Au=C uj---+f

. . . . . . . . . . . . . . 399 ax, YING Lung-an and TENG Zhen-huan: A Hyperbolic Model of Combus409 tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . proximation for the System

ZHOU Yu-lin: Boundary Value Problems for Some Nonlinear Evolutional Systems of Partial Differential Equations . . . . . . . . . . . . . . . 435 ... DIRECTORY OF PARTICIPANTS ................................ xiii

PROGRAM MONDAY, JULY 5 8:45 Opening of Seminar Session 5-1 9:OO-1O:OO Prof. T. Kato (University of California, Berkeley) “Quasi-Linear Equations of Evolution in Nonreflexive Banach Spaces” 10:15-11:OO Prof. K. Masuda (T6hoku University) “Some Remarks on Blow-up of Solutions of Nonlinear Diffusion Equations” 11:05-11:50 Dr. T . Suzuki (University of Tokyo) “Deformation Formulas and their Applications to Spectral and Evolutional Inverse Problems” Session 5-2 13:45-14:30 Mr. H. Okamoto (University of Tokyo) “Stationary Free Boundary Problems for Circular Flows with or without Surface Tension” 14:35-15:20 Prof. Lin Qun (Institute of Systems Science, Academia Sinica) and Prof. Jiang Li-shang (Peking University) “The Existence and the Finite Element Approximation for the System

15:20-16:OO Coffee Break 16:OO-17:OO Prof. H. F. Weinberger (University of Minnesota) “A Simple System with a Continuum of Stable Inhomogeneous Steady States” TUESDAY, JULY 6 Session 6-1 9:OO-1O:OO Prof. R. J. DiPerna (Duke University) “Shock Waves and Entropy” 10:15-11:OO Prof. S. Ukai (Osaka City University) and Prof. K. Asano (Kyoto University) “Stationary Solutions of the Boltzmann Equation” 11:05-1150 Prof. T. Nishida (Kyoto University) and Dr. A. Matsumura (Kyoto University) “Initial Boundary Value Problems for the Equations of Compressible Viscous and Heat-Conductive Fluid” Session 6-2

Program

13:45-14:30 Prof. S. Miyatake (Kyoto University) “Integral Representation of Solutions for Equations of Mixed Type in a Half Space” 14:35-15:20 Prof. H. Fujii (Kyoto Sangyo University) and Prof. Y. Nishiura (Kyoto Sangyo University) “Global Aspects in Bifurcation Problems for Nonlinear Diffusion Systems” (tentative) 15:20-16:OO Coffee Break 16:OO-17:OO Prof. A. C. Newell (University of Arizona) “Two-Dimensional Convection Patterns in Large Aspect Ratio Systems” WEDNESDAY, JULY 7 Session 7-1 9:OO-1O:OO Prof. M. Sat0 (Research Institute for Mathematical Sciences, Kyoto University) “Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifold” 10: 15-1 1:00 Prof. A. C. Newell (University of Arizona) “The Connection between Wahlquist-Estabrook, Hirota, r Function, and Inverse Scattering Methods for the AKNS Hierarchy” 11:05-1150 Prof. Zhou Yu-lin (Peking University) “Some Problems for Nonlinear Evolutional Systems of Partial Differential Equations” THURSDAY, JULY 8 Session 8-1 9:OO-1O:OO Prof. M. Yamaguti (Kyoto University) “‘Chaos’ Caused by Discretization” 10:15-1 1:OO Prof. H. Matano (Hiroshima University) “Asymptotic Behavior of the Free Boundaries Arising in One Phase Stefan Problems in Multi-Dimensional Spaces” 11:05-1150 Dr. E. Hanzawa (Hokkaido University) “Nash’s Implicit Function Theorem and the Stefan Problem” Session 8-2 13:45-14:30 Prof. H. Kawarada (University of Tokyo) “New Penalty Method and its Application to Free Boundary Problems” 14:35-15:20 Prof. T. Ushijima (University of Electro-Communications) “On the Linear Stability Analysis of Magnetohydrodynamic System” 15:20-16:00 Coffee Break 16:OO-17:OO Prof. G . Papanicolaou (Courant Institute of Mathematical Sciences, New York University)

Program

“Modulation Theory for the Cubic Schrodinger Equation in Random Media” FRIDAY, JULY 9 Session 9-1 9:OO-1O:OO Prof. R. V. Kohn (Courant Institute of Mathematical Science, New York University) “Partial Regularity for the Navier-Stokes Equations” 10:15-11:OO Mr. Y. Giga (Nagoya University) “The Navier-Stokes Initial Value Problem in Lp and Related Problems” 11:05-1150 Prof. J. G. Heywood (University of British Columbia) “Stability, Regularity and Numerical Analysis of the Nonstationary Navier-Stokes Problem” Session 9-2 13:45-14:30 Prof. Ying Lung-an (Peking University) and Prof. Teng Zhen-huan (Peking University) “A Hyperbolic Model of Combustion” 14:35-15:20 Prof. T. Miyoshi (Kumamoto University) “Yielding and Unloading in Semi-Discrete Problem of Plasticity” 15:20-16:OO Coffee Break 16:OO-17:OO Prof. G. Strang (Massachusetts Institute of Technology) “Optimization Problems for Partial Differential Equations” 17:05 Closing of Seminar

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DIRECTORY O F PARTICIPANTS OF US-JAPAN SEMINAR '82 IN APPLIED ANALYSIS GUESTS OF SEMINAR Professor Tetsuichi Asaka Science University of Tokyo (Emeritus Professor of University of Tokyo) Dr. Tatuo Simizu Laboratory of Shimizu Construction Co. Ltd. Professor Shoji Tanaka University of Tokyo Dr. Hajimu Yoneguchi Nippon UNIVAC Sogd Kenkyusho, Inc. FOREIGN PARTICIPANTS (US Delegates) Ronald J. DiPerna Tosio Kato Robert V. Kohn Alan C. Newell George Papanicolaou Gilbert Strang Hans F. Weinberger

Duke University University of California, Berkeley Courant Institute of Mathematical Sciences, New York University The University of Arizona Courant Institute of Mathematical Sciences, New York University Massachusetts Institute of Technology University of Minnesota

(Special Participants from Third Countries) John G. Heywood Lin Qun Ying Lung-an Zhou Yu-lin

The University of British Columbia Institute of Systems Science, Academia Sinica Peking University Peking University

xiv

Directory of Participants

JAPANESE PARTICIPANTS Rentaro Agemi Kiyoshi Asano Hiroshi Fujii Hiroshi Fujita Daisuke Fujiwara Isamu Fukuda Yoshikazu Giga Ei-Ichi Hanzawa Masayoshi Hata Imsik Hong Yuzo Hosono Atsushi Inoue Hitoshi Ishii Nobutoshi Itaya Masayuki Ito SeizB It6 Tatsuo Itoh Takao Kakita Hideo Kawarada Shuichi Kawashima Fumio Kikuchi Hikosaburo Komatsu Yukio K6mura Yoshio Konishi Takeshi Kotake ShigeToshi Kuroda Kyoya Masuda Hiroshi Matano Akitaka Matsumura Akihiko Miyachi Isao Miyadera Sadao Miyatake Tetsuhiko Miyoshi Sigeru Mizohata Ryuichi Mizumachi Hiroko Morimoto

Hokkaido University Kyoto University Kyoto Sangyo University University of Tokyo Tokyo Institute of Technology Kokushikan University Nagoya University Hokkaido University Kyoto University Nihon University Kyoto Sangyo University Tokyo Institute of Technology Chuo University Kobe University of Commerce Hiroshima University University of Tokyo University of Tokyo Waseda University University of Tokyo Nara Women’s University University of Tokyo University of Tokyo Ochanomizu University University of Tokyo Tohoku University University of Tokyo Tohoku University Hiroshima University Kyoto University University of Tokyo Waseda University Kyoto University Kumamoto University Kyoto University Tohoku University Meiji University

Directory of Participants

Katsuya Nakashima Yoshimoto Nakata Takaaki Nishida Yasumasa Nishiura Hisashi Okamoto Shin Ozawa Mikio Sat0 Yasuko Sat0 Norio Shimakura Taira Shirota Takashi Suzuki Masahisa Tabata lzumi Takagi Hiroki Tanabe Seiji Ukai Teruo Ushijima Shigehiro Ushiki Masaya Yamaguti Kiyoshi Yoshida KGsaku Yosida

Waseda University Science University of Tokyo Kyoto University Kyoto Sangyo University University of Tokyo University of Tokyo Kyoto University Ryukyu University Kyoto University Hokkaido University University of Tokyo The University of Electro-Communications Tokyo Metropolitan College of Aeronautical Engineering Osaka University Osaka City University The University of Electro-Communications Kyoto University Kyoto University Kumamoto University University of Tokyo

xvi

Directory of Participants

INSTITUTIONAL PARTICIPANTS Co., Ltd. AISl 4 SE AISIN-WARNER Ltd. B UN-EIDO Energy Research Laboratry, Hitachi, Ltd. FACOM HITAC FUJIFACOM CO. Fujitsu Ltd. FUKUMOTO-SHOIN, Ltd. Hitachi Ltd., Central Research Laboratory Hitachi Ltd., Software Works Hitachi Ltd., System Development Laboratory Hitachi Software Engineering Co., Ltd. HOKUSHIN ELECTRIC WORKS Ltd. IBM Japan, Ltd. Institute of Japanese Union of Scientists and Engineers Ishikawajima-Harima Heavy Industries Co., Ltd. Japan Advanced Numerical Analysis. Inc. Japan Process Development Co., Ltd. Japanese Standards Association Kajima Co. Kawasaki Steel Co., Chiba Works KENBUN SHOIN Co., Ltd. Kyoei Information Processing Service Center Ltd. Maeda Construction Co., Ltd. Mitsubishi Central Research Laboratory Mitsubishi Heavy Industries Co., Ltd. Mitsubishi Research Institute Inc. N.C.R. Japan Ltd. NIPPON BUSINESS CONSULTANT Co., Ltd. Nippon Electric Company (C & C Systems Research Laboratories) Nippon Sheet Glass Co. Nippon UNIVAC Kaisha, Ltd. Oki Electric Industry Company, Ltd. PANAFACOM Ltd. Shimizu Construction Co., Ltd. Surugadai Gakuen

Directory of Participants

TOKYO SHUPPAN Co., Ltd. Tokyo Shoseki Co., Ltd. Toshiba Research and Development Center YAZAKI Corporation YOYOGI SEMINAR

xvii

xviii

Directory of Participants

COMMITTEES AND STAFFS 1 . Coordinators

Japanese Coordinator Hiroshi Fujita University of Tokyo U.S. Coordinators Courant Institute of Mathematical Peter D. Lax Scicences, New York University Massachusetts Institute of Technology Gilbert Strang 2. Local Organizing Committee Hiroshi Fujita* Univ. of Tokyo SigeToshi Kuroda Univ. of Tokyo Shigeru Mizohata Kyoto Univ. Masaya Yamaguti Kyoto Univ. Emeritus Professor of Univ. of Tokyo KBsaku Yosida

3. Scientific Committee Hiroshi Fujii Hideo Kawarada Teruo Ushijima

Hiroshi Fujita* Takaaki Nishida Masaya Yamaguti

4. Executive Committee

Hiroshi Fujii Hiroshi Fujita* Asako Hatori Hideo Kawarada Katsuya Nakashima Teruo Ushijima 5 . Working Committee

Akihiko Miyachi Shin Ozawa Kunihiko Takase

Chairman or Chief

Hisashi Okamoto Takashi Suzuki* Masahiro Yamamoto

Lecture Notes in Num. Appl. Anal., 5 , 1-15 (1982) Nonlinear PDE in Applied Science. U . S .-Jripun Seminur. Tokyo, 1982

CONSERVATION LAWS AND THE WEAK TOPOLOGY

Ronald J . D i P e r n a Duke U n i v e r s i t y Durham, N o r t h C a r o l i n a

27706

W e s h a l l d i s c u s s some r e s u l t s c o n c e r n i n g t h e c o n v e r g e n c e o f

a p p r o x i m a t e s o l u t i o n s t o h y p e r b o l i c s y s t e m s of c o n s e r v a t i o n laws. The g e n e r a l s e t t i n g i s p r o v i d e d by a s y s t e m o f

conservation

laws i n o n e s p a c e dimension,

where

Rn.

u = u ( x , t ) € Rn

and

W e assume t h a t

t h a t i t s Jacobian has

i s s t r i c t l y hyperbolic i n t h e sense

i s a smooth n o n l i n e a r map from

r e a l and d i s t i n c t e i g e n v a l u e s

...

A1 With r e g a r d t o a p p r o x i m a t i o n , o n e

A n u)

s i n t e r e s t e d i n s e q u e n c e s of

a p p r o x i m a t e s o l u t i o n s g e n e r a t e d by p a r a b o l i c s y s t e m s

€ ( u ) ~=

D uXx,

u = u (x,t)

and by f i n i t e d i f f e r e n c e schemes

atu + a X f ( u ) = 0,

u = uAx(x,t),

which a r e c o n s e r v a t i v e i n t h e s e n s e of Lax and Wendroff 1

IS].

Ronald J. DIPERNA

standard strategy for convergence seeks to establish uniform estimates on both the amplitude and derivatives of the approximate solutions in appropriate metrics and then appeal to a compactness argument to produce a subsequence that converges in the strong topology. One may regard convergence of the entire sequence as a question of uniqueness of the limit.

We recall that in the setting of hyper-

bolic conservation laws the maximum norm and the total variation norm yield a natural pair of metrics in which to investigate the stability of the solution.

The

norm measures the solution

amplitude and the total variation norm measures the solution gradient.

Their relevance for conservation laws is established by the

following theorem of Glimm [51 dealing with the stability and convergence of the approximate solutions generated by his random choice method applied to the Cauchy problem. Theorem 1.

If the total variation of the initial data

uo(x)

sufficiently small then a sequence of random choice approximations converges pointwise almost everywhere to a globally defined uAx distributional solution u maintaining uniform control on the amplitude and spatial variation:

TV uAx(-,t)

5 const. TV

The constants are independent of the mesh length and depend only on the flux function f. The proof is based on a general study elementary wave interactions in the exact solution and in the random choice approximations

uAx.

It remains an open problem to prove or disprove the

corresponding estimates for conservative finite difference schemes and parabolic systems.

In the latter direction we refer the reader

Conservation Laws and Weak Topology

to 131 which contains an analysis of discrete wave interactions in conservative schemes together with a stability and convergence theorem for a class of methods involving the hybridization of the random choice method with first order accurate conservative methods. Here we shall discuss new compactness theorems for sequences of approximate solutions generated by diffusive systems and conservative difference schemes.

The proof involves the theory of com-

pensated compactness which originates in the work of Tartar [111 and Murat [9,lO] and the main step provides a proof of a conjecture of Tartar [ll]. The analysis appeals to the weak topology and averaged quantities rather than the strong topology and the fine scale features.

Regarding the weak topology and the elliptic conserva-

tion laws of elasticity we refer the reader to the work of Ball [l]. The principle statement is that for a class of approximation methods , which respect the entropy condition, Lm implies convergence.

stability alone

Gradient estimates are not required to pass

to the limit in the nonlinear functions. We shall first recall some background involving Tartar's work on weak convergence and compensated compactness.

Consider a se-

quence of functions un(y) : Rm which is uniformly bounded in

Lw.

It is well-known that one may

extract a subsequence which converges in the weak-star topology of LOD:

for all bounded un

B c Rn.

We recall that in general the sequence

need not contain a strongly convergent subsequence, i.e. a sub-

sequence converging pointwise a.e. to

In particular, if

Ronald J. DIPERNA

a real-valued map on

However, after passing to subsequence, composite weak limits may be represented as expected values of associated probability measures in the following sense. denoted here by

There exists a subsequence of

(still

un) and a family of probability measures over the

range space R",

such that for all continuous

g: Rm

The limit on the left hand side is taken in the weak-star topology of

and equality holds for almost all

notes a generic point in the range space

Rn.

Rm.

Here

de-

This result stems

from the work of L. C. Young and was first used in the setting of conservation laws by Tartar [ll].

It is not difficult to show that

strong convergence corresponds to the case where the representing measure

reduces to a point mass concentrated at

u(y):

More generally, the deviation between weak and strong convergence is measured by the spreading of the support of

Lipschit z then lg(lim un)

lim g(un) I m

2 const. max diam spt w Y

In the framework of conservation laws, the goal is to show that the representing measures associated with a family of exact or approxi-

Conservation Laws and Weak Topology

mate solutions reduces to a point mass or is contained in a set whose geometry allows one to deduce the continuity of the special nonlinear maps appearing in the equations.

In the case of a scalar

conservation law Tartar [ll] has shown that

mass if

is convex and that, in general,

interval where of

is affine.

Y Y

reduces to a point is supported on an

Here we shall discuss the reduction

for strictly hyperbolic systems of two equations with non-

degenerate eigenvalues.

The analysis is based on a study of the

Lax progressing entropy waves in state space [ 7 1 ,

specifically on

connections between their structure and the structure of wave patterns in the physical space, cf. I 2 1 for details and additional We also refer the reader to Lax [ 6 1 which contains a

references.

discussion of the scalar conservation law and the viscosity method in the setting of the weak topology. Before discussing the general case we shall cite an example. Consider the equations of elasticity in Lagrangian form with artificial viscosity

and assume that in

> 0

while

u(v)x = uX

= E V

xx xx '

sgn v u " > 0.

Lm, there exists for each fixed

Given initial data

a globally defined solution

the amplitude of which remains uniformly bounded as the viscosity parameter

vanishes,

Here the constant depends only on data.

and in the

norm of the

The bound follows from the presence of invariant regions in

the state space [14]. We claim that by appealing only to the

stability and the entropy condition, one may extract a subsequence

Ronald J. DIPERNA

(uE ,uE ) which converges pointwise a.e. to a globally defined k k distributional solution of the associated hyperbolic system u

o(v)x = 0

A similar result can be established for a class of first order finite difference schemes which are based on averaging the Riemann problem, e.g. the Lax-Friedrichs scheme and Godunov's scheme. The source of the compactness in the strong topology lies in the nonlinear structure of the wave speeds and in the dissipation of generalized entropy along propagating shocks,

We shall first

recall the notion and some basic properties of generalized entropy as for mulated by Lax [ 7 ] . laws (1).

Consider a system of

conservation

A pair of real-valued mappings on the state space rl:

-f

q: Rn

is called an entropy pair if all smooth solutions of (1) satisfy an addition conservation law of the form

For the purposes at hand we shall restrict our attention to the

class of systems having an entropy pair with

strictly convex.

As observed by Lax and Friedrichs [151 this class includes the

basic systems of continuum mechanics.

Furthermore, Lax [ 7 ] showed

that all strictly hyperbolic systems of two equations has at least a locally defined strictly convex entropy and that a broad class has a globally defined strictly convex entropy. bility condition which links the entropy be derived as follows.

Suppose

u(x,t)

The basic compati-

to its flux

is a

may

solution and

Conservation Laws and Weak Topology

consider the quasilinear forms of the systems of conservation laws

(1) together with the extension ( 3 ) u t

+ Of(u)ux

By replacing the time derivative of

= 0

by the spatial derivative

we find that ( 3 ) is equivalent to

Hence the condition (5)

Vn(u) Vf(u) = Vq(u),

€

is a necessary and sufficient for the existence of an entropy pair. We observe that (5) represents a system of

linear, variable

coefficient partial differential equations in two unknowns q.

n > 2

and

it is formally over determined but fortunately has

a (convex) solution in the setting of mechanics.

Concerning the

structure of ( 5 ) we recall the observation of Loewner that the compatibility condition ( 5 ) retains the same classification as the original system (1). In our setting the demonstration that ( 5 ) is hyperbolic is straightforward: consider the right eigenvectors of the Jacobian of

f Vf(u) r .(u) = A . (u) r.(u) 3

Taking the inner product of ( 5 ) with

r. 3

immediately yields the

characteristic form of ( 4 ) : (A.

Vq)

-r

= 0,

j = 1,2

In the following discussion we shall be mainly interested in the determinate case

n = 2 which can be illustrated with a variety of

Ronald J . DIPERNA examples.

In particular,

it i s u s e f u l t o k e e p i n mind t h a t t h e

s m o o t h m o t i o n o f a n e l a s t i c medium w h i c h c o n s e r v e s mass a n d momentum also c o n s e r v e s m e c h a n i c a l e n e r g y .

The c o n v e x f u n c t i o n

F o r s y s t e m ( 2 ) o n e may t a k e

serves as a generalized entropy f o r (2)

with generalized entropy flux q = u C(v)

The i d e n t i t y ( 5 ) s t a t e s t h e t i m e r a t e o f c h a n g e o f m e c h a n i c a l e n e r gy i s b a l a n c e d by t h e r a t e a t w h i c h t h e s t r e s s t e n s o r p e r f o r m s w o r k . Within t h e class of c o n s e r v a t i o n l a w s w i t h a convex e x t e n s i o n

i t i s s t a n d a r d t o impose t h e Lax e n t r o p y i n e q u a l i t y

o n weak s o l u t i o n s

u(x,t)

f o r t h e p u r p o s e of d i s t i n g u i s h i n g t h e

p h y s i c a l l y r e l e v a n t weak s o l u t i o n s f r o m t h e s e t o f a l l p o s s i b l e w e a k solutions.

Solutions s a t i s f y i n g (6) are c a l l e d admissible.

t h a t t h e d i s t r i b u t i o n a l i n e q u a l i t y i s meaningful i f l o c a l l y bounded f u n c t i o n .

We n o t e

is merely a

F o r o u r c u r r e n t p u r p o s e s w e s h a l l re-

s t r i c t o u r a t t e n t i o n t o weak s o l u t i o n s w h i c h l i e i n t h e s p a c e Lm n B V .

Here

denotes t h e class of f u n c t i o n s of s e v e r a l v a r i -

a b l e s w h i c h h a v e bounded v a r i a t i o n i n t h e s e n s e of C e s a r i , 1 . e . f i r s t o r d e r p a r t i a l d e r i v a t i v e s r e p r e s e n t a b l e a s l o c a l l y bounded Bore1 m e a s u r e s [ 4 , 1 2 1 . m

E x p e r i e n c e w i t h c o n s e r v a t i o n l a w s h a s shown

is a n a t u r a l function space f o r t h e s o l u t i o n opera-

that

tor.

I n t h i s connection w e n o t e t h a t s o l u t i o n s constructed by t h e

3 BV

random c h o i c e m e t h o d l i e i n t h e s p a c e s t a b i l i t y e s t i m a t e s o f t h e o r e m 1.

strate t h a t t h e measure

Lm n BV

Within

by v i r t u e o f t h e BV

o n e c a n demon-

Conservation Laws and Weak Topology

i s c o n c e n t r a t e d o n t h e s h o c k s e t of

r(u)

u , i.e.

the solution

t h e set o f p o i n t s of d i s c o n t i n u i t y a n d c o n s e q u e n t l y t h a t t h e e n t r o p y i n e q u a l i t y ( 5 ) h o l d s i f and o n l y i f a l l shock waves i n

dissipate

generalized entropy:

f o r a l l Bore1

E C r(u).

This i n e q u a l i t y reduces t o t h e second l a w F i n a l l y , we s h a l l

o f t h e r m o d y n a m i c s i n t h e s e t t i n g of f l u i d f l o w .

r e s t r i c t a t t e n t i o n t o systems w i t h non-degenerate e i g e n v a l u e s , 1.e. s y s t e m s f o r w h i c h t h e wave s p e e d s a r e m o n o t o n e f u n c t i o n s of t h e wave a m p l i t u d e s .

T e c h n i c a l l y w e assume t h a t

i s monotone i n

t h e corresponding eigendirection: r . 3

(7)

- BX. # 3

W e n o t e t h a t t h e g e n u i n e n o n l i n e a r i t y c o n d i t i o n ( 7 1 i n t r o d u c e d by

Lax [16] i s s a t i s f i e d b y s e v e r a l s y s t e m s o f i n t e r e s t :

the isentro-

p i c e q u a t i o n s of g a s d y n a m i c s f o r a p o l y t r o p i c g a s , t h e e q u a t i o n s

of s h a l l o w w a t e r waves, t h e e q u a t i o n s of e l a s t i c i t y i f Theorem 2 .

0''

# 0.

Consider a s t r i c t l y h y p e r b o l i c g e n u i n e l y n o n l i n e a r sys-

t e m of two c o n s e r v a t i o n l a w s w i t h a s t r i c t l y convex e n t r o p y . pose

i s a s e q u e n c e of a d m i s s i b l e s o l u t i o n s i n

where t h e c o n s t a n t

i s independent o f

sequence t h a t converges pointwise a.e.

Lm n BV.

SupIf

n , t h e r e e x i s t s a sub-

t o an admissible s o l u t i o n .

Thus t h e e x a c t s o l u t i o n operator r e s t r i c t e d t o a d m i s s i b l e s o l u t i o n s forms a compact m a p p i n g f r o m

1 Lloc.

The s o u r c e

Ronald .I.DIPERNA

of t h e c o m p a c t n e s s l i e s i n t h e l o s s o f i n f o r m a t i o n a s s o c i a t e d w i t h

a d m i s s b l e s h o c k waves a n d i n t h e n o n l i n e a r s t r u c t u r e o f t h e e i g e n W e emphasize t h a t t h e compactness i s e s t a b l i s h e d w i t h o u t

values

d e r i v a t i v e estimates. Next, w e s h a l l d i s c u s s t h e compactness of s o l u t i o n sequences g e n e r a t e d by d i f f u s i o n p r o c e s s e s

(8)

where f o r s i m p l i c i t y

(

~ = )F ~Du

is a c o n s t a n t

xx' n

I n order to

matrix.

e n s u r e correct e n t r o p y p r o d u c t i o n i n t h e l i m i t a s

vanishes, it

i s s u f f i c i e n t (and n e a r l y n e c e s s a r y ) t o r e q u i r e t h a t t h e d i f f u s i o n matrix

b e non-negative w i t h r e s p e c t t o t h e second d e r i v a t i v e o f

n, i . e . 2 0 r l D ~ O .

With r e g a r d t o t h e g e n e r a l q u e s t i o n o f a d m i s s i b i l i t y s h o c k s t r u c t u r e and p r o p e r d i f f u s i o n matrices w e r e f e r t h e r e a d e r t o R.

Peg0

[17,18].

-___ Theorem 3.

Suppose

l i n e a r map o n

i s a s t r i c t l y h y p e r b o l i c g e n u i n e l y non-

w i t h a s t r i c t l y convex e n t r o p y

t h a t the diffusion matrix V

uE.

and suppose

i s p o s i t i v e d e f i n i t e w i t h respect t o

i s a s e q u e n c e of s m o o t h s o l u t i o n s t o ( 8 ) s a t i s f y i n g

t h e r e e x i s t s a subsequence which converges p o i n t w i s e a . e . admissible solution

o f t h e a s s o c i a t e d h y p e r b o l i c system ( 1 ) .

Hence t h e s o l u t i o n o p e r a t o r s

of t h e p a r a b o l i c s y s t e m ( 8 )

p r o v i d e a f a m i l y o f m a p p i n g s w h i c h i s compact f r o m uniformly with t o

t o an

Lloc

The c o m p a c t n e s s p r e s e n t a t t h e h y p e r b o l i c

l e v e l is preserved uniformly i n

provided t h a t t h e d i f f u s i o n

Conservation Laws and Weak Topology

matrix enduces favorable entropy production in the limit. In the setting of continuum mechanics we recall that the standard diffusion matrices are merely positive semi-definite because mass diffusion is neglected.

However with additional work one can

establish the corresponding result. Theorem 4.

Suppose that

(p,,uE)

is a sequence of smooth solu-

tions of compressible Navier-Stokes

for a polytropic gas

If the flow is uniformly

p = A p Y , y > 1.

bounded and avoids the vacuum state, i.e.

then there exists a subsequence which converges pointwise a.e. to an admissible solution of the compressible Euler equations.

We note

that the compressible Euler equations losses its strict hyperbolicity at the vacuum state.

It is an interesting open problem to

establish the corresponding result without the hypothesized uniform lower bound on the density

At a more fundamental level, it re-

mains an open problem to prove uniform circumstances.

estimates in general

For example in the case of hyperbolic systems (l),

it remains an open problem to prove that

for admissible solutions in

with small data.

The esti-

mate (9) is motivated by physical considerations but has only been verified for solutions constructed by the random choice method. The proof of the theorems described above utilizes the theory

Ronald J . DIPERNA

of compensated compactness.

In this connection we refer the reader

to the work of Tartar [ll] and Murat [ 9 , 1 0 1

and to the forthcoming

Proceedings of the NATO/LMS Advanced Study Institute on Systems of Nonlinear Partial Differential Equations held at Oxford 1982 and organized by J. Ball et al.

Here we shall simply mention one of

the problems which the theory addresses: functions g(u) : Rn

characterize the nonlinear

which are continuous in the weak topology

when restricted to sequences of functions

un(y) : Rm

which

satisfy linear constant coefficient partial differential constraints. As an example we mention a result from electrostatics which historically motivated the general theory.

Consider vector

fields z

n *' R3

converging weakly in

and

wn: R3

-f

L2 2

n +z;

w n + w .

is controlled as well as the Zn to the extent that both the sequences of distri-

Suppose that the expansion in rotation in

but ions div zn

and

curl wn

-1 Here Hloc. distinguished linear combinations of partial derivatives are com-

lie in a compact subset of the negative Sobolev space

pact after the loss of one derivative.

Under these circumstances

there is precisely one smooth real-valued function is continuous in the weak topology, i.e. satisfies

and its given by the inner product

@(z,w)

which

Conservation Laws and Weak Topology

@(Z,W) = . Although, in general, the individual terms

zjwj

of the inner pro-

duct are not weakly continuous, there exists compensation among the terms of the sum 3

zjwj

j=1

which allows for weak continuity. From the point of view of electrostatics, there is precisely one quantity, the electrostatic energy density which can be measured, provided one agrees that the process of measurement is modeled by averaged quantities. For the purpose of applications to conservation laws, let us recall the duality between the divergence and the curl in theplane, div z = curl z* where

denotes the orthogonal complement of

and consider

the basic entropy inequality formulated with respect to two distinct entropy pairs

(qj,qj)

j = 1,2.

