NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN APPLIED SCIENCE PROCEEDINGS OF THE U.S. - JAPAN SEMINAR, TOKYO, 1982
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NORTH-HOLLAND
MATHEMATICS STUDIES
81
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 ...............................
17
Yoshikazu GIGA: The Navier-Stokes Initial Value Problem In Lp . . . . . 37 Ei-Ichi HANZAWA: Nash’s Implicit Function Theorem and the Stefan Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Tosio KATO: Quasi-linear Equations of Evolution in Nonreflexive Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
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 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 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 17 1 of Mixed Type in a Half Space ............................. Tetsuhiko MIYOSHI: Yielding and Unloading in Semidiscrete Problem of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Alan C. NEWELL: Two Dimensional Convection Patterns in Large Aspect Ratio Systems ................ ..................... 205 Hisashi OKAMOTO: Stationary Free Boundary Problems for Circular 233 Flows with or without Surface Tension ...................... 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 Magnetohydrody333 namic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hans F. WEJNBERGER: A Simple System with a Continuum of Stable Inhomogeneous Steady States .............................. 345 Masaya YAMAGUTI and Masayoshi HATA: Chaos Arising from the Discretization of O.D.E. and an Age Dependent Population Mo361 del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
X
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
xi
“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
xv
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
n
conservation
laws i n o n e s p a c e dimension,
where
to
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
f
n
i s a smooth n o n l i n e a r map from
f
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
ut
+
€ ( u ) ~=
E
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].
A
Rn
2
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
Lm
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)
is
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
uo
.
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
3
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.
+
Rn
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
u.
In particular, if
g
is
Ronald J. DIPERNA
4
a real-valued map on
Rm
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
un
(still
un) and a family of probability measures over the
range space R",
such that for all continuous
g: Rm
+
R,
The limit on the left hand side is taken in the weak-star topology of
m
L
and equality holds for almost all
notes a generic point in the range space
y
in
Rn.
Rm.
Here
X
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
u
Y
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
u
Y'
If
is
g
Lipschit z then lg(lim un)
-
lim g(un) I m
2 const. max diam spt w Y
Y
.
In the framework of conservation laws, the goal is to show that the representing measures associated with a family of exact or approxi-
5
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
v
Y
mass if
v
Y
f
is convex and that, in general,
interval where of
v
f
is affine.
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 references.
We also refer the reader to Lax [ 6 1 which contains a
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'
> 0
Ut
-
vt
-
while
u(v)x = uX
E
u
= E V
xx xx '
sgn v u " > 0.
Lm, there exists for each fixed
E
Given initial data
a globally defined solution
the amplitude of which remains uniformly bounded as the viscosity parameter
E
vanishes,
Here the constant depends only on data.
IJ
and in the
Lm
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
cu
L
stability and the entropy condition, one may extract a subsequence
6
Ronald J. DIPERNA
(uE ,uE ) which converges pointwise a.e. to a globally defined k k distributional solution of the associated hyperbolic system u
t
-
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
n
conservation
A pair of real-valued mappings on the state space rl:
Rn
-f
R;
q: Rn
Rn
R
+
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.
n
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)
n
The basic compati-
to its flux
is a
C1
g
may
solution and
7
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
u
by the spatial derivative
we find that ( 3 ) is equivalent to
Hence the condition Vn(u) Vf(u) = Vq(u),
(5)
u
Rn
€
is a necessary and sufficient for the existence of an entropy pair. We observe that (5) represents a system of
n
linear, variable
coefficient partial differential equations in two unknowns q.
If
n > 2
n
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
3
3
Taking the inner product of ( 5 ) with
r. 3
.
immediately yields the
characteristic form of ( 4 ) : (A.
3
On
-
Vq)
-r
j
= 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
8
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
rl
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 .
u
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
BV
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
3 BV
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
L
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
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
Lm
by v i r t u e o f t h e BV
o n e c a n demon-
9
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
u
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 .
X
T e c h n i c a l l y w e assume t h a t
i s monotone i n
j
t h e corresponding eigendirection: r . 3
(7)
- BX. # 3
0
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
un
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
M
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
Lw
to
1 Lloc.
The s o u r c e
Ronald .I.DIPERNA
10
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
+
ut
(8)
D
where f o r s i m p l i c i t y
f
(
~ = )F ~Du
is a c o n s t a n t
xx' n
x
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
E
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
D
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
RL
f
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
2
n.
If
uE.
D
n
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
u
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
SE
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
E
~
t o an
Lm
to
1
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
E
provided t h a t t h e d i f f u s i o n
Conservation Laws and Weak Topology
11
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
p.
At a more fundamental level, it re-
mains an open problem to prove uniform circumstances.
Lm
estimates in general
For example in the case of hyperbolic systems (l),
it remains an open problem to prove that
for admissible solutions in
Lm
BV
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
12
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
R
when restricted to sequences of functions
un(y) : Rm
-+
Rn
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
R3
wn: R3
-f
R3
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
wn
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
13
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
z
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
z*
denotes the orthogonal complement of
z
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
un
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
if
rl
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
14
The continuity of the inner product yields a commutativity relation for the representing measure
For all entropy pairs
(nj,qj),
we have
j = 1,2
where
u.
11
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
15
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
where I = ( O , n ) , and a 1
+
f(u,v)
in (t,x)
E (O,+m) x
I,
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
1
-v c1
xx
+
f(u,v) = 0,
+
g(u,v) = 0,
in I,
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
I8
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,,
ci
C 0). His limit system is
The first author has developed a new numerical algorithm
called the shadow system.
to detect and trace all bifurcating branches using a group theoreticmethod [ Z Fujii, Mimura and Nishiura
1.
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
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 =
G
=
(6,i)
> 0 of
(SP). See,
Fig. 0.1. (ii ) 0 is a stable solution of the kinetic system of (P). matrix at
u,
B
= {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
19
sign
=
[
bll
b12
b21
bz2]=[:
I]
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
v
E
A
..
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
<=
uo(v)
When i t i s s o l v e d w i t h r e s p e c t t o u , i t h a s t h r e e b r a n c h e s
J(v) =
(A.4)
5
G+(v) = g ( h + ( v ) , v )
E
C1(A).
Then,
n*.
d(J(v))/dv < 0 a t v =
dG+(v)/dv < 0 f o r a l l v
E
A.
-1
The re a r e a number o f examples w i t h i n t h e s e t t i n g ( A . 1 ) - ( A . 4 ) .
See, [ l ] .
The Mu.ay-Mimrumode2 f o r d i f f u s i v e p r e y and p r e d a t o r sy st em i s an ex amp l e, i n which
where
f ( u ) = (35
+
L
16u - u ) / 9 ,
and
g,(v)
=
1 + (2/5)v
T h i s model h a s been u s e d f o r n u m e r i c a l t r a c i n g o f b i f u r c a t i n g b r a n c h e s i n [ l ] .
Fig. 0.1.
51. -
View n e a r d o u b l e b i f u r c a t i o n p o i n t s The c o n s t a n t s t a t e U = 0 h a s an i n f i n i t e number o f h y p e r b o l i c c u r v e s
1 , 2 , . . . ) on t h e (d , d ) - p l a n e , d e f i n e d by 1 2
r
(n =
Hiroshi F U J I and I Yasumasa N I S H I U R A
20
(1.1)
r
( n = 1 , 2 , . . .) i s t h e s e t o f (d
d ) 1' 2
E
2
lR+ where t h e c o n s t a n t s t a t e
u.
e i g e n v a l u e s , and hence c o r r e s p o n d t o p r i m a r y b i f u r c a t i o n p o i n t s o f Each
and [ l ] .
r n,m
=
r nn rm
(n
r ,m
From each p o i n t s o f
= 1,2,
...,
n # m) a r e d o u b l e b i f u r c a t i o n p o i n t s o f
r,',
'
n'
D
n
=
rn/
See, F i g . l . 1 .
X i s D -symmetric we mean t h a t i t s even e x t e n s i o n
d e r e d a s a p e r i o d i c f u n c t i o n d e f i n e d on t h e c i r c l e [-.,n], group D
r;
Let
[3
1
... Namely,
0, where
the
,,$nl'n,m.
t h e r e a p p e a r s a b i f u r c a t i n g s h e e t o f s o l u t i o n s of ( S P ) ,
which c o n s i s t s o f f u n c t i o n s w i t h D -symmetry ( [ l ] ) . E
See,
has i n t e r s e c t i o n s with t h e o t h e r r m ' s , m # n , m = 1 , 2 ,
l i n e a r i z e d o p e r a t o r o f (SP) h a s two d i m e n s i o n a l k e r n e l .
function U
has zero
8 to
Here, by a [-~1,71],
consi-
i s i n v a r i a n t under t h e
i s t h e d i h e d r a l group which sends a r e g u l a r n-polygon o n t o i t s e l f .
s Fig. 1.1. The l o c a l s t r u c t u r e n e a r t h e d o u b l e b i f u r c a t i o n p o i n t s i n t o s e v e r a l t y p e s a c c o r d i n g t o t h e symmetry groups D
r
may be c l a s s i f i e d n,m and D o f t h e k e r n e l a t Tn,,,.
Using t h e group r e p r e s e n t a t i o n o f t h e Lie group Dm, a number o f new b i f u r c a t i o n diagrams have been o b t a i n e d i n [ l ] . An example i s i l l u s t r a t e d i n F i g . l . 2 , which i s t h e c a s e f o r t h e May-Mimura model ( 0 . 2 ) . The d o u b l e s i n g u l a r i t y a t bifurcation equations a t (1.211
r
1,2
r
may d e s e r v e a f u r t h e r i n v e s t i g a t i o n . S i n c e t h e 1,2 t a k e t h e form (due t o a g r o u p t h e o r e t i c argument)
2 2 p , I - ( a o - a o ) + p11p2 + p30p1 + p12p2 + ( h i g h e r o r d e r t e r m s ) = 0 , 1 1 2 2
21
Global Bifurcation Diagram -(blul- b 2 0 2 ) 0 2
U.2I2 where
G =
-
d- -
rl,2,
are a l l constants;
+
2 q20p1
+
2 qZ1p1p2
+
3 qo3p2 + ( h i g h e r o r d e r t e r m s ) = 0,
t h e l o c a l b i f u r c a t i o n p a r a m e t e r s , and a i , b . , p . . and q . . I s 1 1 9 1 1,J
p_
= (p1,p2)
E
Rf = t h e k e r n e l s p a c e a t r l , 2 , i s t h e a m p l i t u d e
vector of t h e bifurcation equations.
The b i f u r c a t i o n d i ag r am n e a r
F i g . l . 3 f o r t h e c a s e p11q20 > 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
r
I'
(p11420
' O)-
22
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
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
ll
I
< 6
0
(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 ) .
Z
1
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
1
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
''
23
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]),
1.
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
2n
-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 .
0.
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
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
24
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 ] .
nues t o mist as dl
i 0.
t h e n-mode branches
b emanating from (2'n' D). Fig.2.1.
es
En
in
fl)
ii) The one-mode bifurcating branch b, emanating from Id;,
Theorem 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
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 . ,
o f ( S P i f o r smaZl .a
holds as w e l l f o r the bifurcating branches Dn's In = 1 , 2 , . . )
Namely, every bifurcating branch h i t s the sing u l a r wall dl = 0 f o r small
> 0.
3
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
J.
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
X
(where v =
a reduced shadow s o l u t i o n , where Xo
rl
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,
2
HN(I).
x
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
E
I.
Let I e ( q ) = t x
E
I
I u(n;x)
=
5 a. ( n ) ) ,
9. = 0 ,
+l.
Then, (RSS)2
25
Global Bifurcation Diagram reduces t o +I
e (Q) I
where 11
= measure o f I
Thus, f o r any
E
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.
=
(n
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)
1
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
I
=
0,
{ 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
7-
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*) =
1.
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.
26
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
/
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 +
x
Xo
( 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)
1
- v xx ‘y v
=
+
0,
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
21
consists of a one-parameter family of singular solutions). Let S
denote the set of singular solutions. We associate t o So the followFor U = (u,v)
ing asymptotic norm.
/ / u / (=~ lim /I ESO
where (u(E;,x),v(E;,x))
E
S0 ’ the asymptotic norm l l U / l s is defined by
(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)
a
(x)
E
1 C (I), 0
<
= Y
< rt,
A, firstly; solve f(u,v) and = h
+1
(v) for v
>
= 0 to
rt(Fig.4.2).
h(n;v) into (RP)2 leads to a scalor equation for v: v
xx
+
where G(n;v) = g(h(n;v),v).
Eq. (4.1) with vx
ri E
G(ri;v) = 0 ,
x
E
I,
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
E
A.
Let Un’a(x) 1
=
h(n;V:s
(x)).
According to
Hiroshi FUJIIand Yasumasa NISHILIRA
28
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:"
E
I.
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),
E
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
E-]IQf)
x0,
: PI(a)
0 < a < z;).
,
,
I
I
& ' c+llll') u
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)
f
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
or
n
I
(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 wh 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 an sw 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 .
29
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
,V:*'
See, Fig.4.4.
t h a t the depths
(c-l -
J
t~h e s l i t
b
5 :.
There a r e t h r e e such branches, coro 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
:
b
(4.3)
.
e x i s t s : a t t h e l e f t ( #,,),
t h e b o t h ends ( 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
=
0,
f
andJ
f ( s , v M ) ds
=
0.
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
30
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
lo
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 ) .
31
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 ) , $)
E
2 R+
x
X
(§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
on
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
32
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.
13
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
it
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.
A
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
3
> 0.
We note that such a global picture is supported by ilucrical computations in
33
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
r
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
J
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
34
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,
5,
-
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
35
[ 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
Lp
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
D
(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
=
f
x
(O,T),
x
(O,T),
div u = 0
in
D
u
on
S x (O,T),
in
D
= 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
37
.
Here
S. We
u = (u
1
,-*.,
38
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
0
such that W (D) = L (D). onto
X2.
Let
Using P, we can trans-
x2
Here the operator A = A
2
=
-PA is called t h e Stokes
with the domain
D(A) = l u
F o r simplicity we assume
E
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
E
X2
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
E
D(A1'4),
39
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 <
Fujita. When
p
a)
=
is regular. To do this, we develop
which extends the corresponding
n, o u r main result reads that a
e x i s t e n c e o f a unique strong s o l u t i o n . is just a result of [ 8 ] .
L
P
L2 theory of Kato and Ln(D) n X2 impZies t h e
E
With a particular choice of p
2 this
=
On the other hand Serrin I121 raised a question how to
show the existence of strong solutions in n
4. Our
>
L
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’
-A
The operator
P
X
P
and
A
P
are
I, -analogues of
P
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
p
=
ilfll
C , where
O(A‘)
f in
X
P
.
i s t h e complex i n t e r p o l a t i o n space
P
0 _<- a _<- 1.
2 , we easily get the above properties by using the abstract
functional analysis, because A2 Remark.
denotes t h e norm o f
is a strictly positive self-adjoint operator.
Solonnikov 1131 has proved the first part of Theorem 1 when
n
=
3,
although his method is different from o u r s . Remark.
For the Laplace operator, the corresponding results are known by
Fujiwara 121 and Seeley [ l l ] . However, we cannot apply their results to o u r case since P
does not commute with
A.
40
Yoshikazu CIGA To prove the theorems above we construct the resolvent of the Stokes operator,
using pseudodifferential operators; see [3].
Since we use technique of pseudo-
differential operators instead of integral kernels, our argument is more clear than the classical potential-theoretic discussion. Here our symbol class of pseudodifferential operators differs from that of Fujiwara [2] and Seeley [ll] because the Stokes system is elliptic not in the sense of Petrowsky but in the sense of Agmon, Douglis and Nirenberg. Using
L estimate for pseudodifferential P operators, we can prove Theorems 1 and 2; for the detail, see [3,4].
Theorem 2 is useful in estimating the nonlinear term Fu in (11) Lemma 1 ([S]).
Let
1
1 1
-.
0 5- 6 < 2 + n(1 - -) P 2
6
llAp P(u,grad)v// with constant
M = M(6,8,p,p)
I n particular, i f
i f
p = n
6+8+p
We have
e 5 MIIApUII llA;vlI n + -, 1 - 2p 2 0
> 0, p > 0, p+6 >
-.21
the above estimate i s v a l i d f o r
1 6 = -4,
We now consider (11) in
X
P
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]).
y
Fix
n/2p - 1/2 -5
such t h a t
(i)
[o,T)
to
D(A~),
i s continuous from
(0,T)
to
D(Aa)
IIAau(t)ll
=
1. Assume t h a t
a
E
> 0,
u i s continuous from
(ii) u
Moreover, u
T
For some
<
o(tY-")
i s smooth i n
D
us
x
extended t o a global s o l u t i o n .
t
(0,T).
-+
P
0 for some
and a, y
<
CL
< 1.
If IIAyaII i s sma7.2, then u can be
P
D(Ay).
u of (11) with t h e foZlowing
Then there e x i s t s a unique locaZ strong s o l u t i o n properties.
y
P
Navier-Stokes Initial Value Problem in Lr
41
One can easily see that Theorem 3 includes the results of Kato and Fujita [8] as a particular case p When
Remark.
data is a
E
the norm of
=
p = n,y
.
X a
2 , n = 2 , 3.
can be taken to be zero and the assumption on initial
Kohn [15] pointed out this assumption is reasonable because in
Xn
has zero dimension in "dimensional calculus".
Even when the zero-boundary condition in ( I ) is replaced by some first
Remark.
order boundary condition, we can prove similar results; see [6].
We next discuss the analyticity of the solution u (i)
Theorem 4 ( [ 7 ] ) . (0,T)
Let
u
be as i n Theorem 3.
u(t)
is anaZytic in
2
w i t h vaZue in Wp(D).
(ii) Suppose that S is anaZytic at xo.
at
Then
of (11)
(xo,t), t
6
Then, u(x,t)
is anaZytic in
(x,t)
(0,T).
The first part implies the time-analyticity of the solution, while the second part implies the spatial analyticity up to the boundary S. In [ 7 ] we extend the results of Masuda [lo]. analyticity in
He discussed the time-
spaces and the interior spatial analyticity.
L2
I n the following sections we give heuristic arguments to prove the foregoing
Theorems. We omit proof unless it is very short and understandable even to non-specialists. For the technical detail, see author's papers [3-71.
1.
The r e s o l v e n t of t h e Stokes o p e r a t o r .
We investigate the way of 1-dependence of the resolvent
(1
+
AP)
-1
.
this we construct the resolvent. We begin with transforming the equation (1 + A ) u = f
P
in X
P
into the following Stokes equations
To do
42
Yoshikazu CICA (A - A)u + grad p = f
div u
where p
0 in
D,
u = 0 on
S,
is some scalar function. Since
we denote u 1'.
by u
= G
A
in D,
=
f determines u
f. Our plan to construct
for A
E
E \ (-m,o),
is
GA
Reduce the problem to the Dirichlet problem with tangentiaZ boundary data.
2'.
Solve an integral equation on the boundary for large A.
To do Step 1' we recall the hydrodynamic potential. Set
where SiJ
is Kronecker's delta and
151
2
=
C12
+
+
2 Sn. The hydrodynamic
potential of f is
where F
is the Fourier transformation with respect to x. The definition of
KAf implies that u ' = KAf satisfies the equations (A
- A)ul
+
grad p ' = f in Rn, div u ' = 0 in Rn,
where p'
is some scalar function on Rn. Using K A f, we reduce the problem
to the Dirichlet problem. More explicitly, w
(A
- A)w
+
grad p" = 0 div w w
= 0 =
=
KA f
-
GAf satisfies
in
D,
in
D,
KAf on
S,
43
Navier-Stokes Initial Value Problem in L" where
i s some s c a l a r f u n c t i o n on
p"
D.
We now reduce t h e problem t o t h e D i r i c h l e t problem with t a n g e n t i a l boundary data.
Let
z = NQ
satisfy
Az = 0
in
az _ av -
D,
cp
on
z(x)dx = 0 ,
S,
JD
where
d e n o t e s t h e u n i t i n t e r i o r normal v e c t o r t o
v
be t h e s t a n d a r d i n n e r product i n iRn.
v = w
-
at
S
x
E
S.
Let
<
,
>
D e f i n i t i o n shows t h a t
grad N,
q
=
p" + A N w , yKhf>
satisfy
(A - A)v
where
yw
denotes t h e t r a c e o f
w
+
grad q = 0
in
D,
div v = 0
in
D,
v = g
on
S,
= 0
on
S,
on
S.
We c a l l t h i s problem t h e DirichZet
probZem w i t h tangentiai! boundary data and denote that the projection
P
v = PK f - G f h
A
and
G f = PK f A A
Since
Remark.
P
and
Kh
by
v = VAg.
I f we n o t i c e
i s d e f i n e d by
Pf = f - grad N < v , y f > we see t h a t
v
g = yPKhf.
-
V M f A h
with
for
div f = 0,
We t h u s have
Mhf = yPKAf.
a r e w r i t t e n e x p l i c i t l y , a l l we have t o do i s t o c o n s t r u c t
We have reduced t h e o r i g i n a l problem t o t h e D i r i c h l e t problem with
t a n g e n t i a l boundary d a t a n o t with g e n e r a l boundary d a t a .
This i s because t h e
e f f e c t o f t h e normal component d i f f e r s from t h a t o f t a n g e n t i a l component.
V
A'
44
Yoshikazu CIGA We now give a rough outline of Step 2'.
to construct V h .
By Step 1' our problem is reduced
be a pseudodifferential operator of order one on S
Let Y h
We consider the single layer potential
KX(Gs
@
YAh).
We see
Whh = PK (6
A S 0 Yhh)
satisfies the Dirichlet problem with tangential boundary data YWhh; we denote yW h by h
Shh.
Our problem is now to solve the integral equation g
=
Shh
f o r given tangential boundary data g.
has the inverse for large
In [ 3 ] we construct Y h
We thus have V h = WAS;'.
A.
such that
S
This yields
Ghf = PKhf - WhSilMhf. We thus have constructed the resolvent. To construct
YA
the author introduced a symbol class Smik (see below) of
pseudodifferential operators.
Definition ( [ 3 ] ) . Let m and
k
set of all p h
6 (I:
E
C"mn x R")
a,@ and positive numbers
i s valid with constant
and
Remark.
(A E,W
be real numbers. Then we denote by \ (--,O])
Sm;k the
such that for all multi-indices
the estimate
M = M(a,@,~,w); here
<X;O
denotes
(/A\ +
1Ci2
+
= .
Grubb [16] introduces a similar symbol class. In o u r situation her
symbol class reads that the estimate above is replaced by
I f we use this symbol class, o u r formulation will be more clear than that of [3,41.
1) 112
45
Navier-Stokes Initial Value Problem in Lp
2.
A nalyticity o f the semigroup
e-tA and domains o f .'A
In section 1 we obtain an explicit form of G A , i . e . ,
(A + A)-'f
Since the spectrum of A
= G f = PKAf - W S-IM f. A h
A
is contained in
(-m,O),
A
to prove Theorem 1 it suffices
to prove the estimates
with constant C = C ( E )
for large
A.
To show Step 1 is easy, so we give a
proof. Proof of S t e p 1.
I t is known that
P is a bounded operator in
(Lp(D))n,
s o it
suffices to prove
Since the symbol of K A satisfies the estimate
this follows from Mihlin's Remark.
I n principle
LP-boundedness theorem.
LP-boundedness of operator follows from estimates for its
symbol. Classical results are the Calderch-Zygmund inequality and Mihlinls theorem for convolution type operators. In recent years LP-boundedness is proved for more general class of operators, namely, pseudodifferential operators. We often use LP-boundedness theorem to construct and estimate the resolvent; see [ 3 , 4 ] .
Yoshikazu CICA
46
We see Step 2 follows from the following three estimates:
where
Ihl
denotes the norm of h
easy to prove.
in
(Lp(S))n.
The first two estimates are
The last one follows from the construction of Y ; for the detail
x
see [ 3 ] .
By Theorem 1 we can define fractional power
(Re z < 0), using the
A'
P
Dunford integral. To prove Theorem 2 it suffices to prove
for all
z
such that
-a < Re z < 0, where
is the operator norm of A'
1lA;ll
P'
We prove this estimate, using the Dunford integral A
= 2:1
S,
(-1)'
GAdh.
The expression of Gx enable us to estimate the right side. The estimate
is easy to prove l i k e the proof of Step 1.
is not easy.
While
1
We have to study VAMh = WAS; M A
more carefully.
In [ 4 ] we
decompose VXMA into main term and error term to get the estimate.
Remark.
Theorem 2 is very similar to the case o f the Laplace operator
B
P
47
Navier-StokesInitial Value Problem in Lp with zero boundary condition. Fractional powers of both operators are closely related.
Indeed, in [4] we show D(A")
P
3.
=
D(Bn) n X
P
P'
0 < u < 1
Solutions t o the evolution equation (11).
We now consider the nonlinear equation (11) and discuss only the existence In Theorem 3 the reader may have a canonical question.
results.
What is
T ?
Are there any estimate f o r
T from below ?
To answer this question we restrict ourselves in a simple case, for example, p = n
and give a rough outline of proof of Theorem 3; for the detail see [ 5 ] .
We begin with estimates for nonlinear term
Fu
in (11).
To avoid technical
difficulties we only give a proof to the last part of Lemma 1. That is
in a word this estimate follows from Sobolev's inequality and Theorem 2 .
Proof. 2
Let H '(D) P
be the space of Bessel potentials, that is, 2
HzU(D) = [L (D), Wp(D)I,.
P
P
Theorem 2 now implies that the canonical injection D(Ai) Sobolev's inequality now implies that the injections
are continuous. The second inclusion ( b ) yields
c
H2"(D)
P
is continuous.
Yoshikazu CICA
48
11
C 11 P ( u , grad) v 11 2n/3.
P (u,grad)v/l
Applying Hb'lder's inequality, we have
='
/lP(u,grad)vl12n/j since P
to L P
is continuous from L
P
follow from (a) and D(A;l2)
CIlul12n IIgrad
.
Vlln
9
The estimates
Combine the above to get the result.
c W;(D).
To solve (11) we consider its integral form
We use Theorem 1 and Lemma 1 to prove existence theorem f o r the integral equation (111)
in Xn. We construct approximate solutions by the iteration scheme
=
e-tA a,
t) =
uo(t)
We will estimate IIA"u,(t)
Fum(s) ds, m _>- 0.
+
11
, where
11 fll
denotes the norm of f in Xn.
Theorem 1 implies that
This yields the estimate /IACluo(t)ll with
2
KaO t-a, a 1 0
49
Navier-Stokes Initial Value Problem in L'
We consider the following problem:
by
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
of
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,
5
Kam t-'
p
= 1/2
for all
a _>- 0.
implies
We thus have
with
Ka,m+l where
2
KaO
+
Ca+&M B(1-6-a,6) K
@m
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'
K
pm
u,(t)
Xn) n C ( ( O , T ] ,
K
pm'
u,(t) D(Aa))
satisfies
1 and note the definition of K am to get
is well-defined for all
a,
50
Yoshikazu CICA
where C
is a constant depending only on A.
An elementary calculation shows
that if (C)
k
0
< -
1
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
2
Ka,
We thus have
/IAa~m+l(t)ll
2
Ka t-a,
0 _< a
<
3/4, 0 - t 5- T.
Using again Theorem 1 and Lemma 1, we can prove
Since 2CK u(t)
<
1, this implies um(t)
converges a solution u(t)
of (111) and
is eventually a unique strong solution of (11); see [1,5,8]. We consider the meaning of (C).
In (C)
ko depends on
T and a, so (C)
is a condition for initial data and the length of time interval where the
solution u(t)
of (111) exists. There are at least two types of sufficient
conditions for (C). (i)
Conceptually speaking, these are
T is fixed and a is taken
so
that ]la11 is sufficiently small.
(ii) a is fixed and T is sufficiently small. Let u s explain (i). B
=
Suppose IlalI is small, say I(a(/ <
1/2, 1/4. Then clearly ko
solution u(t)
<
1/2C for uZZ
T. This implies that the
of (111) exists for all time if llall 11/2CaC
Namely, there exists a gZobu2 solution of (111) if We next explain (ii).
First, we prove that
1/2CaC for
((a((
for a = 1/2, 1/4.
is smuZZ enough.
Navier-Stokes Initial Value Problem in Lp
ta//Aae-tAa / /+ 0 (t
If a
Cm
E
0,a
=
(C;(D))”
+
0)
n X 2 , we see
b
tallA’e-tA Suppose now a in Cm
090
X
090
such that a
m
2 taIIA e -tA (a C(
is independent of m
( a > 0).
is in Xn. This yields +
0
(t
in Xn as m +
2- Calla - a,//
Since C a
A’a
Xn
E
+
0).
is dense in Xn, we can take a sequence
tends to a
ta(/Aae-tAall
=
a / /= ta//e-tAb / /
Since Cm
n*
for all a
51
- am)//
+
+
m.
Using
{am}
{am}, we have
tallAae-tA am/\
tal/Aae-tAam//.
and since the result is valid for am
E
CE,a,
this estimate now implies tal\Aae-tAa / /+ 0
(t
+
0).
This result implies that we can take T sufficiently small so that TallAae-tA all 5- 1/2C (a = 1/2, 1/4) a
E
X
there is a solution on
f o r fixed
a. I n other words for every
[O,T] for small T. Note that this T
heavily depends on a.
4.
Analyticity of the solution of ( 1 1 ) . We can prove time-analyticity of the solution u of (1x1) as the proof of
Theorem 3 .
Here we discuss spatial analyticity f o r
u
which is analytic in time.
To prove (ii) in Theorem 4 we consider the elliptic system of n+2 variables as in Masuda [lo], where v
(O,T], u
of
is the vorticity, i.e., v = curl u .
Let us derive the elliptic system. For simplicity we assume n = 3 . is analytic in
(u,v)
Since u
is holomorphic in some complex neighborhood U of
52
Yoshikazu CIGA
(O,T].
This implies u ( x , -r+ia) is harmonic on U
c
= R2
for fixed x.
That is, 2 2 z+z=O a u a u in G = D x U . ar
Since div u
=
0, we have
Au - curl v = 0 in G. Sum up both sides to get A u - curl v = 0 G
where
AG
i s the Laplacian on
in G ,
G.
We derive the equation for the vorticity v.
For simplicity we assume
f = 0. Apply curl both sides of (I) to get
Since v
is holomorphic in U, we have 2
2
a-r
ao
a v 2) a v = 0. -(2+
Summing up both sides, we get AGv -
av
The foregoing argument shows that
=
curl(u,grad)u
av AGv - -a-r= g
G
G.
satisfies
(v,u)
A u - curl
in
in G,
v
= 0
u = O
in G, on S x U ,
v - curl u = 0 on
S x U.
53
Navier-Stokes Initial Value Problem in Lr where g = curl(u,grad)u.
If the right hsnd side is regarded as a given data, this system is a linear elliptic system with complementary boundary conditions on
S
x
U; see [ 7 ] .
Moreover, this itself is a nonlinear elliptic boundary value problem for
(v,u).
Apply regularity theorems for elliptic system to get spatial analyticity of u .
Acknowledgments I am grateful to Professor Robert Kohn for useful discussions during his stay in Japan.
I am also grateful to Professor Gerd Grubb for discussions about
symbol class of pseudodifferential operators.
Footnotes 1.
Current Address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, N.Y. 10012, U.S.A.
2. Author partially supported by THE SAKKOKAI FOUNDATION.
References [l]
H. Fujita and T. Kato, On the Navier-Stokes i n i t i a Z value problem I, Arch. Rational Mech. Anal. 16 (1964), 269-315.
[2]
D. Fujiwara, On the asymptotic behaviour of the Green operators f o r e Z l i p t i c boundary probZems and t h e pure imaginary powers of some second order operators, J. Math. S O C . Japan 21 (1969), 481-521.
[3] Y . Giga, A n a l y t i c i t y of the semigroup generated by t h e Stokes operator i n Lr
spaces, Math. Z. 178 (1981), 297-329.
54
Yoshikazu CICA Y. Giga, Domains o f fractioiml powers o f t h e Stokes operator i n
Lr spaces,
Arch. Rational Mech. Anal. (to appear).
Y. Giga and T. Miyakawa, SoZutions i n Lr t o t h e Nauier-Stokes i n i t i a l value problem, ibid. Y . Giga, The nonstationary Navier-Stokes system with some f i r s t order
boundary condition, Proc. Japan Acad., 58 A (1982), 101-104. Y. Giga, Time and s p a t i a l a n a l y t i c i t y of solutions o f the Navier-Stokes
equations, Comm. in Partial Differential Equations. (to appear).
T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova 32 (1962), 243-260. 0. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Revised English ed. New York-London: Gordon and Breach 1969. K. Masuda, On t h e a n a l y t i c i t y and t h e unique continuation theorem f o r
solutions of the Nauier-Stokes equation, Proc. Japan Acad. 4 3 (1967), 827-832. R. Seeley, Norms and domains of t h e complex powers
A,:
Amer. J. Math. 93
(1971), 299-309. 3 . Serrin, The i n i t i a l value problem f o r t h e Navier-Stokes equations,
Nonlinear problems, R.E. Langer ed. University of Wisconsin Press, Madison, 1963, 69-98.
V.
A . Solonnikov, Estimates f o r solutions o f nonstationary Navier-Stokes
equations, J. Sov. Math. 8 (1977), 467-529. R. Temam, Navier-Stokes Equations, Amsterdam-New York-Oxford, North Holland 1977. R. V. Kohn, Partial r e g u l a r i t y f o r t h e Navier-Stokes equations, in this volume. [16] G. Grubb, ProbZDmes a m Zimites pseudo-diffErentiels dgpendedant d ' u n
paramgtra, C. R. Acad. Sci. Paris 292 (19811, 581-583.
Lecture Notes i n N u m . Appl. Anal., 5 , 55-60 (1982) Nonlineur PDE in Applied Science. U.S.-JLlpan Semincir, To!iyo. 1982
iJash's Implicit Function Theorem and. The Stefan Problem
Ei-Lchi Hanzawa Department of Mathematics Faculty of Science Hokkaido University SAPPORO 060, JAPAN
W e sketch out a proof of t h e l o c a l existence of the
c l a s s i c a l solutions f o r the mltidimensional Stefan problem and i t s relevance t o Nash's implicit function theorem.
!the Stefan problem i s a mthematical model of t h e melting of a body of i c e ,
where we suppose that a body of i c e melts, at each point of the surface, with velocity i n proportion t o t h e n o w 1 gradient of t h e thermal d i s t r i b u t i o n i n W e show the
water and that t h e thermal d i s t r i b u t i o n satsfies t h e heat equation.
l o c a l existence of t h e c l a s s i c a l s o l u t i o s for t h e i n i t i a l value problem of t h e Stefan problem, by using Nash's i m p l i c i t function theorem, which enables u s t o reduce the s o l v a b i l i t y of a nonlinear problem t o that of t h e l i n e a r i z e d problem even i f a loss of r e g u l a r i t y of t h e solution f o r given data occurs. Here we suppose that t h e initial surface
C"
hypersurface i n Rn, the e x t e r i o r of
i n t e r i o r of
To
the domin Ro
of a body of i c e i s a closed
To i s the i c e part, a heater i s i n the m
whose surface J i s a l s o a closed bounded by
r0
and J
C
hypersurface i n Rn
i s t h e Hater part.
and
(Except the
conditions of regularity, these assumptions, e.g., th2t t h e water p a r t i s i n the i c e p a r t , are not e s s e n t i a l . )
The locus of a surface of a body of ice, which i s
t h e free boundary t o be determined, i s denoted by T
@,T
= {(x,t)
E
R"
X
[O, TI;
56
Ei-Ichi H A N Z A W A
0(x, t) = 0}, where t is the time variable. We denote the free domin with
water by RO,T.
See Figure 1.
Figure 1.
0
?T
t
Cur unknowns are the definlng function
0
and the t h e m 1 distribution u
in water in QQ,T" Our equations and result are as follows.
muat ions,
(1) QltZO = a0,
~
(3) u = b on J
(4)
u
=
0 on
l
x
=
a, ~ where = ~ {x; 0 0(x) = 01 = r 0'
[0, TI.
rO,T.
51
Nash's Theorem and Stefan Problem
(5) at@
n - kCiZl(ax d(ax i
@) = 0
on rQYT, where k i s a positive (from a
i
physical reason) constant.
Theorem.
Suppose t h a t
a and b
m
are nonnegative C
s a t i s f y the compatibility conditions up t o
m
(which are necessary conditions of the existence of a
( 5 ) ) . Then, for sufficiently small T > m
sense) e x i s t a C
ro
order on m
C
b e t i o n s and and J
at
solution f o r
t = 0
(1)-
0, there uniquely ( i n the e s s e n t i a l
function 0 and a Cm
function u on fl
@YT
which satsfy
(1)-(5).
1. For the one-dimensional Stefan problem, it i s well known t h a t
Remk
the unique global c l a s s i c a l solutions a r e obtained (see Rubinstein
[51). For t h e
mltidimensional problem, the unique global weak solutions a r e obtained by Kamenomostskaja [2].
m
The C -ness of the solutions i n Theorem i s r e w k e d by M.
R e m k 2. Tanigawa [6].
The author's original theorem i s t h a t for solutions with f i n i t e
d i f f e r e n t i a b i l i t y of any order. technical.
The reason f o r giving this limitation was purely
That i s , the author dld not know whether there a r e smoothing
operators up t o
O3
order on a scale of Banach spaces which is used i n [I].
'They
a r e constracted by Tanigawa.
Remk
3. For the general existence theorem of t h e c l a s s i c a l solutions f o r
the multidimensional Stefan problem, it seems that Nash's implicit f'unction theorem i s necessary, i . e . , we encounter an e s s e n t i a l loss of regularity. occurs because t o solve the single first order equation (5) f o r
@
on
It OYT
does not cover the l o s s of regularity of t h e normal derivative of t h e t h e m 1 distribution u, that is, the former gains the regularity only along the
Ei-lchi H A N Z A W A
58
characteristic curves although t,he l a t t e r loses the regularity i n every direction.
See Figure
The situation of t h i s phenomenon is clearly
2.
recognized when we linearize the problem (1)-(5). Note that t h i s d i f f i c u l t y does not occur i n the one-dimensional Stefan problem, because the f r e e boundary
i s one-dimensional so that it i s covered by the characteristic curve.
Figure
2: The reason why the loss of regularity occurs.
the f r e e boundary
I
r
@YT
the characteristic curves
t h e directions of loss of regularity of t h e normal derivative of
(at -
Remark 4.
on
ro
When a body of i c e melts rapidly, e .g.
A)U =
u with
o
, when I grad
at t = 0, we do not need Nash's implicit f'unction theorem.
Kinderlehrer and Nirenberg [3] and M e i m o v
[41.
a1
2
E > 0
See
G . Komtsu suggests t o t h e
author that the essential reason why we can get around the d i f f i c u l t y of the l o s s
of regularity i n this case i s i n the f a c t that a heat potential cover losses of regularity i n the time direction when t h e melting i s rapid.
Remark 5.
The assumption that t h e i n i t i a l data
a and b a r e nonnegative
(which i s natural in physics) enables us t o solve t h e linearized problem of (1)-
(5). We resolve the linearized problem i n t o a parabolic mlxed problem and an i n i t i a l value problem i n
R
@
,T
for a first order operator which has t h e form
59
Nash's Theorem and Stefan Problem
3,
-
k f2(avu)av on Yo,T,, where v
{x; O(x, t)
= 01 in
is an outward unit normal to the surface
Rn and u is the t h e m 1 distribution at which we
liearize the problem. We can solve the latter problem if the' characteristic curves starting f r o m Ro at t = 0 cover the domain RO,T. satisfied because u = a on
r
2
This requirement is
0 on Qo at t = O , u = b -> O on Jx[O,T],u=O
and we have the mximum principle for the heat equation. See Figure
@ ,T
3. This fact is the core of the present work.
Figure 3: The reason why we can solve the linearized Stefan problem.
!he characteristic curves
of the operato r
\
J
l
References.
[l] E. Hanzawa, Classical solutions of the Stefan problem, T6hoku Math. J.
33 (1981), 297-335. [2] S. L. Kamenomostskaja, On Stefan's problem (in Russian), Nau&
Dokl.
Vysg. skoly 1 (1958), No.1, 60-62.
[3] D. Kinderlehrer and L. Nirenberg, The smoothness of the free boundary in the one phase Stefan problem, Corn. Pure Appl. Math. 31 (1978), 257-282.
[4] A. M. Meymnov, On the classical solution of the multidimensional Stefan problem for quasilinear parabolic equations, Math. USSR Sbornik
Ei-Ichi HANZAWA
60
40 (1981), 157-178.
[51 L. I. Rubinstein, The Stefan problem, Translations of Mathematical Monographs, Vol. 27, h e r . Math. Soc., Providence, R . I . ,
[61 M. Tanigawa, The elsewhere.
1972.
m
C -ness of solutions of t h e Stefan problem, t o appear
Lecture Notes in Num. Appl. Anal., 5, 61-76 (1982) Norilirieur PDE in Applied Science. U . S . - J q m r i Semirzur, Tokyo, 1982
Quasi-linear equations o f e v o l u t i o n
i n n o n r e f l e x i v e Banach spaces
T o s i o Kato
1
Department of Mathematics University o f C a l i f o r n i a , Berkeley
An e x i s t e n c e - u n i q u e n e s s t h e o r e m and a r e g u l a r i t y t h e o r e m a r e g i v e n f o r t h e Cauchy p r o b l e m f o r 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 i n n o n r e f l e x i v e Banach s p a c e s . As an a p p l i c a t i o n , C1-solutions a r e c o n s t r u c t e d 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 d i f f e r e n t i a l e q u a t i o n s i n t h e "Schauder c a n o n i c a l form" ( w h i c h i n c l u d e g e n e r i c e q u a t i o n s i n two i n d e p e n d e n t variables. )
1. I n t r o d u c t i o n
I n p r e v i o u s p u b l i c a t i o n s [7,9,10], we c o n s i d e r e d t h e Cauchy p r o b l e m f o r abstract quasi-linear
e q u a t i o n s o f t h e form
I n t h i s t h e o r y ( a t l e a s t ) two Banach s p a c e s tion
u(t)
is in
Y
and
du(t)/dt
is in
X, Y X.
a r e used, such t h a t t h e solu-
The o b j e c t o f t h e p r e s e n t p a p e r
i s t o g i v e a n o t h e r v e r s i o n of t h e t h e o r y which i s more f l e x i b l e a n d c o n v e n i e n t i n some a p p l i c a t i o n s . The main d i f f e r e n c e s o f t h e new v e r s i o n from t h e p r e v i o u s o n e s a r e ( a ) e l i m i n a t i o n o f t h e r e f l e x i v i t y a s s u m p t i o n s on t h e b a s i c Banach s p a c e s
(b) e l i m i n a t i o n of t h e isomorphism t h e domain of t h e map
o f t h e growth r a t e .
w w A ( w )
S between t o a l l of
Y
Y
and
X, Y,
X, and ( c ) e x t e n s i o n o f
w i t h o u t i m p o s i n g any r e s t r i c t i o n
I n t h e s i m p l e form p r e s e n t e d h e r e , t h e new r e s u l t w i l l n o t
c o m p l e t e l y c o v e r t h e p r e v i o u s o n e s , b u t it w i l l b e a p p l i c a b l e t o p r o b l e m s t h a t were n o t a c c e s s i b l e t o t h e l a t t e r .
62
Toshio KATO I n p a r t i c u l a r , we h a v e i n mind a p p l i c a t i o n s t o f i r s t - o r d e r h y p e r b o l i c s y s t e m s
(HS)
a.(u)du/dx. = f ( u ) , J J
du/dt + j=1
i n which t h e unknown a.(z) 3
u = {u ( t , x ) , ...,%( t , x ) } m a t r i c e s d e p e n d i n g on
i s a s i m i l a r N-vector f u n c t i o n .
manner.
that
q(z)
a.(z) J
d e p e n d i n g on
u E C ( [O,T]
u(0) = B E C1(Rm;RN). 0(x)
---t
0
as
'1 m
N
;C ( R ;R ) )
f o r some
f(z)
are s i m u l t a n e o u s l x
zE R
N .
in the C
1
-
T > 0, given a n i n i t i a l
[Here and i n what f o l l o w s
1x1 3 w , and s i m i l a r l y f o r
B E C(Rm;RN) implies
'1 m. N C (R ,R 1 . 1
Our old t h e o r y i s n o t a p p l i c a b l e t o (HS) s i m p l y b e c a u s e ive.
and
Our a b s t r a c t r e s u l t s w i l l b e a p p l i c a b l e t o s o l v e t h e Cauchy problem f o r
(HS) i n t h e c l a s s
value
1 i n t h e C -manner,
z € RN
I t i s assumed t h a t t h e
d i a g o n a l i z u b y a common r e a l m a t r i x
x E Rm,
i s a r e a l N-vector f u n c t i o n , t h e
1
N x N
are real
0,
t
'C
i s not reflex-
I t may b e remarked t h a t what i s r e a l l y r e q u i r e d i n t h e o l d t h e o r y i s n o t
n e c e s s a r i l y t h e r e f l e x i v i t y of X : Yo = Y
Y
but r a t h e r t h a t
i n t h e n o t a t i o n i n t r o d u c e d below.
f i e d by t h e p a i r
be l o c a l l y closed i n
Even t h i s c o n d i t i o n i s n o t s a t i s -
'I , X = C .
Y = C
1 m = 1, c l a s s i c a l C - t h e o r i e s f o r (HS) were
I t w i l l be r e c a l l e d t h a t f o r g i v e n by D o u g l i s
Y
[41 and Hartman and W i n t n e r [6]. ( F o r e a r l i e r works i n t h i s
d i r e c t i o n , s e e S c h a u d e r [14] and F r i e d r i c h s
[5].)
A generalization t o t h e case
m > 1 was g i v e n by C i n q u i n i C i b r a r i o [ 3 ] . A l l t h e s e p r o o f s make e s s e n t i a l use of t h e moduli o f c o n t i n u i t y o f t h e f i r s t d e r i v a t i v e s o f t h e f u n c t i o n s i n v o l v e d , i n d i c a t i n g t h a t t h e r e c a n b e no c h e a p way t o c o n s t r u c t a C l - t h e o r y f o r ( H S ) .
In
f a c t o u r p r o o f depends on t h e u s e of s e m i g r o u p s a c t i n g on s p a c e s o f f u n c t i o n s w i t h f i x e d moduli o f c o n t i n u i t y ( s e e s e c t i o n 5 ) . I n view o f t h i s example, it a p p e a r s t h a t t h e f o l l o w i n g i s a n a t u r a l and i n e v i t a b l e procedure.
I n t h e a b s t r a c t t h e o r y , we s h a l l be c o n t e n t w i t h c o n s t r u c -
t i n g o n l y a weak s o l u t i o n
u(t)
to
(Q)
mined by t h e i n i t i a l v a l u e
u(0) = 0
is t h e l o c a l c l o s u r e of
in
Y
If we want t o show t h a t
X. u(t)
E
which, however, w i l l b e u n i q u e l y d e t e r -
Yo and which w i l l s t a y i n
(For t h e d e f i n i t i o n of
Yo, where
Yo
Yo, s e e b e l o w . )
i s i n f a c t a " s t r o n g s o l u t i o n " which s t a y s i n
63
Quasi-linear Equations of Evolution A , .
Y
provided
(X,Y)
B f Y , we s h a l l c o n s t r u c t a weak s o l u t i o n i n a n o t h e r p a i r ,.
B E Y o c Y.
of s p a c e s s u c h t h a t
I n view o f t h e u n i q u e n e s s , t h i s w i l l show t h a t
u(t) E Y.
[O,T] of t h e s o l u t i o n i s n o t
To e n s u r e t h a t t h e i n t e r v a l of e x i s t e n c e , . A
d i m i n i s h e d when we go o v e r t o t h e p a i r
, we
(X,Y)
would n e e d a r e g u l a r i t y t h e o r e m A
t o t h e e f f e c t t h a t a weak s o l u t i o n i n
(X,Y)
with
u ( 0 ) E Yo
i s a weak s o l u t i o n
*,.
on a l l o f i t s i n t e r v a l o f e x i s t e n c e .
(X,Y)
in
X =
I n t h e a p p l i c a t i o n t o ( H S ) , we c h o o s e =
cp+"
. l+p+O .
A
Y = C
and
a l l functio,is i n
C
Here
.
p
w i l l h e d e t e r m i n e d by t h e i n i t i a l f u n c t i o n
in
cp , c1
t h e s e t of
a. , q
,I
and
f.
0E
c1
with
and t h e c o n t i -
(see section 5 f o r details.)
The e x i s t e n c e t h e o r e m
We s t a r t from a p a i r norms
1
IX
I I y , with
Y C X
X
and t w o norms
I lxw
o f r e a l Ranach s p a c e s , w i t h t h e a s s o c i a t e d
t h e i n c l u s i o n c o n t i n u o u s and d e n s e .
\de assume t h a t . f o r e a c h in
b'
is the s e t of a l l v E
p , and
n u i t y p r o p e r t i e s of t h e f u n c t i o n s
2.
i s t h e closure o f
bp+O
a s above and t h e n
w i t h modulus o f c o n t i n u i t y dominated by a c o n s t a n t m u l t i p l e
of a f i x e d modulus f u n c t i o n 'p+O d.v E C 3
6, Y = b l
w E
,
Y , a vector
I lyw
, pquivalent t o
a r e given with t h e following p r o p e r t i e s . then
(N2)
If
Y, a
f(w) E
w, v E Y
1
Ix
l i n e a r operator
,
with
1
Iy lwly
~ ( w )
, respectively,
5
r, /vIy
r,
X1 ,..., B, ,... ,
Here
( a ) The d i s t a n c e
a C --?migroup
{e-tA;
0
space
w i t h t h e norm
or
\Wly
1
on
X
b e t w e e n two e q u i v a l e n t norms i s means t h a t
/e-tk/x
such t h a t
A e B(Y,X), and
IXw
R+ , a n d
5
-A
generates
eat. We d e n o t e by
Xw
.
lAIX = IA(X,X
(b) I n (N1) - - - (f2), t h e argument
i s t h e Banach
r
in t h e
hl(r), . . . , p 4 ( r ) a r e ( f o r s i m p l i c i t y a n d g e n e r a l i t y ) assumed to b e
parameters
r =
t > 01
t h e o p e r a t o r - n o r m of
X
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(
to
(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
.
a
(We w r i t e
vb
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
/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
{u }
[O,T] i f t h e
un
a r e bounded i n
lwlx)
(Of
C([O,T];Y
, thereby
approximate solu-
n Lip,([O,T];X)
and
(2.1) Here
d u n / d t + A(un)un
u
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
u
u
in
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
u
[O,T]
closed subinterval
E
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
.I
E Lip( [O,Tl;X)
DEFINITION 2 . 4
el (B
and
X
Y
Let
(I))
REMARK '2.5
By(')
Y
he t h e b a l l i n
X
i t s closure i n
.
Yo
As is easily seen,
r > 0
d e f i n e d as t h e infimum of
x
includ<.ng t h e norm.
Y
In general
1 Y = C [O,l] ; then
clX(By(r))
E
If
.
clX(By(r))
.
X
in
i s a Banach space w i t h t h e norm such t h a t
1x1
Yo Obviously we have
Yo = Y ,
i s r e f l e x i v e one has
Y
Yo.
need n o t he dense i n
Yo = L i p [ O , l ] , and
Y
r > 0,
and r a d i u s
0
t h e union of
Yo
We denote by
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
65
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
u
(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)
on
[O,T]
s u b s e t of C ( [O,T];X)
H .
The map
O
H u
i s bounded on a bounded
B ( [O,T];Yo), and i s continuous from t h e X-topology t o
to
Yo
u(0) =
with
to
w i t h i n a bounded s e t of
bounded f u n c t i o n s on
to
I
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
w
t o all
I--$
of
A(w)
Y C X.
w
E
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
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 ,
X
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
may be expected from t h e t y p i c a l example (HS) i n which
Yo = L i p
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 .
If
Y
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
u
in
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
u
u
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)
66
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
Y
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"
B
Y
Al(r), and
u
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
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
( b ) Let
(3.2) where
/@Iy
0 b u t n o t n e c e s s a r i l y on
o n l y on
E
T
and
For e a c h
L
v(t) E B
(R'),
ys
v E C([O,T];Y)
/v(t)-v(s)/,
v E E, l e t
v
depended
5
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)}
T
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
7
X
v ( b y which v ).
It
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
67
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
-
4
u :U v ( . ,010E E
L
if
and
T
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
v.
T
and
u E O?
V
Moreover, t h e map
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
of
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
u
Sin
function approximation
to
Q
L ([O,T];X)-norm.
to
(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
E
u
such t h a t
m
denotes t h e
{un}
~ =+ OCn~
u
Then
l[
2-n,
where
I/
w i l l b e t h e n e x t e l e m e n t , and
i s shown t o b e a n u l l s e q u e n c e .
fun}
lim u
F i n a l l y we show t h a t
= u
[O,T] w i t h
weak s o l u t i o n t o (Q) on
exists i n
u(0) = 0
Lm( [ O , T ) ; X ) ,
.
so t h a t
is a
u
The p r o o f i s b a s e d on t h e
f o l l o w i n g lemma.
LEMMA 3 . 2 .
If
{un}
X, t h e n
exists in
(9)on
i s a n u l l sequence t o
u = lim u
exists i n
[O,T]
C([O,T];X)
l i m un(0)
and i f
(so that
u
i s a weak
solution).
0 E Yo, we c h o o s e a s e q u e n c e
( d ) I n t h e g e n e r a l c a s e i n which
10.-Olx-+ 0.
\ k 7 j l y = /OlpO and
that
J
u.(O) = 0. , which e x i s t s on an i n t e r v a l J
J
u
Let
J
b e a weak s o l u t i o n t o
j
[O,T]
0. G
independent of
j
Y
such
(4)w i t h
by t h e
previous r e s u l t .
u i s c o n s t r u c t e d a s t h e l i m i t i n La( [O,T];X) o f . a n u l l s e q u e n c e j -1 Thus we c a n f i n d a s e q u e n c e {u. 1 = i v . 1 s u c h t h a t I)uj-vj/lX 5 j
Each
{u.
Jn
and
1.
J 'nj
IIQvj/lX
f o r (Q) on
2
[O,T] such t h a t
it follows t h a t with
Qw = dw/dt
j - l , where
lim v
j
=
+ A(w)w
v . ( O ) = fl.+ J J
J
.
Thus
fl i n
X
{v.) J
.
i s a n u l l sequence
A c c o r d i n g t o Lemma 3 . 2 ,
u e x i s t s and d e f i n e s a weak s o l u t i o n t o (Q) on
[O,T]
u ( 0 ) = 0'. ( e ) The u n i q u e n e s s o f t h e weak s o l u t i o n i s a l s o a d i r e c t c o n s e q u e n c e o f
Lemma 3.2.
68
Toshio KATO
4.
The r e g u l a r i t y t h e o r e m I n g e n e r a l d i f f e r e n t c h o i c e s of t h e s p a c e s
c o n d i t i o n s (Nl)
.
Y
a r c possible f o r a given
(Q). I n o t h e r w o r d s , t h e r e a r e many s y s t e m s
equation
f
X,
---
(X,Y,A,f)
satisfying
(X,Y) b u t w i t h t h e same A
( f 2 ) with d i f f e r e n t p a i r s
[To e x p l a i n t h e l a s t e x p r e s s i o n , we s a y t h a t t h e two s y s t e m s
(X',Y',A',f') = f'(w) E
have t h e same
YnY'
A
f if
and
and
(X,Y,A,f)
A(w)y = A ' ( w ) y E X A X '
and
and f!w)
w, y c Y A Y ' . ]
whenever
P.
(X,Y,A,f!
For s i m p l i c i t y , s u p p o s e t h a t . we have two s y s t e m s
A
(X,Y,A,f)
and
A
Y°C
Y C Y
Then w e have o b v i o u s l y
with t h e i n c l u s i o n s continuous. A
.
Yo
,
X C X
such t h a t
d € Yo
I f we choose t h e i n i t i a l v a l u e
,
Theorem I gives: a weak soluA
tion
u
E
B( [ O , T ] ; Y o )
to
n
e B( [ O , T ] ; Y o )
(Q) and a n o t h e r s o l u t i o n
, both
A
satisfying t h e i n i t i a l condition
u(0) = u(0) = d !
,
r e s u l t , we may assume t h a t
T < T
w h e t h e r o r n o t we c a n t a k e
T = T
and
I n view o f t h e u n i q u e n e s s A
A
u = u
[O,T]
on
.
The q u e s t i o n a r i s e s
.
More g e n e r a l l y , one may a s k w h e t h e r e v e r y weak s o l u t i o n i n t h e s y s t e m L . , .
A
(X,Y,A,f)
with
u ( 0 ) E Yo
i s a u t o m a t i c a l l y a weak s o l u t i o n i n
(X,Y,A,f)
T h i s i s t h e p r o b l e m of r e g u l a r i t y f o r
t h e same i n t e r v a l of e x i s t e n c e .
To answer t h i s q u e s t i o n , w e would n e e d f u r t h e r a s s u m p t i o n s .
(Q)
with
.
To f o r m u l a t e
s u c h a s s u m p t i o n s , we f i n d i t c o n v e n i e n t t o i n t r o d u c e t h e n o t i o n s of
m-
compression and c o m p r e s s i b l e s y s t e m s . A
DEFINITION 4.1. Given two Banach s p a c e s
X c X
with t h e inclusion continuous,
A
we may i n t r o d u c e i n
I lc,;
in
X
,
X
new e q u i v a l e n t norms.
d e p e n d i n g on a p a r a m e t e r
A f a m i l y of e q u i v a l e n t norms
> 0, w i l l h e c a l l e d a compressible
E
*
norm i n
X ( r e l a t i v e t o t h e X-norm) if h
(4.1)
l i m s u p 1x1
E*X
E-90
EXAMPLE 4 . 2 .
Set
A
< (xix
f o r each
-
x E X
.
l x I E 9 i = \ x I X V ~1x1; . I n t h i s c a s e we h a v e
f o r s u f f i c i e n t l y small
E
( c i e v n s i n g on
= /XIX .
x ). A
C o n s i d e r now t h e two s y s t e m s
(XIE
(X,Y,A,f)
and
A
(X,Y,A,f)
for
(Q) m e n t i o n e d
69
Quasi-linear Equations of Evolution h
X C X
above, w i t h
Y C Y
and
1
various spaces:
.
Suppose t h a t w e i n t r o d u c e c o m p r e s s i b l e norms i n
r e l a t i v e t o X -norm, and
1
1
r e l a t i v e t o X-norm,
r e l a t i v e t o Y-norm,
W C Y cY.
r e l a t i v e t o Y -norm, where
1
Assume t h a t w i t h t h e s e new norms, c o n d i t i o n (N1) --- ( f 2 ) r e m a i n s a t i s f i e d w i t h a
X1,&’. . . ,u4,€
s e t of parameters
DEFINITION 4 . 3 .
we s a y t h a t t h e s y s t e m
h
. s t a y bounded as
E
-+
i s compressible t o t h e system
(X,Y,A,f)
f
--+ X
f o r some monotone i n c r e a s i n g f u n c t i o n
5
hl,E(r)
if
0
(X,Y,A,f).
X1,o(r) f o r r > 0 ,
.I
190
D e f i n i t i o n 4.3 i s a d m i t t e d l y r a t h e r i m p l i c i t a n d c o m p l i c a t e d .
REMARK 4 . 4
0,
h
i s bounded as
1 ,E
E
,u4,€
If t h e s e p a r a m e t e r s A
[The p a r a m e t e r
d e p e n d i n g on
But i t
i s n o t u n r e a s o n a b l e , as i s s e e n from t h e example (HS) d i s c u s s e d i n t h e n e x t
A s a m a t t e r o f f a c t , c o m p r e s s i b i l i t y h o l d s i n most w e l l - p o s e d s y s t e m s
section.
as soon as
(X,Y,A,f)
i s a r e a s o n a b l y good s y s t e m , a l t h o u g h t h e d e f i n i t i o n would
r e q u i r e a g e n e r a l i z a t i o n t o systems
X, Y
Y = H2(R)
i n v o l v i n g t h r e e Banach s p a c e s
Among s i m p l e s y s t e m s i n which t w o Banach
i f wider a p p l i c a t i o n s a r e desired.
spaces
(X,Y,Z,A,f)
X = H-l(R),
s u f f i c e , we may m e n t i o n t h e KdV e q u a t i o n , f o r which = HS-l(R),
w i l l g i v e a ”good” s y s t e m , and
Hs+2(R)
=
with
s > 0
For r e l e v a n t r e s u l t s s e e [11,12]( t h o u g h
w i l l give a compressible system.
c o m p r e s s i b i l i t y was n o t f o r m a l l y i n t r o d u c e d t h e r e ) . h , .
THEOREI 11.
(regularity)
compressible t o
(X,Y,A,f).
Suppose t h a t t h e s y s t e m
u
If
for (0)i s
(X,Y,A,f)
is a weak s o l u t i o n
t b
($1
on
,
then
[O,T] i n the
A
system
(so t h a t
(X,Y,A,f)
u(t) E
Yo ) a n d A
s o l u t i o n t o (Q) on
[O,T]
i n t h e system
if
u ( 0 ) E Yo
u
i s a weak A
h
(X,Y,A,f)
(so t h a t
u ( t ) 5 Yo).
n
Proof ( s k e t c h ) . A
system
Since
u ( 0 ) E Yo, t h e r e i s by Theorem I a weak s o l u t i o n i n t h e
h
(X,Y,A,f)
h
on an i n t e r v a l
[@,TI w i t h t h e i n i t i a l v a l u e
u(0).
By t h e
A
uniqueness r e s u l t , t h i s s o l u t i o n coincides with
u
on
[@,TI. If t h e a s s e r t i o n
l u ( t ) ( , must blow up a t some t = s < T . I f we t a k e s’ < s YO s u f f i c i e n t l y c l o s e t o s , and choose t h e c o m p r e s s i b l e norms w i t h s u f f i c i e n t l y
were n o t t r u e ,
70
Toshio KATO
,
.
< lu(s')( + 1 2 K Since t h e E,YO YO p a r a m e t e r s s t a y bounded w i t h c o m p r e s s i o n , t h e weak s o l u t i o n i n t h e s y s t e m ( X , Y ) small
E
/u(s')l
we may a c h i e v e t h a t
A
A
w i l l e x i s t on a n i n t e r v a l
[s',s'+Tl]
have a c o n t r a d i c t i o n by c h o o s i n g
RE.WRK 4 . 5 .
1
.
sl.
Thus w e
s.
But we do h a v e s u f f i c i e n t c o n t r o l o v e r
Yo
h
[O,T] t o show t h a t
t h e i n t e r v a l of e x i s t e n c e
independent of
i s u s e d i n t h e proof, we h a v e no s i m p l e e s t i -
(u(t)/, A
> 0
sufficiently close t o
s'
S i n c e norm-compression
mate f o r t h e growth r a t e o f
T
with
A
T = T.
5. A p p l i c a t i o n t o (HS)
I t i s assumed t h a t t h e
(5.1)
a.(z) = q(z) Ll
where t h e
0 J
a.(z)
in (HS).
f = 0
F o r s i m p l i c i t y we assume a. J
are s i m u l t a n e o u s l y d i a g o l i z a b l e :
-1 0 a.(z)q(z)
(j = 1 ,..., m ; z E R
J
a r e r e a l d i a g o n a l m a t r i c e s and
q(z)
N
1,
i s a r e a l nonsingular
matrix, such t h a t 0 a.
(5.2)
J
,
1
m
2
q E C ( R ;R
We may assume, i f n e c e s s a r y , t h a t
N
).
d e t q ( z ) = 1.
( a ) Let
x
(5.3)
i s t h e s e t o f a l l vector-valued
X as
1x1 + m.
d.u(x)
J
+
0
Y as
The norms
where
u = (u,,
1
j
IX
and
...,u,).
m.
c'1(Rm ;RN ) .
continuous functions
i s t h e s e t of a l l
1x1
Y =
= C* ( R ~ ; RN ) ,
u
E
C1
such t h a t
u
such t h a t
u(x)
( d . = d/dx. ) . J J
1 Iy
a r e g i v e n by
In g e n e r a l we a g r e e t o u s e t h e norm
and d l
u(x)
+0
Quasi-linear Equations of Evolution
(5.5)
/zI = lzll
V . . . V lzNl
for
w h i l e we u s e as u s u a l t h e e u c l i d e a n norm
For e a c h
(5.6)
W EY ,
t h e norms
1 Iyw
for x
1x1 and
,..., 2,)
I lYw
E
RN ,
E Rm.
a r e g i v e n by
,
= /s(w)uI,
/Ulx
z = (z,
71
W
With t h e s e norms, c o n d i t i o n s (N1) t o
r = /wIx
(N4)
are easily verified.
Here we may t a k e
lwly , e t c .
instead of
A(w)
Next we d e f i n e t h e o p e r a t o r
f o r m a l l y by
where
(5.8)
To b e p r e c i s e , t h e s e o p e r a t o r s must b e d e f i n e d by c a r e f u l l y s p e c i f y i n g t h e i r domains.
I n any c a s e it i s o b v i o u s t h a t
0
A (w)
o p e r a t o r a c t i n g s e p a r a t e l y on e a c h component.
is a f i r s t - o r d e r d i f f e r e n t i a l Using t h e well-known
results for
t h e f i r s t - o r d e r o p e r a t o r s i n one unknown, i t i s p o s s i b l e , w i t h some e f f o r t s , t o d e t e r m i n e t h e s e domains a n d v ~ r i f yc o n d i t i o n s (Al) t o ( A 3 ) . Thus we a r e a b l e t o a p p l y Theorem I t o c o n s t r u c t a u n i q u e weak s o l u t i o n
for (HS) i n t h e s y s t e m
(X,Y,A).
behavior at i n f i n i t y ) ,
u(t)
be i n
C1,
e v e n when
€7
Since
Yo = Lip(Rm;RN) ( w i t h a p p r o p r i a t e
i s Lipschitzian f o r each
= u(0) E C
1
.
C e s a r i [l] a n d C i n q u i n i C i b r a r i o [2].
u
t
b u t as y e t unknown t o
T h i s c o r r e s p o n d s t o t h e r e s u l t s p r o v e d by
Toshio KATO
72
( b ) To p r o v e t h e s h a r p e r r e s u l t t h a t A
new s y s t e m
If a!