If, for example,

sequence of admissible weak solutions in

Lm r) BV

is a

it can be shown,

by appealing to Sobolev embedding, that the sequence of distribut ions

lies in a compact subset of

-1 Hloc

is convex and consequent-

ly that rl.

J lies in a compact subset of

(un) + q . (un) t ’ x Hloc

for arbitrary

Thus

(nl,ql) and the curl of the -1 (-q2,n2) both lie in a compact subset of Hloc.

the divergence of the entropy field entropy field

(rlj,qj).

Ronald J. DIPERNA

The continuity of the inner product yields a commutativity relation for the representing measure j = 1,2

where

For all entropy pairs

(nj,qj),

we have

denotes representing measure at an arbitrary point, i.e.

Tartar showed that (9) implies v is a Dirac measure (x,t)for a genuinely nonlinear scalar conservation law. In [21 it is

v = u

shown using the Lax progressing entropy waves that (9) implies that v

is a point mass for general genuinely nonlinear systems of two

equations and for the special case of elasticity which has a linear degeneracy alone an isolated curve.

References

[l]

Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977) 337-403.

[2]

DiPerna, R. J., Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal., to appear, (1983).

[3]

DiPerna, R. J., Finite difference schemes for conservation laws, Comm. Pure Applied Math. 25 (1982) 379-450.

141

Federer, H., Geometric Measure Theory (Springer, New York, 1969).

[51

Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697715.

[61

Lax, P. D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7 (1954) 159-193.

[7]

Lax, P. D., Shock waves and entropy, in Contributions to nonlinear functional analysis, e.d. E . A. Zarantonello, Academic Press, (1971) 603-634.

[8]

Lax, P. D. and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960) 217-237.

Conservation Laws and Weak Topology

Murat, F., Compacit6 par compensation, Ann. Scuola Norm. Sup. Pisa 5 ( 1 9 7 8 ) 4 8 9 - 5 0 7 . Murat, F., Compacit6 par compensation: Condition necessaire et suffisante de continuite faible sous une hypotheses de rang constant, Ann. Scula Norm. Sup. 8 ( 1 9 8 1 ) 6 9 - 1 0 2 . Tartar, L., Compensated compactness and applications to partial differential equations, in Research Notes in Mathematics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 4, Ed. R. J. Knops, Pitman Press, 1 9 7 9 . Vol'pert, A. I., The spaces BV and quasilinear equations, Math. USSR, Sb. 2 ( 1 9 6 7 ) 2 5 7 - 2 6 7 . Dacoroqna, B., Weak continuity and weak lower semicontinuity of nonlinear functionals, Lefschetz Center for Dynamical Systems Lecture Notes # 8 1 - 7 7 , Brown University, ( 1 9 8 1 ) . 1141

Chueh, K. N., Conley, C. C. and J. A. Smoller, Positivity invariant regions for systems of nonlinear diffusion equations, Indiana Math. J. 26 ( 1 9 7 7 ) 3 7 3 - 3 9 0 . Friedrichs, K. 0. and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Mat. Acad. Sci. USA 6 8 ( 1 9 7 1 ) 1686-1688.

Lax, P. D., Hyperbolic systems of conservation laws 11, Comm. Pure Appl. Math. 10 ( 1 9 5 7 ) 5 3 7 - 5 6 6 . Pego, R., Viscosity matrices for a system of conservation laws, Center for Pure and Applied Mathematics, University of California, Berkeley, preprint. Peqo, R., Linearized stability of shock profiles, CPAM, University of California, Berkeley, preprint.

This Page Intentionally Left Blank

Lecture Notes in Num. Appl. Anal.,5 , 17-35 (1982) Nonlinear PDE i i i Applied Science. U S . - J t i p t i n Seininur, Tokyo, 1982

Global Bifurcation Diagram in Nonlinear Iliffision Systems Hiroshi FUJI1 and Yasumasa NISHIURA Department of Computer Sciences Faculty of Science Kyoto Sangyo University Kyoto 603, JAPAN.

51.

Introduction Global phenomena of pattern formation in systems of reaction-diffusion equa-

tions is the main theme of the present paper. The system is written as ut = dluxx

f(u,v) in (t,x)

where I = ( O , n ) , and a 1

E (O,+m) x

its boundary. The system (P) is assumed to possess

Turing's diffusion induced instability, whichappearstypically in mathematical biology [ 7 ] .

I n other words, we are interested in the structure of global bifurca-

2 tion diagram - "global" with respect to the two diffusion parameters (dl,dZ) E R + -

of the following stationary system : d u

1 xx

-v c1

f(u,v) = 0, in I,

g(u,v) = 0,

with the boundary conditions (P), on 3 1 ; here, we put d 2 = l/a. The system (P) has been studied by a number of authors from various kind of viewpoints.

In particular, the bifurcation theoretic work of Mimura, Nishiura and

Yamaguti [ 3 ] has motivated the studies which succeed, such as Mimura, Tabata and Hosono [ 4 ] who studiedthe singular limit dl i Oof (SP) u s i n g t h e s i n g u l a r p e r t u r b a t i o n 17

Hiroshi FUJIIand Yasumasa NISHIURA

technique; the second author [ 5 ] has obtained a complete bifurcation diagram with respect to dl of (SP) in the limit case d2 t +m(i.e,, called the shadow system.

C 0). His limit system is

The first author has developed a new numerical algorithm

to detect and trace all bifurcating branches using a group theoreticmethod [ Z Fujii, Mimura and Nishiura

1 ] studied local structures of (SP) near double bifur-

cation points (adopting a group theoretic argument), and drew a global picture of bifurcation diagram, integrating the above analytical and numerical results. The purpose of this paper is, in part, to give a survey of those works, and in part, to describe new results which have been obtained after the publication of [ 1

Our method is based on the study of:

(1)

local structure at double bifurcation points introducing the Lie group ",I

(2)

the complete bifurcati.on analysis of the shadow system [ 5 1 ,

(3)

the singular-shadow limit of d2 I.

+m,

11,

d 1 C 0 - which we call the singular-

shadow edge, (4) the structure of "singular solutions" at the singular limit dl C 0, and

(5) an integration of these analytical results to have a global picture of bifurcating branches. A key in our paper is the discovery of singular branches which possess both

boundary and interior transition layers, and of singular limit points as its consequence. We shall see in the present paper that t h e s t r u c t u r e of s o k i t i o n s a t t h e singular-shadow edge seems t o play t h e r o l e of "organizing centre" of t h e whole global s t r u c t u r e . The solution space for the system (SP) will be R:

X is the Hilbert space EiN2

2 2

= (HN)

2 = (H ( I )

x X ( 3

((dl,cl) ,U),

where

fl (the boundary conditions (P),)).

We state the assumptions on the nonlinearities f and g. ( A . 1) ( i ) There exists a unique constant solution U =

(6,i)

> 0 of

(SP). See,

Fig. 0.1. (ii ) 0 is a stable solution of the kinetic system of (P). matrix at

= {J(f,g)/>(u,v))lo,

satisfies tr(B) < 0 and

I.e., the Jacobian det(B) > 0.

(iii) ( P ) is an activator-inhibitor system, 1.e.. the elements of B have the

Global Bifurcation Diagram

sign

[

bll

b12

b21

bz2]=[:

The z e r o l e v e l c u r v e o f f ( u , v ) i s S-shaped and f < 0 i n t h e u p p e r r e g i o n

(A.2)

of t h e sigmoidal curve.

u+ 1

(v)

for

Fig .O.1 ; f = 0 h a s t h r e e r e a l r o o t s u - , ( v )

1";:::

f(s,v)ds.

Then, J ( v ) = 0 h o l d s i f and o n l y i f v = q * € 2 , and Let

0 ( [ l ] ) .

r1,2

i s shown in

Note, however, t h a t t h e d e s t i n a t i o n of t h e

pr ima r y D - b r a n c h i s n o t i n d i c a t e d i n t h i s d i a g r a m , as w e l l as t h o s e of t h e seco n 1 dary branches of t h e D -branch. 2

Fig. 1.2.

Fig. 1.3.

Schematic Bifurcation Diagram near

Local Bifurcation Diagram near

r 1,z

(p11420

' O)-

Hiroshi FUJIIand Yasurnasa NISHIURA I f one u n f o l d s ( 1 . 2 ) near t h e d e g e n e r a t e parameter v a l u e s pl1qz0 = 0 , an

r e v e a l s i t s e l f . In f a c t , as may 1,2 be t h e c a s e f o r t h e May-Mimura model ( 0 . 2 ) , l e t u s suppose t h a t q 20= q20 # 0, and i n t e r e s t i n g l o c a l b i f u r c a t i o n s t r u c t u r e near

l e t p l l be t h e u n f o l d i n g parameter as ( p

< 6

(6 : s u f f i c i e n t l y s m a l l ) . 0

The b i -

f u r c a t i o n diagram t h u s o b t a i n e d a r e s h o w n i n F i g s . (1.4), and ( 1 . 4 ) b , which c o r r e s > 0 and pond r e s p e c t i v e l y t o t h e c a s e qo p 20 11

< 0, f o r / p l l l < 6 0 .

Fig. 1.4 Fig. 1 , 4b ,r The 'ID p o t near rl,z; 2 i s a simple degenerate singular p o i n t of e l l i p t i c t y p e ( l a ) , p l 1 q z 0 I 1 0 ) ; of hyperbolic type ((bi, pl1qz0 < 0 ) .

i s a simple d e g e n e r a t e s i n g u l a r p o i n t placed on t h e D2-sheet, r e s p e c t i v e l y

< 0. of e l l i p t i c t y p e when qiopll > 0 , and of h y p e r b o l i c t y p e when qo p 20 11

tends t o zero, Z

approaches t o t h e double b i f u r c a t i o n p o i n t

r 1,2.

A s pll

Thus, one s e e s

t h a t t h e primary s h e e t D2 has a secondary b i f u r c a t i o n l i n e , which p a s s e s Z l and

rl,2.

See, F i g . l . 5 .

I f one l e t t h e parameters ( d l , d Z ) c r o s s t h i s l i n e from t h e

r i g h t t o t h e l e f t , t h e D -sheet recovers i t s s t a b i l i t y 2

hence, t h i s secondary l i n e

i s c a l l e d t h e recovery l i n e of t h e D2-sheet. A remarkable f a c t i n t h e s e diagrams i s t h a t i n b o t h c a s e s t h e r e a p p e a r s a

p o t - l i k e " s t r u c t u r e due t o t h e e x i s t e n c e of

r 1,2.

We n o t e t h a t t h i s p o t - l i k e

Global Bifurcation Diagram s t r u c t u r e has been p r e d i c t e d numerically i n [ l

fold-up p r i n c i p l e ([l]),

A remark i s t h a t due t o t h e

every D -3rimary s h e e t t a k e s t h e p o t - l i k e form, t h e o r i g i n

of which i s a degenerate simple s i n g u l a r i t y Z

l o c a t e d on t h e D

-sheet.

The b a s i c q u e s t i o n i s t h e g l o b a l behavior o f t h i s "pot", and a l s o o f t h e o t h e r secondary branch born a t

r1,*.

I t i s a l s o worthy of n o t i n g t h a t from

r2,3,

there

appears a secondary b i f u r c a t i o n l i n e o f D2 - which a c t u a l l y corresponds t o t h e p o i n t s where D 2 l o s e s i t s s t a b i l i t y a g a i n .

losing l i n e of D 2 .

Between

of t h e primary D - s h e e t . 2

Fig. 2 . 1 .

r 1,2

and

r2,3,

Hence, t h i s l i n e may be c a l l e d t h e one s e e s an o u t c r o p o f t h e s t a b l e r e g i o n

See, F i g . l . 5 .

The Shadow Branches and their extension t o a

Fig. 1 . 5 .

The recovery and losing l i n e s o f the D 2 sheet;

the shaded p a r t shows the s t a b l e region. 12. -

By t h e shadow c e i l i n g we mean t h e l i m i t space IR+

X of a C 0 .

If solutions

o f (SP) a r e uniformly L_-bounded with r e s p e c t t o d l and a, one may have t h e l i m i t system : in I,

with t h e boundary c o n d i t i o n u

0 on 2 1 , where v = n i s a constant f u n c t i o n .

second e q u a t i o n comes from t h e i n t e g r a t i o n of (SP)2 o v e r t h e i n t e r v a l I .

The The

system (SS) i s c a l l e d t h e shadow system f o r ( S P ) . The g l o b a l behavior o f s o l u t i o n s of (SS) with r e s p e c t t o dl C 0 i s t h e f i r s t

Hiroshi

Full1

and Yasurnasa N I S H I U R A

o b j e c t of t h e study h e r e , which i s expected t o approximate t h e g l o b a l behavior o f In f a c t , a complete b i f u r c a t i o n

s o l u t i o n s o f (SP) f o r s u f f i c i e n t l y small a > 0.

diagram f o r (SS) has been o b t a i n e d by t h e second a u t h o r [ 6 ] .

i 0.

t h e n-mode branches

b emanating from (2'n' D). Fig.2.1.

= 2,3,

... )

conti-

Bh t h e fold-up p r i n c i p l e , t h e same c o n c l u si o n holds t o

nues t o mist as dl

fl)

ii) The one-mode bifurcating branch b, emanating from Id;,

Theorem 2 . 1 .

i i i l These shadow branch-

Jo not recover t h e i r s t a b i l i t g on t h e way t o t h e l i m i t dl

and consequently, they have no secondary branches i n a generic sense.

i 0,

Hence, o n l y

the 6 -branch i s the s t a b l e one among t h e branches on t h e shadow c e i l i n g . i Theorem 2.2.

the statement Iil o f Theorem 2.1

The global e x i s t e n c e r e s u l t , i . e . ,

holds as w e l l f o r the bifurcating branches Dn's In = 1 , 2 , . . )

o f ( S P i f o r smaZl .a

Namely, every bifurcating branch h i t s the sing u l a r wall dl = 0 f o r small

> 0.

See, F i g . 2 . 1 .

>. 0 .

I t should be noted t h a t t h e s t a t e m e n t ( i i . ) of Theorem 2 . 1 does no more hold f o r t h e primary branches D ' s of (SP) even f o r s u f f i c i e n t l y small a > 0 , except f o r

n = 1. §3.

The s i n g u l a r shadow edge - Edge c o n t i n u a The s t u d y of t h e l i m i t dl i 0 of t h e shadow system (SS) p l a y s

subsequent d i s c u s s i o n s .

The reduced shadow system a s dl

a key r o l e i n

0 i s d e f i n e d by

f(u,n) = 0, (RSS)

I, A s o l u t i o n (u,q)

g ( u , n ) dx = 0 . E

(where v =

a reduced shadow s o l u t i o n , where Xo

i s a constan t f u n c t i o n ) o f (RSS) i s c a l l e d

2 L (I)

o f s o l u t i o n s a s compared with (SS) o r (SP). Suppose C Q ( n ) ,

Q = 0 , :l,

HN(I).

The system (KSS) has a v a s t

I n f a c t , one takes n~ A a r b i t r a r i l y .

a r e t h e t h r e e s o l u t i o n s of (RSS)l.

See, (A.2).

Let

u ( q ; x ) be any s t e p f u n c t i o n i n which u ( q ; x ) t a k e s e i t h e r of c Q [ q ) , Q = 0 , +1, f o r almost a l l x

Let I e ( q ) = t x

I u(n;x)

5 a. ( n ) ) ,

9. = 0 ,

+l.

Then, (RSS)2

Global Bifurcation Diagram reduces t o +I

e (Q) I

where 11

= measure o f I

Thus, f o r any

e (n), a. =

0 , "1

A , i f one c h o o s e s I L [ q ) , 9. = 0, +1, so a s t o s a t i s f y ( 3 . 1 ) ,

t h e c o r r e s p o n d i n g s t e p f u n c t i o n i s a r e d u c e d shadow s o l u t i o n o f (RSS).

See, Fig.

Note t h a t t h e r e a r e many such s t e p f u n c t i o n s , s i n c e o n l y t h e r a t i o o f 11 ( 0 ) I e

3.1.

' s h a s t h e meaning i n ( 3 . 1 ) .

Fig. 3 . 2 . The S i n g u l a r Shadow L i m i t Solution

Fig. 3. I.

1).

Among t h e reduced shadow s o l u t i o n s , we p i c k up t h o s e which s a t i s f y t h e r e l a tion :

and w r i t e them a s (:*(x)

n*) I I:1

g(E;+l ( n * ) ,

(3.2)

,rl*).

g(F-l

(q*),

n*) I I*1

Let u"i(x) be a f u n c t i o n of

{ u"*(x) }

t e n s i o n of which ( c o n s i d e r e d a s a p e r i o d i c f u n c t i o n o n t h e c i r c l e e x a c t l y n i n t e r v a l s o f I:1 Namely, .";(x)

F i n a l l y , l e t u;(x)

{

be a f u n c t i o n o f

tuated sense. the i n t e r v a l

The shadow branch See, F i g . 3 . 2 f o r n

discontinuity of

(;c~-K,z~+K)

has

u"*(x) } which h a s n boundary d i s c o n t i n u i t i e s . {

v a r i a n t under t h e group a c t i o n D .

Theorem 3.1.

[-T,T])

(For t h e d e f i n i t i o n of q*, s e e ( O . l ) . )

( and o f IT1).

i s a f u n c t i o n of

, t h e even ex-

, t h e even e x t e n s i o n of which i s i n -

&(x) }

We have t h e f o l l o w i n g

converges t o (u;(x),n*) =

f o r any

a s dl c 0 i n t h e punc-

NameZy, it converges to u*(xI uniformZy on K

0, where

I x; 1

irl

are t h e p o i n t s of

u * ( x ) and the Zocation o f each d i s c o n t i n u i t y i s determined by

(3.2).

One may t h u s c a l l (u;(x),q*)

the En-limit

solution.

Hiroshi Full1 and Yasurnasa

NlSHlURA

The s e t o f f u n c t i o n s { .“*(x) } can be o b t a i n e d from t h e l i m i t s o l u t i o n uA(x) by a t r a n s l a t i o n , an extension, o r a c o n t r a c t i o n o f i n t e r v a l s o f t h e b l o c k s o f u:(x),

so l o n g a s such an o p e r a t i o n keeps t h e r a t i o

1 I:1 I

d i v i s i o n o f a b l o c k o f u * ( x ) y i e l d s a f u n c t i o n o f { ;;+2k(x) ‘L

Thus, t h e s e t [ u * ( x )

1 I:1 1 .

(Note t h a t a

1 ( f o r some k 2 1 ) .)

may c o n s i s t of a s e t o f o n e - p a r a m e t e r f a m i l i e s o f f u n c t i o n s ,

i n c l u d i n g t h e l i m i t s t a t e u * ( x ) - which we c a l l t h e edge continua.

An example i s

i l l u s t r a t e d i n F i g . 3 . 3 , where t h e c o n t i n u a o f t h e u * ( x ) and u * ( x ) a r e shown. 2 4

Fig.3.3.

of u $ ( x ) and u i i x ) , formed by t r a n s l a t i o n s of b l o c k s .

Edge continua

Note t h a t t h e terminal s t a t e s ( t h e r i g h t and l e f t p i c t u r e s ) are d i f f e r e n t from u f i x ) ( u p p e r ) , and u 2 l x ) ( l o w e r ) , s i n c e they c o n t a i n “sZitst’ a t x

54.

0, 5 2 or

T .

View on t h e s i n g u l a r w a l l By t h e s i n g u l a r w a l l , we mean t h e l i m i t s p a c e R +

( 3

( ~ , ( u , v ) ) )of d l + 0 .

The g o a l o f t h i s s e c t i o n i s t h e s t u d y o f t h e s t r u c t u r e of s o l u t i o n s on t h e s i n g u l a r wall o f : f ( u , v ) = 0, in I,

(RP)

- v xx ‘y v

g(u,v) = 0, on aI.

The system ( R P ) i s c a l l e d t h e reduced problem o f ( S P ) , and i t s s o l u t i o n s reduced

solutions.

However, we a r e o n l y i n t e r e s t e d i n s u c h r e d u c e d s o l u t i o n s t h a t from

which we can e x t r a c t smooth s o l u t i o n s o f (SP) f o r ( s m a l l ) p o s i t i v e d l 4 0 .

Such

l i m i t s o l u t i o n s w i l l be c a l l e d singular s o l u t i o n s ( , a n d a singular branch i f it

Global Bifurcation Diagram consists of a one-parameter family of singular solutions). Let S

denote the set of singular solutions. We associate t o So the follow-

ing asymptotic norm.

For U = (u,v)

/ / u / (=~ lim /I

S0 ’ the asymptotic norm l l U / l s is defined by

(u(E;x),v(E;,x)

ESO

where (u(E;,x),v(E;,x))

is a family of solutions of (SP) which converges to U as

in the punctuated sense, and

E=%

Mimura, Tabata and Hosono [ 4 ] have found a family of singular branches with interior transition layers for sufficiently small a

0. On the singular wall,

these singular branches correspond to the double solid lines in Fig.4.1. At a = U, they start from the singular-shadow s o l u t i o n s (u;,n*) connect to the shadow branches b

(n=1,2,. . . ) , and

hence

(n=1,2,.. . ) at the edge.

Fig. 4.1.

In the following, we consider only one-mode type of solutions for simplicity of presentation. As in 13, one may choose an have u = h(q;v), where h ( q ; v ) = h-l(v) for v Substitution of u

(4.1)

(x)

1 C (I), 0

= 0 to

rt(Fig.4.2).

h(n;v) into (RP)2 leads to a scalor equation for v: v

where G(n;v) = g(h(n;v),v).

Eq. (4.1) with vx

ri E

G(ri;v) = 0 ,

Note that G(n;v) has a discontinuity in v at v = r t .

0 on 2 1 , has an Y -family of strictly increasing solutions V:” < x1

for each n

Let Un’a(x) 1

h(n;V:s

(x)).

According to

Hiroshi FUJIIand Yasumasa NISHILIRA

h a s a d i s c o n t i n u i t y a t x = x*

t h e c o n s t r u c t i o n , t h e f u n c t i o n U:"(x) c o u p l e (U:"

Then, t h e

0 < a < a l , i s an a - f a m i l y o f r e d u c e d s o l u t i o n s f o r each rl

,V:a),

A q u e s t i o n i s t h a t when a m a l l d i f f u s i o n d l > 0 i s i n t r o d u c e d t o (RP)l, u n d e r

what c o n d i t i o n s t h e d i s c o n t i n u i t y of Uy'acan b e smoothed o u t by a n i n t e r i o r t r a n s i tion layer.

The f o l l o w i n g r e s u l t h a s b een o b t a i n e d i n [ 4 ] :

I f the F i f e condition

a s s uumption ( A . 3 ) , t h e r e e x i s t s a u n i q u e s e p a r a t i o n p o i n t (4.2).

Q=Q*,

which s a t i s f i e s

Hence, f o l l o w s an z - f a m i l y o f s i n g u l a r s o l u t i o n s w i t h an i n t e r i o r t r a n s i -

',v:")

t i o n l a y e r 3 = {(u:'

E-]IQf)

x0,

0 < a < z;).

& ' c+llll') u

PI(a)

ctI(o)

E!I(a)

f:I(a)

Fig. 4 . 2 . As i s remarked i n [ l ] , t h i s s i n g u l a r b r an ch w i t h an i n t e r i o r t r a n s i t i o n l a y e r

c e a s e s t o e x i s t a s a r e a c h e s a:. the nonlinearity 1)

See, F i g . 4 . 1 .

Fig.4.2

i l l u s t r a t e s t h a t p a r t of

which i s a c t u a l l y u s e d by a s i n g u l a r s o l u t i o n If:*'" (x) of ( 4 .

( the solid line ).

Since t h e numerical range

(v,(k),vMQ))

o f V:*'l

monotone i n c r e a s i n g w i t h r e s p e c t t o a 1 0 , i t i s e a s i l y s e e n t h a t , as

one of v

m ( a ) and v M ( 2 ) r e a c h e s f i n a l l y t o t h e extremum v a l u e s

(x) i s increases,

of f .

This

i s t h e r e a s o n why t h e s i n g u l a r b r a n c h c e a s e s i t s e x i s t e n c e a t a = z s . I t i s wondered w h eth er t h i s i s a l l t h e e x i s t i n g s i n g u l a r b r a n c h e s , and what happens a t t h e " c r i t i c a l p o i n t " a = a: which h a s be e n

left

on t h e s i n g u l a r w a l l .

This i s a question

open f o r a lo n g t i m e . The answ er i s t h a t t h e re appears

another singular branch f~orncx; upwards t o a

J- 0. S e e , b r o k en l i n e s i n F i g . 4 . 1 .

Global Bifurcation Diagram T h i s new b r a n c h i s c h a r a c t e r i z e d by :hat tion layers.

i t h a s b o t h boundary and i n t e r i o r t r a n s i -

Hence, we s h a l l c a l l it t h e singular branch o i t h boundary and i n t e r i o r

transition layers.

The c r i t i c a l p o i n t ctc w i l l be c a l l e d t h e singular l i m i t p o i n t . 1

The d e t a i l e d c o n s t r u c t i o n and p r o o f s w i l l be p u b l i s h e d e l s e w h e r e . However, i t s h o u l d be remarked t h a t such a s i n g u l a r s o l u t i o n can not be c o n s t r u c t e d w i t h i n t h e F i f e

s e t t i n g a s i n [4]. Moreover, t h e "boundary l a y e r " t h u s c o n s t r u c t e d i s c o m p l e t e l y d i f f e r e n t i n n a t u r e from boundary l a y e r s o b s e r v e d i n D i r i c h l e t b o u n d a r y - v a l u e p r o b lems. The c o n s t r u c t i o n on t h e s i n g u l a r w a l l o f a s i n g u l a r b r a n c h w i t h boundary and i n t e r i o r t r a n s i t i o n l a y e r s can be performed s i m p l y by a d d i n g boundary l a y e r s ( a c t u a l l y , "boundary s l i t s " ) t o (U:*' r e s p o n d i n g t o where t h e s l i t t h e b o t h ends ( t h a t the depths

,V:*'

t~h e s l i t

5 :.

There a r e t h r e e such branches, cor-

(c-l -

o r r i g h t end (

or a t

The e s s e n t i a l p o i n t i n o u r c o n s t r u c t i o n i s a b c _ l ) and ( c , ~

generalized F i f e condition (See, F i g . 4 . 2 )

- c : ~ ) are

determined by the

(4.3)

e x i s t s : a t t h e l e f t ( #,,),

See, Fig.4.4. L

)

b f ( s , v ) ds

Note t h a t t h e p a r t o f n o n l i n e a r i t y

andJ

f ( s , v M ) ds

Sa;l u s e d by t h i s s i n g u l a r s o l u t i o n i s t h e s o l i d

l i n e p l u s (one o r b o t h o f ) t h e d o u b l e s o l i d l i n e s i n F i g . 4 . 2 .

Fig.4.3.

The Singular Solutions with Boundaryand Interior-2'2 a n s i t i o n Layers.

I t is n o t e d t h a t t h e f o u r b r a n c h e s c o n s t r u c t e d i n t h i s way a r e d i f f e r e n t each

o t h e r when t h e y a r e measured by t h e a s y m p t o t i c norm norm. )

11 I[ .

(They have t h e same X

Hiroshi F U J I and I Yasumasa N I S H I U R A

Fig. 4.4. Of

The Dependency on Nonlinearities

Singular Branches.

To s t u d y t h e i n t e r r e l a t i o n of t h e s e b r a n c h e s , o n e may need t h e q u a n t i t i e s : G(rl*;v)dv.

E*(f,g) = Suppose E * ( f , g ) < 0 .

Then, a s 2 t e n d s t o r

t o zero, while t h e depth a t 5 i f E*(f,g) > 0.)

-1

t h e d e p t h of t h e s l i t a t

remains bounded away from z e r o .

tends

(And, v i c e versa

Thus, t h e f o u r s i n g u l a r b r a n c h e s form two "wedges" on t h e s i n g u -

l a r wall a s i n F i g . 4 . 1 , s i n c e t h e a s y m p t o t i c norms o f t h e b r a n c h e s 3 and 9' t h e same v a l u e a t

U-

a;.

The same i s t r u e f o r t h e b r a n c h e s #$# and

take

What happens when one deforms t h e n o n l i n e a r i t y ( f , g ) s o t h a t E * ( f , g ) changes smoothly ?

See, F i g . 4 . 4 .

The f o u r s i n g u l a r b r a n c h e s move smoothly, and when E * (

f , g ) = 0 , t h e t o p s o f t h e two wedges meet t o g e t h e r . E*(f,g) > 0 .

Then, t h e y s p l i t a g a i n f o r

Note t h a t an exchange o f b r a n c h e s o c c u r s i n t h i s p r o c e s s , s i n c e t h e

d e p t h o f t h e s l i t a t i l remains f i n i t e i n s t e a d o f 5

-1

f o r t h e case E*(f,g) > 0.

I t s h o u l d be n o t e d h e r e t h a t the wedges of singular branches on the ualZ are

traces of " h i t t i n g " and "spZitting" of Zimit points o f some branches o f t h e stationary probZem ( S P ) .

Global Bifurcation Diagram 85. -

Discussions - t h e Global View The purpose of d i s c u s s i o n s h e r e i s t o i n t e g r a t e t h e a n a l y t i c a l r e s u l t s i n 5 1

84 i n t o a u n i f i e d view t o t h e g l o b a l b i f u r c a t i o n s t r u c t u r e f o r o u r n o n l i n e a r

d i f f u s i o n systems.

One o f t h e main i n t e r e s t s i s t o s e e t h e mechanism of s u c c e s s -

i v e recovery and l o s i n g o f s t a b i l i t y observed i n primary branches of ( S P ) . (See,

L11.1 The l o c a l s t r u c t u r e of double s i n g u l a r i t i e s placed on t h e t r i v i a l s h e e t ( ( d l , d 2 ) , $)

2 R+

(§I), t h e g l o b a l s t r u c t u r e o n t h e shadow c e i l i n g ( § 2 ) , and t h e

somewhat complex s t r u c t u r e of s i n g u l a r branches on t h e s i n g u l a r w a l l (14) - they a r e all expected t o r e f l e c t t h e r e a l e x i s t i n g b i f u r c a t i o n s t r u c t u r e of t h e nonl i n e a r d i f f u s i o n system. The f i r s t key seems t o be t h e s t r u c t u r e o f c o n t i n u a a t t h e singular-shadow edge.