J
and
h
A
(X,Y,A)
0 C- Y = 'C q
are
with spaces
,
the
d.B J
,
Y
u(t)
0
if
E
Y,
we i n t r o d u c e a
.
Y C X
r e l a t e d t o t h e moduli o f c o n t i n u i t y .
have u n i f o r m modulus o f c o n t i n u i t y .
C1, t h e i r f i r s t d e r i v a t i v e s have u n i f o r m moduli of c o n t i n u i t y R
on any compact s u b s e t o f
N
,
Yo , we may assume t h a t t h e
a!
f i r s t d e r i v a t i v e s w i t h a common modulus of c o n t i n u i t y on a l l o f these functions i f necessary f o r l a r g e
IzI
C
X = C(Rm;R
A
For e a c h
'1 m
N
)
p
such t h a t t h e
N
w E Y , we now i n t r o d u c e e q u i v a l e n t norms:
A
X, Y
iwl;
.
p
w i t h t h e norm
I t c a n b e shown, v i t h some c o m p u t a t i o n s , t h a t c o n d i t i o n s (Nl) f i e d with
0 %aj
d.O, J
w i t h t h e norm
Y C Y = C (R ;R )
S i m i l a r l y we d e f i n e t h e s p a c e
have
q
RN , m o d i f y i n g
have moduli o f c o n t i n u i t y d o m i n a t e d by c o n s t a n t m u l t i p l e s o f
Then we d e f i n e t h e s p a c e
t o be
and
J
.
Thus w e a r e a b l e t o f i n d a "modulus f u n c t i o n " dkq
u to
S i n c e we have a l r e a d y f o u n d a weak s o l u t i o n
(HS) s t a y i n g i n a bounded s e t i n
and
Since t h e
r e p l a c e d by
rather than
--- (N4) a r e
satis-
h
X, Y , respectively.
Here a g a i n ,
r
may b e t a k e n
lwly , e t c . A
F i n a l l y , c o n d i t i o n s (Al) t o ( A 3 ) c a n b e v e r i f i e d f o r t h e s y s t e m making u s e o f t h e e x p l i c i t f o r m u l a s known f o r t h e o p e r a t o r
0
A (w).
A
(X,Y,A)
by
Before doing
s o , however, we have t o make a small c o r r e c t i o n t o t h e p r e v i o u s d e f i n i t i o n s by replacing t h e space ing t h e space
u^ .
X
w i t h i t s subspace spanned by A
(We d e n o t e t h e s e s p a c e s by
C', .
X = CP+',
and a c c o r d i n g l y modify-
Y,. = C.l + p + 0 . )
,
73
Quasi-linear Equations of Evolution -A(w)
This i s necessary because otherwise t h e operator
would n o t b e d e n s e l y
A
d e f i n e d and t h e r e f o r e n o t g e n e r a t e a C -semigroup on 0
h
X
.
Y
or
0
Note t h a t
h
may b e assumed t o b e i n t h e m o d i f i e d
, by weakening
Y
A
Theorem I can now b e a p p l i e d t o t h e s y s t e m
p
s l i g h t l y i f necessary.
h
(X,Y,A), w i t h t h e r e s u l t t h a t
h
u ( t ) E Yo
at l e a s t f o r a s h o r t t i m e ,
Using t h e s p e c i a l p r o p e r t i e s o f t h e s p a c e
Y , t h e n , it i s n o t d i f f i c u l t t o show t h a t
u
.
C([@,T];Y)
E
u
Thus
i s a st,rong
A
s o l u t i o n i n t h e system
(X,Y,A)
A
To p r o v e t h i s , w e a p p l y Theorem I1 by showing
h
(X,Y,A)
t h a t t h e system
.
'T = 'T
A c t u a l l y we c a n t a k e
[O,T].
on
A
(X,Y,A).
i s compressible t o
I n t h e p r e s e n t problem,
however, c o m p r e s s i b i l i t y i s a l m o s t a b u i l t - i n p r o p e r t y . t h a t d i s t i n g u i s h e s a modulus f u n c t i o n
If we c h o o s e
A1,
choosing
.
u
A
A
...,p3
11.1~ .
Iul; =
any f i n i t e number o f f u n c t i o n s ters
from i t s c o n s t a n t m u l t i p l e
p
Thus i t i s n o t s u r p r i s i n g t h a t t h e parame-
AL,
c a n h e made e q u a l t o
...,p3
f o r any
u ( 0 ) = LfG Y = b l ( R m ; R N ) ,
the
fixed).
t i n u o u s from
Y
to
common t o a l l Since
B. c Y , J
u. and J
u
0:---$ 0 i n
such t h a t a l l
0. J
j = l,2,.
.
d e p e n d i n g o n l y on
..,
such t h a t
L?
and
p
bl,
0. J
u
d
+ E'
--j
u
are in
.l+p+O
A
Y = C
u
the
in
if necessary, t h a t t h e
a p p l i e d t o t h e system
J
a r e bounded i n
Y.
in
in
T
Let
can h e t a k e n
C([O,T];Y).
'a
h
J
q
and
(X,Y,A) ( w h e r e X = c"+')
are in
we may C1+p+o
shows t h a t A
C([O,T];X).
Sinre the h
6.
i s con-
w i t h t h e norms hounded.
h
J
u
i t can h e shown t h a t t h e r e i s a modulus f u n c t i o n
A
u.+
10ly ( f o r
0-
, where
u . ( O ) = flj J
We have t o show t h a t
Y =
a l s o assume, by weakening Then Theorem I
C([O,T];Y)
U E
C([O,TJ;Y).
. C~ ;[@,T];Y) be t h e s t r o n g s o l u t i o n w i t h
J
p
T > 0
with
(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
u
by
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 .
E
( 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
J
p.
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
a.
-1
h
(X,Y,A)
i n t h e system
E
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
E
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
u. a r e bounded i n J
Y ) , it f o l l o w s t h a t
u. + u J
B([O,T];Y) in
(because
C( [@,T];Y).
74
Toshio KATO
6.
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
u
(6.1)
t
+uu
= 0 ,
x
t >- O .
x E R ,
choose X = X = H (R),
(6.2)
A
.
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.
h
A
(X,Y,A)
i s compressible t o
(X,Y,A).
A
parameters we have t o c o n s i d e r a r e A
B2,E
A
l(A(w)+A)ulE,;
, which
= cr
so t h a t
'
Y
_> (A -
c[wlE,~)luls,;
i s independent o f
E
.
A
, and
A3,E
p
A
.
B
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
h
(X,Y,A)
is
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.
AW
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)
Ix
[
A
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)
by
2 = luly
B2,€
+
2
E
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.
2
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
be
I ( A ( ~ ) u , u ) ~ , 2y /c l w x x l ( l u l
5 A
This gives
62,E(r)=
CE
-1 r
2
7 lux/-
+
-1 CE
2,
blows up as
+
E
2
Iuxx/2)
.
IWIE,;l~lE,y
, which
75
E
-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.
( 4 ) 1 ( 1 9 7 4 ) , 311-358.
and C i n q u i n i , S . , E q u a z i o n i a d e r i v a t e p a r z i a l i d i
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1964).
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[4]
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5 ( 1 9 5 2 ) , 119-154.
[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.
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Hartman, ?. and W i n t n e r , A . , i o n s , h e r . J . Math.
[7]
Hughes, T . 3 . R . ,
74
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.
I
( 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
2
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'
n
77
Hideo KAWARADA and Takao HANADA
$14
and
+
0
R
(1x1 = x + x + - ) ,
Here n i s t h e outw,ird n o r m a l on
r
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.
on
r
when
E
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.
2.
Theorems
2.1
We p u t
are r e f o r m u l a t e d as follows:
Then ( 1 . 1 ) - ( 1 . 5 ) Find QEC K
such t h a t
=
1
fvdx
,
'v C K.
OO
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
E:
CH-'(O0).
Putting
79
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
E
( n = l , 2,
...)
such t h a t
1
weakly i n H (0 ) .
(2.4)
0
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 .
1
(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
+
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 ) ,
0
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 $
in
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.
E
I f 5 > la\, t h e n
0
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
in
$o0 s a t i s f i e s ( 2 . 1 2 ) .
H
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
k
Suppose f & H
(no) ( k 1 5 ) a n d l e t
E
81
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
m
(a0 ) ( m L O )
and l e t
E
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
r.
b e s m a l l enough.
82
Hideo KAWARADAand Takao HANADA Preliminaries
3.
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.
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
3.1 (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
0
(iii)
no
(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)
in
$ = f
2+
( E ~ - ~
w i t h f z 0 and t h e boundary c o n d i t i o n
6)
Ir
= 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
in
ill
r.
83
Asymptotic Behaviors of the Solutions
$Ian
(3.9)
W e d e n o t e T,:
= 0
and
JI * 0
(1x1
+
-).
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
S:
1 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 ) 0 f
( 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
Y
-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
-
in
no,
b.
Then
2 1
(3.14)
an
= To(a-b)
r
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
I
l i n e a r and Tf is n o n - l i n e a r .
Lemma 2
1 T
and S f l
m+T H
(r)
a r e homeomorphic from :-'(F)
m-to H
1 '(r)
f o r a n y m,O.
1 m+to H
'(r)
and R
i s homeomorphic from
84
Hideo K A W A K A Dand A Takao HANADA
Proof -
m+-
1'
T
i s i n j e c t i v e from H
f
-
T (a) f
=
2(r)
m-into H
1
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).
0
1
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
0
0.
This is
a contradiction.
2"
T
i s s u r j e c t i v e from H f 1
m + 1T (r)
1
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)
+
*
h
0
= f
no
in
(3.17)
has a unique s o l u t i o n @
m+-
E H
(3.18)
3"
b
Hm+'(O
) if h
0
> 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
0
and
-1
b = Tf(Jlblr).
m+are c o n t i n u o u s between H
1
1
2(r)
a n d)T('-;
[ll). m+-1
4"
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
2
1 onto
E
;-'(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
(r)
2(r)
m-and H
-1
and
1 2
I
(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
E
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
85
1
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)
86
Hideo K A W A R A D A and Takao H A N A D A
1
m+Here we have used t h e continuity of To from H 2(r)
Lemma 4
(b)
If
Let
a + @>
E
1
to
g-'(r).
I
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
(a
2
0
= 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
88
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
r
i s mapped i n t o t h e u - a x i s of w-plane.
2
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
E
-to
( n = l , 2,
... ) .
Then
I
Asymptotic Behaviors of the Solutions
89
1 (3.42)
y E n -t
yo
weakly i n H
E
(3.43)
Tf(9 n,
(r),
--1
0
-t
2
weakly i n H
Tf(Y )
2
(r). I -
Proof
Recalling (2.4),
Then, we d e n o t e by $ (3.44)
-A$
Y
(y = a , b )
+ XoJ'
Let a , b be a r b i t r a r y i n B
2
(r).
t h e s o l u t i o n of t h e problem: in
= 0
$1,
(3.45)
(3.42) i s obvious.
iZ
0'
= Y.
By u s i n g G r e e n ' s f o r m u l a , w e have
(3.46)
($bA$a-$aA$b)dx
Using ( 3 . 1 0 ) and t a k i n g a=;"-$P0
=
0.
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 -
SE
(r)
as
E
( 1 . 4 ) i s r e w r i t t e n as f o l l o w s :
(YE).
E
Taking a = y
n
and
E = E
n
-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 h a v e
90
Hideo KAWARADA and Takao HANADA
I
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 ) .
4.
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
0
= 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 ) .
I
Asymptotic Behaviors of the Solutions
91
Proof of Theorem 2
5. 5.1
Using ( 3 . 4 9 ) ,
m+, Find a t H
t h e problem ( 1 . 1 ) - ( 1 . 5 )
1
(r)
such t h a t
T (a) = f
(5.1)
E
2a . S
E
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
1
T (a-b) 0
(5.2)
-
E
2a
.S
E
(a) = -Tf(b)
t
H
-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
$o
(5.2),
OI
=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)
or
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
92
(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
E
( > 0) be s m a l l enough.
m+from H
Then
E
-2a
E
.(S )
-1
T
1
2(r)
onto i t s e l f i f a
<
0 and a + B > 0.
0
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 ) }
1
+ 0(EB-3a + Ea+35)
= -,BaTf(o)
in
H
m - 2(I'),
i f a + B > 0 a n d a > 0.
1
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.
0
0
+qo= 0
Theorem 1 f o l l o w s Tf ( J i o )
0 T0(a-qO)
(5.11)
-
E
+
a
2a
on
r.
Choose b =
E
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
E
E
2a
E
E
-
q00
= E
2a
E
.Sl(a)
By u s e of RE w i t h a =
5,
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
R
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
a
0 ( c ) o f Theorem 1 g i v e s u s T f(JIO) = 0.
O(E
4a
m+)
in
H
2(r).
1
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 =
6.
$o0
+
O(P ---B)
in
H
2(r).
Proof o f Theorem 3
6.1
Assume f C H
k
(no) (k,?)
5
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)
r;
(1.3) a nd (1.5) a n d
I
Asymptotic Behaviors of the Solutions
r.
where s is t h e arc l e n g t h of By ( 6 . 3 ) ,
95
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 )
o
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
in
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
96
Hideo KAWARADA and Takao HANADA
(6.13)
7.
Proof o f Theorem 4 I n the final section we give the proof of Theorem 4 under the drastic
assumption.
Suppose n -
2
1 - R+.
fk
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
91
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 )
1"
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). -
1
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 :
I
L e t J,:
be the
Hideo KAWARADA and Takao HANADA
98
in
(A.5)
S o l v i n g (A.5,6)
(A. 7 )
-E
2
R+
and computing
2 ( a + B ) - ~ ~+1 a(u1,u2)?$:
=
E
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 ,
2"
W e put
(A.10) and (A.11)
=
w
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),
=o
=
0
1
we h a v e
+ a ( u 1 , u 2) 2WE and
wE
=
o(E
+
0
(n+2) (a+B))
(u2++m).
in
R+2 ,
...).
Asymptotic Behaviors of the Solutions
an d mo r e o v e r (A. 16)
3"
W e compute
wh e r e
On t h e o t h e r h a n d , from ( A . 1 6 )
1
1
(or n L -m - 1 ) , I f we c h o o s e n + - > m - 2= 2
t h e n w e ha ve
(A. 20)
By n o t i n g
d l an
r
(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
99
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
R
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
on
Qx(O,T)
and consider the
Robert V . KOHN
102
(1.2)
u(x,O) = uo(x)
on
R,
u(x,t) = 0
on
aRx(0,T)
where uoI
=
0, V*uo = 0,
and V - f = 0
.
aR
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
E
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 <
(1.3a)
u
(1.3b)
equations (1.1), (1.2) hold weakly;
E
1,
(1.3~) /uI2dx + 2 0x1 il OR
2 IVuI dxdt 5 2
t
!j
u-fdsdt +
OR
In relation (1.3~) the "energy inequality", we write
I VU
j n
m:
2
luol dx.
103
Partial Regularity and Navier-Stokes Equations
A sufficiently regular "strong solution" is known to be unique
in the class of Leray-Hopf weak solutions [13].
However, weak
solutions are not known to be unique. The fundamental regularity problem for the Navier-Stokes equations in three space dimensions remains open: even if f = 0, one does not know whether weak solutions of (1.1) remain smooth for all time.
The work presented here achieves a much more limited goal:
we show that a "suitable weak solution" can be singular only on a rather small set.
Results of this type, called partial regularity
theorems, are well-known in the theory of minimal surfaces and quasilinear elliptic systems.
It was Scheffer's remarkable idea to study
the Navier-Stokes equations from this point of view.
Basic tools.
Section 2 .
The proof of Theorem 1 makes extensive use of the fo lowing four (a) Interpolation inequalities; (b) Solving for p in terms
tools:
of u; (c) Dimensional analysis; and (d) The generalized energy inequality.
Of these, (a)-(c) are quite standard, while
introduced by Scheffer in [ 1 6 1 .
d) was
We review each briefly.
Interpolation inequalities The energy (or generalized energy) inequality gives information about
],I2
and 1Vul
2
.
To draw conclusions about other Lp norms one
uses the well-known relation
(2.1)
J Br
I V ~ I ~ ) ~
1 u 1 9< Br
L
L
where 2'S56
3
a = T (q-2)
,
Robert V . KOHN
104
Br is a ball of radius r in R
3
,
and C does not depend on r [4,121. A typical use of (2.1) is this: if u : R x ( O , T ) + R3 , uI = 0, and aR a.e.t
lj
(2.2b)
lVul 2dxdt 5 M
OR Then
IuI
(2.3)
bh
5 -
10
I
dxdt 5 CM3
.
To prove (2.3), extend u by zero off R and apply (2.1) with 10 , a = 1, r + m to see that q = F
( 2 . 3 ) follows by integration in time.
Solving for the pressure. The generalized energy inequality qives information about u, but
To draw conclusions about the pressure, one uses the
not about p. relation
Ap =
(2.4)
-
i,j=1
which follows from (1.1) by differentiation.
If R = R
3
then (2.4)
determines p explicitly as a sum of sinqular inteqral operators i j acting on u u
.
For partial regularity theory, it is more important to represent p locally, using only local information about u. with supp $I
C
Br and $I = 1 on B
_.
2
.
Then
Let o ( x ) be Cm,
Partial Regularity and Navier-Stokes Equations
105
Substituting (2.5b) into (2.5a), one obtains a formula for p on B
-
as a sum of harmonic functions and integral transforms of u.
2
Dimensional analysis. Though elementary, the scalinq properties of the equations are of fundamental importance.
(2.6)
for any X
If (u,p) solve (l.l), then so do
2 2 PX(X't) = h p(hx,X t)
.
We encode this information by assiqning a scaling
dimension to each quantity: x.
has dinension
1
t
has dimension
2
ui
has dimension
-1
p
has dimension
-2
so that each term in (1.1) has dimension - 3 .
The fact that u has
dimension -1 is consistent with its interpretation as a velocity, __ i.e. dimensionally as space/time.
In view of ( 2 . 6 ) , it is natural to use parabolic cvlinders in space-time instead of Euclidean balls; we therefore define
Using ( 2 . 7 ) ,
one can assiqn a dimension to any integral
Robert V . KOHN
106
involving u and p .
For example, the estimates
have dimension one: this corresponds to the fact that
The relevance of this "scalinq dimension" will become clearer as we proceed. dimension
We note here, however, that various estimates of
5 0 imply regularity. For example, every known existence
theorem for a strong solution requires a hypothesis of dimension 0 on the initial data.
-
is
m
C
(2.10)
and
I( I
Moreover, Serrin has proved that if
2 lulqdx)q dt <
m,
3 q
.-
2 + s
< 1
f
,
then u is Cm in space: the estimate (2.10) has dimension < 0 1141. It is thus not surprising that the estimates (2.9), which have scaling dimension one, lead to an estimate of the one-dimensional measure of the sinqular set.
Generalized energy inequality. We shall work only with weak solutions that satisfy the following generalization of the standard energy inequality: if $I
(x,t) is Cm, compactly supported, and (I > 0 then
Partial Regularity and Navier-Stokes Equations
107
One obtains this relation formally, with 5 replaced by =, by multiplyinq (1.1) with u$ and inteqratinq by parts. is, of course, not admissible €or a weak solution.
That procedure There is, however,
at least one weak solution of (1.1) - (1.2) which satisfies the inequality (2.11) [I]. The advantage of (2.11) over ( 1 . 3 ~ )should be clear: by choosing @ appropriately in (2.11), one can obtain local or weiqhted
estimates of u. We close this section with some definitions. We call (u,p) a suitable weak solution of the Navier-
Definition 1:
Stokes equations with force f if i)
u, p, and f are defined on a space-time cylinder D f E L~(D)
5 for some q > 2
\lVu12dxdt < C 1
DI
B x
P ii)
V-f = V*u = 0, and ui t
+ 1V. j
’
(uiuJ) - Aui
+ Vip
= fi
=
B x(a,b)
Robert V . KOHN
108
in the sense of distributions. The generalized energy inequality (2.11) holds for every
iii)
0
0, Cm and compactly supported in D.
Definition 2:
If (u,p) is a suitable weak solution, we call a point
(x,t) regular if u is LToc in a neighborhood of (x,t). Other points are called singular.
s
(2.12)
Notice that
{singular points of ul
=
is by definition a closed set (relative to the domain of u ) . Definition 3 : zero if -
A set X in R 3 xR has parabolic k-dimensional measure
€or every
> 0 there exists a finite family of parabolic
E
N cylinders, {Qi = Qr,(xi,ti)}i=l , with 1
and
How big can X be andstill have parabolic one-dimensional measure zero?
Certainly it cannot contain a smooth arc; indeed, its one-
dimensional Hausdorff measure must be zero.
Moreover, it is easily
shown that the projection onto the t-axis
must have one-half dimensional Hausdorff measure zero.
Section 3 .
Sufficiently small solutions are regular.
We have already noted that “sufficiently small“ solutions of the initial-boundary value problem are regular.
The key to provinq a
partial regularity theorem lies in showinq an analogous local result:
Partial Regularity and Navier-Stokes Equations
Proposition 1:
There are absolute constants
E
~
,C1
109 > 0, and a
constant ~ ~ ( depending q ) only on q , with the followinq property. (u,p)
% 5 suitable weak solution
If
of the Navier-Stokes equations on
Q1 with force f E Lq(q > T5 ) , and if
(3.2)
then -
Proposition 1 is a local version of the main lemma in [161. To understand it, one should ignore f and p; heuristically, it says 3 that if IuI is small enough in the L -norm on a unit-sized cylinder Q1, then u is regular on Q1
.
2 One proves Proposition 1 by a variant of Scheffer's clever inductive procedure. lu(G,O) I equation:
by using a fundamental solution of the backward heat suppose 4
*
0
Substitution of
12;
/u(O,O)
As motivation, consider trying to bound
*
is defined for t < 0,
(-,O)
=
6(*)
@ * into (2.11)
leads formally to a bound for
but one lacks sufficient information to estimate the
right hand side.
I10
Robert V . KOHN
Next, notice that the worst term,
!I
(lul2+2p)u.V@, is cubic
in u, while the left side of (2.11) is quadratic (pis"1ike lu/2" by (2.5)).
Moreover, using (2.1)
for any r > 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
El
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, __
if
J J
and -
Qr
Qr
fIq
(3.8)
5
E2
then a.e. -
on Qr 2
.
Again, the way to understand Corollary 1 is to ignore f and p . says,
(3.9)
It
in essence, that if the dimensionaless quantity
R(r;x,t) =
I!
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
3
,
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:
3
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
Q
C
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
1
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
E~
> 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)
E
3 R xR.
E r > O , % R(r) = r-2
jj i W i 3 Qr
Qr
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
+
E
> 0 there exists 6 > 0 such that
y(r) < 6 * lim inf R(r) < r+ 0
Indeed, (5.3) implies
whenever R(r)
2
E.
1
E
.
3
Choosing 0 < 0 < 1 so that C 2 0
3
<
a1
, then
6 > 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
ii
the generalized energy inequality lets one bound the osc llation in time of
i
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
or
j
R3
/u0l2/xl-l sufficiently small.
1
R3
2
luol 1x1 <
O0r
Partial Regularity and Navier-Stokes Equations
117
REFERENCES Caffarelli, L.; Kohn, R.; Nirenberq, L; "Partial regularity of suitable weak solutions of the Navier-Stokes equations," Comm. Pure Applied Math., to appear. Fabes, E.; Lewis, J,; Riviere, N.; "Boundary value problems for the Navier-Stokes equations," h e r . J. Math., vol. 9 9 , 1 9 7 7 , pp. 6 2 6 - 6 6 8 . Fujita, H.; Kato, T.; "On the Navier-Stokes initial value problem I," Arch. Rat. Mech. Anal., ~ o l .1 6 , 1 9 6 4 , pp. 2 6 9 - 3 1 5 . Gasliardo, E., "Ulteriori urourieta di alcune classi di funzioni in pi; variabili,i' Richerche di Mat. Napoli, vol. 8, 1 9 5 9 , pp. 2 4 - 5 1 . Giqa, Y.; Miyakawa, T.; "Solutions in Lr to the Navier-Stokes initial value problem." to appear. Hopf , E.; "Uber die Aufangswertaufqabe fur die hydrodynamischec Grundqleichungen," Math. Nachr., vol. 4 , 1 9 5 0 - 5 1 , pp. 2 1 3 - 2 3 1 . Kaniel, S. and Shinbrot, M., "Smoothness of weak solutions of of the Navier-Stokes equations," _Arch. Rat. Mech. Anal., v o l . 2 4 , 1 9 6 7 , pp. 3 0 2 - 3 2 4 . Kato, T.; Fujita, H.; "On the nonstationary Navier-Stokes system," Rend. Sem. Mat. Padova, vol. 3 2 , 1 9 6 2 , pp. 2 4 3 - 2 6 0 . Ladyzhenskaya, O.A.; The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 1 9 6 9 . Leray, J.; "Sur le mouvement d'un liquide visquex emplissant l'espace," Acta Math., vol. 6 3 , 1 9 3 4 , pp. 1 9 3 - 2 4 8 . 111
Lions, J.-L.; Quelques Methodes de Rgsolution des Probl2mes aux Limitex non Lin@aires. Dunod-Gauthiers-Villars, Paris, 1969.
121
Nirenberg, L.; "On elliptic partial differential equations," Ann. di Pisa, vol. 1 3 , 1 9 5 9 , pp. 1 1 6 - 1 6 2 .
131
Serrin, J., "The initial value problem for the Navier-Stokes equations," in Nonlinear Problems, R.F. Langer, ed., Univ. of Wisconsin Press. 1 9 6 3 , pp. 6 9 - 9 8 , Serrin, J., "On the interior regularity of weak solutions of the Navier-Stokes equations," Arch. Rat. Mech. Anal. vol. 9 , 1 9 6 2 , pp. 1 8 7 - 1 9 5 . Scheffer, V., "Partial regularity of solutions to the NavierStokes equations," Pac. J. Math., vol. 6 6 , 1 9 7 6 , pp. 5 3 5 - 5 5 2 . Scheffer, V., "Hausdorff measure and the Navier-Stokes equations," Comm. Math. Phys., vol. 55, 1 9 7 7 , p p . 9 7 - 1 1 2 . Scheffer, V., "The Navler-Stokes equations in space dimension four," Comm. Math. Phys., vol. 6 1 , 1 9 7 8 , pp. 41-68.
1 I8 [18]
Robert V. KOHN
Scheffer, V., "The Navier-Stokes equations on a bounded domain,"
Comm. Math. Phys., vol. 73, 1980, pp. 1-42.
[19]
Scheffer, V., to appear in Comm. Math. P h y s .
[20]
Temam, R., Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam and New York, 1 9 7 7 .
Lecture Notes in Num. Appl. Anal., 5 , 119-131 (1982) PDE in Applied Science. U.S.-JapunSeminar, Tokyo, 1982
Nonlineur
Blow-up
o f S o l u t i o n s o f Some N o n l i n e a r D i f f u s i o n E q u a t i o n s
h
Kyuya
Masuda
Mathematical I n s t i t u t e Tohoku U n i v e r s i t y S e n d a i , J a p a n 980
Abstract ut = Au t u2
We p r o v e t h a t a n y s o l u t i o n o f
t ) through t h e upper
a n a l y t i c a l l y continued ( i n ( and lower) h a l f -
interval
1.
(t*,
Introduction
m
c a n be
complex p l a n e i n t o some i n f i n i t e
.
t* > 0
),
Consider t h e nonlinear d i f f u s i o n equation of
t h e form
(1)
u
t
=
h u t
2
X E R ,
u ,
t > O
w i t h t h e homogeneous Neumann b o u n d a r y c o n d i t i o n :
and t h e i n i t i a l c o n d i t i o n :
(3)
Ult=O =
where and
R
u0
uo
i s a b o u n d e d domain ( i n
Rn) w i t h smooth b o u n d a r y
r ,
i s a g i v e n s m o o t h f u n c t i o n s a t i s f y i n g t h e homogeneous
Neumann b o u n d a r y c o n d i t i o n ;
a/an
d i r e c t i o n o f t h e e x t e r i o r normal t o 119
denotes d i f f e r e n t i a t i o n i n t h e
r
.
120
KyDya MASUDA
(l), ( 2 ) , ( 3 )
t ) s o l u t i o n of
W e know t h a t t h e l o c a l ( i n
On t h e o t h e r h a n d , i t i s e a s y t o see t h a t
e x i s t s and unique.
any s o l u t i o n o f (1),(2),(3) d o e s n o t e x i s t g l o b a l l y , a n d b l o w s up in a finite time if
i s p o s i t i v e , where
Pug
1
if
R
: t h e volume o f
Is21
(
uo
) ;
H . F u j i t a [l].
see
In particular,
is a positive constant function, then the solution u ( t ) = uo /(l- u o t )
u ( t ) i s e x p l i c i t l y given by : blows up a t
Pug = u o .
t = l/uo ; note
,
u =
and s u r e l y
Moreover, t h i s l o c a l
s o l u t i o n can be a n a l y t i c a l l y continued i n t o t h e i n f i n i t e i n t e r v a l (l/U0,
.
m )
More g e n e r a l l y we s h a l l show t h a t t h e l o c a l
(1),(2),(3)
solution of
can b e a n a l y t i c a l l y c o n t i n u e d t h r o u g h
t h e upper (and lower) half-(comp1ex)plane interval Fix in
LP(Q)
that
p > n
.
0 <
uo
.
a =
< la1
6
complex-plane
We s u p p o s e t h a t
Pug #
with
:
.
t* > 0
( t * , a),
.
0
Let
Put
.
D1
Li
(cl.
means t h e c l o s u r e ) ; n o t e t h a t
See
l/Puo
,
function
and l e t
b e suck
6
a n g u l a r domain i n t h e
i s t h e a n g u l a r domain:
< ?n/4
a r g ( t-a-61)
( i =
).
i s a real-valued
be t h e
Do
l a r g tl < n/4
5n/4 <
i n t o some i n f i n i t e
Set
D,
= D- =
Lp(Q)
by
Do i f
a < 0.
F i g . 1.
We d e f i n e t h e o p e r a t o r the identity operator i n
Q
LP(Q)).
in
Q = I
- P
( I ;
121
Blow-Up of Solutions of Diffusion Equations
Im
__
Re
0
Fig. 1
O u r r e s u l t s now r e a d a s f o l l o w s .
Theorem 1
/I
1 ) Quo11
Pug I
Let
p > n.
Let
uo E LP(n)
i s s u f f i c i e n t l y small.
,
real.
Suppose t h a t
?hen t h e i n i t i a l - b o u n d a r y
LP
v a l u e problem (1),(2),(3) (resp. a
u- = u - ( t ) ) t h a t i s a n a l y t i c f o r
W1”(R)-valued Theorem 2
Pug
f u n c t i o n ( W1”(S2); Let
uf(t) = u-(t)
r e a l constant. u+(t) = u-(t)
in
LP
/IPuo12
f o r some r e a l
t :
uo t >
(resp.
D,
D-) as
t h e u s u a l Sobolev s p a c e ) .
(IQuoII
Conversely, i f for a l l
t
be as i n Theorem 1.
u o , u+
i s p o s i t i v e , and t h a t
?hen i f
t
ut = u ( t )
has one and o n l y one s o l u t i o n
t : t >
Suppose t h a t
i s s u f f i c i e n t l y small. ci t
6
,
i s a r e a l constant, ci t
6
.
then then
uo
is
122 2.
Kyuya MASUDA
We have decompos tion
Proof of Theorem 1
~ ~ ( =n PLP(~) ) t Let
y = y(t)
condition
F o r any
b
be a solution of (1),(2)
QLP(R), we set
(6) If
$(b) = a
u
= $(b),
with the initial
, i.e.
y(0) = Pug
in
Q L ~ ( ~
+
is a solution of (1),(2) then the function
a2b
.
with the initial condition
v = ( u
-
ultZO
is a solution of the
y ) / y2
equation
(7)
vt =
Av
+ c(t)v 2 ,
(
c(t) = y(t)
2
),
satisfying the conditions:
(8)
av/anlr= 0 ;
and vice versa.
Using the theory of equations of evolution, we
shall show that the initial-boundary value problem (7),(8)y(9) one and only one solution that is analytic in W1”(R)-valued in
LP(n)
by
Dt
D-) as a
function. To this end, we define the operator Au =
-
Au
A
with domain D(A) ;
D(A) = {u cw2~P(n) ; au/an,, = ( W2yp(n) is the usual Sobolev space).
known.
( and
has
o I .
The following is then well
Blow-up of Solutions of Diffusion Equations
Proposition 3
The s p e c t r u m of
c o n s i s t s o n l y of non-nega-
-A m
t i v e and d i s c r e t e e i g e n v a l u e s
,
{ hj}j,l
Moreover, t h e f i r s t e i g e n v a l u e i s zero
123
,
tending t o t h e i n f i n i t y .
which i s a s i m p l e o n e ; t h e
(4).
p r o j e c t i o n c o r r e s p o n d i n g t o t h e e i g e n v a l u e z e r o i s g i v e n by
Setting
we have Propostion 4 semi-groups -tA
The o p e r a t o r
{e-tA 1 t>O
in
i s holomorphic i n
CE
-A
LP(f2)
g e n e r a t e s t h e holomorphic such t h a t
f o r each
E
> 0
Moreover, i t s a t i s f i e s t h e e s t i m a t e s :
,
(10)
(11)
for
t (
in
CE,
ME
and
0 being p o s i t i v e constants.
F o r t h e p r o o f , we r e f e r t h e r e a d e r t o
I n what f o l l o w s we f i x Using a p r i o r i e s t i m a t e :
(see
(13)
=
E
(
n/4
M
and
[31)
H . Tanabe
.
C 2
denotes various constants)
1 3 1 ) , we g e t
II
Ve-tAll
5
M It1
-1/2 ,-BRe
b y t h e p r o p o s i t i o n 3 and ( 1 2 ) , and by n o t i n g
t
,
t
E
C
124
Kyiiya MASUDA
y
Let
,
0 < y < B
be s u c h t h a t
b e i n g as i n P r o p o s i t i o n
B
we i n t r o d u c e f u n c t i o n s p a c e s
Yo,
Y1.
holomorphic f u n c t i o n s
D,
with values i n
Y1
and
values i n
u
in
Yo
is the s e t of a l l
i s t h e s e t o f a l l holomorphic f u n c t i o n s WlYp(Q)
4, a n d
LP(Q) such t h a t
u
D+
in
with
such t h a t
111
E q u i p p e d w i t h t h e norm
u
111
\(Io
(resp.
of
QLP(n) x Y 1
u
IllI
),
Yo ( r e s p .
yl)
i s a Banach s p a c e .
We d e f i n e t h e map
(14)
( t
F(b,w) =
F
w(t)
-
eWtAb
- J
into
t .-(t-s)A 0
Y1 b y : 2
c ( s ) w ( s ) ds
ED+) where t h e p a t h o f i n t e g r a t i o n is t a k e n i n t h e i n t e r i o r o f
D,.
Then: Lemma 5
The
F
i s a n a n a l y t i c map : Q L P ( Q ) x Y 1
+
Y1
,
satisfying
(15)
F(O,O) = 0 ;
(16)
Fw(O,O) = I
( Fw(O,O)
denotes the Frechet derivative with respect t o
b = 0, w = 0). Proof
:
The map
I1
d e f i n e d by :
w
at
Blow-up of Solutions of Diffusion Equations
Il[bJ is analytic map of Next the map
QLP(Q)
-tA
into
I2 defined by
is an analytic map of
e
=
, Y1
125
b EQLP(fi)
by Proposition 4 and (13)
:
Yo into
Y1 , where
g ( s ) = ((l+s)/s)1’2c(s).
In fact, we have the inequalities:
( t E D + ) by
(lo), (11) and (13) from wh ch it follows that
is an analytic map : Yo
+
Y1.
We finally define the map
I$Wl Then the
I 3
=
I3 by: (t/(ltt))1/2
is an analytic map of
Y1
w2. into Yo
since we have
the following inequalities by the Sobolev inequality
and
by the decomposition
I2
I26
Kyuya MASUDA
Since F(b,w) = ( 12013
IILbl + 1 2 0 1 3 [ W l
i s t h e composition of
i s a n a n a l y t i c map of
F
( 1 5 ) and ( 1 6 ) .
I2 a n d I
3
Q L P ( ~ ) x Yi n~ t o
it follows t h a t
),
yl.
It i s e a s y t o s e e
Thus t h e p r o o f o f t h e l e m m a i s c o m p le te d .
Now we a r e i n a p o s i t i o n t o a p p l y t h e i m p l i c i t f u n c t i o n t h e o r e m , a n d c a n see t h a t t h e r e is o n e a n d o n l y o n e s o l u t i o n t
v ( t , b ) of
t
function with values i n
+
(17) ( t ED+).
v (t,b) =
of
Y1
e
-tA
b
b
t
(0,O) t h a t i s an a n a l y t i c
near
I0t
e
The
0.
-(t-s)A
v
t
satisfies 2
c ( s ) ( v + ( s , b ) ) ds
By t h e s t a n d a r d a r g u m e n t in t h e t h e o r y o f e q u a t i o n s o f
e v o l u t i o n , i t follows t h a t strongly i n
near
F(b,v ( . ,b)) = 0
w =
LP(n).
continuous f o r
t in
vt(t,b)
converges t o
We c a n a l s o see t h a t D+ ( i n
Lp(s2)),
v
t
and s o i s
b
as
t
+
0,
i s l o c a l l y Holder
(vt)*
,
since
Blow-up of Solutions of Diffusion Equations
+
( we s i m p l y w r i t e v ( t )
+
for
v ( t , b ) ) ; note t ED+ ( i n
i s continuously d i f f e r e n t i a b l e i n
derivative
+E
Y ~ . Hence
v+(t)
and i t s
LP(Sl)),
i s l o c a l l y HGlder c o n t i n u o u s ( s e e , e . g .
dv+(t)/dt
T . Kato [ 2 ] ) .
v
127
We a l s o s e e
v+(t)
is in
Av+(t)
D(A),
is locally
H g l d e r c o n t i n u o u s , and s a t i s f i e s
with
+
v (0) = b. By t h e r e g u l a r i t y t h e o r e m on s o l u t i o n s of p a r a b o l i c e q u a t i o n s ,
t ED+
u+(x,t) =
y(t)
+
as a
W’”(Sl)-valued
+
c(t) v (x,t)
u-
solution
W1”(R)-valued
u+(x,c),
if
t
E
of ( l ) , ( 2 ) , ( 3 ) function; D-.
u-(t)
uo
W1”(R)-valued
that is analytic for
u-(x,t)
=
If
uo
does not coincide with
D- as a
E
i s a r e a l constant, then
u-(t)
i s n o t ( r e a l ) c o n s t a n t , and i f
v ( t , b ) and
t
t h e complex c o n j u g a t e o f
t a k e t h e same r e a l v a l u e s on
s u f f i c i e n t l y small. t
t E D + as a
T h i s c o m p l e t e s t h e p r o o f o f Theorem 1.
P r o o f o f Theorem 2
u+(t)
Hence
S i m i l a r l y w e c a n show t h e r e i s a u n i q u e c l a s s i c a l
function.
and
function.
i s a unique c l a s s i c a l s o l u t i o n
o f (l), ( 2 ) , ( 3 ) t h a t i s a n a l y t i c f o r
3.
t
(7),(8),(9) t h a t i s
i s a c t u a l l y a unique c l a s s i c a l s o l u t i o n of
analytic for
v
v-(t,b)
Let
ro
.
( a , - )
We s h a l l show
t ( > a + 6)
a ny r e a l
for
IlQu,((
LP
ut(t)
/ (PuOl2 is
b e a p o s i t i v e number s u c h t h a t
are both a n a l y t i c i n
b
,
11
b
llip
<
17 0
’ .
128
Kyuya MASUDA
Here a n d i n what follows we s h a l l u s e t h e same n o t a t i o n s a s i n t h e
vt(t,b)
ro s o small t h a t
Furthermore we can t a k e
p r o o f o f Theorem 1.
i s l o c a l l y bounded i n
Ilb(( LP
<
ro
as c a n b e e a s i l y s e e n f r o m t h e p r o o f o f Theorem 1.
cLP(Q); N = 1 , 2 , . . . )
.
Hence
aN b v+ ( . , b ) [ h , h , . - . , h ]
h a s i n f i n i t e l y many F r G c h e t d e r i v a t i v e s
Y1)
( i n t h e norm of t
v (.,b) ( h
I n p a r t i c u l a r we h a v e t h e e x p a n s i o n o f t h e
form :
+
(18)
v t (*,O) t
v (*,b) = t
abV
t
(.,O)[bJ
2 + + abv ( . , O ) [ b , b ]
Rt(*,b)
where
(19) Clearly v t ( t ) E vt ( t , O ) 0
(20)
BY (17)
t
vlit)
3
t
abv ( t , O ) [ b ]
= 0
.
satisfies
a n d h e n c e , by ( 2 0 ) ,
(21) We c a n also see
t
v,(t)
=
e
-tA
v t2 ( t ) e a 2 b vt ( t , O ) [ b , b ]
satisfies
9
Blow-up of Solutions of Diffusion Equations by (20) a n d (21);
Dt:
i s taken i n If
v,(t),
t h e path of i n t e g r a t i o n
joing
0
to
t
The p r o c e d u r e s a b o v e c a n b e e a s i l y j u s t i f i e d .
v;(t),
V g ( t )
Set
rt
129
v;(t),
=
0 ;
R- ( t , b )
are s i m i l a r l y d e f i n e d , we have
v1 -(t) = e
h ( s ) = (e-SAb)2 ( s E D + ) ,
and l e t
-tA b
r
be a s shown i n F i g . 2
below.
Im
Re
Fig.
2
K y l y a MASUDA
130
Since t 7
- ,
c(s)h(s)
is holomorphic in a complex domain bounded by -T+
we have, f o r any fixed
=
t ( > a + 6 ),
4ni e -(t-c~)A(~~-aA~)2
by the Cauchy integral theorem and formula.
Hence
Consequently,
and hence
If
uo
is not real constant, then
e-ffAb is not real constant, and so
b is not real constant. 2
(Ve-aAb)
Hence,
is non-negative,
and does not vanish identically. This gives the p r o o f of Theorem 2 by (22).
Blow-up of Solutions of Diffusion Equations
4. Remark
131
We c a n a l s o c o n s i d e r d i f f u s i o n e q u a t i o n s of more
general type :
ut =
(23)
where
f = f(z)
+ f(u)
Au
i s an e n t i r e f u n c t i o n of
z
w i t h some c o n d i t i o n s .
The r e s u l t s c o n c e r n i n g t h e e q u a t i o n s above s h a l l b e p u b l i s h e d elsewhere.
References
[l]
Fujita, H.,
On some n o n e x i s t e n c e and n o n u n i q u e n e s s t h e o r e m s
f o r nonlinear parabolic equations, i n Proceedings of the
Symposium on P u r e M a t h e m a t i c s , American Math. S o c . , P r o v i d e n c e , R.I., [Z]
1 9 7 0 , v o l . 1 8 , P a r t 1, p p . 105-113.
K a t o , T., P e r t u r b a t i o n t h e o r y f o r l i n e a r o p e r a t o r s . B e r l i n Gottingen-Heidelberg:
[3]
Tanabe, H . ,
Springer 1966.
E q u a t i o n s o f e v o l u t i o n . London-San F r n c i s c o -
N e l b o u r n e : Pitman 1979-
This Page Intentionally Left Blank
Lecture Notes in Num. Appl. A n a l . , 5 , 133-ISl(1982) Nonlinear PDE in Applied Science. U . S . - J a p a n Seminar. Tokyo. 1982
Asymptotic Behavior o f the Free Boundaries A r i s i n g i n One Phase S t e f a n Problems i n Multi-Dimensional Spaces Hiroshi MATANO Department of Mathematics Faculty of Science Hiroshima University Hiroshima 730
JAPAN
We discuss the asymptotic behavior of weak solutions to one phase Stefan problems in exterior regions. It is shown that any weak solution eventually becomes classical after a finite period of time and that the shape of the free boundary approaches to that of a sphere as t + + - .
1.
Introduction. A Stefan problem is a mathematical model for describing the melting of a
body of ice in contact with a region of water.
I n one phase Stefan problems,
the temperature of ice is supposed to be maintained at
0°C.
Hence the
unknowns are the temperature distribution of water and the shape of the interface ( free boundary ) between ice and water.
In the case where the space dimension i s one, the problem has been completely solved; and it is well known that, for any bounded smooth initial and boundary data, the solution is unique and exists globally in time in the classical sense ( see Friedman [ 4 ] ) . However, in the case of multidimensional spaces, a classical solution cannot always be extended globally over the time interval 0
2 t < f-,
appear on the free boundary.
since cusp-like singularities sometimes
Such singularities occur, for instance, when a
portion of ice is in the process of being separated from the rest of the ice
133
134
Hiroshi MATANO
by the growing water region. Duvaut [31 has introduced a weak formulation of the above problem in terms of variational inequality, for which the solution exists globally in time. Following his formulation, Friedman, Kinderlehrer, Nirenberg, Caffarelli and others have studied the properties of weak solutions and obtained a number of regularity theorems. These studies have revealed that any singular point appearing on the free boundary should be a cusp-like singularity [ i ] and that the solution is sufficiently smooth, say Cm
,
inside the water region and also
up to the free boundary provided that this free boundary forms ( locally ) a
[a].
Ci
hypersurface
It is also known that the free boundary is Ce
C’
or even simply Lipschitz ( [ E l , [I] ) .
if it is
Friedman and Kinderlehrer [5] have
given an example in which the free boundary possesses no singular point for 0 5 t < + m , in other words, the solution is classical for all time.
This
example, however, requires a strong geometric assumption on the initial data. The aim of this paper is to prove that any weak solution is ”eventualZy”
classical. That is, the interface between the ice and water regions is sufficiently smooth for all large t ; hence so is the temperature distribution of water.
It will also be shown that the shape of the free boundary
the interface ) approaches to that of a sphere in a certain manner as
Of course the radius of this sphere goes to infinity as t
-+
+ m a
(
i.e., t
-+
f m .
Note that we
shall impose no specific hypothesis on the geometric features of the initial and the boundary data.
The only assumption required in this paper is that the
temperature of water averaged on the prescribed boundary ( that is, the surface of a heat supplying obstacle located in the midst’of the water region ) be bounded from below by a time-independent positive constant. To state the above results more precisely, let us introduce some notation. First, Let
G1
be a bounded domain in lRn
smooth and that l R n \ G 1
E,
.
The boundary of G
is connected. Let
such that
aG,
is sufficiently
G be a bounded domain containing
-
is also assumed to be smooth. G, represents the
135
Asymptotic'Behavior of the Free Boundaries fixed obstacle, G \El
is the initial water region at
t = 0 and En\G
denotes the initial ice region. A s is mentioned above, the temperature of ice is supposed to be maintained at
O°C
.
The water region at each t > O and the
interface between ice and water are denoted by
Q(t)
and
respectively.
r(t)
From the definition it follows that
The temperature of water will be denoted by function defined on the closure of the
set
8 = e(x,t). {
(x,t)
1
This is a nonnegative
xEQ(t),
t > = O}
and
supposed to satisfy the following initial and boundary conditions:
here h
and
g are given positive smooth functions satisfying
The equations governing the phenomenon are as follows:
et
(5)
where k and
r(t)
- i;lvee/ 1
2
,
x~r(t) , t > o ,
is a given positive constant. Thus the problem is to find satisfying ( 1 )
-
(5) and
e(x,t)
136
Hiroshi MATANO Although the problem ( 1 )
-
(6) does not necessarily have a global classical
solution, it always has a weak solution ( in the sense of Duvaut ) that exists
In his weak formulation, the temperature 9
globally in time.
given by the form 0
=
ut
, where
u = u(x,t)
is a solution of a certain
variational problem ( see Section 3 for details 1. have proved that
0
that
for x$Q(t) U c ,
8(x,t) = 0
Caffarelli and Friedman [ 2 ]
(Rn\G,)x LO,
is continuous in
, tZO.
is locally
+ m )
, where we understand
It is not difficult to see that
8
is sufficiently smooth in the interior of the water region, since it satisfies there the heat equation Bt regularity of
=
A8
in the sense of distributions. Further
near the free boundary
8
depends on the regularity of
r(t)
In other wards, a weak solution i s classical
r(t).
singular point (
Cil, [El
so
long as
r(t)
has no
).
2. Main Theorems.
In what follows the pair the problem (1)
-
(6).
0(x,t), T(t)
will denote the weak solution to
The notation in the preceding section will be used
freely. The main theorem in this paper can now be stated as follows:
Theorem 1 .
Suppose there e x i s t p o s i t i v e constants 6 6
(7)
I
g(x,t)dx
6
and
M
such t h a t
To
such t h a t
5 M
acl
holds f o r a l l
r(tl
02
Then there i s a p o s i t i v e nwnber
t < + m .
i s s u f f i c i e n t l y smooth f o r a l l
.
Let t h e same assumption as i n Theorem I hold.
Theorem 2 .
n t 3.
tLTo
Then there e x i s t a p o s i t i v e nwnber
nJ"
such t h a t f o r any
to
r(tl
at
xo
t
2
To
and any point
intersects the s e t
W
.
To
Assume f u r t h e r t h a t
and a bounded convex s e t
xo E
W
r ( t ) t h e inward normal l i n e
in
Asymptotic Behavior of the Free Boundaries
137
Combining Theorem 2 and the fact that the free boundary away from every finite region ( that is, any point outside
r(t)
is moving
will eventually
GI
be swallowed by the water region 1 , we get
Corollary 3 . B"
, there
n t 3
Let
and Let (7) hold.
Corollary 4 .
Let
n 2 3
A
R ( t l U 5,
TA such t h a t
e x i s t s a p o s i t i v e number
w i t h r e s p e c t t o any p o i n t o f
Then, f o r any bounded s e t
f o r each
t 2 TA
and l e t ( 7 ) hold.
in
A
i s star-shaped
. xl be any p o i n t i n IRn
Let
and
Put Ix-xIl ,
m ( t ) = min
Mfti = m a
x~r(t) Then
M(t)
Remarks.
- m(tl (i)
t
remains bounded a s
+ +m
r w
13:-xlI
.
.
The condition (7) can be relaxed somewhat. For example, even if
the integral of remains true
XE
so
g
tends to zero as
t
-f
+-,
the assertion of Theorem 1 still
long as the rate of decay is moderate.
decays very rapidly as
t +
+ m ,
However, if this value
then the total amount of the heat energy to be
supplied to the water region through the surface aG,
becomes finite, which
implies that only a finite portion of the ice will be melted.
In such a case,
as it seems, the conclusion of Theorem I i s not likely to hold. (ii)
In the case n
= 2
, Corollary 3 still remains true, but the assertions in
Theorem 2 and Corollary 4 should be weakened slightly. For example, the boundedness of M(t)-m(t)
However, if
g(x,t)
=
must be replaced by
constant
on
corollaries all hold true even for n (iii)
In the case where
radius of
r(t)
g
, then Theorem
aG, x [ O , + - )
2 and its
= 2.
is independent of
t
, the growth order of the
can easily be calculated; namely we have
138
Hiroshi MATANO
Corollary 5 .
Let
n 2 3 and l e t
remains bounded as
t
-t
+a,
g
i n ( 2 ) be independent of t
where M l t )
i s a s i n Corollary 4 and
.
C
Then
is a
constant defined by
i s as i n ( 5 ) ,
i n which k that i s , a/ay
i s the volume o f t h e n-dimensional u n i t b a l l ,
bn
r(-)
bn = nn/2/r(n/2+1) w i t h
being the usual g m a function,
is t he outward normal de riv ativ e on
and, finalZy,
aG,
e^
=
6(x)
is
the s ol ut i on of t h e boundary value problem
18
on
= g
aGI
,
3 . Preliminaries - Existence and Regularity Theorems.
In the weak formulation due form
0 = u
t
, with
u
to
is locally given by the
Duvaut, e(x,t)
being a solution to a certain variational inequality.
More precisely, let T be any positive number and R positive number.
Set BR = I xEIRn
1
1x1
<
RI
,D
= BR\E1
t
@(x,t)
g(x,.r)dr,
=
xEaG1
0
(9)
f(x)
h(x) =
{ -k
(
xEG )
(xED\G
)
,
be a sufficiently large
,
, o< t
139
Asymptotic Behavior of the Free Boundaries where k
is the constant given in ( 5 ) and Hm is the Sobolev space of all
2
L functions whose derivatives up to order m belong to L 2 2 a solution u E K n L (0,T;H (D))
2
.
We are to find
to the variational inequality in the pointwise
form ( - A u + u )(v-u)
(10 a)
t
Z
f(v-u)
on D x { O }
a.e.
for v E K
~
,
i u = O
( u =
Given a solution u
o
on
.
aBRx ( 0 , ~ )
to ( l o ) , we put
(13)
for O S t S T . It is known that the problem (10) admits a unique solution ( see Proposition 3.1 below ) . Moreover, one can check that the above-defined 8(x,t), R
are actually independent of the choice of T, R
r(t)
is chosen sufficiently large
for 0 6 t S T O6t<+m.
.
so
provided that
that
We therefore can define 8,
r
globally on the time interval
Hereafter by a weak s o l u t i o n we mean the solution pair
8,
r
defined in the above manner. Let us now consider the problem (10) in more detail. that (10) possesses at most one solution.
It is easily seen
The existence of a solution can be
I40
Hiroshi MATANO
shown by approximation with a suitable penalty function. The following construction of approximate solutions is due to Friedman and Kinderlehrer [S] For
Let
f(x)
E
>0
, define
a function
be as in (9) and let
uniformly bounded, decreasing to bounded variation in D
m
n
where n E Co( IR )
.
Be(t) E Cm( W")
fE(x) f(x)
with the properties
be a sequence of functions smooth in as
E
-t
0
, and
5,
having uniformly
Set
is a function satisfying
for some positive constant a with
3a
.
The approximate equation for (10) is an initial-boundary value problem of the form -Au + u
u = n
t
+ kBE(u)
fE
in D x (0,T) , in D x IOl on
( u = o
where k
is as in ( 5 ) .
,
aG, x (0,T)
,
Asymptotic Behavior of the Free Boundaries Proposition 3 . 1 ( Friedman and Kinderlehrer ) .
unique solution
a.e.
t 2- 0
i n D x (0,T)
i s the Sobolev space of
$.p(D)
order 2 belong t o
there e x i s t s a
t o the problem (10) with the properties
u
u
where
T >0 ,
For
141
Lp(Dl
t o the probZem ( 1 5 J E
.
Lp
Furthermore, i f
,
functions whose derivatives up t o uE ,
E
0
, denotes the solution
, then
uE + u
and weakly i n b?'pID)
(
as
f o r each
E
+
0
)
weakly in
t E (0,TI
and
~ ' " ( D x (0,TII
1
m
; hence
u
+
u
uniformly i n D x ( 0 , T ) .
Remark 3 . 2 . (i) Actually it can further be shown that u€Lm(O,T; W2>"(D)) that u
t
and
is continuous ( [2] ) .
(ii) The region n ( t )
defined by ( 1 1 ) is easily seen to coincide with the set
It therefore makes sense that we have called Q(t)
Definition 3 . 3 .
Let A C R n be a measurable set and let x o
in IR"
.
We say xo
where
!J
is the Lebesgue measure and
be any point
if
has p o s i t i v e Lebesgue density with respect to A
Combining the Cm and the C 1
the water region.
B (x,) P
=
I x E Rn I ( x - xoI
< p
1
smoothness criterion of Kinderlehrer and Nirenberg [S]
smoothness criterion of Caffarelli [ l ] , we get
Hiroshi MATANO
142
Proposition 3 . 4 .
on
iet
xo has p o s i t i v e Lebesgue density w i t h respect t o t h e s e t
r ( t o l . Suppose
Bn\(Q(tol
U
cl).
Then there e x i s t s an x t
(xO,tO1 such t h a t
r ( t ) n V is an
sal t o the hyperplane
I(x,t)
I
X E
nltl
xo be any p o i n t
to be any p o s i t i v e number and l e t
u Tft),
t = to
.
t>01
.
- neighborhood
n - dimensional
Moreover,
9
is
C"
hypersurface trunsverin
C"
of t h e p o i n t
V
V fl
4 . Proof of the Main Theorems.
In order to prove that the solution is "eventually" classical, we must has positive Lebesgue density with respect to
show that each point on T(t)
U z,)
the ice region En\(Q(t) of
r(t)
if
t
is sufficiently large.
Such a property
does not follow simply from local regularity analysis; studies of the
geometric features of
T(t)
in some global aspect are needed.
The main tool employed is the "plane-reflection" method first introduced by Serrin [9] and later developed by Gidas, Ni and Nirenberg [6] and Jones [ 7 ] . In particular, we owe much of the discussion below to Jone's work [ 7 ] ,
in which
he has investigated the asymptotic behavior of radially expanding wave front solutions to the equation u
t
=
Au + f(u)
, x E lRn , t > 0
.
Of course his
argument does not apply to our problem automatically, partly because of the existence of the obstacle G, face r(t)
through which
0
and partly because of the presence of the intergains derivative gaps.
We begin with some notation and lemmas:
Notation.
Let
P be an (n-1) -dimensional hyperplane with
( 1 6 a)
S+: the half space with boundary
(16 b)
x : reflection of a point
A
x
Emn
P
PnG, =
such that
4;
SC=G,
in P.
The following is the key lemma for the proof of the main theorems:
set
,
143
Asymptotic Behavior of the Free Boundaries Lemma A .
Assume
nwnber Ro
n > 3 and t h a t ( 7 ) holds.
such t h a t f o r any hyperplane
Then there e x i s t s a p o s i t i v e 2 Ro
with dist(P,G,l
P
A
it holds
that e(x,ti
(17)
, t 2 0 , where
for a l l
x € Si\C
for a l l
x€SfnCl(tl\GJ
Corollary A .
Lemma B.
t > O
S'
,
are as i n (16).
0
Assume ( 7 ) .
Moreover, we have
.
Assume n 2 3 and that ( 7 ) holds; and l e t
I vX e ( x , t l l #
Then
e ( x,t) ~
2
f o r any x € P n R ( t l
,t
> O
P
, where
be as i n Lemma A .
Ox =
(a/axlJ...,a/axnl
Then
The proof of Lemma A will be carried out in the next section.
Corollary A
follows immediately from Lemma A and the strong maximum principle, since satisfies the heat equation Bt = A0
in the water region n(t)
, t>O
.
0
For
the proof of 'Lemma B, see [ 5 ] .
Proof of Theorem 1.
For simplicity we assume
n t 3
.
The case n = 2
follows from a similar ( but slightly modified ) argument. Let
where
be as in Lemma A and set
R >O 0
co
denotes the closed convex hull of a set. By virtue of Lemma B,
there exists a positive number To r(t)nw for all
t 2 To
.
=
0
such that
.
44
Hiroshi MATANO Take any
tOEITo.+=)
and fix it.
And let xo be any point on r(to).
In order to apply Proposition 3 . 4 , we must check that density with respect to the region Rn\a(t ) . 0
constructing an open cone KO
with vertex
xo
has positive Lebesgue
This will be shown by
xO.
Put
K~ = where
-
z E I R ~I (z-xO)*(y-xO) <
denotes the Euclidean inner product in Rn
convex set and since xo that is, Kg Let
{
6W
, KO
o
for all Y E W } ,
.
Since W
is a compact
is a non-empty open cone with vertex
is an open set such that r(KO-xO) = K O - x O for all
xo;
.
r >0
z be any point in KO and set
where
5 = z
find that
- xo .
Applying Lemma A to P = P
8(xO+aC,t0)
for all a 2 0
.
for each a 2 0 , we easily
is monotone non-increasing in a t 0 , Consequently
This shows that
8
vanishes everywhere in KO
, hence
xO
has positive Lebesgue density with respect to Rn\\n(t). The conclusion of Theorem 1 now follows by applying Proposition 3 . 4 . Let W
Proof of Theorem 2.
for each hence
rc
E >
0
.
is an
By virtue of Corollary A,
V e(x,to)
does not vanish on
rE
(n-1) -dimensional smooth hypersurface. Moreover, as is
easily seen, rEfl aW
= a if
any ray inward normal to intersects the set W E +
be as in (19) and set
.
r
F.
is sufficiently small. We shall show that
( i.e. the direction parallel to
Vxe(x,t,)
)
The conclusion of Theorem 2 then follows by letting
0 and using the fact that
8(x,t0)
is smooth in Q(to)
up to the
'
Asymptotic Behavior of the Free Boundaries
145
boundary '(to).
r E and let L be a ray inward normal to rc at
Let x 1 be any point on x1
.
Assuming that
enw
(20)
=a,
we shall derive a contradiction. Denote by &
5,
a non-zero
n-vector parallel to
By the definition of
&.
, we have 51
(21)
for some c > O
.
x
Since W
exists an n-vector ( 2 2 a)
= cv 6(x,,t0)
5
is a compact convex set, ( 2 0 ) implies that there
such that
5.(y-x1)
<
0
for all Y E W
,
Arguing as in the proof of Theorem 1, we see from ( 2 2 a) that non-increasing in the direction of monotone non-increasing in C . V e(x
x
hence
<*cl
ct2
1
,t ) 0
0 2
. 0
2 0 by virtue of ( 2 1 ) .
5 ; more precisely,
6
is monotone
6(x,+ag,t0)
is
In particular, we have
, But this clearly contradicts ( 2 2 ) , thus
showing that the supposition ( 2 0 ) is false. This completes the proof of Theorem 2 .
We omit the proof of Corollaries 3 , 4 and 5 .
5. Proof of Lemma A . In this section we prove Lemma A , which, although simple, played a key role
Hiroshi MATANO
146
in the proof of the main theorems. We begin with some auxiliary lemmas:
Lemma 5.1.
Let
, e(x,t)
8(x,t)
be solutions t o the following i n i t i a l -
boundary value problems:
I et
ZP\c1 , t > O ,
X E
=
ae’
( e = g ,
XEaG,
e = a -e , -t e(x,Ol
=
,t>O,
xEG\zI, hlx)
,
xEc\G1 X E
t > O ,
,
acl , t > o .
Then B(x,t) t
-e ( x , t )
0(X,t)
5 B(x,t)
Recalling that IRn\G1 Dn(t) = G 0 = 0
on
,
T(t)
for
x:En\Gl’
for
xEG\Gl
El
for
t2 0
t > O
,t
L O
,
.
and that
Bt
= A8
one can easily verify the assertion of Lemma 5.1.
in n(t),
This lemma
is a special case of a more general comparison principle in one phase Stefan
problems; see C5;Lemma 2.51 and its subsequent remark. ( 2 3 ) ( resp. ( 2 4 ) ) derives from the Stefan problem ( 1 ) ( resp.
k =
-
G1 c
Assume (71, and l e t G cc2cG 2
.
61 5
,t t0
e(x,t)
.
G2
( 6 ) if one sets k = 0
be a domain with a smooth boundary
Then there e x i s t constants
that
f o r aZZ x E aC2
-
) in ( 5 ) .
Corollary 5 . 2 .
satisfying
Suffice it to say that
=
<
MI
61 > 0
,M1 > 0
such
Asymptotic Behavior of the Free Boundaries
Corollary 5.3.
Assume
n23
and t h a t (7) holds.
Then there e x i s t s a p o s i t i v e constant
5.2.
where
M2
Let
147
be as i n CoroZlary
G2
such t h a t
i s t h e s o l u t i o n t o t h e problem
$
i
o
A# =
on
$ = 1
lim $ ( x ) = 0 lxl+-
I n p a r t i c u l a r , we have
,
i n IR'\G~
,
aG2
.
lim e ( x , t ) = 0 uniformly i n t >0 1x1 + -
.
The proof of these corollaries are straightforward, so w e omit it. that the assumption
n t 3
in Corollary 5.3 cannot be dropped, since the
n
problem (25) has no solution in the case
Lemma 5 . 4 . 8
= au
f, and
Proof.
where
Let
u
E '
E >
.
BE
are chosen appropriately.
Define
B(s)
BE
= 1
or
n = 2
.
0 , be t h e s o l u t i o n t o t h e problem (15)€ and s e t
/at
Then
Note
uE + u
, eE
+.
0
decreasingly as
by
is a function with the properties B(s) = 0
if
s>1
,
E
4 0
, provided
that
Hiroshi MATANO
148 It is clear that fE
BE
has the properties required in Section 3. Next we choose
appropriately so that
EATI + fE decreases as
the conditions listed above ( 1 5 ) € . The decreasing property of of I$ +
E,
rlE
decreasing as
E
i
0
0 and that
The existence of such ( as
u
, fE and -BE(s)
3
E
.I
E
.
( s>O )
fE satisfies
fE is obvious.
0 ) follows immediately from that
eE is
In order to show that
, we
differentiate ( 1 5 ) c by
-A0
+ 2 + kB'(u
t
, to
get
ae
at
E
E
BL(uE)
eE
.
,
e = o
on
a B R x (0,T)
.
E
that the convergence O C
5
3 0 ; +.
0 and since u
3 0
E
the initial data
.
Let u
is decreasing as
E
3 0
,
BE
9(x,O)
.
for some x E G
be as in Lemma 5.4.
;
in
.)
By virtue of Corollary
such that
By a similar argument, we see that there is
a positive constant, again denoted by
E
,
( Note
does not take place everywhere on D x [ O , T ]
9
i s as in Corollary 5.2.
for any sufficiently small
decreases as
this completes the proof of Lemma 5.4.
5.3, there exists a positive constant R,
h1
9 (x,O)
Applying the maximum principle, we see that
fact, 8 (x,O) does not converges to
where
,
(0,T)
is increasing as
Proof of Lemma A .
in D x ( 0 , T )
0
aG, x
Moreover, since B"
is decreasing as
E
on
By virtue of the assumption above, J. 0
)€I=
g
BE =
E
E
R,
,
such that
149
Asymptotic Behavior of the Free Boundaries Now take a positive constant Ro
contains both dist(P,G,)
and
G
2 Ro
such that the set
x E Rn 1 1x1 5 R, 1
{
; and set
D'
{ xE
=
+
S
.
And let P be a hyperplane with
A
1
x E D 1 , where x
x , S+
are as in
We shall show that
(16).
for all x E D' \G2
, where
G2
is as in Corollary 5.2.