I n f a c t , an i n t e g r a t i o n of a l l t h e above r e s u l t s , t o g e t h e r with t h e numerica

evidences r e p o r t e d i n [I], may l e a d u s t o a working h y p o t h e s i s

the

edge

continua.

Fig. 5 . 1 Fig. 5.1 Extension o f t h e Edge Continurn; The l i n e - - - - - - - s b ~ s t h e r be c o v e v l i n e of D2,- and D which tend t o t h e Shadow Singular L i m i t s . The two SinguLar L i m i t P o i n t s o f the D s h e e t 2,+ appear here. 1

Two s i n g u l a r branches with boundary s l i t s a r e s a i d t o be terminal branches of D

n,+

(or D

n,-

) i f a t J = 0, they a r e connected

continuum which i n c l u d e s t h e l i m i t s t a t e D

terminal singular branches of DIZ,+,

n,+

by a (one-parametrized) (or D

n,-

Then,

fGr

edge

a pair o f

tUG

t h e r e may e x i s t a sh e e t o f s o l u t i o n s o f ( S P )

Hiroshi FUJIIand Yasurnasa N I S H I U R A

whi-12 connect the two terminal branches, and t o which i n t e r s e c t s t h e primary sheet transuersalZy.

n, f

See, Fig.5.1 (a) for the case of D2,-, and Fig.S.l (b) for

D2,+. Assuming this, the global picture of the Dl-sheet looks like Fig.5.2. The pot-like structure in Fig.l.4 which begins at Z1 on the D2 sheet extends, and as a becomes snmaller, the “loop” expands until it hits the singular wall at Y = a:, where

yields two singular limit points.

As3

< a;,

the loop splits into two

arcs, one is, of course, a cross-section of the primary D1-sheet, and the other is the branch connecting the two terminal singular states of D2, which have a boundary and interior transition layers.

As a

tends to z e r o ,

the latter arc shrinks to the

edge continuum of D 2 , while the former remains as the primary fi

shadow branches.

1,:

Fig. 5.2. The Global Picture of t h e D I p o t .

An important consequence of this picture is that the outer surface of this U1pot is the s t a b l e region of (P), while the inner surface corresponding to the boundary- and interior-layered solutions is the unstable region.

remark should

be made here. There remains a possibility that this stabZe region may have some isolated “unstable islands” encircled by a Hopf secondary bifurcation line. However, it is shown in [ 5 ] by an a p r i o r i estimate that such Hopf points, if exists, cannot exist for sufficiently small

> 0.

We note that such a global picture is supported by ilucrical computations in

Global Bifurcation Diagram [ l ] , and i n f a c t , t h i s has been e s s e n t i a l l y p r e d i c t e d t h e r e .

In [ l ] , we were

not aware of t h e n a t u r e of t h e i n n e r s u r f a c e of t h e p o t , s i n c e t h e boundary- and i n t e r i o r - l a y e r e d s o l u t i o n s and t h e e x i s t e n c e of s i n g u l a r l i m i t p o i n t were not d i s covered y e t . F i g . 5 . 3 shows t h e recovery of s t a b i l i t y of t h e D2-sheet.

T h i s p i c t u r e shows

t h e mechanism of a r e c o v e r y of s t a b i l i t y ; t h i s r e c o v e r y i s a c t u a l l y performed by a s e p a r a t i o n o f a sub-branch which has both boundary- and i n t e r i o r l a y e r s .

a = o

X”

F i g . 5.3.

The Singulur L i m i t Points of D I p -+,and t h e Recovery of D2,_+-branches. I t should be remarked t h a t , a s i s mentioned i n 5 1 ( s e e , a l s o [l]),

the

primary b i f u r c a t i n g D 2 - s h e e t l o s e s s t a b i l i t y on t h e way t o t h e s i n g u l a r wall d 1+ 0. One s e e s an o u t c r o p of t h e s t a b l e r e g i o n o f D2 between

i n Fig.l.5. 233 The above a n a l y s e s suggest t h a t t h e recovery l i n e of s t a b i l i t y of D Z , which s t a r t s

5,2J starts

e n t e r s i n t o t h e singular-shadow edge

r2,3,

e n t e r s i n t o t h e s i n g u l a r wall

192

and

= d l = 0 , while t h e l o s i n g l i n e which

dl = 0 a t Z a t e s t a t t h e s i n g u l a r l i m i t

Hiroshi FUJIIand Yasurnasa N I S H I U R A

p o i n t of D ~ namely, , at3

‘2 = 4 < .

As a r e s u l t , t h e t h e s t a b l e r e g i o n of t h e D 2 -

s h e e t occupies a band-shaped r e g i o n of t h e D2-pot. g e n e r a l t o t h e Dn-sheets (n

2 2).

S i m i l a r s t a t e m e n t s hold i n

See, Fig5.4 f o r t h e c a s e of t h e D2-sheet.

‘2

Fi9.5.4.

The Stable Region of D2-pot.

A s a c o n c l u s i o n , o u r s t u d y s u g g e s t s t h a t t h e r e a l organizing c e n t r e which

control t h e whole b i f u r c a t i o n s t r u c t u r e may l i e on t h e singular w a l l , and e s p e c i a l l y i n t h e singular-shadow edge. tion.

Further studies a r e necessary t o c l a r i f y the s i t u a -

Though t h e r e remains many q u e s t i o n s which a r e n o t answered, we b e l i e v e t h a t

our study may s e r v e a s a f i r s t s t e p

towards t h e g l o b a l b i f u r c a t i o n s t u d y of non-

l i n e a r d i f f u s i o n systems.

Acknowledgements We owe a s p e c i a l thank t o our c o l l e a g u e P r o f . Yuzo HOSONO f o r h i s d i s c u s s i o n . References [ l ] H . F u j i i , M.Mimura and Y.Nishiura, A P i c t u r e of t h e Global B i f u r c a t i o n Diagram i n Ecological I n t e r a c t i n g and D i f f u s i n g Systems, Physica D

Phenomena,

Nonlinear

No.1, 1 - 4 2 (1982).

[ 2 ] H . F u j i i , Numerical P a t t e r n Formation and Group Theory, Computing Methods i n

Applied Sciences and Engineering, Eds. R.Glowinski and J . L . L i o n s - Proc. o f t h e Fourth I n t e r n a t i o n a l Symposium on Computing Methods i n Applied S c i e n c e s and Engineering, North-Holland, 63-81 (1980).

Global Bifurcation Diagram

[ 3 ] M.Mimura, Y.Nishiura and M.Yamaguti, Some Diffusive Prey and Predator Systems and Their Bifurcation Problems, Annals of the New York Academy of Sciences, 316, 490-510 (1979). [ 4 ] M.Mimura, M.Tabata and Y.Hosono, Multiple Solutions of Two-Point Boundary Value Problems of Neumann Type with a Small Parameter, SIAM J . Math. Anal. 11,

613-631 (1980).

[ 5 ] Y.Nishiura, Global Structure of Bifurcating Solutions of Some ReactionDiffusion Systems and their Stability Problem, Computing Methods i n Applied Sciences and Engineering, V , Eds. R.Glowinski and J.L.Lions, North-Holland, 185-204 (1982). [ 6 ] Y.Nishiura, Global Structure of Bifurcating Solutions of Some Reaction-

Diffusion Systems, SIAM J . Math. Anal.

,e, 555-593 (1982).

[ 7 ] A.M.Turing, The Chemical Basis of Morphogenesis, P h i l . Trans. Roy. S o c . , 8237, 37-72 (1952).

This Page Intentionally Left Blank

Lecture Notes in Num. Appl. Anal., 5 , 37-54( 1982) Noti/itieLir PDE in Applied Science. U.S.-Japrin Seminor, Tokyo, 1982

The Navier-Stokes l n i t i a l Value Problem i n

and R e l a t e d Problems

Yoshikazu Giga

Department o f Mathematics Faculty of Science Nagoya University Furo-cho, Chikusa-ku Nagoya 464 JAPAN

We discuss the existence of a strong solution of the nonstationary Navier-Stokes system in Lp spaces. Our results generalize Lz results of Kato and Fujita. To establish Lp theory we study the Stokes system and construct the resolvent of the Stokes operator.

I n t r o d u c t i o n and summary of r e s u l t s .

This is an introduction to the articles [3-71 which concern the Stokes and the Navier-Stokes equations. Let

(n 2 2 )

be a bounded domain in Rn

with smooth boundary

consider the Navier-Stokes initial value problem concerning velocity un)

and pressure p:

-au _ at

i;u + ju,grad)u

grad p

(O,T),

div u = 0

S x (O,T),

= 0

u(x,O) = a(x)

with given external force

in D

f and initial velocity a.

(u,grad)

n . C I? a x , j=l 1

Here

S. We

u = (u

,-*.,

Yoshikazu CIGA Many mathematicians, J. Leray, E. Hopf,... have studied the solvability o f

this problem; see Ladyzhenskaya [9 ] and Temam [14] and papers cited there. On the existence of a regular (in time) solution there is a celebrated work

established by Kato and Fujita [1,8]. Let us quickly review their theory. As is well known (see [ 9 ] ) the space (L2(D))"

admits the orthogonal Helmholtz decomposition

where Wm(D)

is the Sobolev space of order m

P be the orthogonal projection from

(L2(D))"

form (I) to the evolution equation in

where Fu

-P(u,grad)u.

operator in X2

such that W (D) = L (D). onto

X2.

Let

Using P, we can trans-

Here the operator A = A

-PA is called t h e Stokes

with the domain

D(A) = l u

F o r simplicity we assume

2 W2(D);

u = 0 on S } n

X2.

Pf = 0. Applying semigroup theory, Kato and Fujita

have proved the existence o f a unique global strong solution of (11) f o r every a

when the space dimension n

is two. While, when n

= 3,

proved the existence of a unique local strong solution of (11) f o r where 'A

denotes the fractional power of A.

they have a

D(A1'4),

Navier-Stokes Initial Value Problem in Lp Our aim is to show the existence o f a unique strong solution without

assuming that the initial velocity a theory

(1 < p

theory also answers

to his problem.

To develop o u r theory the crucial step i s to derive the following two properties of the Stokes operator A . here P’ X 2 and A2, respectively.

Theorem 1 (131). X

P’

The operator

-A

and

are

I, -analogues of

generates a bounded a n a l y t i c semigroup i n

Moreover, t h e estimate

i s v a l i d w i t h constant Theorem 2 ( [ 4 ] ) .

The space

[Xp,D(Ap)Ja, where Remark.

When

C , where

O(A‘)

ilfll

f in

i s t h e complex i n t e r p o l a t i o n space

0 _ 0, p+6 >

-.21

the above estimate i s v a l i d f o r

p = n

1 6 = -4,

We now consider (11) in

e = 4' L

p =

1 -. 2

lhe existence result follows from Theorem 1

and Lemma 1. Our method to prove is similar to that of Kato and Fujita. Theorem 3 ([S]).

Fix

n/2p - 1/2 -5

such t h a t

(i)

For some

[o,T)

D(A~),

i s continuous from

(0,T)

D(Aa)

IIAau(t)ll

1. Assume t h a t

> 0,

u i s continuous from

(ii) u

Moreover, u

- 0.

, where

11 fll

denotes the norm of f in Xn.

Theorem 1 implies that

This yields the estimate /IACluo(t)ll with

KaO t-a, a 1 0

Navier-Stokes Initial Value Problem in L'

We consider the following problem:

i s small enough, i s i t p o s s i b l e t o e st i m a t e //Aaum(t)11 If KaO liam t-a from above w i t h constant Kam =< K < m such t h a t K i s independent

m ?

Theanswer is y e s and we will give a proof; see [ 1 , 5 , 8 ] . some m 1 0, um(t)

satisfies /IAaum(t)I/

Lemma 1 with

Suppose that for

6 = 1/4,

@ =

1/4,

Kam t-'

= 1/2

for all

a _>- 0.

implies

We thus have

with

Ka,m+l where

KaO

Ca+&M B(1-6-a,6) K

is the beta function. This implies that

B(a,b)

for each m

0 as a element of C ( [ O , T ] ,

0 5- a < 3/4 and that

Put km = max{K

Om'

u,(t)

Xn) n C ( ( O , T ] ,

pm'

u,(t) D(Aa))

satisfies

1 and note the definition of K am to get

is well-defined for all

Yoshikazu CICA

where C

is a constant depending only on A.

An elementary calculation shows

that if k

(C)

< -

4c ’

then for each m 2- 1 the estimates

< K Ka,m+l = a0

are valid for constant K.

‘a+6

M B(1-6-a,6) K 2

Ka,

We thus have

/IAa~m+l(t)ll

Ka t-a,

0 _< a

t h e o p e r a t o r - n o r m of

1 \,I 1 ' )

l o g s ~ { l x l / l x / ' , l x l l / l x I } . A E G(X,l,B)

d e f i n e d by

IAly,X

dist(

(X,Y,A,f).

w i l l be c a l l e d t h e parameters of t h e system

REMARK 2 . 1 .

R+

a r e monotone i n c r e a s i n g f u n c t i o n s on

p,,

\wlYv

r =

(vIy

(We w r i t e

f o r sup{a,b} . )

In some

p r o b l e m s , however, (Nl) h o l d s w i t h

xl(

s t r e n g t h e n i n g some of t h e r e s u l t s .

S i m i l a r remarks apply t o o t h e r p a r a m e t e r s .

A sequence

DEFINITION 2 . 2 tions) t o

(Q) on

{u }

/wly)

r e p l a c e d by

A,(

i s c a l l e d a null s e q u e n c e

[O,T] i f t h e

a r e bounded i n

lwlx)

(Of

C([O,T];Y

, thereby

approximate solu-

n Lip,([O,T];X)

and d u n / d t + A(un)un

(2.1) Here

E Lip*([O,T];X)

function

u E Lm([O,T];X)

EC([O,T];X) solutions

f(un) -+ 0

means t h a t

so t h a t

by v i r t u e of (A2) and

i s a n i n d e f i n i t e Bochner i n t e g r a l of a

du / d t = u

(A3).

w i l l be m o s t l y p i e c e w i s e

Lm( [O,T];X)

Note t h a t

A(un)un

[In t h e e x i s t e n c e p r o o f , a p p r o x i m a t e

C1([O,T];X) and n o m e a s u r e t h e o r y w i l l

be required. ]

DEFINITION 2 . 3 . p a r t i t i o n of

[O,T]

closed subinterval

C([O,T];X)

i s c a l l e d a weak s o l u t i o n t o (Q) i f t h e r e i s a

i n t o a f i n i t e number of s u b i n t e r v a l s s u c h t h a t on e a c h

I, u

is the l i m i t i n

C(1;X) o f a n u l l s e q u e n c e .

[Hence

Quasi-linear Equations of Evolution u

E Lip( [O,Tl;X)

DEFINITION 2 . 4

el (B

and

Let

(I))

REMARK '2.5

By(')

he t h e b a l l i n

i t s closure i n

As is easily seen,

r > 0

d e f i n e d as t h e infimum of

We denote by

such t h a t

t h e union of

includ 0,

and r a d i u s

i s a Banach space w i t h t h e norm

Y c Yo c X , w i t h t h e i n c l u s i o n s c o n t i n u o u s .

X = C[O,l],

with c e n t e r

Yo w i l l b e c a l l e d t h e l o c a l c l o s u r e of Y

r > 0.

for a l l

For example, l e t

i s a c l o s e d suhspace of

YO. THEOREM I .

Given

H E Yo

T z 0 , depending o n l y on

t h e r e is

(and t h e parameters o f t h e s y s t e m ) , and a unique weak s o l u t i o n

10lyo (Q)

(existence)

[O,T]

s u b s e t of C ( [O,T];X)

H .

The map

H u

i s bounded on a bounded

B ( [O,T];Yo), and i s continuous from t h e X-topology t o

u(0) =

with

w i t h i n a bounded s e t of

bounded f u n c t i o n s on

Yo.

[Here

B(I;Yo)

denotes t h e s e t o f

Yo. We cannot r e p l a c e it with

Lm(I;Yo) s i n c e t h e

f u n c t i o n s considered may n o t be s t r o n g l y measurable. 1

R W K 2.6

A f t e r i n t r o d u c i n g t h e space

t o all

I--$

A(w)

Y C X.

Yo,

one might t r y t o extend t h e map

Yo, t o he a b l e t o work i n t h e space p a i r

There a r e two d i f f i c u l t i e s i n t h i s a t t e m p t .

g e n e r a l method t o extend

A(w)

instead

F i r s t , t h e r e i s no

i n t h i s manner s o a s t o make

Second, even if t h i s i s p o s s i b l e ,

Y°C

A ( w ) E B(YO,X).

may n o t become a g e n e r a t o r i n

Yo = L i p

may be expected from t h e t y p i c a l example (HS) i n which

Yo.

This

(see section 5).

In f a c t t h e r e a r e no r e a s o n a b l e C 0-semigroups on t h e s p a c e Lip. REMARK 2 . 7 .

i s r e f l e x i v e , we have

Theorem I helongs t o i s a solution t o

s o l u t i o n t o (Q).

Cw([O,T];Y), where

(Q) with

Yo = Y Cw

and t h e s o l u t i o n

i n d i c a t e s weak c o n t i n u i t y , and

d u / d t E Cw([O,T];X).

Thus

i s almost a s t r o n g

I n f a v o r a b l e c a s e s one may be a b l e t o show t h a t

u E C([O,T];Y)

Toshio KATO

( s t r o n g s o l u t i o n ) by a u x i l i a r y c o n s i d e r a t i o n s s u c h a s t h o s e g i v e n i n [8;Remark

5.31 3. S k e t c h o f t h e p r o o f o f Theorem I For s i m p l i c i t y we assume

f = 0.

[7,9,10],we u s e s u c c e s s i v e a p p r o x i m a t i o n b a s e d on t h e

A s i n p r e v i o u s works

[81.

theory of l i n e a r evolution equations given i n ( a ) F i r s t we assume t h a t

0 G Y , and f i n d a b a l l i n

expect t o confine t h e v a l u e s of t h e approximate s o l u t i o n s val

[G,T].

R > IB/,

To t h i s e n d , f i x a n

i n which we can u

f o r a fixed inter-

R', R"

Then w e c a n d e t e r m i n e

such t h a t

R' = R e x p [ X 2 ( R ) ]

I n d e e d , i n view o f (N3) it s u f f i c e s t o s e t [A2(ri)]

A l l approximate s o l u t i o n s

below w i l l t a k e v a l u e s in i n a l l t h e parameters

REhIARK 3.1. R ,

R', R"

Al(r), and

and

and r e l a t e d f u n c t i o n s we i n t r o d u c e

(R"), s o t h a t we s h a l l b e a b l e t o s e t

o n l y on

I,, T ( i n t r o d u c e d b e l o w ) a y e d e t e r m i n e d by

(3.2) where

/@Iy

0 b u t n o t n e c e s s a r i l y on

( b ) Let

and

For e a c h

v(t) E B

(R'),

v E C([O,T];Y)

/v(t)-v(s)/,

v E E, l e t

depended

such t h a t

Llt-sl,

be a step-function approximation for

a r e a s u b s e t of t h e v a l u e s of

f o l l o w s from ( 3 . 2 ) and (N2), ( N h ) , ( A l ) t h a t

AV(t) = A(v(t))

form s t a b l e f a m i l i e s of g e n e r a t o r s i n

w i t h uniform s t a b i l i t y c o n s t a n t s . d

{Uv(t,s)}

a r e constants t o b e determined.

it i s i m p l i e d t h a t t h e v a l u e s o f

A(?(t))

I0Iy

only.

be t h e s e t of a l l functions

v ( 0 ) = M,

r = R"

...,p h ( r ) .

T h i s i s a g r e a t a d v a n t a g e o v e r t h e s i t u a t i o n i n "7,101, where

only.

R" = R ' e x p

associated with

v ( b y which v ).

and A v ( t ) =

as w e l l as i n

Y (see

[8]),

T h e r e f o r e , t h e r e i s an e v o l u t i o n o p e r a t o r

{A ( t ) ] ( s e e [8]; h e r e we may d i s r e g a r d f i n i t e l y many

Quasi-linear Equations of Evolution

IJV(t,s) ) .

d i s c o n t i n u i t i e s f o r t h e derivatives of

I t f o l l o w s from t h e u n i f o r m e s t i m a t e s f o r t h e s t a b i l i t y c o n s t a n t s t h a t

u :U v ( . ,010E E

and

any s t e p f u n c t i o n a p p r o x i m a t i o n

3 of

Moreover, t h e map

and

u E O?

can

Lm( [O,T];X), by r e d u c i n g t h e s i z e

be shown t o b e a c o n t r a c t i o n i n t h e m e t r i c o f

v E E

a r e chosen a p p r o p r i a t e l y , f o r any

i f necessary. ( c ) We can now c o n s t r u c t a n u l l s e q u e n c e

un(0) = 0'. Assuming t h a t

Sin

function approximation

L ([O,T];X)-norm.

(4)on

[O,T]

such t h a t

h a s been c o n s t r u c t e d , we c h o o s e a s t e p

such t h a t

denotes t h e

{un}

~ =+ OCn~

Then

l[ 0

with

are in

6. J

C([O,T];X).

a r e bounded i n

we may C1+p+o

shows t h a t

Sinre the

u. a r e bounded i n J

B([O,T];Y)

(because

the

(For

i s s y s t e m a t i c a l l y used. )

Moreover, we s h a l l show t h a t t h e dependence

To t h i s end l e t

s u f f i c i e n t l y s m a l l , a l t h o u g h t h e p r o o f i s by no means t r i v i a l .

( c ) Thus we have shown t h a t (HS) h a s a u n i q u e s t r o n g s o l u t i o n

T h i s c a n h e done s i m u l t a n e o u s l y f o r

d e t a i l s c f . Nakata [13], where norm-compression

-1

(X,Y,A)

i n t h e system

Iu[[pl i n ( 5 . 9 )

s u f f i c i e n t l y s m a l l , t h e a s s o c i a t e d seminorm

becomes s m a l l so t h a t we h a v e

Indeed, t h e r e i s nothing

Y ) , it f o l l o w s t h a t

u. + u J

C( [@,T];Y).

Toshio KATO

An example of compressible system Let us i l l u s t r a t e t h e n o t i o n o f compressible systems by a simple example.

EXAMPLE

6.1.

Consider t h e f i r s t - o r d e r s c a l a r equation

(6.1)

+uu

= 0 ,

t >- O .

x E R ,

choose X = X = H (R),

(6.2)

It i s known ( s e e [ 9 ] ) t h a t A

YA = H 3( R ) ,

Y = H2 ( R ) , (X,Y,A)

We s h a l l show t h a t

i s a "good" system.

(X,Y,A)

i s compressible t o

(X,Y,A).

parameters we have t o c o n s i d e r a r e A

B2,E

l(A(w)+A)ulE,;

, which

= cr

so t h a t

_> (A -

c[wlE,~)luls,;

i s independent o f

, and

A3,E

1,E

3,E

Among them,

( r ) 2 B,(r)

, etc.

Hence w e can t a k e A

This shows t h a t

(X,Y,A)

It i s i n s t r u c t i v e t o s e e what happens i f i n t h e above example we

REMARK 6 . 2 .

IuI2E,Y

B2,E

Thus t h e only

(X,Y ,A).

compressible t o

replace

may choose t h e norms

, etc.

i s n o n t r i v i a l , s i n c e it i s e a s y t o see t h a t

It follows t h a t

B2,€(r)

[

B1,E

, we

X = X

Since

I n t h i s problem we do not need v a r i a b l e norms

only

A ( w ) = Wdx

H1(R)

2 = luly

B2,€

luxxl

and 2

; by

H 2 ( R ) , with

In t h i s case

could not s t a y bounded as

(X,Y,A) E

-0.

luly = IuI2

3 + IuXI-

and

i s not a "good" system, Indeed, t h e b e s t e s t i m a t e one

Quasi-linear Equations of Evolution can e x p e c t of t h e s o r t o f

(6.4) w i l l

I ( A ( ~ ) u , u ) ~ , 2y /c l w x x l ( l u l

5 A

This gives

62,E(r)=

-1 r

7 lux/-

-1 CE

blows up as

Iuxx/2)

IWIE,;l~lE,y

, which

-0.

Footnotes 1.

T h i s work was p a r t i a l l y s u p p o r t e d by NSF G r a n t MCS 79-02578.

References [l] C e s a r i , L . , A boundary v a l u e p r o b l e m f o r q u a s i l i n e a r h y p e r b o l i c s y s t e m s i n t h e S c h a u d e r c a n o n i c form, Ann. S c u o l a Norm. Sup. P i s a [2]

C i n q u i n i C i b r a r i o , M.

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1964).

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Some e x i s t e n c e t h e o r e m s f o r h y p e r b o l i c s y s t e m s o f p a r t i a l

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[5]

F r i e d r i c h s , K . O., N o n l i n e a r h y p e r b o l i c d i f f e r e n t i a l e q u a t i o n s o f two i n d e p e n d e n t v a r i a b l e s , Amer. J . Math. 70 ( 1 9 4 8 ) , 555-589.

[6]

Hartman, ?. and W i n t n e r , A . , i o n s , h e r . J . Math.

[7]

Hughes, T . 3 . R . ,

On t h e h y p e r b o l i c p a r t i a l d i f f e r e n t i a l e q u a t -

(1952),

834-864.

K a t o , T . , a n d Marsden, J. R . ,

Well-posed q u a s i - l i n e a r

second-order h y p e r b o lic systems w i t h a p p l i c a t i o n s t o n o n l i n e a r elastodynamics a n d g e n e r a l r e l a t i v i t y , Arch. R a t i o n a l Mech. A n a l . 63 ( 1 9 7 7 ) , 273-294.

76 [8]

Toshio KATO Kato,

T., L i n e a r e v o l u t i o n e q u a t i o n s o f " h y p e r b o l i c " t y p e , J. F a c . S c i . Univ.

Tokyo, S e c . I , 1 7

(197O), 241-258.

[ 9 ] K a t o , T., Q u a s i - l i n e a r e q u a t i o n s of e v o l u t i o n , w i t h a p p l i c a t i o n s t o p a r t i a l d i f f e r e n t i a l e q u a t i o n s , S p e c t r a l Theory and D i f f e r e n t i a l E q u a t i o n s , L e c t u r e Notes i n Math.,

448 ( S p r i n g e r 1975, p p . 25-70).

[lo] K a t o , T . , L i n e a r and q u a s i - l i n e a r e q u a t i o n s o f e v o l u t i o n o f h y p e r b o l i c t y p e , C. I . M . E . , I1 C I C L O ,

1976, H y p e r b o l i c i t y , p p . 125-191.

[Ill Kato, T . , The Cauchy p r o b l e m f o r t h e Korteweg-de V r i e s e q u a t i o n , N o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s a n d t h e i r a p p l i c a t i o n s , i n : B r & z i s , H . and L i o n s J . L . ( e d s . ) , C o l l e g e de F r a n c e Seminar. VOL.

( P i t m a n 1 9 8 0 , p p . 293-

307). [12] K a t o , T . , On t h e Cauchy problem f o r t h e ( g e n e r a l i z e d ) Korteweg-de V r i e s e q u a t i o n , Advances i n Mathematics S u p p l e m e n t a r y S t u d i e s (Academic P r e s s ,

t o appear).

[13] Nakata, M . , Q u a s i - l i n e a r e v o l u t i o n e q u a t i o n i n n o n r e f l e x i v e Banach s p a c e s , with a p p l i c a t i o n s t o hyperbolic systems, D i s s e r t a t i o n , University of C a l i f o r n i a , Berkeley, 1983.

[ 1 4 ] S c h a u d e r , < J . , C a u c h y ' s c h e s Problem f c r p a r t i e l l e D i f f e r e n t i a l g l e i c h u n g e n e r s t e r Ordnung.

Anwendurigen e i n i g e r s i c h a u f d i e a b s o l u t b e t r z g e d e r Ldsungen

b e z i e h e n d e n Abschxtzungen, Commentarii Math. H e l v . 9 ( 1 9 3 7 ) , 263-283.

Lecture Notes in Num. Appl. Anal., 5, 77-100 (1982) Nonlinear PDE in Applied Science. U.S.-Japan Seminar, Tokyo, 1982

Asymptotic Behaviors of the S o l u t i o n

of an E l l i p t i c Equation w i t h Penalty Terms

Hideo Kawarada* and Takao Hanada**

* Department of Applied Physics, Faculty of Engineering, University of Tokyo Bunkyo-ku, Tokyo 113, JAPAN ** Department of Information Mathematics, The University of Electro-Communications 1-5-1, Chofugaoka, Chofu-shi, Tokyo 182, JAPAN 1.

Introduction Let 0

satisfy (i)

be connected domain in R with smooth boundary .'I

C22Eo;

(ii) R l = C 2 - Q o

is a connected domain; (iK) the measure of

an is positive or a1 is unbounded; (iv) aC2 is smooth

an:l)r\

so as to

Take

(see

F1g.l).

We shall consider the boundary value problem defined in R for every E > O and a,B ER.

Find $' =

(1.3)

in Q0,

such that

on I'

Hideo KAWARADA and Takao HANADA

$14

and

(1x1 = x + x + - ) ,

Here n i s t h e outw,ird n o r m a l on

t o $and lo ho i s a p o s i t i v e c o n s t a n t .

I t is

fo u n d i n L i o n s ( [ 3 ] , C h a p t e r 1, p.80)

t h a t t h e b o u n d a r y c o n d i t i o n w hic h t h e l i m i t

f u n c t i o n of

i s c l a s s i f i e d i n t o t h r e e t y p e s , which

as E+O

s a t i s f i e s on

depend upon t h e r e l a t i v e v a l u e o f a and 0. I n t h i s p a p e r , we s t u d y a n a s y m p t o t i c b e h a v i o r o f $I' enough.

We now summarize t h e c o n t e n t s of t h i s p a p e r .

Theorems.

when

i s small

Section 2 includes four

I n s e c t i o n 3 , w e p r e p a r e some Lemmas f o r t h e p r o o f s o f Theorems.

S e c t i o n s from 4 t o 7 a r e d e v o t e d t o t h e p r o o f s of Theor'ems.

Theorems

2.1

We p u t

Then ( 1 . 1 ) - ( 1 . 5 )

are r e f o r m u l a t e d as follows:

Find QEC K

such t h a t

fvdx

'v C K.