Put v(x,t)
The functions v
,w
(x,t)
= u
-
UE(X
x ,t) ,
satisfy the degenerated parabolic. system
(28 a)
wt = Aw
-
kB'(us)w
x
-
kOf(x ,t)S(x,t)v
in
(D' \G2)
x
(0,T)
in
(D'\G2)
x
(0,T)
together with the initial and the boundary conditions
o
v 2
(28 b)
i
in D ' \ G ~ ,
w 2 0
in D'\G2
w 2 0
on
a
,
(D'\G2)
x
(0,T) ,
where
Note that the last inequality in (28 b) follows from ( 2 6 ) , Corollary 5 . 2 and the decreasing property of
6
( Lemma 5 . 4 ) .
Since 6" 5 0
, we
have
Hiroshi MATANO
I50
It is now clear from ( 2 8 ) , (29) and the maximum principle that w 2 0 in (D'\G2)
a.e. in
[O,T]
x
, which implies that (27) holds. Letting
(D'\G2)
x
[O,T]
;
hence (30) holds everywhere i n
virtue of the continuity of
Considering that T
0 , we get
(D'\G2)
and R
x
[O,T]
by
( where
can be chosen arbitrarily large, we see that ( 3 0 ) holds for all
D = BR\Gl)
x E S+\G2
0 ([2]).
E J.
, t20
.
Thus the former part of Lemma A is proved. The latter part
is now obvious from the strong maximum principle.
This completes the proof of
Lemma A.
Acknowledgement The author would like to express his gratitude to Professor Ei-Ichi Hanzawa for stimulating discussions.
References
111
Caffarelli, L.A.,
The regularity of free boundaries in higher dimensions,
Acta Math. 139 (1977) 1 5 5 - 184.
[Z] Caffarelli, L.A. and Friedman, A . ,
Continuity of the temperature in the
Stefan problem, Indiana Univ. Math. J. 28 (1979) 5 3 - 70. [31
Duvaut
,
G.,
REsolution d'un probleme de Stefan ( Fusion d'un bloc de glace
1 zero degr6 ) , C. R. Acad. Sci. Paris 276 (1973) 1 4 6 1 - 1463. [41
Friedman, A . ,
Partial differential equations of parabolic type ( Prentice-
Asymptotic Behavior of the Free Boundaries
151
Hall, Englewood Cliffs, 1964 1. [5]
Friedman, A. and Kinderlehrer, D., A one phase Stefan problem, Indiana Univ. Math. J. 24
[6]
(1975)
1 0 0 5 - 1035.
Gidas, G., Ni, Wei-Ming and Nirenberg, L., Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68 (1979) 2 0 9 - 2 4 3 .
[7]
Jones, C. K. R. T.,
Asymptotic behavior of a reaction-diffusion equation
in higher space dimensions, Technical Report, Inst. Appl. Math. Stat. Univ. British Columbia ( Oct. 1980 1. [B]
Kinderlehrer, D. and Nirenberg, L.,
The smoothness of the free boundary
in the one phase Stefan problem, Corn. Pure. Appl. Math. 31 (1978) 2 5 7 282. [9]
Serrin, J., A symmetry problem in potential theory, Arch. Rat. Mech. Anal. 43 (1971) 3 0 4 - 3 1 8 .
This Page Intentionally Left Blank
Lecture Notes in Num. Appl. Anal., 5, 153-170 (1982) Noniinear PDE in Applied Science. U.S.-JupanSeminur, Tokyo, 1982
INITIAL BOUNDARY VALUE PROBLEMS FOR THE EQUATIONS OF COMPRESSIBLE VISCOUS AND HEAT-CONDUCTIVE FLUID
Akitaka Matsumura
and
Takaaki Nishida Department of Mathematics Kyoto University Kyoto 606, Japan
Department of Applied Mathematics and Physics, Kyoto University Kyoto 606, Japan
The equations of motion of compressible viscous and heatconductive fluid are investigated for initial boundary value problems on the interior and exterior domain of any bounded region and also on the half space. The global solution in time is proved to exist uniquely and approach the stationary state as t + +m, provided the prescribed initial data and the external force are sufficiently small.
5 1.
Introduction The motion of a compressible, viscous and heat-conductive Newtonian fluid
is described by five conservation laws:
where
x
=
velocity, 1
(x1,x2,x3)
E
3
R ,
0 is the absolute temperature, p 2
3
f = (f ,f ,f ) coefficients,
i s the external force, K
= K(p,O)
1
t > 0, p is the density,
u
=
p(p,O)
= p(p,O)
2
3
is the
u = (u ,u ,u )
is the pressure, and
u'(p,O)
are viscosity
is the coefficient of heat conductivity, c = c(p,O)
is the heat capacity at constant volume and
153
4' ' =
2
(ui x
j
.
+ uJ )
xi
2
+ p'(uk
l2 Xk
Akitaka MATSUMURA and Takashi
154
NlsHlDA
is the dissipation function. First we assume the following basic assumptions on (1.1): (A.l)
The external force fi i f =-@
is generated by a potential function 0(x),
i.e.,
.
X.
1
(A.2)
u, p', V,
K,
p and c are smooth functions of
K, C , P r
V'
Pp, Pe > 0 ,
2 +
3
u
p , 0 > 0 , and
2 0.
We consider the system (1.1) in the following domain R. Case 1.
R
=
R1
:
a bounded open set in
Case 2 .
R
=
R2
:
compliment of
R1
R3
with
m
C
or the half space
boundary 3
R+
=
{x
an. E
R
3
, x3
> 0).
For both cases we discuss the existence of a global solution in time under the suitable initial and boundary conditions since the local existence theorem is established by Tani [141 under full generality.
I n this note we shall sumarize
the results in [91 and [lo]. Case 1.
(Interior Problem)
Let us consider the system (1.1) in
R1 with the
initial condition
and the boundary conditions
where
8 i s any fixed positive constant. In order to state the theorem
precisely, let u s list up further assumptions:
(~.5)
(compatibility condition) E
uO
H~(R),
a.
-
3 E H,(1R),
155
Compressible Viscous and Heat-Conductive Fluid +
j
-eo(po)ouo ,x
i
where
po
=
H R (R) = {f
x
in
V(n)
where
(C2(G))’
J
t Yo<
Hg(n), 1
J
R
R
“
L2 (R), Dkf=(alalf/axl’ ax2 “2 a x a3 3 , l a ] = k }E L 2 (a),1 6 k 5
t
with the norm Ct(R)
6 ) 0 0 , x . x.
~ ( p ~ , 6 ~ )uo , = ~ (0 ,6 p ) , . - - , and so on. Here H (0) denotes the 0
Sobolev’s space on
of
(K
11 f /I9. = ( 1 H (a). Define
1 lDkf 12dx)2 , and a positive constant
represents the volume of
R. We call
H
1
0
p
!?.I
(n) denotes the completion by
(p(x),
u^(x), ;(x))
an equilibrium state of the problem (1.1)(1.2)(1.3)1
E
C’(3)
---.
when ( p , u , O )
satisfies (1.1) and the additional conditions
Then we have Under the assumptions ( A . l ) ( A . Z ) ( A . 3 ) ( A . 4 ) , ( A . 5 ) , positive constant E, a and C such that if /I p0-P, uo, Oo-SI), + Theorem 1.
the problem (1.1)(1.2)(1.3) unique equilibrium state
there exist
l1Q)l4
6
has a unique global solution in time (p,u,e)
1 -. (p,u,6) = (G,O,z)
E,
then
and a
satisfying
and
Case 2 .
(Exterior or Half-space Problem)
In this case let u s consider the
system (1.1) in Q2 with the initial condition (1.2) and the boundary conditions
IS6
Akitaka MATSUMURA and Takashi
(1.3)~
where
p
u(t,x)
= 0,
u(t,x)
+
-
and
0,
e(t,x) =
3
e(t,x)
3,
-+
for
x
p(t,x)
E
an,
+
5
NlSHlDA
t 2 0,
as
In this case we call
(1.3)2 and (p-p,
t 2 0,
.u, 0-0) 5
E
H
2
(G(x), z(x), G(x))
. 5 - . .
an equilibrium state of the problem (1.1)(1.2)(1.3)2 -
for
+m
are any fixed positive constants. Let us assume that
0
and the compatibility condition (A.5).
5
1x1 +
when (p,u,e) satisfies (1.1)
(n). Then we have
Under the assumptions (A.l)(A.Z)(A.3)(A.4),(A.5), a positive constant E such that if )I p0-p, uo, 0,-%/\, + 1)
there exist
Theorem 2 .
problem (1.1)(1.2)(1,3)
2
unique equilibrium state
(u,e-g)E
@
\I5
5
E,
then the
has a unique global solution in time (p,u,B) and a . z c
(p,u,O)
0
C (O,+m;
=
(z,O,x)
H3(Q)
fl
satisfying
H:(R))
fl
1 C (O,+m;
1
Ho(Q)),
and
The proof of both Theorems are given by an energy method similar to those of our previous papers C71C81 on the initial value problem.
However the initial
boundary value problems require a new a-priori estimates of the solution near the boundary.
In the following sections, we shall show rough sketch of the proof
only for the half space case because the other cases are proved along similar strategy. Details are to be appeared in c101. Finally we should mention that f o r one-dimensional model system of (1.1) we can see more precise results in
Kawashima-Nishida C51 and Okada-Kawashima ClllC121.
Compressible Viscous and Heat-Conductive Fluid §
157
Stationary solution
2.
Let us write the equations and conditions for the stationary solution );,;,(;
:
(2.1)
(2.3)
where
..
-.
p = p(p,8)
etc.
The stationary problem (2.1)-(2.4)
has a unique solution
as Lemma 2.1. and
C
Under the assumptions A . l
such that if
unique solution
)I @ I l k
(;(x),O,e)
- A.3
there exist positive constant
R = 3,4, or 5, the problem (2.1)-(2.4)
5 E,
in a small neighborhood of
(p,O,B)
in
E
has a
(H2)3
satisfying
where
Proof.
-p(x)
is determined by (1.8),
i.e.,
Ip-pl, 1;,
Sobolev's lemma we may suppose
I ;-TI
estimate the equalities:
(2.7)
(p,O,%)
Since we consider a small neighborhood of
j
c2.13 x
I
C2.21'
j ; e r)
X
x d ' ;
= 0,
dq dx
= 0,
<
1
5 min{p,8).
in
(H2)3, by Then we can
Akitaka MATSUMURA and Takashi NISHIDA
158 C2.31
X
(&8)dx
=
0,
where C2.11, C2.21' and C2.31 denote the terms on the left hand side of ( 2 . 1 ) , (2.2)i and (2.3) respectively. Take the sum of (2.7) and integrate it by parts using the mean value theorem and Lemma 4.1.
;=e.
;=o,
(2.8)
We obtain the inequality:
If we substitute (2.8) in ( 2 . 2 ) , we have
which implies ( 2 . 6 ) .
§
3. Local and g l o b a l existence Let u s rewrite the problem ( l . l ) , ( 1 . 2 ) by the change of variables
(p,u,e)
(3.1)'
+
3)
( p + G,u,0 +
using ( 2 . 6 ) as follows:
Lu(p,u) 0 5 pt +
+
UJPX
j
4
L
(u,o)
z
et
-
Kex.x ~j
PUJX. '
f0 ,
=
I
+
' ,J 3 x
where we denote the constant for the function g
= f
4
.
j
of
P
and
0
by
__
= g(P,e),
Compressible Viscous and Heat-Conductive Fluid and a l s o
=
_- --
p3 = ePg/pc
.
i/z, G'
- = U'fP,
___
A
= KfPC,
K
- -
PI = P p / P ,
P2
- Pel0
159 and
The terms on the right hand side of equations (3.4) are nonlinear
and have the form:
(3.4)
Next we choose
a
constant
Then the solution of (3.1)-(3.3) f o r some
where
E 5 Eo
, where for 0
Eo
by use of Sobolev's lemma such that
is sought i n the set of functions X ( 0 , m ; E) S tl 5 t2 <
m,
we define
Akitaka MATSUMURA and Takashi NISHIDA
160
Here and in what follows we do not write R
9. in H (Q).
We will obtain the global solution by a combination of a local existence theorem and some a priori estimates for the solution in X, namely that for the norm N. Proposition 3.1.
(local existence)
Suppose the problem (3.1)-(3.3)
has a unique solution
for some h 2 0 and consider the problem (3.1)-(3.3) exist positive constants T, that if N(h,h),
( 1 Q /I5
2
E~
E~
(p,u,O)
E
for
and C o ( ~ o m5 Eo)
(p,u,O) t 2 h.
E
X(0,h; E O )
Then there
independent of
h
such
, the problem has a unique solution X(h,h+r; CoN(h,h)).
The proof is the same as that for the interior problem in C91 and is omitted. Although the local existence theorem by Tani C141 is more general, we need it in the form of Proposition 3.1 to extend the solution globally in time by use of L
2
energy method. proposition 3.2.
(a priori estimates)
Suppose the problem (3.1)-(3,3)
has a solution (p,u,8)
given h > 0. Then there exist positive constants which are independent of h
such that if N(O,h),
El
and C1
11 0 \ I 5
Z
E
for
X(0,h; Eo)
(El E~
2 Eo,
E
~ 5C Eo) ~
, it holds
If Proposition 3.1 and 3.2 are known, the global existence of unique solution can be proved as follows: Choose the initial data potential function 0
so
small that it holds
(po,uo,Oo) and the
Compressible Viscous and Heat-Conductive Fluid N(O,O)
2 min{E 0’ cl/CO, E
~
/ 1
and C
16
I/~ 0 / I 5 ~ 5
Then Proposition 3.1 with h=O gives a local solution (p,u,O)
t
E~ ,
X(0,T; CON(O,O)).
Since C N ( O , O ) 5 c1 L EO , Proposition 3.2 with h=T implies N(O,T) 5 C N ( 0 , O ) . 0 1 Then Proposition 3.1 with h=T implies the existence of solution (p,u,O)
Hence, since gives
4
N(0,ZT)
C
N(O,T)
N(O,O), 1
E
X(T,~T;C0N(r,r)),
6
X(0,zT;
CIJ<
2
N(0,T)).
2 N(0,O)
5 E~
, Proposition 3.2 with h
and Proposition 3.1 with
h
=
2T
=
gives
Repetition of this process yields Proposition 3.3.
(global existence)
There exist positive constants
I/ 0 ) I 5
2
E,
E
and
C(EC 2 Eo)
such that i f
then the initial boundary value problem (3.1)-(3.3)
solution (p,u,e)
§
4.
E
x(o,~; CN(O,O)).
A priori estimates for the half space case
R
3
= R+ ,
First we recall some inequalities of Sobolev type. Lemma 4.1.
It holds
N(O,O),
has a unique
2.r
Akitaka
I62
Proof.
MATSUMURA
and Takashi NISHIUA
See for example C31, C41.
Next we note some estimates of elliptic system of equations for our domain, when we regard equation (3.1)i,
i
=
1,.--,4, as elliptic with respect to x
variables, i.e.,
u/
=
an
Lemma 4.2.
u/
8) = 81 aR m
= m
We have for k
= 0.
2,3
=
The first estimate is well known, e.g. [l].
The second one i s given in C51.
The last estimate for an elliptic system concerns Stokes equation in
i
which comes from (3.1) ,
-
pu'
=
X.
i = 0,s.. , 3 *
h ,
J
(4.5)
^ i
-~
u
+
~j
i pipxi ~ ,= g ~ ,
uIm = 0.
u I a R = a,
Lemma 4 . 3 .
For k
i
=
2,3,4
it holds
=
1,2,3,
R
Compressible Viscous and Heat-Conductive Fluid
163
where t h e l a s t term on t h e r i g h t hand s i d e i s n e c e s s a r y i n t h e c a s e of e x t e r i o r domain.
Proof.
See f o r example Solonnikov C131 and C a t t a b r i g a c21.
Now we begin t o o b t a i n the energy e s t i m a t e f o r s o l u t i o n of e q u a t i o n (3.4)i ,
i
=
0,.*-,4, with ( 3 . 5 ) .
where
(4.9)
Proof.
Compute t h e i n t e g r a l
I n t e g r a t i o n by p a r t s u s i n g t h e boundary c o n d i t i o n g i v e s
Akitaka MATSUMURA and Takashi NISHIDA
164
where AO
is defined by ( 4 . 8 ) .
obtain (4.7),
P.
=
0
If we use the notation
dpldt
in ( 4 . 8 1 , we can
from this equality. The time derivative can be treated
similarly, because it has the same boundary conditions. Next compute the integra1
Integration by parts gives by use of Schwarz inequality
where
f
=
(fo-ujp
1
2
, we obtain (4.10),
pu;,
3
4
, f ,f ,f , f ) . 2!, = 0 .
If we use Schwarz inequality for the term P.
The estimate ( 4 . 9 ) ,
=
1
is obtained
2
similarly. Since the tangential derivatives of the solution of (3.1) satisfy the same boundary conditions (3.2)(3.3), to the above lemma 4 . 4 .
we can obtain the estimates for these similarly
Let us denote the tangential derivatives by
a=(ax
,ax 1
and integrate the equality for each k = 1,2,3
”z - ak(LO P
Thus we have
-
fO)akp
by use of integration by parts
ak(Li - fi)akui
=
0,
2
165
Compressible Viscous and Heat-Conductive Fluid
where ' 1 k k O
-
a
'
- u J p X )dx
pa ( f
for each
k
=
i,z,3.
j
Then we have to obtain the estimates for the normal derivatives of solution. To do that we use the following equations from (3.1).
If we eliminate the term u from these, we have x3x3
(4.13) 3
A
+ U(UX
1 1
+ u3 ) + j'(ul + u2 ) x2x2 Xl x2 ,3
where we note the second derivatives of
,
u at the last two terms on the right
hand side contain one tangential derivative. Multiply (4.13) by (dpldt)
and 3 respectively and integrate them respectively. We obtain after intep,
x3 gration by parts
=
+
1
dx + -po,x3
:j 1 +
3 {-ut
-
e
1; 1 +
x3
A
A
-( - u i
p,
j
G(U~
+ u
x 1x 1
2b1;' f," + f3)px dxdt ( 7 3 0 3
3
+ 2 u j p, ) p x dxdt 3 j 3
+ u
x2x2
2
166
and
respectively. Thus we have obtained the following Lemma 4.6.
For
k
t 9, =
0,1,2
it holds
where
and here the summation is not taken for k
and
R.
Last we use lemma 4 . 3 for Stokes equation ( 4 . 6 ) with and
g
i
have the following explicit f o r m s .
u
Ian
= O,
where
Compressible Viscous and Heat-Conductive Fluid
167
(4.15) gi -
--u
Lemma 4.7.
For
k
i
A
h
ptiJ'
8
t -
+fl,
i = 1,2,3.
0,1,2, we have
t
R
=
I( D2"aku ( 1
t
1 1 D1+'aki, 11
(4.16)
Now we can combine the above lemmas 4.1
- 4.7 to obtain necessary a-priori esti-
mates. Although we omit the details, by combining step by step lemma 4.4, K.
=
k+R
0, k =
=
0, Q
=
1, lemma 4.2 for
k
0, lemma 4.2 for u ,
k = 3,
K.
=
lemma 4.5, k
=
2,
0, lemma 4.6, k
=
0, R
the
H2 version of norm
6, k
2
lemma 4.5,
2, lemma 4.7, k+R
=
lemma 4.6, k =
=
1 and
N(O,t),
=
1,
Q
lemma 4.7,
=
=
k = 1,
0, lemma 4.2 for
0, lemma 4.7,
k
=
lemma 4.6,
0, R
=
k
=
0,
1,
1, we can obtain
i.e,
To elevate the differentiability once to obtain the estimate of norm
we can repeat the above argument beginning from lemma 4.4, lemma 4.2, ktR
=
2.
k = 3,
lemma 4.5, k
=
3, lemma 4.6,
k+R
=
Therefore we arrive at the estimate for N(0,t).
k
=
1
N(0,t)
and by use of
2 and lemma 4.7,
168
Last we have to show Lemma 4 . 8 .
It is proved by use of lemma 4 . 1 and integration by parts. We show o n l y the term A.
and omit the proof of the other terms which can be treated similarly.
Let u s recall ( 4 . 9 ) and compute the following
Compressible Viscous and Heat-Conductive Fluid
The remaining terms in .A
169
can be treated in the same way as above.
Finally we note that the inequalities (4.18)(4.19) a-priori estimates, in fact, we may choose
E~
easily imply the desired
so small that
Thus the proof of Theorem 2 for the half space is completed.
References c11 Agmon, S., Douglis, A. and Nirenberg, L . , Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Corn. Pure Appl. Math., ,lJ(1964) 35-92. c21
Cattabriga, Su un problema a1 contorno relativo a1 sistema di equazioni di Stokes, Rend. Mat. Sem. Univ. Padova,
C3l
2
(1961) 308-340.
Finn, R., On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rat. Mech. Anal.,
2
(1965) 363-406.
c41 Heywood, J.G., A uniqueness theorem for non-stationary Navier-Stokes flow 6 (1979) 427-445. past an obstacle, Ann. Scuola Norm. Sup. Pisa lV C51 Kawashima, S. and Nishida, T., Global solutions
to
the initial value problem
for the equations of one-dimensional motion of viscous polytropic gases, J . Math. Kyoto Univ.,
21 (1981)
825-837.
Akitaka MATSUMURA and Takashi NISHIDA
170
C6l Ladyzhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. C71 Matsumura, A., An energy method for the equations of motion of compressible viscous and heat-conductive fluids, Univ. of Wisconsin-Madison, MRC Technical Summary Report #2194, 1981.
C81 Matsumura, A. and Nishida, T., The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A , C91
55
(1979) 337-342.
....., Initial boundary
value problems for the equations of motion of
general fluids, Proc. of 5th Internat. Symp. on Computing Methods in Appl. Sci. and Engin., Dec. 1982, INRIA, Versailles, France, (1982) 389-406. ClOl
....., Initial boundary
value problems for the equations of motion of
compressible viscous and heat-conductive fluids, to appear in Communications in Mathematical Physics.
Clll
Okada, M. and Kawashima, S., On the equations of one-dimensional motion of compressible viscous fluids, t o appear in J. Math. Kyoto Univ.
C121
.....,
Smooth Global Solutions to the one-dimensional equations in
Magnetohydrodynamics, to appear in Proc. Japan Acad. C131
Solonnikov, V.A., A priori estimates f o r certain boundary value problems, Sov. Math. Dokl.,
C141
z (1961) 723.
Tani, A., On the first initial-boundary value problem of compressible viscous fluid motion, Publ. RIMS, Kyoto Univ.,
13 (19771
193-253.
Lecture Notes in Num. Appl. A n a l . , 5 , 171-188(1982) Noiiliizeor PDE it7 Applied Science. U . S .- J q m i Semintir. Tokyo. 1982
Integral Representation of Solutions f o r E q u a t i o n s o f Mixed Type i n a H a l f Space
Sadao bliyatake Department o f Mathematics Faculty of Science Kyoto U n i v e r s i t y K i t a s h i r a k a w a Sakyo-ku Kyoto 606 JAPAN
We c o n s i d e r boundary v a l u e problems f o r e q u a t i o n s o f mixed t q(x)utt = 0 i n a h a l f s p a c e ( 0 , w ) x R . IVe type u discussxfhe existence of solutions written i n t h e i n t e g r a l form u ( x , t ) = [eATt v ( x , T' ) g ( - r ) d f , t h e e s t i m a t e f o r v ( x , d ) and f u n c t i o n s p a c e s f o r g ( t ) and u ( x , t ) . We p u t W ~ X . cf.)=v'(x, c c ) / v ( x , c i ) and c o n s i d e r n o n - l i n e a r e q u a t i o n o f t y p e w ' = q ( x ) d - w 2 i n Riemann s p h e r e , where a t o p o l o g i c a l method i s u s e f u l . As f o r e s t i m a t e s we n e e d a s p e c i a l d e v i c e c o n c e r n i n g t h e e n e r g y method.
81. I n t r o d u c t i o n and s t a t e m e n t s o f r e s u l t s . . T h i s n o t e i s c o n c e r n e d w i t h t h e f o l l o w i n g problem
c
(PI
uxx + q ( x ) u t t = 0 u(0,t) = p(t)
where t h e c o e f f i c i e n t (C)
for
( x , t ) E ( 0 , ~ )X R
l i m u(x, t) = 0, x+m
d e p e n d s o n l y on
x
for t
€ R,
and s a t i s f i e s
q ( x ) is a r e a l v a l u e d p i e c e w i s e c o n t i n u o u s bounded f u n c t i o n s a t i s f y i n g
Remark t h a t
q(x)
may change i t s s i g n .
The boundary d a t a
Namely
g(r)
q(x)
and
,
g(t)
g(t)
i s assumed t o h e w r i t t e n i n a form
i s a summation o f e x p o n e n t i a l f u n c t i o n
d e f i n e d on complex p a t h
p.
rwill
eiTt
he d e f i n e d l a t e r .
with d e n s i t y function Naturally t h e
f o r m u l a (1) i s n o t a n y t h i n g h u t L a p l a c e i n v e r s i o n f o r m u l a i f g ( t )
171
and
r
satisfy
172
Sadao MIYATAKE
suitable conditions.
(P)
From t h e l i n e a r p r o p e r t y o f
we s e e k t h e s o l u t i o n
g i v e n by u(x, t ) = Here
v(x,K) is
2 2 x SreiTtv(x,
vxx
q(x)
-
V(O,K) = 1
r
2
2
cc = T
{ ~ d: = T2,
=
.
s o l u t i o n of
il
We hav e de note d
't2)a(t)dT
.
I:
Thus
tE
x E( 0 , ~ )
for
and
v(00,
a)
= 0.
i s a c u r v e i n complex p l a n e s u c h t h a t
i s i n v o l v e d i n t h e f o l l o w i n g domain $ ( q )
which we
d e s c r i b e i n Theorem 1
Theorem 1 .
(p)
The problem
OI} ,
(-%
dj,
where
<
(0
<
dl
d2
changesits sign, while
each
d.
1
for
&
If
I1 { d . ) i s r e p l a c e d by empty s e t j=1 J
q(x) 2 0
If
In c a s e where v ( x , -r2 ) = e -
q(x)
a r e p o s i t ve c o n s t a n t s t e n d i n g t o
r
q(x) 2 0
if
a(q) f o r each
x E [0,
q(x)
in
m).
i s bounded i n a n ei g h b o u r h o o d
Finally
v(x,&)
for
x E (0, oo),
q(x)
ITIX
E
1,
?(t) =
and
we can t a k e
T
a s real
( 2 ) becomes Po isso n f o r mu l a by F u b i n i t h eo r em
changes i t s s i g n , we t a k e
For example
~l E
if
00
> 0.
Remark 1.
with
...)
being a simple pole.
of K = 0
axis.
<
V ( X , K ) is analytic in
Moreover
(0,m ) .
has unique s o l u t i o n
a belongs t o t h e following s e t
i f and o n l y i f
u(x,a)
(C).
Suppose
,j T'
O3
-'"R ( s ) d s -
a s a curve s a t i s f y i n g
r2 c
B(q) U { O }
c o i n c i d e s r e a l a x i s i n a n eig h b o u rh o o d of o r i g i n and a n o t h e r
p a r a l l e l l i n e i n a neig h b o u r h o o d o f i n f i n i t y . Next we e s t i m a t e
v(x,d)
i n o r d e r t o g i v e e x a c t meanings t o t h e f o r mu l a
(2) i n function spaces. Theorem 2 . constant
Cy
Suppose
(C).
s u c h t h a t we h av e
Then f o r any
y
7 0,
t h e r e exists p o s i t i v e
Integral Representation for Equations of Mixed Type
2 Cr = O ( l / l )
I n g e n e r a l we w i l l s e e By Theorem 2
t h r o u g h t h e p r o c e s s of p r o o f s .
we c a n d i s c u s s f u n c t i o n s p a c e s f o r
A t f i r s t we i n t r o d u c e f u n c t i o n s p a c e s
..
B
L2d,k
g(t)
and 1
, I, y , k ’
u(x, t ) . *
-2
Y,k ’
L1y,k
with following norm r e s p e c t i v e l y .
X, k
l lh I i P
r,k
=
2
j=l
{
( O -m
(k If
173
0
<
x1 <
y,
,
then it follows
T=Tl
Then we f i x a smooth p a t h
Denote by
g(t) =
-T
&jfe
= 0, 1 , 2 ,
+
r2 + I-,
+
r4
,
where
r l = {T : I m T = 0 , - a 5 ReT‘6a } , a c d l 1 / 2 rz = 1 1 : - I r n ‘ f = Y , I R e T I 2 2 a ) , r3 c I f : O < - I r n t < Y , a < R e 7 < 2 a ) , r 4 c { t : 0 < - I m t < # , -2a < R e f < - a } . t h e curve
itt
g(O) d T
IS.
.
:
?€TI
Remark t h a t
Now l e t us p u t
.
. .),
and
Sadao M ~ Y A T A K E
174
Using t h e s e n o t a t i o x w e can s t a t e Theorem 3 . 1 Ly, k+4)
g(-t) E
&k
, (resp.
If
(C).
b' >
for certain
:-(x,
k 2 2
In c a s e of
g(t)
0 ,
then
t) = u-(x, -t) E
,
,
u + ( x , t)
(resp.
Y,k
...
K = 0, 1 , 2 ,
Remark 2 .
and
C . ( g , k) 1
'The a n a l y t i c i t y of
i s i n v a r i a n t even i f we change
.
and i R - { O )
u+(x, t )
) ,
belongs t o
(k = 0, 1, 2 ,
.. . ) . (P).
s o l u t i o n of
C'
T
g(t)
i(t) =
by
v(x, T L )
says t h a t
2 v ( x , t ), e
Let
i s t h e boundary of a bounded domain
is a given a n a l y t i c f u n c t i o n . u(x, t )
i (id.1 / 2 ) t J
g(-t). u(x, t )
(P)
In f a c t , since
with
B
1
be smooth c l o s e d
.
Then
g 5 0,
T = + d.1'2 - J
(2)
d .1/2
r
if
u(x, t )
of
E(T)
i s a simple
i s d e s c r i b e d by r e s i d u e c a l c u l u s .
aij(x)
in
i n a bounded domain which excludes
< s i n g e n e r a l non t r i v i a l s o l u t i o n of
i s of type
G-(x, t ) = u-(x, - t )
a r e constants.
I n c i d e n t a l l y w e g i v e a n o t h e r remark.
curve such t h a t
p o l e of
(respectively
kt4,
u - ( x , t ) ) is a
s a t i s f i e s t h e same i n e q u a l i t i e s r e p l a c e d
(2)
1 E Ly,
s a t i s f i e s following e s t i m a t e s i f r i g h t hand s i d e s a r e bounded.
u+(x, t )
where
Suppose
This s o l u t i o n
and does n o t belong t o f u n c t i o n s p a c e s
j
-2 Lr,k
i n Theorem 3 . As f o r e q u a t i o n s of mixed t y p e we know t h e Tricomi e q u a t i o n , f o r which w e r e
i n v e s t i g a t e d u s u a l l y l o c a l s i n g u l a r boundary v a l u e problemssuch as Tricorni
175
Integral Representation for Equations of Mixed Type problem and Frank1 pro b lem .
I n t e n s i v e r e f e r e n c e s a r e found i n [ l ] , [ 2 ] , and
(P)
G loba l t r e a t m e n t s as o u r p r o b lem
131.
c o u l d n o t b e found by t h e a u t h o r .
The method i n t h i s p a p e r i s c o n t i n u e d from
5
[4]
and
1.51.
2. P l a n o f p r o o f s . From t h e c o n d i t i o n
>
q(x)
( 2 , 1)
t h e r e e x i s t s a p o s i t i v e number
(C)
in
mo 7 0
(P)
I f the solution
v ( x , cc)
of
( 2, 2)
1
CL)
w(x,c(
w(x,
we s e e t h a t
=
V"X,
\q(x)I < M
( x , , ~ ) and satisfies
v(x,&)
#
xo
su ch t h a t
x E (0,~).
for all
then putting
0,
/ v ( x , cc)
is a solution of
a()
c Re j:w(s,
Conversely i f
d )ds = - m .
i s a s o l u t i o n of
w(x,cc)
i s a s o l u t i o n of
w(s, d )ds
f o r an y f i x e d
():
then
1ox
v ( x , C L ) = exp
(2, 3)
( P) ,
.
OL
Thus we c o n s i d e r
(P)l
from
t o p o l o g i c a l v i e w p o i n t s a s w i l l b e shown i n lemmas i n n e x t s e c t i o n . Here we e x p a i n o u r s t e p s i n t h e p r o o f s o f Theorem 1.
2.1. (I)
Replacing
solution Put
w(x,
by
(0,eo) d )
v ( x , K ) = e xp
of
(x,,oo),
( P) ,
we show t h a t t h e r e e x i s t s u n i q u e
for all
J:
w(s, d ) d s ,
then
C(
E C
-
( - m , 01.
d(x, c i )
i s unique s o l u t i o n of
0
;I~(X,
e c ) = q ( x ) K G(x,
d E C
-
(-m,
t h u s so i s (111) f o r any
;(x,
w)
E ( x ~ * c o ,) w(x,
satisfying
-
v(xo,
=
is a n a l y t i c i n
c -
1
f o r any
(-w,
01,
K).
Then we e x t e n d d E C
(xo,
01.
For a ny
(11)
in
-
(-m,m),
w(x, d )
to
and d e f i n e
[O, m )
T(x, K )
satisfying =
w'
=
q(x)Cr - w
w(s, k)ds.
ex p 0
2
176
Sadao MIYATAKL
For in
d E (0, w ) ,
we e x t e n d
T(x,
satisfying
d )
= q(x)a
d ' v(0, d j ) # 0,
( 0 , ~ ) . In t h i s c a s e we p r o v e , v ( 0 , d . ) = 0 ,
I
( j = 1, 2 , 3 ,
...) ,
r.
g v ( 0 , ~ )> 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
(2, 4 )
W(X,d)\V(X,d)(
=
1'
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
0 1 x1 4 x
-
.
..
$v"(y, d.)v(y, r ) d y
=
i s s a i d simply t o b e
p -
p
- component of
component o f b o t h s i d e s of (2, 4 ) ,
w
.
.
.
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 )
I
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
( 2 , 2).
Take
1 1 2dY ,
x1
x1
we
2
+
):
lP
2 I ~ ' ( Y , o c dY, )~ 1
where
1,
Imd = -Im(p i )
Im d
> 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 {
<
Re T
1 < -p 1.
ul(x, t)
and
u2(x, t)
-3a
In f a c t we have
Thus As f o r
of
v(x,
C
holds f o r
(E)Z
u3(x, t ) 2
In f a c t f o r
)
2
supp
,
f
r,
4
,
Here we p u t , f o r convenience,
where
a s follows.
I Re f 1 > 2a }
supp
i4 c { T E
1 ~ ,
<
Ref
4
3a
]
f o l l o w from P l a n c h e r e l ’ s theorem.
C I g ( t ) I1 L2
and
k 1 1.
we u s e i n t e g r a t i o n by p a r t s i n view of smoothness
and u 4 ( x , t )
and
E
k=o and s i m i l a r l y f o r
on t h e support of j = 3
we u s e t h e f o l l o w i n g
u.(x, t). j=1 1
smooth f u n c t i o n s defined on
a neighbourhood of o r i g i n . SUPP
To o b t a i n ( E ) Z
d.(z). J
u.(x, t) = 1
K j ( t , s ) g ( s ) d s , where
178
Sadao MIYATAKE
From
( 2 , 11)
we have
f o r any f i x e d ( 2 , 12)
I ;(.,
,(/I
5 Therefore
d ) uj(x, cj(t,. ;
d)
;(.,
for
and
k = 0.
tj(t, . ; x.
x) 11
5 C II
and
=
I1
L1
f . ( . , s;
x)
I[
6 C
n
L
1
Since m -
u . (x, t) 1
e(t,
we have
(Lz)
t, s
1 I
J-,
.)
J
L2
X I [ (L1
u j ( x , .)
(
11
a ( ~ ) }d s
K . ( t , s ; x) { e ( s , h')
I"
L2
1 e(s,
I K j ( t , s; x)I C
-
C2
nz 1
In t h c Same way we have
Y)g(s)[
e ( . , 1 ) g(.)
(E)2
for
11
2
ds)dt
, which means L
k 2 1.
(9. e . d . )
3 3 . Detailed proofs and viewpoints.
A t f i r s t we prove
Lemma 1 .
of
>
0
w' = c(x)
-
w2
Corollary. II"
=
q(x)
Re q"(x)
satisfying w(x)
Let
be complcx valued piecewise continuous f u n c t i o n for a l l
x E
satisfying
Then t h e r e e x i s t s a s o l u t i o n
[O, m ) .
Re w(x)
<
0
for a l l
x E
Suppose t h e same c o n d i t i o n s as i n Lemma 1.
q"(x)u h a s a s o l u t i o n
u(x)
satisfying
lu(x)
> lu(xl)
[O,m).
Then i if
0 6 x
< x1 .
Integral Representation for Equations of Mixed Type
i f we p u t
.
w # 0
i n t h e c a s e where
l/w
=
179
Thus we r e g a r d
w I =
q(x)
-
w2
a s a d i f f e r e n t i a l e q u a t i o n on Riemann sphere with two l o c a l c o o r d i n a t e s : b
Put Q,
{w
=
{
w
: Re w
:
<
w
Kemark t h a t t h e v e c t o r 2 Re (G(x) - w )
>
w(0)
w(x) E
uniquely such t h a t
Denote by f
(xo,oo).
a
goes o u t of
= wo
2
1
c).
, then
Q, i s compact.
f a c e s t h e e x t e r i o r on a s l
>0
for a l l
the solution
.
w
w(t)
, i.e. and
€
of
= F(x)
wI
w(x,) E
i n (0, x o ) ,
3
to
i s i d e n t i t y mapping.
aS2 and w(x) E.
a,Q
and
C
R
2
-
w
2
C$'L
x
0
in
+ w(xo). Then f i s
: wo
T h i s c o n t r a d i c t s t o well-known (q.e.d.)
p r o p e r t y o f t h e continuous f u n c t i o n on t h e c o n t r a c t i b l e domain. Identifying
x & [O,oo).
Namely t h e r e e x i s t s a p o s i t i v e number
t h e mapping from
f!,,
continuous and
< 0
: Re
52
wo E
E
i j = l/w
Re (1 - :(x)G2)
Now assume t h a t f o r e v e r y with
{ G :
q ( x ) - w2
and
0
{G
U
0)
u
c)
E
, we can r e g a r d
w' = q(x) - w2
and
a s a system o f d i f f e r e n t i a l e q u a t i o n s w i t h v a l u e s on a r e a l
= 1-G(x)G2
The above proof g i v e s d i r e c t l y f o l l o w i n g
compact two dimensional manifold. s t s tement s
Let
Lemma A .
M
be a r e a l manifold of
M, where
a d i f f e r e n t i a l e q u a t i o n with v a l u e s on continuous on M
such t h a t
i.e.
Q(x, U)
U.
Let
i s compact and
2.Q
i s a piecewise smooth h y p e r s u r f a c e
be an open c o n t r a c t i b l e s e t i n
a union of a f i n i t e number o f p a r t s o f h y p e r s u r f a c e s .
Q ( x , U) and U E
faces the exterior s t r i c t l y i . e .
a R , where
3
a t l e a s t one s o l u t i o n for a l l
x E
3 . Q ( x , U)
7 0
s t a n d s f o r u n i t o u t e r normals of U(x)
of
U' = Q(x, U)
Assume t h a t on for a l l x E [O,oo)
an.
such t h a t
Then t h e r e e x i s t s
U(x) E
$2
[0, -).
E v i d e n t l y we have Lemma 9. ____ Y.Q(x, U)
>
Suppose same c o n d i t i o n s as i n Lemma A , r e p l a c i n g 0
by
3 . Q ( x , U)
<
0.
is
is piecewise
and smooth on
x
a
8
n-dimensions. U' = Q(x, U)
Then from
U(O)E
a
follows
I80
Sadao
U ( X ) EQ ,
for all
x E
MIYATAKE
(0,m).
A s f o r Lemma A we can f i n d general s t a t e m e n t s i n
dimen i o n a l s p ce
n
of T. Wazewski [6] by v i r t u e of r e t r a c t method, ( c f . f o r example [ 7 ] ) .
Mere we s t a t e a n o t h e r d i r e c t e x t e n s i o n , which i s u s e f u l i n our r e a s o n i n g .
x.
Lemma ___-
one s o l u t i o n
x
E
x
U(x)
of
stays in
U ' = Q(x, U)
x1
x.
Proof o f Lemma
>
J.Q,(U)
on
0
as?, .
Let
U,(x) E $2
such t h a t
fi
U(x) E
>
U(x)
or
and
Then t h e r e e x i s t s a t l e a s t
such t h a t
x E [U,w)
for a l l
[ x , , M ) f o r some
E
u.Q(x, U) 2 0
by
0
for a l l :
A t t h a t time we have t h e a l t e r n a t i v e a s f o l l o w s
[O,CQ).
U(x)
>
p.Q(x, U)
replacing
M , U ' = Q(x, U)
Suppose t h e same c o n d i t i o n s on
aa
belongs t o
for all
0.
Take a smooth v e c t o r f i e l d (x)
U
U'
be a s o l u t i o n o f
x 6 [0,
f o r all
= Q(x.
fi
Since
m).
Q,(U)
satisfying
U)
€Q1(U)
+
i s compact
Uf,(0) 1
has a l i m i t of
a s a s u i t a b e sequence
Uo
U' = Q(x, U)
with
U(x) = lim U
since
E: J
t e n d s t o zero.
E. 1
satisfies
U ( 0 ) = Uo
U(x)
E
The s o l u t i o n
x
for
E
U(x)
[O,m)
(x).
In t h e same way we have t h e above a l t e r n a t i v e . Remark
In o u r problems t h e l a t t e r c a s e of t h e above a l t e r n a t i v e does
n o t occur.
Now we make Lemma 1 more p r e c i s e . Lemma 2 .
Let q ( x )
Then t h e r e e x i s t s a s o l u t i o n Re w
<
-m1'2
and
This solution
w(x)
Let
Proof.
S
2 3
{ G
0
< m < Re q ( x )
of
for a l l
x E
is unique one s a t i s f y i n g
and
-
Re w(x)
S3 : S1 ={w
t)itl , 0 L t
:
w" = t ( - r ) + (1 1
= l/w
:
G = t ( - e ) - (1 - t ) i c l 1
,0
I<(x) I
<
M.
satisfying
[O, O D ) .
be an open bounded s e t i n { w S1, S2
and
w ' = T(x) - w2
l/w
= { i j = =
w(x)
I w I < (2M)1'2
surrounded by t h e following S
satisfy
5
E
<
0
for
x
C : Re w < 0
c
}
: Re w = -m 1 / 2
1)
6 t I l ) ,
[O,m)
}
I
and where
l1 = (2M)
-1/2
.
Integral Representation for Equations of Mixed Type
.I81
Now we i d e n t i f y Riemann sphere and r e a l compact manifold of two dimension. Then 5;!
and
can show
Re ( q ( x )
-
s a t i s f y a l l t h e c o n d i t i o n s i n Lemma A , s i n c e we
- w2
w ' = {(x)
2 w )
>
on
0
S1
and
1-Mf
-G,
by
v" = q ( x ) v
given by
<
m0 ( q ( x )
(q. e. d . )
(2.3).
g"(x) = q(x)o( , q(x)
Suppose t h a t a r e a l valued f u n c t i o n
<
Mo
cc
and t h a t
satisfying
w(x)
of
(m0Reo()1'2(Rep)~1~ S ( X , O L31)
Im(w(x, d ) d )
1w1
p
x E [ o , m ) , where
=
w
number
where
r
w
<
0
2 and
g d
s ( x , d ) . Remark t h a t any complex
i s d e s c r i b e d uniquely w = ro(-
(3.4)
Im w
-
and
a)
cc1/2 , a r g p b e i n g a 21r
At f i r s t we must g i v e an e x p l a n a t i o n o f
>0
w' = q(x)u
(2Mol&~)1'2 ,
< -(moReq) 1/2 ,
satisfies
Re ci
<
Re w
Im ( a
for a l l
b e i n g a r e a l valued
q(x)
i s a number s a t i s f y i n g
Then there e x i s t s a s o l u t i o n
I m d 3 0.
(3.3)
2
Then we have more p r e c i s e r e s u l t s .
Lemma 3. 0
w ' = q"(x) - w
and t a k e account o f t h e l i n e a r p r o p e r t y o f s o l u t i o n s o f
Now w e proceed to 34r c a s e where function.
3 '
'
1
The uniqueness f o l l o w s i f we apply Limma B t o replacing
S20S
on
= I
and
s
sp,
a r e r e a l numbers.
In o r d e r t o show
w ( x , g ) # 0,
We have
we w i l l v e r i f y
>
0
in our
be an open bounded domain i n { w
: Re w
s(x,d)
arguments Proof o f Lemma 3. surrounded by S4 =
{w
s,, S2,
: (moRe fc )"'(Re
a
Let
S3
=
-f
p)-l
<0)
w : Im w = 0 } and =
Im(w
2)
Im(d
p)
} , where
S1
and
S2
a r e t h e same
I82
Sadao MIYATAK E m = m Re 4
ones a s i n t h e proof o f Lemma 2 with a p p l y Lemma A
x E [x,,
and
w)
Rea
>
0
(2.1)
and apply Lemma 2
w ' = q(x)or - w2
then
(xo, oo),
( 0 , ~ ) by
0 Thus we have Lemma 3.
a s i n t h e c a s e of Lemma 2 .
Now r e t u r n t o our assumption
. Then
M = M la1
and
0
we can
(q.e.d.1
and Lemm 3
replaced
has a s o l u t i o n s a t i s f y i n g , f o r
,
< -(moRe O L )
, J w 1 < (2M
i)
Re w
ii)
(moReci)1/2(Rep)-1 .C s ( x , a )
IN1 ) l l 2
(3.5)
< (2M 1x1 ) 1/2 2
if
Ima+ 0 .
Im(wj) In f a c t
(3.5) i i )
follows from
(3.4)'
-
-1 =
and
Im(otj) Here we c o n s i d e r t h e c a s e where
Re d C 0
Im(o(i)
Im CC # 0 .
and
B
At first
we s t a t e Lemma 4 .
Im {(x) 5
-
6
Let
<
0
q(x)
be a bounded piecewise continuous f u n c t i o n s a t i s f y i n g
, ( r e s p e c t i v e l y Im
<(x)
Then t h e r e e x i s t s a s o l u t i o n
w(x)
of
>
I m w(x)
<
Im w(x)
6,
and
(resp.
> 6, M.
surrounded by
Here
0 , tl,
=
-6,(O
and
)
< Let
0 )
for a l l
x E
satisfying J W \ <
t 2 and
v ( t ; El,
f o r some c o n s t a n t s
M
Im w(x)
>0
,
[O,m).
9 be an open bounded domain i n { w
S1, S 2 , S3 and
x E [O,m).
for a l l
q(x) - w2
The s o l u t i o n i s unique one s a t i s f y i n g
Im w(x)
Proof.
, (resp.
0
w'
2 6 >O )
>
,
: Im w
S 4,
where
i2)
s a t i s f y the followings:
ti
0
> 0)
( i = 1, 2 , 3)
and
183
Integral Representation for Equations of Mixed Type
, t2)=
(ii)
v ( t ; &1
(iii)
Im(tl + iE3)
(iv)
I m ( t 2 + iE3)
F i r s t we s e e t h a t
Take
-1
1
= - E ~
, tl<
= -E2
t2
9
r \ a G and
S3
the condition.
0
>
0.
E 2 = 2E1
aG
, so t h a t
i n t e g r a t i o n by p a r t s of
(v" - c(x)v)G
Take t h e imaginary p a r t o f
c3
where
l i m I V ( X I) = 0
and
w,(x)
= q"(x)
v
v ( x ) = exp
such t h a t The uniqueness h o l d s
as follows.
The
jr w(s)ds
and
-9
is a s o l u t i o n s t a y i n g i n
Then we s e e (3.7)
1
v(s)
ds
<
00
.
X+O
and
w1
l i m Iv, (x)
x-?m
-
-
wz
exp
1,:
= q(x)
v,(x)
=
1
=
00
.
such t h a t
w,(x)
If t h e r e E i s t s another solution
I m w2
)
0
w Z ( s ) d s must s a t i s f y
x E [O,m),
for
r:
2 I V ( S )1 d s
Therefore we have unique s o l u t i o n in
(xo,
m)
f o r a l l cc E C -
(-00,
01
of
then from
, which
(3.6) is a
1
c o n t r a d i c t i o n s i n c e t h e s o l u t i o n space i s two dimensional. w(x, a) of satisfying
L
A.
satisfies
yields
with
(3.6)
j," w l ( s ) d s ,
v"
S1 n 2 R
x C [O,ao).
for a l l
from t h e l i n e a r p r o p e r t y o f s o l u t i o n s of
v (x) = exp 1
(1 - t ) E 2 ( c o t e - i )
F i n a l l y we can choose p o s i t i v e c o n s t a n t
f a c e s t h e e x t e r i o r on
- w2
+
n 2 R s a t i s f y t h e c o n d i t i o n i n Lemma
s u f f i c i e n t l y small and p u t
El
c(x)
S2
-1
- i)
t & (-cot 0
(q.e.d.)
w ' = q ( x ) d - 'w
Sadao M I Y A T A K E
I84 (3.8)
l i m v"(x,
c()
, where
= 0
G(x,
= exp
4 )
w(s, d ) d s ,
X -too
>
x
xo
.
0
Here we have Suppose
P r o p o s i t i o n 1. C
-
Then above
w(x0,a)
A t f i r s t l e t us s e e t h a t
Proof.
W ( X , K ) is analytic i n
x E [x,,~).
€ o r any f i x e d
(-PO, 01
(2.1)
is continuous.
In f a c t i t s u f f i c e s t o a p p l y Heine-Bore1 theorem on {w
: Re w
<
0
1 u{
i n c a s e where
:
Re&
>
.
0
w(x,x)
CC.
Re
Re 4:
If
compact s e t s i m i l a r l y u s i n g u n i q u e n e s s of
,
= l/w
Im w
<
r
0
0
1
- {w
and
: (w(xo,
Im
instead of
compact s e t
B
do)
-
w
I
# 0 we d e f i n e t h e above
c(
.
Re w
We u s e o n l y t h e
and t h e c o n t i n u i t y of s o l u t i o n s f o r d a t a and p a r a m e t e r
To p r o v e a n a l y t i c i t y we p u t
d which s a t i s f i e s w h ( x , d ) = q ( x ) - f w ( x ,
Q
-
+ h)
W(X,LY))
wh(x,d)
.
Thus i t f o l l o w s wh(x,
(3.9)
c()
\* vh(s,
d + h)?(x, d ) ) - '
= -{?(x,
+ h ) G ( s , GC)q(s)ds.
X
T h e r e f o r e we can p r o v e from (3.10)
a
w(x, a )
=
w(x,
w ,
- v ( x , cc)
E v i d e n t l y ;(x,cz) Now w e e x t e n d
(3.6)
9)
-* 1:
and
(3.7)
v"(s, o() 2 q ( s ) d s .
is a n a l y t i c i n d to
x
[O, OD)
for a l l
(q.e.d.) x
e
[x,,~).
as a solution of
w ' = q(x)M
I n o r d e r t o show t h i s p o s s i b i l i t y we s t a t e
Suppose
Lemma 5. w ' = q ( x ) g - w2
(3.11)
G'
= q(x)o(
Then f o r a l l c( E {.l : I m d # 0
h a s u n i q u e bounded s o l u t i o n
Im w ( x , o ( ) g
Proof,
(C).
20
for all
w(x,d)
x E [O,M)
if
I t s u f f i c e s t o prove i n t h e case
-
w2
f o l l o w s from
w ' = q(x)c(
-
w2
.
Let
}
satisfying Imd 2 0 Im
o(
> be
. 0, s i n c e
-
w
2
Integral Representation for Equations of Mixed Type [w
>
: Im w i
before.
O} U
G d
< 0
The uniqueness follows from
Taking account o f we s e e t h a t T(x,
Im
l/w ;
=
w(x,d)
= exp
4 )
i:
Lemma 3 in
}.
Then we can a p p l y Lemma
Lemmas 4 , and
Lemma B .
and t h e uniqueness i n
Lemma 5
185 as
(q.e.d.)
Lemma 2
i s the desired extension.
and
Lemma 4
'Thus
i s bounded, non z e r o and a n a l y t i c i n K
w(s,sc)ds
2
for
0
x E
if
[O,oo)
Im
For r e a l p o s i t i v e d
= q(x)q;
as a solution of Then
# 0.
c(
with
c ( x , ~ ) is analytic in
Since
v(x,d)
C -
K C
Lemma 6.
= 1
c()
G T ( x o ,K
f o r any f i x e d
01
(-m,
and
Suppose t h a t
equals
"hen
x1 E ( 0 , x o ) .
{d
= dj ; 0
w(xo, d).
x E [O,m). w
q(x)
satisfies
(C)
and
q(xl) 4 0
I
for certain
) =
, w e need a lemma concerning z e r o s of v ( 0 , K ) .
;(x, & ) / ; ( O , < )
=
v(xo,
$(x, 4 ) d i r e c t l y
we extend
{d : 0
< d l < d2 < d3
(
d . , ( j = 1, 2, . . . ) i s determined by I a t each d = d . and s a t i s f i e s
< cC , v ( 0 ,
...,
l i m d. = 1
q(x).
]
ct) = 0
;(O,
o()
Then
v
m )
, where
has a simple z e r o
I
(3.12)
<
0
< lim v ( 0 , a )
l i m v(0,q)
qmi &
Proof.
Put
.w
v ( 0 , ~ )= 1
.
(00
d.lO
v(x,
and
c()
=
v(xo +
v(x,d) = 0
lim
$ , <)
>
f o r any
satisfies
q ( x , d ) = q(xo
0, where
+
X+W
(3.13)'
Applying
mo
(
qfx, d)
Lemma A
to
we have a s o l u t i o n 1
v ( x , ~ )= exp
c
<
'*.
w'
w
M
for
=
q(x,
w(x, d )
x E [O,m)
- -2w
with
staying in
for
c()
and a l l
>
dGY 0 ,(= v)(O, 6 ) = w(0,
.
= (-M112, -m,, ' I 2 )
x 6 [O,m).
w ( s , ~ ) d s from t h e uniqueness i n
Remark t h a t f o r a l l CC
OL 7 0
Q()
Then we have
Lemma 2 .
E. (-M
1/2
in
, $2). o
R,
$1
,
I86
Sadao MIYATAKE Iq(x,d)
Since (3.12).
1
cc
If
{ct : 0
0:
E c -
Then
6)= 0
Therefore
l;lcl(s,
d ) =
2
(rq(s)c(s,d)
.--
E
C - (-w,w).
Thus we have From x
e,
X.
1
=
'4
v(x,p: From
d) # 0.
cc) s a t i s f i e s
oscillates in the h i s f a c t we can s e e l a t e r
i s an i n f i n i t e s e t .
d T I ( 0 , d) = - T(0, dx
= V'
c(
}
= ?(-axo,
Now suppose
Similarly t o
(3.6)
we have f o r
(-m, 01
-;to, o c ) ?(O,
for
01 , T(o, Q )
[-dXo,
i s n e g a t i v e valued.
q(x,o:)
< CC , ?(O,
.
G(0, d) = 0
x
for
becomes l a r g e r t h e s o l u t i o n
i n t e r v a l where that
L M
{x
Making
&)I
2 ds
c(
+
q(s)l;(s,
d )
2 ds = - d - l { 3 v ( s , d) ds f 0 .
(0, d ) d
I 'ds
On t h e o t h e r hand,
d d v(o, d ) - dec ' v l (0,cC ) G ( O , d )
tend t o
d
we have
Lemma 6.
Lemma 6 : x 7 0
(9. e .d. )
v ( x , a ) has simple p o l e s a t
d = d. I
for
, ?(x, d j ) = O} , which con t i t u t e s o f
j-1
Combining t h e s e arguements we have Theorem 1. Proof o f Theorem 2 .
W e show t h e f o l l o w i n g two e s t i m a t e s .
points.
187
Integral Representation for Equations of Mixed Type for
a:
f
{ a €C
(E)
we o b t a i n t h e e s t i m a t e Re PC
1 Im p 1
and
, 1Imo:I
= O( IdL)
d dx
- s ( x , cc).
O(1)
=
From
}.
O
:
(3.141, (3.15)
i n Theorem 2 , i f we use =
,
O(l@l)
.
d
f o r fixed (2.8)
From
IG-
(dl =
(2.7),
- il
I2
= O( la?*),
IRe /31 = O ( i W )
I n o r d e r t o prove
w2 = ( r d - S P ) ~
and
and
,
we c o n s i d e r
(3.14)
follows
2 ( l / s ) ' 2 -IcCI(Irnp)(Ima)-l
Thus from
1
1
we have
!a!Img ( x , - x )
+
Im
,
oc
O L X 4 X
(3.5),
l/s(x)
I (m, R e a ( ) - 1 / 2 ( R e p ) , f o r
Thus we have
(3.14)
.
From
following
6
To show
x E [O,oo)
for all
surrounded by
S1 = {w
s2 = { i.
,. w
= l/w
= E
;
0
x E [x,,
(3.15)
we prove t h a t
.
fi
Here
s = -Im(wk)/Im(pd)
(-i?) + r;
, r
6
a,)
(= =
R)
s
0
m).
w(x, a )
stays in the
i s an open bounded domain
) and
, where we t a k e
Eo
s u f f i c i e n t l y small so t h a t f o l l o w i n g two c o n d i t i o n s a r e f u l f i l l e d : (I)
The v e c t o r f i e l d
(11)
-
f a c e s t h e e x t e r i o r on
involved t h e domain d e f i n e d by
The c o n d i t i o n s
l a r g (1- q(x)&C2) 1
<
E S n 2a a r e s u f f i c i e n t . 2
for a l l E0 =
1 - q(x)s?
'Irnd I m B '
2M
ld13
sm
.
Then we have
(3.5)
larg
2 I
35 .
S2
fl
<
(2M 1 4 )
i). and
1;
I
-1/2
By elemental c a l c u l u s w e can t a k e w(x, cc) E
3
for a l l
x E [O,oo),
Sadao
188
-1
-1
2h1 ld12
MlYArAKE
s
-1
.
This completes the proof of
Theorem 2. Final comments. v"
=
q(x) v
It is interesting that unstable solutions of
are useful to construct the solution of
the uniqueness of
(P)
(P).
As for Theorem 3 ,
should be discussed more precisely.
is useful to see the singularities of solutions of
(P).
The formula
The argument in this
paper will be valid for another type of problems, for example,
(P) with q ( x ) by A
in Rn
satisfying
.
lim q(x) < x+
0 and
(P)
replaced
m
a2 a x2
They will be discussed elsewhere.
Bibliography
L. Bers, Mathematical aspects of subsonic and transsonic
gas
dynamics, Wiley, New York, (1958) A . R . Manwell, The tricomi equation with applications to the
theory of plane transsonic flow, Pitman Adv. Publ. (1979) M.M. Smirnov, Equations of mixed type, American Math. SOC. Monographs. Sl(1978)
S. Miyatake, Construction of eigenfunctions f o r Sturm-Liouville operator in
[-m, co).
to appear Kyoto .J. Math.
S. Miyatake, Th6or;me d'existence des solutions des syst$mes d ' equations differentielles et applications, Seminaire J.vaillant Univ. Paris V I
(2)
(1980-81)
T.Wazewski, Une methode topologique de l'examen du ph&nom$ne asyrnptotique relativement aux &pat ion diff6rentielles ordinaires, Lincei-Rend. Sc. vol 111, 1947 S.R.Bernfeld-V. Lakshmikantham, An introduction to nonlinear boundary value problems, Academic Press, 1974.
Lecture Notes in Num. Appl. Anal., 5 , 189-204 (1982) Nonlirieur PDE in Applied Sciei1t.e. U.S.-Jupun Semintrr, Tokyo. 1982
Yielding and Unloading in Semidiscrete Problem of Plasticity
Tetsuhiko Miyoshj Department of Mathematics Kumamoto University Kumamoto 860, JAPAN
Introduction In formulating the elastic plastic problem it is usually assumed that each element of the material is either in the e l a s t i c s t a t e or in the pZastic s t a t e and that these states continue for a while after they have been chosen. The transition from the elastic state to the plastic state is called the yielding. The reverse change is a case of the unloading.
Therefore the material under-
goes the elastic and plastic deformations repeating yielding and unloading alternately.
Mathematically, this is nothing but to assume the existence of
the classical solution, which is proved only f o r some special cases. For the semidiscrete system, that is, for the case that only the spatial region is discrete,it is possible to show the existence of the classical solution. In this paper we discuss the way to get this solution.
This kind of systems
is essential for deriving the approximate methods in engineering. the e z p Z i c i t integration schemes start from this system.
Especially,
A l s o , if we want the
solution of the fully continuous problem, it is only necessary to pass to the limit with respect to the spatial approximation. The key to prove the existence of the classical solution is to guess the behaviour of the solution after the time at which the state of the element may change.
Assume that there are N elements which may yield o r unload at time t=
I89
190
Tetsuhiko MIYOSHI
The number of t h e combinations of t h e p o s s i b l e s t a t e s a f t e r t
to.
0
Hence i f t h e r e e x i s t s a unique c l a s s i c a l s o l u t i o n , then o n l y one i n 2
N
i s 2N.
possibili-
t i e s must t a k e p l a c e , and what w e want t o show i s t h a t t h i s i s a c t u a l l y r e a l i z e d .
We w i l l show t h a t t h e next s t a t e of each element i s uniquely determined by u s i n g t h e d a t a of t h e s o l u t i o n b e f o r e t o and t h e g i v e n d a t a a t t o which i s independent of t h e s o l u t i o n . The mechanism of t h i s d e t e r m i n a t i o n i n dynamic problem i s d i f f e r e n t from t h a t in thequasi-staticproblem.
I n t h e former t h e e x i s t e n c e of t h e i n e r t i a term i s
t h e key ( s e e [l] f o r t h e d e t a i l s ) .
In t h e l a t t e r , a c e r t a i n p o t e n t i a l
r e l a t i n g t o each o r d e r of d e r i v a t i v e s of t h e s o l u t i o n p l a y s t h e e s s e n t i a l r o l e .
I n t h i s p a p e r we d i s c u s s t h e q u a s i - s t a t i c c a s e .
We w i l l e x p l a i n our b a s i c i d e a
t a k i n g a f i n i t e element approximation of a 2-dimensional problem a s an example. The r e s u l t of t h e p r e s e n t paper i s announced i n [ 2 ] with a b r i e f proof.
1.
S e m i d i s c r e t e f i n i t e element approximation Let
R be a r e g i o n i n x=(x1,x2) p l a n e which
i s composed of t h e f i n i t e number
Each t r i a n g l e i s c a l l e d a f i n i t e element o r simply an element.
of t r i a n g l e s .
We s e e k t h e
Let { @ ] be t h e u s u a l piecewise l i n e a r f i n i t e element b a s i s . P f u n c t i o n ui ( i = 1 , 2 ) of t h e form
Here P denotes t h e set of a l l t h e
v e r t e x e s of t h e t r i a n g l e s of R e x c e p t i n g P
which t h e boundary v a l u e s of ui a r e given. simplicity, P
U
We assume t h a t , f o r t h e sake of
i n c l u d e s a t l e a s t two a d j e c i e n t v e r t e x e s and u =O on P i
.
{ u i ( t ) } are determined by t h e following system of e q u a t i o n s .
(l.l)b
b
=
D;,
(l.l)c
6=
(D
& = 0
-
D');.
< 3, o r f ( u - a )
if f ( u - c I )
h
= (U
-
a)
af*ir
L_
f
= 5 and af
i f f(u-ci)
=
5 and
*.u
< 0,
af*3 2 0,
on
Yielding and Unloading in Semi-Discrete Plasticity where (
,
) denotes the
L2 (Q) inner product of functions.
for both the single and the vector functions.
19 1
We use this notation
The above system is derived from
the Prandtl-Reuss flow rule and the Ziegler's hardening assumption. NOTATIONS :
a = u(t,x) =
(
cr 11'
E = E(t,X) = ( E
a
=
"1
a(t,x) = (
=
f
=
n
;
?,1
11' E 2 2 ' €12)
0.
11' a 2 2 2
€22
f(O-a),
( u21 = 0 1 2 )
O22' O12
= u2,2
2
f(T)
= Tll
'12
=
+
TZ2
positive function
Y,2
- Tll
2
(
+
u2,1 T22
+ 3
2
T12
assumed to be constant )
D ; elastic stress-strain matrix
uli= w a x i ,
-
cr Now, since u element.
i
=
du/dt,
bi
=
bi(t,x) ; given function
: given positive constant.
is piecewise linear with respect to x,E is constant on each
Hence u and a which are determined by (l.l)b or (l.l)c are also
constant on each element.
We assume that b. are continuous and piecewise ana
lytic with respect to t.
Under these assumptions we seek a continuous (u,u,c()
satisfying the equations (1.1) in I 2.
=
-
[O, T] and the initial condition (u,u,a)=O.
Determination of the first derivative We say that an element is elastic ( resp. plastic ) if (l.l)b( resp. (1.llc)
i s satisfied on this element.
Let E be the set of all elements of R.
Since
Tetsuhiko MIYOSHI
192
we started from the zero initial condition, all elements of E are elastic until
=a at,
some elements satisfy f(o)
for example, t
=
to.
It is clear that o u r
problem has a unique piecewise analytic solution ( u ,u ,O ) in [ 0 , to). at t=t o.
the set of all elements which satisfy f(o)=E
0
The key is to guess the sign of
Hence the next
yield at t=tO or still remain elastic.
af*6 at
( resp. negative ) then the stress point
of the y i e l d surface
t
0
f(o)
=
t=t +0, since if this is positive 0
u moves to the outside ( resp. inside
We note that the signs of
0.
0
E- E0 is clearly still
elastic after to, since the solution must be continuous. problem is whether the elements of E
Let E be
af*6
)
and af*Di for t >
are same, since
af*b
= af"D6
( = af*D;
(
1-
+
))
af*D af
if elastic (if plastic).
Hence we shall consider the following system which must be satisfied by the first derivatives of the solution ( if it exists ) .