T h e r e e x i s t s a u n i q u e s o l u t i o n Q E (K t ) of ( 2 . 2 ) v = q E i n (2.2),

(2.3)

w e see t h a t

f o r 'f

i s u n i f o r m l y bounded in

CH-'(O0).

Putting

Asymptotic Behaviors of the Solutions where C d e p e n d s upon o n l y t h e d a t a f . When E t e n d s t o z e r o , w e c a n e x t r a c t a s e q u e n c e

( n = l , 2,

...)

such t h a t

weakly i n H (0 ) .

(2.4)

Then

1 0 0

L e t v C H (Q ) and i n (2.2)

b e t h e z e r o e x t e n s i o n of v t o R .

(R) y i e l d s

f o r :t.H

(2.6)

from which, w e have

- A Q 00

(2.7)

m-1

I f we assume f t H

0 0

(a,)

= f

(mLO),

i n H-’(.Q~).

t h e n we h a v e

By t h e t r a c e t h e o r e m (NeFas [ 5 1 ) ,

Moreover, Q0 s a t i s f i e s on

Theorem (a)

d3) Suppose

If 6 >

(2.10)

101,

then

r: m- 1

f tH

(Q,) (m,O).

Passing t o the l i m i t

Hideo KAWARADA and Takao HANADA

80 (b)

I f 6 = a >0 , t h e n

(c)

I f B < a and a > 0, t h e n

= o

(2 .1 2 )

Remark 1

There a l s o h o l d s $

1 m-2 H (r).

= O i n t h e case a + 6'0

a nd a < 0 u n d e r t h e same

ass ump t i o n .

2.2

We now s t a t e o u r mai n r e s u l t as f o l l o w s .

Theorem 2 (a)

S u p p o se f E H m ( Q ) (m,O) 0

and l e t

b e small enough.

I f 5 > la\, t h e n

where JI0 s a t i s f i e s (2.10).

(b)

I f 5 = a > 0, t h e n

6,0

where

(c)

s a t i s f i e s (2.11).

I f 161 < a , t h e n

1 m+(2.15)

where

$o0 s a t i s f i e s ( 2 . 1 2 ) .

(r)

Asymptotic Behaviors of the Solutions 2.3

By u s i n g ( a ) of Theorem 2 , w e h a v e t h e r e g u l a r i t y r e s u l t s a b o u t

Theorem 3

Suppose f & H

(no) ( k 1 5 ) a n d l e t

Ji'.

b e s m a l l enough.

I f 8 > / a / , then

2.4

The m o t i v a t i o n of t h i s p a p e r c o n s i s t s i n t h e i n t e g r a t e d p e n a l t y m e t h o d

p r e s e n t e d by o n e o f t h e a u t h o r [2]. The m a t h e m a t i c a l j u s t i f i c a t i o n of t h i s method was d o n e i n t h e s e n s e o f d i s t r i b u t i o n . prove t h e key-point Theorem 4

I f w e u s e ( a ) o f Theorem 2 , w e c a n

o f t h i s method i n t h e framework o f t h e S o b o r e v s p a c e .

Suppose f t H

(a0 ) ( m L O )

and l e t

I f @ > la/,t h e n

Here s s t a n d s f o r t h e l e n g t h of t h e a r c a l o n g

b e s m a l l enough.

Hideo KAWARADAand Takao HANADA Preliminaries

The a i m o f t h i s s e c t i o n i s t o g i v e some p r e p a r a t o r y lemmas w h i c h w i l l b e n e e d e d i n t h e p r o o f s of Theorems. 3.1

We f i r s t i n t r o d u c e some o p e r a t o r s d e f i n e d b e t w e e n t r a c e s on

(i)

D e f i n e t h e mapping

$a i s t h e s o l u t i o n o f t h e p r o b l e m ;

(3.2)

-A$

where f 6 H

(ii)

m-1

(no)

+ A

(iii)

(m2O).

D e f i n e t h e mapping

i s t h e s o l u t i o n of (3.2)

(3.5)

$ = f

( E ~ - ~

w i t h f z 0 and t h e boundary c o n d i t i o n

= b.

D e f i n e t h e mapping

(3.6)

i s t h e s o l u t i o n o f t h e problem;

(3.7)

- - E

a+B

.A $ + $ = O

ill

Asymptotic Behaviors of the Solutions

$Ian

(3.9)

= 0

and

JI * 0

(1x1

-).

1 m +W e d e n o t e T,:

and RE by t h e r e s t r i c t i o n of T f , SE and R" t o H m

'(r).

But we

abbreviate the s u f f i x m hereafter,

3.2

1 m +-j Lemma 1

T (a) f

(3.10)

where To i m p l i e s T L e t JI

Proof --

T (b) = T ( a - b ) f 0

( Y = a , b ) b e t h e s o l u t i o n o f ( 3 . 2 ) u n d e r t h e boundary c o n d i t i o n

-Jib.

Then

f=O'

$Ir

(3.11) Put Y = $

(r).

L e t a , b be a r b i t r a r y i n H

= Y.

I satisfies

(3.12)

-AY

(3.13)

Ylr

+ A = a

0Y = 3

no,

Then

2 1

(3.14)

= To(a-b)

On t h e o t h e r h a n d ,

(3.15)

From ( 3 . 1 4 ) and ( 3 . 1 5 ) f o l l o w s (3.10).

Here w e s h o u l d n o t e t h a t To i s

l i n e a r and Tf is n o n - l i n e a r .

Lemma 2

1 m+-

1 T

and S f l

a r e homeomorphic from :-'(F)

m--

m+T H

(r)

to H

1 '(r)

f o r a n y m,O.

to H

'(r)

and R

i s homeomorphic from

Hideo K A W A K A Dand A Takao HANADA

Proof -

m+-

i s i n j e c t i v e from H

T (a) f

2(r)

m-into H

2(r).

In f a c t , l e t a , b C H

m+- 1 2

(r)

Then, by (3.10)

Suppose T ( a ) = T f ( b ) . f

(a\ b).

T (b) = T ( a - b ) k 0 f 0

b e c a u s e of t h e s t r o n g maximum p r i n c i p l e u n d e r t h e a s s u m p t i o n h

This is

a contradiction.

i s s u r j e c t i v e from H f 1

m + 1T (r)

onto

c-'(r).

I n f a c t , w e c h o o s e any

m+-

bt H

'(r).

Then, t h e f o l l o w i n g p r o b l e m :

-a*

(3.16)

= f

(3.17)

has a unique s o l u t i o n @

m+-

E H

(3.18)

Hm+'(O

) if h

> 0, w h i c h s a t i s f i e s

1 2(1')

I t is c h e c k e d t h a t T f and ( T f )

(see

and

-1

b = Tf(Jlblr).

m+are c o n t i n u o u s between H

2(r)

a n d)T('-;

[ll). m+-1

Summing up l o , 2" a n d 3 " , w e see t h a t T f i s a homeomorphism f r o m H

(r)

1 onto

;-'(r).

The r e p e a t e d u s e of t h e a b o v e a r g u m e n t s g i v e s t h a t ( R )

1 m+SE are a l s o homeomorphic b e t w e e n H

3.3

2(r)

m-and H

-1

and

1 2

(r).

Here w e g i v e t h e e s t i m a t e s o f t h e norm of RE a n d SE, w h i c h are c r u c i a l f o r

t h e p r o o f of Theorems 1 a n d 2 . Lemma 3

Let

b e s m a l l enough a n d s u p p o s e B l a a n d m 2 0 .

Then

Asymptotic Behaviors of the Solutions

and

Proof

U si n g G r e e n ’ s f o r m u l a i n t h e probl em d e f i n i n g R E , we h a v e

From (3.2 2 ) i t f o l l o w s

and

(3 .2 4 )

Usi n g t h e s t a n d a r d t e c h n i q u e t o raise up t h e r e g u l a r i t y p r o p e r t y of t h e s o l u t i o n o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s , w e o b t a i n (3. 19) and ( 3 . 2 0 ) . m+-1 2 R e w r i t i n g ( 3 . 5 ) w i t h an a i d o f To a n d R E , w e ha ve f o r V a E H (r)

Hideo K A W A R A D A and Takao H A N A D A

m+2(r) Here we have used t h e continuity of To from H

Lemma 4

(b)

Let

a + @>

g-'(r).

be small enough and m,O.

0, then

'bCH Proof -

We prove this lemma in the two cases.

2 special case R = R = (x,,x,) 1 c

10 < xl,

-- < x2 < + -1

1 m+2

In the first case, we prove the

= R ) by using the fourier -

transformation. Subsequently, we give the plan of the proof in the general geometry

1" Let

and

(r).

Asymptotic Behaviors of the Solutions Here I)~(x,,x,)

i s t h e s o l u t i o n of t h e problem (3.7)-(3.9).

Solving (3.30) and (3.31), we have

From (3.32)

From (3.34), i t follows

87 4 Then JI s a t i s f i e s

Hideo K A W A R A D and A Takao HANADA

Hence w e o b t a i n

R e p e a t i n g t h e s i m u l a r a r g u m e n t s as a b o v e , w e c o n c l u d e ( 3 . 2 7 ) - ( 3 . 2 9 )

The domain R1 i s a r e g u l a r s i m p l y

2" L e t u s now d e a l w i t h t h e g e n e r a l c a s e .

c o n n e c t e d domain; t h e n t h e r e e x i s t s a ( f i x e d ) r e g u l a r c o n f o r m a l mapping

w = f ( z ) = u + i u 2 ( z = x + i x ) which maps R1 i n t o R t . 1 1 2

i s mapped i n t o t h e u - a x i s of w-plane.

yE = *€(f-l(w))

A s a matter of f a c t ,

Then t h e t r a n s f o r m e d s o l u t i o n

satisfies

(3.37)

By means o f t h e i t e r a t i v e method p r o p o s e d i n t h e t h e o r y of s i n g u l a r p e r t u r b a t i o n ( s e e [ 3 ] ) , Y E is a s y m p t o t i c a l l y d e v e l o p e d i n t h e f o l l o w i n g way:

Using ( 3 . 3 9 ) , w e o b t a i n (3.26)-(3.29)

3.4

(see the appendix).

Define

Then w e h a v e

Lemma 5

Let

-to

( n = l , 2,

... ) .

Then

Asymptotic Behaviors of the Solutions

1 (3.42)

y E n -t

weakly i n H

(3.43)

Tf(9 n,

(r),

--1

-t

weakly i n H

Tf(Y )

(r). I -

Proof

Recalling (2.4),

Then, we d e n o t e by $ (3.44)

-A$

(y = a , b )

+ XoJ'

Let a , b be a r b i t r a r y i n B

(r).

t h e s o l u t i o n of t h e problem: in

= 0

$1,

(3.45)

(3.42) i s obvious.

= Y.

By u s i n g G r e e n ' s f o r m u l a , w e have

(3.46)

($bA$a-$aA$b)dx

Using ( 3 . 1 0 ) and t a k i n g a=;"-$P0

i n (3.46),

I Suppose u = a + 8 + p ( a - 8 ) > 0 ( p t R)

Lemma 6

n(P-1) ( a - 8 )

(3.48)

Then

E E ( ~ - ~ ) *-t G0 ~

--I strongly i n H Proof -

By u s i n g t h e d e f i n i t i o n of T f and SE, Tf(yE) =

(3.49)

E 2a -

(r)

( 1 . 4 ) i s r e w r i t t e n as f o l l o w s :

(YE).

Taking a = y

and

E = E

-to.

i n ( 3 . 2 6 ) f o r m = O and s u b s t i t u t i n g ( 3 . 4 9 ) ,

w e have

Hideo KAWARADA and Takao HANADA

Let E ~ + O . Then w e c o n c l u d e ( 3 . 4 8 ) w i t h a n a i d o f ( 2 . 3 ) .

Proof o f Theorem 1 L e t p = 0 and 8 > / a / i n t h e a s s u m p t i o n of Lemma 6 .

(a)

Then

= a + B > 0 and

( 3 . 4 8 ) becomes

By ( 3 . 4 2 ) and ( 3 . 4 3 1 ,

(4.2)

(b)

L e t a = B > 0.

Then

0 =

2a > 0 and ( 3 . 4 8 ) becomes

(4.3) which i m p l i e s

(4.4)

by t h e d e f i n i t i o n of T f and

(c)

90 .

Let p =I, a > 6 a n d a > 0.

T h e n u = Z a > 0 a n d ( 3 . 4 8 ) becomes

(4.5) from which

(4.6)

Combining ( 2 . 8 ) w i t h t h e r e s u l t s o b t a i n e d a b o v e , w e c o n c l u d e ( 2 . 1 0 ) - ( 2 . 1 2 ) .

Asymptotic Behaviors of the Solutions

Proof of Theorem 2

5. 5.1

Using ( 3 . 4 9 ) ,

m+, Find a t H

t h e problem ( 1 . 1 ) - ( 1 . 5 )

(r)

such t h a t

T (a) = f

(5.1)

2a . S

H e r e a f t e r we c a l l (5.1) s o l u t i o n of (5.1)

i s t r a n s f o r m e d i n t o t h e f o l l o w i n g one:

(a).

the transmission equation.

is equal t o the t r a c e

A s a m a t t e r of f a c t , t h e

o f t h e s o l u t i o n of t h e p r o b l e m

(1.1)-(1.5). 1

m+5.2

'(r).

L e t b be a r b i t r a r y i n H

Then, combining (5.1)

and ( 3 . 1 0 ) , w e h a v e

T (a-b) 0

(5.2)

(a) = -Tf(b)

-7( r ) .

L e t u s b e g i n t o p r o v e ( a ) , i n which B > (a1 i s assumed. have

(5.2),

=O.

T h e r e f o r e we c h o o s e b = O i n ( 5 . 2 ) .

By ( a ) of Theorem 1, w e

On s u b s t i t u t i n g ( 3 . 2 7 )

we g e t

(5.3)

The d e f i n i t i o n of RE a l l o w s u s t o r e w r i t e ( 5 . 3 ) by

Let E ( > O )

be s m a l l enough i n ( 5 . 4 ) .

Using Lemmas 2 , 3 and 4, we s e e t h a t t h e mapping fU+B.RESi becomes t h e 1 m+c o n t r a c t i o n mapping from H '(r) o n t o i t s e l f i f E i s s m a l l enough and

1 3

B>a>--

Indeed,

into

Hideo K A W A R A Dand A Takao HANADA

(by 3.19)

(by 3.27)

On t h e o t h e r h a n d , by ( 3 . 2 1 )

m+y1 Here w e n o t e t h a t T (0) s h o u l d be i n c l u d e d i n H f m assume f t H (no). Summing up ( 5 . 4 ) ,

( 5 . 5 ) and ( 5 . 6 ) ,

(r).

T h e r e f o r e , we have t o

have

WE:

1 3

if 5>a>--B. We remove i n t o t h e c a s e - a < 6 2 - 3

O p e r a t i n g E - ~ ~ * ( S ~ ) on - ' b o t h s i d e s of

(5.2), we h a v e

Let

( > 0) be s m a l l enough.

m+from H

Then

-2a

.(S )

-1

2(r)

onto i t s e l f i f a

becomes t h e c o n t r a c t i o n mapping I n f a c t , t h e b o u n d e d n e s s o f T0

and (3.28) y i e l d s

Therefore, by u s i n g ( 5 . 8 ) , (5.9)

and ( 3 . 2 9 ) , we h a v e

Asymptotic Behaviors of the Solutions a =

(5 .1 0 )

$€II'

{ I - €-2a (S E ) -1T o ) - 1 { - ~ 8 d T f ( 0 )

93 O ( E a+38 ) }

+ 0(EB-3a + Ea+35)

= -,BaTf(o)

m - 2(I'),

i f a + B > 0 a n d a > 0.

Combining ( 5 . 7 ) and ( 5 . 1 0 ) , we o b t a i n ( 2 . 1 3 ) .

5.3

We s h a l l p r o v e (b) of Theorem 2 , i n whi ch a = B > 0 i s assumed.

+qo= 0

Theorem 1 f o l l o w s Tf ( J i o )

0 T0(a-qO)

(5.11)

Choose b =

S ( a ) = -T

0 f (I0)) =

Then w e h a v e

I' i n ( 5 . 2 ) .

$o0

From ( b ) of

By ( 3 . 2 7 ) , we h a v e

0 TO(a-JIO)

(5 .1 2 )

wh ere SE ( a ) = S E ( a ) + c-2aa. 1

0 a - JI, =

(5 .1 3 )

Then

a > 0 an d

q00

= E

.Sl(a)

By u s e of RE w i t h a =

2 a E E

.R S , ( a ) .

m+l E

- R S1 becomes t h e c o n t r a c t i o n mapping from H

i s small enough.

In f a c t , by (3.19) and ( 3 . 2 7 ) ,

2(1') o n t o i t s e l f i f

we have

T h e r e f o r e we h a v e 1

(5 .1 5 )

5 .4

$'I,

= (I-E

0 sl) JI, = $o +

2 a E 6 - 1 0

Now w e a r e i n t h e f i n a l s t e p t o p r o v e ( c ) .

assumed. have

0 ( c ) o f Theorem 1 g i v e s u s T f(JIO) = 0.

O(E

m+)

2(r).

I n t h i s case, a > 8 a n d a > 0 a r e Put b =

i n (5.2).

Then we

Hideo KAWARADA and Takao HANADA

94 O p e r a t i n g (To)

-1

on b o t h s i d e s of ( 5 .1 6 ) , we h a v e

R e p e a t i n g t h e s i m i l a r a r g u m e n t s as i n t h e p r o o f s of ( a ) a nd ( b ) , E2a*(To)-1SE m+-1 2 ( r ) o n t o i t s e l f i f a > B a n d E i s small becomes t h e c o n t r a c t i o n mapping from H enough.

Then w e h a v e

m + -1 =

$o0

O(P ---B)

2(r).

Proof o f Theorem 3

6.1

Assume f C H

(no) (k,?)

and f i >

10.1.

Then, from ( 2 . 7 ) and ( 2 . 1 0 1 , we h a v e

(6.2) By u s i n g (2.13) a n d ( 6 . 2 ) , w e h a v e

By a p p l y i n g t h e maximum p r i n c i p l e t o t h e probl em (1.2), u s i n g ( 6 . 3 ) , we o b t a i n

(6.4)

We compute on

(6.5)

(1.3) a nd (1.5) a n d

Asymptotic Behaviors of the Solutions

where s is t h e arc l e n g t h of By ( 6 . 3 ) ,

we have

(6.6)

From t h e d e f i n i t i o n of S E , we have

(6 .7 )

By (3 .2 7 ) a n d ( 2 . 1 3 ) ,

(6.8)

Combining ( 6 . 6 ) a n d ( 6 . 8 ) ,

(6 .9 )

S i m i l a r l y , we have

(6 .1 0 )

b e c a u s e o f ( 6 . 4 ) , J, P u t Y'=VJ,;.

(6 .1 1 )

and -

an an

E H

'(an) nck-2s6(an).

Then YE s a t i s f i e s

--E

2(a+B)*yE

= 0

From t h e maximum p r i n c i p l e t o g e t h e r w i t h (6.9)

Ql.

and ( 6 . 1 0 ) ,

i t follows

(6 .1 2 )

Here w e h a v e t o assume

k24

s i m i l a r a r g u m e n t , w e have

t o o b t a i n t h e good r e g u l a r i t y of J,".

Repeating t h e

Hideo KAWARADA and Takao HANADA

(6.13)

Proof o f Theorem 4 I n the final section we give the proof of Theorem 4 under the drastic

assumption.

Suppose n -

1 - R+.

In the same way as in 1" of the proof of Lemma 4 , we transform J,f into J, Then

satisfies

(7.1)

By ( 2 . 1 3 )

of Theorem 2, we have

By substituting (7.2) into (7.1), we have

(7.3)

We compute

(7.4)

Asymptotic Behaviors of the Solutions

From ( 7 . 4 ) , w e h a v e

Appendix m ,

F o r s i m p l i c i t y , we assume A(u2) t- C o ( r )

a n d rewrite

Here w e s t a t e how t o c o n s t r u c t Jln ( n = O , 1, Z , . . . )

1 d z ~ = a(U1,u2). 1 i n (3.39).

s o l u t i o n of t h e f o l l o w i n g o r d i n a r y d i f f e r e n t i a l e q u a t i o n :

Solving (A.1,2),

JI 0

(A. 3 )

we g e t

= A(u2).exp(-E

a(O , u 2 ) u 1

W e compute

(A.4)

= E 2(a+B). -

L e t J,

g; (Ill, u 2 ) .

be t h e s o l u t i o n of t h e p r o b l e m :

L e t J,:

be the

Hideo KAWARADA and Takao HANADA

(A.5)

S o l v i n g (A.5,6)

(A. 7 )

-E

R+

and computing

2 ( a + B ) - ~ ~+1 a(u1,u2)?$:

we can c o n s t r u c t t h e e q u a t i o n which JIz s a t i s f i e s i n t h e f o l l o w i n g way:

Using t h e c a s c a d e s y s t e m d e f i n e d a b o v e , we c a n o b t a i n $* ( n = O , 1, 2 ,

W e put

(A.10) and (A.11)

IJE

- 8.

Then wE s a t i s f i e s

(A. 12)

--E

2(*+B)-~w

(A.13) w-EIu

From (A.12,13),

we h a v e

+ a ( u 1 , u 2) 2WE and

o(E

(n+2) (a+B))

(u2++m).

R+2 ,

...).

Asymptotic Behaviors of the Solutions

an d mo r e o v e r (A. 16)

W e compute

whe r e

On t h e o t h e r h a n d , from ( A . 1 6 )

I f we c h o o s e n + - > m - (or n L -m - 1 ) , 2= 2

t h e n w e ha ve

(A. 20)

By n o t i n g

d l an

(3.26)-(3.29).

=-*

a(0,u2)

IE'a

aul

and using t h e d e n s i t y a r g u m e n t , w e c o n c l u d e

Hideo KAWARADA and Takao

100

HANADA

Footnotes

an s t a n d s f o r t h e boundary of R.

I I v ~ / ~ s, t a~ n d s

f o r t h e norm of v i n Hm(E).

T h i s theorem w a s p r o v e d i n [ 3 ] f o r t h e c a s e a > 0 a n d B > O .

I n t h i s paper, we

g i v e a n o t h e r p r o o f , which i s s i m p l e r t h a n i n [ 3 ] .

[l] S. Agmon, A. D o u g l i s and L. N i r e n b e r g , Estimates near the boundary f o r

solutions of e l l i p t i c p a r t i a l d i f f e r e n t i a l equations s a t i s f y i n g general boundary conditions I , [2]

Comm. P u r e Appl. Math.

H. Kawarada, Numerical methods for f r e e surface problems by means of penalty,

L e c t u r e Notes i n Mathematics, [3]

1 2 (19591, 623-727.

704,

Springer-Verlag,

1979..

J . L . L i o n s , Perturbations singuliares duns l e s problsmes aux Zimites e t en

contrble optimaZ, S p r i n g e r - V e r l a g , 1973. [4]

J.L.

L i o n s and E. Magenes, ilonhomogeneous boundary value problems and

Applications, S p r i n g e r - V e r l a g , [5]

B e r l i n , New York, 1972.

J. NeEas, Les metizodes d i r e c t e s en tkgorie des gquations e l l i p t i q u e s , Masson, P a r i s , 1967.

Lecture Notes in Num. Appl. Anal., 5 , 101-118(1982) Nonlinear PDE in Applied Science. U S . - J a p a n Seminar, Tokyo, I982

PARTIAL REGULARITY AND THE NAVIER-STOKES EQUATIONS

Robert V. Kohn Courant Institute of Mathematical Sciences

It is a pleasure and an honor to participate in this U.S.-Japan Seminar.

My talk concerns recent joint work with L. Nirenberg and

L. Caffarelli, in which we prove Theorem 1:

The singular set of a "suitable weak solution" of the

Evier-Stokes equations has "parabolic one-dimensional measure zero" in spacetime. I shall explain what we mean by a "suitable weak solution," and by the phrase "parabolic one-dimensional measure zero"; and I shall describe the structure of the proof, avoiding the more technical parts.

A fully complete discussion can be found in [l].

Theorem 1 extends and strengthens results of V. Scheffer 115-191, and our arguments draw extensively from his ideas.

Scheffer has

recently proved a result on "partial regularity at the boundary" [191 i here and in [l] we consider only the interior problem. Section 1. Let

Remarks on existence and regularity. be a smoothly bounded domain in R ' ,

initial-boundary value problem

(1.1)

ut + U - V U

- AU +

Vp = f

A-u = 0 10 1

Qx(O,T)

and consider the

Robert V . KOHN

102

(1.2)

u(x,O) = uo(x)

u(x,t) = 0

aRx(0,T)

where uoI

0, V*uo = 0,

and V - f = 0

The function u = (u1,u2,u3) represents the velocity of an incompressible fluid with unit viscosity; p is the pressure; and f is a nonconservative force. It is well-known that if uo and f are Cm then (1.1), (1.2) has a unique Cm solution on Rx(0,T) for some T > 0 [ 7 1 .

There is also

an extensive theory of strons solutions with less regular data [9,11,20]. if u o

If, for example, uo has "one-half derivative in L2" or

L3, one can still show the existence of a unique stronq

solution locally in time [ 2 , 3 , 5 , 8 1 .

One might conjecture that the

strong solution exists for all time; but this has been proved up to now only when the data uo, f are sufficiently small. The concept of a weak solution of (l.l), (1.2) was introduced by J. Leray, in order to obtain an existence theorem that is global Pioneerinq work of Leray [lo] and Hopf [ 6 1 showed the

in time.

existence of a function u and a distribution p such that

Lm(O,T;L2 (a))n L2 (0,T;H1 ( n ) )

for each T
0, where

is roughly the information available on the left of (2.11). The inductive argument, then, uses not @

but a sequence of

test functions{$n}, 9, being essentially a mollification of order 2-".

@ *of

As one enters the nth staqe, one knows

(3.5) Qr and an analogous bound for the pressure.

Therefore, using (3.5),

The function $,, satisfies

111

Partial Regularity and Navier-Stokes Equations One bounds the other terms in (2.11) similarly, to obtain

Using ( 3 . 4 ) , and assuming that

is small, it follows that

One proves a corresponding estimate for D using (2.5) (this is the most technical part), and the induction continues.

The key is the

different homogeneity on the left and the riqht in (2.11), which allows the smallness hypothesis to be useful in (3.6). One can rescale Proposition 1 to obtain a result on Q

Qr(x,t)

for any r, using (2.6): Corollary 1: -

For _any r > 0, __

J J

and -

fIq

(3.8)

then a.e. -

on Qr 2

Again, the way to understand Corollary 1 is to ignore f and p . says,

(3.9)

in essence, that if the dimensionaless quantity

R(r;x,t) =

meas (Qr)

3 (rlul) dxdt Or (x,t)

is small enough, then u is regular on Qr. 2

One may view R(r) as a

local Reynolds number for the flow on the cylinder Qr.

Robert V . KOHN

112 5

A dimension result. 3-

Section 4 .

If ( u , p ) is a suitable weak solution on all of R

then by

(2.3) and ( 2 . 4 )

As Scheffer observed in 1161, Corollary 1 and (4.1) imply an estimate 5

for the parabolic --dimensional measure of the sinqular set S. 3 idea is simply this:

The

10 ~~

, then "at most points" the average of u on

if u E L

Q (x,t) will

not be too large; at such points R(r;x,t) + 0 as r -+ 0.

To quantify

this, one uses the following Vitali-type coverinq lemma.

Let J be any set of parabolic cvlinders Qr(x,t) contained

Lemma 1:

in 5 bounded subset of R xR. -

There exists a finite or denumerable

family J' = {Qr,(xi,ti)} such that 1

(4.2)

the elements of J ' are disjoint;

(4.3)

for each Q E J there exists

Q5ri (Xi,ti)

Given Lemma 1, we argue as follows. 5

F i x & > 0 ; since f E L4

and q > , we may assume that (3.8) holds whenever r < 6, by choosing 2 6 small enough. By (3.7) and Holder's inequality, there exists E;

> 0 such that

whenever r < 6

Partial Regularity and Navier-Stokes Equations

113

Let V be any open, bounded subset of R 3x(O,-), and let 3 consist of all cylinders Q,(x,t)

such that

(4.5) Qr By (4.4), J covers S n V.

If J ’ is as in Lemma 1 then

by (4.3), and

’Qr . by (4.1), (4.2), and (4.5). Lebesque measure zero.

As & + 0, we conclude that

Since UQ,,

is contained in a

S n

V has

6-neiqhborhood

of S, the right side of (4.6) actually tends to zero as & + 0, so 5 the --dimensional measure of S is zero. 3

Section 5.

The dimension 1 result.

5 The argument in section 4 gives a 3-dimensional estimate

for S because it uses the global estimate (4.1), which has scaling 5 dimension To prove a dimension-one result by this method, one 3 must use the dimension-one estimates (2.2) instead of (4.1).

Returning to Corollary 1, suppose that the point (x0,t0) is singular.

Then (3.7) must fail for every sufficiently small r > 0.

Heuristically, this means that R(r;xo,tO) is bounded away from zero, i.e. that

Robert V . KOHN

1 I4

Thus Corollary 1 specifies a minimum rate at 1 which singularities can develop. If ] u I qrows as r , it is natural 1 to guess that IVul should grow as . These considerations motivate r "in the L3-mean".

Proposition 2:

There is an absolute constant

following property.

If (u,p)

Navier-Stokes equations

then (x,t) -

near

> 0 with the

5 suitable weak solution of the

(x,t) and if

d regular point.

Proposition 2 implies Theorem 1 by the coverinq arqument of section 4 , using (5.2) in place of ( 4 . 4 ) . The essential idea in the proof of Proposition 2 is contained in the following calculus lemma

Lemma 2:

Let w(x,t) be 5 W1I2 function defined near

(0,O)

3 R xR.

E r > O , % R(r) = r-2

jj i W i 3 Qr

Notice that R(r) , B(r) and y(r) are dimensionless in the sense of ( 2 . 7 ) .

Proving the lemma is an amusinq exercise, using the

115

Partial Regularity and Navier-Stokes Equations

interpolation inequality ( 2 . 1 ) , Holder's inequality, and the fundamental theorem of calculus.

The conclusion (5.3) is a sort of

decay estimate for R(p) : Corollary to Lemma 2:

(5.4)

For any

lim sup O(r) r-+O

> 0 there exists 6 > 0 such that

y(r) < 6 * lim inf R(r) < r+ 0

Indeed, (5.3) implies 1

whenever R(r)

Choosing 0 < 0 < 1 so that C 2 0

0 so that

we conclude

whenever R(r)'E

and B(r)

y(r) < 6

The assertion (5.4) follows,

with this choice of 6. The proof of Proposition 2 is rouqhlv parallel to the above 3 ut]T , but argument. For a weak solution u, one has no bound on

the generalized energy inequality lets one bound the osc llation in time of

IuI2

One proves a "decay estimate" like (5.3), for

Br the entire left side of (3.7) instead of for R ( p ) .

Robert V . KOHN

1 I6

Section 6.

Concluding remarks.

One reason for studying partial reqularity is the hope of settling, by this method, certain classical open questions about weak solutions.

Miqht one prove uniqueness or stronq continuity, Theorem 1 alone

for example, without actually provinq reqularity?

does not suffice; one appears to need information about the maximum rate at which singularities can develop.

We note in this context

a qualitative difference between Corollary 1 and Proposition 2: and the conclusion r' u : the hypothesis of the latter concerns all

the hypothesis of the former concerns a fixed Q asserts a bound for

Qr, and the conclusion gives no explicit estimate. Might similar methods be used to prove an estimate of the singular set of dimension less than one?

This would require a

global estimate with scalinq dimension less than one.

Provinq such

an estimate would take, it seems, a fundamental new idea. It may be, of course, that weak solutions are not reqular. An attractive scheme for constructing a solution with a self-similar singularity is proposed in [lo]. Finally, I note that the generalized energy inequality may have uses other than for partial reqularity theory.

In [l], for

example, it is used to prove weighted norm estimates f o r the Cauchy problem, in case the initial velocity satisfies

j R3

/u0l2/xl-l sufficiently small.

luol 1x1
0

(VI)

V(X,d) =

Put

-l i-m v ( x , d ) < m

and

*LO

dLO

?@-?-d for d E &(q). "vo, 4

Then

v(x,d)

s a t i s f i e s the desired conditions. 2.2. -

In o r d e r t o o b t a i n t h e e s t i m a t e i n Theorem 2 , we u s e t h e i d e n t i t y W(X,d)\V(X,d)(

(2, 4 ) =

I v ' ( Y , CO 1

2 2

- w(xl,4)lV(xl,d) I dy

d l x q(Y)lv(Y,d

which f o l l o w s from

$v"(y, d.)v(y, r ) d y

0 1 x1 4 x

p -

- component of

component o f b o t h s i d e s of (2, 4 ) ,

Denote

s(x,d) = - ( W ( X , ~ ) ) ~

then it follows

2 2 s ( x l , ~ ) I ~ ( x l , ~= ) s\ ( x , d ) I v ( x , a ) l

(2, 6 )

d l q ( y ) I v ( y , , ) I 2 d y , and

Now we i n t r o d u c e f o l l o w i n g c o o r d i n a t e s depending on CC

i s s a i d simply t o b e

Take

1 1 2dY ,

( 2 , 2).

2 I ~ ' ( Y , o c dY, )~ 1

where

Im d

Imd = -Im(p i )

> 0.

Therefore

la\Imfi

Hence i t s u f f i c e s t o e s t i m a t e

s(0,d)

and

l/s(s,d)

w e can u s e t h e e q u a t i o n

s'(x,

OL) =

Im(w';)/Im(ap)

= -Im(w2d)/Im(4~).

The d e t a i l s w i l l be shown i n n e x t s e c t i o n .

For t h i s p u r p o s e

Integral Representation for Equations of Mixed Type 2.3. __

Here we p r o v e Theorem 3 i n view of Theorem 1 and 2 . f o l l o w s d i r e c t l y from t h e e s t i m a t e

(E)l

lim

(E).

177

The e s t i m a t e

u(x, t ) = 0

X+ m

f o l l o w s from Lebesque theorem. l o c a l i z a t i o n of

Here

u(x, t) : u(x, t ) =

x j ( ~ )a r e

i3C

{I€

r‘

The e s t i m a t e s f o r

XI2 c {

2a }

supp

i4 c { T E

1 ~ ,

are same, since

af*b

= af"D6

( = af*D;

(

))

if elastic (if plastic).

af*D af

Hence we shall consider the following system which must be satisfied by the first derivatives of the solution ( if it exists ) .

(2.Ub

for E- E

= DE

= DEO

o 3

in D- =

uo ; af*(tO)DEo

o I 0 1,

Ep- Ee 1.

Let (u,u,c() be its s o l u -

is same to that of the solution of the

By Theorem 2.3, (;,6,&)(t0+0)

problem (2.3) with o l d Ep. Let ( u o , a')

Theorem 3.2. (i) (ii)

(UO,

uO)

be the solution of (3.1).

Then hold

(ii,3 )(tO+O),

Let E be the set of elements of Ep such that 2 1 af*(to)o

( 3.6)

= 0.

Then, for every element, (ii,g,E)(t +0) is determined independently of the choice 0

of the next

Proof.

relation of E2'

By (2.3);')

and (2.3)L1) we denote the equations obtained by differ

enciating twice the both sides of (2.3)

and once the 6-k relation of (2.3)b with

respect to time t, respectively, where Ep is replaced by the new Ep. (2.3)")

we denote the system of these equations.

satisfy [2.3)('),

Then (L,y)(to+O)

where [ ) has the same meaning as before.

By and (uo,oo)

Also the solution

Tetsuhiko MIYOSHI

200

of [2.3)(')

is unique at t +O.

) Hence (Iholds.

exactly the same to that of Theorem 2.3.

The proof of (ii) is

Note that for the element of E2 the

equa1ity D ' Eo

holds for the solution (uo,

0')

d (D'); 1 +dt to+o

= 0

of ( 3 . 1 )

If, furthermore, E is not empty, we repeat this discussion until E becomes 2 K empty for a certain K

+ k2(A2)"

2A2(A2)"k2

0) i n t e r a c t s with and

2 k ) l y i n g on the u n i t

41 -

The second c l a s s of i n s t a b i l i t i e s occur f o r A2(A2)'

and

o r e q u i v a l e n t l y when (7)

$kB>O

where B = A 2 dA2/dk2 = -2(k2

1)(R

(k2

. This

is t h e Eckhaus

i n s t a b i l i t y and occurs when a r o l l has too s m a l l a wavelength.

It is useful t o

summarize these r e s u l t s by way of f i g u r e 2.

k8, A

R k Figure 2: Graphs of A , kB and R Busse Balloon

VS.

k. and t h e

212

Alan C . NEWELL

S o l u t i o n s can e x i s t f o r k

< k

d 0, =(kB)

E l l i p t i c unstable

for B

a% +

k B ) T

ax 0, $(kB)>

0 t h e u n s t a b l e modes are p a r a l l e l t o

( k , 0).

The a d d i t i o n of t h e c 4 t e r m i n ( 2 6 ) which i n v o l v e s h i g h e r d e r i v a t i v e s o n l y s e r v e s t o c o n t r o l t h e growth of t h e i n s t a b i l i t i e s a f t e r t h e y b e g i n .

It d o e s n o t

i n h i b i t them a l t o g e t h e r nor d o e s it of i t s e l f t r i g g e r any new i n s t a b i l i t y . The r e a d e r might l i k e t o compare t h i s r e s u l t w i t h what h a p p e n s in n o n l i n e a r wavetrains.

and

T h e r e , t h e a n a l o g u e of e q u a t i o n ( 2 6 ) is a s e c o n d o r d e r s y s t e m i n and s o it i s t h e e l l i p t i c i t y o r h y p e r b o l i c i t y of t h e s e c o n d o r d e r

o p e r a t o r which d e t e r m i n e s i n s t a b i l i t y o r ( n e u t r a l ) s t a b i l i t y of t h e w a v e t r a i n .

For example, f o r a t r a i n of g r a v i t y waves on t h e sea s u r f a c e , t h e h y p e r b o l i c n a t u r e of ( 2 6 ) c h a n g e s t o e l l i p t i c when t h e r a t i o of d e p t h t o w a v e l e n g t h is l e s s t h a n 1.36. (b)

The Newell-klhitehead-Segel

limit.

To t h i s p o i n t , we have t a k e n v a r i a t i o n s i n t h e d i r e c t i v e s p a r a l l e l t o and

p e r p e n d i c u l a r t o t h e l o c a l r o l l t o be of t h e same o r d e r of m a g n i t u d e .

It i s

c l e a r t h a t i f f o r some r e a s o n t h e l o c a l wavenumber is f o r c e d t o s t a y a p p r o x i m a t e l y c o n s t a n t , t h e v a r i a t i o n s in wavenumber of o r d e r IJ p a r a l l e l

to (e.g.

a r e accompanied by v a r i a t i o n s of o r d e r J p i n t h e p e r p e n d i c u l a r d i r e c t i o n (kc

+ pL) 2 +

( J I J M ) = kc

)

Near k = 1,

we f i n d t h a t v a r i a t i o n s

p e r p e n d i c u l a r t o t h e r o l l a r e of a n o r d e r of magnitude g r e a t e r t h a n t h o s e p a r a l l e l t o t h e r o l l a n d t h i s l e a d s t o a b a l a n c e between t h e t e r m

k B and

218

Alan C . N F W LII

some of t h e

terms i n th e phase e q u a tio n ( 2 6 ) .

T h i s s i t u a t i o n c e r t a i n l y o b t a i n s when

i s s u f f i c i e n t l y small, f o r t h e n

( s e e F i g u r e 2) t h e bandwidth of wavenumbers p a r a l l e l t o t h e r o l l i s O(JR) a n d

4 -

t h e bandwidth p e r p e n d i c u l a r t o t h e r o l l i s O( JR). A s we h a v e m e n t i o n e d , i n t h i s l i m i t t h e a m p l i t u d e n o l o n g e r f o l l o w s t h e p h a s e g r a d i e n t a s i n (25) b u t t h e

terms on t h e RHS of t h e a m p l i t u d e e q u a t i o n (27) became e q u a l l y i m p o r t a n t t o T h i s b a l a n c e i s a c h i e v e d when R =

t h e s e on t h e L.H.S.

For r o l l s which a r e

and

a =

where x = direction.

+(x,

X a s before and

(34) = y/s =

EY, t h e new s c a l i n g i n t h e p e r p e n d i c u l a r

It is now e a s y t o show from (11) t h a t

kaa

ax + $_a_ ,

La8

1 / ;a~

= at =

2 E

Y Y

kq8 =

+yy

and

-kqa

-~+,;j-

E+-$-

(35)

where we have u s e d s u b s c r i p t s i n o r d e r t o d e n o t e p a r t i a l d e r i v a t i v e s . S u b s t i t u t i o n of (35) i n t o (26) a n d d i v i d i n g by

g i v e s (we d r o p t h e t i l d e on

A$t

2(+x + '2 6 y ) ( 2 a x

+yy)A

2(2ax

+(zax + q Y a y + $ y y ) ~ y , ,+ ay 2(2ax + z$ a + + y y ) ~= o

tJYY)(OX

7 $y)A

(36)

Y Y

I t i s r e a d i l y shown t h a t , i f W = Aei$ i n ( 9 ) , e q u a t i o n (36) i s p r e c i s e l y t h e i m a g i n a r y p a r t of e q u a t i o n (9).

C a r r y i n g out t h e same c a l c u l a t i o n on ( 2 7 )

219

Two Dimensional Convection Patterns (recall A

A) g i v e s t h e r e a l p a r t of e q u a t i o n ( 9 ) .

T h e r e f o r e t h e e q u a t i o n s ( 2 6 ) , ( 2 7 ) c o n t a i n a l l t h a t was p r e v i o u s l y known about r o l l solutions.

They a l s o c o n t a i n some new i n f o r m a t i o n .

New r e s u l t s ; some a n s w e r s , more q u e s t i o n s . I n what f o l l o w s we s h a l l t a k e

t o be of o r d e r one a n d t h e r e f o r e ( 2 7 ) can

be r e p l a c e d by ( 2 5 ) a l m o s t e v e r y w h e r e .

The e x c e p t i o n s a r e t h o s e r e g i o n s We w i l l c o n c e n t r a t e on t h e

where V = O ( E - ~ ) b u t t h e s e p o i n t s a r e i s o l a t e d . phase e q u a t i o n (26),

ae + si;1 V at

(kB)

( D 1 * D 2 + D2*D1)A

= 0

which may b e r e w r i t t e n i n a v a r i e t y of ways.

V(+kB) = k

kBII

(37)

I n p a r t i c u l a r we may w r i t e

o r i n a more r e v e a l i n g way a s V(

0 i n the

= 0 axis.

These

s o l u t i o n s seem t o g i v e a f a i r l y a c c u r a t e p i c t u r e of t h e r e a l d i s l o c a t i o n s seen i n experiments. F i n a l l y , w e i n d i c a t e how t o i n c l u d e mean d r i f t terms i n t h e model. Consider

$ + (V2+1) 2w where

+ w2w* + u

vw = 0

u = Vx TZ (z i s t h e u n i t v e c t o r p e r p e n d i c u l a r t o

X,Y)

and

F o l l o w i n g t h e p r e v i o u s a n a l y s i s , we f i n d t h a t t h e s l o w e q u a t i o n f o r t h e phase is

kt a kB -A % 3-

+ Akll 3aTT +

O(c4) = 0

where

2 a a uL p V ~ = k t = k l l ~

(70)

In (68), t h e p a r a m e t e r l i p mimics t h e e f f e c t of low P r a n d t l number s i t u a t i o n s where mean d r i f t i s c a u s e d by t h e n o n l i n e a r a d v e c t i o n t e r m s i n t h e momentum equations.

I n (70),

refers

t o t h e slow d e r i v a t i v e s wi t h r e s p e c t t o

6. SUMMARY. I n t h i s p a p e r we have p r e s e a t e d a m a t h e m a t i c a l framework f o r d e s c r i b i n g

230

Alan C. NEWELI

c o n v e c t i o n p a t t e r n s which i n c l u d e s a l l p r e v i o u s t h e o r i e s a n d from i t we have

made s e v e r a l p r e d i c t i o n s a b o u t t h e manner in which t h e p a t t e r n s e v o l v e . p a r t i c u l a r , we s u g g e s t t h a t on t h e h o r i z o n t a l d i f f u s i o n time s c a l e

TH, t h e

c o n v e c t i o n f i e l d d e v e l o p s p a t c h e s , o f t e n of a c i r c u l a r n a t u r e s u r r o u n d i n g a s i n k , i n which t h e wavenumber is c o n s t a n t .

The i n c o m p a t i b i l i t y of t h e s e p a t c h e s

is i r o n e d o u t o v e r t h e l o n g e r t i m e scale of t h e a s p e c t r a t i o t i m e s

and the

p r o c e s s i n v o l v e s a g l i d i n g motion (compare F i g u r e s 3 a n d 5) i n which r o l l d i s l o c a t i o n s move i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e r o l l a x i s .

The c l i m b

m o t i o n , where t h e d i s l o c a t i o n s move a l o n g t h e r o l l a x i s , o c c u r on t h e s c a l e TH

as t h e i r r o l e is t o a d j u s t w a v e l e n g t h , a l t h o u g h small a d j u s t m e n t s of o r d e r w i l l be made on t h e

E-%'~

scale.

While we b e l i e v e w e h a v e made a s t a r t , many q u e s t i o n s s t i l l remain open. Some of t h e s e are. 1.

F o r what c l a s s of models i s t h e f l o w on t h e h o r i z o n t a l d i f f u s i o n time s c a l e a g r a d i e n t o n e ; e q u i v a l e n t l y , f o r which models does ( 4 4 ) o b t a i n ?

What is t h e e f f e c t of t h e mean d r i f t term?

What p a r a l l e l c o n c l u s i o n s can we

draw?

Do t h e p a t t e r n s e v e r s e t t l e down o r do t h e y a l w a y s remain n o i s y ?

If the

f o r m e r is t h e c a s e , i s i t a consequence of geometry where t h e d i s l o c a t i o n s g e t stuck i n corners?

I n a c i r c u l a r g e o m e t r y , one m i g h t a r g u e t h a t t h e

g l i d e motion n e v e r s t o p s .

I f t h e l a t t e r is t h e c a s e , d o e s t h e r e s u l t i n g

c h a o t i c motion l i e on a low d i m e n s i o n a l s t r a n g e a t t r a c t o r , one w h i c h , f o r example, mimics t h e v e r y g e n t l e h e a v i n g of t h e g l i d e m o t i o n as i t r o t a t e s a r o u n d t h e box?

23 1

Two Dimensional Convection Patterns REFERENCES 1.

G o l l u b , I. P. a n d McCarriar A. R.

A h l e r s G. a n d Walden R. W.

Busse F. H. turbulence. Verlag

1982.

Phys. Rev. A

1980 Phys Rev. T a t t .

44, -

z,347O.

445.

1980 Hydrodynamic i n s t a b i l i t i e s a n d t h e t r a n s i t i o n t o 97-136. Eds. H. L. Swinney a n d J. P. G ollub. Publ. S p r i n g e r -

1970.

J. F l u i d Mech

Whitham G. B.

N e w e l 1 A. C. a n d W hi t head J. A . 1969 J. F l u i d Mech. 203.

Pomeau Y. a n d M a n n e v i l l e P.

S i g g i a E. a n d Z i p p e l i u s A.

S t u a r t J. T. 1960. Mech. 371-389.

Busse F. H. a n d W hi t ehead J. A.

10.

Chen M. M. a n d W hi t ehead J. A .

11.

G r e e n s i d e H.

12.

Ekrge , P. 1980. S p r i n g e r - Ve r la g

1969.

3, 373.

1981. 1982.

Phys. Lett

S e g e l , L. A .

40,1067.

J. F l u i d Mech. t o a p p e a r .

J. F l u i d Mech. 9 , 353-370.

1971. 1968.

Watson J.

J. F l u i d Mech.

1960.

J. F l u i d

47, 305-320.

J. F l u i d Mech.

S . , Coughran W . M. a n d S c h r y e r N . L.

38, 279.

J. F l u i d Mech.

2,1.

1982. P r e p r i n t .

Chaos a n d Order i n N a t u r e pp, 14-24.

Ed. H. Haken.

Publ.

This Page Intentionally Left Blank

Lecture Notes in Nurn. Appl. Anal., 5, 233-251(1982) Notditie(ir PDE in Applied Science. U . S . - J a p c i n Seminar. Tokvo. 198?

Stationary free bo-

problem for circular flaws

with or without surface tension

Departnmt of bbthemtics Faculty of Science vniversity of Tokyo H o n g o Bmyo-ku Tokyo 113 Japan

Free boundary problems f o r flows c i r c u l a t i n g around a c i r c l e or sphere a r e considered. It i s r e v e a l e d t h a t t h e s u r f a c e t e n s i o n p l a y s a c r u c i a l r o l e concerning p e r t u r b a t i o n s and b i f u r c a t i o n s of a t r i v i a l flow. Main t o o l s are i m p l i c i t funct i o n theorems ( c l a s s i c a l or g e n e r a l i z e d ) and b i f u r c a t i o n t h e o r y due t o S a t t i n g e r or G o l u b i t s k y & S c h a e f f e r . Therefore a l l t h e c l a s s i c a l s o l u t i o n near t h e t r i v i a l one are d e a l t w i t h .

§I. Physical maning. Consider a f l u i d around a p l a n e t .

We consider a plane p e r p e n d i c u l a r t o t h e a x i s of r o t a t i o n and we r e g a r d

mind.

We assume t h a t t h e flow is e n c i r c l e d w i t h two

t h e flow as a two dimensional one. c l o s e d Jordan curves

t h e planet,whence

i s a given curve.

unit circle i n R The o u t s i d e of

denoted by

tional.

and

The i n n e r curve

The o u t e r curve

r e p r e s e n t s t h e s u r f a c e of

For s i m p l i c i t y we assume t h a t

i's t h e

y r e p r e s e n t s a f r e e boundary t o b e sought.

i s assumed t o be a vacuum or t o b e f i l l e d w i t h a p e r f e c t f l u i d

whose p r e s s u r e i s given.

and

We keep a f i g u r e l i k e t h e J u p i t e r i n

Hence we t r e a t a one phase problem.

QY , i . e . , we denote by

The flow r e g i o n i s

t h e doubly connected domain between

F i n a l l y w e assume t h a t t h e f l u i d i s i n c o m p r e s s i b l e . i n v i s c i d and i r r o t a Then t h e problem is formulated by t h e stream f u n c t i o n

PKBLENA.

Find a c l o s e d Jordan curve

y outside

such t h a t

233

as f o l l o w s .

and a f u n c t i o n

i n fi

234

Hisashi OKAMOTO

(1.1)

AV = 0

in Cl

(1.2)

v = o

(1.3)

V = a

Y ,

$10~1' + Q +

(1.4)

Is2

(1.5)

= w

= unknown constant

0 '

The quantities appearing above are defined below. a , u0 ; prescribed positive constants, the surface tension coefficient (

0 )

Q ; a given function defined outside

;

... given ,

. the curvature of y ,the sign of which is taken to be positive if y is Y'

convex,

IQyl

;

the area of 12

REMAIX 1.1.

The equation (1.4)is a consequence of Bernoulli's law and the

Laplace equation arising in the theory of surface tension.

In fact,Bernoulli's

law asserts that

1 F I w ~+ ~p

(1.6) where p

+ Y = unknown constant,

is the pressure of the fluid and

on y

is a potential of the volume force.

On the other hand,the Laplace equation is expressed as (1.7) where peXt

P = Pe*

is the known pressure of the external atmosphere.

pext , we obtain (1.4) from (1.6)and (1.7).

Putting Q = p +

In this regard, Q E 0 o r

Q =-g/r

Free Boundary Problems for Circular Flows ( g ; a constant

= (x2

y 2 ) l i 2 ) i s an i n t e r e s t i n g case.

Trivial solution.

t r i v i a l solution.

Take a number

of radius

235

i s r a d i a l l y symmetric,then t h e r e e x i s t s t h e f o l l o w i n g

> 1 such t h a t nr;

= wo.

men a circle

rO w i t h t h e o r i g i n as i t s c e n t e r i s a s o l u t i o n f o r any

I n f a c t t h e corresponding stream f u n c t i o n

The unknown c o n s t a n t i n ( 1 . 4 ) i s

i s r e p r e s e n t e d as

1 2 p(a/rologro )

Q(ro)

+ a/ro.

O u r a i m i s t o s t u d y p e r t u r b a t i o n s and b i f u r c a t i o n s of t h i s t r i v i a l s o l u t i o n . Our a n a l y s i s i s based on c l a s s i c a l o r g e n e r a l i z e d i m p l i c i t f u n c t i o n theorems and

t h e b i f u r c a t i o n t h e o r y due t o S a t t i n g e r [ 5 ] o r Golubitsky and S c h a e f f e r [ 2 ] . Now let us c o n s i d e r t h e case where t h e f l u i d i s governed by t h e NavierStokes e q u a t i o n :

PIiaBLEM B.

Find a c l o s e d Jordan curve

y and f u n c t i o n s

(V V

R Y'

1' 2

that

(1.10)

div

= 0

(1.11)

Y ,

(1.12)

(1.13) (1.14) The q u a n t i t i e s appearing above are d e f i n e d below.

V =

(V,,V2)

; t h e v e l o c i t y vector

P ; the pressure

Y ,

such

236 v

Hisashi OKAMOTO

; t h e kinematic v i s c o s i t y

; t h e outward normal v e c t o r on

t ; a t a n g e n t v e c t o r on y , T ( 1 ) ; t h e stress t e n s o r

,t h e

components of v h i c h a r e

b_ ; a p r e s c r i b e d R 2-valued f u n c t i o n on

1r -b-- n d r

satisfying

= 0.

A t h r e e dimensional analogue o f PROBLEM B i s a l s o c o n s i d e r e d ( s e e 53 ) .

Mathematical Formulation and results for PROBLEM A.

52.

We p r e p a r e some symbols.

Cm+"(B)

( m = 0,1,2,**. , 0 < c1 < 1 ) ; t h e HElder spaces w i t h u s u a l

, Cm+'(S1)

We f i x a number

(0,l) and a f u n c t i o n

c1 E

Qo = Qo(r) = -g/r

When a small

or t

C2+'([l,

m)).

Q0 -= 0.

C3+'(S1)

H e r e a f t e r w e i d e n t i f y a f u n c t i o n on

R . We denote a domain between denoted by

(

KU.

i s g i v e n , we denote by

curve which i s parametrized i n t h e p o l a r c o o r d i n a t e s as

< 271 ) .

The t y p i c a l c a s e

and

a c l o s e d Jordan

( r O + u ( o ), 0 )

( 0

S1 w i t h a 2n-periodic f u n c t i o n on

fiU.

The c u r v a t u r e of

It i s r e p r e s e n t e d as

means t h e d i f f e r e n t i a t i o n w i t h r e s p e c t t o

a t i o n along t h e outward normal v e c t o r on t h e D i r i c h l e t problem

0 ). Vu

a/aUu

means t h e d i f f e r e n t i -

d e n o t e s t h e unique s o l u t i o n of

231

Free Boundary Problems for Circular Flows (2.1)

(2.2)

vUJr= o

= 0

v I

= a

yu For

C3+a(S1) , Q

C2+a-

(a)

Using a c a n o n i c a l pull-back,we r e g a r d it i s e a s y t o see t h a t

for

i f and only i f

F(a,*;-,*)

6 c R , we

and

F (a,Q;u,S)

as a f u n c t i o n on

F(a,QO;O,O) = ( 0 , O )

{yu, V

and t h a t

F(a,Q;u,c) = ( 0 , O )

put

f o r some

IR.

i s a continuous mapping from a neighborhood of

xc3+ci ( s1) x R

into

Cl+"(

s1)

sl.

Then

is a solution

Note t h a t

(Q,;O,O)

C2+"(E)

Now p e r t u r b a t i o n of t h e t r i v i a l s o l u t i o n i s p o s s i b l e i n t h e f o l l o w i n g s e n s e .

THEOREM 1.

Asswne t h a t u > 0 .

there e x i s t s a p o s i t i v e constant

11 9 - 90 112+a,R
1 i s determined by

For s u f f i c i e n t l y small for u

b_ = ( O , O , b )

( b

Then we put i t

Z Vu

and

we s o l v e ( 3 . 9 ) through ( 3 . 1 2 )

This i s u n i q u e l y determined f o r s m a l l

, Pu.

Note t h a t

and

b c 2

and

a r e independent of Q.

24 1

Free Boundary Problems for Circular Flows A

Now w e d e f i n e a mapping

REMARK.

i n a way s i m i l a r t o t h e c a s e of

The mean c u r v a t u r e

i s represented as

i s a continuous mapping from a neighborhood of

H Y

(O;O,O)

Z x Xx

IR i n t o

Then w e have t h e f o l l o w i n g

?HEOREM 6.

If b

is sufficiently small, then there exists

{u,c)

such that H(b;u,E) = ( 0 , O ) .

The solution is unique in some neighborhood of the

origin.

54.

Outline of the proof.

4.1.

THEOREM 1 i s proved i f we have shown t h a t t h e F r e c h e t d e r i v a t i v e o f

{u,c}

with r e s p e c t t o for

{an}n

i s an isomorphism from

C3+&(S1)

cl+ct

To show t h i s we have t o c a l c u l a t e t h e d e r i v a t i v e of

itly:

Claim.

onto

F i s a C'-mapping

and i t s d e r i v a t i v e i s given by

(s ) X W explic-

Hisashi OKAMOTO

242

(4.1)

Here w e have p u t

(

means t h e d i f f e r e n t i a t i o n with r e s p e c t t o

The f u n c t i o n

C(Vu)

i s d e f i n e d by

C(vU) =

(4.51

The o p e r a t o r

€I.)

[(ro+u

i s d e f i n e d by

?uw =

,using t h e solution in

AU = 0

The proof of ( 4 . 2 , 3 , 4 ) i s s t r a i g h t f o r w a r d .

RU ,

To show ( 4 . 1 ) it i s s u f f i c i e n t t o

prove t h a t

(4.6) where

D ~ T ( U ) W=

T(u) = / W u / /

? u ~ + C ( VU )W

The formula above i s proved i n [ 3 ] .

Here w e o n l y g i v e

a formal c a l c u l a t i o n t o d e r i v e ( 4 . 6 ) . f u n c t i o n on some neighborhood of

"i,.

F i r s t l y we extend Secondly n o t e t h a t

to a

lvVul I

= wu'Vu YU

since

i s a c o n s t a n t on

y,.

Then we have

class

Free Boundary Problems for Circular Flows

I1 + I2

U = vu+w

Putting

- Vu

, we

I3.

AC =

obtain

= v (

= a -

I1 = aUw modulo

v Urw

Since I

= 0.

- vu

= t

11 W I ~ + ~ ) .

11 W I ~ + ~ ) with

Claim.

Proof.

Assume t h a t

{an)n

A(a) 5 D

, then

u,s

A(a)

and

-- - a r avu w.

a tangent vector

+ t

u '

re o b t a i n

(4.6). i s an isomorphism f o r

{anIn.

i s injective.

1bnsinnB n=l

Then we have

y,+,

F(a,Q ;O,O)

A(a)(w,A) = (0,O).

w =

U = 0

It i s e a s y t o s e e

From t h e s e c o n s i d e r a t i o n s w e f i n d

Now we show t h a t

GU+,nGu

- v I

Hence

243

b S(a, n ) = cnS(a,n) = 0

We r e p r e s e n t W

c cosne n=O ( n

by t h e F o u r i e r s e r i e s :

. = cox{something)

and

co = 0,

where w e have put

Since if

S(a,n)

IanIn.

vanishes i f and o n l y i f

a = a

, we

see t h a t

A(a)

is injective Q.E.D.

244

Hisashi OKAMOTO

On the other hand,it holds that A(a) =

". A(a)

a compact operator

Using the claim above and the Riesz-Schauder theory we can conclude that

4 {anIn.

is an isomorphism for a

= 0, D

4.2. Proof of THEOREM 2. When

phism from C3+a ( S j X R

cl+a 1 (S )

$+a

an isomorphism

(5 )

X ~ R

onto

c2+a 1

(S ) x

onto

F(a,Q ;O,O) is no longer an isomoru,s 0 x R . However,it is an isomorphism from

From this fact one observes that we are in a

position to use a generalized implicit function theorem. Among others we use a one due to Zehnder 181.