(2.Ub
u uo
0
0
for E- E
= DE
= DEO
0
o 3
in D- =
I
uo ; af*(tO)DEo
<
in D+
I
uo ; af*(t0)DE0
2o I
for E - E
(2.1)c Uo =
( D - D ' )Eo
D'
D'(t
=
where
Theorem 2.1.
=
0
).
Problem (2.1) has a unique solution (uo,
minimizes the functional
0').
This uo
0
I93
Yielding and Unloading in Semi-Discrete Plasticity under the subsidiary conditions (2.1) Proof,
(i)
Let (
,)
b
and (2.1)c.
be the LL(e) inner product of (vector) functions
( e E E ) and define
7 l (DO,
€o)e
-
Fe is a C -class function of uo and 1
so
Fe
e E E - Eo.
=
( 6(to+O),
is F 1
=
Let e be an element of Eo.
u ' ) ~ .
C Fe too.
This is clear for
Then in D- we have
and similarly in D+
These derivatives coincide on the plane between D- and D+ since
on this hyperplane. (ii)
-
D
This proves the C1 continuity of Fe.
D' is positive definite.
q is positive constant and
This is easily proved by the facts that
c < af*Daf $ c2 '1
c1 and c2 are positive constants.
,
(Dafaf*DE,E)<
af*Daf(DE,E),
where
Therefore F (uo) is bounded below by the 1
Korn's inequality ( we do not need this inequality as far as we treat the semidiscrete system ) .
Hence F (uo) has a minimum point which is also a stationary 1
However, if uo is a stationary point of F1, then at this point
point.
o0(uo) determined by (2.1) or (2.1) b
is equivalent to (2.1) mizes F (iii)
.
Uo =
must satisfy the stationary condition which
Hence the problem (2.1) has a solution which mini-
1'
To prove the uniqueness of the solution it suffices to show the unique-
ness of the stationary point of Fl(uo). the hyperplane in uo-space :
For each element e
E
Eo we consider
Tetsuhiko MIYOSHI
I94
TI
Let {R EO
x1
i
=
uo ; af*(tO+O)DEO
= 0
1 In each R
be the partition of the uo-space by these planes. Also, in R
relation is definite for all elements of E.
x
the
Uo-
F (uo) is a posi1 1
tive definite quadratic form of uo and the stationary point is at most one. Now assume that there are two stationary points u
1
E
Rx
and u
2
E
R
li
(1
!f
u
).
Consider the line
2 : u1
+
t ( u2- u1 )
t E[0,11.
This line goes through at least two regions of {R
x 1 when
t moves from 0 to 1.
Then the function g(t)
=
Fl( u
1
+
2 t ( u - u1 ) )
i s smooth, and a non- degenerate quadratic on t in each region.
1
if u1 is a stationary point, that is, if F 1 (u ) i s
Therefore
the minimum, then g(t) must
be strictly increasing in [0, 11, which contradicts that u2 is another stationary
PO int
.
This completes the proof.
We want to show that the solution (uo,ao) of the problem (2.1) is the first derivative at t=t +O of the solution of (1.1) provided (1.1) has a solution. 0
By Theorem 2.1 we can determine the sign of af*(t
0
)D?
for the element of Eo.
We denote by Ee and Ep the set of all elements of Eo for which this sign is negative and nonnegative, respectively, and solve the following initial value problem set up at t=t
0'
where
k= € ( A ) , D'
=
D'(t),
f
=
f((7-a) and (u,U,a)(to)
=
(u,U,a)(tO-O).
I95
Yielding and Unloading in Semi-Discrete Plasticity
Theorem 2.2.
The initial value problem (2.3) has a unique analytic
solution (u,u,c1) in a certain neighborhood of t Proof.
and (h,b)(to+O)=(uo,
Differentiate the both sides of (2.3)
Substituting the
and ( 2 . 3 ) b .
h-; relation
we denote the system of (2.3)b
and solving the resulting equation with respect to h, we have function of u ,
c1
0').
with respect to t and
By (2.3)'l)
denote the resulting equation by (2.3)'l). ( 2 . 3 ) : '
0'
and t in a certain neighborhood of to.
into (2.3), (1)
A as an analytic
Therefore (2.3)
b can
be regarded as a system of ordinary differential equations of the form
24 dt
c1
=
X(U,CY,t)
where X is an analytic vector function.
This system has a unique analytic
solution under the given initial condition.
Hence the problem for (2.3) has a
unique analytic solution ( u , u , a ) in a certain neighborhood of to. the solution (uo,)'a [2.3)(')
of the problem ( 2 . 1 ) satisfies [2.3)(')
Furthermore,
at t 0+O.
Here
is the system composed of (2.3f1) and the 5-k relation of (2.3)b.
Since [2.3)(')
has a unique solution on (;,6)(t
0
+O),
we have the theorem.
Let E be the set of all the elements of Ep for which af*(t )Dt(tO+O)=O 1 0 holds for the solution of ( 2 . 3 ) .
If El is empty, then the next state is
completely determined for all the elements.
Because, the elements of E-E
0
are
still elastic after to and for those of Ee holds
for small 6 and for those of Ep we have
for a while after to, where O
In other words, the 5-5 relation of the
elements of Ee and Ep are already chosen correctly and we could determine which elements yielded at t o.
We here emphasize that this determination is depend-
Tetsuhiko MIYOSHI
196
ent only on the data at t=t - 0 and the given function b. 0 If, however, E i s not empty we have to guess the sign of d/dt(af*& 1 In this case the following theorem is important.
tO+O.
elements of E
1
Replace some
from Ep to Ee and solve the initial value problem ( 2 . 3 ) for this
Let the new system be denoted by 1 2 . 3 ) .
new Ep.
) at t =
Then { 2 . 3 } has a unique
solution ( u , o , a ) under the same initial condition at t=tO, and moreover we have For every element of E, the value (A,6,&)(to+O)
Theorem 2 . 3 .
independently of the choice of the next
6-
Samely as before let [2.3}(')
Proof.
{2.3}L1) and the
5-6
relation of i 2 . 3 1
t = t
0
+o.
relation of E
1'
be the system w h i c h is composed of Since the solution (u0,oo) of(2.1)
b'
satisfies D'(tO)~O =O for the elements of El,
is determined
(uo,UO)
is a solution of [ 2 . 3 1 ( 1 ) at
But the solution of [2.3f(l) is unique with respect to ( h , b ) ( t o + O ) .
Therefore (&,h )(to+O)
= (
u',oo)
and follows the theorem since &(t
+O)=O 0
for E
Thanks to this theorem it i s assured that the elements of E - E behave 1
so
1'
as
to satisfy the subsidiary condition of (1.1) f o r any choice of the next state of In other words, the next state of the element of E-E
El. mined.
1
i s already deter-
Hence we can exclude them from our Consideration. Determination of the higher derivatives
3.
The next state of the element of El is determined by the sign of d/dt(af at t=t +O.
It is easy to see that this sign and that of
0
at t +O for the elements of E1. 0
*. d/dt(af DE)
*.5
)
is same
Hence we consider the following problem which
must be satisfied by the second derivatives of the solution.
1 (Goij,
(3.1),
j oo =
@p,j)
= (
d2 - b (t + O ) , $ ) dt2 P
p
E
P
for the elastic element of E - E
DEO
1
(3. 5'
= {(
D
-
D'
)E0
- &D')i]to+o
for the plastic element of E
-
El
Yielding and Unloading in Semi-Discrete Plasticity
197
and for the element of E
1
where
u o , af and D' are defined as before, and
EO,
r1
=
{c(3f*)D;}to+o. dt
+0) is the derivative of the solution of ( 2 . 3 ) .
Note that (;,6,&)(t
0
Theorem 3 . 1 .
Problem (3.1) has a unique solution (uo,ao).
minimizes the following functional F ( u o ) and (3.1)
2
. F (uo) =-(a',~') 1 2
where A2
=
{dD'/dt i j
r is
E:
an
in D-
TI
be the e clear.
-
-(1
2
A
2
,E
)
-
(- d2 b(to+O),
uo),
dt2
for the elastic (resp. plastic ) element of E-E
Eo)
1
i
for the element of E 1'
arbitrary fixed vector included in the hyperplane of uO-space
(3.3)
Proof.
-
under the subsidiary conditions ( 3 . 1 ) b
tO+O and
0 ( resp. E:
Here
2
This uo
(i)
: {u0; 3f*(t0)DEo
+
rl = 01.
F2(uo) is a continuous function.
part of F2 as before ( e EE ) .
For e
E
E-E
To prove this let Fe
1
its continuity is
For e EE the discontinuity of Fe might appear across the plane ve. 1
However, at t=tO+ 0 it holds that on ve
i (3.4)
= 0.
198
Tetsuhiko MIYOSHI
Hence Uo i s continuous w i t h r e s p e c t t o uo and t h e f i r s t term o f Fe i s continuous The jump a t E'
belong t o
X2,
of t h e second term of Fe i s 1 / 2 (
TI
re, then D'(
=
E z -E')
& -:
c0)
But i f E z and
e'
0 a t t = t O + O by ( 3 . 4 ) .
Therefore, s i n c e
D' i s symmetric, we have a t t = t 4-0 0
( A2,
Ez-
E0)
e
,
= ( -(D')E d
dt
=
-(
D'
&O,
Ez
-
Eo)e
c i - c0) e = - (
Eo,
D'(
E ' ) ) ~ = 0.
&-:
T h i s i m p l i e s t h e c o n t i n u i t y of t h e second term of Fe and hence of Fe i t s e l f .
( i i ) F,(uo)
is a C - c l a s s f u n c t i o n .
1
To prove t h i s w e check t h e t h r e e c a s e s .
F i r s t l e t e be an e l a s t i c element of E - E
Then Fe is smooth, s i n c e
l d2 Fe = F ( D E O , Eo) - ( - b ( t +O), dt2 Secondary l e t e be t h e p l a s t i c element of E - E 1.
uo)
e'
Then
Theref o r e
(3.5)
Since
u0
I s continuous, t h i s e q u a l i t y i m p l i e s t h e smoothness of Fe.
l e t e be a n element of E 1.
Then t h e f o l l o w i n g I s c l e a r i n D-
.
Finally
Yielding and Unloading in Semi-Discrete Plasticity On the other hand, the relation (3.5) is valid in D+.
199
Hence the C continuity 1
of F (uo) is proved.
2
(iii) F2 ( u o ) is a positive definite quadratic form which is bounded below. Hence the minimizing point uo of F
2
0
the unique u
-E
0
exists and (uo,oo), where
relation, is the solution o f (3.1).
the same way as in Theorem 2.1.
Uo
is determined by
These are proved by just
This completes the proof of the theorem.
The solution ( uo, uo) of the problem (3.1) is the second derivatives of the true solution. theorem.
Strictly speaking, this is in the sense of the following E;
+
EY, where
Devide El
=
EE
=
I e
EY
=
{ e €El ; af*(tO)D
C E ~; ar*(tO)u
Eo
+
r1 <
E'
+
rl
and solve (2.3) replacing Ep by the neu Ep tion.
By Theorem 2.3, (;,6,&)(t0+0)
=
>
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
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)
Eo
+
rl
= 0.
Then, for every element, (ii,g,E)(t +0) is determined independently of the choice 0
of the next
6-
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. )
0
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 <
It might happen that there are some elements for
m.
which the equality
t ldk c ( af *6))t o+o dt
holds for all k.
=
o
But this means that the stress point moves along the yield
surface for a while after to: f(o) relation to these elements.
=
5.
Hence we assign the plastic 6 - i
For the completeness we descrive below the
procedure to determine the derivatives of the solution at t +O when Ek( ka 2 ) is 0
not empty. Assume that the derivatives of order k are already determined independently of the choice of the 6 - k relation of Ek:
Let us define ( formally )
Xk+l
{-(Ddk 'L)
=
-
d tk
rk
=
I,(af
dk
dt Let (uo,oo)
D'
dk
,
dtk
*DZ) -
* dk af D -dtk4 1 to+()'
be the solution of the following problem set up at t=tO+O.
Yielding and Unloading in Semi-Discrete Plasticity
(3*7)b
{
uo
= DEO
Uo
-
= (D
- Ak+l
D')E'
20 1
for the elastic element of E
- Ek
for the plastic element of E
- Ek
and for the element of E k
Theorem 3.3.
The problem (3.7) has a unique solution
uo minimizes the following functional F
This
(uo,Oo).
under the subsidiary conditions
k+l
(3.7)b and (3.7)c.
where
o
(resp.
f o r the elastic (resp. plastic) elements of E
EO)
- Ek
for the element of E k in D
+
and
E :
is an arbitrary vector i n uo-space included in the hyperplane : {uo ;
~i
Classify Ek
= EE
non-negative for :E
+
+
af*(tO)DEo
rk
=
0
EE in such a way that af*DEo
1.
+
rk is negative for EE and
at t=t +0, and solve (2.3) of the preceding stage replacing 0
Ep by the new Ep = Ep - EE.
the derivative of order k+l of
Let (u,U)
(u,U,a)
at t +O. 0
Summarizing the above results we have
(uo,Uo)
is
The derivative of order k+l of
(u,u,a) is determined independently of the choice of the next
Ek+l *
Then
be the solution.
6-i
relation of
Tetsuhiko
202
The b
Theorem 3 . 4 .
-
MrYOsHl
relation of each element of E is determined
uniquely after to and the problem (1.1) has a unique analytic solution in a certain time interval [t So
t +6) ( 6 2 0 ) . 0' 0
far we have discussed only the initial yielding.
However, the above
procedure and the results are valid almost word for word to the subsequent yieldings and also to the case that the unloading may occur.
Since the boundedness
of the solution is assured by the energy inequality we can continuate the s o l u tion over the given time interval I. Theorem 3.5.
In fact we have
Let (u,a,a) be the solution of (1.1) in a time interval
Then there is a constant C which is independent of I' such that
I'C I. (3 .8 )
Proof.
In I' we have
Therefore hold ( 3.9)
The first three inequalities of (3.8) matrix D
-
D'.
thus follows from the positivity of the
To prove the last inequality of ( 3 . 8 ) , use the following
relation which holds f o r any t in I' at which the derivatives exists.
Remark.
Since we assumed that the function b is piecewise analytic, the
number of the changes of the state will be finite in any finite time interval. If this is the case, the continuation of the solution is obvious.
However,
203
Yielding and Unloading in Semi-Discrete Plasticity this is not yet proved.
Hence we have to consider the case that such points
accumulate to a certain t o < T. at t
=
In any case, however, the value of (u,a,a)
t -0 becomes definite by the uniform boundedness of the solution, and 0
our procedure to continuate the solution is completely valid to such to too.
It
is clear that there is no bound beyond which this continuation is impossible. The conclusion is Theorem 3.6.
There is a unique absolutely continuous function ( u , E , ~ , c ~ )
which satisfies (1.1) except at most countable t Proof.
= {T ;
If (u,~,u,a)satisfies (l.l), then u
&
However we shall prove the
absolutely continuous on tEI and f(T-a)
- C6, =
T
-
0)dt
qs-1(: - c6
E
6
a
+
BB
where f(@)
for all T
0
a.e. I
)
- a*
JI(i*- c&*, a* + u =
6
Since u
K, ( C=D-l ) .
E
K can be written as
,< 1, we have
JI(t - cG, a + u*
Define (U,E,X,A)
3 1.
K and the followings hold.
Assume that (u,E,~,c~)* satisfies (1.1) too. =
A l s o , the
Let K=Ka be defined by
uniqueness by another method.
u
I.
The existence of the solution is already proved.
above discussion shows the uniqueness too.
K
E
-
- a
u)dt 6
-
u,)dt
(u,~,u,a)- ( u , E , u , ~ ) * .
o 6 0. Adding these inequalities, we have
Tetsuhiko MIYOSHI
204
0 3
I
I
(i - C t ,
-
A
Z)dt
from which the uniqueness follows.
Now as is already seen in the existence proof of the solution, the exceptional t is always an end of a time interval of positive length. number of such time intervals of length
>
l/n (n 2 1) is finite.
countability of the exceptional t follows.
Hence the
This completes the proof.
In this paper we considered only the kinematic hardeninn
Remark. problem.
But the
The i s o t r o p i c hardening can also be treated in the same way [ 3 ] .
References
[l] T. Miyoshi,
On the existence
proof i n p l a s t i c i t y theory, Kumamoto J . Sci. 14
(1980). 18 - 3 3 .
[2] T. Miyoshi, A note on the c l a s s i c a l s o l u t i o n s of semi-discrete q u a s i - s t a t i c
p l a s t i c i t y problems, Kumamoto J. Sci. 15 (1982). 7 [ 3 ] T. Miyoshi, On the convergence
-
10.
of the f i n i t e element s o l u t i o n s for
p l a s t i c i t y problems ( in Japanese ), in RIMS reports No. 430 (1981). 3
-
12.
Lecture Notes in Num. Appl. Anal., 5, 205-231 (1982) Nonlineur PDE in Applied Science U.S.-Jupun Seminur, Tokyo, 1982
Two Dimensional Convection Patterns i n Large Aspect Ratio Systems*
Alan C . N e w e l l Program i n Applied Mathematics University of Arizona, Tucson
85721
This work was supported i n part by DOA Contract DAAG29-82-C-0068.
205
206
c. NEWEI.1.
Alan
F i g u r e 1 below i s t a k e n f r o m a v i s u a l i z a t i o n of a n e x p e r i m e n t of G o l l u b a n d M c c a r r i a r 111 in which t h e y f o l l o w t h e t i m e development o f a c o n v e c t i n g h o r i z o n t a l l a y e r of f l u i d of d e p t h techniques.
h
t h r o u g h t h e u s e of l a s e r D o p p l e r
The d o t s a r e p o i n t s where, a t a depth of
h/4
from t h e t o p of t h e
l a y e r , t h e v e l o c i t y component p a r a l l e l t o t h e l a r g e r s i d e of t h e box is z e r o a n d The e x p e r i m e n t is c o n d u c t e d i n a
mark t h e b o u n d a r i e s of t h e c o n v e c t i v e r o l l s .
r a n g e of R a y l e i g h number where n o n l i n e a r s t a b i l i t y t h e o r y g u a r a n t e e s t h a t i n a h o r i z o n t a l l a y e r of i n f i n i t e e x t e n t , s t r a i g h t p a r a l l e l r o l l s b o t h e x i s t a n d a r e stable.
.. .. .. .. .. .. .. .. .. .. .. .. .. .<..'.' : ... ... ... ........... ... .! t.' ., *. .....'..' .. '...** . ...... .*:. . . . . . . .: .* ..; . ......... ... 1,;:::::
10
.
.
.
.
*
.
.
I
.
.
.
.
::::;!;;:*.s.:
... ...:...:...:...:...:...:...i .:. I. ... .,... ...: ..:. :;:,: . . . . .., .,,. .. .. .. .. .. ..... . . . . ........ .t*,..*'
::..;,t
2 .
i .
... I ................. ............. ....... , ... .. . . ...... . . .r.t.! .
, . . !: , ::::.' :/a. ! :!:::::::;:I:.+, ,.* ..;:::;::;:::;::.+* . : : : : :: : : : : ;!::;!;/*. I
0
0
5
10
1s
x (cm) Figure 1 :
D o p p l e r Map Of The V e l o c i t y F i e l d F o r A S t a b l e C o n v e c t i v e Flow 256 Hrs. After R a y l e i g h Number Was I n c r e a s e d To 2.05 Rc. ( H o r i z o n t a l D i f f u s i o n Time, 40 HKS.)
However, t h e e x i s t e n c e of o r i e n t a t i o n a l d e g e n e r a c y a n d a band of s t a b l e r o l l wavenumbers t o g e t h e r w i t h t h e f a c t t h a t t h e rolls a l i g n t h e m s e l v e s p e r p e n d i c u l a r t o a l l l a t e r a l b o u n d a r i e s makes t h e s e s o l u t i o n s u n a t t a i n a b l e .
The p a t t e r n s t h a t
a r e s e e n a r e much more c o m p l i c a t e d n o t only c o n t a i n i n g Curved r o l l s b u t a l s o e x h i b i t i n g many d i s l o c a t i o n s .
F u r t h e r m o r e , i t is n o t c l e a r w h e t h e r a n d , i f s o ,
u n d e r what c i r c u m s t a n c e s t h e p a t t e r n a c h i e v e s a t i m e i n d e p e n d e n t e q u i l i b r i u m . I n d e e d , i n some cases, G o l l u b a n d M c c a r r i a r h a v e s e e n p a t t e r n s which are s l o w l y t i m e dependent o v e r many h o r i z o n t a l d i f f u s i o n t i m e s .
The f a i l u r e t o r e a c h a
207
Two Dimensional Convection Patterns s t e a d y s t a t e was a l s o noted e a r l i e r by Ahlers and Walden [Z] who observed t h a t the e f f e c t i v e thermal c o n d u c t i v i t y of a l a y e r of convecting f l u i d helium remained n o i s y f o r a l l Rayleigh numbers above
RC
f o r sufficiently large aspect
ratios. Our g o a l i n t h i s paper i s t o develop a t h e o r y t o d e s c r i b e t h e s e p a t t e r n s . We s t a r t with the o b s e r v a t i o n t h a t almost everywhere i n the convection f i e l d a l o c a l wavevector is d e f i n e d and v a r i e s slowly o v e r the box. the wavevector is tangent.
A t the boundaries,
Therefore we e x p e c t t h a t t h e r e e x i s t s l o c a l l y
p e r i o d i c s o l u t i o n s defined by f ( 0 ; A , R) when
f
is 2n - p e r i o d i c i n 0 and
VO = k(X, Y , T)
(1)
( V = (a/aX, a/aY)) is a slowly varying f u n c t i o n of p o s i t i o n c o o r d i n a t e s and t i m e .
X, Y, T, t h e h o r i z o n t a l
Indeed we know from the work of Busse 131 t h a t
such s o l u t i o n s a s f u n c t i o n s of 0 e x i s t and, a s we d e s c r i b e l a t e r , have c e r t a i n s t a b i l i t y properties.
I n o t h e r words, while t h e f i e l d v a r i a b l e
d i s t a n c e s of the o r d e r of the roll s i z e , t h e parameters and
d
(wavevector), which t o g e t h e r with a knowledge of
f
A (amplitude) f as f u n c t i o n of
0 d e s c r i b e t h e p a t t e r n , vary o v e r d i s t a n c e s of t h e s i z e of t h e box.
a s p e c t r a t i o c 2 , t h e r a t i o of t h e r o l l wavelength d dimension
L
of t h e box, is the
The Rayleigh number
RC,
R
only small
(2.
v a r i e s over
The i n v e r s e
h) to the l i n e a r
parameter which e n t e r s t h e theory.
can be an o r d e r one amount above i t s c r i t i c a l value
the value a t which the p u r e l y conductive s t a t e becomes u n s t a b l e , although
i t must be less than t h e Rayleigh number a t which t h e s t r a i g h t p a r a l l e l r o l l
s o l u t i o n becomes u n s t a b l e .
The i d e a s we a r e about t o d e s c r i b e a r e c l o s e l y
r e l a t e d t o those of Whitham (41 who i n t h e l a t e s i x t i e s developed a theory t o d e s c r i b e f u l l y n o n l i n e a r , almost p e r i o d i c w a v e t r a i n s . A s a f i r s t a t t e m p t we s h a l l use model e q u a t i o n s , t h e use of which
f a c i l i t a t e s the a n a l y s i s by making c a l c u l a t i o n s e x p l i c i t b u t which c a p t u r e what
we b e l i e v e a r e some of the e s s e n t i a l f e a t u r e s of the Overbeck-Boussinesq equations.
We d e r i v e e q u a t i o n s f o r the slow v a r i a b l e s
d,
A which
( i ) show
t h a t a l l t h e s t a b i l i t y c r i t e r i a d e r i v e d by Busse and h i s c o l l e a g u e s [31 f o r
208
Alan C. N ~ W E L I . ( i i ) r e d u c e i n t h e l i m i t of small
s t r a i g h t p a r a l l e l r o l l s h o l d l o c a l l y and
R-RC
[ 5 ] equations.
t o t h e Newell-Whitehead-Segel
I n a d d i t i o n we show t h a t t h e
e f f e c t of c u r v a t u r e is t o d r i v e t h e r o l l p a t t e r n s toward a s t a t e in which t h e l o c a l wavenumber assumes a c o n s t a n t v a l u e a l m o s t e v e r y w h e r e , a r e s u l t c o n s i s t e n t w i t h a r e c e n t r e s u l t of Pomeau a n d M a n n e v i l l e [ 6 ] f o r a x i a l l y s y m m e t r i c r o l l s . T h i s is done by ( i ) p r o v i n g t h a t u n d e r c e r t a i n n a t u r a l boundary c o n d i t i o n s , t h e macroscopic e q u a t i o n s f o r t h e phase, v a l i d f o r times up to t h e h o r i z o n t a l d i f f u s i o n t i m e , are d e r i v e a b l e from a Lyapunov f u n c t i o n a l e v e n though t h e f u l l m i c r o s c o p i c e q u a t i o n s f o r t h e model n e e d n o t b e a n d ( i i )
examining t h e n a t u r e
of t h e s t a t i o n a r y s t a t e s which would be r e a c h e d on t h i s t i m e . But t h i s is n o t t h e whole s t o r y a s such p a t t e r n s i n a n d of t h e m s e l v e s c a n n o t s a t i s f y a l l t h e boundary c o n d i t i o n s .
I n between t h e p a t t e r n s ,
In
d i s l o c a t i o n s are formed and s o l u t i o n s d e s c r i b i n g t h e s e s t r u c t u r e s a r e g i v e n . l o o k i n g a t t h e s e s o l u t i o n s , i t is c l e a r t h a t t h e t i m e s c a l e which t h e t o t a l
s y s t e m would n e e d t o r e l a x to e q u i l i b r i u m is n o t s i m p l y t h e h o r i z o n t a l d i f f u s i o n time L
2
/V
b u t is t h i s time s c a l e m u l t i p l i e d by t h e a s p e c t r a t i o
L/d.
Finally,
we i n c l u d e i n t h e model a t e r m which t a k e s a c c o u n t of t h e "mean d r i f t " which o c c u r s i n s y s t e m s of low t o m o d e r a t e P r a n d t l number.
The p r e s e n c e of t h i s
e f f e c t was f i r s t p o i n t e d o u t by S i g g i a a n d Z i p p e l e u s 171, a n d c a n c a u s e marked change i n t h e b e h a v i o r of t h e s y s t e m .
I s h o u l d p o i n t o u t t h a t t h e i d e a s we d i s c u s s h e r e c a n b e e x t e n d e d t o i n c l u d e s i t u a t i o n s where t h e b a s i c f l o w
is c o n s i d e r e d t o b e a
f
is m u l t i p e r i o d i c .
2n p e r i o d i c f u n c t i o n i n e a c h of
p h a s e s 0 where 80 = $ a n d which a l s o depends on i i i p a r a m e t e r (or set of p a r a m e t e r s )
n
In t h a t case
f
n
amplitudes
Ai
and a
R.
A more comprehensive p a p e r is b e i n g w r i t t e n j o i n t l y w i t h my c o l l e a g u e Mike
C r o s s who s t a r t e d me t h i n k i n g a b o u t t h e s e problems a g a i n d u r i n g a most p l e a s a n t l e a v e s p e n t a t t h e I n s t i t u t e of T h e o r e t i c a l P h y s i c s a t t h e U n i v e r s i t y of C a l i f o r n i a i n Santa Barbara.
I am most g r a t e f u l f o t h e i r h o s p i t a l i t y a n d t o t h e
e n e r g e t i c l e a d e r s h i p of P i e r r e Hohenberg who was a c o n s t a n t s t i m u l u s .
209
Two Dimensional Convection Patterns 2.
The p r e d i c t i o n s of p r e s e n t t h e o r i e s :
Consider
w(X, Y , T )
g i v e n by
a ----w aT
on
-a)
<
X, Y
<
+
(V2
+
1)2w
-
Rw
+
w2w*
=
0
( 2)
-, The p r i n c i p a l r e a s o n f o r c h o o s i n g t h i s model is t h a t it
possesses a simple f u l l y nonlinear periodic s o l u t i o n e
io,
+ + X, X = ( X , Y ) .
+
0 = k
The " c o n d u c t i o n " s o l u t i o n
w = 0
is u n s t a b l e t o
modes of t h e form
w(X,
whenever R
>
Y, T) = We
ic
k
(3)
2
f i i n ( k - 1 ) 2 = 0 , which v a l u e i s r e a l i z e d when
n e u t r a l s t a b i l i t y c u r v e in t h e
( R , k)
p l a n e is g i v e n by
($1 R
=
k = 1.
The
We
(k2-1)2.
o b s e r v e t h e s y s t e m i s d e g e n e r a t e in t h e s e n s e t h a t any mode of t h e form ( 3 ) with
lcl
= 1 w i l l grow a t t h e same r a t e when R
>
0.
The n o n l i n e a r s a t u r a t i o n
of t h e l i n e a r l y u n s t a b l e modes i s d e s c r i b e d by t h e S t u a r t - W a t s o n [ 8 1 e q u a t i o n
- w%*
WT = Rhl
(4)
which in t h i s c a s e is a n e x a c t s o l u t i o n f o r ( 2 ) b u t which g e n e r a l l y i s o n l y t r u e f o r v a l u e s of
R
p r o p o r t i o n a l t o JR
near its c r i t i c a l value
- RC
. Because
RC
=
0
and f o r amplitudes
W
of t h e o r i e n t a t i o n a l d e g e n e r a c y , i t i s i n d e e d
n a t u r a l t o look f o r s o l u t i o n s of t h e form
a n d , in t h e small
R
limit, we c a n show
It is a n e a s y m a t t e r t o s e e t h a t t h e o n l y s t a b l e s o l u t i o n of t h i s set is t h e
single r o l l s o l u t i o n
- iZI
W1 = JR
e
k ,
w
j
= O , j # I .
(6)
210
Alan C. NEWELL
So, j u s t a s i n t h e case of the Rayleigh-Benard problem under a v e r t i c a l l y
symmetric, a p p l i e d temperature f i e l d with moderate P r a n d t l number u / K > 0 ( 1 ) , s i n g l e , p a r a l l e l r o l l s a r e p r e f e r r e d . pointed o u t , no d i r e c t i o n i s chosen a p r i o r i .
But, a s we have
Therefore when
R
is suddenly
r a i s e d from s u b c r i t i c a l t o s u p e r c r i t i c a l v a l u e s , the system a c t s a s a n o i s e A t any p a r t i c u l a r l o c a t i o n i n a l a r g e h o r i z o n t a l l a y e r , i t w i l l
amplifier.
choose among t h e wavelengths of the noise f o r one c l o s e t o n o t make any choice among d i r e c t i o n s .
kC
1
but i t
W i l l
Therefore u n l e s s the experiment i s
c a r e f u l l y c o n t r o l l e d ( a s i n t h e case of the Busse-Whitehead [ 9 1 and WhiteheadChen [ 101 e x p e r i m e n t s ) , r o l l s of approximately the c r i t i c a l wavelength b u t of d i f f e r e n t d i r e c t i o n s w i l l s p r i n g up in d i f f e r e n t p l a c e s .
This d i r e c t i o n a l
d i v e r s i t y is even more a c c e n t u a t e d due t o t h e i n f l u e n c e of a c l o s e d boundary.
(i is t h e
= 0
A t each boundary l o c t i o n ,
u n i t normal t o t h e boundary) and
t h e r e f o r e i f n i s continuous, r o l l s of a l l d i r e c t i o n s k a r e e x c i t e d .
Now, once
t h e r o l l d i r e c t i o n is chosen a t a p a r t i c u l a r l o c a t i o n , i t becomes more s t a b l e a g a i n s t l i n e a r d i s t u r b a n c e s of r o l l s of o t h e r d i r e c t i o n s t h e more i t grows So, f o r e a r l y times, one f i n d s a f l u i d l a y e r
towards i t s s a t u r a t i o n amplitude.
resembling a sea of q u a s i s t a b l e patches and somehow t h e f l u i d has t o f i n d a way t o resolve the i n c o m p a t i b i l i t i e s between them. There is a l s o a bandwidth degeneracy. f i n i t e bandwidth of wavenumbers
WT and W
+
JR
-
(k2
-
1)’
k
= (R
It i s e v i d e n t t h a t f o r R
can be e x c i t e d .
-
(k2
-
Indeed f o r
>
0, a
t 1,
- W%*
1)’)W
a s y m p t o t i c a l l y i n time.
We may t e s t t h e ( l i n e a r )
s t a b i l i t y of these s o l u t i o n s by s e t t i n g
w = eikX (W
+ b l e ILX+ IMY
+
-iLX
-
b2e
iMY
1
whereupon we f i n d a f t e r some c a l c u l a t i o n t h a t t h i s s o l u t i o n is u n s t a b l e i f
2A2(A2)’(L2~2)+”2(A2)’’(
2k2L2)+4k2L2(A2)’2 > 0
(5)
Two Dimensional Convection Patterns where A 2 = R
-
(k2
-
1)*
and
(A2)'
. The f i r s t
dA2/dk2
=
21 1 c l a s s of
i n s t a b i l i t i e s occurs f o r
L = 0,
M
f
0
2 2 ' B = (A ) ( A )
and
These correspond t o t h e zig-zag a wavevector with wavenumber k
>
0.
i n s t a b i l i t i e s discovered by Busse and a r i s e when
<
1 (in which case ( A 2 ) '
g i v e s i t s energy t o e i t h e r one of two modes ( k , f circle.
+
M = 0
>
+ 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
0,
o r e q u i v a l e n t l y when (7)
$kB>O
-
where B = A 2 dA2/dk2 = -2(k2
1)(R
-
(k2
. This
-
is t h e Eckhaus
i n s t a b i l i t y and occurs when a r o l l has too s m a l l a wavelength.
It is useful t o
summarize these r e s u l t s by way of f i g u r e 2.
k8, A
I
I
I
I
I
R k Figure 2: Graphs of A , kB and R Busse Balloon
VS.
k. and t h e
212
Alan C . NEWELL
S o l u t i o n s can e x i s t f o r k
< k <
L
k
R
b u t a r e s t a b l e o n l y when kC
s h a d e d a r e a i s known a s t h e B u s s e b a l l o o n .
<
k
<$
When d e r i v e d i n t h e c o n t e x t of t h e
The r i g h t
e q u a t i o n s i t i s somewhat more c o m p l i c a t e d .
f u l l Overbeck-Boussinesq
. The
hand c u r v e which is t h e boundary w i t h t h e Eckhaus i n s t a b i l i t y i s r e p l a c e d f o r l a r g e P r a n d t l number by a boundary t o a n i n s t a b i l i t y t o r o l l s i n t h e
On t h e o t h e r h a n d , f o r s m a l l e r P r a n d t l numbers, t h e
perpendicular direction.
Eckhaus s t a b i l i t y boundary becomes l i n k e d w i t h t h e s k e w - v a r i c o s e i n s t a b i l i t y which i s a v a r i a n t of t h e f o r m e r when mean d r i f t e f f e c t s are i n c l u d e d . show how t o i n c l u d e t h e s e i n t h e model.
k
boundary need n o t remain a t
=
We w i l l
We a l s o remark t h a t t h e l e f t hand
kC = 1 b u t can bend l e f t w a r d d e p e n d i n g on
The B u s s e b a l l o o n ( o r Busse w i n d s o c k ) , t h e r e g i o n of s t a b l e
P r a n d t l number.
p a r a l l e l r o l l s i n the R, p Since f o r R
>
R
C'
-
v/K, k
p l a n e i s g i v e n in r e f e r e n c e [ 3 1 .
there i s an order JR
i n a direction parallel t o
-
$ a n d a n o r d e r 4JR
R
C
band o f a l l o w a b l e wavenumbers
- RC
band 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 , i t i s n a t u r a l t o i n c l u d e a r i c h e r c l a s s of s o l u t i o n s which h a v e a n a l m o s t p a r a l l e l r o l l s t m c t u r e by l e t t i n g
-Y. -t ) e i x
w ( x , Y, T) = WC;,
2 4 where x = u X, y = uY, t = p T w i t h N
(8) 4
x
=
R
-
RC
<<
1. This g i v e s t h e N e w e l l -
Whitehead-Segel e q u a t i o n (51 which f o r t h e model ( 2 ) i s
aw
4.
a7
(21
a+ a;
a
2
2 w 3)
=
xw - w%*
.
(9)
This e q u a t i o n i s c a n o n i c a l f o r d e s c r i b i n g s i t u a t i o n s in which t h e r o l l s a r e a l m o s t p a r a l l e l and
3.
R
i s c l o s e t o i t s c r i t i c a l value.
A new a p p r o a c h :
We w i l l now g i v e a d e s c r i p t i o n t h a t a l l o w s t h e l o c a l w a v e v e c t o r undergo o r d e r one changes c o n t i n u o u s l y b u t s l o w l y o v e r t h e box. w(X, Y , T) where
W
and
ie(X, = We
Y, T)
= VO a r e f u n c t i o n s of t h e s l o w v a r i a b l e s
to
Let ( 10)
Two Dimensional Convection Patterns 2
213
4
2
x = E X , ~ = E Y , ~ = E T
and I / E
2
is t h e a s p e c t r a t i o .
I t i s u s e f u l t o write 9 =
1 7 O(x,
y , t ) whence
E
V O = V+9 x G
T
=
k’
= (m,
n)
(k cos JI, k sin J I )
=
x
(11)
2
2
= E B
t
= E D
We w i l l a l s o f i n d i t u s e f u l t o i n t r o d u c e new c o o r d i n a t e s 9 = a(x, y ) ,
B = B(x, Y )
d e f i n e d by
ax = k Cos $
Bx = -P Sin J,
ay = k S i n JI
By = II Cos JI
The J a c o b i a n of t h e t r a n s f o r m a t i o n from (a, B )
to
( x , y)
is
kll
and
The curves a ( x , y ) = c o n s t a n t a r e . of course, t h e l o c i of c o n s t a n t phase 8 while the B c o o r d i n a t e s a r e the orthogonal t r a j e c t o r i e s and measure d i s t a n c e a l o n g t h e rolls.
by ll
The c u r v a t u r e K
$ . Tne
a
of the
a ( x , y)
=
c o n s t a n t curves is given
c u r v a t u r e of t h e c o n s t a n t curves B
is
K
B
=
-k
3.
I n t h e s e c o o r d i n a t e s , the c o m p a t i b i l i t y c o n d i t i o n s ( 1 1 ) g i v e us t h a t
and
Note t h a t
ma Aeie = e i e ( i e 2 u V 2 Aeie = eie(-k2
+
E4
&)A
+ ie2Dl +
,
r4D2)A
,
214
Alan C . NEWELL
where
a +
D ~ A=
=
2k2
=
2k
2n
+ Ak
+ a + (2+ % ) A ak
+ Akll
a$
,
,
2 aA ak 2 ae ~ + A k ~ - A k / L ~ ,
and
a2
a'
Q 2 A = 2 + 7 A
We now proceed t o determine the e q u a t i o n s s a t i s f i e d by the slow v a r i a b l e s and
the l a t t e r being t h e l e a d i n g approximation t o
A,
loss of g e n e r a l i t y , can be taken t o be r e a l .
u = a. R-A'
W
which, w i t h o u t
Let
... , = R + eS2 + ... . 0 +
E'O
+
We choose t h e sequences [ a n } , {R,) appearing i n the expansion f o r
W.
i n a manner so a s t o e l i m i n a t e s e c u l a r terms In t h i s c a s e , s e c u l a r means t h a t no s o l u t i o n
W2, W 4 e t c .
e x i s t s t o the a l g e b r a i c e q u a t i o n s f o r t h e 0 v a r i a b l e and expanded w = Aeie terms were removed, no 2ll
(20),
(21),
2 [ic a
+
c4
E2W2(o)
+
..., t h e n , u n l e s s
periodic solution f o r
( 1 6 ) , (17) i n t o ( 2 ) g i v e s
(22),
2 at +
+
(k
2
-
I f we had l e f t i n
2 2 2 i e (Dl(k - l ) * + ( k - l ) D l * )
w2
would e x i s t .
the secular Substituting
Two Dimensional Convection Patterns
+
- c4(D2(k2-l)*
-R
0
- r%, -
(k2-1)D2.
+ D12) + i E 6 ( D 1 - D 2 + D 2 * D I ) +
€k4 ....+ k 2 A W 2
What does t h i s e q u a t i o n t e l l u s ? R
0
+ W22)...](A
ic4(2AW4
At
+
215
68D 2
c?W2
+
2
...) = 0 .
O(1)
(24)
= ( k 2- 1 ) 2
and so, t o leading order, the amplitude
is d e t e r m i n e d from t h e " e i k o n a l "
A
equation A A t order
2
= R
E~
2
-
( k -1)
2
.
(25)
we have t h a t
+
(-Ro
(k2-1)2)W2
+ a%, =
R.H.S.
Note t h e f o l l o w i n g i n t e r e s t i n g f e a t u r e .
For
Ro = ( k 2 -
t h e r e f o r e , e v e n though
R
- RC
= 0(1),
A2
is f i n i t e and
t h e o n l y t e r m s on t h e RHS w h i c h
g i v e r i s e t o s e c u l a r b e h a v i o r a r e t h o s e which a r e p u r e l y i m a g i n a r y .
were s m a l l , we would h a v e t o remove a l l t h e terms on t h e
A2
o t h e r hand, i f
On t h e
RHS.
I n o t h e r words, t h e n u l l s p a c e of t h e l i n e a r i z e d e q u a t i o n i s c u t i n h a l f
when
A = O(1).
f o r the amplitude
What t h i s means i s t h a t i n s t e a d of h a v i n g t w o e q u a t i o n s , one
A
a n d t h e o t h e r f o r t h e p h a s e of t h e c o n v e c t i v e p a t t e r n , we
s i m p l y have a s i n g l e e q u a t i o n f o r t h e p h a s e . a l g e b r a l c a l l y from ( 2 5 ) . of small
R
-
RC
The a m p l i t u d e is d e t e r m i n e d
T h i s a l s o means, of c o u r s e , t h a t t h e l i m i t t o t h e c a s e
and small a m p l i t u d e
A
is f a i r l y s u b t l e .
[For workers i n
n o n l i n e a r wave t h e o r y , t h e r e i s a d i r e c t a n a l o g u e between t h i s l i m i t p r o c e s s and t h e l i m i t p r o c e s s one e n c o u n t e r s when one a t t e m p t s t o o b t a i n t h e 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 from Whitham's t h e o r y . ]
We w i l l c a r r y o u t t h e p e r t u r b a t i o n
a n a l y s i s i n such a way s o a s t o f a c i l i t a t e t h i s l i m i t p r o c e s s .
What we do is t o
u s e t h e e x p a n s i o n ( 2 2 ) which i s a l s o a n a m p l i t u d e e x p a n s i o n t o r e e x p a n d n e c e s s a r y so t h a t we can s i m p l y set W t h e following equations,
2j
= 0, j
>
A
1. We f i n d d i r e c t l y f r o m
if
23)
Alan C. NEWELL
216 uA
- Dl(k2-l)A -
R-A*
(k2-1)
-
2
=
(k2-l)DIA
E4 ii ( A t -
-D$
+ E 4 (D1*D2 + D2*Dl)A = 2 (k -1)Df
+E'D~)
-
0,
(26)
2
D2(k - l ) A
.
Because of t h e s i m p l i c i t y of t h i s model, t h e s e e q u a t i o n s a r e e x a c t .
W e will
f i r s t examine t h e s e e q u a t i o n s w i t h a veiw t o making c o n t a c t w i t h known r e s u l t s a n d t h e n we w i l l discuss aome new consequences.
4.
Connections with previous t h e o r i e s . F o r v a l u e s of
and then
A
t h e phase u
of o r d e r u n i t y , we c a n n e g l e c t t h e RHS of e q u a t i o n ( 2 7 )
R
is g i v e n as f u n c t i o n of
-
k
by ( 2 5 ) .
Equation (26) tells u s about
4
8, a n d i g n o r i n g t h e O(E ) terms can be w r i t t e n as
Az Eae +
g m ~ +
an^=^,
7
or
where
B(k)
-
A2(k) dA2/dk2
.
(30)
I want t o rernark a t t h i s p o i n t t h a t t h e f a c t t h a t t h e s p a t i a l t e r m s h a v e c o n s e r v a t i o n form is n o t a c o n s e q u e n c e of t h i s p a r t i c u l a r model n o r t h e f a c t t h a t i t can be d e r i v e d from a Lyapunov f u n c t i o n a l . (a)
The Busse B a l l o o n h o l d s l o c a l l y .
E q u a t i o n ( 2 9 ) i s e l l i p t i c s t a b l e o r u n s t a b l e ( i n t i m e ) or h y p e r b o l i c u n s t a b l e d e p e n d i n g o n which one of t h e f o l l o w i n g f o u r c a s e s o b t a i n s : 1)
2)
B B
3)
B
4)
B
<
d 0, =(kB)
Elliptic stable
>
< >
0;
d 0, =(kB)
0;
E l l i p t i c unstable
> <
d 0, x ( k B )
<
0;
Hyperbolic unstable
d 0, &kB)
>
0;
Hyperbolic unstable.
217
Two Dimensional Convection Patterns
These r e s u l t s are s i m p l y t h e same r e s u l t s which a r e d i s p l a y e d in F i g u r e 2,
x, y
e x c e p t now a l l t h e v a r i a b l e s a r e f u n c t i o n s of
and
t.
T h e r e f o r e w e can
s a y t h a t a l l t h e s t a b i l i t y f e a t u r e s we h a d f o u n d when l o o k i n g a t t h e s t a b i l i t y of s t r a i g h t p a r a l l e l r o l l s c o n t i n u e t o h o l d l o c a l l y .
Case (1) above i s t h e
Busse b a l l o o n ; c a s e ( 2 ) i n v o l v e s i n s t a b i l i t i e s which have wavenumber dependence
in b o t h t h e a l o n g a n d p e r p e n d i c u l a r t o t h e r o l l d i r e c t i o n s ; c a s e ( 3 ) i s t h e z i g zag i n s t a b i l i t y a n d c a s e ( 4 ) is t h e Eckhaus i n t a b i l i t y . by t a k i n g t h e l o c a l r o l l w a v e v e c t o r t o be
ae + xd (
A x
Hence f o r B (k, 0);
>
for B
a% +
k B ) T
ax
0, $(kB)>
<
0,
B
a% = -
0
( k , 0)
T h i s can be e a s i l y s e e n
i n which c a s e ( 2 9 ) becomes
.
(32)
aY
0, t h e i n s t a b i l i t y h a s a w a v e v e c t o r p e r p e n d i c u l a r t o
arrd 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.
x
and
t
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 +
-
2
( J I J M ) = kc
2
)
.
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
1
V
+
k B and
218
Alan C . N F W LII
some of t h e
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
R
-
( 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.
E
4
x.
For r o l l s which a r e
and
e
a =
where x = direction.
E
2
=
x
+
E
2
7)
+(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~
,
(I
= at =
2 E
$t
Y Y
K,
=
kq8 =
+yy
and
K
e
=
-kqa
=
-~+,;j-
E+-$-
Y
(35)
w'
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
E
2
g i v e s (we d r o p t h e t i l d e on
Y)
A$t
-
1
2
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
1
+
2
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
+
2
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 ) .
E
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.
5.
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
R
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),
A
ae + si;1 V at
+
+
(kB)
E
4
( 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.
a
V(+kB) = k
+
kB
kBII
,
(37)
I n p a r t i c u l a r we may w r i t e
2
o r i n a more r e v e a l i n g way a s V(
applying k
= k2
a i
-aa-A ,
ak a t + k -(aa
a kB -aa L
.
(39)
a i 2 - - t o ( 3 7 ) g i v e s u s two e q u a t i o n s f o r ag A
ai A z a k ~ kkB B + ~+ e~~k~ G ): [ D l * D 2
k
and
+ D2*D1)A
J,
= 0
, (40)
and
For t h e f i r s t s t e p , l e t u s assume t h a t a l l d e r i v a t i v e s a r e of o r d e r one a n d consequently ignore the
E~
t h e Busse b a l l o o n 1 = kC
<
terms. k
<
k
W e w i l l assume t h a t e v e r y w h e r e
kE(R) a n d prove t h a t i n a r e g i o n
R
belongs t o
with c e r t a i n
c o n d i t i o n s on t h e b o u n d a r y a R , t h e s y s t e m r e l a x e s t o a s t a t i o n a r y s t a t e w i t h wavenumber
k
t a k i n g on t h e v a l u e which makes
B = 0.
T h i s r e s u l t does n o t
depend c r i t i c a l l y on t h e f a c t t h a t t h e p r e s e n t model i s d e r i v e a b l e from a Lyapunov f u n c t i o n ; i n d e e d i t i s a l s o v a l i d f o r s y s t e m s which do n o t h a v e t h i s property.
The r e a d e r might l i k e t o v e r i f y t h a t t h e model
a (m
2 2 (V - l ) ) ( V -1)'w
+
( R - ww
*
* 2
2
- ww V ) V w
=
0 ,
which does n o t d e r i v e from Lyapunov f u n c t i o n a l , g i v e s t h e p h a s e e q u a t i o n
(k2
+
1)2A2etk2(1
-
vk2)
+
V*CB
+
O ( E ~ )= 0
(43)
220
Alan C . N E W E L L
-
.
4 2 2 k (1-uk ) a n d A 2 = R
- ( k 2 -t 1)3 I t is c r u c i a l , however, dk2 k2 t h a t t h e o r d e r one s p a t i a l d e r i v a t i v e t e r m s i n t h e phase e q u a t i o n h a v e
where B
A2
c o n s e r v a t i o n form a n d a l t h o u g h I h a v e n o t y e t c a t e g o r i z e d t h e c l a s s , t h i s We now prove o u r r e s u l t .
happens f o r a l a r g e c l a s s of problems.
For p o s i t i v e
J ~ ,l e t J
2
et + v
o
*(CB) =
(44)
and c o n s i d e r
F =
//
G(g) dxdy
(45)
R
where 2 G = -1/2,fk B ( k 2 ) d k 2
i s p o s i t i v e a s we i n s i s t
k
>
0
<
b e l o n g s t o t h e Busse b a l l o o n i n which B
0.
Then,
dF
- =
dt
n
-/
ds
Bt h . n
- //
aR
2
J Bt
2
dxdy,
(47)
R
t h e outward u n i t normal t o t h e boundary a R , where we h a v e u s e d t h e f a c t
that V G =
i:
-b.T h e r e f o r e ,
if on a R e i t h e r (i) $*n = 0 ( t h e r o l l a x e s are
p e r p e n d i c u l a r t o t h e f i x e d b o u n d a r y ) o r (ii) B
-dF
G
k
decreases. f o r which
model ( 4 2 ) ,
0 ( a p o r t i o n of aR may b e a
B = O),
fluid boundary where
and
-
But
is minimum o n l y when
G
dA2/dk2 = 0.
+
B = 0
or
k = kC
For model ( 2 ) t h i s is t h e point
kC l i e s t o t h e l e f t of
The f a c t t h a t k
I
k
-
1
the value
k = 1; f o r
by a n amount d e p e n d i n g on v
.
kC on a f r e e boundary on p o r t i o n s of aR i s c o n s i s t e n t
w i t h t h e a n a l y s i s of t h e s t a t i o n a r y e q u a t i o n ,
This mans that the quantity 2 kB/e = -H ( 8 )
(49)
i s c o n s t a n t a l o n g t h e o r t h o g o n a l t r a j e c t o r i e s of t h e c o n s t a n t phase c o n t o u r s . R e c a l l in t h e i n t e r v a l 1
<
k
<
kE, kB
<
0. T h i s in t u r n means t h a t i f
t h e 6 c o n t o u r s c o n v e r g e , which t h e y w i l l do in p a t c h e s where t h e c u r v a t u r e of
22 1
Two Dimensional Convection Patterns t he phase contours i n c r e a s e s torwards a c e n t e r , L i n c r e a s e s .
h
t h e f l u x of
I t a l s o means t h a t
between two 6 c o n t o u r s is i n d e p e n d e n t o f a. I t may be u s e f u l f o r
t h e r e a d e r t o keep i n mind t h e a x i a l l y s y m m e t r i c c a s e where a = I k ( r ) d r ,
r
6
the r a d i a l coordinate,
c o o r d i n a t e , whence L =
. Now
=
JI
=
Q,
4 the azimuthal
i n o r d e r f o r t h e s o l u t i o n t o remain s t a b l e we
must have t h a t l < k < k E which imp Les t h a t 0
where IkB
i s t h e a b s o l u t e v a l u e of
0
<
LH2(6)
and t h e r e f o r e a s L
< +
c4
before the aa
2
0
H (5) = kB
(E
E
-2
K(B)
<
lkBIE T h e r e f o r e we must have t h a t
kE B(kE).
-
(50)
along a
6
8
contour,
H2(6) m u s t tend t o zero.
Since i t
c o n t o u r s , i t must become a s small a s i t c a n
terms i n ( 3 7 ) e n t e r t h e p i c t u r e , which t h e y w i l l do when and
)
2
lkBl
lkBIE
is c o n s t a n t a l o n g c o n s t a n t
a = -
<
L = O(E-’).
a n d hence
is small only n e a r
Thus, in o r d e r t h a t t h e i n e q u a l i t y (50) h o l d s , kB = O(E 2) e v e r y w h e r e t h a t
kc = 1,
we must have
k
=
1
L = O(1).
+
But, s i n c e
2 O ( E ). Near t h e s i n k , we
can f i n d s o l u t i o n s of ( 3 7 ) i t e r a t i v e l y in t h e form 2 2 2 k = 1 += + , ,-+ =- 3y + 1 L
where y 2 =
4R
..
2 r + ~ ~ 4v K ( B ) , which i n d i c a t e s t h a t a s r
v a l u e somewhere b e t w e e n
kC = 1
and
e2, k
+
g o e s from
kC = 1
to a
kE.
Thus o u r f i r s t p r e d i c t i o n i s t h a t on t h e time s c a l e
E - ~
,
the horizontal
d i f f u s i o n t i m e , p a t c h e s form which s a t i s f y t h e boundary c o n d i t i o n s a l o n g p o r t i o n s of t h e box b o u n d a r y , a n d in which
k
+
kc
k’
n = 0
almost everywhere.
Examine t h e n u m e r i c a l e x p e r i m e n t s of G r e e n s i d e , Coughran a n d S c h r y e r [ 111 ( f i g u r e 3 ) c a r r i e d o u t on e q u a t i o n ( 2 ) f o r r e a l b r g e [121 ( f i g u r e 4 ) .
J,
a n d t h e real e x p e r i m e n t s of
Alan C . NtwEl
222
F i g . 3:
Fig.
4:
I_
-
Numerical I n t e g r a t i o n of ( 2 ) , J, R e a l H o r i z o n t a l D i f f u s i o n Time F o r Time
.
From a n e x p e r i m e n t of P. Berge Contours of C o n s t a n t Downward V e l o c i t y . A s p e c t R a t i o of 1 6 . R 2Rc.
-
223
Two Dimensional Convection Patterns Note i n t h e r e c t a n g u l a r geometry of f i g u r e 3 , t h a t p a t c h e s w i t h c i r c u l a r symmetry form a b o u t t h e c o r n e r s A a n d C.
I n t h e c i r c u l a r geometry o f
Berge ' s r e a l e x p e r i m e n t , one a g a i n s e e s c i r c u l a r p a t c h e s f o r m i n g a b o u t s i n k s which a r e a t t a c h e d t o t h e boundary.
Moreover, i t i s a b u n d a n t l y c l e a r from t h e s e
f i g u r e s t h a t t h e box c a n n o t be t i l e d w i t h t h e s e p a t c h e s .
Certain areas, f o r
example t h e c o r n e r B a n d D i n f i g u r e 3 , a r e q u i t e i n c o m p a t i b l e . compensate f o r t h e s e mismatches,
the
In order to
terms of t h e phase e q u a t i o n ( 3 7 ) must
E~
be i n c o r p o r a t e d i n t he a n a l y s i s . One way i s t o t a k e a / a a , 3 / 3 8 t o the microscopic theory.
t o be
O(E-')
b u t t h i s simply b r i n g s u s back
A n o t h e r way, which r e t a i n s t h e f u n d a m e n t a l i d e a t h a t
t h e c o n v e c t i o n f i e l d c a n b e d e s c r i b e d by a s l o w l y v a r y i n g w a v e v e c t o r
c,
is t o
.
This
recognize t h a t t h e terms
can b a l a n c e when t h e
B
derivatives are
r e d u c e s b o t h terms i n ( 5 3 ) t o
O(E-')
and
k = 1
+
O ( E ~ ) a n d s i n c e w e saw t h a t t h e main e f f e c t of
t h e dynamics 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 i s t o d r i v e kc, if
t h i s approximation is not a t a l l unreasonable. R = 0(1),
k = 1
+
2
O(E )
O ( E ~ ) , then
A
is
k
towards
A l i t t l e a l g e b r a shows t h a t
fi t o w i t h i n
O ( E ~ )a n d t h e
s t a t i o n a r y p a t t e r n s which one might e x p e c t t o r e a c h on t h e c-6 t i m e s c a l e (which i s 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 m u l t i p l i e d by t h e a s p e c t r a t i o
g i v e n by
and t h e c o m p a t i b i l i t y c o n d i t i o n s ( 1 1 ) a r e
c-')
are
224
Alan C. NEWELL
E q u a t i o n s (54), (55) show i t s e l f i s of o r d e r straight.
US
that
is a t most o r d e r o n e a n d t h e r e f o r e
JI
For t h e s e s o l u t i o n s , t h e n , t h e r o l l s a r e l o c a l l y a l m o s t
E.
S i n c e we a r e now working i n d i s t a n c e s of o r d e r
E
( a s measured in box
i n r o l l w a v e l e n g t h u n i t s ) we c a n make t h e f o l l o w i n g l o c a l u n i t s ; I/E a p p r o x i m a t i o n s , which a r e v e r y s i m i l a r t o t h o s e made when we were d e r i v i n g t h e Newell-Whitehead-Segel
equations (36).
Let
( 6 , ~ ) be l o c a l l y t h e ' a c r o s s a n d
along' r o l l cordinates and
a sa = - - i a E
as
1
a n d e q u a t i o n (53) i s ( u s i n g s u b s c r i p t s f o r p a r t i a l d e r i v a t i v e s )
which is p r e c i s e l y t h e Newell-Whitehead-Segel amplitude A h e l d constant.
equation ( 3 6 ) and (9) with
We a r e now g o i n g t o d i s c u s s s o l u t i o n s of t h i s
e q u a t i o n which l e n d some i n s i g h t i n t o F i g u r e 5 which is t h e a s e q u e l t o F i g u r e
3
225
Two Dimensional Convection Patterns
B
C
A
Figure 5:
Numerical I n t e g r a t i o n of (z), $J real For Time >> H o r i z o n t a l D i f f u s i o n Time.
Notice t h a t i n o r d e r t o compensate f o r t h e i n c o m p a t i b i l i t i e s i n t h e c o r n e r B of Figure 3, the c i r c u l a r r o l l s emanating from AD have undergone a change and have R o l l number 7, counting from A a l o n g
i n t r o d u c e d d i s l o c a t i o n s along t h e w a l l AB.
AD, doubles i n width a s i t approaches t h e s i d e AB and undergoes a d i s l o c a t i o n . Roll number 9 detaches from AB a l t o g e t h e r and forms a series of d i s l o c a t i o n s along AB (which we c a l l a g r a i n boundary) and then a t t a c h e s i t s e l f t o the upper
wall
BC.
R o l l s 10 through 25 t a k e on an
S
shape i n which t h e approximate
d i s t a n c e o v e r which s i g n i f i c a n t changes occur i s t h e square r o o t of the box dimension, o r the 'along the r o l l ' s c a l i n g in e q u a t i o n (58).
226
Alan
c. N E W E l 1
Figure 6:
Dislocation
W e f i r s t n o t e t h a t a property of t h i s e q u a t i o n is t h a t i f
$ ( 5 , 1 ; ) solves
Also observe t h a t
o
=
(581,
0 = ~
5 + e(c,s),
-
6
so does
-@(-L,T,).
(59)
~i s 9g i v e n by
= E
2?
5.
The shapes of the phase contours near d i s l o c a t i o n s s u g g e s t that w e search f o r s e l f s i m i l a r s o l u t i o n s of the form
227
Two Dimensional Convection Patterns
F(z)
s a t i s f i e s the equation,
F' =
which i s r e a l l y a n e q u a t i o n f o r
G
=
dz
'
F'.
It h a s t h e symmetry p r o p e r t y t h a t i f
s o l v e s ( 6 3 ) , s o does
( F ( z ) , G(z))
E q u a t i o n ( 6 3 ) h a s a one p a r a m e t e r f a m i l y of s o l u t i o n s F ( C ; z ) w i t h d e r i v a t i v e s 2 G(C,z) which decay as Ce-' a s z + -This can b e s e e n by l i n e a r i z i n g ( 6 3 )
.
which t h e n h a s e r r o r f u n c t i o n s o l u t i o n s .
For
C
v e r y small, t h e s e s o l u t i o n s
behave v e r y much l i k e t h e e r r o r f u n c t i o n s o l u t i o n s ; t h e y a r e s y m m e t r i c a b o u t z = 0
and l e a d t o a jump i n
nonlinear and the bigger (actually
G
F
-
Iln
F
Co
of
C,
F
approaches its pole s o l u t i o n s
1 2-zo
(65) and 6 F f 2 F " .
Thus t h e r e i s a c r i t i c a l
which from n u m e r i c a l c a l c u l a t i o n s i s a p p r o x i m a t e l y . 5 6 4 ,
above which t h e s o l u t i o n s do n o t e x i s t o v e r t h e l i n e approaches
Co from below,
( z n)
v a l u e of 3.14 F(z),
Y(z)
for C =
0 =
5
However ( 6 3 ) i s
= 6 C .
has a logarithmic s i n g u l a r i t y )
r e p r e s e n t i n g a b a l a n c e between F"" value
AF = F(m) - F(-)
g e t s the c l o s e r
C
has the pole;
F(z)
of
TI
at
AF
-m
<
z
<
i s very s e n s i t i v e to changes i n
C = .54 t o 7.93 a t C = .565.
m
.
As
C
C , g o i n g from a
I n f i g u r e 7, w e g r a p h
a n d i n F i g u r e 8 , we draw t h e c o n t o u r of c o n s t a n t p h a s e 0
- + P (%T)
.
228
Alan C. NEWELL
I
/---
-F(T,z),
F(n,z).
Figure 7 :
Graphs of
Figure 8:
Constant Phase Contours
0
.
Two Dimensional Convection Patterns
229
F o r v a l u e s of 0 s l i g h t l y g r e a t e r t h a n n, t h e c o n t o u r s are d e f i n e d f o r a l l values l e s s than
For
<
TI.
t h e phase c o n t o u r s i n t e r s e c t t h e
'5
7. For
= 0 a x i s a t the o r i g i n .
0, we u s e t h e symmetry p r o p e r t y (60) t o i n f e r t h a t t h e p h a s e c o n t o u r s i n
t h i s r e g i o n a r e s i m p l y a r e f l e c t i o n of t h o s e f o r
>
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
-
Rw
+ 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),
V
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
In
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
TH
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
c2
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 ?
2.
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?
3.
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.
2.
A h l e r s G. a n d Walden R. W.
3.
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
4.
Whitham G. B.
5.
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.
6.
Pomeau Y. a n d M a n n e v i l l e P.
7.
S i g g i a E. a n d Z i p p e l i u s A.
8.
S t u a r t J. T. 1960. 371-389. Mech.
9.
Busse F. H. a n d W hi t ehead J. A.
10.
Chen M. M. a n d Whi 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.
2,
9,
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?
problem for circular flaws
Stationary free bo-
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
r
t h e planet,whence
i s a given curve.
unit circle i n R The o u t s i d e of
y
2
.
denoted by
y.
tional.
and
y.
The i n n e r curve
The o u t e r curve
r
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
r
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
Y
r
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
r
V
as f o l l o w s .
and a f u n c t i o n
V
i n fi
Y
234
Hisashi OKAMOTO
(1.1)
AV = 0
in Cl
(1.2)
v = o
on
r
(1.3)
V = a
on
Y ,
on
Y ,
$10~1' + Q +
(1.4)
Is2
(1.5)
Y
UK
1
Y'
Y
= w
= unknown constant
,
0 '
The quantities appearing above are defined below. a , u0 ; prescribed positive constants, the surface tension coefficient (
2
0 )
Q ; a given function defined outside
r
,
;
c7
... given ,
. the curvature of y ,the sign of which is taken to be positive if y is Y'
K
convex,
IQyl
;
the area of 12
Y
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
'Y
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*
+
Y'
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
,r
= (x2
+
y 2 ) l i 2 ) i s an i n t e r e s t i n g case.
Trivial solution.
If
t r i v i a l solution.
Take a number
yo
of radius
Q
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
r
0
> 1 such t h a t nr;
-
TI
= 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
V
U
2
0.
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
1=
(V V
,P
in
R Y'
in
R Y'
on
r
1' 2
that
(1.10)
div
= 0
(1.11)
Y ,
(1.12)
(1.13)
on
(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
fl
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
r
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
is
When a small
u
or t
Q
0
E
C2+'([l,
m)).
Q0 -= 0.
C3+'(S1)
denoted by
(
'
KU.
and
yu
U
a c l o s e d Jordan
( r O + u ( o ), 0 )
( 0
5
0
S1 w i t h a 2n-periodic f u n c t i o n on
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
y
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
fiU.
by
The c u r v a t u r e of
y
is
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
y
U
.
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 = 0
(2.1)
AV
(2.2)
vUJr= o
U
in
v I
,
Ru
,
= a
yu For
u
E
C3+a(S1) , Q
E
C2+a-
(a)
Using a c a n o n i c a l pull-back,we r e g a r d
F (a,Q;u,S)
F(a,QO;O,O) = ( 0 , O )
for
F(a,Q;u,c) = ( 0 , O )
i f and only i f
F(a,*;-,*)
put
as a f u n c t i o n on
1
it i s e a s y t o see t h a t Q
6 c R , we
and
{yu, V
and t h a t f o r some
U
6
E
IR.
i s a continuous mapping from a neighborhood of
xc3+ci ( s1) x R
into
Cl+"(
s1)
x
IR
1
sl.