In verifying several assumptions of the generalized im-

plicit function theorem,we use a priori estimates of Schauder type which are borrowed from Schaeffer [ T I . THEOREM

For the details,see Okamoto [ 3 ] in which the proof of

is included.

4.3. Proof of THEOREM 3.

From now on we put

G(a;u,c) = P(a,Qo;u,S).

proof of 'THEOREM 1 we have shown that Aia) 5 D

In the

F(a,QO;O,O) is an isomorphism

u>s

for a

{a 1 n n

and that A(an)

(cos n9

has a null-space spanned by

(sinno , 0 ) . ( Here we have used (2.7). )

and

In order to use a theory of bifurca-

tion from simple eigenvalue we use tha following Banach space:

with the norm

IL+,+,. Let

G* denote the restriction of G on X3+cLx R

Then it holds that the range of G*(a;*,.)

is included in X1+a

IR and the null-

space of D

G*(a ;O,O) is spanned only by (cosn9, 0 ). Consequently we can u,S n apply THEOREM 1.7 of Crandall and Habinowitz [l]. The details are in [41. To see whether the bifurcation occurs supercritically or subcritically,we

proceed as follows. space of

( The details are also in

c ~ ( S + ) ~spanned by

and

sinno

denote a canonical projection from C3+a(S1)

onto

Then we define functions @

(4.7)

cosne

[4].) Let a two dimensional

and

be C

< cosn9, sinno >

cos ntl

, sinn9

by the equations below.

(I -P)G (a;xcosn9 + ysinntl + $(a;x,y) , S(a;x,y)) = 0 , 1

sub-

by P.

Free Boundary Problems for Circular Flows

(4.8)

G (a;xcosne

+ ysinne

245

$(a;x,y) , < ( a ; x . y ) ) = 0.

The assumption ( 2 . 7 ) and t h e c l a s s i c a l i m p l i c i t f u n c t i o n theorem e n s u r e s t h a t and

5 are

well-defined i n some neighborhood of ( a ; O , O )

ranges a r e i n

(4.9)

(I -P)C3+'(S1) F(a;x,y)

, IR , r s s p e c t i v e l y .

E 3 , and t h a t t h e i r

Then t h e e q u a t i o n

PG1(a ; xcos no + y s i n no + @ ( a ; x , y ) , S ( a ; x , y ) )

i s a bifurcation equation.

= o

If we write

F ( a ; x , y ) = F l ( a ; x , y ) c o s nB + F ( t i ; x , y ) s i n n6

then t h e solution s e t near

(an;O,O)

is i n a one-to-one

{ (a;x,y) ; Fl(a;x,y) = F2(a;x,y) = 0

correspondence w i t h

1 .

Since t h e o r i g i n a l problem i s O ( P ) - c o v a r i a n t , w e have

The bifururcation func tion

PRDPOSITION 4.1. 2.c

C -mapping

F is a C -mapping.

F* defined i n some neighborhood of

(4.11)

F ( a ; x , y ) = yF*(a ; x 2

(a ; O )

There e x i s t s a such that

+ y ).

By t h i s p r o p o s i t i o n we o b s e r v e t h a t t h e s o l u t i o n s e t i s composed of x = y = 0 j

and

2 F*(a; x2 + y ) = 0

1 . Of

c o u r s e t h e former corresponds t o

To d e a l w i t h t h e n o n t r i v i a l ones we expand

the t r i v i a l solution.

F*(a ; x + y ) = A ( a - a ) + B ( x + y )

+ h i g h e r o r d e r terms.

By t h e result of Golubitsky and S c h a e f f e r [ 2 j , i t h o l d s t h a t equivalent t o

F = 0

O(2)-

246 if

Hisashi OKAMOTO An # 0

and

# 0.

Therefore t h e d i r e c t i o n of t h e b i f u r c a t i r i g branch i s

determined by t h e s i g n of

Bn.

and

Since

and

are given by

F , hence of

w e have t o compute t h e t h i r d o r d e r d e r i v a t i v e of

G1. To t h i s end

we p r e p a r e some symbols.

NOTATIchl.

For w

U(w)

we denote by

C3+'(S1)

AU = 0

uJr =

t h e s o l u t i o n of

i n l < r < r 0 ' 0

U/r=r

aw -r logr ' 0

For

w, z

1 C4+"(5 )

,w2 , w3

t h e symbol

denotes t h e s o l u t i o n of

AY = 0

l < r < r o ,

Y = O

r ,

c C

?+a( S1)

t h e symbol

ax=o

1 2 3

Wi'

denotes t h e s o l u t i o n of

0 '

2aw w w

w i+3 =

X(wI,w2,w3j

i n l < r < r

- r3logr

where we p u t

Y(w,z)

Free Boundary Problems for Circular Flows

NOTATION '

241

. L

B(w,z) =

ay(W,z) ~

ati(w) = r

3+a

( W € C

2 r logr 0 0

a2ti(w) +ar

a 2U(Z)

2 ar

Zawz r3logro

Mow the third order derivative of G1

aw'z' r3logr 0 0

( w

( S ) ) ,

4+a

(S1) ) .

is given by

au(wi)

+a 3 1 arw;+1w;+2 r logr i=l 0

9[ rO

3 -w w w - 2 1 w;wi+lwi+2 12 3 i=l

1 w"w' w' 1 WiWf+lWf+2 + i=l 1 i+l i+2 i=l

On the other hand,we have 68 cos n6 = PD3G ( a ;O,O)(cos n0 ,cos no ,cos no ) u l n

Therefore,in principle,we can compute Bn by the formula above. However,it is very difficult to decide its sign,since it is very complicated. But we have

248

Hisashi OKAMOTO

Hence any

i s negative for a l a r g e

In f a c t we have

o-ro

r0l o g r 0

Thus we see t h a t t h e b i f u r c a t i o n is s u b c r i t i c a l f o r a l a r g e

4.4.

i s negative f o r

On t h e o t h e r hand, A

Proof of THEOREMS 5 and

It i s not s o hard t o v e r i f y t h a t

Hence t h e proof of THEOREM 5 or

is a CL-mapping.

H and

6 a r e completed by checking

that

H(O;O,O)

or D

u.E, a t i v e s a r e given below.

D H (O;O,O)w

u l

H(O;O,O)

-0 ( 2

is an isomorphism,respectively. The deriv-

w + w" ) +

( w

c3+a ( s1) x R

we f i r s t show t h a t it i s i n j e c t i v e .

C 3 + W ) ),

In a way similar t o t h e proof of THEOREM 1 we can prove t h a t an isomorphism from

onto

c'+~(s~)

X B .

D H(O;O,O) u,s

treat

D U,S

H(o;o,o),

Free Boundary Problems for Circular Flows

H (O;O,Oj(w,h) = (0,O) , then w

If D

and

249

satisfy

U,S

(4.131

J-n/2

We change the variable from 8

-n/2

s E

sine

Then w(s) 5 w(0)

satisfies

Expanding w

by the series of the Legendre polynomials,we see that w

must be

a constant. Then (4.14)implies that w E 0. Consequently '(w,X) = (0,O). A

To show Chat Range D U,S

fine operators A

and B P

H(O;O,O)

= Y x H ,we do as follows. Firstly we de-

by the equalities below.

v=-(-w"+w'tane 2 2r 0

(4.15)

+pw)-h--+~w-X, 2r2 1-I

(WCX),

(4.16)

(

1-1

is a positive parameter ) .

Then B

t L ( X X R , Y x R ) is a compact operaP tor. By the Riesz-Schauder theory it is sufficient to show that A is an iso-

morphism from X x IR

onto

Y x R , we have to find a

Y x R I'or some 1-1.

, we

employ the Tollowing

NOTATION.

(w,h) c X x R satisfying (4.15) and (4.16). Observe

that we have only to show the surjectivity of

Therefore,for a given (v, O ,

(3)

r > 0 ,

C a r e f u l n u m e r i c a l c o m p u t a t i o n s [ 4 1 i n d i c a t e t h a t i n d e e d t h e blowup o c c u r s w i t h t h e power 2/3 a s o b t a i n e d by Zakharov and Synakh. Concerning t h e n a t u r e of t h e s i n g u l a r i t y i n t h e s u p e r c r i t i c a l c a s e l i t t l e seems t o be known. W e have l o o k e d i n d e t a i l i n t o t h e p r o b l e m of u n d e r s t a n d i n g t h e

form ( 2 ) o f t h e blowing up s o l u t i o n i n t h e c r i t i c a l c a s e .

For tech-

n i c a l r e a s o n s w e have so f a r r e s t r i c t e d a t t e n t i o n t o t h e c a s e

N = l

a = 2 . W e h a v e shown t h a t i n t h i s case t h e r e i s a f u n c t i o n z ( t , x ) 1 1 i n H (IR ) f o r - ~ ~ i t < O O<E,,, , s u f f i c i e n t l y small, such t h a t f o r each

Xo # 0

Focusing Singurarity Nonlinear Schroedinger Equation

is a solution of (1) in

t < 0

sup /z(t,x)j

(-t)2'7

(4)

255

and as

In other words we have shown that singular solutions of the form (3) exist with

being a lower order correction in view of

We do not know why solutions of the form ( 3 ) arise as singular

(4).

solutions for a broad class of initial data as has been observed numerically. The main tool in the analysis is the study of the linearized problem about

2iwt + Aw

(5)

Schroedinger equation

(cr+l)R2"w

OR'";

= 0

then we may rewrite ( 5 ) i n system form

u+iv

L+ = -A

L-

+ 1-

(2a+l)R2"

-A + 1 - R 2 0

On pairs of functions

. H1 x H1

define the bilinear form

(7)

One verifies easily that this bilinear form is invariant for solutions of ( 6 ) .

However, B

is not an inner product in

cause it is not positive definite owing to the nullspace that has. One easily finds that the function

satisfy

be-

256

D. MCLAUGHLIN, G . PAPANICOLAOU and M. WEINSTEIN

(9)

Lnl

= Ln2 = 0

L2n 3 = L2n 4 = 0 .

Moreover these null vectors are associated with the classical symmetries of our problem that take a solution

where

@(t.x)

into

(x,g,xo,to) are four parameters.

One might expect that the bilinear form tions in

n2, n 3 ,

restricted to func-

n1 ,

that are orthogonal to four function pairs

(the biorthogonal basis for example) is positive. 2 u < ( N = l in the present N

This is true in the subcritical case discussion) and in fact H 1 X H1

inner product in

becomes then equivalent to the standard

But this is not true in the critical

case!

In the critical case there is one more symmetry to the problem (N=l, u = 2 )

where

X =

a-l,

T =

linear function. solutions with

jA2ds, 8 = Ax

and

a = 0

, i.e.

is a

(Notice that this transformation leads to singular

fi singularity; they have never been observed in Therefore, in addition to ( 8 ) we have

numerical experiments.)

But, without having another classical symmetry, we also have (11)

(:)

It can be shown that space of

and that

, L+p = -x2 R , n1,n2,...,n6 B

with

L4n 6 = 0

span now the (generalized) null-

restricted to functions orthogonal to six

Focusing Singurarity Nonlinear Schroedinger Equation

function pairs Hl

nl,

... ,ri6

257

is an inner product equivalent to

One now looks for solutions of the form ( 3 ) and one must show a

z(t,x) with the correct properties exists.

The power

2/7

emerges as the only suitable candidate for this purpose and the structure of the nullspace discussed above is essential. *Supported by Air Force grant AFOSR-80-0228.

REFERENCES Zakharov, V.E. and Synakh, V.S., The nature of the self-focusing singularity, JETP 41 (1976) 465-468. McLaughlin, D., Papanicolaou, G. and Weinstein, M., Focusing and saturation in nonlinear beams, to appear. Weinstein, M., N.Y.U. Dissertation, 1982. Sulem, P.L., Sulem, C. and Patera, K., Numerical investigation of focusing singularities, to appear. Zakharov, V.E. and Shabat, A.B., Exact theory of two-dimensional self focusing and one-dimensional self modulation of waves in nonlinear media, JETP 34 (1972) 62-69. Ginibre, J. and Velo, G., On a class of nonlinear Schrodinger equations I, 11, J. Funct. Anal. 32 (1979) 1 - 3 2 , 33-71. Glassey, R.T., On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrodinger equation, J. Math. Phys. 18 (1977) 1794-1797.

This Page Intentionally Left Blank

Lecture Notes in Num. Appl. Anal., 5 , 259-271 (1982) N o t i l i n e a r PDE it1 A p p l i e d Scieiice. U.S.-Jrrpan S e m i t w r . Tokyo. 1982 Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifold

Mikio Sato RIMS, Kyoto University, Kyoto 606 Yasuko Satc Mathematics Department, Ryukyu University, Okinawa 903-01

In the winter of 1980-81 it was found that the totality of solutions of the Kadomtsev - Petviashvili equation as well as of its multi-component generalizati.on forms an infinite dimensional Grassmann manifold [l].

In this picture the time

evolution of a solution is interpreted as the dynamical motion of a point on this manifold.

A generic solution corresponds to a generic point whose orbit (in the

infinitely many time variables) is dense in the manifold, whereas degenerate solutions corresponding to points bound on those closed submanifolds which are stable under the time evolution describe the solutions to various specialized equations such as KdV, Boussinesq, nonlinear Schrodinger, sine-Gordon, etc. We foresee that a similar structural theory should hold also for multidimensional 'integrable' systems.

§1. The universal Grassmann manifold F o r a vector space

Grassmann manifold

V=V(N)

(say, over

GM(m,V) (=GM(m,n))

C) of dimension N (=m+n)

is by definition the parameter space for

the totality of m-dimensional subspaces in V. GM(m,V)

the

We can write

{m-frames in V} / GL(m)

where an m-frame means an m-tuple of linearly independent vectors. homogeneous space of the general linear group GL(V).

259

GM(m,V)

is a

Mikio SATO and Yasuko SATO

260

Further, itis viewed as an algebraic submanifold (of dimension mn) of the N

dimensional projective space P(/\%)

(m)-l

projective embedding). denote a basis of

by letting an m-frame

( 5 ( 0 ) ,. . . ,

V, then

antisymmetric in suffixes) satisfy the Plccker's relations:

and vice versa; i.e. a point in the ambient IP(/\?V)

lies in the embedded

if and only i f its projective coordinates E;,o.*.

, 05!2.. Y ;

and, denoting by

Y GM (m,n)

those points

corresponds, we have a cellular decomposition GM(m,n) = 1 Y Y = g o + * * . + i lm-1 --m(m-l) GM (m,n) (Cmn-IY! / Y I = size of 2

'mn (resp.

UGM'). Y

Consider the infinite dimensional vector space V (resp.

consisting of

262

Mikio SATOand Yasuko SATO

elements

( 5 u ) u E z , with c u e & , cu

(Setting e,, = ( d , , v ) u e Z t ~

for v 0).

one also writes

subject to the condition z

In any case we have, for the PlGcker coordinates 5 Y (t)

where

denotes the empty Young diagram, x,(t)

for the general linear group, and obtained from xy(t)

xY (2t )

by replacing tv

one has

< a-'.) of d;(t),

denotes the character polynomial

denotes the differential operator

i a ~7 .

(After H. Weyl, xy(t)

admits various expressions, one of which is

v. v,

v v where

is the irreducible character of the symmetric permutation

ny(l '2

group of

/ Y / letters, labeled by the Young diagram Y

v1 cycles of size

conjugacy class consisting of

We call

(t)

the T

above formulae show that

function of

T(t)

1, v2

and evaluated at the cycles of size 2 , etc.)

(Notation: T(t; 6 ) or

T(t)).

The

plays the role of generating function for Plccker

coordinates:

dratic dif-

and

ferential equations, or, what amounts to the same, the form of 'bilinear' equations of R. Hirota. Summing up, we have Theorem 1.

Although any

f(t)c

E[[tl,t2,***]]

admits the formal expansion of the

form: f(t) = Lc (t), where the coefficients are uniquely given by c = Y Y Y Y function of some e; if and only if its xY ( a t )f(t) 1 t ~ it, represents the T

coefficients cy

satisfy the PlGcker's relations.

265

Infinite Dimensional Grassmann Manifold Theorem 2.

f(t)E(C[[tl,t2,*-*]]

is the

function of some 6 6 GM

if and

only if it satisfies theHirotabilinear equations of the form

Moreover these exhaust all the Hirota equations to be satisfied by

These quadratic differential equations are also equivalent to the quadratic difference equations. Namely, Theorem 3. SO

that

(Addition formulae)

t+[a]

SQ0..

For any a t (c

(tl+a, t2+!jCt2,*-*).Let a i € E for

.‘m- 1 (t)

1 3 [a] = (a,?1 2 ,y ,*

we set

i = O,-.-,N-l

and define

a )‘c(t+[aa ]+**-+[a I), O S R . < N ‘m-l,*”, to 0 ‘m-1

A(a

II ( a . - a . ) . Then (t) satisfy the PlGcker’s m>i>iZo ‘m- 1 _.relations for GM(m,V(N)). This property again characterizes the function T .

with

A(am-l,.-.,ao)

’

E.g. we have

Denote by

UiJ

the linear operator on V

sending e

and

all the

e K # u , to 0 ( i . e . EVu&eK = c,ev), and by L the vector field K’ uu induced by E (i.e. EL IF(O -Res P*dx. Thus

{differential oprators}

fcL,x,,

as a subring. We have:

;

&[[XI],

as left

class of

similarly with

Res Pdx

mod &x

P+ + P-

with

P+

&(m)

and

& .

Then V

of 5 1

by its maximal left ideal

1 cveuEV correspond to the residue -m,

with

l(-)vS-y-let

6 /Ex.

V*,

Further we write

V =

that we have:

< S ' , P C > = 0 s.t. av(x)

are holomorphic in

1x1 < 6,

"/FvV

d v

av(x)(z)

-m(y<m

-v-1

(x) I

v * ==,

are bounded as

1x1 < 6 1,

both uniformly in ={

'Jv! la

0 and

3k61N s.t. a (x)

are polynomials of

degree 2 k}, and get:

Vans(') =

that W

d -V eZ(') 1 wv -(-)dx 6?

v20

-1

d -V 1 (z) -wc

v20 : W E )&

with w* = 1. 0

with

Let

C((x))

denote the totality of such nionic operators the field of formal Laurent series in x, which

is the field of quotients of

C[[x]]),

there exists m , n E N s.t. x%

and

E E[[x]]

=fl)x.

which is monic (i.e. w =1) 0 is again an operator of the same kind which we shall write W-l =

Consider the operator W so

I lanax,

&ana

for y = l , Z ,

...).

satisfying the additional condition that

W-'xn

Set 'V

both @

For W

we set y(W)

[ ~ ~ ~ x l l This . definition of (This is because xV' Theorem 5 .

For

and

xnw:

C e v C V ; VrD is a l s o characterized by

1 or 0 according as Y=$

-1 n 41 (W x )V , where

y(W)

is s o chosen that W-'xnE

does not depend on the choice of such n .

).'V

tv, y(W)

and this map is bijective, namely

In this correspondence, the inverse images of

vfin and Wana('),

m (i.e. x wv

v 2.

far, accounts are given for the 1-component case. To generalize it to

the r-component case we shall modify the notations a s follows. For U 6 Z and the O-i

1, and otherwise

p o i n t ( k g r a n g i a n ) form i s

The f i n a l minimum over

it i s

Suf

as i n 1 B .

-m

The o p t i m a l i t y c o n d i t i o n s are

which gives an i n t e r e s t i n g form f o r

It i s the character-

i s t i c f u n c t on, or more e x a c t l y a m u l t i p l e

of t h e

l//P-Q

c h a r a c t e r i s t i c function, of t h e s e t bounded by t h e l i n e between

and

I-

--where t h e s e a r e t h e p o i n t s ( n o t

n e c e s s a r i l y unique) a t which Thus

except a c r o s s t h i s l i n e ; on t h e l i n e ,

s i n g u l a r measure of mass one and magnitude

Q = (0,-r).

i s t h e normal v e c t o r of

on t h e c i r c l e

The Lipschitz c o n s t a n t i s

and

1/2r

$,J -trx)

An optimal

and

(nfn,).

( i )and ( i i ) of Theorem 15 a r e g e n e r a l i z e d as f o l l o w s by t h e G

equation.

Recall

t h e s e t G d e f i n e d b e f o r e Theorem 2: m

Theorem 16 ( [ 3 6 ] ) . Let (p,h,H,H*) (H#H*)

b e given w i t h {An}n=O=a(Ap,h,H)and

{X~}n=O=o(Ap,h,HY).L e t , f u r t h e r m o r e , N b e f i n i t e and Ozn1< n2 E, 2 0

1 1 = - - - > q P

Put for

vc(S)x,

Let

Theorem 2 . 4 . p

and

jG1

pj(0,c).

ao, c o ,

[0,1), R = 0 , 1

2-R n e

-1< I - - - 2.

n+A

[0,1]

c1 E

v(X,c)

Pc =

(A-Bc)-'(I-Pc)

RAcu(c) a

> 0

satisiy

he us before.

Suppose

q 5 2

324

Seiji UKAIand Kiyoshi ASANO

(iii) Under t h e c o n d i t i o n o f ( i ) ( i i i w i t h a y-v(O,c) c Y p ' - ,

-M0v(O,c)

and -Bcv(O,c)

13 = P = 0, v(0,c)

= u(c)

Combining this with the estimate in [ 5 ] for of

Proposition 2.5. Let

n 2 3

Then t h e r e is a constant

A(l

and noting that the nullspace

(2.11)

(0)hc + @ OC

and Go

C 2 0

be as b e f o r e .

and for any

Let

Brio],

-1

Lp.

2 + -).

The inverse Bc @c

Wp,

is invariant in c , we have the

with

holds i n

obtained in Theorem 2.4 is also useful to solve ( 2 . 4 ) .

i s substituted, it is reduced to

Rc(0)hc

Proposition 2.6. L e t

B c 6 = -KcRc(0)hc.

Bc-lKcRc(0)hc.

and

be as before and 2et

Therefore

Stationary Solutions of the Boltzmann Equation These two propositions enable us to (2.5). when

-L

0, and that if

8 > 0, (2.11) becomes meaningless

u = jclav

in (2.5) to write

c by

3 , 8 = 0 is excluded since then (2.10) becomes vac-

But this difficulty can be removed as follows.

then a1

for

apply the contraction mapping principle

It should be noted, however, that if

u o u s for

-G[v]

for

Put

-G[v](c)

0. For functions v

a such that a1
0

and

> 0

and Proposition 2.6

is sufficiently small.

(ii), G

maps

Proposition 2.5 and writing the norm of

where

0 =

al > 0, T

LY.

By virtue of Theorem 2 . 4 (i)

Vi,E

into itself, and by the aid o f

B ,E

111 111,

> 0, whence the desired conclusion readily fol-

lows.

Now

has a unique fixed point

solves (2.5) uniquely.

Theorem 2.4(iii)

the proof of the

Theorem 2 . 7 .

Let

n 2 3

and l e t

v = v(c)

V;,€

and

lc/"v(c)

and Proposition 2.6 (i) then complete

326

Then there e x i s t s a p o s i t i v e number

and a constant

BIEO], (2.1) has a unique LP-s,Zution

> 0

Moreover

C 2 0

such t h a t f o r each Vp

B ,E

for1 any

and

Obviously fc

go ( c

f c = gc

g;/'u

i s a d e s i r e d s t a t i o n a r y s o l u t i o n t o (1.1) and

0).

3 . S t a b i l i t y of S t a t i o n a r y S o l u t i o n . I n ( l . l ) , put

f = fc

Then

g ol/'w

w = w(t,x,S)

= g

go1'2(uc+"),

should s o l v e

If we would have a n i c e decay e s t i m a t e i n l i n e a r i z e d e q u a t i o n t o (3.1),

of t h e l i n e a r semigroup f o r t h e

then we could prove t h e e x i s t e n c e i n t h e l a r g e i n

time f o r t h e n o n l i n e a r problem (3.1) by t h e technique developed f o r t h e c a s e

Stationary Solutions of the Boltzmann Equation [1,91).

c = 0 ( s e e e.g.

327

However such an e s t i m a t e i s d i f f i c u l t t o deduce because

of t h e presence of t h e term

l'[u

which i s a n o p e r a t o r w i t h " v a r i a b l e co-

e f f i c i e n t " , and s o w e have t o l i n e a r i z e (3.1) i g n o r i n g a l s o t h i s term. a g a i n meet t h e o p e r a t o r

E (t) = e

w = w(t)

Then i f

of ( 2 . 3 ) .

tBC

Thus w e

Suppose i t g e n e r a t e s a semigroup

is a solution t o (3.1), i t s a t i s f i e s t

(3.2)

w(t) = Ec(t)wo

+ .f

Ec(t-s){2r [u~,w(s)]+~[w(s).w(s)]]~~

uc = 0

When

as i s t h e c a s e with

c = 0, t h e e x i s t e n c e i n t h e l a r g e i n

(3.2) can be shown i f t h e decay r a t e

decay with

l ' [ u c , w]

y > 1 i s r e q u i r e d a s w e l l a s t h e s m a l l n e s s of

when u

for

y > 1/2,

is available with

E c ( t ) = O(t')

b u t i n o r d e r t o d i s p o s e of t h e e x t r a l i n e a r term

the

The d e s i r e d decay s h a l l be found s t a r t i n g from t h e semigroup

:t B Em(t) = e

g e n e r a t e d by

of ( 2 . 6 ) .

t r a n s f o r m J-l[(A-A)-l] m

Be.

R e c a l l t h a t a semigroup

etA

of t h e r e s o l v e n t o f t h e g e n e r a t o r

is t h e i n v e r s e Laplace A.

Apply t h i s t o

By v i r t u e of Theorem 2 . 1 , w e have t h e o r t h o g o n a l decomposition

and can f i n d t h a t

while for

1 5 j

n+2,

Suppose

Proposition 3.1.

Iq 5

p s

and

m = 0, 1. Put

n l - -). 1 2 q P

= -(-

Then there is a constant

C >_ 0

and for all

c E lRn, t 2 0

Theorem 2.2, in this time with

1 -P

Take the inverse Laplace transform of (2.8) to obtain

(3.4)

Ec(t)

where, writing

and

j < n+2,

It suffices to proceed exactly in the same way as in the proof of

Proof.

where

and

rEc(t)e

(y-rE:(t)*e)**Dc(t)*iorEz(t)e

T a c k ) = Tc(X)(1-Tc(A))-',

* means the convolution in

329

Stationary Solutions of the Boltzmann Equation

No c o n f u s i o n s a r i s e w i t h t h e a d j o i n t symbol

Let

P r o p o s i t i o n 3.2.

[ 0 , 1 ) , there is a c o n s t a n t

with y 2

1 y(n-l+F))

Note t h a t 3.1,

i f

= $n-1)

s u b s t i t u t e d i n t o (3.4),

only

rewrite

Let

y > 1/2

and

p, a

For 8

is even.

8 > 0.

Propositions

g i v e d e s i r e d estimates of

and

Ec(t)

More p r e c i s e l y i f w e

H [ w ] ( t ) , w e g e t t h e f o l l o w i n g estimates. p o , cx0

B[co],

by t h e a i d of t h e scheme i n [S] s t a t e d e a r l i e r .

Theorem 2 . 7 ,

Then

be a s in P r o p o s i t i o n 2.3.

and f o r any

i s odd and

w r i t e t h e r i g h t s i d e of ( 3 . 2 ) as

po < n .

co >

> 1 is possible for

3.2 a n d (3.3),

Ec(t)*,

and

n 2 3

and i mpose t h e a d d i t i o n a l c o n d i t i o n

Seiji UKAIand Kiyoshi ASANO

330 where a

IIuAl

and xpo

-0

c(c( O

0 (c

111.11='sup(l+t)Yl\w(t)[/ . XP a

By Theorem 2.7, Icl-Oa

0).

-f

It then follows that if wo

is small in Xp

Zq, and

is small, H

XP ) D is the desired solution to B 0([O,,);

traction map on a ball o f the Banach space of functions w(t) such that

Illdl\
K~/P,

i s positive definite, satisfying

L e t a(A ) be t h e s p e c t r u m s e t o f A P

w h i c h i s t h e complement of t h e r e s o l v e n t

s e t P ( A ) o f A i n t h e complex p l a n e C : P P p ( A p ) = {AcC: t h e r e i s a bounded i n v e r s e (A-A ) - ' P

from X

i n t o Xp}.

By P r o p o s i t i o n 2 o(A ) i s a p a r t o f r e a l l i n e w h i c h i s bounded below. P

Let

i n f a(A ) . P

Then we have

342

Teruo

USHlJlMA

An e q u i l i b r i u m {p,v=O,P,J,B,E=O}

Definition 3

and u n s t a b l e i f i > O , i = O , and

X c(6)

6 > 1, there exist

and X,Y

B(F,r)

r > 0 and c ( 6 ) > 0

where

B(?,r)

and its radius r. Proof. Because of our conditions (1.151, we get

such that

is a ball whose center is

367

Chaos and Age Dependent Population Model

Therefore we can show easily

(1 DF($x/~ Here

min

JA,in(Ix (1

( for all

means the minimum eigenvalue of DF(G)*DF(c).

R~ By the continuity,

where At

2 (1 + -

c(6) :

c.q.f.d.

6).

Jhmin

Lemma 2. For sufficiently small open neighhourhood W bounded set B, there exists a positive constant c(W,B)

?? and any

such that the equation

GAt(w) = b

has at least one solution w

for any At > c(W,B)

and for any h

Using these lemmas we can constract a snap-hack repeller. Thus we can prove the conclusion of the theorem.

An age dependent population model.

We consider here an age dependent population model which is described by the following equation:

Masaya YAMAGUTI and Masayoshi HATA

368

( I: b(k)uF)(R

k=l

N C b(k)ut) k=l

(2.1)

where we denote

O such that

t 0 for any n

2 0.

3 represent a homoclinic orbit of ha which is found

Chaos and Age Dependent Population Model

313

by starting in a fixed point and iterating backward.