Then
is a solution
Note t h a t
(Q,;O,O)
in
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 .
a
=
l
+
z
there e x i s t s a p o s i t i v e constant
11 9 - 90 112+a,R <
Define
r
n 0
6
6 we h a m a soZution
+ r
a
n
by
r logr
-n
0
0
0
such t h a t for any {u,[}
Q
c
C2+"(n)
of t h e equation
?'he solution i s unique in some neighborhood o f t h e origin.
'IHEOREM 2.
Asswne t h a t
u = 0.
Let
Qo
E
C19+ci(n) s a t i s f y
satisfying
F ( a , Q ; u , E ) = (0,O).
238
Hisashi OKAMOTO
(2.6)
Then there e x i s t s a p o s i t i v e constant satisfying
11 Q - Q, /llo+(1/2)+a, n
<
E
E
such t h a t f o r any
we have a solution
Q
E
C
lo+( 1 / 2 ) + y n )
t u , ~ I of the equation
F ( a , Q ; u , E ) = (0,O). The next two theorems s t a t e uniqueness o r nonuniqueness of t h e s o l u t i o n . We put
G(a;u,C) = F ( a , Q o ; u , C ) .
Fix a nuturaZ number
lHEOREM 3.
am
(2.7)
f
n.
Asswne t h a t
0
for a l l
an
Then there e x i s t s a branch of nontriviaZ soZutions of If
(a,;O,O).
'IREOREM 4.
n
> 0.
Assume aZso t h a t
m # n.
c(a;u,S)
= (0,0)
through
i s s u f f i c i e n t l y large, then the bifurcation occurs subcriticaZZy.
Asswne t h a t
0 = 0
and ( 2 . 6 ) . Then, in some c2+"-neighborhood of
yo, there e x i s t s no solution other than
yo.
We prove 'IHEOREM 1 by means o f a 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.
On
t h e c o n t r a r y we use a 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 theorem due t o Zehnder [81 i n o r d e r t o prove W O R E M 2 .
?HEOREM 3
i s proved by a b i f u r c a t i o n t h e o r y due t o
[1,2,51. 'I€IEoREM 4 i s a consequence of t h e m a x i m u m p r i n c i p l e . g i v e o u t l i n e s o f t h e proofs.
33.
I n 14 w e w i l l
For t h e d e t a i l s , s e e Okamoto [3,4].
W s u l t s for the Navier-Stdces prdlem. Using t h e n o t a t i o n i n t h e preceding section,we formulate PKBLEMB as f o l -
lows.
(3.1)
F i r s t l y , f o r a given
u c C3+a(S1),we
- VAU + (pV)g= -
(3.2)
div
(3.3)
u=b_
= 0
consider t h e s o l u t i o n of
Vq + V(g/r)
in
Qu
,
in
Qu
,
on
r ,
239
Free Boundary Problems for Circular Flows
Y, ,
on
where
E
X 5 {
E
C3+a(r)2
;
1.
! 6.n dT = 0
r--
The boundary c o n d i t i o n (3.4)
c o n s t i t u t e s a complementary c o n d i t i o n i n t h e s e n s e o f Agmon,Douglis and Nirenberg. T h e r e f o r e , f o r s u f f i c i e n t l y small
u
E
u
where
E
C
3+0
1
(
(S )
H
X,such a s o l u t i o n
E
u. We denote i t by
i s determined uniquely and continuously from we can d e f i n e a mapping
b
and
C3+a(S1)
V
-
g
,Pu.
,q
Hence
by t h e e q u a l i t i e s below.
11 U((3+0
<< 1 )
,5
E
x
R ,b
(
11 b113+a<<
1 )
3
5 , :( u - g ) / r o . Obviously
b E
-
H(b;u,S) = (0,O) for some
i f and only i f
X
{yu,Vu, Puj i s a s o l u t i o n f o r
H(O;O,O) = (0,O). Furthermore
5
E
R .
S i m i l a r l y t o THEo17EM
1 we o b t a i n t h e following
5. isfying
= (0,o).
We can choose a p o s i t i v e constant
IIg113+a<
rl
,there e x i s t s a
{u,C)
I?
suck t k a t , f o r m y
< C3*(S1)xR
The soZution i s unique i n some neighborhood o f
b_
which solves (0,O)
in
E
X
H(b_;u,C)
c3+0 ( s1) X I ? .
In o r d e r t o t r e a t t h e t h r e e dimensional v e r s i o n of PmLEMB,we employ polar coordinates
r
,'a
,$
determined by
z = rsin8.
sat-
240
Hisashi OKAMOTO
The problem t o b e considered i s w r i t t e n as 2 Find a S - l i k e s u r f a c e y
PmLEM B ' .
1=
and f u n c t i o n s
(V,,V
8'
V ),P
4
such
that
(3.9)
- WAX
(3.10)
div
(3.11)
V = b -
+ (pV)v =
1=
-
VP + V ( g / r )
0
in
R
in
R
on
r = 1 ,
Y' Y'
(3.14)
R is a Y
where
domain between
t h e mean c u r v a t u r e of
as
T(1);
y.
S2
and
,
y
i s t h e volume o f
nY' HY
is
The second e q u a t i o n i n ( 3 . 1 2 ) should b e i n t e r p r e t e d
= [cT(x)clg .
If a small
u
E
X
i s given,we denote by
yu
a s u r f a c e p a r a m e t r i z e d as
x = ( r +u(9))cosOcosQ 0 Ifere r
0
> 1 i s determined by
For s u f f i c i e n t l y small for u
E
b_ = ( O , O , b )
X.
( b
Then we put i t
E
Z Vu
1.
-
b
E
2
and
u
E
X
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
Vu
-
and
P
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
HU
of
y,
x
W
.
H.
i s represented as
i s a continuous mapping from a neighborhood of
H Y
H
in
(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
If b
?HEOREM 6.
E
Z
is sufficiently small, then there exists
{u,c)
E
X
x
IR
A
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
a
4
{an}n
.
i s an isomorphism from
C3+&(S1)
x
W
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
F
F
1
(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
ou
€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 .
U
of
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
YU
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
vU
to a
-+
lvVul I
= wu'Vu YU
since
Vu
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
0
= v (
= a -
I1 = aUw modulo
+
v Urw
Since I
3
= 0.
+
+
- vu
= t
+
o(
o(
11 W I ~ + ~ ) .
11 W I ~ + ~ ) with
4
Claim.
If
Proof.
Assume t h a t
a
{an)n
A(a) 5 D
, then
u,s
A(a)
r
and
+ t
on
y
u '
re o b t a i n
(4.6). i s an isomorphism f o r
0
a
4
{anIn.
i s injective.
W
1bnsinnB n=l
Then we have
on
-- - a r avu w.
a tangent vector
F(a,Q ;O,O)
A(a)(w,A) = (0,O).
w =
y,+,
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
in
- v I
y,
Hence
243
b S(a, n ) = cnS(a,n) = 0
We r e p r e s e n t W
+
1
c cosne n=O ( n
2
1)
,
w
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
a
S(a,n)
4
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
U
phism from C3+a ( S j X R
cl+a 1 (S )
$+a
an isomorphism
"
1
(5 )
X ~ R
onto
c2+a 1
onto
(S ) x
IR
.
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
4
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
for a
4
In the
F(a,QO;O,O) is an isomorphism
u>s
{a 1 n n
and that A(an)
has a null-space spanned by
(sinno , 0 ) . ( Here we have used (2.7). )
(cos n9
,0
and
In order to use a theory of bifurca-
tion from simple eigenvalue we use tha following Banach space:
with the norm
/I
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
x
.
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
5
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.
We
Free Boundary Problems for Circular Flows
(4.8)
G (a;xcosne
2
+ ysinne
+
245
$(a;x,y) , < ( a ; x . y ) ) = 0.
The assumption ( 2 . 7 ) and t h e c l a s s i c a l i m p l i c i t f u n c t i o n theorem e n s u r e s t h a t and
5 are
well-defined i n some neighborhood of ( a ; O , O )
ranges a r e i n
(4.9)
(I -P)C3+'(S1) F(a;x,y)
f
, IR , r s s p e c t i v e l y .
in
E 3 , and t h a t t h e i r
Then t h e e q u a t i o n
PG1(a ; xcos no + y s i n no + @ ( a ; x , y ) , S ( a ; x , y ) )
i s a bifurcation equation.
= o
If we write
F ( a ; x , y ) = F l ( a ; x , y ) c o s nB + F ( t i ; x , y ) s i n n6
2
then t h e solution s e t near
c$
(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
PRDPOSITION 4.1. 2.c
C -mapping
The bifururcation func tion
03
F is a C -mapping.
F* defined i n some neighborhood of
(4.11)
F ( a ; x , y ) = yF*(a ; x 2
2
(a ; O )
in
B2
There e x i s t s a such that
2
+ 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.
2
2
2
2
F*(a ; x + y ) = A ( a - a ) + B ( x + y )
as
+ 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*
F = 0
is
O(2)-
246 if
Hisashi OKAMOTO An # 0
and
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
# 0.
Bn
An
determined by t h e s i g n of
Bn.
and
A
Since
B
and
n
n
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.
For w
NOTATIchl.
C3+'(S1)
t
U(w)
we denote by
AU = 0
uJr =
t h e s o l u t i o n of
i n l < r < r 0 ' 0
3
=
U/r=r
aw -r logr ' 0
For
For
w, z
w
E
1 C4+"(5 )
,w2 , w3
t h e symbol
denotes t h e s o l u t i o n of
AY = 0
in
l < r < r o ,
Y = O
on
r ,
c C
?+a( S1)
t h e symbol
ax=o
1 2 3
0
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)
0
0
Free Boundary Problems for Circular Flows
NOTATION '
241
. L
B(w,z) =
w
ay(W,z) ~
ar
ati(w) = r
--
aw
3+a
( W € C
2 r logr 0 0
a2ti(w) +ar
+
2
a 2U(Z) 2
ar
+
Zawz r3logro
Mow the third order derivative of G1
3
aw'z' r3logr 0 0
+
0
( w
,z
1
( S ) ) ,
E
c
4+a
(S1) ) .
is given by
au(wi)
+a 3 1 arw;+1w;+2 r logr i=l 0
+
9[ rO
0
3 -w w w - 2 1 w;wi+lwi+2 12 3 i=l
3
-
3
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
B
n.
i s negative for a l a r g e
n.
In f a c t we have
r0l o g r 0
o-ro
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
6.
H 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.
n. H
6 a r e completed by checking
n
that
D
H(O;O,O)
or D
u.E, a t i v e s a r e given below.
4
D H (O;O,O)w
u l
H(O;O,O)
=
-0 ( 2
is an isomorphism,respectively. The deriv-
w + w" ) +
rO
5
( 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 ) ),
rO
In a way similar t o t h e proof of THEOREM 1 we can prove t h a t an isomorphism from
E
onto
c'+~(s~)
X B .
TO
D H(O;O,O) u,s
treat
D U,S
is
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
h
U,S
(4.131
J-n/2
J
We change the variable from 8
to
-n/2
s E
sine
h
.
Then w(s) 5 w(0)
satisfies
A
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
P
and B P
H(O;O,O)
= Y x H ,we do as follows. Firstly we de-
by the equalities below.
0
v=-(-w"+w'tane 2 2r 0
(4.15)
CI
+pw)-h--+~w-X, 2r2 1-I
(WCX),
0
(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-
(
u
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.
E
(w,h) c X x R satisfying (4.15) and (4.16). Observe
that we have only to show the surjectivity of
YP
Therefore,for a given (v,<)
Y
P'
To show the surjectivity of
Hisashi OKAMOTO
250 V = { f c H ;
f ’ c H } ,
I n v i r t u e o f t h e Lax-Milgram theorem t h e r e e x i s t s , f o r a g i v e n
w
e V
f
E
H
,a
unique
such t h a t
This e q u a l i t y f o r m a l l y i m p l i e s
Yw P
Then w e show t h a t
= f.
This i s shown i n an elementary way.
Hence w e omit t h e p r o o f .
w c X
if
f
E
Y.
By t h e procedure
above we a r r i v e a t THEOREM 6 .
The a u t h o r wishes t o e x p r e s s h i s deep g r a t i t u d e t o Profes-
Acknowledgment.
sor H . F u j i t a f o r h i s unceasing encouragement.
He i s a l s o g r a t e f u l t o P r o f e s s o r
H. Weinberger for h i s important a d v i c e and h e l p f u l d i s c u s s i o n d u r i n g h i s s t a y i n U n i v e r s i t y of Tokyo.
P a r t i a l l y supported by t h e F u j u k a i .
REFERENCES
M.G.
C r a n d a l l and P.H. Rabinowitz
4 (1971) 321-
J. Func. Anal.,
M. Golubitsky and D . S c h a e f f e r
Comm. Math. Phys.,
symmetry, H. Okamoto,
,
3hO.
,
Imperfect b i f u r c a t i o n i n t h e p r e s e n c e of
67 (1979)205-232. ( preprint ).
B i f u r c a t i o n phenomena i n a f r e e boundary problem f o r a c i r c u -
l a t i n g flow with s u r f a c e t e n s i o n ,
D.H.
B i f u r c a t i o n from simple e i g e n v a l u e ,
On t h e s t a t i o n a r y f r e e boundary problem f o r a c i r c u l a t i n g
flow w i t h o u t s u r f a c e t e n s i o n , H. Okamoto
,
Sattinger
,
( preprint ).
Group T h e o r e t i c Methods i n B i f u r c a t i o n T h e o r y ,
Springer
Free Boundary Problems for Circular Flows L e c t u r e Notes i n Math., No. [6]
762 (1979).
D.H. S a t t i n g e r , On t h e f r e e s u r f a c e of a v i s c o u s f l u i d m o t i o n , SOC. London A
[7]
.251
D. Schaeffer
,
,
Proc. R .
(1976 ) 183-204. The c a p a c i t o r problem,
I n d i a n a Univ. Math. J . ,
& (1975)
-
1143 1167.
[8]
E . Zehnder
,
Generalized i m p l i c i t f u n c t i o n theorems w i t h a p p l i c a t i o n s t o
some small d i v i s o r problems, I
, Comm. Pure
Appl. Math.,
8 (1975)91 - 140.
This Page Intentionally Left Blank
Lecture Notes in Num. Appl. Anal., 5 , 253-257 (1982) Nonlinear PDE in Applied Science. U . S . - J a p a n Seminar, Tokyo. 1982 FOCUSING SINGULARITY FOR THE NONLINEAR SCHROEDINGER EQUATION G. Papanicolaou,* Courant Institute, New York University
D. McLaughlin,* University of Arizona M. Weinstein," Stanford University
We summarize recent results on the focusing singularity of the nonlinear Schroedinger equation. In this note we shall give a brief account of some recent work that we carried out motivated by the observations and calculations of Zakharov and Synakh [l]. and in [ 3 ] .
A detailed exposition is given in [ 2 ]
In [ 4 ] the results of careful numerical computation are
reported. The nonlinear Schroedinger equation 2i$t
+
A$
+
1 $ 1 ~ =~ o$ $(O,X) =
,
N
t > O ,
XElR
$,(XI
arises as a canonical problem in which focusing (the nonlinearity with the plus sign in (1)) competes with dispersion (the Laplacian in (1)).
Among the many specific contexts where this occurs we men-
tion nonlinear optics, where N = 3 ,
N = 2
a = 1 , water waves, etc.
and
u = 1 , plasma problems,
The case
N
=
1, u
=
1
has been
studied extensively and was first shown to be solvable by the inverse scattering method by Zakharov and Shabat [ 5 ] . In the analysis of (1) three cases with distinct behavior arise. The subcritical case
u < 2/N
where dispersion dominates and a 253
D.
254
M C L A U G H L I N , G. P A P A N I C O L A O U
global solution i n
C([O,m);H1 (JRN)
)
and
exists.
M . WEINSTEIN
H 1(JRN )
Here
denotes
t h e u s u a l S o b o l e v s p a c e of f u n c t i o n s w i t h s q u a r e i n t e g r a b l e d e r i v a tives.
T h i s r e s u l t i s p r o v e d i n d e t a i l i n [6].
2 a = -
case
s o l u t i o n s of
and i n t h e s u p e r c r i t i c a l c a s e
In the critical 2
a > N
i t i s known t h a t
(1) w i l l blow up i n a f i n i t e t i m e , i . e . t h e i r ' H1
norm
w i l l become i n f i n i t e [ 7 ] . Based on n u m e r i c a l e v i d e n c e and some h e u r i s t i c c a l c u l a t i o n s , Zakharov and Synakh [ l l c o n c l u d e t h a t i n t h e case
N = 2
,
a = 1
( c r i t i c a l c a s e ) i f a n a x i a l l y s y m m e t r i c s o l u t i o n becomes s i n g u l a r a t t = t*
where
then near
R(r)
t*
it h a s t h e form
i s t h e "ground s t a t e " s o l u t i o n o f
R > 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
E
5
~
t < 0
sup /z(t,x)j
(-t)2'7
(4)
-
+
X
o
255
and as
t
+
o.
In other words we have shown that singular solutions of the form (3) exist with
z
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)
If
Schroedinger equation
R
w
=
u+iv
-
w
+
(cr+l)R2"w
+
OR'";
= 0
.
then we may rewrite ( 5 ) i n system form
L+ = -A
L-
=
+ 1-
(2a+l)R2"
-A + 1 - R 2 0
On pairs of functions
f1
9'
in
. 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
H1
x
H1
cause it is not positive definite owing to the nullspace that has. One easily finds that the function
satisfy
be-
L
D. MCLAUGHLIN, G . PAPANICOLAOU and M. WEINSTEIN
256 (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
H1
n2, n 3 ,
n4
X
H1
B
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
B
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.
a
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)
n6
=
(:)
It can be shown that space of
L
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
x
H1
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 ) ,. . . ,
If
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..
GM(m,V)
'm- 1
Plccker's relations (i.e. are PlGcker coordinates). To each set of suffixes
diagram Y
(20,*-*,Lm-l), O
we associate a Young
-(m-l),**-,il-l,ko, respectively
consisting of rows of length
'm- 1
(cf. H. Weyl, The Classical Groups, Princeton, 1939) and often identify them;
cy,
e.g. Plccker coordinates are also written ed in the m x n
rectangular diagram
'mn
the diagrams Y being those contain-
*
After Weyl's celebrated work Young diagrams (of vertical size5 N ) classify irreducible tensor representations of
GL(V).
Denoting by
R.
1j gredient of the irreducible representation space labeled by the
diagram A , .
our
GM(m,V)
13'
the contraix j
rectangular
is the projective algebraic manifold corresponding
m
to the graded algebra
B
Rmj.
(Here multiplication is unambiguously defined
j =O
because R . m1
where
€4
Rmj.
containes Rm,i+j
GM(m,V)
=
(G(m,V)
&(m,V)
=
((5
Let
m6m'
-
) Y YcAmn
and
n
exactly once.) We can also write:
[O})
/ GL(l),
15,
satisfy the PlGcker's relations)c/2%.
Then: (i) if
relations, so does its restriction to Y ' s &(m,n)).
On the other hand, (ii)
(5')
Y YCAmini within Amn
(cy)ycAmn
satisfies the PlGcker's (whence %(m',n')
+.
satisfies the Plccker's relations
26 1
Infinite Dimensional Grassmann Manifold if and only if
(;.;'ycAm,n,
according as Y C A m n
does,
E;
= Q
being defined by
or not (whence a ( m , n )
G
%(m',n').
or
=
0
(i) and (ii)
combined give the commutative diagram
Hence, defining the universal Grassmann manifold
GM
fin dense submanifold GM
by
-
-
(&fin
- (01) / GL(1)
=
(%-{O})
/ GL(1)
and its
respectively, we have
63 dense
surjective
Ti
To each g, tGM(m,n)
(resp. an unrestricted
5, #
0
c,
while
= 0
to which the given Y
u
+ G(m,n)
GMY(m,n), with
(resp. CGM)
uniquely corresponds a diagram Y C Amn
Y ) in such a way that, f o r the Plicker coordinates of
unless Y ' > 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.
GM
=
8,
UGM'). Y
Consider the infinite dimensional vector space V (resp.
c)
consisting of
262
Mikio SATOand Yasuko SATO
elements
5
( 5 u ) u E z , with c u e & , cu
=
(Setting e,, = ( d , , v ) u e Z t ~
=
0
for v<> 0).
5
one also writes
1
=
<,eU
(resp.
5
=
-wu<m
1
Further, by introducing the dual (or contragredient) basis
-<warn
(ep,
E
1
z to (e,,)p E z and the dual space V*
= {<* =
1
-m
G)
to W (resp. to
Ete$lct€K})
by the effectively finite sum:
=
15;Cy,
= {<* = c$e$1~$ e -ca
~ }(resp.6" is given
our vector space naturally acquires
the weak topology (or rather, S . Lefschetz's linear topology, in which our space i s locally linearly compact).
(Any locally convex topology on a vector space
induces via its dual a linear topology there, and its subsapce is closed by the latter if and only if it is Define subspaces (resp.
6(m))=
so
by the former.)
of V
{ ( 5 v ) u E Z ~ V (resp.
GM(resp. GMfin )
=
of
C) 1 5 ,
Then we have
{closed subspaces V
=
0 for vim}.
of V (resp.
Ker and Coker of the natural map
$/+(o)) =
9 ) 1 The dimensions V
+
W/V(O)
(resp.
of
+
are both finite and coincide.}
{closed subspaces V (resp. dim
where the closedness of V
W (m)
i ) ,m EZ, by
(resp. subspaces fl(m)
VnC('))
of W =
IuI
(resp.
i )1
dim
VTiV(")
for vc;; 01,
is a consequence of the other conditions and the
qualifier is dispensable for V C W , while i t is not for V C G . Also we have, for any diagram Y
Y GM
parametrized by
= {VCW
understanding that
I dim
VnW(u);
Ry = u
and for O'a<m,
k
if and only if u > -m+R
m-1-k (u E Z , k e N )
for u < 0.
Between these extremes, GM example we define, for
(Eo,-..,Rm-l),
and
r=1,2,...,
fin GM , come various intermediates.
For
>
,
263
Infinite Dimensional Grassmann Manifold I.,
I
so that we have
Then GMana ( r ) (resp. GMexp (a) ) Vexp(a))
I The
=
{closed subspaces V
of V ana(r) (resp. of
V
dimensions of Ker and Coker of the natural map
Vana(r) /(vana(r))(0)
(resp.
-f
v exp(a)/(Vexp(a))(0))
-f
are both finite and
coincide. 1 where
92.
Time evolution on Denoting by
GM
A the shift operator:
we define for c E ' % 2 t A+t2A 1 = e b;.
its evolution in time variables
by
t = (t,,t,;--)
[(t)
+..a
-fin In the case of [ € G M , &(t) For general
E G G ,
however, E ( t )
is again i n
t
E (c,
v=1,2,-..
should b e understood a s a generalized element
whose components are formal power series in numbers.
sfin for any
rather than complex
(t,,t,,...)
-ana(r)
(In the case of 5 GGana(r) , one has c(t)c GM
if
ltul
is
264
Mikio SATO and Yasuko SATO
sufficiently small for v = r h(t) E % ~ ~ ~ ( for ~ ) t E P:
and are
0
for v > r.
For CeGM
subject to the condition z
v
m
In any case we have, for the PlGcker coordinates 5 Y (t)
where
I$
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
by
i a ~7 .
(After H. Weyl, xy(t)
admits various expressions, one of which is
v. v,
where
v v
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
5
We call
4
(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
x
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
Ea
coefficients cy
satisfy the PlGcker's relations.
265
Infinite Dimensional Grassmann Manifold Theorem 2.
f(t)E(C[[tl,t2,*-*]]
An
is the
T
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
T.
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
E
UiJ
the linear operator on V
sending e
iJ
to
e
and
all the
e K # u , to 0 ( i . e . EVu&eK = c,ev), and by L the vector field uu K’ induced by E (i.e. EL IF(<) Z F((1+EEu,,)6) mod E 2 for any
other on
W
function F
6,
6,’s
of
‘ao..
.p+m . .
diagram Y
on 6%). Luv
.‘m- 1
PV
Since any
F(C )
is a function of the PlGcker coordinates
i s also characterized by:
assuming u+m
labelled by
also labels the same Y
and
‘m-1
u+m 2 0.
(ko,***,Rm-l) since for any
For the shift operator
=
Osi<m 1 Gv+m,a.x
(This poses no restriction on the (O,l,~~-,k-l,~O+k,~~~,Rm~l+k)
kclN.)
A we have: An
=“Ic tu,u+n,
n E 2.
Further, define
Mikio SAio and Yasuko SATO
266 the operator K
1
f
(v)Ev, v+n
1 (;)Ev,
s . t. AK-KA = 1 by
Klcvev
for any polynomial f(v)
of
lvcv-lev to have
v+n + l+cf(KA)An,
For nf0, the infinitesimal operator
(l+tU(b))F(c )
U (k)
=
defined by
I(:)Lv+n
ME;
=
5,
mod
E ~ ,induces
the well-
&, and one can write
defined infinitesimal transformation on
with
f (KA)An = Kk k+n v, and in particular, =
F ( ( ~Kk+ Ak+n)&) E ~
,vl while for n=O
mod
E2
we introduce another vector field M
1
1
(i)(Lvv-M) + (l)Lvv to have an wellv
and set: U(k)
=
O
defined vector field on &. R.-m
1
((
Ik
0 .i<m
and
U(k)'s
Theorem 4 .
)
-
('km)).
In particular U(O) 0
=
0.) M
commutes L
W
and
(k) v '
satisfy the commutation relation
-r(t;b; ) , as the function of
t
and E t G , satisfies, and is charac-
terized up to an arbitrary constant factor by, the following holonomic system of linear differential equations:
Indeed, the first equations (which are of the second order) restrict the Lcy(t)c, of the PlGcker coordinates 5, while the Y remaining (first order) equations fix the coefficients cy(t) to c.xY(t).
solution to a linear form
Here we see that the holonomic system of these linear equations on
{t)x&
produces no linear equation but the system of non-linear (quadratic o r Hirota) equations of Theorem 2 , upon elimination of the variables the direct image by the projection finite dimensional case 121.
{t)x%
+
It)),
E e 6 i i (i.e. upon taking
in a sharp contrast to the
261
Infinite Dimensional Grassmann Manifold
Also remarkable is the close resemblance between this nolonomic system and (Theorem 3 also s u g g e s t s analogy
the system characterizing theta functions [ 3 ] . between r
and
0.)
The holonomic system generated by these equations in Theorem 4 contains
(U(t)-T(:))r
also the equations of the form: differential operator in
=
of the form:
t
0, ktl", V C Z , where
p=F
+ terms of lower degree, with
*..s
sv =
1
a , 0, or
lultlvl
according
a tV
v k-1
as v > 0, =0, or < O .
53.
is a
k! vO,"',vk~l€Z,VO+-..+vk~l= v
v
SvoSv1
T(k) V
(
):(T
0,):'T
=
= sv.)
Soliton equations and their solutions Consider the totality
category P
1
=
av(x)(z)
&
of the microdifferential operators in the formal
d v
, where the coefficients au(x)
are taken from
-m
a given differential ring derivation with
P
dx d
+
:
its adjoint
& (i.e. an associative algebra endowed with the If
@ ). P* =
a (x) d
1
for V > m
= 0
( )'av(x)~
PC&E'.
we write
, and
is again in
Together
for P , Q E
&&
their product PQ E &R is well-defined by employing the Leibniz rule a')&( = =
for v e Z .
(z)a(k)(x)(-&)v-k k>O -Res P*dx. Thus
&
{differential oprators}
fcL,x,,
as a subring. We have:
&
=
E
;
=
&[[XI],
P
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
modules, by letting 5
Icu(&)-'-'
Res P dx, we have
the ring of formal power series in x,
is canonically isomorphic to the quotient module of &x
=
constitutes a (non-commutative) ring including &&
I n the following we choose
and simply write
Setting a (x) -1
( x)
=
Mikio SATOand Yasuko SATO
268 by
5
V*
by identifying lSvev' V
=
-
++
P(K,A)C.
Hereafter we identify them: V with
l(-)vS-y-let
=
6 /Ex.
E
V*,
so
Further we write
V =
that we have:
< S ' , P C > = .
,
We also set
Lana= {1aV(x)(-&)'
136 > 0 s.t. av(x)
are holomorphic in
1x1 < 6,
V
"/FvV
+
0 and
1
d v
av(x)(z)
-m(y<m
I
-v-1
(x) I
v * ==,
are bounded as
1x1 < 6 1,
both uniformly in ={
'Jv! la
3k61N s.t. a (x)
V
are polynomials of
x
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
a;
I lanax,
&ana
for y = l , Z ,
...).
satisfying the additional condition that
W-'xn
Set 'V
=
both @
v
the property that its P l k k e r coordinates not.
For W
Ep
we set y(W)
[ ~ ~ ~ x l l This . definition of (This is because xV' Theorem 5 .
For
W
=
=
5,
=
and
1 or 0 according as Y=$
-1 n 41 (W x )V , where
y(W)
m (i.e. x wv
xnw:
C e v C V ; VrD is a l s o characterized by
n
or
is s o chosen that W-'xnE
does not depend on the choice of such n .
).'V
tv, y(W)
C
GM
and this map is bijective, namely
In this correspondence, the inverse images of
vfin and Wana('),
(0) < &C,(xll
GMfin and
respectively, where wana(l)
GMana(l) are given by
= ~Eana(l) n and
269
Infinite Dimensional Grassmann Manifold Theorem 6. Let &(t)
E
2 t A+t2A t... 1 e 6 as in 52, and let W
=
=
W(t)
Y-’(C
=
(t))
be the corresponding microdifferential operator. Then the evolution of
W(t)
in
is given by
t
-
[WI
-
B W
=
atn
d n W(-) , with dx
Bn
=
d %-1
(W(z)
)+.
i s a differential operator of the n-th order.)
(Bn
Theorem 5 tells that conversely any solution of [ W] More explicitly we have
in a unique way.
w
where
p,
=
is given in the above form
p,(-at)
T(t
;a
T(t;t)
w*
’
v
=
P”( at) T(t;6)
represents the character polynomial
d -1 Put L = w--W dx
.
Then
L
=
d 1-V
lu,(z)
u 20
T(t;b;)
x
I
tl++ X+tl
for Y = A
with
l,,’
u =1, u =0, and 0
’
1
u,
are
differential polynomials of W~,.*.,W,+~,and the above system of evolution equations for W
F1
-atn aL -
immediately implies that for L
- B L - LB,, n
with
as follows:
Bn = (Ln)+
which is also equivalent t o the following system:
[B]
aBm - aBn t atn
BmBn
-
BnBm
= 0,
atm
which constitutes the integrability condition for
(Incidentally, the explicit solution for where a;’
is any element of
[L] (or [B])
GM
$
is given by
containing &
(as subspaces in V) .)
gives infinite number of non-linear equations for U2’ U3’”‘,
known as the equations of Kadomtsev (-4u +u t12u u ) 2,t3 2,t t t 2 2,t t 1 1 1 1 1
-
Petviashvili hierarchy (e.g.
3u2,t t + 2 2
= 0)
Explicit forms of the equations are easier to obtain for v2,v3,.** than for
Mikio SATOand Yasuko SATO
270 u2,u3,*** where v's (vn
are defined as coefficients of
is a differential polynomial of
u2,'**,u
v2,.-.,vn;e.g. v2 = -u2, v3 = -u3, etc. and
dx
=
L + v '-,I 2
and conversely
u
+ v L-2+..* 3
is that of
= -v3, etc.)
u2 = -v2, u
Namely
we have
[$'I
pn(-at)$
=
vn$, for
n
2 2,
and its integrability condition ~,(-a~)v~+~t (Jn,m[v])x
IVl
=
0, for
m,n 2 1,
with
z
i+jvil+j'Vi"+j"+"' i,i',i'',j,jl,jllz; j+ j ' +j''=n i+i 1 and [L], respectively.
+7
as the equivalents of Again, v 50
[$I
is explicitly given by
v
=
n
a (pn-l (-2 t) log T at,
)
Itlr
t1+x
, for n > 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
Erv+i,ru+i as A
U t L
and
E.ii' respectively, s o that we now have
Let the Young diagram Y be labelled by (Lo,~~~,Lmr-l), and for each i = O , . . . , r-1, suppose that there are m. of (i)
as
(ku
r+i)u,o
,...,m. -1'
Young diagram labelledlby
Set m!
1
L =
I s
m.-m 1
(i) (L(i) ,...,Lm.-,).
s.t.
L
to have
i (mod r), whom we rewrite lm! = 0, and call
Yi
the
Then we see that the single diagram
1
Y
and the composite object
((Yo,m~),...,(Yr-l,m:-l))
correspond to each other
.
Infinite Dimensional Grassmann Manifold in 1 to 1 manner.
Accordingly, we rewrite Y'
as
27 1
i'(yo,m~),...,(~r_l,m~-l)'
with the possible change of sign caused by rearrangement of the suffixes il If m!
=
0
for
i=O,...,r-l,
it
is simply written as
'6, o '
.
--
* ,yr-l. All the results for the 1-component case are, mutatis mutandis, generalized
to the r-component case.
For example,
References
[l]
M. Sato, Soliton equations as dynamical systems on an infinite dimensional Grassmann manifold, RIMS Kokyuroku 439 (1981), 30-46.
[2]
M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, Lecture Notes in Math. 287, Springer (1973), 265-529.
[3]
M. Sato, Pseudo-differential equations and theta functions, Astsrisque 2 et 3 (1973).
286-291.
This Page Intentionally Left Blank
Lecture Notes in Num. Appl. Anal., 5 , 273-288 (1982) Nonlinear PDE in Applied Science. i7.S.-Japan Seminar, Tokvo, 1982
L1
and
Approximation of Vector F i e l d s i n t h e Plane
L"
Gilbert Strang Massachusetts i n s t i t u t e of Technology
ABSTRACT
We s t u d y f o u r problems, two i n whose analogues i n
L2
L1
and two i n
a r e t h e f a m i l i a r minimum
p r i n c i p l e s which l e a d t o t h e Laplace equation.
One
p o s s i b i l i t y is t o be given t h e boundary v a l u e
6 = Q
and t o minimize (x,y)
point
IIVJrlll
in
Cl
Lmr
or
llV@~lrn;
i s measured by
the gradient a t a lVJrI2
2 = f,
+ Q Y2 '
In t h e o t h e r problems we a r e given a v e c t o r f i e l d v: R
+
R2,
and minimize e i t h e r
IIVw-vlll
or
~IoW-v!lm.
i n each c a s e we use t h e d u a l i t y t h e o r y of convex a n a l y s i s t o g i v e e q u i v a l e n t s t a t e m e n t s of t h e problem, o f t e n with an i n t e r p r e t a t i o n i n mechanics and o f t e n partly solved.
Nevertheless some q u e s t i o n s s t i l l
remain open.
213
274
Gilbert S T R A N G
L2 norm l e a d s t o l i n e a r equations
Approximation i n t h e
(which a r e forbidden a t t h i s c o n f e r e n c e ) .
In
<
LP, 1
P
<
(0,
t h e equations become n o n l i n e a r but much of t h e a n a l y s i s continues t o apply.
In
i s entirely different:
L1
and
Lm, however,
the s i t u a t i o n
it i s not t h e d i f f e r e n t i a l equation
b u t t h e underlying v a r i a t i o n a l p r i n c i p l e t h a t l e a d s t o an e x i s t e n c e theory, and suggests how t o c o n s t r u c t t h e optimal solution. This note s t u d i e s f o u r t y p i c a l problems for s c a l a r valued f u n c t i o n s on a simply connected domain
r.
s u f f i c i e n t l y smooth boundary
fl
C
R2
with
Each of t h e problems has
a dual--a maximization i n s t e a d of a minimization--and a p p l i c a t i o n s t h e dual i s of equal importance.
i n the
Where one
v a r i a t i o n a l statement i s t h e " s t a t i c " formulation of a problem i n mechanics, with s t r e s s e s a s t h e unknown, t h e o t h e r i s t h e "kinematic" form i n terms of v e l o c i t i e s .
We
w i l l study s e v e r a l combinations of boundary c o n d i t i o n s and
inhomogeneous terms, b u t n o t every p o s s i b l e combination, because a l r e a d y we ask t h e r e a d e r ' s consent about one more thing.
In a d d i t i o n t o t h e dual of each problem, t h e r e i s
another p a i r of optimizatlons ( e q u i v a l e n t t o the given one) c r e a t e d by a s p e c i a l s i t u a t i o n i n s o l u t i o n of
div n
= 0
is
T
=
($
R2,
t h a t the general
- $ x ) for some
Y9
4.
Therefore it w i l l happen t h a t each of our problems has four e q u i v a l e n t forms, and t h a t one of them i s simpler t o s o l v e than t h e o t h e r s .
(For t h e o t h e r s we may l e a r n only t h e value
of t h e maximum o r minimum, by d u a l i t y , without f i n d i n g t h e f u n c t i o n which achieves t h a t v a l u e . )
Some q u e s t i o n s w i l l
remain unsolved even w i t h four a l t e r n a t i v e s .
L' and L" Approximation of Vector Fields
275
The problems arise i n t h e s t u d y of p l a s t i c i t y and optimal p l a s t i c design, and elsewhere; we w i l l g i v e r e f e r e n c e s r a t h e r t h a n a complete d e s c r i p t i o n of t h e s e a p p l i c a t i o n s .
And we do
t h e same f o r t h e proofs of d u a l i t y ; i n our problems t h e y come d i r e c t l y from t h e techniques of Ekeland-Temam [ll, who a p p l i e d t h e Moreau-Rockafellar t h e o r y t o a sequence of important examples i n 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 .
Our c h i e f purpose
i s t o c o n t r i b u t e some a d d i t i o n a l examples, and t h e y have
developed from our j o i n t work w i t h Robert Kohn and Roger Temam. We mention t h a t optimal d e s i g n l e a d s t o more complicated v a r i a t i o n a l problems ([2-31
i s p a r t of a l a r g e l i t e r a t u r e ) ,
and a l s o t h a t p e r f e c t p l a s t i c i t y i n
has r e q u i r e d a new
R3
space of vector-valued displacements and a corresponding analysis
[4-51. The problems i n
t h i s note a r e e a s i e r , b u t
t h e y have n a t u r a l i n t e r e s t and it may be u s e f u l t o organize them more s y s t e m a t i c a l l y . A t t h e end we d i s c u s s some a p p l i c a t i o n s i n optimal d e s i g n
and t h e nonconvex problems t o which t h e y l e a d .
1. The minimum of
l/Vt!!m
w i t h Dirichlet data
It i s t h i s form which can be solved d i r e c t l y , b u t we
begin w i t h t h e f o u r e q u i v a l e n t problems: 1A.
M I N I!n(lm s u b j e c t t o
1B.
MAX
1C.
MIN
1D.
MAX
r
uf
j1VdrIlrn
subject t o subject t o
div o = 0 JslVul 0
rlr = g
=
and
1
on
r
o.n = f
on
r
Gilbert S I R A N G
216
The conversion between 1 A and 1C i s by r o t a t i o n through f = n.n = V $ . t
Therefore
=
of
s/2
It follows t h a t
VQ.
&/&, t h e t a n g e n t i a l d e r i v a t i v e of
6 = g = s f ds
assumed t h a t
a
fl = ( l y J - ? b x ) ,
Jf ds
= 0
on t h e c l o s e d curve
S i m i l a r l y 1B and 1 D a r e connected by i n t e g r a t i o n by p a r t s on
rj r.
up t o a c o n s t a n t on
r.
1B
7
1.
it i s
= (uy,-ux),
and
i s dual t o lA, and 1D t o 1C.
The s o l u t i o n of 1 C comes from an e x t e n s i o n lemma proved independently by McShane and Kirszbraun f o r I d p s c h i t z f u n c t i o n s on a metric space.
Their construction applies
e q u a l l y t o Hblder continuous f u n c t i o n s , and was followed by deeper r e s u l t s of Whitney and Calderon.
In our case t h e
metric i s t h e s h o r t e s t d i s t a n c e w i t h i n
0; i t is Euclidean
distance, i f
Q
i s convex.
Then t h e q u a n t i t y
minimized i s t h e Lipschitz c o n s t a n t f o r g
on
to
l on
Ca
1.
t o be
IIVJIII,
The lemma extends
w i t h no i n c r e a s e i n t h e L i p s c h i t z
c o n s t a n t , by t h e simple c o n s t r u c t i o n
Therefore t h e minimum in 1C i s immediate: Lipschitz c o n s t a n t f o r
it e q u a l s t h e
g, and Problem 1 i s solved.
T h i s provides a n a t u r a l analogue f o r continuous flows of
t h e max flow-min c u t theorem of Ford and Fulkerson.
Instead
of a f i n i t e network with c a p a c i t y c o n s t r a i n t s on t h e edges, 0
t h e flow through i t s c a p a c i t y by
capacity
In1
5
i s described by a v e c t o r f i e l d
la1
c(x,y)
fl a r e included i n
0
and
1. The p o s s i b i l i t i e s of varying and nonzero sources
div u
=
F
[6]. I r i c r e a t e d a s i m i l a r t h e o r y
n e a t a p p l i c a t i o n t o t r a f f i c i n Tokyo [ 7 ] .
within
with a
L' and L' Approximation of Vector Fields
211
We g i v e a b r i e f b u t very informal d e r i v a t i o n of t h e o p t i m a l i t y c o n d i t i o n s t h a t connect 1A t o lB, w i t h
u(x,y)
as
div n
=
0.
SSIVuI
>
1, and otherwise
Lagrange m u l t i p l i e r for t h e c o n s t r a i n t
The saddle-
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 .
TT
is
if
-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
u.
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
l//P-Q
of t h e
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
P
Q
on
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 Vu
Thus
=
0
g
>
Q = (0,-r).
0
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
x
n
and
1/2r
c1
$,J -trx)
u
An optimal
and
n
r
<
1/2.
P = (0,r)
is zero i n the semicircle
i n t h e complement
= (l/r,O),
of r a d i u s
l/r, between t h e p o i n t s
x
1. 0.
( n o t t h e same as McShane's e x t e n s i o n ) i s =
is a
Iluli,.
Example 1: g = s i n 0
and
Vu
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
a t t a i n s i t s Lipschitz constant
1
An optimal =
y/r,
t
with
i s normal t o t h e diameter
FQ.
A l e s s t r i v i a l example, with more uniqueness, would s t a r t w i t h an e l l i p t i c a l
n.
278
Gilbert STRANG
2.
The minimum of
with D i r i c h l e t d a t a
\lVllrlll
Again w e g i v e t h e f o u r e q u i v a l e n t extremal p r i n c i p l e s : PA.
SsIflI
MIN
subject t o div
and
n = 0
a-n
=
f
n
2B.
MAX s r u f
2C.
MIN J S l V S l
2D.
MAX
subject t o
1 1 V ~ := ) ~1
subject t o subject t o
1
div
T
r
on
g
=
= 0
and
!\T'la
= 1.
The connections among 2A-2D a r e t h e same a s f o r 1A-1D.
I n t h i s case i t i s a g a i n 2 C ( t h e least g r a d i e n t problem
181) t h a t can be solved most d i r e c t l y , using t h e coarea formula f o r a function
where
yt
length.
4(x,y)
of bounded v a r i a t i o n :
is t h e l e v e l s e t where More p r e c i s e l y , s i n c e
1b
s e t of p o s i t i v e area, we c o n s t r u c t
and then
yt = a E t \ r .
Jr = t
and
is its
IYtl
could be c o n s t a n t over a E~ = ( ( x , y ) E Q: llr(x,y)
This coarea formula and i t s g e n e r a l i z a -
t i o n s a r e a v a l u a b l e t o o l i n geometric measure t h e o r y for a smooth f u n c t i o n
t]
$,
[9-lo];
or a piecewise l i n e a r f u n c t i o n , t h e
proof is s t r a i g h t f o r w a r d . To minimize
of
Yt,
llV@\\l
i s t o minimize f o r each
s u b j e c t t o t h e requirement t h a t
boundary p o i n t s a t which
g
may be required; t h e s e t
Et
g
2
t.
We note t h a t f o r
L1(r), t h e f u n c t i o n
g
in
t
and is i n t e g r a b l e . )
$
=
t.
Yt
t
the length
connects t h e
(Again a more p r e c i s e form
must c o n t a i n t h e s e t on which of bounded v a r i a t i o n , w i t h t r a c e IYtl
i s d e f i n e d for almost a l l
L’ and L L Approximation of Vector Fields
279
0. The L2 norm i s s m a l l e s t f o r t h e harmonic f u n c t i o n 6 = r 2 cos 20,
Example 2 . of
g = cos 28
on t h e u n i t c i r c l e
z e r o o n l y on t h e f o u r r a y s t o
0 =
norm i s minimized by a f u n c t i o n whole i n s c r i b e d square of s i d e diagonals.
2
\b
J=’
g =
cOS
=
L1
w i t h t h e s e r a y s as
Between t h e s q u a r e and t h e c i r c l e , t h e l e v e l l i n e s
of t h e square, f o r example,
x
But t h e
which v a n i s h e s over t h e
4 = c o n s t a n t a r e s t r a i g h t , t o minimize
lines
2 3s/4.
IT/&,
$ =
To t h e r i g h t
lYt(.
2x2-1:
c o n s t a n t on v e r t i c a l
r
c o n s t a n t , and a g r e e i n g a t t h e boundary
=
1 with
28 = 2 cos2 0 - 1 .
I n t h e d u a l i t y with 2B, t h e o p t i m a l i t y c o n d i t i o n i m i t a t e s
1B above t o g i v e
Therefore
Vu
i s a u n i t v e c t o r i n t h e d i r e c t i o n of
n
# 0, and elsewhere
llr
= 2x2-1
circle. fx u =-
and
~y
< 1.
n = (0,-4x)
Therefore
or
IVul
I n o u r example we had
i n the section
Vu = ( 0 , l )
and
The
L’
x
u = -y.
2
l/n
of t h e
Similarly
i n t h e f o u r q u a r t e r s of t h e c i r c l e - - b u t n o t
uniquely so i n t h e i n s c r i b e d s q u a r e where
3.
where
0
$ = 0
and
CT =
0.
approximation of a v e c t o r f i e l d by a g r a d i e n t
The given v e c t o r f i e l d i s
i s i t s e l f a gradient i f
F
=
v = (v,(x,y),v2(x,y)),
dvl/dy
-
&,/ax
= 0.
and i t
Otherwise
t h e maxima and minima w i l l exceed z e r o i n o u r f o u r e q u i v a l e n t problems :
3A.
280
Gilbert STRANC
3s.
MIN ~ ~ I V w - v I
3C.
MAX JJFb
3D.
MIN
c1
In 3 D ,
subject t o
iSl1-1 s u b j e c t -F.
=
=
7
Cl
-F.
Ow-v
= (-wy,wx)
Ow'
-aiv v
7 =
div
i s t h e r o t a t i o n of
7
divergence of div
to
T, lv~lrl 5 1 i n
on
= 0
1C
through
~ / 2 ;t h e
i s a u t o m a t i c a l l y zero, s o t h a t
R2.
Again we use a p r o p e r t y s p e c i a l t o
In t h i s c a s e it i s 3 C which can be solved a t s i g h t ,
F
provided we assume i s t o maximize lvllll
5
0.
To make
as l a r g e a s p o s s i b l e
//Fib
among f u n c t i o n s v a n i s h i n g on
$
with
r,
Q = distance t o
1. The extremal f u n c t i o n i s
and
t h i s choice g i v e s t h e optimal values i n 3 A - 3 D . of varying s i g n t h e problem i s much more d i f f i c u l t
F
For
and i n t e r e s t i n g ; we b e l i e v e it t o be unsolved. in
R1
Q
t h e optimal
has
1lr1
On an i n t e r v a l
-
= +1, and t h e "breakpoints"
can be determined. A c o n s t r u c t i o n of t h e optimal
w
(assuming
F
1. 0 )
was
given by Mosolov [U].The computation i s more d e l i c a t e t h a n t h a t of
and i s guided by t h e o p t i m a l i t y c o n d i t i o n
111,
n =
vw-v , F a t every p o i n t
Example 3 .
v
t h i s domain n =
( ey'
-$x)
=
and
F = 2
Vw
# v.
on t h e u n i t c i r c l e .
Ib = I-r, t h e d i s t a n c e t o t h e boundary.
On
Therefore
i s t h e u n i t v e c t o r f i e l d t a n g e n t t o t h e con-
centric circles instance
y,-x)
where
w c 0.
i s s a t i s f i e d by
r = constant.
It happens t h a t i n t h i s
The o p t i m a l i t y c o n d i t i o n d i s p l a y e d above
u
=
-v/IvI,
and t h e r e f o r e
"CaMOt be approximated by a g r a d i e n t . I' mation i s null..
v = (y,-x)
The c l o s e s t approxi-
L ' and L' Approximation of Vector Fields T h i s choice of
c y l i n d r i c a l rod.
a r i s e s n a t u r a l l y i n t h e t o r s i o n of a
v
The v e c t o r
gives t h e shearing s t r e s s and
d
is t h e a s s o c i a t e d moment r e s i s t i n g an e x t e r n a l f o r c e
-slv.o
t h a t t w i s t s the bar. tion
28 I
161
5
1
The maximum i n 3A, s u b j e c t t o t h e l i m i t a -
f o r a p l a s t i c m a t e r i a l , i s t h e l i m i t moment of a
b ar w i t h cross-section
61.
T h i s i s t h e l a r g e s t torque the b a r
can resist; t h e angle of t w i s t approaches i n f i n i t y and t h e bar becomes f u l l y p l a s t i c The d u a l v a r i a b l e
Q).
( I n 1 = 1 throughout
w(x,y)
measures t h e "warping" of each
c r o s s - s e c t i o n out of i t s o r i g i n a l plane--and
f o r the circular
b a r of Example 3 each c r o s s - s e c t i o n remains plane and
w = 0.
The minimization 3D has a f u r t h e r mechanical i n t e r p r e t a tion.
It a g a i n r e f e r s t o a b a r w i t h c r o s s - s e c t i o n
subject t o the axial force
F(x,y) .
r e s i s t e d by s h e a r s t r e s s e s
T
expresses e q u i l i b r i u m .
= (7
ss
Then
This body f o r c e i s T
xz' yz
1
17
0 , b u t now
), and
div
7
=
-F
is proportional t o the
minimum weight of a b a r which can withstand t h e load
It
F.
i s a s p e c i a l i z a t i o n of t h e Michell-Prager t h e o r y of optimal design t o t h e c a s e of pure s h e a r [12].
4.
The
Lm
For given
approximation of a v e c t o r f i e l d by a g r a d i e n t
v, w i t h
F = -div v'
and
T
as
= (OW-v)'
before, t h e e q u i v a l e n t problems a r e
4A. MAX - s s d , v 4B.
subject t o
d i v n = 0, n.n
=
0
on
M I N IIVw-v/lrn
4 ~ .MAX
JJF@ s u b j e c t
4D. MIN
/\7/Irn
to
$ =
o
subject t o
div
7
on =
-F.
r, IIv$Il,
= 1
r,
IIulll = 1
282
Gilbert STRANG
Our method i s t o apply t h e coarea formula t o 4 C . show t h a t t h e o p t i m a l function
Q
i s a m u l t i p l e of t h e c h a r a c t e r i s t i c
of some s u b s e t
C
We want t o
E
C
Q.
The coarea formula can be
written as m
where
is t h e c h a r a c t e r i s t i c f u n c t i o n of
Ct
Et = [ $
2. t l .
It i s n o t d i f f i c u l t [ 6 ] t o show t h a t a l s o m
JJF$
n
Suppose
(JSFCt) d t .
=
n
,m
i s t h e maximum i n QC,
M
and suppose t h a t
f o r every c h a r a c t e r i s t i c f u n c t i o n
f o r t h e optimal
t
C.
Then choosing
( o r more p r e c i s e l y
C = Cnt
C = Ct
for a
t n ) w e would c o n t r a d i c t t h e
maximizing sequence of f u n c t i o n s
t.
previous e q u a t i o n s by i n t e g r a t i n g over
Therefore t h e
maximum ( o r supremum) i s a t t a i n e d i n 4C by a normalized characteris tic function
~
= C/\\VC\ll.
I n o t h e r words, Problem 4 C i s e q u i v a l e n t t o t h e s i m p l e r problem
Whenever
F
i s c o n s t a n t , we have an i s o p e r i m e t r i c problem:
n.
maximize area/perirneter w i t h i n weighted i s o p e r i m e t r i c problem.
For
And f o r
F
> F
0
it i s a
of varying s i g n ,
I t i s a g e n e r a l i z e d i s o p e r i m e t r i c problem which i s new t o We have not mentioned t h e boundary c o n d l t i o n seems t o be v i o l a t e d i f
aC
n
r #
0.
JI
US.
= 0, which
Nevertheless t h e a n a l y s i s
L ' and L ' Approximation of Vector Fields
c a n be j u s t i f i e d ;
it i s the condition
t
283
which must be
= 0
r e l a x e d [13], and t h e c o r r e c t form of 4 C i s
S i m i l a r l y Problem 2 C can be r e l a x e d by t h e boundary i n t e g r a l The e f f e c t i n Problem 4 C i s t h a t t h e l e n g t h of
I$-gI.
i s i n c l u d e d i n t h e p e r i m e t e r of
aC
n
r
and we r e a c h t h e i s o p e r i -
C,
m e t r i c problem d e s c r i b e d above. We show by example t h a t t h e o p t i m a l
(at least for
can be computed
~
F = 1 and f o r simple domains).
Q
Since
is
piecewise c o n s t a n t , changing o n l y a t t h e boundary of t h e o p t i m a l set
i n t h e i s o p e r i m e t r i c problem,
C
a C --a
s i n g u l a r measure supported on
= ($
CT
Y' " l i n e of
The o p t i m a l i t y c o n d i t i o n c o n n e c t i n g it t o
T
-$x)
is a
& - f u n c t i o n s .I '
gives only
moderate i n f o r m a t i o n :
We have no method f o r computing
Example 4:
v = (y,-x)
and
n.
in
T
n.
on t h e u n i t c i r c l e
F = 2
T h i s i s t h e o l d e s t of a l l i s o p e r i m e t r i c problems, and t h e s u b s e t which maximizes a r e a / p e r i m e t e r i s field
T
is radial, w i t h
gradient t o
Example 5 :
v
is
Ow =
v = (y,-x)
with v e r t i c e s a t
div
T
=
-2,
C =
a.
The v e c t o r
and a g a i n t h e n e a r e s t
0.
and
(+ 1/2,+- 1 / 2 ) .
n,
on t h e u n i t s q u a r e
F = 2
The o p t i m a l s u b s e t
n e i t h e r t h e whole square n o r t h e i n s c r i b e d c i r c l e .
C
Instead
it i s a compromise [6,14] whose boundary c o i n c i d e s w i t h except f o r q u a r t e r c i r c l e s of r a d i u s
(2
f
fi)"
is
r
i n the four
284
Gilbert
c o r n e r s of t h e s q u a r e . i s smooth.)
STRANG
aC
(They a r e t a n g e n t t o t h e square, s o
We have s o f a r been u n s u c c e s s f u l i n determining
a corresponding v e c t o r f i e l d
with
T
div
7
= (2
+
fi)!lT\lm.
It e x i s t s , by d u a l i t y t h e o r y , and we have o f f e r e d a modest
p r i z e (10,000 Yen a t t h e U.S.-Japan Seminar) f o r i t s d i s c o v e r y .
L1
It would be i n t e r e s t i n g t o compare t h e s e
and
L”
o p t i m i z a t i o n s with t h e corresponding d i s c r e t e problems i n and
a”.
4’
There t h e o p t i m a l i t y c o n d i t i o n s a r e c l a s s i c a l , and
t h e b e s t approximations must approach o u r s o l u t i o n s ( i n c l u d i n g t h e c h a r a c t e r i s t i c f u n c t i o n s ) i n an i r r e g u l a r b u t c o n s i s t e n t way.
The r a t e of convergence, and t h e p a t t e r n of error i n t h e
d i s c r e t e problems, should be v i s i b l e from numerical experiments-s i n c e this i s a c l a s s of d i s c r e t e problems i n which t h e cont i n u o u s l i m i t can be s o l v e d .
We h e s i t a t e t o propose a more complete l i s t of d u a l v a r i a t i o n a l problems of t h e same t y p e . appear f o r o p t i m i z a t i o n i n
L2,
Harmonic f u n c t i o n s w i l l
and t h e s p e c i a l p r o p e r t y of two
dimensions (which produced a l l t h e second p a i r s of e q u i v a l e n t problems) i n t r o d u c e s t h e conjugate harmonic.
Within t h e l i s t
above t h e r e a r e combinations of c o n d i t i o n s t h a t e a r l i e r e n t e r e d only s e p a r a t e l y , f o r example t h e combination MAX JJF$ s u b j e c t t o
P = g
on
f
and
! l V $ ! l w = 1.
This corresponds t o a p p l y i n g both s h e a r and t o r s i o n t o a c y l i n d r i c a l rod, and it i s solved ( i f t h e c o n s t r a i n t s a r e compatible and
F
>
0) by t h e maximal f u n c t i o n
3i(P) = min [ g ( Q ) + d ( P , Q ) I . A r e l a t e d problem mixes
L2 and
Lm, arid has become a
fundamental example i n t h e t h e o r y of v a r i a t i o n a l i n e q u a l i t i e s :
L’ and L* Approximation of Vector Fields
285
The d u a l minimizes a combined norm
This seems a p p r o p r i a t e a l s o f o r ” r o b u s t s t a t i s t i c s , “ i n which t h e l e a s t squares model (Gauss-Markov l i n e a r r e g r e s s i o n ) i s
natural--except t h a t it a s s i g n s t o o much weight t o o b s e r v a t i o n s t h a t l i e f a y o u t s i d e t h e normal range. 1
significant i n
L
, and
These o u t l i e r s a r e l e s s
a mixed norm i s more r e a l i s t i c .
F i n a l l y we mention optimal design, which i s s u b j e c t t o a l l t h e s e c o n s t r a i n t s and one more:
it begins a s a nonconvex
problem, t o minimize t h e support of
A. t y p i c a l case, f o r
n.
longitudinal shear i n a p l a s t i c cylinder, i s
The i n t e g r a n d jumps from to
at
00
(01
= 1
0
to
1 at
>
(since
u = 0, and from
1 i s inadmissible).
1
The
e q u i v a l e n t “ r e l a x e d problem” r e p l a c e s t h i s i n t e g r a n d by the l a r g e s t convex f u n c t i o n which does n o t exceed it:
10)
for MIN
lul
SJ’IUI
5
1 and
subject t o
m
for div
Id1
CI =
0,
>
1.
n-n
it e q u a l s
I n t h i s new problem = f,
11U11,
2
1
t h e r e e x i s t s an optimal s o l u t i o n (which i s a weak l i m i t
Of
t h e h i g h l y o s c i l l a t o r y minimizing sequences i n t h e o r i g i n a l problem).
The s o l u t i o n can a c t u a l l y be computed by modifying
t h e c o n s t r u c t i o n i n 2 C t o account f o r t h e new c o n s t r a i n t
101
= (V$(
2
1.
It g i v e s t h e admissible s t r u c t u r e of minimum
weight. There i s a l s o a more d e l i c a t e c l a s s of nonconvex problems,
286
Gilbert STRANC
whose relaxed form f a i l s t o be convex:
i n s t e a d i t i s polyconvex.
In a forthcoming paper w i t h Kohn w e s t u d y e l a s t i c design s u b j e c t t o two loads, l e a d i n g t o t h e new minimum p r i n c i p l e
I n t h i s case
n , ~ represents a
loads It would be
2
by
n.
2
by
2
matrix; with
Only t h e c a s e of a s i n g l e l o a d
l e a d s t o an e q u i v a l e n t convex problem; for
n = 2
the
relaxed integrand i s polyconvex--a convex f u n c t i o n of and t h e i r determinant
D =
n
d.7.
a,~,
The underlying theory was
developed a b s t r a c t i y by Morrey [ 1 5 ] , and more r e c e n t l y b y
Ball [16] and Dacorogna [ 1 7 ] .
Perhaps our example i s t h e
f i r s t involving a l l t h r e e arguments i n which t h i s polyconvexif i c a t i o n has been found.
ACKNOWLEDGEMENT W e g r a t e f u l l y acknowledge t h e support of t h e National
Science Foundation (MCS 81-02371 and INT 81-00464) and t h e Army Research Off i c e (DAAG 29-8O-KO033).
L ' and L' Approximation of Vector Fields
287
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Bombieri, E., DeGiorgi, E . and G i u s t i , E . , Minimal cones and t h e B e r n s t e i n problem, I n v e n t i o n e s Math. 7 (1969) 243-268. Fleming, W . and R i s h e l , R . , An i n t e g r a l formula f o r t o t a l g r a d i e n t v a r i a t i o n , Archiv d e r Mathematik 11 (1960) 218-222. Federer, H., Geometric Measure Theory (Springer-Verlag, New York, 1 9 6 9 ) . M o s O ~ O V J P.P.,
On t h e t o r s i o n of a r i g i d - p l a s t i c c y l i n d e r ,
PMM 4 1 (1977) 344-353.
S t r a n g , G . and Kohn, R . , Optimal d e s i g n of c y l i n d e r s i n s h e a r , i n : Whiteman, J . ( e d . ) , The Hathematics of F i n i t e Elemcnts and n p p l i c a t i o n s TV (Academic Press, Iondon, lU8?). S t r a n g , G . , A f a m i l y of model problems i n p l a s t i c i t y , i n : Glowinski, R . , Lions, J . L. ( e d s . ) , Proceedings of t h e Symposium on Computing Methods i n Applied S c i e n c e s , Lecture Notes i n Mathematics 704 (Springer-Verlag, New York, 1 9 7 9 ) . S t r a n g , G . , A minimax problem i n p l a s t i c i t y theory, i n : Nashed, Z . ( e d . ) , J?unctional Analysis Methods i n Numerical Analysis, Lecture Notes i n Mathematics 701 (Springer-Verlag, New York, 1 9 7 9 ) .
Gilbert STRANC
288
[15] Morrey, C.B., Multiple Integrals in the Calculus of Variations (Springer-Verlag, Berlin, 1966). [16] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Nech. Anal. 63 (1977)
337-403.
[17] Dacoro$na, B., Weak Continuity and Weak b w e r Semi-
continuity of Nonlinear Functionals, Springer Lecture Notes in Mathematics 922 (Springer-Verlag, Berlin, 1982)
Lecture Notes in Num. Appl. Anal., 5, 289-31 1 (1982) Norilrrieur PDE
rti
Applied Screrice. U.S.-Jczprin Sernrncir, Tokyo. 1982
Deformation formulas and t h e i r a p p l i c a t i o n s t o s p e c t r a l and e v o l u t i o n a l i n v e r s e problems"
Takashi SUZUKI Department o f Mathematics F a c u l t y of Science U n i v e r s i t y o f Tokyo
We d e s c r i b e our r e c e n t r e s u l t s on s p e c t r a l and e v o l u t i o n a l i n v e r s e problems. O u r main i n t e r e s t l i e s i n t h e uniqueness of t h e problems and 24 theorems w i l l b e s t a t e d .
§1. Summary. I n t h i s a r t i c l e , two t o p i c s a r e t a k e n up:
i n v e r s e problems f o r e v o l u t i o n e-
q u a t i o n s and i n v e r s e problems f o r s p e c t r a l t h e o r i e s .
These two are a s s o c i a t e d
with each o t h e r , and our methodology o f t h e s t u d y i s t h e same.
That i s , t h e de-
formation formulas, which a r e simple and are e a s i l y proven. O u r f i r s t t o p i c , t h e i n v e r s e t h e o r y for e v o l u t i o n e q u a t i o n s , studies t h e f o l lowing problem:
By observing w i t h i n a c e a t a i n a r e a t h e v a l u e s o f t h e s o l u t i o n of
an e v o l u t i o n e q u a t i o n , can we determine t h e c o e f f i c i e n t s or t h e i n i t i a l v a l u e of t h e equation ?
Many a u t h o r s have been i n t e r e s t e d i n such a problem, and some of
t h e i r works are r e f e r r e d t o i n Suzuki [29,30,31].
W e h e r e refer t o L a v r e n t i e v [13
,1111, Nakagiri [ 251 and Kohn-Vogeiius [ 42 1, and a l s o Kitamura-Nakagiri man [ 2 8 ] , P i e r c e [ 2 7 ] , Suzuki-Murayama [41] and Murayama [ 2 3 ] .
t e r s are r e l a t e d t o t h e problems which we study h e r e .
[121, Seid-
Actually, t h e lat-
I n t h i s a r t i c l e , we consid-
e r p a r a b o l i c equations on compact i n t e r v a l s and on c i r c l e s , and g i v e c o n d i t i o n s f o r t h e equations t o b e determined through v a r i o u s o b s e r v a t i o n s .
Details w i l l b e
d e s c r i b e d i n 5 8 2 and 3. O u r second t o p i c , t h e i n v e r s e s p e c t r a l t h e o r y , determines a d i f f e r e n t i a l op-
e r a t o r by i t s spectrum.
This kind of problem h a s been i n v e s t i g a t e d f o r Sturm-
289
290
Takashi S U Z U K I
L i o u v i l l e ' s o p e r a t o r by V . Ambarzumian, G . Borg, N . Levinson, I . M . L<evitan, M . G .
Gasymov, V . A .
Marchenko, B.Y.
G e l ' f a n d , R.M.
Levin, H. Hochstadt and B. Lieberman
([l], [2], [171, [5], [la], [20], [16], [lo], [ll]), and f o r H i l l ' s e q u a t i o n by G. Borg, H. Hochstadt, P.D. Lax, W. Goldberg, H. F l a s h k a , B.A.
Dubrovin, V.A.
Mateev,
S.P. Novikov, H.P. Mckean, P. van Morebeke, E. Trubowitz and o t h e r s ([2], [ g ] , [l 51, [6,71, C41, 131, 1211, [22]).
In
§ 5 4 and 5 of t h e p r e s e n t a r t i c l e , w e re-
examine t h e s e s t u d i e s from t h e viewpoint of our deformation formulas, and come up w i t h some new r e s u l t s , among which a r e involved c r u c i a l improvements on t h e i r t h e orems. Throughout t h e p r e s e n t a r t i c l e , t h e "deformation formulas" p l a y an important role.
The f i r s t and t h e second ones w i l l b e s t a t e d i n 5 5 2 and 3, r e s p e c t i v e l y .
These connect e i e e n f u n r t i o n s o f two s e p a r a t e d i f f e r e n t i a l o p e r a t o r s by i n t e g r a l t r a n s f o r m a t i o n s , of which k e r n e l s s a t i s f y c e r t a i n h y p e r b o l i c e q u a t i o n s . i n s p i r e d by t h e Gel'fand-Levitan t h e o r y
[5] i n d e r i v i n g t h e s e formulas.
We a r e By u s i n g
t h e s e deformation formulas, t h e answers t o our i n v e r s e problems a r e sometimes arr i v e d at a c e r t a i n non-linear e q u a t i o n , which we c a l l t h e " G
-
equation".
A typ-
i c a l example of t h i s can b e s e e n i n Theorem 2 of 42. This a r t i c l e i s made up o f f i v e s e c t i o n s .
A s mentioned above, we show i n 52
some theorems on i n v e r s e problems f o r p a r a b o l i c e q u a t i o n s which are proven by t h e f i r s t deformation formula,
Theorems on similar problems whose p r o o f s are based
o n t,he second deformation formula w i l l b e s t a t e d i n $3.
55
4
and 5 are devoted
t o t h e study of i n v e r s e s p e c t r a l problems f o r S t u r m - L i o u v i l l e ' s o p e r a t o r and f o r H i l l ' s equation, respectively.
I n t h i s a r t i c l e , p r o o f s o f theorems are not e x p l i c i t l y s t a t e d , except f o r a few a l l u s i o n s t o them.
12.
See t h e papers r e f e r r e d t o t h e r e , f o r t h e p r o o f s .
I n v e r s e problems f o r e v o l u t i o n e q u a t i o n s (I).
1
2
For peC [0,1], hER, HER and UEL (O,l), Let ( E
P,h,H,a
tion
) b e t h e p a r a b o l i c equa-
Deformation Formulas and Inverse Problems
29 1
w i t h t h e boundary c o n d i t i o n
and w i t h t h e i n i t i a l c o n d i t i o n
A s i s well-known, i f t h e i n i t i a l v a l u e a and t h e c o e f f i c i e n t s (p,h,H) are g i v e n ,
.
I n p a r t i c u l a r , t h e values
of t h e s o l u t i o n on t h e boundaries 5=0,1 a r e determined.
Hence w e o b t a i n t h e map-
t h e r e e x i s t s a unique s o l u t i o n u = u ( t , x ) of ( E
P,h 9 H - a
)
ping
1 F = F1 T .T . 1
f o r some T1,T2
(p,h,H,a)\+
{
[
Tl(tT2j.
2
i n O
<m.
-1 2 1 2 Let ( p , h , H , a ) ~ C [0,1lxRxRxL (0,l)b e g i v e n w i t h t h e s o l u t i o n u = u ( t , x ) o f ( E
P¶
h,H,a
) , and c o n s i d e r t h e s e t
I t denotes t h e t o t a l i t y o f e q u a t i o n s (E
s.j ,J,b
) whose s o l u t i o n s v = v ( t , x ) have t h e
same boundary v a l u e s as t h o s e of u:
S i n c e u and v a r e a n a l y t i c i n t E ( O , m ) ,
1 so t h a t M p.h,H,a
( 2 . 4 ) i s equivalent t o
i s independent o f T or T 1 2'
Takashi S U Z U K I
292
on t h e o t h e r hand, t h e s e v a l u e s { u ( t , c ) ('p, h ,I3 ,U
).
1
5=0,11
Tl(t
However, ( 2 . 5 ) does not always h o l d .
determine t h e e q u a t i o n
For i n s t a n c e 1.120 follows from
a30 for any ( p , h , H ) , hence
M1
p,h,H,O
for each ( p , h , H ) .
1 '
-
{(q,j,J,O)
1 qEC1[0,11,j E R ,
JER) To
We wish t o g i v e a c o n d i t i o n on ( p , h , H , u ) f o r (2.5) t o h o l d .
t h i s end we i n t r o d u c e t h e f o l l o w i n g
Notation.
2
The r e a l i z a t i o n i n L (0,l) of t h e d i f f e r e n t i a l o p e r a t o r p(x)-*
with t h e boundary c o n d i t i o n ( 2 . 2 ) i s denoted by A e i g e n f u n c t i o n s of A
* * I , respectively,
P*h,H
are denoted by {An
Furthermore, n o t i n g t h a t each A
Definition.
I n=0,1,2,***}and )=l.
i s simple (-a
0 1
{@(*,An)
1 n=0,1,2,.
0 we g i v e
2 For each UXL (O,l), we c a l l
t h e "degenerate number" of a w i t h r e s p e c t t o A P,h,H'
2
The e i g e n v a l u e s and t h e
p,h,H'
t h e l a t t e r b e i n g normalized by $(O,x
a2
ax
t h e L -inner product.
where (
,
)L2(0,1) d e n o t e s
0
Then we have Theorem 1 (Murayama [ 2 3 ] , Suzulri [32]).
( 2 . 5 ) h o l d s i f and only i f N=O,
where N i s t h e degenerate number o f a w i t h r e s p e c t t o A 'Fnis theorem i s shown by t h e f o l l o w i n g a s s e r t i o n :
1,2,***}c h a r a c t e r i z e s t h e o p e r a t o r A
p,h,H'
1
p,h,H'
0
The s e t {An, @(l,A,)
Murayama [231 found t h a t
n = 0 , 1 , 2 , * * * } corresponds one t o one t o t h e s p e c t r a l f u n c t i o n of A
I n=O,
{An, $ ( l , h n )
p,h,H'
and
293
Deformation Formulas and Inverse Problems showed t h e a s s e r t i o n by t h e Gel'fand-Levitan t h e o r y [ 5 ] .
Suzuki [321 proved t h e
a s s e r t i o n d i r e c t l y b y t h e f o l l o w i n g f i r s t deformation formula:
L e t @=@(x,A)eC
2
[0,
11 b e t h e s o l u t i o n o f
f o r each AER. y<x
( T h i s n o t a t i o n i s compatible t o t h a t of
@(.,A
).)
S e t D={(x,y)
1 O<
men,
1 2 Lemma 1. For each p,qEC [0,1]and h,jER, t h e r e e x i s t s a unique K=K(x,y)eC (
D) such t h a t
Lemma 2 ( f i r s t deformation formula, Suzuki ( 3 2 1 ) .
For K i n Lemma 1,
satisfies
f o r each AER.
n
The p o i n t i s t h a t K i s indepenent of A .
An)
I n=0,1,2,.*.)
On account of i t , c o n d i t i o n s on { A n ,
can be c o n c e n t r a t e d i n t o t h o s e on K through ( 2 . 8 ) .
$(I,
1321 proved
t h e a s s e r t i o n mentioned above by t h e s t u d y of t h e s e c o n d i t i o n s o n K , t o g e t h e r w i t h (2.7').
On t h e o t h e r hand, t h e p r o o f s of Lemmas 1 and 2 are elementary.
I n fact,
Lemma 1 i s o b t a i n e d by t h e method of P i c a r d 1261 ( t h a t i s , " i t e r a t i o n " ) , wh-ile Lemma 2 f o l l o w s immediately from t h e i n t e g r a t i o n by p a r t s .
See [ 321.
1 Furthermore b y t h i s method, [32] gave t h e f o l l o w i n g c h a r a c t e r i z a t i o n of M Pih,
294
H-0
Takashi SUZUKI
by t h e " G
-
equation", i n c a s e of 1zN<m.
Namely, assume Ncm and
w e c o n s i d e r t h e f o l l o w i n g non-linear N-simultarieous o r d i n a r y d i f f e r e n t i a l equation (C
-
equation)
and set
and I denotes t h e i n n e r product and t h e u n i t m a t r i x i n RN, r e s p e c t i v e l y .
where
W e s e t , furthermore,
Then,
Theorem 2 (Suzuki [ 3 2 ] ) . a,
Under t h e s e assumptions, f o r each (q,j,J)E;l
2 1 b € L (0,l)with (q,j,J,b)EMp,h,H,a i s unique.
M1P ,h 9 1 1 a i s homeomorphic 9
3
(s,j,J)
++
G E G
through t h e r e l a t i o n s
(2.17)
q = p + 2 -d- - ( G * Q ) , dx
j = h
+
(G.Q)(O),
J = H
-
(G*@)(l).
0
p,h,H, t o G:
Deformation Formulas and Inverse Problems
-
I n p a r t i c u l a r , M~ h a s 2N P,h,H,a
degrees of freedom.
G
-
295
e q u a t i o n ( 2 . 1 1 ) i s ob-
t a i n e d by e l i m i n a t i n g ( q , j , J ) i n t h e r e l a t i o n s ( 2 . 7 ) and i n some e q u a l i t i e s on K d e r i v e d from (9, j ,J ) E M
^l p,h,H,a’
See [32], f o r t h e r e l a t i o n between b and G , and
f o r a p p l i c a t i o n s o f Theorem 2 . Now i t i s n a t u r a l t h a t we n e x t l y c o n s i d e r t h e mapping
for x ~ ( 0 , 1 ) u , = u ( t , x ) b e i n g t h e s o l u t i o n of (E ). 0 p,h,H,a
S i m i l a r M can b e consid-
ered:
-
1
-1
1
(FT T x (p,h,H,a)) 1’ 2’ 0 1’ 2’ 0 = {(s,j,J,b)~C’[o,l]xRxRxL~(o,l) I t h e s o l u t i o n v = v ( t , x )
M1p,h,II,a,xo =
T
x
of t h e e q u a t i o n ( E 9 r j , J , b ) s a t i s f i e s
(2.13)
M1
p,h,H,a,xo
( T l t T 2 ; E;=O, xo ) 1
v(t,S) = u(t,E)
coincides with
M
1 p,h,H,a
.
i n c a s e x =l. However, we u n f o r t u n a t e l y have 0
Theorem 3 (Suzuki [ 3 6 ] ) . I n c a s e of x #1, we always have 0
Namely, non-uniqueness h o l d s even i f N = O ,
unless x =1. 0
I n view o f t h i s , we n e x t l y c o n s i d e r t h e mapping
and o b t a i n t h e f o l l o w i n g theorem, where
(2.15)
Takashi
296
SUZUKl
( Tl
4
Theorem to A
P,h,H'
(Suzuki [33,36]).
3:
Let N b e t h e d e g e n e r a t e number o f a w i t h r e s p e c t
Then,
I n t h e c a s e of x =1, 0
(i)
(2.16)
= {(p,h,H,a) 1
M2
P 9h 9H ,a9 xo
h o l d s i f and only i f N=O.
I
(ii)
I n t h e c a s e o f 4 x (1, (2.16) h o l d s whenever N<m.
(iii)
In t h e c a s e of x =-, 0 2
(iv)
I n t h e cas'e of O<x <-, 0 2
2
0
1
( 2 . 1 6 ) holds i f and o n l y i f N(1. -
1
2
w e always have M {(p,h,H,a)!. P ,h ,H,a,xo$
Thus, t h e p o s i t i o n xo p l a y s an important r o l e as w e l l as t h e number N .
3 and 4 a r e a l s o proved by f i r s t deformation formula.
0 Theorems
Since t h e equation (2.7.1)
i s of hyperbolic t y p e , having t h e p r o p e r t i e s o f t h e domain o f dependence and so
1 0 2
o n , t h e p o i n t x =- comes t o b e important i n Theorem
4.
Before concluding t h i s s e c t i o n , we b r i e f l y mention t h e connection between ( i ) of Theorem
4
and Theorem 1.
I n t h i s case ( x o = l ) , (2.15) i s equivalent t o ( 2 . 4 ) 2
1
i s nothing b u t M with J = H u n l e s s a f 0 , s o t h a t M r e s t r i c t e d t o J= P,h,Hi,a,xo P ,h ,His 2 H . By Theorem 2, M1 has 2N - degrees of freedom, hence t h o s e o f M p,h ,H ,a P ,h r H , a . X O w i l l b e 2N-1 i n c a s e x = l . I n p a r t i c u l a r , 1 - degree of freedom remains even i f
0
N=l, which e x p l a i n s why ( i )o f Theorem 4 h o l d s .
13.
I n v e r s e problems f o r e v o l u t i o n e q u a t i o n s (11).
Let us now c o n s i d e r t h e mapping
w i t h similar M:
Deformation Formulas and Inverse Problems
297
In view of Theorem 4 , w e s e e t h a t 3 = {(p,h,H,a)j MP,h 9 H r u 9 xo
(3.2)
1 holds only i f x0=5 and
Nzl.
I n f a c t , i f o<x
0 2
d o e s n ' t hold even i f we assume v ( t , O ) = u ( t , O ) (Tl$tLT2) Theorem 4.
(q,j,J,b)=(p,h,H,a)
b e s i d e s ( 3 . 1 ) , by ( i v ) o f
S i m i l a r l y , if 41 x <1, uniqueness d o e s n ' t h o l d even i f w e assume v ( t , l ) 2 0
= u ( t , l ) (T1LtT2) besides (3.1).
1 Therefore, ( 3 . 2 ) i m p l i e s xo=z, hence a l s o i m -
p l i e s N L l b y ( i i i ) o f Theorem 4 . F o r t u n a t e l y we have Theorem 5 (Suzuki [34,37]). 1
1
If x0=: and
N=O, (3.2) holds.
For t h e case x =- and N = l , which i s d e l i c a t e , s e e [ 3 7 ] . 0 2 Theorem 6 ( [ 3 4 , 3 7 ] ) .
Furthermore, we have
1 1 Let x #- and assume --<x <1 without l o s s of g e n e r a l i t y . 0 2 2 0
Then ( 3 . 1 ) i m p l i e s
whenever
N<m.
0
Theorem 7 ( [ 3 4 , 3 7 ] ) .
if and o n l y i f N
0
0
1 Under t h e same s i t u a t i o n > x o < l ,
298
Takashi S U Z U K I
1 S i m i l a r theorems a l s o hold f o r t h e c a s e of O < x <-. 0 2
I n view of Theorems 6 and 7,
1 we c a l l ( x o , l ) t h e "domain of uniqueness" i n t h e case of +x
P
O
<1,which comes t o
b e ( 0 , x ) i n t h e case of O<x <&. 0 0 2 These theorems cannot be proved by f i r s t deformation formula. Lemma 2
$=@(-,A)
i s requested t o s a t i s f y t h e boundary c o n d i t i o n s @(O,X)=l
O,A)=h, which are independent o f A . u ( t , l ) , w e cannot apply t h i s formula. Set D~=I(X,Y)
Lemma 3.
In f a c t , i n
1 l-x
and
@ I (
Without observing boundary v a l u e s u ( t , O ) or Another deformation formula i s needed:
$+
1 1 1 For each peC [0,11 and ~ E C[--,1],t h e r e e x i s t s a unique K = K ( x , ~ ) E 2
2-
C (Dl) such t h a t
Lemma
4
(second deformation formula, Suzuki [34]).
2
If @ = $ ( x ) E C [ O , l ] s a t i s -
fies
satisfies
The p o i n t i s t h a t no boundary c o n d i t i o n on $ i s assumed i n (3.6) and t h a t i n s t e a d
1 (3.8)holds only on [--,l] i n s p i t e of ( 3 . 6 ) on [0,1]. To g e t a similar r e l a t i o n 2 1
-
t o ( 3 . 8 ) on [0,7],another K h a s t o b e c o n s t r u c t e d on D p , where D2={(x,y)
I x
Deformation Formulas and Inverse Problems
299
I n v i r t u e o f second deformation formula, we can a l s o s t u d y i n v e r s e problems Henceforth S1 denotes t h e compact i n t e r v a l
f o r e v o l u t i o n e q u a t i o n s on c i r c l e s . t h end p o i n t s i d e n t i f i e d .
, we
on S :
S 1 denotes t h e r e a l i z a t i o n i n L2(S ) o f t h e d i f f e r e n t i a l o p e r a t o r P The eigenvalues of As are denoted by {An n = 0 , 1 , 2 , . * * ) (-m
A
I
P(X)-;1;;2.
[
1
consider t h e f o l l o w i n g p a r a b o l i c e q u a t i o n (Es P,a
Notation.
w)
F o r pEC 1( S1) ( i . e . , pcC1(R), p ( x + l ) = p ( x ) ) and
i s denoted by a ( n ) . Note t h a t a ( n ) = l o r 2.
and t h e m u l t i p l i c i t i y o f h
l$Rza(n)} denotes t h e e i g e n f u n c t i o n s of As
P'
malized by
corresponding t o
A
"nk
and b e i n g nor-
/ I $ nR 1 1 L2 ( S1) =l. 0
) a r e more d i f f i c u l t t h a n t h o s e of ( E ), p a r t l y P *h,H,a because of t h e e x i s t e n c e o f double eigenvalues of As I n o r d e r t o overcome t h i s I n v e r s e problems f o r (Es
P,a
P'
d i f f i c u l t y , we c o n s i d e r s e v e r a l s o l u t i o n s of (Es ) according t o t h e i d e a o f NakaP .a g i r i [241.
We t h u s extend t h e n o t i o n of t h e "degenerate number'' as
Definition. <jza}, -
P u t Emax a ( n ) ( = lor 2).
For t h e s e t of i n i t i a l v a l u e s {aJ 11
l e t us c o n s i d e r t h e m a t r i x
f o r each n=0,1,2,...
.
Then w e c a l l
N = #{
t h e "degenerate number" of f a J
. .
1
rank A
< a(n)}
I lLjza} with
S
respect t o A P'
0
S J ) ( 1 z j L a ) . For s i m p l i c i t y , we h e n c e f o r t h P ,a S assume N=O, where N i s t h e degenerate number of {a' 1 lzjza} w i t h r e s p e c t t o A P' S 1 1 L e t v = v J ( t , x ) b e another s o l u t i o n o f ( E j ) f o r some qEC (S ) and bJ~L2(S1). s,b
L e t u J = u J ( t , x ) be t h e s o l u t i o n of (E
Then,
Takashi
300 Theorem 8 (Suzuki [35,37]).
SUZUKI
1 Let x 5.5 and x eS1 satisfy the central symme1 2
try, say x 2 and x =1(=0). Then the equalities 1 2 2
imply
1 Theorem 9 (135,371). Suppose that xlcS1 and x2€S don't satisfy the central symmetry, and let xlrES1 and xgiES1 be the symmetric points of x1 and x 2 , respectively, say x =L bx2
A
x1x2, x1'x2',
1 and x '=x 2 2 2
Gl' and x2'x1, respectively as n
--.
Let A, A', B and B' be the arcs
i n Fig. 1. Then (3.10) implies
Fig. 1
Under the same circumstances as those of Theorem 9 ,
Theorem 10 ([35,37]).
the equalities (3.10) combined with either q(x)=p(x) (xEB) or q(x)=p(x) ( x E B ' ) imply (3.11).
0
In virtue of Theorems 8-10, we can call AuA' the domain of uniqueness for this problem. Finally we have Theorem 11 ( [ 3 5 , 3 7 ] ) .
1 In case of p(x%)=p(x)
qualities
1
imply (3.111, where x ES 1
.
0
1 and q(x%)=q(x)
(xER), the e-
30 1
Deformation Formulas and Inverse Problems In this case, one point x cS1 is enough f o r the uniqueness. 1
Inverse spectral problems for Sturm-Liouville’s operators.
54.
In this section, we consider the so-called inverse Sturm-Liouville problem. The first deformation formula will give crucial answers to them. Recall that for 2 1 denotes the realization in L (0,l)of the differPEC [0,1],hER and HER, A P&,H d2 d ential operator --2+p(x) with the boundary condition ( c h ) . I~=~=O. dx lx=O=(-%H).dx
Let {An
1 n=0,1,2,-**)(-&A
be cr(A
P.h,H
), the eigenvalues of A
p,h,H‘
Firstly, Theorem 12 (Suzuki [361).
(4.1)
’n
Suppose q(x)=p(x)
‘(*q,,j,J)
1 (OLxk-), j=h and
( n#nl )
1
for qEc [0,1],jER, JeR and nl~NE{0,1,2,*-*}. Then
Theorem 12 is an improvement of Hochstadt-Lieberman [ll]. Actually they derived ( 4 . 2 1 , assuming J=H and
1 X nE O ( A s,j,J ) (n=O,l,2,***),besides q(x)=p(x) (O<x<-) and =;2
j=h. It is important that the converse of Theorem 12 holds: Theorem 13 ( [ 36 3 1. (i)
For each A
p.h,H’
n ER and xo in O x <-,1 there exist q#p, j and J such 1
-02
that q(x)=p(x) (O<xlxo), j=h and (4.1). For each A
and nl,n2EN with nl#n2, there exist q#p, j and J such that P,h,H 1 q(x)=p(x) (Ozxs), j=h and (ii)
Nextly, let us consider another Sturm-Liouville operator 4 with ‘(A J,h,H* p,h, m m and its eigenvalues {Xnjn=i,. H,)={AE}n=o for H*#H, along with A P,h,H
302
Takashi S U Z U K I Theorem
(i)
14 (Suzuki [ 361).
Suppose
follows.
(ii)
Suppose
( i i )o f Theorem 14 i s an improvement of Borg [ 2 ] , Levinson [17] and Hochstadt
[lo].
I n f a c t , t h e y d e r i v e d ( 4 . 5 ) assuming j = h , J=H, J*=H*
(4.6')
A
=
un
(n=1,2,**.),
m
where {unjn=O=o(Aq, ,J found i n Levin [ 1 6 ] .
n=O,l,Z,.*.)
) (--
0
:A
<*-*-m).
1
E
)
o(A
q,j
and (n=0,1,2,.-.),
,tJ*
A proof of (i) o f theorem 14 can b e
Levitan-Gasymov [lS] r e c o n s t r u c t e d (p,h,H,H*) from
{A n,A*n 1
under s u i t a b l e c o n d i t i o n s .
The converse o f Theorem 14 i s o b t a i n e d : Theorem 15 ( [ 36 ] 1.
(i) that
(ii)
For each (p,h,H,H*) and n EN, t h e r e e x i s t ( q , j ) # ( p , h ) , J#H and J*#H* 1
such
(4.6). For each (p.h,H,H*)
J* such t h a t J = H , J*=H* and
and nl.n2EN
w i t h nl#n2,
t h e r e e x i s t ( q , J ) f ( p , h ) , J and
303
Deformation Formulas and lnverse Problems
X
(4.7) (iii)
E U(A
s,j ,J
)
h;
(n#nl, n 2 ) ,
For each (p,h,H,H*) and nl,n2EN,
a(A
E
)
(nd).
s,j .J*
t h e r e e x i s t ( q , j ) # ( p , h ) , J and J* such
t h a t J = H , .J*=H* and
(4.8)
An
o(A q , J., J )
E
(nfn,),
E U(Aq , j , J * )
'
(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
m
{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<"'<%<m gers.
Then, ( q , j , J , J * )
Xn
(4.9)
be inte-
satisfies
E o(A . ) 9.J ,J
(n#na; l(kZN),
E
5(A
s,j .J* )
(nd"
i f and only i f t h e r e e x i s t s GEG w i t h
(4.10)
G'(1)
+ (H*
-
(G.O)(l))G(l)
=
0
such t h a t
(4.11)
d q = p + 2-(G.O), dx
j =
h + (G*@)(O
J* = H*
-
,
and G i s homeomorphic.
Hochstadt [lo] showed more weakly t h a t i f j = h , J = H , J*=H*
X
= 1-1,
(n#nk;
IZkZN),
1;
(G-O)(l),
(G*@ (1).
Furthermore, t h e correspondence between ( q , j ,J,J*
(4.9')
-
J = H
E U(A
s,j ,J*
0
and
1
(ndJ),
t h e n q s a t i s f i e s t h e f i r s t e q u a l i t y of ( 4 . 1 1 ) f o r Some GEG. Finally, s e t 1
1
cs[o,l] = { PEC [0,1] 1 P(1-X) = p(X) W e say t h a t A
P,h,H
(ozX$l)}.
i s ( s p a t i a l l y ) symmetric i f f p ~ C ; [ O , l l and h=H.
Suppose t h a t
304
Takashi S U Z U K I
implies
The converse of Theorem 17 h o l d s . Theorem 18 m
{Anjn,0
([%I).
Let n EN. 1
*
More p r e c i s e l y ,
Let a symmetric o p e r a t o r A Then a symmetric o p e r a t o r A
P ,h ,h
b e given with I J ( A ~ , ~ , ~ ) = m
s,j $ 5
w i t h U ( A ~ ,, j ~)={pnjn=O sat-
isfies
un
(4.14)
=
in
(nfn,) 2
i f and only i f t h e r e e x i s t s gEC [0,1] s a t i s f y i n g ( G
-
equation)
such t h a t
(4.17)
q = p
d
+
",,(&(-,An
)),
j = h + g(0).
1
Furthermore, t h e correspondence between ( q , j ) and g is homeomorphic. Here t h e n o t a t i o n ( 2 . 6 ) of $(.,A)
i s adopted.
t u a l l y e x i s t s , f o r each symmetric o p e r a t o r A erator A
s,j , J
satisfying
(4.14) i n
S i n c e such g$0 as
P,h,h
s p i t e of A
qsj,j
(4.15)-(4.16)ac-
and n EN w e have a symmetric op-
1
#A
P,h,h'
m
In t h i s way, {An}n=O c h a r a c t e r i z e s a symmetric o p e r a t o r A we wonder if
0
p.h,h'
Naturally,
(4.14) combined w i t h j = h i m p l i e s q=p. A c t u a l l y , t h e s t u d y
of
(4.15)
Deformation Formulas and Inverse Problems
305
gives
Theorem
19
([38]).
I n t h e following cases,
( 4 . 1 4 ) with j = h i m p l i e s (4.13)
f o r symmetric o p e r a t o r s A p,h,h and A q,J . j . (i)
n
(ii)
Xn
1
= 0
E
1
u(A"). P
2 d2 Henceforth A' denotes t h e r e a l i z a t i o n i n L (0,l) of --2+p(x) P dx boundary c o n d i t i o n :
*
lx=0=. I x = l = O .
with Di r i chl et
0
Borg [ 2 ] , Levinson [17] and Hochstadt
[lo]
proved Theorem 19 f o r t h e c a s e o f ( i ) .
Recently, we have succeeded in proving t h e converse of Theorem
Theorem 20 (Suzuki [ 3 9 1 ) .
19. Put
U(A")
P
and an i n t e g e r n > 1= P,h,h m . 1 be given w i t h O(Ap,h,h)={Xn}n=O. Suppose An #Ax , where { X ~ ~ n = l = u ( A o )Then, P 1 1 Let a symmetric o p e r a t o r A
m
03
~ = ~ that w i t h o(Aq, j , ) = ~ I . A , } such q,J ,J 1 Furthermore, such q€CS[O,1] i s given by
t h e r e exists a unique symmetric o p e r a t o r A q#p,
( 4 . 1 4 ) and j = h .
where
(4.18)
L = L(x) =
@*'(X,h"
)@(x,An ) 1
- @*(x,x; )@'(x,Xn )
nl
Henceforth
(4.19)
@*=@*(.,A)
(>O).
1
1
@"'(O,X)
= 1. 0
i s t h e solution of
d2 ( - z z + p ( x ) ) @ *=
A@*,
@*(O,h) = 0 ,
1 Namely, f o r each pcC [0,1],hER and n ~N!{0,1,2,...), 1
t,he set
c o i n c i d e s w i t h {p} i f and only i f e i t h e r n =O o r X =Xz ( n l z l ) , and o t h e r w i s e 1 1 c o i n c i d e s w i t h {p, p - 2 ( L ' / L ) ' ) . For given pEC [ 0 , 1 ] and n >1, F i g . 2 d e s c r i b e s 1=
306
Takabhi S I I / I J K I m
m
the set of (q,h) which satisfies ( 4 . 1 4 ) for {Pn)n=O=U(Aq,h,h ) and iAnIn=n=u(Ag,h, d h). A bifurcation structure can be seen. Here h =s'(O)/s(O) with s f 0 , (--2+ dx "1 if and only if h-h p(x))s=X" s and s(l-x)=(-l)nls(x). It holds that A n =A: 1 1 "1 "1 h
.
Fig. 2
The key to the proof of Theorem 20 is the following theorem, which states a ) and U(A") and by itself is interesting: relation between O(A P P,h,h
if a.nd only if there exists hER such that
We conclude this section by the following Theorem 22 ( 1391)
.
Let. A and A be symmetric operators with P,h,h q,j ,j m =o(A?,~,~) and {pnjn=O=G(A9,J. .J. ) . Assume (4.14) for n1EN and also (4.23)
1
-(q(O)-p(O)) 2
= (j+h)(J-h).
Then, ( 4 . 1 3 ) follows. 0 Thus, ( 4 . 2 3 )
m
{Xn}n=o
is more powerful than j=h to get uniqueness.
Similar results to Theorems 19-22 are expected for the problems considered
Deformation Formulas and Inverse Problems
307
in Theorems 12-13 and Theorems 14-16. Furthermore, it would be interesting to consider
for (1.14) and to study similar problems to those in Theorems 19, 21 and 22. We shall discuss them in a forthcoming paper.
55. Inverse spectral problems for Hill's equations. In this section, we refer to our results on Hill's equations obtained by second deformation formula. Recall that 'S denotes the compact interval [0,1] with end points identified.
For pEC1(S1) , let us consider Hill's equation
for XER.
The discriminant, the trace of the monodromy matrix M ( A ) (see Magnus-
Winkler 1191, f o r example), is denoted by A ( A ) . noted by
{i } with n n=O Do
d2 ) of --2+p(x), dx
plicities counted in. malized by
I
A
h
,
.
,
.
c1
^S 23 coincides with U(A 1, the eigenvalues of A with multiP Let he the eigenvalues of corresponding t,o A n' norP
m
It is known that {An}n=o
h
.
-m<X
where h
The solutions of A(A)=+2 are de-
3
,
;"
1 L2( "1 s 1=1. h
is
Then, it is also known
h
,.
A
(mod h ) ) , 6 ( x + l ) = -@
@ n ( x + l )= @ (x) (n:0,3 h
(x) (nt1,P (mod 4)).
h
See [191, for these facts. Set I =(A 2n-1,A2n) (n=1,2,-**). Then, p is said to be of N - "finit,e hand" iff I =Q (n#ne; l(R<_N) for some O
Let pECm(S h
+-).
1
be of N h
-
A
finite band with
h
(A:)={An}n10
h
(-..
h
h
*
.
h
0
A
Set In=(A2n-1,A2n) and [In1=[A2n-1,X2nj for n=-1,2,.-., and assume
<.*a
308
Takashi SUZUKI
Theorem 23 (Suzuki [40]).
Suppose t h a t f o r each n#nQ
(ILQZN)t h e r e
some m(n)EN* such t h a t
Then,
Hochstadt [ 81 showed t h a t conversely
m
1
i m p l i e s ( 5 . 4 ) , assuming qsC (S ) a l s o t o b e o f N
-
f i n i t e band.
We conclude t h e p r e s e n t a r t i c l e by c h a r a c t e r i z i n g t h e s e t
and
exists
309
Deformation Formulas and Inverse Problems
In ( 5 . 7 ) , the notation n(O)=O is adopted. Then, Theorem 24
(5.9)
([40]).
qd! if and only if there exists GFG such that
d q = p + 2--(G*'$). dx
Furthermore, the correspondence between q and G is homeomorphic. Mckean-Moerbeke 1211 showed that G is homeomorphic to rus.
F, the N -
0 dimensional to-
We conclude that so is G by Theorem 24. However, the direct proof of
@?
has not been obtained yet.
Footnote
*
This work was supported partly by the Ffiju-kai.
References
[l] Ambarzumian, V., Uber eirie Frage der Eigenwerttheorie, Z. Phys., 53 (1929) 690-695. [2] Borg, G., Eine Umkehrung der Sturm-Liouvilleshen Eigenwertaufgabe, Acta Math.
, 78 (1946)1-96.
[31 Dobrivin, B.A., Mateev, V.B., Novikov, S . P . , Nonlinear equations of Kortewegde Vrie type, finite band operators and Abelian varieties, Russian Math. Surveys, 31 (1976) 59-146.
[4] Flashka, H., On the inverse problem for Hill's operator, Arch. Rat. Mech. Anal., 59 (1975)293-304. [ 5 ] Gel'fand, I . M . , Levitan, B.M., On the determination of a differential equation by its spectral function (English translation), A.M.S. Transl. ser. 2., l (19 55) 253-304.
[6] Goldberg, W., Hill's equation for finite number of instability intervals, J. Math. Anal. Appl., 51 (1975) 705-723. [ 7 ] -, Necessary and sufficient condition for determing a Hill's equation from its spectrum, J. Math. Anal. Appl., 55 (1976)549-554.
[8] Hochstadt, H., Function theoretic properties of the discriminant of Hill's equation, Math. Z., 82 (1963) 237-242. [9] -, On the characterization of Hill's equation via its spectrum, Arch. Rat. Mech. Anal., 19 (1965)353-362.
310
Takashi
[lo] -, 715-729.
sU711KI
The inverse Sturm-Liouville problem, Corn. Pure Appl. Math., 26 (1973)
[ll] -, Lieberman, B., An inverse Strum-Liouville problem with mixed given data, SIAM J. Appl. Math., 34 (1978)676-680. [12] Kitamura, S., Nakagiri, S., Identifiability of spatially-varying and constant parameters in distributed systems of parabolic type, SIAM J. Control & Optimi zation. 1131 Levrentiev, M.M., Some improperly posed problems of mathematical physics ( English translation, Springer, Berlin-Heidelberg-New York, 1967). Integral geometry and the problems in mathematical physics, to appear 1141 -, in: Glowinski, R. and Lions, J . L . (eds.) Computing Methods in Applied Sciences and Engineering, V. (Noth-Holland, Amsterdam, 1982). [ I 5 1 Lax, P.D., Periodic solutions of the KdV equations, in: Lectures of Applied Mat,h., 15, pp.85-96, (A.M.S. Providence, Rhode Island, 1974).
[16] Leviri, B.Y., Distribution of zeros of entire functions (English translation, A.M.S. Providence, Rhode Island, 1964). [17] Levinson, N., The inverse Sturm-Liouville problem, Mat. Tidsskr, B., (1949) 25-30. [la] Levitan, B.M., Gasymov, M.G., Determination of a differential equation by two of its spectrs, Russian Math. Surveys, 19 (1964)1-63.
[19] Magrius, W., Winkler, S., Hill's equation, (Dover, New York, 1979). [20] Marchenko, V . A . , Some questions of the theory of one-dimensional difqgrential operators of the second order, 1. (in Russian), 'TrudyMoskov. Mat. Obsc., 1 (1952) 327-420. [21] Mckean, H.P., van Morebeke, P., The spectrum of Hill's equation, Inventiones Math., 30 (1975)217-274. [22] --, Trubowitz, E., Hill's operator and hyperbolic function theory in the presense of infinitely many branch points. Corn. Pure Appl. Math., 29 (1976) 143 -226.
[23] Murayama, R., The Gel'fand-Levitan theory and certain inverse problems for the parabolic equation, J. Fac. Sci. Univ. Tokyo 28 (1981) 317-330.
1241 Nakagiri, S., Identifiability of linear systems in Hilbert spaces, to appear in SIAM J. Control & Optimization. [251
1982.
-,
On the identifiability of parameters in distributed systems, preprint,
[26l Picard, E., Lecons sur quelque types simples d'equation aux dGriv6es partielles, (Paris-Imprimerie Gauthier-Villars, Paris, 1950). [271 Pierce, A., Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control & Optimization, 17 (1979)494-499.
[28] Seidman, T.I., Ill-posed problems arising in boundary control and observation for diffusion equations, in: Anger, G. (ed.), Inverse and Improperly Posed
Deformation Formulas and Inverse Problems
31 I
Problems in Differential Equations, pp.233-247, (Akademie, Berlin, 1979). [291 Suzuki, T., Inverse problems for the heat equation (in Japanese), Newsletter from C&A seminar, 16 (1981)13-20, 17 (1981)10-11. [30]
-,
55-64.
Inverse problems for the heat equaiton (in Japanese), SGgaku, 34 (1982
Uniqueness and nonuniqueness in an inverse problem for parabolic equa[31] -, tions, in: Glowinski, R. and Lions, J.L. (eds.), Computing Methods in Applied Sciences and Engineering, V., pp.659-668 (North-Holland, Amsterdam, 1982). Uniqueness and nonuniqueness in an inverse problem for the parabolic [32] -, equation, to appear in J. Differential Equations. Remarks on the uniqueness in an inverse problem for the heat equation, [33] -, I., Proc. Japan Acad. ser. A., 58 (1982) 93-96. [34] -,
ditto, II., ibid, 58 (1982) 175-177.
[35] -,
On a certain inverse problem for the heat equation on the circle, ibid
, 58 (1982)243-245.
Inverse problems for heat equations on compact invervals and on cir[36] -, cles, I., submitted to J. Math. S O C . Japan. [37] -,
ditto, II., in preparation.
On the inverse Strum-Liouville problem for spatially symmetric opera[ 381 -, tors, submitted to J. Differential Equations. 1391
-,
[4O] -,
ditto, II., in preparation. in preparation.
[41] -, Murayama, R., A uniqueness theorem in an identification problem for coefficients of parabolic euqations, Proc. Japan Acad., 56 (1980)259-263. [42] Kohn, R., Vogelius, M., Determing conductivity by boundary measurements, preprint, 1982.
This Page Intentionally Left Blank
Lecture Notes in Num. Appl. Anal., 5, 313-331 (1982) Nonlinecrr PDE in Applied Science. U.S.-Jupan Seminar, Tokyo, 1982
STATIONARY SOLUTIONS OF THE BOLTZMANN EQUATION
*
Seiji Ukai
*
**
and Kiyoshi Asano
Department of Applied Physics, Osaka City University.
** Institute of Mathematics, Yoshida College, Kyoto University.
1. Introduction We consider a gas flow having a prescribed constant velocity c infinity and passing by an obstacle
o c
Rn.
6
R"
at
Such a flow has been discussed by
setting an exterior problem for the Euler or Navier-Stokes equation.
There are
many literatures on the existence and stability of stationary solutions for the incompressible case, but few for the compressible case, which gives a better description of gas flows. We mention the works [ 2 ] , [ 8 ] in which the compressible Euler equation is solved for small c on the existence of two-dimensional isentropic, irrotational stationary flows, whose stability, however, is still open. We should also mention [ 7 ] which solves the compressible Navier-Stokes equation in the large in time for
c
=
0.
The aim here is to discuss the exterior problem for the Boltzmann equation describing our gas flow, and specifically to show for n 2 3
that if
c
is
small, then stationary solutions exist and are asymptotically stable in time. The special case c = 0 has been solved in [l], [ 9 ] in the large in time, for
which stationary solutions are trivially given by Maxwellians. When
c f 0 , non-
trivial stationary solutions appear. Put n
=
Rn\8.
We assume that
piecewise smooth boundary
30
=
an.
8
is a bounded convex domain in %Zn with
Let
313
f
=
f(t,x,c)
denote the (probability)
Seiji UKAIand Kiyoshi ASANO
314
density of gas molecules having the position x E and velocity 5 E lRn at time t t R+. Then we shall study the following nonlinear initial boundary value problem on
f.
The equation (I.la) is the Boltzmann equation, where !Rn while
product o f
to
denotes the inner
Q, called the collision operator, is a quadratic integral
5
operator in the velocity variable specific
*
whose kernel is the collision cross section
the intermolecular potential.
We assume the cutoff hard potential in
the sense of Grad, see [ 5 ] . In the boundary condition (l.lb) at infinity, gc(5)
is a Maxwellian which
describes an equilibrium state o f a gas moving with the mean velocity The boundary condition on
afi
=
is (l.lc), which expresses reflections
20
ab. Let n(x)
of gas molecules by the wall
aR at x
E
' Then y
are the trace operators yff
denote the unit outward normal to
8 2 , and put
=
fl
+,and M
is an operator which maps
S-
funcrions on
into those on
S+
If the reflection at
S-.
terministic change of molecular velocity from m(x,S) map
c.
S- 3 (x,S)
+
(x.m(x,C))
E
+ S
and
to
x c
aR causes a de-
5 , then we are given a
315
Stationary Solutions of the Boltzrnann Equation For example, m(x,S) -E,
5-2(€,+n(x))n(x)
=
for the specular reflection and
m(x,c)
=
for the reverse reflection. We employ the regular reflection law of [ 6 ] for
the function m(x,E).
Non-deterministic reflection laws are also possible phys-
ically, an example of which is the diffuse reflection
This was discussed in [ 4 ] , and we impose on the kernel m(x,c,c') alogous to those given there,
(1.3).
so
conditions an-
as to include a wider class of M
than that of
may he any convex linear combination of (1.2) and (gener-
Furthermore, M
alizations of) ( 3 ) . Besides (l.l), we shall study the corresponding stationary problem:
That for c
=
Q[gc,gc]
=
0 for all
0 but not for
flections.
c
is known [ 5 ] , while in general, y-gc
as
c
t
+gc
c # 0, as is the case with the specular and reverse re-
Therefore go
solves (1.4) for
c
=
0, and it is natural to expect
solutions to ( 1 . 4 ) for small c # 0 which slightly differ from
go
= My
gc
and tend to
0.
The existence of such stationary solutions has been shown in
[lo],
using a
classical implicit function theorem for n 2 4 , but a theorem of Nash-MoserNirenberg type for n
=
3.
I n 5 2 , we will give a simplified proof showing that a
classical one is also useful for n
=
uniform estimates in the parameter
c which diverges as c
3.
pensated by thc closeness of solutions to
This became possible by getting non-+
0 but can be com-
gc'
Further, it will be proven in 1 3 that whenever the initial fo
of (l.ld) is
sufficiently close to a stationary solution, (1.1) has a unique solution in the large in time, which approaches the stationary solution as
t
-+
m
in the order
Seiji U K A Iand Kiyoshi ASANO
316
+
of (1
c1
t)*,
mates i n
t
> 1/2.
The proof i s c a r r i e d o u t w i t h t h e a i d of n i c e decay e s t i -
of t h e semigroup f o r a l i n e a r i z e d e q u a t i o n of (1.1). The p r o o f s
w i l l be only o u t l i n e d .
See [ll] f o r d e t a i l s .
2 . E x i s t e n c e of S t a t i o n a r y S o l u t i o n s Putting
f = gc
+
g:l2u,
w e rewrite ( 1 . 4 ) as
where we have d e f i n e d
The o p e r a t o r
Lc
have been i n v e s t i g a t e d e x t e n s i v e l y i n [ 5 ] f o r t h e c a s e
c = 0 , most r e s u l t s of which remain v a l i d f o r
decomposition
m
where
Uc(5)
t
Lloc(RE)
and
c f 0.
For example, Lc
has the
Stationary Solutions of the Boltzmann Equation with some constants vl, v2 > 0 and
1
L‘(!R~),
on
5
p
5
0
5
y
5
1, while K
317
is a compact operator
5 a.
In order to state the precise definition of the solution to (2.1), we need the spaces Lp = LP(Q
x
IR;)
and
t
Using ( 2 . 2 ) , we can show the unique existence of the trace operators y-
such
that
Here and hereafter lB(X,Y)
into another Banach space Y.
a Banach space X
Mo
for all
B(Yp’+, Yp’-),
E
p
E
[l,-j,
Definition. Let
i)
u E
iii)
u
p
E
hc
E
By our assumption on
M,
Yp’- with llhcll + 0(/c[ + 0).
the following definition makes sense.
so
[l,~). u
t
wp, y-u
ii) r [ u , u ]
will denote the set of linear bounded operators from
E
is said to be an LP-solution
to
(2.1) if
yp.-,
L‘,
satisfies (2.la) and (2.1~) in the spaces Lp and
Yp’-
respectively.
Note that if
p<
a,
u E Lp
satisfies (2.lb) in a generalized sense.
Seiji UKAIand Kiyoshi ASANO
318
Define the linearized Boltzmann operator
B
C
=
-c'V
Bc by
x + Lc
-1 Suppose Bc possess an inverse Bc . Suppose further the linear inhomogeneous
boundary value problem -E*Qx$
@
(2.4)
+
Mo$
0
+
LcO
=
in R
0,
(1x1 * "),
5'
on
OC.
=
IR~,
in iRn
= hc,
possess a solution $
x
S
-
,
Then (2.1) can be rewritten as
It i s this equation to which the implicit function theorem is to be applied. The existence of
BC1
shall be established with the aid of the p e r t u r b a t i o n
technique and the limiting absorption principle. we employ is Bc
m
B
= -5.0
m
for the case
+
The unperturbed operator which
Rn, denoted as Bc, namely,
=
Lc.
For later purposes we prepare notations. We denote by the resolvent set and spectrum of
m
Bc
respectively, and f o r ao,
Put
-
c ~ + (=5(A ~ t)
C( Reh
z(ao,
6
Go)
=
{A
2 do],
2+ (4 )I o
-Reh
5
m
p(Bc)
2
a o ~ I i d1~,
co,
and
O(B:)
uo > 0, we
Stationary Solutions of the Boltzmann Equation B[co] = (c
E
Further, i f
E
c/
!Rni
5
c
1.
i s a m e t r i c space and
t h e Banach space of
3 19
X
0 i s a Banach s p a c e , B ( E ; X )
w i l l denote
E.
.<-valued, bounded and continuous f u n c t i o n s on
We need
a l s o t h e space
A s p e c t r a l a n a l y s i s of
BZ
i s found i n [ 3 ] f o r t h e s p e c i a l c a s e
can be c a r r i e d o u t s i m i l a r l y f o r
Theorem 2 . 1 .
u0
and
ii.)
Let
p = 2.
For any
and
c )F 0, y i e l d i n g t h e
co z 0 , t h e r e are p o s i t i v e numbers
such t h a t t h e followings hold for a l l
BZ
c = 0
c
ao,
K~
B[co].
has t h e orthogonal decomposition such t h a t
Here
and f o r
where
yx
1
s n+2,
means t h e Fourier transformation w i t h r e s p e c t t o
variabZe t o and
5 j
x,
x (1 kl SK
)
x, k
E
Rn
the c h a r a c t e r i s t i c f u n c t i o n f o r t h e domain
a dual
1 kl
5
ice,
Seiji U K A Iand Kiyoshi ASANO
320 h.(k,c) = p . ( / k ] ) J J
-B
a . K
.(K)
J
J
pjpQ =
f o r any
p ? 2
ik-c
2
+
J
a . E IR. B
w i t h coefficenta
iff;),
.K
+
J
o
(j \
3
)
(K
-f
o), P.’s are orthogonu2 pro,jections on
> 0 , while
3
Q,) with
6
and
o(/K/
j
,/-1
i =
E
R.
m
Bc
I n view of (i) of t h e above from ( i i ) t h a t t h i s i s because
U. J
i s n o t i n v e r t i b l e on
(A
m
+
+
0)
in
2
B(L )
2 IB(L )
for
and i t i s s e e n
15 j
n+2.
However we can e s t a b l i s h t h e l i m i t i n g a b s o r p t i o n p r i n c i p l e i n t h e s e n s e t h a t
A
l i m i t as
has a well-defined
+
0
i f d i f f e r e n t spaces a r e chosen.
l y w e need t h e space
Theorem 2 . 2 .
Put
(2.7)
Let
y =
Let
1S q
2
5
p 5
m,
8
[0,1),
II
Then f o r
1
E
=
0 , 1.
1 1 - - and suppose q P
-
Y
2-R ’ n+F3
*
ao, co,
u0 be as i n Theorem 2 . 2 .
Proof.
Write
G =.jk,u.
j
It i s n o t hard t o s e e t h a t
5
n+2,
U
j
More p r e c i s e -
32 I
Stationary Solutions of the Boltzmann Equation
holds i f
p t 2
. . and
+ I, = P
P
1.
T h e r e f o r e , by v i r t u e of Theorem 2 . 1 ,
where
The l a s t i n e q u a l i t y i s v a l i d o n l y i f ( 2 . 7 ) i s f u l f i l l e d . and
c
The c o n t i n u i t y i n
c a n b e proved s i m i l a r l y .
In o r d e r t o l i n k
m
B
to
Bc, we need t h e s o l u t i o n o p e r a t o r
Rc(h)
to the
inhomogeneous boundary v a l u e p r o b l e m
I t c a n b e shown t h a t a u n i q u e S(Yp’-,
X
Wp).
Let
e,r
LP-solution
u = R (h)h
e x i s t s and
b e t h e e x t e n s i o n and r e s t r i c t i o n o p e r a t o r s ;
Rc(X)
E
322
Seiji U K A Iand Kiyoshi ASANO eu
=
R,
in
u
ru = u/
u
n'
=
o
IR~\R, u
in
LP(Q),
c L'(R;).
Define t h e o p e r a t o r
Tc(h)
= Gor(h-Bc) m -1eKcRc(h),
0 B (X(ao, ao)xB[co]:k3(Yp'-))
which can be found t o be i n where
a o , c o , a.
a r e t h o s e of Theorem 2 . 2 .
for a l l
that
1
F
p(Tc(X))
n
2
f o r a22
p c [2,m], h
E
[2,-]
and
E ( a o , Go)
-
-
ao , c o . a o ,
There e x i s t p o s i t i v e numbers
3.
E
Furthermore, w e can s e e t h e
Let,
Proposition 2.3.
p
c
E
B[co].
such
According
7-11 there e x i s t s
T h i s f a c t i s e s s e n t i a l f o r t h e proof of t h e e x i s t e n c e o f solve the equation
(A-Bc)u
=
f
by l e t t i n g
m
u = r(X-Bc)
-1
ef
(A-Lc)
+
ul.
-1
.
Indeed,
A f t e r some
manipulations, w e g e t
where
*
denotes adjoints.
T h i s i s a s u b s t i t u t e f o r t h e second r e s o l v e n t equa-
t i o n i n t h e p e r t u r b a t i o n t h e o r y of l i n e a r o p e r a t o r s .
Combined w i t h Theorems 2 . 1 ,
2 . 2 and P r o p o s i t i o n 2 . 3 , i t p r o v i d e s n e c e s s a r y i n f o r m a t i o n s of
see t h a t i n
L
2
,
Bc.
F i r s t , we
Stationary Solutions of the Boltzrnann Equation c
for
Moreover, a p p l y i n g t o
B[co].
E
Uo
o f Grad [ 5 ] w h i c h makes p o s s i b l e t o d e d u c e a i d of n i c e p r o p e r t i e s of t h e o p e r a t o r
Bc
tion principle for
(2.9).
at
X
=
0
of
323
Theorem 2 . 1 t h e i t e r a t i v e scheme
m
L -estimates f r o m
2 L -ones w i t h t h e
Kc, w e c a n e s t a b l i s h t h e l i m i t i n g a b s o r p -
and o b t a i n s t h e i n v e r s e
-1 , r e g a r d l e s s of
Bc
Put
=
I;i"'" " LP,m 1 B-
'
-
P
n+2 and d e f i n e
Ac
s m ,
Y
n
B > E, 2 0
1 1 = - - - > q P
Put for
vc(S)x,
Let
Theorem 2 . 4 . p
=
and
3
2
jG1
-
=
pj(0,c).
-
ao, c o ,
[0,1), R = 0 , 1
2-R n e
'
-1< I - - - 2.
p
n+A
[0,1]
c1 E
v(X,c)
E
Pc =
(A-Bc)-'(I-Pc)
RAcu(c) a
Go
> 0
satisiy
he us before.
Suppose
1
5
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)
0
=
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
al
=
A(l
and noting that the nullspace
R
c
(2.11)
(0)hc + @ OC
=
and Go
C 2 0
be as b e f o r e .
and for any
c
6
Let
Brio],
P
-1
=
Lp.
2 + -).
The inverse Bc @c
Wp,
is invariant in c , we have the
Pc
with
r
holds i n
E
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.
n
2
3
and
co
be as before and 2et
If
Therefore
Stationary Solutions of the Boltzmann Equation These two propositions enable us to (2.5). when
c
-L
0, and that if
n
=
8 > 0, (2.11) becomes meaningless
a2
u = jclav
in (2.5) to write
c
for
Put
0.
by
3 , 8 = 0 is excluded since then (2.10) becomes vac-
But this difficulty can be removed as follows.
p.
then a1
for
apply the contraction mapping principle
It should be noted, however, that if
u o u s for
-G[v]
to
-G[v](c)
p
2
=
0. For functions v
If
a such that a1 <
2, and we can choose an
G(v,O)
325
v(c),
=
E
[0,2/7),
LY.
< a2.
Put
define the nonlinear map N
=
G(v(c),c).
What is to be proved is that
G
is a contraction
on a ball of the space
if
E
(ii)
> 0
and
if
c
> 0
and Proposition 2.6
is sufficiently small.
(ii), G
maps
Proposition 2.5 and writing the norm of
where lows.
0 =
a
Now
-
*
G
al > 0, T
=
a2
-
LY.
Theorem 2.4(iii)
the proof of the
Theorem 2 . 7 .
Let
n 2 3
and l e t
Vi,E
into itself, and by the aid o f
Vp
as
B ,E
111 111,
> 0, whence the desired conclusion readily fol-
has a unique fixed point
solves (2.5) uniquely.
By virtue of Theorem 2 . 4 (i)
v = v(c)
6
V;,€
and
uc
=
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
c
E
E
> 0
f
u
.
Moreover
C 2 0
u
c
E
such t h a t f o r each Vp
B ,E
for1 any
and
Obviously fc
Co
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),
t
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
C'
'1
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 ) .
Bc
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)]]~~
0
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
uc
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
t
0,
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
m
of ( 2 . 6 ) .
Bc
t r a n s f o r m J-l[(A-A)-l] m
Be.
on
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
L
2
,
and can f i n d t h a t
while for
1 5 j
S
n+2,
Proposition 3.1.
Suppose
1
Iq 5
2
5
p s
m
and
m = 0, 1. Put
n l - -). 1 2 q P
y1
= -(-
Then there is a constant
C >_ 0
and for all
c E lRn, t 2 0
Theorem 2.2, in this time with
6
=
4
1 -P
.
Take the inverse Laplace transform of (2.8) to obtain
(3.4)
Ec(t)
m
=
where, writing
and
1
5
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
t;
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,
y
i f
n
C
2
1
= $n-1)
n
s u b s t i t u t e d i n t o (3.4),
=
3
c
t
if
only
rewrite
Let
y > 1/2
and
p, a
as
For 8
n
is even.
if
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
E
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.
0
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 .
0
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
In
Seiji UKAIand Kiyoshi ASANO
330 where a
=
IIuAl
and xpo
5
B
-0
c1
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
B
is small, H
c
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\ <
Its unique fixed point w
m.
=
w(t)
is a con-
In this way we prove the
(3.2).
Theorem 3.3.
Let
n
z 3 and
p , q,
8,
be a s ubovs.
ao, al, co such t h a t f o r each
c
t h e n (3.2) hus u unique gZobaZ soZution
w
numbers
Obviously
f(t)
=
f c -t g:''
w(t)
w(t)
11 woIIxp a
and if
B[co] =
Then t h e re is p o s i t i v e
c
Bo([O,
solves (1.1) and
showing the asymptotic stability of the stationary solution
5
ao*
t
+ to,
Xp) hiith
m ) ;
f(t)
n zq
5
L
fc
as
fC.
References Asano,K.; On the iniLial boundary value problem of the nonlinear Boltzmann equation in an exterior domain, to appear. Brezis,H. and Stampacchia,G; The hodograph method in fluid dynamics in the light of variational inequalities, Arch. Rat. Mech. Anal. 6 1 (1975) 125156.
Ellis,R.S. and Pinsky,M.A.; The first and second fluid approximations to the
linearized Boltzmann equation, Giraud,J.P.;
An
H
J.
Math. Pures et Appl. 54 (1975) 125-156.
theorem f o r a gas of rigid spheres, Theories Cinetiques
Classiques et Rilativistes, C. N. R. S., Paris (1975). Grad,H. ; Asymptotic theory o f the Boltzrnann equation, in Laurmann, J . A . (ed.), Rarefied Gas Dynamics, I (Academic Press, New York, 1 9 6 3 ) .
Stationary Solutions of the Boltzmann Equation
331
Kanie1,S. and Shinbrot,M.; The Boltzmann equation I, Comm. Math. Phys. 57 ( 1 9 7 8 ) 1-20,
Matsumura,A. and Nishida,T.; Global solutions to the initial boundary v a l u e problem f o r the equation of compressible, viscous and heat conductive fluids, to appear. Morawetz,C.;Mixed equations and transonic flows, Rend. Math. 25 (1966) 482509.
Ukai,S. and Asano,K; On the initial boundary value problem of the linearized Boltzmann equation in an exterior domain, Proc. Japan Acad. 56 (1980) 1217. and
; On the existence and stability of stationary solutions
of the Roltzmann equation for a gas flow past an obstacle.
Research
notes in Math. 60 (Pitman 1 9 8 2 ) 350-364. and
; Stationary solutions of the Boltzmann equation for a
gas flow past an obstacle, I Existence and I1 Stability, preprints.
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Lecture Notes in Num. Appl. Anal., 5 , 333-344 (1982) Nonlinear PDE in Applied Science. U.S.-JapanSeminar, Tokyo, 1982
On t h e L i n e a r S t a b i l i t y A n a l y s i s o f Magnetohydrodynamic System * )
Teruo Ushi j i m a Department o f I n f o r m a t i o n Mathematics The U n i v e r s i t y o f Electro-Communications 1-5-1 Chofugaoka, Chofu-shi, Tokyo, 182
JAPAN
A mathematical d e s c r i p t i o n o f l i n e a r s t a b i l i t y a n a l y s i s o f magnetohydrodynamic (MHD, i n s h o r t ) system o f equations i s presented. A way o f l i n e a r i z a t i o n o f i d e a l MHD system i n a v i c i n i t y o f an e q u i l i b r i u m was p h y s i c a l l y e s t a b l i s h e d . We show t h a t t h e o b t a i n e d l i n e a r i z e d MHD system o f equations can be t r e a t e d as t h e 2nd o r d e r 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 a H i l b e r t space under a p p r o p r i a t e c o n d i t i o n s on t h e equilibrium. A j u s t i f i c a t i o n o f energy p r i n c i p l e i s a l s o established.
01
THE INITIAL BOUNDARY VALUE PROBLEM OF THE LINEARIZED MAGNETOHYDRODYNAMIC
SYSTEM OF EQUATIONS. The governing system o f equations i s t h e f o l l o w i n g i d e a l magnetohydrodynamic system. I n t h e plasma r e g i o n R (1) (2)
(3)
P'
= - p d i v v, p
(4)
+
= -7P
Dt
D
JxB,
(PP-Y) = 0, = -rot
B
E,
(5)
div
(6)
rot B = pJ,
(7)
E + vxB = 0.
= 0,
T h i s work was p a r t l y supported by G r a n t - i n A i d f o r S p e c i a l P r o j e c t Research ):; on Energy (Nuclear Fusion) o f t h e M i n i s t r y o f Education, Japan.
333
Teruo USHIJIMA
334
I n t h i s model t h e plasma i s c o n s i d e r e d t o be a f l u i d s p r e a d i n g o v e r t h e bounded The i n t h e space R 3 w i t h t h e d e n s i t y p and t h e v e l o c i t y v e c t o r v . P s c a l a r f u n c t i o n P i s t h e p r e s s u r e , and t h e 3 - v e c t o r f u n c t i o n s B, E and J a r e t h e
domain R
m a g n e t i c f l u x d e n s i t y , t h e e l e c t r i c f i e l d and t h e c u r r e n t d e n s i t y , r e s p e c t i v e l y . The p o s i t i v e c o n s t a n t 1 ~ , and y a r e t h e m a g n e t i c p e r m e a b i l i t y o f t h e vacuum, and the s p e c i f i c heat r a t i o .
Here we used t h e m a t e r i a l d e r i v a t i v e :
w i t h t h e g r a d i e n t o p e r a t o r V and t h e 3 - d i m e n s i o n a l i n n e r p r o d u c t : ( symbol
x
denotes t h e e x t e r i o r p r o d u c t o f 3 - v e c t o r s .
=
aa t +
(v,
,
).
The
v)
The e q u a t i o n ( 1 ) i s t h e
c o n s e r v a t i o n o f mass, ( 2 ) i s t h e e q u a t i o n o f m o t i o n , ( 3 ) i s t h e a d i a b a t i c c o n d i t i o n coming f r o m t h e c o n s e r v a t i o n o f energy.
And ( 4 ) , ( 5 ) and ( 6 ) a r e d e r i v e d
f r o m t h e Maxwell e q u a t i o n u n d e r t h e assumption o f t h e s u p e r i o r i t y o f t h e m a g n e t i c field.
The c o n s t i t u t i v e
l a w ( 7 ) i s Ohm's l a w w i t h i n f i n i t e e l e c t r i c
conductivity. Now we c o n s i d e r t h e c l o s u r e o f plasma r e g i o n R bounded r e g i o n R, whose boundary
i s t h e vacuum r e g i o n i$. L e t r be t h e P P Then t h e b o u n d a r y aRv o f nV i s t h e u n i o n o f r and r
conducting w a l l . boundary o f
R
The domain
n-5
P
P'
(see F i g . 1 ) .
r
i s completely contained i n a P i s c o n s i d e r e d t o be a c o m p l e t e l y e l e c t r i c a l l y
H e r e a f t e r Rv i s assumed t o be connected.
I n t h e vacuum r e g i o n Rv, t h e f o l l o w i n g c o n d i t i o n s a r e assumed
t'
t o hold:
(8)
p = P = O , v = J = O ,
(4)v
at =
(5)v
d i v B = 0,
(6)v
r o t B = 0.
- r o t E,
On t h e boundary
r,
the
f o l l o w i n g boundary c o n d i t i o n i s assumed:
(9)
nxE = 0
where n i s t h e u n i t normal
Figure 1.
I l l u s t r a t i o n o f TOKAMAK t y p e plasma c o n f i n e m e n t
Linear Stability of Magnetohydrodynamic System
r
vector o f
T h i s r e p r e s e n t s t h e e f f e c t o f complete c o n d u c t i v i t y of t h e w a l l
335
r.
I t i s noted t h a t t h e boundary c o n d i t i o n (10)
=o
(B,n
i s compatible w i t h ( 9 ) s i n c e ( 9 ) i m p l i e s ( r o t E,n) = 0 which may i m p l y
a at (B,n)
=
0 by ( 4 I v .
r
On t h e boundary
P
t h e f o l l o w i n g 3 connection c o n d i t i o n s a r e imposed:
( B p 3 n ) = (Bv,n)
(lO)p,v (11)
(EP + vp
(12)
p
+
P
1
B
Bp)
x
2
211 P
=
0,
= x
L
n =
(E +
B
,
2
v
V
P
x
Bv)
x
n,
2P v Here t h e s u b s c r i p t s p, and v, r e p r e s e n t t h e l i m i t i n g values o f t h e s u b s c r i p t e d q u a n t i t i e s from t h e i n t e r i o r o f t h e plasma r e g i o n 0 and t h e vacuum r e g i o n R V $ P' (lO)p,v corresponds t o the requirement t h a t r should be respectively. P (11) represents t h e c o n t i n u i t y of t h e t a n g e n t i a l
a magnetic surface.
p a r t o f t h e e l e c t r i c f i e l d i n t h e c o o r d i n a t e system a t t a c h e d t o t h e plasma. (12) i s c a l l e d t h e pressure balance c o n d i t i o n .
1 The sum P t - B2 i s s a i d t o be
2u
t h e t o t a l pressure, where B 2 = (B,B). Thus we have an e v o l u t i o n problem: (1).,(12)
f o r t h e i d e a l MHD system.
A t i m e independent s o l u t i o n o f t h i s system w i t h z e r o v e l o c i t y and zero
e l e c t r i c f i e l d i s s a i d t o be an e q u i l i b r i u m : (13)
{p,v=O, 'P,J,B,E=O}.
Then an e q u i l i b r i u m t r i p l e {P,J,BI
satisfies
VP = JxB, d i v B = 0, r o t B = p J
I
\ p + l B 2 211 p
= 1 B
zu
v
2
in R
on
r
P'
P'
I n t h e work of B e r n s t e i n e t a l . [l],a way o f l i n e a r i z a t i o n o f i d e a l MHD system i n a v i c i n i t y o f an e q u i l i b r i u m was f o r m u l a t e d .
The o b t a i n e d l i n e a r i z e d
magnetohydrodynamic (LMHD, i n s h o r t ) system o f equations d e s c r i b e s t h e time e v o l u t i o n of t h e Lagrangean displacement ( ( t , r )
RP a t t i m e t - 0 .
o f a p o i n t r o f t h e plasma r e g i o n
We encounter t h e f o l l o w i n g i n i t i a l boundary value problem:
336
Teruo USHIJIMA
PROBLEM 1
I__-
Find a p a i r
C = C(t,r):
I
o f vector valued functions
{C,cl}
[O,m)xlp + R 3
and a = a ( t , r ) :
(O,m)xEv + I R 3
s a t i s f y i n g the following conditions:
(15)
pF = -KC
i n (O,m)xfip,
a
C(O,r) = 0 , L(5,a) = 0
C(0,t-l
, -(5,n)Bv
=
=
in R
v(r)
on
vxa
P’
rP’
rot rot a = 0
i n Rv,
nxcl = 0
on
r.
In t h e above problem, we used t h e n o t a t i o n s :
t} -
1 - { r o t B x r o t ( S x B ) + [ r o t rot(CxB)]xB} ,
(16)
KS = -V{(c,VP)tyP
(17)
L(S,cc) = - r P d i v 5 + ; 1 (Bp. rot(CxBp)+(C,V)B
div
1-1
w i t h s p e c i f i e d e q u i l i b r i u m q u a n t i t e s P and B.
1- 1 1 (Bv,rOt cx + (S.V)Bv) P Operators K, and L, a r e formal
and on r respectively. The boundary c o n d i t i o n : P’ P’ i s d e r i v e d from t h e pressure balance c o n d i t i o n ( 1 2 ) . And t h e
operators defined i n R
L(E,cx)=O on
rP ’
boundary c o n d i t i o n : -(C,n)Bv=nxa,
comes from t h e c o n t i n u i t y c o n d i t i o n (11).
The v e c t o r r o t cx r e p r e s e n t s t h e v a r i a t i o n o f magnetic f i e l d 6, region.
The requirement: r o t r o t a = 0 corresponds t o ( 6 ) v .
i n t h e vacuum Finally the
boundary c o n d i t i o n : nxa-0 corresponds t o ( 9 ) .
Hereafter we f i x an e q u i l i b r i u m {P,J,BI
s a t i s f y i n g t h e f o l l o w i n g Assumption 2.