Figure 3. Homoclinic orbit of h

. 1

Using a transversal homoclinic orbit

transversal homoclinic orbit

P - ~ =

‘P-n’ n>-O of ha, we construct a

{P-n} n ~ Oof Ha as follows;

( p-n ,p-n

,..., p-,

for

2 0,

since from (2.141,

Since the existence of a snap-back repeller is a stable property under small CLperturbations and the orbit

“-n’

n>O of

is contained in the interior of Q,

E
0

, satisfies

sup

I/Dw//

1 i s equivalent t o the

A c c o r d i n g t o Lemma 2 , one may g u a r a n t e e t h a t

to 5 t 2 t

inequality

by t a k i n g

small.

//Vwo[(

I / w o ( ) 5 c / / V w o \ / , we s e e t h a t i f

condition of Definition 1 ensures t h a t and h e n c e Lemma 3 e n s u r e s t h a t

Mindful of P o i n c a r e ' s

is taken small, then the

//vwot]

]/w(t)Il

//Vw(t)

/ / V w ( t ) / l is s m a l l ,

is s m a l l f o r a l l

t 2 t

t t t

0 '

Thus

t h e c o n d i t i o n of Theorem 1 i s s a t i s f i e d . Next we check t h a t t h e c o n d i t i o n o f Theorem 1 i m p l i e s t h a t of D e f i n i t i o n 1.

A c c o r d i n g t o Lemma 1 , one may g u a r a n t e e t h a t to 5 t

1 , by t a k i n g

a c c o r d i n g t o Lemma 3 . time

implies

small.

/lw,II

]/w(t)

But t h e n ,

is s m a l l , for I/Vw(to+l)

i s also small,

Hence t h e c o n d i t i o n o f Theorem 1, c o n s i d e r e d w i t h s t a r t i n g IIVw(t)

is small for

Poincare's inequality, (/w(t)I/ is small for

t 2 t

Thus, remembering

This completes

the proof. The n e x t t h e o r e m i s more c o m p l i c a t e d , b u t proved by a s i m i l a r t y p e o f argument

I.he s t a b i l i t y

Theorem 2 .

D e f i n i t i o n 2 & e q u i v a l e n t t o anu one

condition

o f t h e f o l l o w i n s conditions

(i) There e z i s t numbers Atk

and 1 -

/lvw0ll < 6

i i i ) There e z i s t numbers

6,T>O

such t h a t every p e r t z e b u t i o n

, satisfies

6,a,A>O

such t h a t every p e r t u r b a t i o n

386

John G . HEYWOOD

with __

(iiil w

E 0

[lwo(( < 6

woe J

, satisfies

Thcre e x i s t nunhers

such t h a t every perturbation

G,cx,A>O

w , li)ith

Ilvw0II < 6 , s u t i a f i e s

and 1 -

One o f o u r p r i n c i p a l r e s u l t s a b o u t t h e n u m e r i c a l a n a l y s i s of p r o b l e m (1) is t h a t t h e e r r o r c o n s t a n t s of P r o p o s i t i o n 2 r e m a i n hounded a s solution

Theorem 3.

t 2 0

K , ho > 0

__ Proof.

-+ m

, if

the

b e i n g approximated i s e x p o n e n t i a l l y s t a b l e .

If. , and

u,p

uh,ph

g g continuous

is exponentially

4d i s c r e t e

solutions defined

s t a b l e , then there e x i s t constants

such t h a t

Rather than a c t u a l l y choosing

as i n D e f i n i t i o n 2 , i t w i l l h e more

6,T

c o n v e n i e n t t o c h o o s e them i n a c c o r d a n c e w i t h Theorem 2 , so t h a t f o r a n y s o l u t i o n v

of ( 6 ) s a t i s f y i n g

IIV(v-u)(to)

< 6

, there holds

The main p o i n t t o b e e s t a b l i s h e d i s a n i n d u c t i o n s t e p f o r t h e v e l o c i t y e r r o r estimate. of

h < ho

We c l a i m t h e r e e x i s t c o n s t a n t s and

to 2 0

, if

and

such t h a t , f o r any c h o i c e

Numerical Navier-Stokes Problem

387

(11)

then

Since

sup((VulI < t>O

where

depends on

i t i s c l e a r t h a t (11) i m p l i e s

and

only through t h e i r product

Using

P r o p o s i t i o n 3 , one sees t h a t ( 1 3 ) i m p l i e s

with

a l s o d e p e n d i n g on

and

only through t h e i r product

Clearly

F u r t h e r , using ( 3 ) , i t is seen t h a t ( 1 4 ) i m p l i e s

with

a g a i n d e p e n d i n g on

and

only through t h e i r product

F i n a l l y , t a k i n g (11) and ( 1 6 ) t o g e t h e r , i t i s e v i d e n t t h a t

provided Let

h K and v

a r e s m a l l enough.

be t h e s o l u t i o n of (6) s a t i s f y i n g

s a t i s f i e s ( 9 ) and ( l o ) , p r o v i d e d

In view o f ( 1 5 ) , (lo), an e r r o r e s t i m a t e

and

v(to)

R \(to)

Then

a r e s m a l l enough t o e n s u r e ( 1 7 ) .

( 1 6 ) and ( 1 4 ) , w e c a n a p p l y P r o p o s i t i o n 2 t o o b t a i n

John G . HEYWOOD

388

between C2

and

C3 , i . e . ,

l a r g e and a l l

with constants

d e p e n d i n g on

only through t h e i r product

and

o n l y through

Thus, f o r

sufficiently

s u f f i c i e n t l y s m a l l , t h e r e will h o l d

K e K T < i K , C3 5 K w h i l e a t t h e same t i m e b o t h

hK w i l l be s m a l l enough t o e n s u r e ( 1 7 ) .

and

Now ( 1 6 ) and (11) i m p l y (18) imply ( 1 2 ) .

\\\(v-u)(to)\llh 5 2 h K

s o t h a t t o g e t h e r ( 9 ) and

This completes t h e proof of t h e v e l o c i t y e r r o r estimate ( 8 ) .

The p r e s s u r e e r r o r estimate (8) i s a r e l a t i v e l y e a s y c o n s e q u e n c e o f i t . Much o f t h e e x i s t i n g t h e o r y o f h y d r o d y n a m i c s t a b i l i t y r e s t s upon t h e " p r i n c i p l e of l i n e a r i z e d s t a b i l i t y " .

This is a general a s s e r t i o n t h a t i n

d e t e r m i n i n g t h e s t a b i l i t y of a s o l u t i o n

it sufficestoconsider the linear-

ized perturbation equation

(19)

- bW

+ u-Dw- + w-vu

= -Dq

i n place of the f u l l nonlinear perturbation equation ( 7 ) .

In the lollowing

t h e o r e m we g i v e a p r e c i s e s t a t e m e n t o f t h e p r i n c i p l e o f l i n e a r i z e d s t a b i l i t y a p p r o p r i a t e in t h e g e n e r a l c o n t e x t of t h e n o n s t a t i o n a r y p r o b l e m .

The , > r o o f

i s a d i r e c t and s i m p l e o n e , e n t i r e l y b y p a s s i n g s p e c t r a l m e t h o d s , a s i n d e e d one must i n t h e n o n s t a t i n a r y case.

Theorem 4 .

The s o l u t i o n

if there e x i s t numbers

problem (1)

a,A > 0

, such

& exponetially

t h a t every s o l u t i o n

s t a b l e i f and onLg

of t h e l i n e a r i z e d

Erturhation equation ( 1 9 ) s a t i s f i e s

m. L e t respectively,

J, =

w-w

, where

satisfying

;(t

and

are s o l u t i o n s o f (19) a n d ( 7 ) ,

) = w ( t ) = wo

S u b t r a c t i n g ( 7 ) from ( 1 9 ) g i v e s

Numerical Navier-Stokes Problem

f o r some s c a l a r f u n c t i o n

- A$

$It

u*v*

$.VU

M u l t i p l y i n g by

389

w.vw = -vq

and i n t e g r a t i n g ,

this leads t o

U s i n g G r o n w a l l ' s i n e q u a l i t y now y i e l d s

T > 0

f o r any f i x e d

Thus, i f

cM T

sup [to,to+Tl

IIVwo/!

to+T

I I ~ ~ I It I

I I ~ W I I

is s u f f i c i e n t l y s m a l l , depending on

T , Lemmas 2 a n d 1 i m p l y

Now s u p p o s e the c o n d i t i o n o f Theorem 4 h o l d s .

Choose

above s u c h t h a t

(20) i m p l i e s

Il;(to+T)

Then, a l s o , p r o v i d e d

IIVwo//

5 ~llwolI

is s u f f i c i e n t l y small, (21) i m p l i e s

!lWo+T)

5 ;llw,ll

Combining t h e s e g i v e s

s h o w i n g t h a t c o n d i t i o n ( i ) o f Theorem 2 i s s a t i s f i e d , i m p l y i n g t h e e x p o n e n t i a l s t a b i l i t y of

To show t h a t e x p o n e n t i a l s t a b i l i t y i m p l i e s l i n e a r i z e d s t a b i l i t y , w e a r g u e similarly, s t a r t i n g again with (21).

This completes t h e proof.

I n [l], w e a p p l i e d Theorem 4 t o show t h a t t h e s e t o f i n i t i a l v a l u e s f o r

u , w h i c h g i v e r i s e t o s o l u t i o n s t h a t are e x p o n e n t i a l l y s t a b l e a n d h a v e bounded

390

John G. HEYWOOD

D i r i c h l e t norms, i s open w i t h r e s p e c t t o t h e D i r i c h l e t norm.

All s o l u t i o n s

s t a r t i n g w i t h i n a common c o n n e c t i v i t y component of t h i s s e t converge t o g e t h e r a s t +

We a l s o showed t h a t an e x p o n e n t i a l l y s t a b l e s o l u t i o n n e c e s s a r i l y t e n d s

t o a steady o r time p e r i o d i c motion, i f t h e f o r c e s and boundary c o n d i t i o n s a r e s t e a d y o r time p e r i o d i c .

These r e s u l t s combined w i t h Theorem 3 were shown t o

p r o v i d e a j u s t i f i c a t i o n of t i m e s t e p p i n g a s a means of c a l c u l a t i n g s t e a d y o r t i m e periodic solutions.

Quasi-Exponential S t a b i l i t y Below,

w i l l r e p r e s e n t t h e a n g u l a r v a r i a b l e about an a x i s o f s y m e t r y

common t o both

f , i f t h e r e i s one.

and

For s i m p l i c i t y , we w i l l w r i t e

u = u ( $ , t ) , s u p p r e s s i n g i n o u r n o t a t i o n t h e u s u a l l y n o n t r i v i a l dependence u

on t h e o t h e r s p a t i a l v a r i a b l e s .

The symbol

w i l l a l s o denote an a n g l e

about t h e a x i s o f symmetry, thought of a s a r o t a t i o n .

possess a common a x i s of symmetry, i t w i l l be understood t h a t

R ,if

f o r any s

w = 0

D e f i n i t i o n 3.

Ve say u

& guasi-exponentiaZZy

such t h a t j-or e u e r x p e r t u r b a t i o n

6,T,B > 0

there exists a --___

do n o t

Further,

i s t i m e independent w e w i l l c o n s i d e r t i m e s h i f t s denoted by

i s n o t t i m e independent, i t w i l l be understood t h a t

€

and

time shift

s = 0

stable i f there are numbers

and a spatial r o t a t i o n

wo w

J &g

//woI/ < 6

satisfying

(22)

where __

i s t h e solution of the perturbed probZern ( 6 ) corresponding t o the

perturbation

;(x,t)

= u($+w,t+s)

A s i m p l e example of q u a s i - e x p o n e n t i a l s t a b i l i t y o c c u r s i n t h e Taylor

experiment.

A t c e r t a i n r o t a t i o n a l speeds of t h e c y l i n d e r s , t h e convection c e l l s

l o o s e r o t a t i o n a l symmetry, t a k i n g on a wavy appearance i n t h e a n g u l a r v a r i a b l e . C l e a r l y , i f t h e boundary v a l u e s and f o r c e s a r e r o t a t i o n a l l y symmetric, a small a n g u l a r s h i f t i n t h e p a t t e r n of waves w i l l c o n s t i t u t e an a d m i s s i b l e p e r t u r b a t i o n

39 1

Numerical Navier-Stokes Problem

w i t h no tendency t o decay.

However, t h e same r e a s o n i n g t h a t l e a d s one t o

b e l i e v e simple Taylor c e l l s a r e e x p o n e n t i a l l y s t a b l e l e a d s t o t h e c o n c l u s i o n t h a t wavy Taylor cells a r e q u a s i - e x p o n e n t i a l l y s t a b l e "modulo s p a t i a l r o t a t i o n s " , meaning t h a t t h e r e i s a f i x e d l e n g t h of t i m e between a s l i g h t l y d i s t u r b e d flow

d u r i n g which t h e d i f f e r e n c e

and a s l i g h t l y r o t a t e d image

u(++w,t)

of t h e o r i g i n a l undisturbed flow w i l l decay t o h a l f t h e s i z e of t h e i n i t i a l perturbation

w =v(to)-u(to)

, and f u r t h e r t h a t t h e r e q u i r e d r o t a t i o n

should be l e s s than a f i x e d c o n s t a n t

times t h e s i z e of t h e i n i t i a l p e r t u r -

I n t h i s case the t i m e s h i f t

i n D e f i n i t i o n 3 i s taken t o be z e r o .

bation.

A l t e r n a t i v e l y , i f t h e waves a r e p r e c e s s i n g about t h e a x i s of symmetry, and i f t h e f o r c e s and boundary v a l u e s a r e time independent, t h e flow can be considered a s q u a s i - e x p o n e n t i a l l y s t a b l e "modulo time s h i f t s " , meaning t h a t t h e r e e x i s t s a time shift

such t h a t t h e d i f f e r e n c e between

t o h a l f t h e s i z e of

i n time

and

u ( + , t + s ) decays

An important example of a flow which i s

q u a s i - e x p o n e n t i a l l y s t a b l e modulo time s h i f t s , b u t n o t modulo r o t a t i o n s , i s provided by von-KQrmdn v o r t e x shedding behind a c y l i n d e r .

Small p e r t u r b a t i o n s

decay modulo s l i g h t s h i f t s i n t h e t i m e phase. D e f i n i t i o n 3 p e r m i t s c o n s i d e r a t i o n of q u a s i - e x p o n e n t i a l s t a b i l i t y modulo both t i m e s h i f t s and s p a t i a l r o t a t i o n s s i m u l t a n e o u s l y .

An example o c c u r s i n

t h e Taylor experiment, when a t c e r t a i n r o t a t i o n a l speeds of t h e c y l i n e r s wavy c e l l s a r e observed t o undergo a f u r t h e r time p e r i o d i c o s c i l l a t i o n , odd and even numbered c e l l s a l t e r n a t e l y expanding and c o n t r a c t i n g .

Though t h e s e c e l l s a r e

sometimes r e f e r r e d t o as doubly t i m e p e r i o d i c , i t i s c l e a r t h a t t h e second time p e r i o d i c i t y i s p o s s i b l e only because t h e f i r s t one i s e q u i v a l e n t t o a s p a t i a l periodicity. I n [l] w e proved a r e s u l t concerning t h e d i s c r e t e approximation of q u a s i e x p o n e n t i a l l y s t a b l e s o l u t i o n s , analogous t o Theorem 3 .

I t s conclusion d i f f e r s

from t h a t of Theorem 3 i n t h a t i t provides e r r o r e s t i m a t e s modulo r o t a t i o n s and time s h i f t s .

More p r e c i s e l y , i t a s s e r t s t h e e x i s t e n c e of time dependent

John G . HEYWOOD

392 rotations ho

and

and t i m e s h i f t s

uh(t)

0 < h

such t h a t f o r

sh(t) h

, in

addition t o the constants

t > 0

and

there holds

2 ~ ~ ~ ( ~ - ~ h5 )h ( Kt I) ~ ~ ~ h

I / ( p - p h ) ( t ) l l ,. L'/Nh wh ere

L($,t)

Mo reo v e r , u h ( o )

u ( $ + wh(t ) , t + sh t ) ) =

and

hKmax(l,t-1/2)

p ( $ , t ) = p($+w,(t)

t+sh(t))

s h ( o ) = 0 , and t h e i r t i m e d e r i v a t i v e s s a t i s f y

Thus t h 2 rates o f a n g u l a r p r e c e s s an and of d r i f t i n t h e t i m e p h a s e , of t h e

d i s c r e t e s o l u t i o n r e l a t i v e t o t h e c o n t i n u o u s s o l u t i o n , a r e of o r d e r

s t a b i l i t y h a s b e e n d e v e l o p e d i n [l]

The t h e o r y of q u a s i - e x p o n e n t i a l

s i m i l a r l y to t h a t e x p o n e n t i a l s t a b i l i t y , w i t h s i m i l a r c o n s e q u e n c e s f o r d i s c r e t e approximations.

We w i l l o n l y s t a t e h e r e t h e c o r r e s p o n d i n g p r i n c i p l e of l i n e a r -

ized stability.

To u n d e r s t a n d t h e m o d i f i c a t i o n ne e de d i n Theorem 4 , n o t e t h a t

i s independent of t i m e , a n d / o r

and

p o s s e s s a common a x i s of

r o t a t i o n a l symmetry w i t h t h e c o r r e s p o n d i n g a n g u l a r v a r i a b l e derivatives

and/or

t h e n the

a re n e c e s s a r i l y s o l u t i o n s of t h e l i n e a r i z e d p e r t u r -

bation equation (19).

Theorem 5.

The soZution

problem ( 1 )

and only if t h e r e e x i s t numbers

cr,A,B > 0

& quasi-exponentiaZZy , such

s t a b l e if

t h a t every s o l u t i o n

;(t)

of t h e l i n e a r i z e d perturbation equation (19) s a t i s f i e s

@ t 2 t

, where

Nonzero _ _ _ _muZtipZiers _

are scaZar m u l t i p l i e r s s a t i s f y i n g

are required in

( 2 4 ) if and only

if nonzero

393

Numerical Navier-Stokes Problem __ time s h i f t s 5.

& nuntriuial

rotations

, respectively,

required

(23).

C o n t r a c t i v e S t a b i l i t y t o a Tolerance and Long Term A P o s t e r i o r i Error Estimates We t u r n now t o t h e q u e s t i o n of whether t h e "global e x i s t e n c e " of a smooth

s t a b l e s o l u t i o n of problem (1) can be v e r i f i e d by means o f a numerical experiment.

There i s a known argument f o r bounding a s o l u t i o n ' s D i r i c h l e t norm

(and thus o b t a i n i n g i t s f u l l r e g u l a r i t y ) " l o c a l l y " v i a a numerical experiment combined with an

posteriori e r r o r estimate.

I t goes roughly a s f o l l o w s .

Suppose t h e D i r i c h l e t norm of t h e d i s c r e t e s o l u t i o n , f o r a given mesh s i z e

i s found t o remain l e s s than some number M > Nh M

Choosing a second number

t h e D i r i c h l e t norm of t h e smooth s o l u t i o n c e r t a i n l y remains l e s s than

on some unknown i n t e r v a l

[O,th]

Using t h e l o c a l e r r o r e s t i m a t e (Proposi-

t i o n 2 ) which h o l d s on t h e b a s i s of t h e assumed bound

one then o b t a i n s an

e x p l i c i t e s t i m a t e ( e x p o n e n t i a l i n time) f o r t h e s o l u t i o n ' s D i r i c h l e t norm on

[O,t,] th

Equating t h e r i g h t s i d e o f t h i s e s t i m a t e w i t h

o r more p r e c i s e l y , a lower bound f o r

which

numbers

does indeed bound t h e D i r i c h l e t norm. remain bounded as

h + 0

i.e.,

one may s o l v e f o r

an i n t e r v a l of t i m e d u r i n g

A t b e s t , i f t h e computed

one f i n d s t h a t

of t h e e x p o n e n t i a l growth of t h e l o c a l e r r o r e s t i m a t e .

- - 1 o g h , because

I n o t h e r words, t o v e r i f y

e x i s t e n c e t h i s way on an i n t e r v a l [O,T] r e q u i r e s a numerical experiment w i t h mesh s i z e

exp(-T)

The p o i n t of Theorem 6 below i s t o demonstrate t h a t i n v e r i f y i n g e x i s t e n c e over t i m e i n t e r v a l s of any l e n g t h , i t s u f f i c e s t o work w i t h a s i n g l e s u f f i c i e n t l y small c h o i c e of t h e mesh s i z e , provided t h e d i s c r e t e s o l u t i o n i s found t o be

s t a b l e a s w e l l a s of bounded D i r i c h l e t norm. This r a i s e s t h e q u e s t i o n of whether i t i s p o s s i b l e t o v e r i f y numerically t h e s t a b i l i t y of a d i s c r e t e s o l u t i o n .

I t c e r t a i n l y i s n o t i f one has i n mind

t h e u s u a l n o t i o n s of s t a b i l i t y , which s e t a c o n d i t i o n t o be s a t i s f i e d by a l l p e r t u r b a t i o n s , no m a t t e r how s m a l l .

For t h i s reason we i n t r o d u c e , f o r u s e

394

John G . HEYWOOD

a s a h y p o t h e s i s i n Theorem 6 , a n o t h e r n o t i o n o f s t a b i l i t y which w e c a l l I n Theorem 7 i t i s shown t h a t t h e

“contractive s t a b i l i t y t o a tolerance”.

q u e s t i o n of whether a d i s c r e t e s o l u t i o n p o s s e s s e s t h i s type of s t a b i l i t y can be answered through a f i x e d , f i n i t e amount of computation p e r u n i t of time.

The

q u e s t i o n of whether t h e d i s c r e t e approximations of an e x p o n e n t i a l l y s t a b l e s o l u t i o n i n h e r i t t h e p r o p e r t y of b e i n g c o n t r a c t i v e l y s t a b l e t o a t o l e r a n c e i s answered a f f i r m a t i v e l y i n Theorem 8.

I t i s shown, moreover, t h a t t h e s t a b i l i t y

parameters of t h e d i s c r e t e s o l u t i o n are bounded uniformly i n

0 ,

s o t h a t t h e hypotheses of Theorem 6 are n e c e s s a r i l y s a t i s f i e d f o r all s u f f i c i e n t l y s m a l l v a l u e s of

Together, Theorems 6 , 7 and 8 imply t h a t t h e e x i s t e n c e

o f a s t a b l e smooth s o l u t i o n can be v e r i f i e d ( a t l e a s t i n p r i n c i p l e ) through a

f i x e d , f i n i t e amount of computation p e r u n i t of t i m e . i n [l].

The p r o o f s a r e s u p p l i e d

Below, f o r s i m p l i c i t y , we d e f i n e c o n t r a c t i v e s t a b i l i t y t o a t o l e r a n c e

r e l a t i v e t o t h e i n f i n i t e time i n t e r v a l

t
0, v

0, K > 0

represent the binding energy, viscosity and

the rate of chemical reaction respectively, 11 is a lumped variable representing

409

410

YiNc

Lung and TENZhen-Hum

density, velocity and temperature, z

is the fraction of unburht gas.

Majda 131

has investigated the travelling wave solutions of (1) and explained some interesting phenomena from it, such as strong and weak detonation waves. The properties of (1)when

v = +O

and K =

are o f most interest because

the mathematical shock waves and mathematical detonation waves are involved in the solutions at this case. We will prove the global existence of the weak solutions f o r the initial value problems under some hypotheses. The relationship between system (1) and the reacting fluid dynamic system is just the same as that between Burgers' equation and the fluid dynamic system. But system (1) is much more complicated than Burgers' equation, because first of all it is a system, not a sinele equation, secondly,

because many properties of Burgers' equation, for

example, the order principle, are violated here, another example is that there is no "overshot" of shock waves in the solutions of Burgeis' equation, while it is just normal w i t h discontinuous solutions of (1). Many difficulties in analysis arise from this. We will give some hypotheses and two definitions of weak solutions: Problem

P and Problem Q, discuss the strong discontinuous curve and the Riemann Problem in the first section, the formulation of Problem Q is stronger than that of Problem P since it determines the state at critical point

u = 0. We will prove the

global existence of Problem P at the second section if, roughly speaking, the initial values are functions with bounded variation. Under an additional hypothesis on the points where the initial value u,(x)

assumes the value

(Hypothesis A), we will prove the global existence of Problem Q at the third section. 51. The definitions of solutions. We always assume that the function f(u) is sufficiently smooth and f" > 0

Function

is defined as

4(u)

{

u > 0,

f' > 0 ,

Hyperbolic Model of Combustion where

is the "ignition temperature", which is a critical point, we will

u = 0

assume that

41 1

$(0) = 1 at the following Problem Q. Clearly

z 5 1, according

to its physical background. Let

+0, K *

in system (l), we obtain a formally classical formula-

+a,

tion as

a ax

(u + q z ) + - f(u) = 0,

z = 0,

u > 0,

az _

u < 0.

(3)

at - O,

The Rankine-Hugoniot condition is also obtained as

(4)

[ u + q z l o = Lfl.

where [ ] denotes the jump of function, 5 is the slope of the discontinuity curve. If the limit of curve are denoted by

u, z

from the left and right sides of the discontinuous

and

+ ,z

respectively, then it is easy to

classify the discontinuous curves into five classes:

a) shock waves (abbr. S), either u , u

< 0 or u , u

b) strong detonation waves (abbr. SD), u- > 0 , u+ < 0, and c) weak detonation waves (abbr. WD), u- > 0 , u+ < 0, and

z =

> 0, and

f'(u-) > 0,

f'(u-)
0, u+ < 0, and

f'(u-)

= o,

e) contact discontinuities (abbr. C ) , u

= u

# z

, where -

Some other cases are possible, for instance the case when

U = 0.

< 0 , u+ >

but we assume that the Lax condition of stability

hT1 -> o is satisfied, where

1. x;, h (u) 1

for

i = 1

0, A2(u) = f'(u),

A f ( u ( x + 0, t)), then neither the case

xi =

xi(U(X

t)),

u- < 0 , u+ > 0, nor weak detonation wave

are admissible. We will assume that only cases a,) b) d) e) are admissible in the following.

YING Lung and TENZhen-Huan

412

There are some other critical cases, for instance u will assume in the following that u- > 0 or ut > 0 to u-

z = 0 when

or u+

= 0

or u

= 0. We

u = 0, hence we may change

at the above inequalities.

For the convenience of following discussion, two auxiliary functions are defined. Function u* = g ( u , z )

Lemma 1.

is defined by

exists uniquely and

> 0,

> 0.

Proof. Set

then (5) is equivalent to

Y(u*) = 0, u*

?'(u*)

It is easy to verify

> 0

and

But = 0.

v ( t m )

But

+m,

z 2 0, hence u*

> 0 and

such that

y(u*)

> 0 can be verified from ( 5 ) directly.

By (5) it is easy to see that

corresponds to u

u + qz

therefore there exists a unique u*

SD corresponds to u- > g(u , z )

and

$(O) =

= g(u , z ).

The second auxiliary function is w = $ ( u ) , satisfying

w =

tu,

It is easy to see that $

is continuous, monotonous and one-to-one,

$'(u) 1.0.

Lemma 2. If uo < 0, z1 > z2 > 0, g ( u o . z2) 2 0, then

Hyperbolic Model of Combustion

413

(7) If u1 < u2 < 0 , zo = 0 , p(ul,

Proof.

On the

2,)

0 , then

(u, f) plane, the straight lines

are the tangent lines of curve

f = f(u)

these two lines with horizontal line

by ( 5 ) .

The intersection points or'

f = f(0) are

respectively, hence

i'" > 0. From (5)

(6) we

w2 < q(zl -

2,)

know w1 = $ ( g ( u o , zl)), w2 = $(g(uO, z2)), there-

fore (7) is proved. The proof of inequality (8) is similar. We consider the initial value problem of ( 2 ) ( 3 ) with initial values

where

satisfies 0

5 z0 -
u r , b u t

are a

and a

as t h e s o l u t i o n of e q u a t i o n

z0 ( X I , t h e n t h e s o l u t i o n of t h i s case is

i n the solution.

uI1 < 0 .

We can c o n s t r u c t t h e s o l u t i o n as c a s e a ) .

There

i n t h e solution.

uI1 > g ( u r , z r ) .

SD, it d e g e n e r a t e s t o a

uI1 > ur, uI1 1. 0 , b u t

There i s a

u(x, t )

z(x, t ) E

Set

uI1 > u r , uI1 1. 0 , and

There is a d)

T h e r e a r e four c a s e s :

are constants.

when

5 g(ur,

zr).

Let

z = 0. r

Let

CJ.

T h e r e f o r e , t h e Riemann p r o b l e m i s always s o l v a b l e .

But it s h o u l d b e n o t i c e d

t h a t even t h e c o n d i t i o n o f s t a b i l i t y i s s a t i s f i e d , t h e s o l u t i o n s a r e s t i l l n o t unique.

F o r example, when

u (x) E uo < 0 , z 0 ( x ) f z

besides t h e t r i v i a l solution t i o n as:

u Euo

and

> 0 , g(uo,

2,))

z 1 zo, we may a l s o c o n s t r u c t a s o l u -

415

this solution corresponds to the case when one fires a match in a space filled with combustible gas and oxygen. We conjecture that the solutions obtained in the following are not those solutions of "catastrophe".

For the general initial value problem, u,(x),

zo(x)

are assumed to be

bounded measurable functions. Two formulations of weak solutions are given.

Problem P. To f i n d bounded measurable functions u(x, t), z(x, t) defined in t > 0

such that for all

and

exists,and for any smooth function (p(x, t) with compact support on

holds, moreover, for any non-negative smooth function

'9'". t)

with compact

support on t L 0 ,

holds, and finally such that with

we have

z(x, t) =

Problem Q.

{

O' zo(x), if

v(x, t) > 0 , v(x, t) < 0.

To find bounded measurable functions u(x, t), z(x, t) defined in

416

YING

t > 0

s a t i s f y i n g (11) ( 1 2 ) and

where

v ( x , t)

Lung and TENZhen-Huan

i s d e f i n e d by (lb).

(16)

The formulation of Problem Q i s s t r o n g e r t h a n t h a t of Problem P , because i m p l i e s ( 1 3 ) and determines t h e s t a t e as

52.

u = 0.

The e x i s t e n c e of t h e s o l u t i o n s o f Problem P. F i r s t of a l l , l e t us c o n s i d e r a c l a s s of s p e c i a l i n i t i a l v a l u e s and d i s c u s s

t h e p r o p e r t i e s of t h e s o l u t i o n s f o r t h e s e s p e c i a l i n i t i a l value problems. Lemma 3.

(-m,

= constant, u ( x ) 0

c o n s i s t s of a f i n i t e number o f i n t e r v a l s , and

+m)

zo(x)

does n o t d e c r e a s e on each of them, t h e n t h e s o l u t i o n of Problem

Q exists.

Proof.

Suppose t h e r e a r e

When

u(x, t )

N = 1, a s o l u t i o n

i s c o n s t r u c t e d as t h e s o l u t i o n o f e q u a t i o n

u o ( x ) , u(x, t )

(10)w i t h i n i t i a l v a l u e f o r each

intervals.

I t is s u f f i c i e n t t o s e t

does n o t d e c r e a s e as a f u n c t i o n o f

z(x, t ) f z (x). 0

N = 2 , we may suppose t h e two i n t e r v a l s a r e

out losing generality.

Let

= u (+O) O

c :;:

zo(x) =

and

x > 0

with-

uk = uo(-O),

and

x > 0,

There a r e f o u r c a s e s ( c o n s u l t with t h e Riemann problem):

> 0

uR 5 ur, c o n s t r u c t

uy,> u r , b u t

and

x < 0

uk < 0 .

as t h e c a s e Construct

z ( x , t ) 5 zO(x).

N = 1 and s e t

u(x, t ) with t h e i n i t i a l v a l u e on

separately j u s t l i k e t h e case

N = 1, t h e n c o n s t r u c t a discon-

t i n u i t y through t h e o r i g i n d e f i n e d by dx _ dt

f(u(x - 0 , t ) ) - f ( u ( X + 0 , t ) ) u ( x - 0 , t ) - u(x + 0 , t )

Hyperbolic Model of Combustion u(x

u(x - 0, t )

0 , t ) i n c r e a s e s and

always unequal. c)

z(x, t ) 5 z (x)

We h a v e

' u r , ue -> 0 ,

d e c r e a s e s as

increases, but they a r e

i n t h i s case.

uR > g ( u r , z r ) .

and

417

u(x, t ) separately

Construct

l i k e b ) , then c o n s t r u c t a d i s c o n t i n u i t y d e f i n e d by

_ dx - f ( u ( x - 0 , t ) ) - f ( u ( x + 0, t ) ) a t u ( x - 0 , t ) - u ( x + 0 , t ) - 9zr

t i n u o u s v a r i e s from

u Q > u r , uQ

S t e p 1. x

S t e p 3.

N = 1 we o b t a i n t h e s o l u t i o n on

ur(x, t )

i s c o n t i n u o u s on

on t h e s e c t o r

Q(T),

and

uQ(x, t )

f'(ur)t

respectively.

g(ur, zr)

2 uQ x

t > 0 , we o b t a i n a smooth s o l u t i o n

and t h e s l o p e of

x(t)

increases, t h e curve

f'(uk)t.

Construct a solution

u(x, t ) = ( f ' )

d), i . e . t h e d i s c o n -

up.5 g ( u r , z r ) .

0, but

always l i e s i n r e g io n

i n region

u(x - 0, t )

S o l v e t h e i n i t i a l v a l u e problem of o r d i n a r y d i f f e r e n t i a l e q u a t i o n :

Because

Step

vary as t h e previous, i f

CJ.

Using t h e s o l u t i o n o f

ur(x, t )

Since

=x(t)

f ' ( u R ) t , d e n o t e them by

Step 2.

= x(t).

0, t )

to, t h e n it becomes t h e c a s e

a t some

g ( u ( x + 0, t o ) , z r )

and

increases, u ( x + 0, t ) , u ( x

f'(uR)t < x

-1 x ($

z = 0

f ' ( g ( u r , zr))t.

Construct c h a r a c t e r i s t i c s

f'(g(u,,

zr))t < x < x ( t ) , then define

it i s e a s y t o p r o v p

Q(T)

c o v e r t h e whole r e g i o n and

(u, z )

is a

YING Lung and TENZhen-Huan

418 is a

solution, x = x ( t )

CJ.

For t h e g e n e r a l c a s e when N

Remark.

( u ( x , t ) , z ( x , t ) ) i s t h e s o l u t i o n o b t a i n e d b y Lemma 3 , t h e n t h r o u g h

(xo, t ) 0

any p o i n t

there is a characteristic

*= dt u ( x , t) 5 u ( x o , t 0 )

Identity

x-axis, o r a

holds on t h i s c h a r a c t e r i s t i c . if

u ( x o , t o )5 0 ,

u ( x , t ) > 0 , it i n t e r s e c t s e i t h e r t h e

CJ.

3 satisfies

If t h e s o l u t i o n by Lemma

( x , t ) , t h e n f o r any

at a point

t < t

f'(u(x, t ) ) ,

this l i n e must i n t e r s e c t t h e x - a x i s ,

Lemma 4 .

2 , it i s e a s y t o p r o v e by i n d u c t i o n .

5
0

{zn(x, t ) }

L1(-M, M)), we s t i l l d e n o t e t h e s e s u b s e q u e n c e s by

space

Let

w(x, t )

and

u ( x , t ) = $-'(w(x,

z(x, t )

has a s o l u t i o n

t)).

and

T > 0 , t h e correspond-

converge i n space

un)

and

C([O,

TI;

{zn} f o r con-

and

We change t h e v a l u e of

w(x, t )

t , w ( x , t) and

l e f t c o n t i n u o u s and t h e v a r i a t i o n o f them i s bounded, t h e n u ( x , t )

url(x, t )

t h e l i m i t f u n c t i o n s . They b e l o n g t o

on a n u l l measure s e t s u c h t h a t f o r e v e r y

tinuous too.

z,,(x)

z n ( x , t ) by Lemma 3 , and t h e e s t i m a t i o n i n Lemmas 5-10 h o l d s .

and

venience.

converges t o

and

z(x, t )

are

i s l e f t con-

z n ( x . t ) a r e a l s o l e f t c o n t i n u o u s by t h e P r o o f o f

Lemma 3.

Now we p r o v e t h a t

(u(x, t ) , z ( x , t ) )

C l e a r l y it s a t i s f i e s ( 1 2 ) and ( 1 3 ) b e c a u s e

We h a v e t o v e r i f y ( 1 5 ) . L e t

i s t h e s o l u t i o n of Problem P .

(un(x, t ) , zn(x, t ) ) are solutions.

425

Hyperbolic Model of Combustion t > 0, we take a subsequence of

For any and

zn(x, t)

converge to w

and

{(un, zn)}

again such that

almost everywhere as the functions of

{(un, z n ) } .

independent variable x, the subsequence is still denoted by also converges almost everywhere by the continuity of {(un, z

where

( u , z)

does not converge to

Suppose that

-1

is denoted by

u ( 5 , tl) 2

such that w(x, tl)

where, if un(5, t,)

XI.

But

J,(un(x, t ) ) 1

converges to

vn(5, t) > 0, z

(5,

t) = 0 by (15). Let

Therefore z(c, t) = 0

N1.

+a,

If not, then

> 0 and every

-2

{T~}.

-rn < t, such that un(x, Tn)

vn(x, t)

That is, for

-E.

We take a subsequence, still denoted by

We take

+ m.

If T > 0 , then for sufficiently large n, T

We have

for all

by Lemma

15 - x /
- 2 ~uniformly with respect to n. { $ uniformly with respect to

almost every where.

Hence

we get

for all sufficiently large

n, there exists a

accumulation point of

{ T ~ I ,such that T

vn(x, t)

holds almost everywhere on

z(x, t) = 0 by the left continuity of

Thus

If v(x, t) < 0, we prove that there is a subsequence such that < 0.

h > 0,

> 0 and

is a point where it converges, then for sufficiently large

> 0, hence

h, x].

for

such that

norm, we take a subsequence such that it converges almost every-

z ( 5 , t) = 0 only if

N1’

< t

u(x, tl) > 0 by (14). By the left continuity, there are

The set of points

If v(x, t) > 0, there exists

Nl.

J,(un(x,t))

v(x, t)

un}

TE/2C,

then we get

converges to

uni5, Tn)

L1 norm

t, hence

By the left continuity

-2~, but

is arbitrary, hence

V(X,

t) > 0, which contradicts

YINC Lung and TENZhen-Huan

426 v(x, t) < 0. If

T =

0, then we may construct a characteristic or a curve con-

CJ, which intersects the x-axis at

sisting of piecewise characteristics and

E [x

> -E E

f'(u,)Tn,

X I , and

uo(x)

2 0, so

v(x, t)

Therefore, there is a subsequence such that = z(")(x).

Because

we get

u 0(x) and the uniform convergence of

by the left continuity of

is arbitrary, hence

u(n)(~ni 2 - E . 0

En*

uo(x) u("), but 0

0, it is also a contradiction.

vn(x, t) < 0, hence

zn(x, t)

N1,

z(x, t) = lim z (x, t) = zO(x). nTherefore (15) holds for almost every x.

But

almost everywhere. We can change the value of

is arbitrary, thus (15) holds

z(x, t) on a null set

such that (15) holds everywhere.

53.

The existence of the solutions of Problem Q. First of all, let us introduce a definition and a hypothesis.

Definition. It is

u (x) assumes the value 0

said that

at point

x, if one

of the following holds: u (x 0

0 ) = a;

u (x + 0 ) = a;

u (x + 0 ) > u (x 0 0

Lemma 11. Suppose if

u(xo, t ) < 0 0 -

- o),

a € (uo(x - o ) , uo(x + 0)).

(u(x, t), z(x, t)) at a point

is the solution obtained by Theorem 1,

(xO, t ) , then the straight line, which is called 0

the characteristic,

has the following two properties:

a) u (x) assumes the value u ( x o , t ) 0

at point

x = x0

f'(u(xo, to))to

which is the intersection of this characteristic and x-axis; b)

if u(xo, to) # u(x13 t,) and

)z0, then the downward

u(xl, t 1

Hyperbolic Model of Combustion c h a r a c t e r i s t i c s through p o i n t s

421

( x ~ t, 1)

( x o , t o ) and

do n o t i n t e r s e c t on

> 0.

For any

Proof.

h > 0 , E > 0 , t h e r e i s an i n t e g e r

En E

and

[x,

h , x,],

such that

b e c a u s e i f n o t , t h e n t h e r e were

h > 0,

> 0 , such t h a t

for sufficiently large

n , wliicn c o n t r a d i c t s t h e

t h e l e f t c o n t i n u i t y of

L1 c o n v e r g e n c e of

A c c o r d i n g t o t h e above p r o p e r t y , t h e r e e x i s t s a s u b s e q u e n c e o f denoted by

{ u n l , and a s e r i e s o f p o i n t s

We c o n s t r u c t a downward c h a r a c t e r i s t i c o f

cri

(En, t ) , i f

< 0 , it i n t e r s e c t s t h e x - a x i s , t h e i n t e r s e c t i o n p o i n t i s - f'(un(Cn, tO))tO and u

(Cn, t 0 )

E [C, uo(x)

and

{unl, still

xn =

un(Cn, t o )

Cn at

xn ' i f

> 0 , we c a n c o n s t r u c t a p i e c e w i s e c h a r a c t e r i s t i c and

c u r v e as i n

assumes t h e v a l u e

u n ( C n , to)

u(")(x)

Lemma 9 which i n t e r s e c t s t h e x - a x i s , u ( " ) ( x ) point

such t h a t

+ m,

through

assumes a v a l u e a t t h e i n t e r s e c t i o n

which i s n o n n e g a t i v e and n o t g r e a t e r t h a n

f'(un(Cn, t O ) ) t O En, assumes t h e v a l u e

- f'(o)t,I.

u(xO, t )

Let

u n ( c n , t o ) ,hence x

at t h i s p o i n t because

u n i f o r m l y and t h e l e f t and r i g h t l i m i t o f

uo(x)

f'(u(x,,

{uAn'l

x to))to.

converges

e x i s t s at each p o i n t , t h u s a )

i s proved A s f o r b ) , we can t a k e a s u b s e q u e n c e

{un}

and t w o s e r i e s o f p o i n t s

l e a s t one o f them i s n e g a t i v e , t h e r e f o r e t h e c h a r a c t e r i s t i c s o r p i e c e w i s e

{En]

YINGLung and TENZhen-Hum

428 characteristic and

(Ln,

curves through points

(CA, tl)

to) and

do not

t > 0. These two families of curves converge to their limit posi-

intersect on tions as n

-+a,

which do not intersect on t > 0 either.

We make the following hypothesis on the initial values: Hypothesis A.

If u0( x ) assumes the value

[b, c] 3 x

interval

such that

u,(x)

0 at

0 on

point

x, then there is an

( b , c).

Lemma 12. F or the solution obtained by Theorem 1, if there are t > to > 0

__.

x E

(-a,

+m),

of those T

such that v(x, t)

0, z(x, t) > 0 , and

and

is the supremum

satisfying

u(x, to) = 0.

then

Proof. Because

is a supremum, there are only two possibilities:

a) there is a series T

-f

to, such that u(x, Tn)

-f

b) u(x, to) = 0. If possibility a) holds, and if

decreases montonously, u(x, Tn ) -< 0

v ( x , t) = 0. We construct a downward characteristic of u (x,

7n),

= x

which intersects the x-axis at

f'(0)t

as n

+ m.

5'.

= x

f'(u(x, T ~ ) ) ? ~ . x n

( 5 , to) to the x-axis, let the intersection point

( 5 , to) is on the left side of characteristic through (x, Tn) for

sufficiently large n, by Lemma 11, 5'

xn.

Let

the s l o p e f'(U(5, to)) =

5 5-5'>-

that is

through the point

If u(5, to) 5 0 for some 6 < x, then a characteristic

can be constructed from point be

because

we get

5' 5 xo, hence

429

Hyperbolic Model of Combustion u ( 6 , to)

The above inequality still holds if

20.

Let

-f

x, u

is left contin-

uous, hence

i.e. u(x, to)

2 0.

u(x, t ) 0

But

characteristic through point

2 xo

as n

f'(0)to.

(x, t0 )

But we have

0, we can construct a

and intersect the x-axis et point

xn by Lemma 11 for any

XI,

x n

u(x, t0 ) = 0.

therefore x' = xo, we also get

u(x, t ) = 0. 0

0, hence

increases monotonously, because u(x,t ) 0

If Tn

Therefore we get u ( x , t0 ) = 0 at any case. We know by Lemmas 11 and 12 that u o ( x ) assumes the value

0 at point

xO'

Moreover, we can prove

Lemma 13. Under the conditions of Lemma 12, if Hypothesis A holds and U(")(X) 0 >

uo(x), lim z (x, t) = z(x, t), x, = x n-

> 0, such that

point

f'(0)tO, then there is a constant h

5 E (xo, xo

0 for every

u,(6)

5 in (x, - 6, xo), such that

u,(t)

h), and there is always a

< 0 for any

6 > 0.

Proof. We prove it by contradiction. If the conclusion were false, then there

6 > 0 , such that u o ( 5 )

would be a

sufficiently small, such that T

E [to, t

6/f'(O)],

Since u(x,

Lemma 11, 5

5 xo. If

5
0

for

< 0

value

such t h a t

f o r any

at p o i n t u 0 ( 5 ) -> 0

This i s a contradiction.

We have proved t h a t t h e r e i s always a p o i n t

u,(t)

FIGURE 1

(xo - 6, xo), s u c h t h a t

By H y p o t h e s i s A a n d t h e f a c t t h a t

u,(x)

assumes t h e

xo, t h e o n l y p o s s i b i l i t y i s t h a t t h e r e i s a c o n s t a n t for every

5 E (xo, xo + h ) .

h > 0,

Hyperbolic Model of Combustion

43 1

lim zn(x, t) = ‘ z ( x , t) at the above lemma can be n* relaxed to the effect that this limit holds only for a subsequence.

Remark.

Clearly the condition

By the convergence of sequence

{zn(x, t)}, such that it converges

a subsequence from it again, still denoted by to

almost everywhere on

z(x, t)

such that any subsequence of It is obvious that

Lemma 14. x C

1. 0.

{zn(x, t)}

and

We define N1

as a set of

(x, t)

z(x, t).

does not converge to

is a null measure set.

If Hypothesis A holds and

+m)

(-m,

proved in Theorem 1, we can take

{zn(x, t)}

s > so > 0 , y

u(”)(x) > u O - 0

(-m,

(XI,

there are

s u c h that

+m)

t, to, x

(x, t)

satisfy the conditions of Lemma 12 respectively, and

t > to > 0, and

s, so, y

N1, (y, s)

N1.

Set

xo = x

Yo = y

f’(0)tO’

f’(0)so

xo # y o , if x f Y.

then

Proof.

x < y. Take a subsequence of

We may assume that

{zn}

converges to

Thus

uo(x) < 0.

z at point

(x, t).

We obtain

{(u,,

zn)}

zo(x) > 0 from

such that z(x, t) > 0.

If u (x + 0) > 0, then there is a constant 6 > 0, such that uo(()

0 for

( E (x, x + 6), hence

u(”)(() 0

If u (x + 0) < 0, by the same reason, there is a constant 0 u O ( c ) < 0 for

5 E (x, x + 61, hence uin)(()

If u (x + 0 ) = 0, then uo 0

< 0.

assumes the value

< 0, by Hypothesis A, there is also a

6 > 0 , such that

0 at point

x, but u,(x)

6 > 0, such that uin)([)

0 for

E (x. x + 6). It is known that and that is

x
((,

T).

Therefore

432

Y I N G Lung and TENZhen-Huan

( 5 , T)

through any point

fi we can always construct a downward character-

istic of u . If it intersects line

6 = x,

11
tl.

& [ 0 , t o ) ,t h e n

which c o r r e s p o n d s t o a n

tl.

Set

NX.

u(x, t )

and

N2 = N

[0, +-),

u(x, t ) , z(x, t ) z(x, t )

also s a t i s f y

Taking an a r b i t r a r y

+@), we

x E

satisfying

and

s a t i s f y i n g t h e above c o n d i t i o n .

If t o i s t h e supremum o f t h o s e

(x, t ) p l a n e .

v(x, t ) = 0.

i n t h e case of

and o b t a i n t h e s o l u t i o n

Problem P by Theorem 1. We p r o v e t h a t e q u a t i o n (16)

xO w h i c h s a t i s f y t h e c o n c l u s i o n

Hence t h e r e a r e a t most c o u n t a b l y

Denote t h e s e t o f them by

i s a n u l l measure set, o n t h e

u, z & ‘ BV.

By Lemma 1 3 , x

f i e s t h e c o n c l u s i o n of Lemma 1 3 , h e n c e

satisfying

c o r r e s p o n d s t o an xo

( x , t ) E N2.

We may

v(x,

T )

< 0,

which satis-

This i s a contradiction.

T h e r e f o r e t h e s e t of p o i n t s which s a t i s f y ( 1 9 ) i s o f m e a s u r e z e r o , which i s d e n o t e d by

Nlu N;,

%. UN3

i s a n u l l m e a s u r e s e t , we d e f i n e t h e v a l u e of

z(x, t )

Y I N CLung and TEN Zhen-Huan

434

according to (16) on this set, then

(16) is satisfied everywhere. The obtained

u(x, t), z ( x , t) is the solution of Problem Q.

References [I] Williams, F.A., Combustion Theory, Addison-Wesley, Reading, Mass., 1965. [PI

Courant, R. and Friedrichs, K.O.,

Supersonic Flow and Shock Waves (Inter-

science Publishers, Inc., New York, 1948).

[3] Majda, A , , A qualitative model for dynamic combustion, SIN4 J. Appl. Math. 41, 1 (1981)70-93.

[h]

Volpert, A . I . ,

The space

and quasilinear equations, Math. USSR Sb., 7

(19671 257-267. [5J

Dafermos, C.M.,

Characteristics in hyperbolic conservation laws, a study

of t h r structure and the asymptotic behavior of solutions, in Nonlinear Analysis and Mechanics, Vol. 1, Pitnian, London, 1977.

Lecture Notes in Num. Appl. Anal., 5 , 435-457 (1982) Nonlineur PDE in Applied Science. LI.S.-Jupun Seminar. Tokyo, 1982

Boundary Value Problems f o r Some Nonlinear Evolutional Systems of P a r t i a l D i f f e r e n t i a l Equations

Zhou Yu-lin

Department of M a t h e m a t i c s Peking U n i v e r s i t y Beijing CHTNA

I n t h e p a p e r , t h f boundary v a l u e problems f o r t h e n o n l i n e a r systems of t h e Schrodinger t y p e , t h e pseudo-parabolic t y p e and t h e p s e u d o - h y p e r b o l i c t y p e o f p a r t i a l d i f f e r e n t i a l equat i o n s a r e c o n s i d e r e d . The g e n e r a l i z e d g l o b a l s o l u t i o n s and t h e c l a s s i c a l g l o b a l s o l u t i o n s f o r t h e boundary v a l u e problems o f t h e s e n o n l i n e a r s y s t e m s a r e o b t a i n e d .

51.

Systems o f S c h r o d i n g c r Type. The n o n l i n e a r S c h r o d i n g e r e q u a t i o n s

- iu

+ fi/uIpu =

(1.1)

and t h e n o n l i n e a r S c h r z d i n g e r s y s t e m s

o f complex v a l u e d f ~ l n c t i o n s [ l - ~ ’r,e g a r d e d a s t h e s y s t e m s o f r e a l v a l u e f u n c t i o n s ( o f r e a l p a r t s and i m a g i n a r y p a r t s ) a r e c o n t a i n e d i n t h e g e n e r a l s y s t e m

ut - A ( t ) u x x =

a s s i m p l e s p e c i a l c a s e s , where functions,

A(t)

and

f(u)

(1.3)

f(ti)

are N-dimensional v e c t o r v a l u e d

i s a n o n s i n g u l a r and n o n n e g a t i v p l y d e f i n i t e m a t r i x .

problems o f t h e t h e o r e t i c a l p h y s i c s , c h e m i c a l r e a c t i o n s e t c . ,

435

In the

it i s v e r y o f t e n

ZHOUYu-Lin

436

For t h e systems of

t h a t t h e r e appear t h e e q u a t i o n s and systems o f such k i n d .

form (1.3) e f h i g h e r o r d e r , t h e p e r i o d i c boundary problems and t h e i n i t i a l v a l u e problems have been s t u d i e d i n [ 5 - 7 ] and t h e g e n e r a l i z e d g l o b a l s o l u t i o n s and t h e c l a s s i c a l global solutions a r e obtained. Now i n t h e p r e s e n t s e c t i o n , w e a r e going t o c o n s i d e r t h e f i r s t boundary

v a l u e problems

i n t h e r e c t a n g u l a r domain

= {O 6 x 6

L, 0

t h e S c h r a d i n g e r t y p e o f second o r d e r , where

u o ( x ) 6 WLl’(0,

a unique g l o b a l s o l u t i o n

i s bounded, t h e J a c o b i

t h e vector valued function

t h e r e e x i s t s a constant

E RN

A(t)

i s semibounded,

f(u)

s u c h t h a t f o r any N-dimensional v e c t o r s

IuI
0.

T a k i n g t h e s c a l a r p r o d u c t of t h e v e c t o r grating the resulting relation for

{uE(x, t ) }

and t h e s y s t e m (1.5) a n d i n t e -

in the interval

[O, I!],

we g e t

By making u s e o f t h e boundary c o n d i t i o n (1.4),t h e s e c o n d find t h e t h i r d t e r m s of t h e l e f t hand s i d e o f t h e above e q u a l i t y t a k e t h e forms

Znou Yu-Lin

438

rcsyectively.

B;y v i r t u e o f t h e semi boimdcdness o f t h e J a c o b i d e r i v a t i v e m a t r i x

t h e l a s t t e r m o f t h e above e q u a l i t y c a n be w r i t t e n a s

Then t h e above, mentioned E q u a l i t y becomes

Ry means o f Gronwall't: lemma, the f o l l o w i n g lemma h o l d s .

Lemma I.2.

Under t h e c o n d i t,i o n s of Theorem 1.I, t h e a p p r o x i m a t e s o l u t i o n s

{uE(x, t)}

o f t h e problem (1.5), ( 1 . 1 4 ) have t h e e s t i m a t i o n

wherc

f(0) = 0

is i n d e p e n d e n t o f

and d i r e c t l y d e p e n d e n t on

is a z e r o v e c t o r o r t h e s y s t e m ( 1 . 5 ) i s homogeneous,

elf(0)

1 2.

When

i s a l s o indc-

L > 0.

p e n d e n t of Iri

order t o estimate t h e derivative

(x, t ) , we make t h e s c a l a r product,

w i t h t h e s y s t p m (1.3) and i n t e g r a t e t h e r e s u l t i n g r e l a t i o n for

interval [0,

by p a r t s .

qdt. u x (

where t h e s y s t e m

Then we h a v e

. y t ) 1 ' L 2 ( 0 , P . )+

2 2 4 "xx ( . *t)II L2( 0 , ).P

< Zbll ux( . , t

( 1 . 5 ) i s assumed t o be homogeneous, i . e . ,

by v i r L u e of t h e boundary c o n d i t i o n ( l . j l ) ,

t h e n we have

)I1L 2 ( " ,

f(0) = 0 .

(1.9)

In fact,

Nonlinear Evolutional Systems

439

On the other hand,

Under the assumption

f(0) = 0, this becom?r;

From the inequality ( I . Y ) , we have the following lemma.

Lemma 1.3.

For the homogeneou, system (l.>), i.e.,

tions of Theorem 1.1,

t.!ip

approximate soluLionu

K3 is independent of

where

> 0

and

{uE(x, t )

[O,

a],

have t,he estimation

k > 0.

Differentiating the system (1.5) with respect to product of the resulting relation and

f(0) = 0, uricier the assump-

xxx'

x, making the scalar

then integrating for

in interval

we have

x = 0, L

On the lateral boundaries

(0,

of the rectangular domain

Q,,

the relations

t) = uxx(9". t,) = 0

(1.12)

follow immediately from the system (1.5). In fact, on account of the nonsingularity of the matrix = A(t) + EE

expressed as

A(t),

is bounded for

the invcrse matrix E > 0

and

1 A- (t,) of the matrix

AE(t)

0 6 t 6 T, then the system (1.5) can be

ZHOUYu-Lin

440

Thus t h e c o n d i t i o n s ( 1 . 1 2 ) are o b v i o u s l y a v a i l a b l e .

where E

> 0

and

K1, K2

a r e c o n s t a n t s dependent on

and a r e i n d e p e n d e n t o f

P. > 0.

and

Lemma 1 . 4 . -

B e s i d e s t,he c o n d i t i o n s o f Lemma 1 . 3 , assume t h a t

t i n u o u s l y d i f f e r e n t i a b l e and { u E ( x , t)}

uo(x) E. WL2)(0,

e).

f(u)

i s t w i c e con-

The a p p r o x i m a t e s o l u t i o n s

s a t i s f y the inequality

where

Also

is i n d e p p n d e n t of

> 0

and

> 0.

By means o f t h e above e s t i m a t i o n s we can c o n s t r u c t t h e g l o b a l s o l u t i o n o f problem (1.31, ( 1 . 4 ) from t h c s e t of a p p r o x i m a t e s o l u t i o n s t h e a s s u m p t i o n s o f Lemma

1.4

{ u E ( x , t ) ) and

T); L2(0,

a))

for

> C

I t c a n b e s e l e c t e d from

{u,(x,

{uE,(x,t ) ) , t h a t t h e r e e x i s t s a v e c t o r v a l u e d f u n c t i o n when

that, and

-f

t h e sequences

{uE,(x, t ) }

and

> 0.

Then

u(x, t )

and

ux(x, t )

t)},

a sequence

u(x, t ) , s u c h t h a t

{ U ~ . ~ ( t)} X , a r e uniform-

ly c o n v e r g e n t t o

{ u E x ( x , t ) ) a r e u n i f o r m l y bounded i n t h e s p a c e o f H 6 l d e r con-

tinuous f u n c t i o n s f o r

Under

i s u n i f o r m l y bounded i n t h e f u n c t i o n a l

{uE(x, t ) }

T); W2(2)(0, 1 ) r\ W h ” ( ( 0 ,

L,((O,

space

{uE(x, t ) } .

respectively i n

QT.

Hence i t i s c l e a r

{ f ( u E (x, t ) ) } u n i f o r m l y c o n v e r g e s t o f ( u ( x , t)) a n d a l s o c u E , x x ( x ’ t ) } i 1 { u E i t ( x , t)} c o n v e r g e weakly t o u x x ( x , t ) a n d u t ( x , t ) r e s p e c t i v e l y .

For any t e s t f u n c t i o n

$(x, t ) , t h e r e i s t h e i n t e g r a l r e l a t i o n

I! QT$[uEt

A(t)uEXX -

E UEXX

f ( uE ) ] d x d t

From t h e e s t i m a t i o n formular (l-lO), we know t h a t

(1.14)

Nonlinear Evolutional Systems t e n d s t o z e r o as

44 1

Therefore passing t o t h e l i m i t f o r t h e ( i . h ) , w e get

u(x, t ) s a t i s f i e s (1.3) a l m o s t e v e r y w h e r e , i . e . ,

T h i s means t h a t

g e n e r a l i z e d g l o b a l s o l u t i o n o f t h e boundary v a l u e p r o b l e m

u(x, t )

is a

( 1 . 4 ) f o r t h e degener-

a t e system (1.3) of t h e Schrodinger t y p e .

Theorem 1 . 2 .

Under t h e c o n d i t i o n s o f Lemma

1 . 4 , t h e boundary v a l u e problem ( 1 . 4 )

f o r t h e s y s t e m (1.3)o f S c h r a d i n g e r t y p e h a s a u n i q u e g l o b a l s o l u t i o n

T); WL2)(0, a ) )

u(x, t ) E L,((O,

nWp)((O.T); Lp(O, a ) ) .

S i n c e t h e e s t i m a t i o n s g i v e n i n t h e l a s t t h r e e lemmas a r e a l l i n d e p e n d e n t o f t h e width

for

L > 0

+ m,

Q,,

o f t h e r e c t a n g u l a r domain

by t a k i n g t h e l i m i t i n g p r o c e s s

we can o b t a i n the s o l u t i o n o f t h e boundary v a l u e p r o b l e m

u ( 0 , t) = 0 ,

i n t h e i n f i n i t e domain

Q; = {O \< x < -,

O < t , < T ;

Nonlinear Partial Differential Equations in Applied Science: Seminar Proceedings (Mathematics Studies) - PDF Free Download (2024)

References