ASSUMPTION 2.
(A.l)
R,Rp and Rv a r e bounded connected domains w i t h Cm c l a s s boundaries.
(A.2)
B I Q p , and 81
QV’
have extensions b e l o n g i n g t o
C2(6) , P
and C 2 ( n v ) ,
respectively. (A.3)
PIQp
has t h e e x t e n s i o n b e l o n g i n g t o C’(5 ) , being p o s i t i v e on R whose P P’ c r i t i c a l p o i n t s e t Cp=IrcR .(VP)(r)=O} i s composed o f t h e f i n i t e number P’ of connected components, each o f which i s e i t h e r a C 3 - c l a s s s i m p l e curve,
or a C3-surface a t every p o i n t o f which B i s c o n t a i n e d i n t h e t a n g e n t i a l plane.
Linear Stability of Magnetohydrodynamic System Either B
(A.4)
rP'
never vanishes on
P
constant i n
o r P i s bounded b e l o w by a p o s i t i v e
R
P' a F o r t h e e x t e r i o r normal d e r i v a t i v e Fn on
(A.5)
337
n from R t o
Qv,
P
a Bzp I - 2an
t h e gap: G=
rP
a l o n g t h e u n i t normal v e c t o r
a
8' an (% +p)IP
-
i s a nonnegative C'
c l a s s f u n c t i o n on I' where t h e f i r s t , and t h e second, terms a r e t h e one P' s i d e d d e r i v a t i v e s f r o m RV, and f r o m R respectively. P'
52
THE PLASMA ENERGY BILINEAR FORM AND THE LINEARIZED MAGNETOHYDRDDYNAMIC
OPERATOR.
L e t Hm(n) be t h e u s u a l m - t h o r d e r Sobolev space o f t h e r e a l v a l u e d s c a l a r f u n c t i o n s on t h e domain R i n R 3
.
The t o t a l i t y o f 3 - v e c t o r f u n c t i o n s whose
components b e l o n g t o Hm(Q) i s d e n o t e d by Mm(R). A
*
p a i r s {F,,a} o f 3 - v e c t o r f u n c t i o n s , W , V and (1)
i
=
{i={t,a}FH ' ( R ~ ) x M ~ R ~ ,n)x:a
A
(2)
v
h
=
i ~ = { ~ . a ~ c ri oi :t r o t
c1
=
o
=
6 as
- (S,n)Bv
Now we i n t r o d u c e t h e spaces o f follows. on
rP'
nxa
=
0
on
r].
i n a,,}.
6 = { ~ = { S , ~ I ~ V ct: I H Y RP 1, L ( s , ~ ) = D e f i n e t h e b i l i n e a r f o r m $(i,G)f o r = (3)
on
r PI .
{C,CX},
6=
o
{n,B}< 'H1(Rp)XM'(Qv) as
follows :
a(i,fi)= ap(S,n)+a5(S,n)+Sv(",~), ap(S,q) = al (S,n)+a2(c,n), al(c,q)
=
(4)/ a2(t,q) =
as(c,n)
=
dv(a,B) =
1
{yP d i v 5 d i v q
RP 1 jn T(S,BP)div n - - ( r o t ( S x B ) , n x r o t B ) } d r , u B2 + P)lp}(5,n)(q,n)dr, I rP pa { B Z~ I ~v -- a ( -
1
1
- ( r o t a,rot R)dr, Rv lJ
where n i n t h e e x p r e s s i o n o f a s ( c , q ) from
n t o Rv. P
+1 (rot(C,xB),rot(nxB))}dr u 2u
i s t h e u n i t o u t w a r d normal v e c t o r on
It i s e a s i l y checked t h a t S ( i , R )
r
P
i s well-defined f o r
[ , f i ~ H'(Qp)x .H'(Rv). Using t h e p r o p e r t y o f t h e e q u i l i b r i u m i n (1.14), i t can be p r o v e d t h a t
and t h e i n t e g r a t i o n b y p a r t ,
Teruo U S H I J I M A
338 /,p(KS,n)dr
i + swith
f o r any
s(i,G) + lrP L ( S , a ) ( q , n ) d r
=
F,r
M2(n ) a n d P
GE~.
From t h i s formula we can conclude the
following P r o p o s i t i o n 1 .
Now we i n t r o d u c e f u n c t i o n spaces V (8)
Vo =
(9)
Do
=
0 and D0 a s f o l l o w s . IF,+ H’(62 ) : there i s an nt,H’(itv) such t h a t = { C , u } c i } . P iCclH2(Q ) : t h e r e i s an ar.H’(nv) such t h a t
P
= t c , a} c b} .
From t h e theory of harmonic v e c t o r f i e l d s w r i t t e n i n t h e book of Morrey [31, we have t h e following f a c t .
Proposition 2
For F,.V0,
Vo = IH’(n ) . P
r o t a i s uniquely determined.
form a ( c , n , ) d e f i n e d on (10)
a(t,n) =
2(i,G),
VoxVD
Hence we can d e f i n e a symmetric
by the r e l a t i o n :
E.rlcVo.
For n R - C d e f i n e 3-vector f u n c t i o n s E = ( r o t B ) x e , F=Z(B,V)e+Bxrot e where P P’ e=VP/IVPI. For S , q c M1(n ) and NCR -C let P P P’ 1 ( 1 1 ) Ap(S,ri) = ;( r o t ( E x B ) + ( c . e ) E , r o t ( n x B ) + ( q , e ) E ) + yP d i v 5 d i v n 1 -(E,e)(q,e)(E,F). 1-1 Then we can show (12)
ap(C,n)
1
A ( E , n ) d r f o r e,rie M ’ ( R p ) . RP From t h i s r e p r e s e n t a t i o n of Plasma energy, the following P r o p o s i t i o n 3 can be =
deduced.
Proposition 3
Let
Linear Stability of Magnetohydrodynamic System
COROLLARY 4 (15)
Assume t h a t
a G=-an
8’ 211
1 v - -aa(n - +Bz 2p
P)lp ’ 0
Then the symmetric b i l i n e a r form (16)
339
aK(E,n)
=
aK(c,q)
a(5,n) + K(S,I))~,
on
r
P‘ d e f i n e d on Vo:
5.n6Vo,
i s positive definite i f (17)
K>KO.
The following P r o p o s i t i o n 5 i s fundamental t o o u r d i s c u s s i o n , t h e proof of
which was given i n
Proposition2
Let X be the H i l b e r t space {L2(np)}3.
i s closable in X i f V
0
141.
K > K ~ .
Namely, i f the saquence {$,:
Under Assumption 1 . 1 , a m = 1 , 2 , . . . } c o n t a i n e d in
satisfies
II
(18)
5,I
x
= 0
and (19)
-
liT
m,m
aK(Sm-E;,-) = 0,
then we have (20)
where
lim
mt-
/I 511
aK($,,) = 0, =
(5,C)x1’2
and a K ( E ) = a K ( 5 . 5 )1 / 2
.
The preceding argument can be transformed i n t h e frame work o f complex H i l b e r t space t h e o r y .
Under t r i v i a l m o d i f i c a t i o n s , we can c o n s i d e r the h e r m i t i a n
form a K ( 5 , n ) with t h e domain Vo=iH‘ln,)
in the complex H i l b e r t space X={L2(np))’.
Teruo USHIJIMA
340
Let
then a K i s c l o s a b l e by P r o p o s i t i o n 5 .
K’KO
T h e r e r f o r e we have the f o l l o w i n g
s t a t e m e n t s ( 2 1 ) and ( 2 2 ) f o r t h e completion V of Vo by aK-norm. (21)
V i s densely imbedded i n X .
(22)
a K ( S , n ) , which i s considered t o be d e f i n e d on V by c o n t i n u i t y , i s a p o s i t i v e d e f i n i t e c l o s e d hermitian form.
From Theorem 2.23 o f Chapter VI in Kato [2], there e x i s t s the unique p o s i t i v e d e f i n i t e s e l f a d j o i n t o p e r a t o r AK s a t i s f y i n g t h e following p r o p e r t i e s ( 2 3 ) ~ ( 2 5 ) . (23)
The domain D ( A K ) i s dense i n V with aK-norm.
(24)
(AKS,n)X= a K ( E , q ) f o r S t D ( A K l . ncV.
(25)
D(AK”‘)
= V,
and ( A K 1 ’ 2 5 , A K 1 / 2 ~ ) x
=
aK(S,q)
f o r S,ncV.
Now we d e f i n e the LMHD o p e r a t o r A by A = AK -
(26)
K,
D(A) = D(AK).
By a s t a n d a r d argument, t h i s d e f i n i t i o n ( 2 6 ) i s independent of t o be noted t h a t a K a r e mutually e q u i v a l e n t i f
K~K,,.
if
K
K>K,,.
I t is
On the o t h e r hand we can
d e f i n e t h e o p e r a t o r KO by KoS = KS , ScD(KO) = Do,
(27)
which i s symmetric by ( 7 ) o f P r o p o s i t i o n 1 . Theorem 6
_ I _
A i s an e x t e n s i o n of KO s a t i s f y i n g t h e following p r o p e r t i e s .
(28)
I f F,cDO, then ScD(A) and AF, = K05.
(29)
I f F , c D ( A ) n l H 2 ( Q ) , then S E D ~ . P
Namely we have Do = D(A) n H 2 ( Q p ) .
(30)
Now Problem 1.1 can be regarded
x
as an e v o l u t i o n e q u a t i o n i n t h e H i l b e r t space
= {L2(s? 1 1 3 :
(31)
{
M
:2s dt2 +
AF, = 0 , t > O ,
5 ( 0 ) = 5’,
$ (0) = E0,
with t h e i n i t i a l d a t a IF,’,5°1 = ( 0 , ~ )under Assumption 1 . 2 on the e q u i l i b r i u m quantites.
Here M i s t h e m u l t i p l y i n g o p e r a t o r with m u l t i p l i e r
LMHD oDerator.
p
and A i s t h e
Linear Stability of Magnetohydrodynamic System
93
34 1
A REMARK ON THE ENERGY PRINCIPLE.
Assumption 1
L e t {P,v=O,
P,J,B,E=Oj i s an e q u i l i b r i u m o f i d e a l MHO system ( 1 . 1 4 ) b e i n g measurable i n fi P'
w i t h t h e p r o p e r t i e s t h a t f o r t h e plasma d e n s i t y p = p ( r ) , t h e r e a r e p o s i t i v e c o n s t a n t s p and
such t h a t
-
-
O < -~ < p ( r ) < ~ < on m R
P' and t h a t Assumption 1 . 2 i s s a t i s f i e d f o r {P,J,B}.
L e t us d e f i n e
m(S,o)
M5
=
=
1
%
p(r)(S,n)C3dr
(R,,) 1 3 ,
f o r S,IIC{L'
PS
for b{L2(op)13.
Denote t h e H i l b e r t space i L 2 ( Q ) I 3 w i t h t h e i n n e r p r o d u c t m ( c , r l ) by X P P' Define t h e operator A
A ~ F ,=
AS
?
acting i n X
? by
f o r C?D(A~)=D(A),
where A i s t h e LMHD o p e r a t o r e s t a b l i s h e d i n Theorem 2.6.
Proposition 2 ___-__
2) Let
K~
1 ) The o p e r a t o r A
be as i n ( 2 . 1 3 ) . KP)l/')
t
=
For
P
i s s e l f a d j o i n t i n t h e space X
A +
K~>K~/P,
?
K
P
P
.
i s positive definite, satisfying
v,
L e t a(A ) be t h e s p e c t r u m s e t o f A P
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
P
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. Let
X=
P
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
i s s a i d t o be s t a b l e , m a r g i n a l ,
respectively.
The f o l l o w i n g Theorem may be c o n s i d e r e d as a j u s t i f i c a t i o n o f t h e e n e r g y p r i n c i p l e f o r m u l a t e d by B e r n s t e i n e t a l . [l].
=
A
An e q u i l i b r i u m i s s t a b l e o r m a r g i n a l i f
Theorem 4
^
i~,a)ci.
And i t i s u n s t a b l e i f
A
A
A
a(E,c)
110
f o r any
=
{S,afei.
h
a(c,c) c 0
f o r some
Here
i s defined i n (2.1).
To p r o v e t h i s Theorem we u t i l i z e t h e n e x t Lemma w h i c h can be o b t a i n e d f r o m Theorem 7 . 8 . 2 of Morrey [3]. Lemma 5
1
r o t r o t a.
(2)
E,
F o r any =
t
IH '(R ) , t h e r e i s an
a.
P
0
6
M '(Rv) s a t i s f y i n g
i n Rv,
n x u 0 = -(n.C)Bv
on
rP'
nxao = 0
on
r.
__ P r o o f of Theorem 4
The f i r s t s t a t e m e n t o f Theorem f o l l o w s f r o m t h e l a s t e q u a l i t y
i n ( 1 ) s i n c e we have
ici
statement, l e t
i
and a(c,C) = ;(;,<)
i(f,i)< 0.
satisfy
= {c.a}ifi
f o r
To p r o v e t h e second
By Lemma 5, t h e r e i s a n A
h
satisfying (2).
a. c M ' ( a v )
Let
= a.
CY
+ aI.
iV(ao.al)
= =
Let Lo
=
I 5 ,sol.
Then
cot
h
V.
Then 1 1-I
-
1-I
1Qv( r o t 1Rv ( r o t
a o . r o t al)dlr
r o t a0,a l ) d r +
P
Ir p u r ( r o t u 0 , n x a l ) d r
= o s i n c e r o t r o t a. = 0
a
V (ol,a) =
on Rv and nxal = 0
on
r ur. P
Hence
av(aO.CLo) + a ^ v ( a l , a l ) .
Therefore , , A
a(5,c)
,.
= a(So,h) = ap(c,t)
+
as(c,<)
+ ;,,(a0,ao) 5
aP(E,s)
+
a S ( c , c ) + 2,,ia,a)
Linear Stability of Magnetohydrodynamic System =
a^(i,t)<
343
0.
The l a s t e q u a l i t y i n ( 1 ) again i m p l i e s t h e a s s e r t i o n .
F i n a l l y we add t h e following P r o p o s i t i o n 6 which may g i v e an e x p l a n a t i o n of t h e energy p r i n c i p l e . , . A _
I f the plasma energy a ( 5 , S ) a t t a i n s t h e minimum
Proposition 6
in = {etfi
:
m(E,,[)
In t h i s c a s e
X
=
on t h e s e t
11, then we have
i s an e i g e n v a l u e o f t h e o p e r a t o r A
The corresponding
P'
e i g e n f u n c t i o n 9 belongs t o V a determined by the following g e n e r a l i z e d e i g e n v a l u e
pro bl em :
Proof for
such t h a t a ( $ , c ) = &n($,C)
Find rg,Vo-{O},
(4)
The i n e q u a l i t y
ScVo.
A ? .
i'h
A
follows from the f a c t s VcW and
h
_
a(C,F,) = a ( S , C )
ccV0.
The i n e q u a l i t y minimum (5)
for
of
i:A
:(:,I
in
a(t,i) $(fi,G)
i s shown as f o l l o w s .
e
= {S,a}t
P
g i v e the
6 P .. for h B
5
Let
Choose B C IH'(RV) such t h a t
P
.
nxa = nxa
on
r u r ci
.
Then
=
( C . R ) ~ ~ P.
By ( 5 )
we have +,a)
5
av(a,B).
For any $ c { H A ( Q ~ ) and }~ t
c
R
',
we can choose B a s R=a+t$.
For t h i s B, i t holds
i V ( a t t $ , cr+t$) = a V ( $ , $ ) t 2 + 2Re i v ( $ , a ) t + S v ( a , a ) .
So we have a s t h e s t a t i o n a r y c o n d i t i o n
(6)
iv($,ct) = 0
f o r any 9c{H;(Rv)l3 with
I1 r o t $ 1 1
L2(Rv)
+
0.
On t h e o t h e r hand, we have t h e i d e n t i t y : ~ , ( $ , c c )= 1
1
Rv
( r o t r o t $, n ) d r -
lanv(rot
@,
axn)dr
f o r any $tt.D(S2,,)j3
,
where t h e boundary i n t e g r a l i n t h e r i g h t hand s i d e i s understood t o be the d u a l i t y p a i r o f r o t $ t M 1 / 2 ( 2 2 v ) and a x n e M - 1 / 2 ( a Q v ) . (7)
1 i v ( @ , a )= tJ
1Qv ( r o t
rot
$,
a)dr
Hence we have
f o r any $ C I D ( ~ , , ) } ~ .
344
Teruo USHIJIMA
Then
Let & 8 ( Q V ) - I O } .
@ =
(@,O,O) c {t( Q v ) ) 3 s a t i s f i e s t h e c o n d i t i o n :
O ( I n f a c t , u n l e s s t h e c o n d i t i o n h o l d s , we have xax,= * =ax, Hence @ = @ ( x l ) . And 0 vanishes on t h e boundary o f s u p p o r t of @.
I/ r o t $11
f 0.
L2(RV)
i n a,,.
Hence @ = O . )
From ( 6 ) and ( 7 )
( r o t rot @,a)dr= 0
(8)
V
f o r t h i s @= ( @ , O , O ) .
By t h e same reason, ( 8 ) holds f o r Namely [B(Qv)}’.
Hence i t holds f o r any @ A
h
which implies C t V and CCV,,. Since we have
=
X, t h e
T h u s we o b t a i n
we have
@= ( O , @ , O )
and @= (O,O,@).
r o t r o t a= 0 in
.d(f&),
p X. 2
second e q u a l i t y of ( 3 ) i s v a l i d .
e q u a l i t y of ( 3 ) i s o b t a i n e d from t h a t o f ( 1 ) .
Hence the t h i r d
The f o u r t h e q u a l i t y and the
remaining s t a t e m e n t of t h e present P r o p o s i t i o n f o l l o w s from s t a n d a r d arguments.
I t i s noted t h a t A. =
4)EV
9
- I01
implies t h a t @ s a t i s f i e s (9)
a ( @ , C ) = Am(@,<),
GV.
Using t h e d e n s i t y argument, we have ( 4 ) from ( 9 ) .
References [l]
B e r n s t e i n , I . , Frieman, E . , Kruskal, M . , and Kulsrud, R . ,
An energy
p r i n c i p l e f o r hydromagnetic s t a b i l i t y problems, Proc. Royal SOC. A.
244
( 1958), 17-40.
[2]
Kato, T.,
Perturbation theory f o r l i n e a r operators (Springer, Berlin,
1966). [3]
Morrey, C . , J r . ,
Multiple i n t e g r a l s i n t h e c a l c u l u s
of v a r i a t i o n s .
(Springer, Berlin, 1966). [4]
Ushijima, T.,
On the l i n e a r i z e d magnetohydrodynamic systems of e q u a t i o n s
f o r a contained plasma i n a vacuum r e g i o n , i n : Glowinski, R. and Lions, J . L. (eds)
Computing Method i n Applied Sciences and Engineering, V,
(North-Holland, Amsterdam, 1982).
Lecture Notes in N u m . Appl. Anal., 5, 34.5-359 (1982) Nonlinear PDE in Applied Science. U.S.-JupunSeminur, Tokyo, 1982
A s i m p l e system w i t h a c o n t i n u u m o f s t a b l e inhomoqeneous s t e a d y s t a t e s
H.F.
Weinherqer
I n s t i t u t e f o r Mathematics and i t s A p p l i c a t i o n s l l n i v e r s i t y of M i n n e s o t a
I.
Introduction The system u
v
= { ( l t ~ v ) u +) ~(R1~
t
= (R2
t
-
bu
-
-
au
-
hv) u
av)v
{(I + av)ulx = 0 a t
x = G
and
x = 1
with (1.2)
and
i ;
>
a(a2' ZahRl
was c o n s i d e r e d b y
-
-
b2)
(1.3)
(a2 + b2)R2
M. Mimura [ Z ] as a model f o r t h e p o p u l a t i o n d e n s i t i e s o f
t w o c o m p e t i n q s p e c i e s , one o f w h i c h i n c r e a s e s i t s m i q r a t i o n r a t e i n r e s p o n s e t o c r o w d i n q hy t h e o t h e r species. model o f N. Shiqesada, K.
I t i s a s p e c i a l case o f t h e
Kawasaki, and E. Teramoto [3].
34s
H . F.
346
Numerical c o m p u t a t i o n by D.G.
WElNBtKCEK
Aronson and P.N.
Brown seems t o i n d i c a t e
t h a t t h e s o l u t i o n converqes t o a s t e a d y s t a t e i n w h i c h
v
has one o r more
d i s c o n t i n u i t i e s , and t h a t t h e s e d i s c o u t i n u i t i e s move c o n t i n u o u s l y w i t h chanqes i n t h e i n i t i a l c o n d i t i o n s . The e x i s t e n c e o f a c o n t i n u u m o f d i s c o n t i n u o u s s o l u t i o n s of t h e s y s t e m (1.1) was p r o v e d by Mimura.
The purpose o f t h i s l e c t u r e i s t o p r o v e t h a t
t h e r e a r e , i n d e e d , whole one p a r a m e t e r f a m i l i e s o f d i s c o n t i n u o u s s o l u t i o n s which are s t a b l e i n a s u i t a h l e topoloqy. The f a m i l y o f p i e c e w i s e c o n t i n u o u s s t e a d y s t a t e s i s d e s c r i b e d i n S e c t i o n 2. We s h a l l show i n S e c t i o n 3 t h a t a somewhat unusual t o p o l o q y i s needed f o r t h i s p r o b l e m and p r o v e t h a t l i n e a r i z e d s t a h i l i t y i m p l i e s s t a h i l i t y i n t h is t o p o l oqy. I n S e c t i o n 4 we q i v e a s u f f i c i e n t c o n d i t i o n f o r s t a h i l i t y and show t h a t
a c o n t i n u u m of t h e d i s c o n t i n u o u s s t e a d y s t a t e s s a t i s f i e s t h i s c r i t e r i o n . S e c t i o n 5 d i s c u s s e s t h e e v o l u t i o n a r y consequences o f t h e e x i s t e n c e of s t a b l e nonconstant steady states. T h i s work i s a p a r t o f o n q o i n q j o i n t r e s e a r c h w i t h D.G.
Aronson and A.
Tesei.
I am q r a t e f u l t o Don Aronson f o r q e t t i n q me i n t e r e s t e d i n t h i s p r o h l e m and f o r a q r e a t d e a l o f u s e f u l d i s c u s s i o n and c r i t i c i s m .
2.
The s t e a d y s t a t e . M. Mimura
w i n (1.1)
=
“f!]
i n t r o d u c e d t h e new i n d e p e n d e n t v a r i a b l e
(1 + a v ) u
t o o b t a i n t h e system
C o n t i m u u m of Stable Inhomogeneous Steady States v
w
G(v,w)
=
t
= (1 + av)(wxx + H(v,w))
t
341
(2.1)
where
G(v,w)
v(R2
=
-
bw
- --
av
1
t
1
9
av
The n o - f l u x boundary c o n d i t i o n s a r e
= O
w
at
x = O
and
x = l .
(2.3)
The c o r r e s p o n d i n q s t e a d y - s t a t e e q u a t i o n s a r e
w"
n ,
I;(v,w)
=
H(v,w)
= 0
t
w' = 0
at
,
x = 0,l
(2.4)
.
When R2/b t h e equation v
0
<
w
<
G(v,w)
wm : (aR2 + a f 2 / 4 a a h = 0
(2.5)
has t h e t h r e e n o n n e q a t i v e branches o f s o l u t i o n s
E O
I f we s u b s t i t u t e t h e s e i n t h e second e q u a t i o n of (2.4), t h r e e ordinary d i f f e r e n t i a l equations
we f i n d t h e
H.F. WHNBIIHGEK
348 w"
when
H(O,w)
t
0
=
w " + H(vl(w),w)
= 0
w"
= 0
H(v2(w),w)
t
(2.7)
lies i n the interval
VJ
f i r s t e q u a t i o n o f (2.1) v = vl(w)
for
w
E
that
.
(R2/h,wm) v
(R2/h,wm)
I t i s easy t o see f r o m t h e
t e n d s t o move away f r o m t h e h r a n c h v = 0
and f r o m
w
for
<
Rp/h
.
Thus o n l y
t h e f i r s t and l a s t o f t h e s e e q u a t i o n s can y i e l d s t a h l e stead,y s t a t e s . Eas,y c o m p u t a t i o n s show t h a t
>
H(v2(w),w)
0
w
for
<
H(O,w)
0
w > R2/h
for
.
( 0 , ~ ~ ) I f we i n t e q r a t e t h e f i r s t and t h i r d
E
e q u a t i o n s o f ( 2 . 7 ) and use t h e h0undar.y c o n d i t i o n s
wx = 0
n e i t h e r one can have a s o l u t i o n w i t h
.
w
(R2/h,wm)
E
One can, however, o b t a i n s o l u t i o n s by l e t t i n q v
0
t o t h e branch
tinuous.
v2
while
and hack a q a i n w h i l e k e e p i n q
v
, we
see t h a t
.jump f r o m t h e h r a n c h
w
and
wx
con-
T h i s can he seen f r o m t h e method o f f i r s t i n t e g r a l s (see
[Z])
or
The p o i n t s o f d i s c o n t i n u i t y a r e r a t h e r
f r o m phase p l a n e diaqrams.
a r b i t r a r y , so t h a t one o b t a i n s a l a r q e c o n t i n u u m o f s t e a d y - s t a t e s o l u t i o n s with
w
(R /h,w ) and d i s c o n t i n u o u s 2 m
E
.
v
I n p a r t i c u l a r , by i n t r o d u c i n q
s u f f i c i e n t l y many d i s c o n t i n u i t i e s one can keep constant
in
(R2/b,wm)
w
a r b i t r a r i l y c l o s e t o any
.
These s o l u t i o n s w i t h d i s c o n t i n u o u s
v
a r e t h e only candidates f o r
stahle solutions.
3.
The s t a b i l i t y o f d i s c o n t i n u o u s s o l u t i o n s . We s h a l l i n v e s t i q a t e t h e s t a h i l i t y o f a s t e a d y - s t a t e s o l u t i o n
(y(x),$(x)) Section.
w i t h piecewise continuous
v
, as
discussed i n t h e precedinq
I t i s n o t d i f f i c u l t t o p r o v e t h e a s y m p t o t i c s t a b i l i t y o f such a
s o l u t i o n i n t h e norm
IIVII~
+ iiwiiHi
when t h e l i n e a r i z e d o p e r a t o r i s
m
stahle.
u
(See Remark 2 a f t e r Theorem 1.)
Contimuum of Stable Inhomogeneous Steady States
349
T h i s f a c t seems s u r p r i s i n q because t h e p o i n t s o f d i s c o n t i n u i t y of c a n be chosen r a t h e r f r e e l y . ween a f u n c t i o n w i t h a jump a t
However, i n t h e
x
0
L_
norm t h e d i s t a n c e h e t -
and one w i t h o u t a jump a t
l e a s t h a l f t h e maqnitude o f t h e jump.
v
xo
i s at
T h e r e f o r e i n t h i s norm s o l u t i o n s
w i t h d i s c o n t i n u i t i e s a t d i f f e r e n t p o i n t s a r e i s o l a t e d f r o m each o t h e r . F o r t h e same r e a s o n a s u f f i c i e n t l y s m a l l n e i q h b o r h o o d of a d i s c o n t i n u o u s f u n c t i o n c o n t a i n s no continluous f u n c t i o n s . s y s t e m (2.1),
(2.3)
Since a s o l u t i o n o f t h e
w i t h smooth i n i t i a l d a t a remains smooth,
verqe t o a discontinuous s o l u t i o n i n t h e
Lm
norm,
i t cannot con-
Thus t h e
Lm
toooloqv
on t h e v-component o f t h e s o l u t i o n i s n o t a p p r o p r i a t e f o r t h i s problem. We s h a l l , i n s t e a d , use t h e weaker t o p o l o q y w i t h t h e n e i q h h o r h o o d base
f o r t h e v-component,
The c l o s u r e o f t h e s e t o f c o n t i n u o u s f u n c t i o n s i n
t h i s t o p o l o q y i s t h e s e t o f f u n c t i o n s w h i c h a r e a l m o s t everywhere continuous. It i s usual t o r e l a t e t h e s t a h i l i t y o f t h e steady-state s o l u t i o n o f t h e
system (2.1)
t o t h e spectrum o f t h e l i n e a r i z e d operator
w i t h t h e houndary c o n d i t i o n s
5' = 0
at
0
and
1
.
We s h a l l show t h a t
t h i s can a l s o he done i n o u r t o p o l o q y hy p r o v i n q t h e f o l l o w i n q r e s u l t . THEOREM I
(0,G)
Let
he a s t e a d y - s t a t e s o l u t i o n o f ( Z . l ) , (2.3)
u
piecewise continuous, w continuously d i f f e r e n t i a b l e , 0 < R2/h
<3 <
wm
.
7<
5uppose t h a t t h e s p e c t r u m of t h e o p e r a t o r
RZ/a
L
,
with and
i n (3.1)
l i e s uniformly i n the l e f t half-plane Re
x
< -2k
<
0
.
(3.2)
d
v
350
H . F. WEINBERCER
Then t h e r e e x i s t p o s i t i v e c o n s t a n t s i s a s o l u t i o n o f (2.1),
(v,w)
-
Ilv(x,n)
Y(X)II'
where t h e measure
IS'
I
L-6)
(2.3)
E~
and
A
with the property t h a t i f
such t h a t
t ilw(x,n)
- G(x
0
< v(x.0)
2
6
R2/a
,
2
II~IC
(3.3)
5
o f t h e complement o f t h e s u h s e t
of
[0,1]
satisfies
IS'I < and i f
E
<
E~
(3.4)
, the
inequality
- C(X)II'
iiv(x,t)
L,(S)
i s valid for a l l
Proof -
,
ELI
t
>
0
+ ilw(x,t)
-
G(x)li2 H'
<
AE
2
.
(3.5)
.
We f i r s t observe t h a t
when
It f o l l o w s [ 4 ]
(2.1),
(2.3).
t i e s , so does
t h a t t h e s e t (3.6) That i s , i f (v(x,t),w(x,t))
We aqree t o choose (3.3) (3.6).
c0
(v(x,O)
i s an i n v a r i a n t s e t f o r t h e system
, w(x,O))
for all
t
s a t i s f i e s these i n e q u a l i -
>
0
so s m a l l t h a t f o r
. E
<
E~
the inequalities
i m p l y t h a t t h e i n i t i a l v a l u e s , and hence a l s o t h e s o l u t i o n , s a t i s f y (The i n e q u a l i t y (1.3)
implies that
w,,
>
wm .)
Contimuum of Stable lnhornogeneous Steady States
35 I
We now d e f i n e
-
n(x,t) = v(x,t)
-
n 0 ( 4 = V(X,O)
v(x) v(x) (2.3)
and w r i t e t h e system (2.1),
a n
rl
-
(c)
L(J
S X ( W
n(x.0)
=
=
=
,
c(x,t)
,
c 0 ( 4 = w(x.0)
-
= w(x,t)
-
w(x)
W(X)
i n t h e form
P
(J
(3.7)
SX(l,t) = 0 llo(x)
S(Xl0) = S0(X) where
L
i s d e f i n e d h.y (3.1)
and
We f i r s t t r e a t (3.7) as a l i n e a r system.
Because t h e spectrum of
i s i n t h e h a l f - p l a n e ( 3 . 2 ) , a standard e s t i m a t e (see e.q.
L
[ l , Theo. 1.3.43)
shows t h a t llll(.,t)Il
t
lIC(..t
(3.9)
L1 t Jo[llP(*,T
Here and i n a l l t h a t f o l l o w s on hounds f o r and on
k
G
and
H
c
stands f o r any constant which depends o n l y
and t h e i r p a r t i a l d e r i v a t i v e s on t h e s e t (3.6),
.
because t h e second e q u a t i o n o f (3.7) i s p a r a b o l i c , we can f i n d a hound of t h e form
H. F. WEINBERGER
352
The f i r s t e q u a t i o n of (3.7) terms o f
p
5
and
.
V
c -2k
G
implies that
h.y q u a d r a t u r e s i n
It i s e a s i l y seen t h a t t h e c l o s u r e o f t h e ranqe o f
l i e s i n t h e spectrum o f
G (v,w)
n
can he s o l v e d f o r
.
L so
that, t h e s p e c t r a l hound (3.2)
C o n s e q u e n t l y we f i n d t h a t
I t follows that
We combine t h i s i n e q u a l i t y w i t h (3.9)
and (3.10)
t o ohtain
(3.12)
We see f r o m (3.8)
llP1I2
f
L2
Il01l2
that
<
C[llIl112
L2
Jl
and f o r
L2 (2.1)
i n t h e form
1511~
L4
Lz
and (3.13)
I n o r d e r t o use (3.12) 115 II
ll< IN2
.
+
IIrlII
4
L4
+
4
Il
4
1
.
(3.13)
we need bounds f o r an i n t e q r a l of
F o r t h i s p u r p o s e we w r i t e t h e second e q u a t i o n of
353
Continiuum of Stable Inhomogeneous Steady States We m u l t i p l y by (3.13)
(ct
f
71 k g ) e k t , i n t e q r a t e
hy p a r t s , and u s e (3.12)
and
t o find that
Suppose t h a t on a t i m e i n t e r v a l c111nl12 < 1 12 Then (3.14)
we have
[O,t,)
.
(3.15)
shows t h a t on t h i s i n t e r v a l
2 II~II
<
2
2
H'
L2
C ~ { [ I I ~ ~ I I i. I I ~ )II
H'
]e
t ~ g t [ i i v i i ~ + iir;ir4
14
-kt
le-k(t
(3.16)
.
-
L4
We now o b s e r v e t h a t hecause of t h e hound (3.6) llllllL P
P
<
11q11
We f i n d f r o m (3.11)
Lm(S)
+
.
(R2/a) I S ' /
(3.17)
and S o b o l e v ' s i n e q u a l i t y t h a t -2kt +
I l e combine t h i s w i t h (3.16)
,;
[ll~ll
and (3.17)
H'
i .
11<112
H'
+
11rl11 2
le-2k(t-~)
Lm(S)
t o ohtain the inequality
dT}
.
354
H. F. WEINBERGER Choose any A
where
c2
>
A
such t h a t
max(c2,1)
i s t h e c o n s t a n t i n (3.18)
.
Let
E~
he so s m a l l t h a t t h e
inequalities
are valid. t
.
i m p l i e s (3.5)
Then t h e i n e q u a 1 i t . y (3.3)
Moreover, i f (3.5)
h o l d s i n an i n t e r v a l
i s valid for a l l positive
1.
a constant
C
t
, which
The f i r s t e q u a t i o n o f (2.1)
such t h a t i f t h e i n e q u a l i t y
whole i n t e r v a l
[O,l]
If
IS'(
= 0
f o r one v a l u e o f
, that
t = t
t
and (3.5) v < v,(w)
, it
imply that there i s t
CE
i s v a l i d on t h e
i s valid for a l l larqer
i s , i f one works i n t h e norm
t
iir,iiL
+ iiciiHi
t h e bounds
topol0q.y.
so t h a t
(v,w)
t
. W
f o l l o w f r o m (3.18),
1 ' That
p r o v e s t h e Theorem.
and t h a t t h i s i n e q u a l i t y h o l d s f o r a l l s u f f i c i e n t l y l a r q e
2.
i s valid
h o l d s i s h o t h open and c l o s e d .
Thus t h e maximal i n t e r v a l where (3.5)
REMARKS:
t h e n (3.15)
implies that t h i s inequality i s s t i l l valid a t
t h e r e and (3.18)
i s , (3.5)
[O,tl),
for s u f f i c i e n t l y small
i s asymptotically stahle i n t h i s
,
,
355
Contimuum of Stable lnhomogeneous Steady States 4.
A sufficient condition for stahility.
We w i s h t o d e r i v e a s u f f i c i e n t c o n d i t i o n f o r t h e s p e c t r u m of t h e o p e r a -
L d e f i n e d hy (3.1) t o be hounded away f r o m t h e r i q h t h a l f - p l a n e , so
tor
t h a t t h e c o n c l u s i o n o f Theorem 1 i s v a l i d . We c o n s i d e r t h e system
whose s o l u t i o n q i v e s t h e i n v e r s e o f set.
L - XI
a t points of i t s resolvent
As we have a l r e a d y mentioned, t h e c l o s u r e o f t h e r a n q e o f
shown t o l i e i n t h e spectrum.
(3.2)
G
V
can he
Consequently, a n e c e s s a r y c o n d i t i o n f o r
is
Gv(T,G)
c
-2k
<
0
.
(4.2)
Our c r i t e r i o n w i l l i n v o l v e t h e s o l u t i o n o f t h e i n i t i a l v a l u e p r o b l e m
(4.3)
r(0) = 0 r'(0) = 1 As u s u a l ,
CSl+ =
THEOREM 2.
o
if
s c n
s
if
s > O .
Suppose t h a t
and t h a t t h e s o l u t i o n o f (4.3)
GV
s a t i s f i e s an i n e q u a l i t y o f t h e f o r m (4.2)
has t h e p r o p e r t i e s
356
H . F. W E I N H E H G E R
(4.4)
Then t h e s t e a d y - s t a t e s o l u t i o n
~PROOF.-
- x
If
+
i s s t a h l e i n t h e sense o f Theorem 1.
i s o u t s i d e t h e c l o s u r e o f t h e ranqe of
f i r s t equation f o r
C''
v, w
v
, we
can s o l v e t h e
and s u h s t l t u t e i n t h e second t o o h t a i n t h e p r o h l e m A
-
LHW
Gv
c'(0)
=
(4.5)
r'(1) = 0
T h i s e q u a t i o n can always be s o l v e d u n l e s s t h e r e i s a n o n t r i v i a l s o l u t i o n o f t h e equation w i t h c l o s u r e of t h e ranqe o f s p e c t r u m we s e t
p
Gv
p = u = 0
= u = 0
.
Therefore t h e spectrum o u t s i d e t h e
i s discrete. i n (4.5),
To l o c a t e t h i s p a r t o f t h e
m u l t i p l y by t h e complex c o n j u q a t e
7 ,
and i n t e q r a t e hy p a r t s t o f i n d an e q u a t i o n on whose r e a l and i m a q i n a r v p a r t s are
and
The second e q u a t i o n shows t h a t any complex s p e c t r u m i s c o n f i n e d t o t h e u n i o n as
qoes f r o m
x
(A
-
GV\'
G
-(
w h i c h i s a hounded set. no eiqenvalues w i t h
Re
t o 1 o f the discs
0
+
av)GwHv
,
T h e r e f o r e i t i s s u f f i c i e n t t o show t h a t t h e r e a r e
x>o.
Contimuum of Stable Inhomogeneous Steady States If
x > 0 , we
Re
see f r o m (4.2) t h a t
Ix -
+
Thus (4.6)
x - GV)GwHV
(Re
Re A
-m
351
Gvl
-c GwH v
<
2
-G
1,
-
5
t o find that
yields the inequality
We now d e f i n e
i n t e q r a t e (4.8)
hy p a r t s , and s u h s t i t u t e f o r
Since t h e eiqenfunction lq(1)I2
>
0
, and
cannot s a t i s f y ~ ( 1 =) ~ ' ( 1 =) 0
5
>
r(l)r'(l)
since
0
,
t h i s leads t o a contradiction.
We c o n c l u d e t h a t i f t h e s o l u t i o n o f (4.3) and i f (4.2)
(4.4),
, since
satisfies the conditions
i s s a t i s f i e d , t h e n t h e s o l u t i o n (v,w) i s s t a h l e , w h i c h
Droves our Theorem.
The c o n d i t i o n s (4.4) i n (4.3)
is n o n p o s i t i v e .
Gv(v,(w),w) Hw
<
a r e o b v i o u s l y s a t i s f i e d when t h e c o e f f i c i o n t o f
<
,
0
and
Because
Hw(O,w)
<
Gw(v2(w),w)
0 on t h e p a r t o f t h e r a n q e o f
shows t h a t
Hv
>
0
,
Hw
<
0
at
0
<
,this
(7,;) w
w >Rl/a
for
i s t h e case i f
where
(v2(wm),wm)
f a m i l y o f s t a h l e s o l u t i o n s by k e e p i n q
0
, so
A ,
v = v,(z)
, Gw(O,w) Hv
.
>
0
=
r
0 ,
and
Computation
t h a t one can c o n s t r u c t a
near a constant j u s t helow
wm
.
F o r t h e l i m i t i n q s o l u t i o n s computed h y Aronson and Brown t h e s u f f i c i e n t
c o n d i t i o n s (4.4)
a r e f o u n d t o be v a l i d i n most thouqh n o t a l l cases.
358
H. F. W E I N B E K G E R
REMARK.
(2.1),
(1 + av)
Replacinq t h e f a c t o r
1 i n t h e second e q u a t i o n o f
by
y i e l d s a s e m i l i n e a r system w i t h t h e same s t e a d y s t a t e s .
The ahove
a n a l y s i s shows t h a t t h i s system a l s o has a c o n t i n u u m of s t a h l e s t e a d y s t a t e s , 50 t h a t q u a s i l i n e a r i t y i s n o t needed t o produce t h i s phenomenon.
5.
A h i ol o q i c51 consequence. The system (1.1) i s a model f o r a p a i r o f c o m p e t i t o r s ,
one of w h i c h
a v o i d s t h e o t h e r t o such an e x t e n t t h a t t h e homoqeneous s t e a d y s t a t e solution
Ria -
-
u -
-
a’
-
R h
2
R2a
-
Rlh
a2
-
b‘
, v =
h2
i s r e n d e r e d u n s t a b l e and inhomoqeneous s t a h l e s t e a d y s t a t e s
(u,v)
occur instead.
I t i s r e a s o n a h l e t o ask w h e t h e r t h i s mechanism i s advantaqcous t o t h e
t w o species. We i n t e q r a t e t h e s t e a d y s t a t e f o r m of (1.1) t o f i n d t h a t
1,1, u ( R 1
-
d .
au
-
b7)dx
= 0
.
The second e q u a t i o n o f (1.1) hecomes while
R2
-
ht
R 2 v>-
on t h e b r a n c h
0
(
-
h?
a
7
(5.1)
R2
-
= 0
.
hz
-
a? = 0
when
= v2(G)
Thus on h o t h hranches
,
(5.2)
so that
1,1, u(R1 - ac - -ha (R2 Since
R1 = a:
+ b?i
and
,
-
hG))dx > 0
R 2 = hG
+
a?i
.
we o h t a i n t h e i n e q u a l i t y
C o n t i m u u m of Stable Inhomogeneous Steady States
359
so t h a t
Thus i f
G(x)
i s n o t c o n s t a n t , we f i n d t h a t
j $ i x
(5.3)
Thus t h e a v o i d a n c e mechanism reduces t h e t o t a l p o p u l a t i o n o f t h e o r q a n i s m t h a t possesses i t . On t h e o t h e r hand,
a(; Thus,
(5.3)
- 7)>
(5.7)
b(u -
d)
can he w r i t t e n i n t h e f o r i n
.
implies that
j o' Vd d X > 7 ,
s o t h a t t h e second s p e c i e s p r o f i t s f r o m t h e nervousness o f t h e f i r s t one. The second s p e c i e s m i g h t t h u s e v o l v e a mechanism t o f r i q h t e n i t s comp e t i t o r s away.
BIBLIOGRAPHY Henry, O., Geometric Theory o f S e m i l i n e a r P a r a b o l i c E q u a t i o n s , N o t e s i n M a t h e m a t i c s #840, S p r i n q e r , New York, 1982).
(Lecture
Mimura, M., S t a t i 0 n a r . y p a t t e r n o f some d e n s i t y - d e p e n d e n t d i f f u s i o n s y s t e m w i t h c o m p e t i t i v e dynamics, H i r o s h i m a Math. J. 11 (19R1), 621-635. Shiqesada, N., Kawasaki, K., and Teramoto, E., S p a t i a l s e q r e q a t i o n of in t e r a c t i n q s p e c i e s , (1. Theor. B i o l . 79 (1979), 83-99. Weinherqer, H.F., e l l i p t i c systems.
I n v a r i a n t s e t s f o r w e a k l y c o u p l e d p a r a b o l i c and R e n d i c o n t i d i Mat. R ( V I ) ( 1 9 7 5 ) , 295-310.
T h i s work was p a r t l y s u p p o r t e d by t h e N a t i o n a l S c i e n c e F o u n d a t i o n t h r o u q h Grant MCS 8102609 and hy a Japan S o c i e t y f o r t h e P r o m o t i o n of S c i e n c e F e l l o w s h i p f o r Research.
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Lecture Notes in N u m . Appl. Anal., 5 , 361-375 (1982) Nonlitieur PDE in Applied Science. U.S.-Jupatr Sernincir, Tokyo. 1982 Chaos arising from the discretization of O.D.E. and an age dependent population model.
Masaya YAMAGUTI and Masayoshi HATA Department of Mathematics Faculty of Science Kyoto University Kyoto 606 ,JAPAN In the first part of our paper, we review our recent studies of some usual finite difference schemes for the autonomous system, which can produce chaotic dynamical system. In the second part, we present an age dependent discrete population model whose solution exhibits some significant chaotic behaviors.
1. Finite difference schemes. 1.1 Definition o f Chaos.
First we begin with a recall for the notion of Chaos. Let us consider most simple dynamical system which is described as follows:
(1.1)
Yn+l =
1
yn
5
1 5)
1 (2. < Yn
5
1)
(0
2yn 2(1
-
Yn)
F(yn) whose graph is shown in Figure 1.
1
Figure 1. Graph of F.
36 I
a
Masaya Y A M A G U and T I Masayoshi HATA
362 We denote
1
t h e domain [O,?]
A
and
F(A)
=
A u B,
This p r o p e r t y means t h a t f o r any p o i n t always
y'
and
y"
such t h a t
t h e domain
F(B)
we remark t h e
3
A u B.
which belongs t o [0,1], t h e r e e x i s t
y
F ( y ' ) = y. y '
Now we c o n s i d e r an o r b i t {y
1 [5,1]. Then
F.
f o l l o w i n g p r o p e r t y of t h e mapping
(1.2)
B
E
A
also
F ( y " ) = y , y"
1 which s t a r t s from yo.
t
B.
We l i s t up t h i s se-
quence of v a l u e s as f o l l o w s :
(1.3)
..., Yn. . . *
YO' Y1* Ye,
And we a l s o l i s t up t h e sequence of t h e symbols of domains t o which
y
belongs.
A,A,B,B,B,.. We w r i t e t h i s sequence as f o l l o w s :
where
w
J
= A or B
corresponds t o
y
2'
Conversely, u s i n g t h e p r o p e r t y (1.2), w e can prove t h a t for any a r b i t r a r y given sequence
{wk~kzO we g e t
(1.5)
This means t h a t i f t h e sequence of symbols
{wkl
i s given by some r e c o r d of
c o i n - t o s s t r i a l s , even s o , w e can d e c i d e an i n i t i a l p o i n t belongs t o
w
f o r any
(1.1) s t a r t i n g from
yo.
n
where
{yn]
(Here we u s e
yn =
w
as t h e name o f domain.)
2 a r c s i n JF
t o t h e system (1.1). Then we g e t
(1.7)
such t h a t
y
i s a n o r b i t of t h e dynamical system
Now we c o n s i d e r a change o f v a r i a b l e s :
(1.6)
yo
x = 4x (1 - x ) . n+l n
363
Chaos and Age Dependent Population Model This mapping maps [0,1] to [O,l].
This is a special case a =
4 of the more
general dynamical system:
x = ax (1 - x ) :Fa(xn) n+l n
(1.8)
which also maps [0,1] into [0,1] f o r
4. We show the graph of this
0 < a 5
dy-
namical system in Figure 2.
1
Figure 2. (1)Graph of F4.
of Fa where ciently near 4.
( 2 ) Graph
a
is suffi-
As is easily seen, the fact (1.5) holds for the system (1.7) by exactly similar reason as in the case of (1.1). But how about for (1.8)? If a
4, then
near
(1.9)
some weak property (1.9) follows. Fa(A)
2
It produces any symbolic sequence
Remark.
is sufficiently
B, Fa(B) {wk}
3
where
A u B. w. =
J
A,
w.
J +1
= A
This simple dynamical system (1.8)was considered by
never arise
R. May who
studied a discrete population model of some insects population which has nonoverlapping generation. We are going to explain this fact introducing an age dependent population model in the second part of this paper.
Masaya YAMACUTI and Masayoshi HATA
364
Now we explain the notion of Chaos mathematically, that is, the definition of Chaos in the sense of Li-Yorke and Marotto.
We consider a dynamical system
which described by (1.10)
= F(Xn)
'n+1
,
where F is a continuous mapping from Rm
Xn c Rm, to Rm.
Definition. We say F is chaotic in the sense of Li-Yorke and Marroto if
F has the following four properties: (1) (1.10) has infinite periodic orbits with distinct periods. (2)
there exists an uncountable set S
x
c
Rm
such that for any
X,Y c S ,
f Y,
1im n-tm
(3) for the
X
same
Y
and
lim I n-
(4)
x
f o r any
t
s, x
as in ( 2 1 ,
F"(x)
- F"(Y) 1
=
o
is not even asymptotically periodic.
Now we can state very briefly the result of Li-Yorke[l] condition (1.9)f o r
R1
has shown this "Chaos"
implies
ler
"Chaos" in the above sense. Also
in Rm ( for any m )
h a s a snap-back repeller.
which is that our Marotto[2]
under the assumption that (1.10)
Here we recall the definition of the snap-back repel-
. Definition. Assume that
F is continuously differentiable.
of (1.10) a snap-back repeller if
a fixed point
2
neighbowhood
U of 2 and there exists a point Xo
= 2
and
(D$(XO)I
Z is expanding in some E
U
with Xo f Z, ?(Xo)
f 0 f o r some positive integer M where
Jacobian determinant of
at
Xo
.
Then we call
]Dp(X,)(
is a
365
Chaos and Age Dependent Population Model 1.2
Chaos arising from the discretization of ordinary differential equations.
Here we mention a review of the results of our group about the
"Chaos"
which are obtained by some simple discretization of ordinary differential equations. O u r first result was that of
Yamaguti-Matano[3]
which is stated as follows.
Theorem 1. For a given differential equation: (1.11)
where
f(y)
is continuous function of y
in R1.
If f(y) has at least two
zeros, one of which is globally asymptotically stable, then the Euler's difference scheme: (1.12)
yn+1 - n'
is chaotic if we take positive values
T
1'
At
+
At f(yn) :FAt(yn)
sufficiently large. such that for At
-i2
More precisely, there exist two
which satisfies
T~ 5
At
5 T ~ ,the
mapping (1.12) maps a finite interval into itself and this dynamical system is chaotic in this interval.
After this result, we study several generalizations of this result. Here, we limit ourselves to list up a series of our recent results. Let us consider the system of differential equations:
_ dU --
(1.13)
dt
G(U),
where U
is unknown m-vector and
from Rm
into Rm.
Then
G
U(0) = Uo is a continuously differentiable mapping
Hata succeeded to prove that the Euler's difference
scheme for (1.13):
(1.14)
Uncl = U
+ At G(Un) :GAt(Un)
Masaya YAMACUTI and Masayoshi HATA
366
is chaotic for sufficiently large At
(1.15)
there exist U and
+v
under the following conditions:
such that C ( C ) =
G(T)
= 0,
IDG(u)l f 0
IDC(v)l f 0.
For the proof of this theorem, see [4]. Nextly, S. Ushiki and Yamaguti[5] studied a central discretization of the following differential equation: -dx = dt
(1.16)
x(l
-
x).
The central difference scheme of this equation is
Xn+l - Xn-l 2At
(1.17)
= xn(l
- xn)
Putting xn-l = yn, we get a mapping from R~ = yn
+ 2ht xn(l
(1.18)
into R~
-
as follows.
xn)
S. Ushiki[6] proved that this dynamical system shows some chaotic behavior for any mesh size At. Similar result as Yamaguti-Matano's has been proved by modified Euler scheme of (1.11). Also the above result of
Y. Oshime for the Hata has been gener-
alized by himself for Runge-Kutta scheme of (1.13). Before finishing our review of the results, we sketch the proof of the above theorem by
Hata.
Lemmal. For any
for any At -
U
> c(6)
6 > 1, there exist
and X,Y
t
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
X
.
min
2
JA,in(Ix (1
x
( for all
E
means the minimum eigenvalue of DF(G)*DF(c).
R~ By the continuity,
where At
t
2 (1 + -
c(6) :
Jhmin
c.q.f.d.
6).
Lemma 2. For sufficiently small open neighhourhood W bounded set B, there exists a positive constant c(W,B)
of
?? and any
such that the equation
GAt(w) = b
has at least one solution w
E
W
for any At > c(W,B)
and for any h
E
B.
Using these lemmas we can constract a snap-hack repeller. Thus we can prove the conclusion of the theorem.
2.
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
N
-
( I: b(k)uF)(R
k=l
N C b(k)ut) k=l
(2.1)
where we denote
<
is the population of k-age animals at n-th year, b(k)
k-age population, d ( k )
is the birthrate of
is the deathrate of k-age population, R
constant which means a saturation, and b(i), d(i)
is a positive
satisfy the following condi-
tion;
(2.2)
0
2
b(i)
<
1 for
1 5 i 2 N, b(N) f 0,
0
5
d(i)
i
1 for
1
2
i
S
N-1.
Now it is convenient to introduce new variables by the following formulae;
for
Then we have new equation;
(2.4)
p+l=
n
G(Vn) =
v1
n
V
where we denote
vn
= t( v;, v;,
...,
N-1
vn ) N
1
5
5
2
N.
Chaos and Age Dependent Population Model
369
Also we have
We assume that AR
5
4 where A =
N C
a(k)
k=l
Under this assumption, it is easily verified that the N-dimensional mapping
G
in (2.4)has the following invarient domain;
Q = [O,l]x[O,l]x
...x[O,1]
c
N
R ,
P
N times
that is, G maps
Q
into itself. Then we find the fixed points of G
in Q
as follows.
(a) For the case i n which the origin 0 = (0, 0, (b) For the case in which
0 < AR I I, the only one fixed point in
Q
is
..., 0 ) . 1 < AR
S
4, the fixed points in Q are the
origin and
-
v
(2.5)
=
4
(E(1-,,I,
1
1 ..., -AR(14 ---)I. AR
The local stability of these fixed point of G
is easily studied by stand-
ard linearization techniques. From (2.41, we obtain
(2.6)
where
DG(V) = D G ( V ~ , V,..., ~ vN) =
1
0
Masaya
310
Especially, for the fixed points of ~ ( 0 =)
and Masayoshi HATA
YAMACUTI
R
G,
we have
-
u(V) =
and
2 n -
R.
It follows easily from (2.6) that the characteristic equation of DG(V)
is
(2.7) Putting
we have the following estimates on the unit circle
Therefore, if
1x1
= 1;
Iu(V)lA < 1, we have
on the unit circle. By the theorem of Rouch6, the all r o o t s of the polynomial f(h)
-
g(A)
lie inside the unit circle. So we have the followings.
(c) For the case in which
0 < AR < 1, the origin 0 is locally stable.
(d) For the case in which
1 < AR
<
-
3, the non-trivial fixed point V
is
locally stable.
Similarly, we can study when all roots of the polynomial f ( A ) outside the unit circle.
Put
1 5 = A
-.
Then from (2.7)we have the following new polynomial of 5;
(2.9)
-
g(A)
lie
37 1
Chaos and Age Dependent Population Model Putting
we have the following estimates on the unit circle
151 = 1;
By the theorem of Rouch6, we have the followings.
N-1 a(j)
DG(0)
exceed
(f) If a(N)
>
3
1
and
2a N)
-A
< AR 5
4, then all eigenvalues of
AR
4, then all eigenvalues of
in magnitude.
N-1 C a(j)
and
4a(N)
-
A
<
5
j=1
DG(v)
exceed 1 in magnitude.
The above conditions (e) and (f) about the distribution of a(j)
include
some insects population which has non-overlapping generation as a special case. Actually, we can prove that
G is chaotic for suffiriently small a(l),
a(Z),.,.
a(N-1)
and some a(N) which satisfy our assumptions by showing that the nontrivial fixed point V of G is a snap-back repeller. First, we choose a special parameters as follows; a(1) = a ( 2 ) = And
Ga
...
= a(N-1) = 0, o
a(N)R
E
(0,4].
denotes the corresponding mapping defined in (2.4)with a parameter a,
that is,
aVN -
a2
v
1
(2.11)
v
N-1
2
VN
372
Masaya Y A M A G U and T I Masayoshi H A T A
Putting h (x) = ax
where : G
-
a2
x
2
, we
get
denotes the N-fold iteration by
Also we obtain from (2.12),
Gas
Therefore
I DHa(vI ,v2,. . . ,vN)I
(2.14)
= ha' (v, )ha' (v,)
. . .ha'(v,
)
.
The condition of a snap-back repeller clealy includes the existence of a homoclinic orbit z,
{ z - ~) k > O
such that
F,
is a fixed point o f
F(z ) = z - ~ + for ~ k -k
z - ~f zo,
1, and F(z-~)
2
-f
zo as k
+ a.
W e n for one-dimensional continuous mapping, it is sometimes simpler to find a homoclinic orbit than to verify (1.9)o r some odd period conditions. In this case, for one-dimensional continuous mapping exists a positive constant . a
<
4
has a transversal homoclinic o r b i t hu'(p-,) The dotted lines in Figure
ha, it is easily shown that there
such that, for any fixed
a
t
[ao,4],
ha
{p-n} n>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.
1
Figure 3. Homoclinic orbit of h
. 1
0
Using a transversal homoclinic orbit
transversal homoclinic orbit
P - ~ =
t
‘P-n’ n>-O of ha, we construct a
{P-n} n ~ Oof Ha as follows;
( p-n ,p-n
,..., p-,
6
R~
for
n
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
Ha
is contained in the interior of Q,
E <
1 such that for any
we have the following.
Theorem. There exists a constant 0 <
which satisfies AR
5
4, the corresponding mapping G in (2.4) is chaotic in Q
in the sense of Li-Yorke.
Finally, by computer simulations ( see Figure 4 ) , we conjecture that G sometimes chaotic even for the case of over-lapping generations.
is
Masaya Y A M A G U and T I Masayoshi HAIA
374
( 4 - a ) T o t a l Population.
Figure
4.
Numerical computations of our model ( 2 . 1 ) where w e p u t
and b i r t h r a t e and d e a t h r a t e a r e shown i n (
4-
Birthrate
0
Age
F i r s t , w e choose a s a t u r a t i o n v a l u e
c
N = 100
1.
Deathrate
100
0
R = 9.0
Age
in (
100
4
-
a ) and presumably
t h e p e r i o d i c s t r u c t u r e i s caused by a Hopf b i f u r c a t i o n from a n o n - t r i v i a l fixed point.
Secondly, we choose a s a t u r a t i o n v a l u e
and t h e c h a o t i c phenomenon w i l l be appear.
R = 11.3
in (
4-
b )
315
Chaos and Age Dependent Population Model
References
[l] T.-Y.
Li and J, A. Yorke, P e r i o d t h r e e i m p l i e s chaos, h e r . Math. Monthly
82 (1975),985-992. [2] F. R. Marotto, Snao-back r e p e l l e r s imply chaos i n R n , J . Math. Anal. Appl. 63
(19781, 199-223.
[ 3 ] M. Yamaguti and H. Matano, E u l e r ' s f i n i t e d i f f e r e n c e scheme and chaos, Froc. Japan Acad. 55A
[4]
(1979),78-80.
M. Hata, E u l e r ' s f i n i t e d i f f e r e n c e scheme and chaos i n R n , €'roc.
Acad. 58A
[5] M.
(1982),178-181.
Yamaguti and S. U s h i k i , Chaos i n numerical a n a l y s i s o f 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 s , Physica 3 D , 3
[6]
Japan
(1981),618-626.
S . Ushiki, C e n t r a l d i f f e r e n c e scheme and c h a o s , Physica
4D (1982), 407-424.
This Page Intentionally Left Blank
Lecture Notes In N u m . Appl. Anal., 5, 377-307 (1082) Notilinear PDE in Applied Scierice. U.S.-JupunSemrriur. Tokyo. 1982
STABILITY, REGULARITY AND NUMERICAL ANALYSIS OF THE QONSTATIONARY NAVIER-STOKES
PROBLEM
John G . Heywood U n i v e r s i t y of B r i t i s h Columbia Vancouver, B . C . ,
Canada
In t h i s paper I w i l l d e s c r i b e some r e s u l t s r e l a t i n g t h e s t a b i l i t y and r e g u l a r i t y of s o l u t i o n s of t h e Navier-Stokes e q u a t i o n s w i t h t h e long term e r r o r and s t a b i l i t y of numerical approximations.
These r e s u l t s were o b t a i n e d j o i n t l y
w i t h Rolf Rannacher and a r e p r e s e n t e d i n f u l l d e t a i l , along w i t h r e l a t e d r e s u l t s , i n P a r t I1 of o u r work on f i n i t e element approximation of t h e n o n s t a t i o n a r y Navier-Stokes
problem [l].
They a r e
of two g e n e r a l t y p e s , both u t i l i z i n g
s t a b i l i t y assumptions t o extend r e s u l t s which were known l o c a l l y i n t i m e t o ones which a r e g l o b a l i n time.
F i r s t , t h a t i f t h e s o l u t i o n of t h e i n i t i a l
boundary value problem i s s t a b l e , then t h e e r r o r i n i t s d i s c r e t e approximation remains s m a l l uniformly i n t i m e , a s
t +
m
.
Second, t h a t from 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 , f o r a s i n g l e s u f f i c i e n t l y s m a l l c h o i c e o f t h e mesh s i z e , one can i n f e r t h e g l o b a l e x i s t e n c e of a c l o s e l y neighboring smooth s o l u t i o n .
The
concepts o f s t a b i l i t y which a r e d e a l t w i t h a r e formulated t o d e s c r i b e t h e s t a b i l i t y o f such phenomena a s Taylor c e l l s and v o n - K Q r m h v o r t e x s h e d d i n g , and a l s o t h e p a r t i a l s t a b i l i t y observed i n some flows e x h i b i t i n g s l i g h t o r i n c i p i e n t turbulence. in
111,
I w i l l i n c l u d e an account of some of t h e s t a b i l i t y theory developed
p a r t i c u l a r l y a s e x e m p l i f i e d by a new proof of 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 appropriate to nonstationary solutions.
377
378
John G . HEYWOOD
1. The Continuous Problem We consider the nonstationary Navier-Stokes problem
in a bounded two or three-dimensional domain fi
.
velocity of a viscous imcompressible fluid, p
the pressure, f
external force, and a
Here u
the prescribed initial velocity.
represents the the prescribed
The boundary values
are zero. The fluid's density and viscosity have been normalized, as is always possible, by changing the scales of space and time. A s usual,
Lp(G)
,
or simply Lp , denotes the space of functions defined
and pth-power summable in n 2
product in L
by
(*,*)
, and
II*IILp its norm. We denote the inner
(I-II
and let
=
(1-/lL2
.
Cm
is the space of
functions continuously differentiable any number of times i n m
consists of those members of space Hm
n
with compact support i n
C
R
.
, and
m
cO
The Sobolev
is obtained by the completion in the norm
expressed in multi-index notation, of those members of
1 is finite. Ha is the closure o f
in H1
12:
.
m
C
for which the norm
Spaces of Rn-valued functions
will be denoted with boldface type. We use
as
inner product and norm for J
=
{$EL* :
0.0
J1 = {gtE: : 0.g of solenoidal functions.
1
Eo
.
= 0
Finally, we need the spaces in fi
= 01
and
g'njaR
= 0
, weakly) ,
379
Numerical Navier-Stokes Problem L2
Denoting t h e o r t h o g o n a l p r o j e c t i o n of t h e "Stokes o p e r a t o r " r e g u l a r , t h e mapping
holds for a l l
V E
A = PA A : J
.I1 n H2
1
.
. n El
onto
J
Assuming t h e boundary 2
+.
J
i s one-to-one
by
P
, we
introduce
is sufficiently
aR
and o n t o , and
We assume t h i s as w e l l a s some r e g u l a r i t y of t h e
p r e s c r i b e d d a t a , namely t h a t
acJlnE m
2
, 2
f , f t c L (0,m;L )
For t h e s a k e of s i m p l i c i t y i n our p r e s e n t a t i o n , we have assumed t h e boundary values vanish.
A l l of o u r r e s u l t s remain v a l i d i n t h e c a s e of inhomogeneous
boundary c o n d i t i o n s i f one assumes an a p p r o p r i a t e degree of smoothness of t h e boundary v a l u e s , as w e l l a s t h e same c o n d i t i o n s of s p a t i a l and temporal i n v a r i a n c e a s may b e r e q u i r e d o f
f
.
F i n a l l y , w e assume t h a t t h e s t r o n g s o l u t i o n
u,p
of problem (1) e x i s t s
g l o b a l l y and s a t i s f i e s
Once t h i s much r e g u l a r i t y i s known o r assumed of a s o l u t i o n , i t s f u l l r e g u l a r i t y can be proved s o f a r as i s p e r m i t t e d by t h e d a t a .
I n p a r t i c u l a r , the following
i s proven i n Theorem 2 . 3 of [2].
Proposition 1 . function
F
&
0 satisfying (Al),
there e x i s t s a continuous increasing
of t h r e e variables, such t h a t e u e q s o l u t i o n
u
(1) s a t i s f i e s
John G. HEYWOW
380
We mention t h a t t o bound h i g h e r o r d e r d e r i v a t i v e s of t
+
0
,
,
u
uniformly as
requires nonlocal compatibility conditions of the prescribed d a t a ,
conditions usually unverifiable i n practice. tend t o i n f i n i t y as
t
+
0
(1
For i n s t a n c e ,
((Vut
unless there e x i s t s a solution
p,
and
(lu/(
of t h e
overdetermined Neumann problem
t
The l o s s of r e g u l a r i t y a s
+
0
complicates t h e proof of e r r o r e s t i m a t e s of
h i g h e r than second o r d e r f o r numerical approximations.
While such e s t i m a t e s a r e
proven i n P a r t 111 of t h i s work, independently of any n o n l o c a l c o m p a t i b i l i t y c o n d i t i o n s , t h e p r e s e n t d i s c u s s i o n w i l l be r e s t r i c t e d t o second o r d e r e r r o r estimates. 2.
-Discrete
Problem
Hh
We suppose c h a t
Lh
and
a r e f i n i t e dimensional subspaces of
L2
and
L2 , r e s p e c t i v e l y , corresponding t o a sequence of v a l u e s , t e n d i n g t o z e r o , o f a d i s c r e t i z a t i o n parameter
Hi
d i s c r e t e analogue of
h
.
, 0
< h S 1
.
such t h a t
Vkl E V
has an e x t e n s i o n
V
1 Bo
on
i s considered a s a
I n o r d e r t o i n c l u d e t h e c o n s i d e r a t i o n o f nonconform-
Hh c E1
i n g f i n i t e elements, i t i s n o t r e q u i r e d t h a t gradient operator
Hh
The space
, and
b u t merely t h a t t h e
1
Ho + Hh
t o t h e a l g e b r a i c sum
Vh
such t h a t
,
I(V1;
/I
i s a norm on
Hh
.
f r e q u e n t l y used norm i s
~
/
A d i s c r e t e analogue of t h e s p a c e
Jh
E
(v,
E
~
' *II'II~
J1
+~ h
~ ~h v *h ' ~ ~
i s i n t r o d u c e d by s e t t i n g
Hi, : (xh,Vh-vh) = 0 , f o r a l l
xh c
L
hI
H 1
.
We a l s o s e t
Nh i Ixh E Lh : (xh,Vh.vh) = O
, f o r a l l vh
t
h
.
9
Another
Numerical Navier-Stokes Problem Ah
A d i s c r e t e analogue
: Eh + H
h
38 1
o f t h e L a p l a c i a n o p e r a t o r i s d e f i n e d by
requiring
,
(A v c$ ) = -(V v V @ ) h h' h h h ' h h
Ph : L
Letting
Ah
to
v ~ , ~ $ ~ E H ~
2
2 +
Jh
d i s c r e t e analogue, of
.
for
A
h
Jh
denote t h e L - p r o j e c t i o n onto
z P hA h , of t h e Stokes o p e r a t o r
, we
introduce a
A = PA
.
The r e s t r i c t i o n
'-1
i s a u t o m a t i c a l l y i n v e r t i b l e , w i t h i n v e r s e denoted by
Jh
i n v e r t i b i l i t y of
Ah
and
.
Ah
p e r m i t s us t o i n t r o d u c e o p e r a t o r s
A
% _ A --I P a h h ~
H2
: J l n
h --1 R E A PAh : Jh
-f
Jh
J l n H
-f
2
, ,
a s s o c i a t i n g d i s c r e t e s o l e n o i d a l f u n c t i o n s w i t h smooth ones and v i c e v e r s a . assume t h e r e a r e c o n s t a n t s and
vh
E
J
h
,
The
c
, independent o f
h
,
such t h a t f o r
We
vcJ1 n H
2
there holds
These i n e q u a l i t i e s were proven i n C o r o l l a r y 4 . 3 o f [2] under t h e d e t a i l e d assumptions of
[Z].
We suppose we have a d i s c r e t e a n a l o g u e o f t h e Navier-Stokes e q u a t i o n s determining, f o r any given and
p h ( * , t ) t Lh/Nh
a
h
E
Jh and
, defined for a l l
t t 1 t
2 0
,
,
unique f u n c t i o n s
such t h a t
\(-,to)
\(* = ah
, t ) E Jh
.
Our
n o t a t i o n h e r e i s t h a t f o r s e m i d i s c r e t e approximation, w i t h t h e time v a r i a b l e remaining continuous and a " d i s c r e t e "
analogue of t h e Navier-Stokes e q u a t i o n s
c o n s i s t i n g o f a s y s t e m 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 s .
However, i n [ l ] we
adapted our n o t a t i o n and argument t o apply t o a f u l l d i s c r e t i z a t i o n of the e q u a t i o n s , a t l e a s t i n a simple case of backward E u l e r t i m e d i s c r e t i z a t i o n of t h e ordinary d i f f e r e n t i a l equations. We assume a " l o c a l " e r r o r e s t i m a t e i s a l r e a d y known, " l o c a l " meaning w i t h
John G . HEYWOOD
382
e r r o r c o n s t a n t s t h a t grow ( e x p o n e n t i a l l y ) w i t h t i m e :
If
proposition 2 .
u,p
&
are continuous & d i s c r e t e [ t o , t , ) , then
u h"h
defined on some time i n t e r v a l
solutions
(4)
hold for SUP
llfll
t
E
+
[to,t*),
Ilftll
[to.t*)
9
with constants SUP
Ilvull
9
t t o, t * )
K
&
We proved P r o p o s i t i o n 2 f o r a c l a s s of s e m i d i s c r e t e f i n i t e element approxThe e x p o n e n t i a l growth of t h e e r r o r c o n s t a n t s i s unavoidable i f
imations i n [ 2 ] . the solution
u,p
under c o n s i d e r a t i o n i s u n s t a b l e .
Our proof o f "global" e r r o r e s t i m a t e s , e x e m p l i f i e d by Theorem 3 below, r e q u i r e s an e s t i m a t e f o r t h e r e g u l a r i t y o f d i s c r e t e s o l u t i o n s analogous t o P r o p o s i t i o n 1.
The following w a s proven f o r a c l a s s o f s e m i d i s c r e t e f i n i t e
element approximations i n Lemma 5.5 of [ 2 1 .
Proposition 3 .
There e x i s t s a continuous increasing f u n c t i o n
v a r i a b l e s , such t h a t every s o l u t i o n
y, of
defined on any time i n t e r v a l [ t o , t , ]
, satisfies
The f u n c t i o n 3.
F
kindependent a
h
, as
of t h r e e
F
t h e d i s c r e t e Navier-Stokes equations,
w e l l as
to
t*
.
Exponential S t a b i l i t y and Global E r r o r E s t i m a t e s Questions about t h e s t a b i l i t y of
u
s o l u t i o n s " , by which we mean any s o l u t i o n
v
t
-
AV
concern t h e behaviour of "perturbed v
+ V*VV +
of t h e Navier-Stokes
Vq = f
,
problem
383
Numerical Navier-Stokes Problem s t a r t i n g a t a n i n i t i a l time
We r e f e r t o
w = v-u
t
L
0 , w i t h an i n i t i a l v a l u e
a s a " p e r t u r b a t i o n " of
u
,
and t o
v t
a s t h e " i n i t i a l time" and " i n i t i a l value" of t h e p e r t u r b a t i o n doubt about t h e g l o b a l e x i s t e n c e of
v
,
and hence of
w
d e f i n e i t f i r s t as a weak s o l u t i o n i n t h e s e n s e of Hopf.
,
near and
w
.
.
u(to)
w =vo-u(to)
To avoid any
i t i s necessary to
We w i l l n o t b e l a b o u r
t h i s p o i n t , a s a p r o o f o f t h e r e g u l a r i t y of any small p e r t u r b a t i o n of a s t a b l e s t r o n g s o l u t i o n i s i m p l i c i t i n Theorem 1 below. The o r d i n a r y , s i m p l e s t , n o t i o n of s t a b i l i t y i s t h e f o l l o w i n g .
D efi n it i on 1 .
every w
x w
E
u
and
J
((wo(( < 6
(1) i s said t o be stable if,
o-fproblem
, t her e e x i s t s a nwnber
> 0 E
The s olution
6 > 0
, satisfies
for
such t h a t e v e w perturbation w
sup ( / w ( ( < [ t o ,m)
Here, i n s p e a k i n g of "every p e r t u r b a t i o n " ,
E
,
.
i t should be understood t h a t
w e a r e r e f e r r i n g t o every p e r t u r b a t i o n , s t a r t i n g a t every i n i t i a l t i m e
to
2
.
0
A s t r o n g e r n o t i o n o f s t a b i l i t y i s r e q u i r e d upon which t o b a s e e r r o r e s t i m a t e s which a r e uniform i n t i m e .
D e f i n i t i o n 2.
*s oZution
i f t h ere e x i s t nmibers
&a
ilwol/ < 6 If
u
u f o
6,T > 0
, satisfies
problem (1) i s said t o be exponentiaZly stable such t h a t every perturbation w ,
/ ( w ( t o + T ) / /5
with
wO< J
1
y/(wo//.
s a t i s f i e s t h e c o n d i t i o n s o f e i t h e r of t h e s e d e f i n i t i o n s w i t h
6 =
m
,
w e say u i s u n c o n d i t i o n a l l y s t a b l e . An example of an e x p o n e n t i a l l y s t a b l e flow i s provided by simple a x i a l l y symmetric Taylor c e l l s o c c u r r i n g i n flow between r o t a t i n g c o a x i a l c y l i n d e r s .
The
s i t u a t i o n i s one i n which, i f t h e d a t a a r e s t e a d y , t h e r e e x i s t m u l t i p l e s t e a d y solutions.
I f t h e d i f f e r e n c e between two such s o l u t i o n s i s c o n s i d e r e d as a
p e r t u r b a t i o n , i t c e r t a i n l y w i l l n o t decay. ally stable.
Thus Taylor c e l l s are n o t uncondition-
F u r t h e r , t h e r e g e n e r a l l y e x i s t even small p e r t u r b a t i o n s whose decay
2 i n t h e L -norm i s n o t monotonic.
However, t h e c e l l s a r e c e r t a i n l y s t a b l e i n some
s e n s e , and i n t u i t i v e c o n s i d e r a t i o n s of l i n e a r i z a t i o n s u g g e s t t h a t t h e decay of
John G . HEYWOOD
384
small perturbations is exponential. Our development of a s t a b i i i t y theory i s based on s e v e r a l lemmas a s s e r t i n g the continuous dependence of s o l u t i o n s on t h e i r i n i t i a l v a l u e s .
M = supllVu// t>O
For ever3 pertur>bution w
for
t
2
For every
perturbation
for
t
w of
5 t 5 t
L m a 3.
is
of
.
t h e r e holds
u
.
t
Lemn 2 .
c
R , and
a g e n e r i c c o n s t a n t depending only on
Lemmn I .
Below,
w
u
, with
w
0
E
exists a nwnber J
and
1
6 > 0
such t h a t every
., s a t i s f i e s
I/Vwo(/ < 6
+ T .
For every
perturbation
, there
T > 0
T > 0
of
u
, there
, with
e z i s t s numhers
I/w(to)
I/
< p
p,B
>fi
such thai; every
, satisfies
To prove t h e s e lemmas, one begins by w r i t i n g t h e p e r t u r b a t i o n e q u a t i o n
f o r the difference
by
w
w
=
v
-
u
of t h e s o l u t i o n s of ( 6 ) and (1). M u l t i p l y i n g ( 7 )
and i n t e g r a t i n g l e a d s t o Lemma 1.
l e a d s t o Lemma 2 . inequality.
M u l t i p l y i n g ( 7 ) by
I n both c a s e s , t h e c o n s t a n t s
c
v
and i n t e g r a t i n g
depend on Sobolev's
Lemma 3 i s o b t a i n e d by combining Lemmas 1 and 2.
need somewhat more p r e c i s e s t a t e m e n t s i f
Aw
A l l t h r e e lemmas
i s understood only a s a "weak
solution." Using t h e preceding lemmas, we can e s t a b l i s h t h e e q u i v a l e n c e of v a r i o u s
3x5
Numerical Navier-Stokes Problem d e f i n i t i o n s of s t a b i l i t y .
We p r o v e t h e f o l l o w i n g s i m p l e t h e o r e m t<>i n d i c a t e t h e
n a t u r e o f argument.
Theorem 1.
I;he s t a b i l i t y c o n d i t i o n & D e f i n i t i o n
following:
For ever3
perturbation
, with
w
0
E
J
,_ t h e_ re _ e x i_ sts g _ number _ and 1 -
j(vwo(( < 6
6 > 0
such t h a t every
, satisfies
sup
I/Dw//
<
E
.
[ t o, m )
F i r s t w e check t h a t t h e c o n d i t i o n of D e f i n i t i o n 1 i m p l i e s t h a t of
Proof.
~
Theorem 1. for
w
> 0
E
1 i s equivalent t o the
+
to 5 t 2 t
inequality
/ / V w ( t ) / l is s m a l l ,
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 1
,
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
]/w(t)Il
//Vw(t)
11
is taken small, then the
//vwot]
is s m a l l f o r a l l
is s m a l l f o r a l l
t 2 t
t t t
+
0 '
.
1
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.
/I
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
t
S
+
1 , by t a k i n g
a c c o r d i n g t o Lemma 3 . time
to
+
1
,
implies
small.
/lw,II
]/w(t)
But t h e n ,
is s m a l l , for I/Vw(to+l)
11
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)
11
is small for
Poincare's inequality, (/w(t)I/ is small for
t
2
t 2 t
t
+
+
1
1
.
.
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
__
w
E
0
J
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
w
,
, satisfies
6,a,A>O
such t h a t every p e r t u r b a t i o n
w
,
386
John G . HEYWOOD
with __
(iiil w
0
J
E
[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 u
solution
Theorem 3.
&
t 2 0
K , ho > 0
__ Proof.
t
-+ 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
&
if
uh,ph
g g continuous
is exponentially
u
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)
I(
< 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
K
and
ho
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
C1
where
depends on
m
h
,
i t i s c l e a r t h a t (11) i m p l i e s
and
K
only through t h e i r product
.
hK
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
C2
a l s o d e p e n d i n g on
h
and
K
only through t h e i r product
hK
.
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
C3
a g a i n d e p e n d i n g on
h
and
K
only through t h e i r product
hK
.
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
h
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
hK
and
h
v(to)
=
h
R \(to)
.
Then
v
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
v
,
and
C3 , i . e . ,
l a r g e and a l l
h
with constants
K
d e p e n d i n g on
only through t h e i r product
.
and
o n l y through
K
K
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
h
,
hK w i l l be s m a l l enough t o e n s u r e ( 1 7 ) .
and
2
Now ( 1 6 ) and (11) i m p l y (18) imply ( 1 2 ) .
hK
h
\\\(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
u
it sufficestoconsider the linear-
ized perturbation equation
-
-
w
(19)
- bW
t
+ 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
u
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
w
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
w
and
w
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
q
.
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
4
2
T > 0
f o r any f i x e d
.
Thus, i f
ce
cM T
2
sup [to,to+Tl
IIVwo/!
2
to+T
I I ~ ~ I It I
I I ~ W I I
d7
3
0
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
T
above s u c h t h a t
(20) i m p l i e s
11
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)
II
1
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
u
.
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.
4.
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
$
a
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
of
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
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 .
If
f
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
Ve say u
D e f i n i t i o n 3.
& 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
f
If
€
R
and
time shift
s
w
s = 0
.
stable i f there are numbers
,
and a spatial r o t a t i o n
wo w
E
J &g
//woI/ < 6
satisfying
(22)
where __
v
i s t h e solution of the perturbed probZern ( 6 ) corresponding t o the
perturbation
w
,
;(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
v
T
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
w
should be l e s s than a f i x e d c o n s t a n t
B
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
s
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
s
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
w
i n time
T
.
v
and
u=
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
5
, in
addition t o the constants
,
t > 0
and
K
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 ) ) =
0
,
5
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
I
m
p
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
h'
.
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
if
f
i s independent of t i m e , a n d / o r
R
and
f
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
u
t
and/or
u
$
$
,
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.
problem ( 1 )
u
The soZution
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
+
1
, where
Nonzero _ _ _ _muZtipZiers _
a
u
&
p
p
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.
s
& nuntriuial
rotations
, respectively,
w
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
d
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
,
Nh
h
,
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
,
M
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
M
numbers
th
'
does indeed bound t h e D i r i c h l e t norm. remain bounded as
Nh
h + 0
,
i.e.,
M
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 .
t
h
- - 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
h
-
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
h
as
h
.+
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
h
.
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
0
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
5
t <
m
and s t a t e our theorems
a c c o r d i n g l y , t h e m o d i f i c a t i o n t o s o l u t i o n s d e f i n e d on f i n i t e time i n t e r v a l s b e i n g obvious. Let
uh
be a s o l u t i o n o f t h e d i s c r e t i z e d Navier-Stokes e q u a t i o n s , d e f i n e d
.
t 2 0
for
of
uh
I n analogy w i t h t h e continuous c a s e , we c a l l
wh = vh - uh
if
a t some i n i t i a l time
to
, where 2
0
,
vh
wh
a “perturbation”
i s a second d i s c r e t e s o l u t i o n , s t a r t i n g
with an i n i t i a l v a l u e
Whenever w e speak below of a p e r t u r b a t i o n
wh
,
vh(to)
near
uh(to)
.
i t i s t o b e understood t h a t t h e
a s s o c i a t e d i n i t i a l time, i n i t i a l v a l u e , and p e r t u r b e d s o l u t i o n a r e denoted by to
,
wh(to)
and
D ef ini t i on 4 .
vh , r e s p e c t i v e l y .
A so Zution
(defined f o r
t L OJ
Uh
is said t o be %ontractiueZy stabZe t o 5 tolerance”
i f there e x i s t p o s i t i o e numbers
any time
t
there holds --
2
o
of the d i s c r e t i z e d Navier-Stokes equations
6 ,p
, A ayld T , with
and any perturbation
wh
uh
p c 6
satisfging
, such t h a t for /Ivhwh(t0)
I/ < 6 ,
Numerical Navier-Stokes Problem
We c a l l time" of
uh
Theorem 6.
,
and
A
Suppose t h a t , f o r some
g
t h e " s t a b i l i t y r a d i u s " and
supllvhuh/l
h
42
t h e "decay
, there i s a d i s c r e t e s o l u t i o n u
a tolerance. %,
h
& sufficiently
, llZhuh(o) /I , and t h e s t a b i l i t y parameters
t 20
$Definition
T
a " D i r i c h l e t bound" f o r i t s p e r t u r b a t i o n s .
i s contractively stable t o relation
6
the "tolerance",
p
395
there e x i s t s a continuous s o l u t i o n
h -
which
small i n p.6,T
of the Navier-Stokes
u
equations s a t i s f 9 i n g
Further, i-f
3p < 6
, and ~
~
h
i s small enough, t h e c o n t i n u o u s s o l u t i o n
w i l l a l s o be c o n t r a c t i v e l y s t a b l e t o ---
u
tolerance.
The proof t h a t a d i s c r e t e s o l u t i o n ' s 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
i s amenable t o numerical v e r i f i c a t i o n depends upon d i s c r e t e s o l u t i o n s e n j o y i n g continuous dependence p r o p e r t i e s analogous t o t h o s e s t a t e d f o r continuous s o l u t i o n s i n Lemmas 1 and 2 .
A s we a r e d e a l i n g a b s t r a c t l y w i t h t h e d i s c r e t i -
z a t i o n o f t h e Navier-Stokes e q u a t i o n s , we must assume such p r o p e r t i e s of c o n t i n uous dependence.
Theorem 7 .
Then,
3
The following then h o l d s .
Suppose
I+,
u,,
&- 44
(&a
sup IIvhuh t 20
I/
< =
.
a tolerance,
t h i s can be v e r i f i e d by
tolerance i n f i x e d t i m e ) $ a
f i x e d f i n i t e number of
& contractively
checking the decay
discrete solution satisfying stable t o
t e s t perturbations per u n i t of time. The assurance t h a t d i s c r e t e s o l u t i o n s approximating 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 w i l l , f o r a l l s u f f i c i e n t l y s m a l l v a l u e s of hypotheses of Theorem 6 i s provided i n o u r f i n a l r e s u l t .
h
,
satisfy the
Contractive s t a b i l i t y
t o a t o l e r a n c e i s d e f i n e d f o r continuous s o l u t i o n s analogously t o D e f i n i t i o n 4 .
396
John G . HEYWOOD
Theorem
a. &&
Suppose
that
& y, & continuous
u
sup/lvu/l < t 20
with parameters
p,6,T
m
and t h a t
&
A
.
u
and d i s c r e t e s o l u t i o n s
& contractively
41s problem
stable t o
Then there e x i s t constants
K
(I),
tolerance,
&
h
0 ’
such that
__.
for a l l
.
h < h
Further,
~
3p < S
, then t h e d i s c r e t e solution y 1 2 s
c o n t r a c t i v e l y s t a b l e t o a tolerance f o r c e r t a i n f i x e d values of t h e s t a b i l i t y parameters, f o r a l l s u f f i c i m t l $ j smali: values of
h
.
We have s t a t e d t h i s l a s t r e s u l t f o r continuous s o l u t i o n s assumed merely
t o be 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 , r a t h e r than t o p o s s e s s t h e s t r o n g e r p r o p e r t y of e x p o n e n t i a l s t a b i l i t y , a s w e t h i n k t h e r e i s a n a t u r a l l y o c c u r i n g and important c l a s s of flows which p o s s e s s t h i s weaker s t a b i l i t y p r o p e r t y w i t h o u t being, i n f a c t , exponentially s t a b l e .
For example, imagine t h a t
R
is a section
of p i p e o r t u b i n g and l e t smooth boundary v a l u e s be p r e s c r i b e d f o r a f l o w e n t e r i n g a c r o s s a n upstream s e c t i o n and e x i t i n g a c r o s s a downstream s e c t i o n . Adjusting t h e r a t e o f flow and t h e l e n g t h of t h e p i p e , one may expect t o observe i n c i p i e n t t u r b u l e n c e i n a flow which i s y e t , i n some s e n s e , s t a b l e t o l a r g e r disturbances. t o grow.
Small p e r t u r b a t i o n s i n t h e n e a r l y uniform upstream flow begin
However, b e f o r e they grow very l a r g e they pass o u t of
downstream boundary.
Yet, t h e i r e f f e c t may n o t decay t o z e r o .
!2
across the
Even a s they
p a s s downstream they i n f l u e n c e t h e upstream f l o w ; t h e flow i s a n a l y t i c a f t e r a l l . T h e i r e f f e c t might be l i k e n e d t o t h e i n t r o d u c t i o n of new p e r t u r b a t i o n s upstream, which i n t h e i r t u r n w i l l grow, pass downstream, and a g a i n c r e a t e new p e r t u r b a t i o n s upstream.
T f a l a r g e r d i s t r u b a n c e i s i n t r o d u c e d , i t s e f f e c t w i l l decay
t o t h e same ambient l e v e l of minor d i s t u r b a n c e s .
Another type of example
probably occurs i n von-KQrmbn v o r t e x shedding, i f t h e r e are s l i g h t i n s t a b i l i t i e s i n the vortices.
S t i l l a n o t h e r i n t h e Taylor experiment, when wavy c e l l s appear
with s l i g b t l y t u r b u l e n t c o r e s .
I f t h e s e flows r e a l l y are c o n t r a c t i v e l y s t a b l e
Numerical Navier-Stokes Problem
397
t o a t o l e r a n c e , t h e e r r o r e s t i m a t e t o a t o l e r a n c e (26) a p p l i e s t o t h e i r d i s c r e t e approximations, a t l e a s t a f t e r t a k i n g account of time s h i f t s and s p a t i a l r o t a t i o n s as was done f o r e x p o n e n t i a l s t a b i l i t y i n s e c t i o n 4 .
References
[l] Heywood, J . G .
and Rannacher R . ,
F i n i t e Element Approximation o f t h e Non-
s t a t i o n a r y Navier-Stokes Problem, P a r t 11: S t a b i l i t y o f S o l u t i o n s and E r r o r Estimates Uniform i n Time, p r e p r i n t , Univ. of B r i t i s h Columbia (October 1982). [2]
Heywood, J . G .
and Rannacher R . ,
s t a t i o n a r y Navier-Stokes
F i n i t e Element Approximation o f t h e Non-
Problem, P a r t I : R e g u l a r i t y o f S o l u t i o n s and
Second-order E r r o r E s t i m a t e s f o r S p a t i a l D i s c r e t i z a t i o n , SIAM .I.Numer. Anal. 19 (1982) 275-311.
This Page Intentionally Left Blank
Lecture Notes in Num. Appl. Anal., 5, 399-407 (1982) Nonlinear PDE in Applied Science. US.-Japan Seminar, Tokyo, 1982
THE ZXISTdNCE AND THb FINITb ELLNI~NTAPPAOXIICATION FOR THE SYSTEM A U = i: u .
+
J axj
Lin Qun
Jiang Lishang
and
Institute of Systems Science
Department of Mathematics
Academia Sinica Beijing,
f
Beijing University
China
Beijing, China
There is no global existence theorem about the nonlinear Navier-Stokes equation un.til now. However, there is an existence theorem for the linear Stokes equation, and there also exists an existence theorem, as we will prove in this paper, for the nonlinear system
N hi=
aui
;uj ax.
ui = gi
J
on
+
fi
in c = R ~ ,
an, lsilN
,
which has the same nonlinearity as the Navier-Stokes equation b u t without the condition
div u = 0. These results mean, as pointed out
by R. B. Kellogg, that the difficulty about the Navier-Stokes equation stems from both the condition div u = 0 and the nonlinearity rather than from only one of them. We will also prove in this paper the convergence theorem f o r the finite element approximation of the system (1) and some acceleration results. 1. Existence Theorem
The systern (1) was discussed by Kiselev and Ladyzenskaya in 1957. It has been pointed by Nirenberg that the proof of existence 399
400
Lin
QUN and
Jinag LISHANG
is incorrect (see MR 20 #6881, by Finn). We first prove that the system (1) always has a classical solution. Theorem 1. and
gi E C2+,(
n
Suppose
E A2+a.
Then f o r any data
an), the system (1) has a solution ui
fi E C,(n)
E C2+a(n),
I s i S N .
The p r o o f is based on Lemma 1. maxn c Jui(x)) 4 maxn
ffi(x)J + rnaxall2' lgi(x)l
.
+ 2 maxn L lxil
Proof.
We write, for fixed i, Ui(X)
=
v(x) + x i '
Hence the ith equation of ( 1 ) becomes
J
L
v = gi(x)
-
x.
1
on an
. (xy,
Now assume for v a positive maximum at Po =
-
V(Po)
v(P,)
5
=
+
maxn lxil
Similarly, assume f o r v a negative minimum at Po
V(P 0)
2
-maxn Ifi(x)I
E n that
,
fi(Po) + xp
maxn Ifi(x)I
..., ):x
-
€
maxn lxil
Finally, assume for v an extreme value at Po E
an
. so
.
that
401
Existence and Finite Element Approximation
Thus in any case we have maxn Iv(x)l
maxn Iui(x)I 5
+
maxn Ifi(x)l
maxn IxiJ +
maxan lgi(x)l
2 maxn / x i
+
and Lemma 1 is proved. Apply ng Lemma 1 and the Learay-Schauder tkleorem we obta n Theorem 1. The existence theorem for the nonstationary proolem correspondir.g to (1) can be treated in a similar fashion. 2. Finite Element Approximation and Its Acceleration Consider the simple solution u of (1) defined by
with scalars ($,rO)
=
SILiPidx ,
(Ic,S)l = 2
alL J
g.) J
We have the following approximation theorem. Theorem 2. and
u
f
H2
fl
Suppose that N = 3 * g = 0, i2 is a convex polygon
i1 is a
simple solution of (1). Then for h small
enough, the finite element equation
in the piecewise linear trial space Sh has a unique solution uh E Sh in a neighborhood of u , which satisfies
(3)
IIuh - uiIl
Furthermore, if
5
Ch IIuIl2
I
lIuh
-
uII0 5 Ch
2
IIui12
.
u E H3, the accuracy of energy norm can be raised by
computing the following quadratic finite element solutioris
Lin QUN and Jinag LISHANG
in the piecewise quadratic trial space Sk with a coarser mesh size k (see the figure). We have
illk - uill
(5)
5
ch2
.
Note that the two finite element equations for uh and
ck have
the same degrees of freedom.
Theorem 2 will be proved by the following abstract operator framework. Consider the abstract operator equation
where K is Frkchet-differentiable in a Banach space E. The following lemma can be found in, f o r example, G. Alefeld, Beitr Numer. Anal. 6(1977). Lemma 2.
If
(I
-
K I ( U ~ ) ) - exists ~
for some
Lipschitz-continuous with constant c1 in the ball
then (6) has a unique solution u in the region r3
= (-1
+
4i-T.E) 3 5 IIU - uoll c4
(1
-1 4 -
and has no solution in the region
Moreover, u is a simple solution: 3 (I - K' ( u ) ) - ' , We now consider the projection equation
uo E E, I('
Ilu
-
uoII
<
is
5
r l , and
=
r4
C
Existence and Finite Element Approximation
PhE = Shc E.
where Ph is a projection with Lemma 2.
403
Suppose that u is a simple solution of ( 6 ) . PhK is
Lipschitz-continuous in a neighborhood of u , and Ilu
- Phuli - 0 ,
ll(I
-
Ph)K'(u)ll
-
0
(as h
-
0).
Then for h small enough, equation ( 7 ) has a unique solution uh in a neighborhood of u, which satisfies
The Newton iterates f o r ( 7 ) exist and converge quadratically. Moreover
Proof.
Replacing K with PhK and choocing uo = u in Lemma 2 ,
one obtains the first part of Lemma 3 . The estimate (8) can be derived by using the following identities
?in operator defined by 0
Ku E HI
et Sh be the piecewise linear trial space with the mesh size h and Ph the standard Ritz-projection defined by
404
Lin
QUN
and Jinag LISHANG
Then (1) and (2) can be written as (6) and ( 7 ) respectively. And the Frkchet derivative K'(u) will be determined by 0
v = K'(u)w E H, ,
We now come to prove
For this we note
Since
so
(9) holds. By means of Lemma 3 and Nitsche's trick we obtain the estimate
(3) and the following estimate
where
is the solutions of the uncoupled Poisson equation
The estimate ( 5 ) can be derived by t h e following Brezzi's trick.
405
Existence and Finite Element Approximation
F i r s t , s p l i t t h e P o i s s o n e q u a t i o n ( 1 0 ) i n t o two p a r t s :
Correspondingly t h e q u a d r a t i c f i n i t e element e q u a t i o n ( 4 ) can a l s o be s p l i t t e d i n t o two p a r t s :
-
('1kSPk
and ( 5 ) f o l l o w s . We would l i k e t o make t h e f o l l o w i n g r e m a r k . R e c e n t l y L i n Qun and Lu Tao p r o p o s e d a " s p l i t t i n g e x t r a p o l a t i o n d i f f e r e n c e method" f o r s o l v i n g t h e l i n e a r e l l i p t i c problem
u = g i n a cubic
on
an
fl c RN. The w e l l known d i f f e r e n c e method c o n s i s t s i n
r e p l a c i n g t h e d i f f e r e n t i a l e q u a t i o n w i t h t h e c e n t r a l d i f f e r e n c e quot i e n t e q u a t i o n , t h e c o n t i n u o u s domain h2,
..., h N ) w i t h
s i z e h l , h2,
n
w i t h t h e d i s c r e t e mesh
..., hN a l o n g v a r i a b l e s
x l , x2,
V(hl,
...
,X
N
.Lin QUN and Jinag LISHANG
406
respectively and replacing the continuous exact solution u with the discrete approximate solution U(hl, h2,
..., hN).
We have, if u is
smooth enough,
u
-
U(hl, h2,
..., h N )
M
N
21M+2)
I: ci2j)h$j + O ( Z hi = ‘Z j=1 i=l i=l
The usual global extrapolation method involves the following: hl h2 hN hl h2 Make the homogeneous refinement meshes V(F, V(r,
r,..., r),
r,
hN ..., ri-), ... with corresponding difference solutions U(F,hl Fh2, ..., hN ) , hl h2 hN F U ( r , T , ..., r), ... and compute the homogeneous extrapolation solutions
F h2, ..., F hN )-
HE1 = $ ( ~ U ( hl T,
r,
1 hl h2 HE2 = ~ ( 6 4 U ( r ,
e..,
..., hN))
U(hl, h2,
hN r) -
-,...,, hN )
hl h2
ZOU(- 2 ’ 2
... hN)) .....................
+
U(hls h2v
U
-
D
Then one has 4
HE1 = O ( Z hi)
;
u
-
6
HE2 = O ( L hi)
;
... .
Our splitting extrapolation method involves the following: hl h2, hN), V(hl, Make the one-variable refinement meshes V(F, h2 hN hl , hN), V(hle h2v F ) : V(F~ h2t * * * P ~ N ) P V(hle h2 hN hN), V(hl, h2, with the corresponding differ-
...,
...,
...,
...,
...
..., r); ... ..., hN), U(hl;
hl ,h2, ence solutions U( F h2,
...,
hN hl y): U ( r , h2,
...,
F h2,
..., hN), U(hl, r, 2
a * . *
* . * o
hN)*
hN)e
* . * r
* * * *
U(hl,
U(hl, h2,
hN ..., r); ... and compute the splitting extrapolation solutions SE1 =
-
1 N‘Z
3iZ1
( 4 U(hl,
(N - l)U(hl,
hi , ..., hN) ..., hN)
U(hl,
..., hN))
Existence and Finite Element Approximation
407
................... Then one has U
-
4 SE1 = O ( C hi)
;
u
-
6
SE2 = O(.Z hi)
;
... ,
Actually,
in the asymptotic sense. It i s easy to see that the splitting extrapolation rnethod will save computational effort in comparison with the homogeneous extrapolation method. le hope that the splitting method w i l l be effective also for the system (1).
Acknowledgments. The authors are greatly indebted to Professors F. Brezzi, J. Frehse, B. IKellogg and G. Strang for their helpful comments.
This Page Intentionally Left Blank
Lecture Notes in Num. Appl. Anal., 5, 409-434 (1982) Nonlinear PDE in Applied Science. U.S.-Japan Seminar. Tokyo, 1982
A hyperbolic model o f combustion
Ying Lung-an and Teng Zhen-huan Department of Mathematics Peking University Beijing
CHINA
If fluid flow is accompanied by chemical reaction, then very complicated wave motion phenomena occur. Chapman and Jouguet used a simple and typical model which showed various waves of combustion: strong detonation wave, weak detonation wave, strong deflagration wave, weak deflagration wave, and their critical states, the so-called Chapman-Jouguet detonation wave and deflagratlon wave [1,2]. Afterwards, many autnors have done various works about the structnre of these waves and their formative conditions using different kinds of models.
More research works
have been don? in the laboratories and by numerical experiments.
It is an interesting problem how a mathematical model can be applied to these phenomena and one may jnvestigate them by the theory of partial differential equations. Some authors have investigated the travelling wave solutions and some Riemann initial value problems for these problems, but up to now these investigations are not so deep as that for the shock waves.
In this paper we consider a model system of combustion as
where constants q > 0, v
2
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 =
+m
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
0
(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
4
is defined as
4(u)
=
{
0,
u
1,
u > 0,
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
0
5
z 5 1, according
to its physical background. Let
v
-+
+0, K *
in system (l), we obtain a formally classical formula-
+a,
tion as
a
a ax
(u + q z ) + - f(u) = 0,
z = 0,
as
u > 0,
az _
as
u < 0.
-
at - O,
(3)
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
-
u
,z
from the left and right sides of the discontinuous
-
and
+
u
+ ,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
+
z
,
f'(u-) > 0,
f'(u-) <
0,
d) Chapman-Jouguet detonation waves (abbr. C J ) , u- > 0, u+ < 0, and
f'(u-)
= o,
e) contact discontinuities (abbr. C ) , u
+
-
= u
,
+
z
-
# z
, where -
Some other cases are possible, for instance the case when
u
U = 0.
< 0 , u+ >
0,
but we assume that the Lax condition of stability
hT1 -> o is satisfied, where
1. x;, h (u) 1
for
f
i = 1
0, A2(u) = f'(u),
A f ( u ( x + 0, t)), then neither the case
or
xi =
2,
xi(U(X
-
0,
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-
2
z = 0 when
or u+
0
+
= 0
-
or u
= 0. We
u = 0, hence we may change
at the above inequalities.
0
For the convenience of following discussion, two auxiliary functions are defined. Function u* = g ( u , z )
Lemma 1.
is defined by
exists uniquely and
u*
> 0,
2
> 0.
Proof. Set
then (5) is equivalent to
Y(u*) = 0, u*
2
u.
It is easy to verify
?'(u*)
> 0
and
But = 0.
v ( t m )
But
aU
=
+m,
therefore there exists a unique u*
z 2 0, hence u*
> 0 and
2
2
corresponds to u
such that
y(u*)
u.
> 0 can be verified from ( 5 ) directly.
By (5) it is easy to see that
-
u + qz
+
+
+
SD corresponds to u- > g(u , z )
and
CJ
$(O) =
0,
+
= g(u , z ).
The second auxiliary function is w = $ ( u ) , satisfying
w =
tu,
It is easy to see that $
'u
0.
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
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
w
by
i'" > 0. From (5)
(6) we
1
-
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.
0
We consider the initial value problem of ( 2 ) ( 3 ) with initial values
where
zo
satisfies 0
5 z0 -<
1 and
z (x) 0
=
0 when
we consider the Riemann problem, that is the case of
uo(x)
2
0. First of all,
YINGLung and TEN Zhen-Huan
414
u R , ur, zk, zr
where
uI1 5 ur.
a)
We c o n s t r u c t
and t h e i n i t i a l v a l u e . There i s a
obtained.
uI1 > u r , b u t
b)
are a
and a
C
c)
C
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)
S
T h e r e a r e four c a s e s :
are constants.
ue
S
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:
0
u Euo
and
0
> 0 , g(uo,
2,))
0,
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
t
and
x,
exists,and for any smooth function (p(x, t) with compact support on
holds, moreover, for any non-negative smooth function
'9'". t)
t
2
0,
with compact
support on t L 0 ,
holds, and finally such that with
we have
z(x, t) =
Problem Q.
{
if
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
u = 0.
i m p l i e s ( 1 3 ) and determines t h e s t a t e as
52.
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.
If
(-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.
N
Suppose t h e r e a r e
When
When
u(x, t )
N = 1, a s o l u t i o n
t.
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
r
= u (+O) O
c :;:
zo(x) =
x
x
50
and
x > 0
with-
uk = uo(-O),
and x > 0,
x
5
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
a)
uR 5 ur, c o n s t r u c t
b)
uy,> u r , b u t
and
x < 0
u
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 )
.
x
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
0
' u r , ue -> 0 ,
UQ
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
t
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
=
g ( u ( x + 0, t o ) , z r )
t i n u o u s v a r i e s from
d)
x
5
=x(t)
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
g(ur, zr)
2 uQ
always l i e s i n r e g io n
S t e p 3.
on t h e s e c t o r
on
Q(T),
and
uQ(x, t )
x
2
f'(ur)t
respectively.
x
2
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
4.
x(t)
x
increases, t h e curve
f'(uk)t.
Construct a solution
u(x, t ) = ( f ' )
i n region
d), i . e . t h e d i s c o n -
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
u(x - 0, t )
up.5 g ( u r , z r ) .
0, but
Using t h e s o l u t i o n o f
ur(x, t )
Since
vary as t h e previous, i f
CJ.
f ' ( u R ) t , d e n o t e them by
Step 2.
= x(t).
2
to
0, t )
to, t h e n it becomes t h e c a s e
a t some
SD
u Q > u r , uQ
S t e p 1. and
-
increases, u ( x + 0, t ) , u ( x
t
As
.
f'(uR)t < x
5
-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
x
YING Lung and TENZhen-Huan
418 is a
solution, x = x ( t )
CJ.
2
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
If
(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
0'
holds on t h i s c h a r a c t e r i s t i c . if
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 .
0
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 <
X,
2 -Mo,
and
and c o n s t a n t
Mo
u(x, t )
u(x, t
< -
0
we h a v e
c
u(x, t ) - u(5, t ) < x - 5 - t' where
Proof.
i s a c o n s t a n t d e p e n d i n g on f u n c t i o n
C
Take any
6<
x.
If
u(5,
t)
5
f
only.
0 , t h e n by t h e Remark a b o v e , t h r o u g h
(5,
t ) t h e r e i s a downward c h a r a c t e r i s t i c which i n t e r s e c t s t h e x - a x i s a t p o i n t
c,
if
x1
t h e x-axis,
i s t h e i n t e r s e c t i o n point of t h e c h a r a c t e r i s t i c through
get x
- 5
= x1 + f l ( u ( x , t ) ) t -
> f'(u(x, t))t=
u
and
then
But t h e s e t w o c h a r a c t e r i s t i c d o n o t i n t e r s e c t by t h e Remark, h e n c e
where
(x, t)
f"(U)(U(X,
5, -
f'(U(S,
ff(u(5, t))t
t))t
t ) - u(5, t ) ) t , I
i s a c e r t a i n mean v a l u e , u
2 -Mo,
so
xi
L 5,.
We
419
Hyperbolic Model of Combustion
If
u ( 5 , t ) > 0 , t h e n t h e above i n e q u a l i t y i s o b v i o u s b e c a u s e
Lemma 5 . __
The s o l u t i o n by Lemma 3 s a t i s f i e s
6.
The s o l u t i o n by Lemma 3 s a t i s f i e s
Lemma
u(x, t)
5
0.
[7
The above two lemmas a r e o b v i o u s b e c a u s e o f t h e s t , r u c t u r e of t h e s o l u t i o n .
Lemma
7.
where
C
)I
o uo
__ Proof.
v a r $ ( u ( . , t ) )< Civar $
uo
o
i s a c o n s t a n t d e p e n d i n g on i s t h e c o m p o s i t i o n of
For a given
$
t , u(x, t )
+ v a r zo},
q , function
and
Let
< z+(xi, t ) , u-(xi, that
0 = z-(yi,
xi
t) < 0
suplu(x, t)l
and
i s p i e c e w i s e monotonous, z ( x , t )
( i = 1,2,*.-),y i t)
is piecewise
u-(x, t ! , z-(x, t ) and
b e t h e d i s c o n t i n u o u s p o i n t s such t h a t
t ) < z+(yi, t!, u-(yi,
only,
u f) *
c o n s t a n t , t h e r e f o r e t h e l e f t and r i g h t l i m i t z+(x, t ) , exists.
f
20
u+(x, t ) ,
z-(xi,
t)
be t h e discontinuous p o i n t s such ( i = 1,2,-..), s e t
F ( t ) = v a r $ ( u ( . , t ) ) + 2qc{z+(xi, t ) i
-
z-(xi,
t)}
+ 2~ r n a x i $ ( g ( u + ( y i , t ) , z + ( y i . t ) ! ) - @ ( u - ( y i , t ) ) , 01.
i
Let u s prove t h a t
F(t)
F i r s t l y we compare
> F(0). F(0)
and
F(+O).
Each t e r m i n
F
does n o t change
l o c a l l y a t t h e i n t e r i o r p o i n t s of e v e r y i n t e r v a l s and t h e d i s c o n t i n u o u s p o i n t s f o r cases a ) b ) c ) .
F o r c a s e d ) , when
continuous points a r e
t = + 0 , t h e jump of
@(u(x, t ) ) at t h e dis-
420
YING
Lung and TEN Zhen-Huan
hence
There are only a finite number of discontinuous curves and the numbers of intersection points are finite, we may assume that the lowest intersection point is at t = to. Next, we consider F(t)
as
t < to. The third term in F(t) disappears
and the second term keeps invariable in this case, therefore we will consider the
variation of $
o
u
only. For the continuous points of u , because u
is
constant along characteristic, the local variation keeps invariable along the characteristic. We will consider the influence of S, SD, CJ
to the l o c a l
variation only. The local variation decreases for S and
SD, hence they does not cause an
increase of F(t). Fur. a
CJ
curve x ( t ) , set b = x(t), construct a characteristic through
(b, t) whlch intersects the x-axis at at
x = x(O), which is denoted by
Hence
x = c, construct a tangent line of x(t)
xl(T), let
a = x (t) as 1
T =
t, then
42 1
Hyperbolic Model of Combustion
therefore F(t)
5 F(+Oj t = t o , we d e n o t e by
The d i s c o n t i n d o u s c u r v e s i n t e r s e c t each o t h e r a t +
C
a
that
c a t c h s up w i t h a
SD
The v a r i a n c e from
C.
F(tO- 0) t o
SD
F(tO)
i s considered i n t h e following c a s e s : SD
+
C
CJ
or
-f
C:
r i g h t and left s i d e s of
SD
l e f t s i d e s of
+ z
If
<
or
z , then
we d e n o t e by
C , by C J , then
and
-
u
u
-
t h e values of
2 g(u+,
t h e r c i s a new
SD
z
t h e v a l u e s of
u
from t h e
from t h e r i g h t and
z-). t = t o . The l a s t t w o terms of
at
have no c o n t r i b u t i o n , t h e local v a r i a t i o n does n o t change.
F(t)
If z
+
-
> z
+
,
t h e n it c o r r e s p o n d s t o a p o i n t
xl, if
u- > g(u
SD, t h e l o c a l v a r i a t i o n does n o t change, and
there is also a
-
If z > z
and
u-
5 ~ (+ ,u z + ) ,
be e l i m i n a t e d and one term to
u+
-
z
and
2'
2{$(g(ut,
t h e n one t e r m 2'))
- $(u-)}
Zq{z+ -
+
,
F(t)
2-1
+
z ), then decreases.
in
will
F(t)
w i l l be added from
t = t -0
0
t = to, b u t by Lemma 2
hence
F(t)
does n o t i n c r e a s e e i t h e r . There are o t h e r c a s e s such as
i s e a s y t o check
F(t)
S
+ C, S
-f
SD,
S
-+
CJ, SD
-+
By
we o b t a i n
CJ * S, it
i s i n v a r i a b l e under t h e s e c a s e s .
There i s no d i f f e r e n c e between t h e a n a l y s i s s t a r t i n g from
F(0).
S,
F ( t0 )
and
from
YINGLung and TENZhen-Huan
422
therefore
~er:una 8.
__ Proof.
var z ( *, t
15
var z
0'
Each i n t e r v a l where
1.
= 0
extends a s
t
increases, while
z
does not
v a r y a t t h e r e s t p a r t , t h u s t h i s c o n c l u s i o n follows.
where
C
depends on
q , function
We may assume t h a t
f and
um = Sup/u(x,t )
I
only.
t > T
w i t h o u t l o s i n g g e n e r a l i t y , c o n s t r u c t a down-
w s r d c h a r a c t e r i s t i c through point
(x, t ) , it i n t e r s e c t s e i t h e r t h e x - a x i s o r a
Proof.
CJ
this
by t h e Remark of Lemma 3. CJ, i f this
If it i s a
stops a t a p o i n t
CJ
constructing a c h a r a c t e r i s t i c through
( x l , t ) , t h e n c o n t i n u e t h i s p r o c e d u r e by
1
(xl, t ) , a f t e r a f i n i t e number of s t e p s , 1
t h i s moving p o i n t w i l l a r r i v e a t a p o i n t t a l l i n e as
(x',
7)
which l i e s a t t h e same h o r i z o n -
(x, T ) . I $ ( u ( x , t ) ) - @(u(x', T ) ) \ , it i s s u f f i c i e n t t o c o n s i d e r
F i r s t l y we e s t i m a t e
u
t h e v a r i a n c e of charact,eristics.
along If
Q,, Q2 b e c a u s e
P1P2
uC(P,) P1 and
CJ
n
c h a r a c t e r i s t i c s through
l i n e s through
C J , t h e n l e t t h e p o i n t move down a l o n g
only berause
is a
CJ
u
d o e s n o t change i t s v a l u e a l o n g
a r c m e n t i o n e d b e f o r e , we c o n s t r u c t downward
PI, Pg. t h e y i n t e r s e c t t h e T - h o r i z o n t a l l i n e a t p o i n t s and P2
u+(P, )
2
a r e n e g a t i v e , we c o n s t r u c t p e r p e n d i c u l a r
which i n t e r s e c t t h e
T - h o r i z o n t e d l i n e at
R1
and
Hyperbolic Model of Combustion
423
There is no overlap hence
Secondly by applying the triangular inequality, we get
K = f'(um), we obtain
taking
The conclusion follows from Lemma
Proof.
~
than
z ( x , t) varies on
7 and Lemma 8.
SD and CJ
only, the slope of which is not greater
f'(u 1, hence in the sense of distributions, m
I-, By Lemma
dx < fl(um)JIz lazldx -
ax
8 the conclusion holds.
Finally, we prove
= f'(u
m
) var z ( * , t).
424
Y I N G Lung and TENZhen-Huan
'Theorem 1.
$(uo(x))
If
We may assume t h a t
Proof.
b e r of i n t e r v a l s ,
1
< .;
If
v(")(x)
0
vLrl)(x) < 0
iiO(x)
$
u, z E BV
0
i s l e f t c o n t i n u o u s , and we may assume t h a t
v ( " ) ( x ) , such t h a t 0
v 0( " ) ( x )
20
([41).
uO(x) < 0.
(-a,
i s ' c o n s t a n t on e a c h i n t e r v a l ,
on some i n t e r v a l , t h e n we d e f i n e
n,
c o n s i s t s of a f i n i t e num-
+m)
on some i n t e r v a l , t h e n s e t
For any i n t e g e r
zLn)(x) E 0
zin)(x)
-
/Vi")(X)
$(uO(x))
I
on i t , i f
on t h i s i n t e r v a l s u c h
z ~ " ) ( x ) a r e p i e c e w i s e c o n s t a n t s on a. f i n i t e number o f i n t e r v a l s , and
that
- zo(x)I < .;1
lzIn'(x! +
a r e f u n c t i o n s w i t h bounded v a r i a t i o n , t h e n
i s l e f t c o n t i n u o u s on t h e s e p o i n t s where
we c a n d e f i n e a f u n c t i o n
n
z,(x)
P e x i s t s , and
t h e s o l u t i o n o f Problem
zo(x)
and
m,
ubn)(x)
Let
uLn)(x) = $-l(v;")(x)).
u (x) 0
converges t o
u n i f o r n i l y and
I t i s e a s y t o s e e t h a t as
z(")(x)
li,
p o i n t w i s e , and it i s e a s y t o make t h e v a r i a t i o n o f By d e f i n i t i o n , z ( " ) ( x ) = 0 0
uniformly.
Problem Q w i t h t h e i n i t i a l v a l u e u n ( x , t)
uAn'(x)
if
uin)(x)
t a k e a s u b s e q u e n c e of i n g subsequences o f
Denote by
BV.
z(x, t )
bounded
o L I ( ~ ) :,2
0
2 0.
and
{n}
such t h a t f o r a l l
z(")(x)
0
{$(un(x, t ) ) }
and
M > 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 ) = $-'(w(x,
z(x, t )
has a s o l u t i o n
t)).
and
T > 0 , t h e correspond-
converge i n space
{$
o
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 )
We
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
0
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 For any and
t > 0, we take a subsequence of
zn(x, t)
converge to w
and
z
{(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
)I
does not converge to
x
Suppose that
( u , z)
-1
.
J,
is denoted by
u ( 5 , tl) 2
such that w(x, tl)
in
5
where, if un(5, t,)
Ll
-
[x
-
h,
XI.
But
E
> 0 and
J,(un(x, t ) ) 1
converges to
vn(5, t) > 0, z
Thus
N1.
5
t) = 0 by (15). Let
(5,
Therefore z(c, t) = 0
n
+a,
z(x, t) = 0 by the left continuity of
If not, then
> 0 and every
E
a
is
>
-.
-2
2
0
T
+
T
{T~}.
n.
-rn < t, such that un(x, Tn)
vn(x, t)
That is, for
2
-E.
If
T
We take a subsequence, still denoted by
as
n
4.
We take
+ m.
If T > 0 , then for sufficiently large n, T
We have
for all
55
x
by Lemma
15 - x / <
> - 2 ~uniformly with respect to n. { $ uniformly with respect to
almost every where.
Hence
we get
z.
for all sufficiently large
n, there exists a
accumulation point of
{ T ~ I ,such that T
vn(x, t)
n,
holds almost everywhere on
If v(x, t) < 0, we prove that there is a subsequence such that < 0.
h > 0,
is a point where it converges, then for sufficiently large
> 0, hence
h, x].
5E
for
such that
-
norm, we take a subsequence such that it converges almost every-
z ( 5 , t) = 0 only if
[x
E
N1’
< t
u(x, tl) > 0 by (14). By the left continuity, there are
un
The set of points
If v(x, t) > 0, there exists
Nl.
J,(un(x,t))
v(x, t)
0
un}
TE/2C,
then we get
converges to
w
in
uni5, Tn)
L1 norm
t, hence
By the left continuity
-2~, but
E
is arbitrary, hence
V(X,
t) > 0, which contradicts
YINC Lung and TENZhen-Huan
426 v(x, t) < 0. If
0, then we may construct a characteristic or a curve con-
T =
CJ, which intersects the x-axis at
sisting of piecewise characteristics and
tn
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
0
x
+
as
n
+
a,
we get
u 0(x) and the uniform convergence of
by the left continuity of
is arbitrary, hence
5,
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,
x
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
t
is arbitrary, thus (15) holds
z(x, t) on a null set
0
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
a
at point
x, if one
of the following holds: u (x 0
-
0 ) = a;
u (x + 0 ) = a;
0
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
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
t
> 0.
For any
Proof.
n
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,
E
> 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
u.
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
u
+
x
as
n
(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
(Cn, t 0 )
E [C, uo(x)
-
{unl, still
xn =
un(Cn, t o )
Cn 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
CJ
c u r v e as i n
0
assumes t h e v a l u e
0
x
and
at
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
u
u n ( C n , to)
- f'(un(Cn, tO))tO and u
a
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 )
0
Let
n
+
m,
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)
xo
-
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 intersect on tions as n
CJ
(Ln,
curves through points
(CA, tl)
to) and
do not
t > 0. These two families of curves converge to their limit posi-+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
2
u,(x)
0 at
0 on
tt
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
to
and
is the supremum
satisfying
u(x, to) = 0.
then
Proof. Because
to
is a supremum, there are only two possibilities:
a) there is a series T
-f
to, such that u(x, Tn)
-f
0;
b) u(x, to) = 0. If possibility a) holds, and if
T
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
-+
xO
( 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'
5
xn.
Let
n
+
m,
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
x
because
xo
'
we get
5' 5 xo, hence
429
Hyperbolic Model of Combustion The above inequality still holds if
u ( 6 , to)
20.
5:
Let
-f
x, u
is left contin-
uous, hence
i.e. u(x, to)
If Tn
2 0.
But
u(x, t ) 0
2 xo
as n
-+
=
0, hence
-
x
f'(0)to.
(x, t0 )
But we have
5
0, we can construct a
and intersect the x-axis et point
5
x'
xn by Lemma 11 for any
therefore x' = xo, we also get
m,
u(x, t ) = 0. 0
increases monotonously, because u(x,t ) 0
characteristic through point x'
5
n.
XI,
x n
+
x
O
u(x, t0 ) = 0.
0
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
t
0
6/f'(O)],
5
Since u(x,
7)
Lemma 11, 5
5 xo. If
>
0.
If
5 <
we also get
20
for
6 E (xo - 6, xo).
6 5 f'(O)(t - to).
We take
Take an arbitrary
then construct a characteristic of u
through
0, this characteristic intersects the x-axis at a point G
[xo
-
6, x,],
6
then we get
u(x,
T )
0
(x, T).
5.
By
from u o ( c )
xo - 6, then from the slope
u(x,
7)
2 0.
But
u(x,
T)
5
0, hence
lirn z (x, t) = z(x, t), zn(x. t) > 0 n-
u(x,
T) =
0. Because z(x, t)
> 0
and
for sufficiently large n.
(u,,
zn) is the solution of Problem 4, so v (x, t) < 0 by (161, that is,
But
YINC Lung and TENZhen-Huan
430 ulI(x, T) < 0 t e r i s t i c of u(")(x) 0
2o
'< t .
T
for all u
through
on
(x,
-
As
T
(x, T)
E [ t o ,t o + 6 / f ' ( O ) ] , we c o n s t r u c t a c h a r a c -
i n t e r s e c t i n g t h e x - a x i s at p o i n t
2 xo.
6, x o ) , 5
We f i x one p o i n t
5.
Because
tl E ( t o ,t o + h / f ' ( D ) ) ,
t h e n t h e s l o p e of c h a r a c t e r i s t i c s a t i s f i e s
x - x
x - x
x - 5 , _ 0 <- ?
for
T >
O
-
f'(0)
-
tl. Hence un(x, T )
for all
[ t l , to + 6/f'(o)
T
< 0,
-T)
$ E ( t l , to + 6 / f ' ( 0 ) ) , t h e n
We Lake a
n.
5
=
u(x,
way as i n t h e p r o o f o f Lemma 11, we may t,ake a s u b s e q u e n c e of p o i n t s
6,
+
x
n
as
f'(0)
n
as
-f
(x, f3)
with slope
f'(O),
-
6, t h e n
b <
through
(En, B),
then t h e slope
f'(-T)), a s t r a i g h t l i n e through t h e point
B)
B c l o s e enough t o p o i n t
such t h a t
u
t h e n t h e two l i n e s i n t e r s e c t , ( s e e H'igurr 1 ) . P u t
rl)(tO - S/f'(O) + f'(-r)) b = f ' ( -fl(0)
Take
{un} and a s e q u e n c e
We c o n s t r u c t a downward s t r a i g h t l i n e t h r o u g h t h e
m.
(x, to + 6 / f ' ( O ) ) w i t h s l o p e
point
Tn t h e s m e
such t h a t
+ m,
We c o n s t r u c t t h e downward c h a r a c t e r i s t i c of tends t o
0.
A;i6/f'(o)
t o + 6/f'(O),
t h e downward c h a r a c b
t e r i s t i c s of
(cn,
and
6)
u
through
(x, t o + 6/f'(O))
----.-___ would i n t e r s e c t on
suffjciently large
n.
t > 0
for
< 0
value
0
such t h a t
f o r any
at p o i n t u 0 ( 5 ) -> 0
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)
0
6>
0.
FIGURE 1
5
in
(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,
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,
N1
t
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.
We may assume that
{zn}
converges to
Thus
uo(x) < 0.
x < y. Take a subsequence of
z at point
(x, t).
We obtain
zn)}
{(u,,
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
0
( E (x, x + 6), hence
u(”)(() 0
L
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)
2
6 > 0, such that uin)([)
0 for
5
E (x. x + 6). It is known that and that is
a
x <
5
u (x,
T)
< 0 for
T
10, t]
and Sufficiently large
u ( ~ )does not change its sign on interval (x,x+6). 0
generalized solution of equation (10) on domain < x + 6, 0 < 7 < t}
with the value on
T
= 0
R
and
un(t,T)
=
{(c,
T);
5
= x
as its initial
azn = value and boundary value, 0, so is the limit function at
Therefore
n,
I>((,
T).
Therefore
Y I N G Lung and TENZhen-Huan
432
( 5 , T)
through any point
in
fi we can always construct a downward character-
istic of u . If it intersects line
6 = x,
then
11 <
0. We can continue this
5'.
characteristic to the x-axis. Let the intersection point be E;'
-E
u(S,
(xo, xo + h). Because if 5' E (x0, xo + h), then u ( 5 , T ) =
0. If
(x, t ) 0
is on this characteristic, then
is at the left side of it, then, by
5'
Therefore, all contradict
5'
Let and
0, hence
= xo. If
< t,),u(E,
T)
(x, to)
20
If
6' 2
is
5 x,.
(x,, xo + h).
be the intersection point of characteristic through point
and the x-axis.
If
(vTl
2
(x, t o ) is at the right side of it, then by Lemma 11, 5'
impossible. If
Let
u ( x , Tl) < 0
5'
T)
By Lemma 13,
(5,
T)
xo + h, then from the slope of it we get
5 ' 5 xo, from the slope we get
( 5 , T)
-+
(x, t0 ) , then the right hand sides have
the limit
(f')-
X-x -h ( 0 7 ) 0
= 0 respectively. Therefore there is a neighborhood of
(fI)-'(F) 0
(x, to) such that (
then a discontinuity curve
' ).-1(5 - xo - ) 2 T
5
= x(T)
-28 < 0,
through point
(x, to) is generated
[?I.
By the Rankine-Hugoniot condition we know the slope of this discontinuity is
The points which lie at the neighborhood of the line
5 - x
= fl(O)(T
-
to)
(17)
433
Hyperbolic Model of Combustion a r e at t h e r i g h t s i d e of
u ( ~ T,
)
x ( T ) , h e n c e t h e v a l u e s of
u
on t h e s e p o i n t s s a t i s f y
5 -2e.
Now we p r o v e
xo
# yo.
If n o t , t h e n l i n e
(17)would c o i n c i d e w i t h l i n e
b e c a u s e t h e y i n t e r s e c t t h e x - a x i s a t t h e same p o i n t and t h e i r s l o p e s a r e e q u a l .
5
But when
( x , x + 6), a t t h e n e i g h b o r h o o d of l i n e ( 1 7 ) we a l w a y s have
u ( c , T)
a
on l i n e (181, t h i s i s a c o n t r a d i c t i o n .
< -28, b u t u E 0
F i n a l l y , we p r o v e t h e e x i s t e n c e t h e o r e m .
Theorem 2 .
$(uo(x))
If
and
z O ( x ) a r e f u n c t i o n s w i t h bounded v a r i a t i o n and $
H y p o t h e s i s A h o l d s , t h e n t h e s o l u t i o n s o f Problem Q e x i s t , and T h e r e are at most c o u n t a b l y many points
Proof.
B y Lemma 14, t h e r e i s a t most o n e
of Lemma 13.
?uch t h a t t h e c o n d i t i o n o f Lemma 1 2 holds. many s u c h p o i n t s
thpn
TIL,
x.
We t a k e
u (x) 0
uF)(x)
p r o v e t h e r e i s a t most one
I f n o t , t h e r e would b e
assume t h a t T
t > tl.
& [ 0 , t o ) ,t h e n
to
t
t
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
X
0
[0, +-),
u(x, t ) , z(x, t ) z(x, t )
of
also s a t i s f y
Taking an a r b i t r a r y
+@), we
x E
satisfying
and
tl
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
5
x
(x, t ) p l a n e .
v(x, t ) = 0.
i n t h e case of
x
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.
0
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
T
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
BV
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
u
- iu
t
xx
+ fi/uIpu =
o
(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)
ti
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
Q
T
= {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
<
t
<
u,(x)
TI for t h r systems ( 1 . 3 ) of i s a n i n i t i a l v e c t o r valued
f unct i o n .
Let u s t a k ? t h e approximate s e m i l i n e a r p a r a b o l i c system
ut
whcre
E
-
A(t)uxx
-
EU
xx
(1.5)
= f(u)
> 0. F i r s t l y we e s t a b l i s h t h e s o l u t i o n s f o r t h e problem (1.5), ( 1 . 4 ) .
And next we g e t t h e s o l u t i o n f o r t h e d e g e n e r a t e problem ( 1 . 3 ) , (1.4) by p a s s i n g t o l i m i t as
E
+
0.
[a],
A s a consequence of t h e r r s u l t , i n
we have t h e f o l l o w i n g lemma f o r t h e
c a s e of t h e l i n e a r p a r a b o l i c systems.
___ Lemma 1.1.
Suppose t h a t f o r t h e l i n e a r p a r a b o l i c systems
ut - A(x, t)uxx
and t h e boundary c o n d i t i o n
(1) The
N
X
N
(1.6)
+ C(X, t ) u = f(x, t)
t)u
( 1 . 4 ) , hold t h e following assumptions.
coefficient matrices
u r a b l e and bounded and (2)
+ B(x,
A(x, t ) , B(x, t )
and
C(x, t )
are meas-
A(x, t ) i s p o s i t i v e l y d e f i n i t e .
The free t e r m v e c t o r v a l u e d f u n c t i o n
f(x, t)
i s quadratic integrable i n
QT.
( 3 ) The i n i t i a l v e c t o r v a l u e d f u n c t i o n
u 0 (x) b e l o n g s t o
wil)(o,
i).
437
Nonlinear Evolutional Systems
(1.6), ( 1 . 4 )
Then t h e boundary v a l u e problem
6 Lm( ( 0 , T ) ; WL”(0,
R) )
nWL2”)(QT),
has a unique s o l u t i o n
u(x, t )
satisfying the estimating relation
(1.7)
where
Kl
i s a constant.
Theorem 1.1. Suppose t h a t t h e c o e f f i c i e n t m a t r i x derivative matrix i.e.,
6
a f ( u ) of au
, (6,
Ts)
valued function
6 b(E,
E >
b
5)
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 <
holds, f o r a l l
L).
m
and t h e i n i t i a l v e c t o r
Then t h e b o u n d a r y p r o b l e m (1.5),
u(x, t ) E Lm!(O, T); Wil)(O,
I!))
( 1 . 4 ) has
nWL2”)(QT),where
0. The p r o o f of t h e e x i s t e n c e o f 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 )
on t h e f i x e d p o i n t t h e o r e m t r e a t m e n t as s i m i l a r l y u s e d i n [ 9 , 101.
is based
The method of
i n t e g r a l e s t i m a t i o n s f o r t h e p r o o f o f t h i s t h e o r e m i s v e r y similar t o t h e way o f t h e e s t i m a t i o n s needed i n t h e l i m i t i n g p r o c e ss o f t h e approximate s o l u t i o n s t o t h e s o l u t i o n o f d e g e n e r a t e problem. Thus we have a s e t o f a p p r o x i m a t e s o l u t i o n s a t e p r o b l e m (1.5), (1.4), where
&
f o r t h e nondegener-
> 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 ) }
x
uE
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
K2
wherc
f(0) = 0
is i n d e p e n d e n t o f
of
u
xx
>0
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)
K?
1 2.
When
i s a l s o indc-
L > 0.
p e n d e n t of Iri
E
order t o estimate t h e derivative
u
EX
(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,
L]
by p a r t s .
qdt. u x (
where t h e s y s t e m
2
in
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
x
2
)I1L 2 ( " ,
f(0) = 0 .
k)
(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
E
> 0
and
{uE(x, t )
[O,
a],
u
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
x
in interval
we have
x = 0, L
On the lateral boundaries
u
(0,
xx
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 .
C1
where E
> 0
C2
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 . -
f(u)
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)}
where
Also
uo(x) E. WL2)(0,
e).
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
is i n d e p p n d e n t of
K4
E
> 0
and
(1
> 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
E
> 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
i
+
-
,
E.
that, and
-f
0,
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-
1
ly c o n v e r g e n t t o
E
{ 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
1
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 ) } .
1
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
0.
(1.14)
Nonlinear Evolutional Systems t e n d s t o z e r o as
E.
+
0.
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
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 ;
0
<
t
<
TI
f o r t h e s y s t e m ( 1 . 3 ) of
Schrudingrr type.
Theorem 1 . 3 . of
P. > 0 by
Suppose t h a t a l l c o n d i t i o n s of Lemma -,
1 . 4 hold with t h e replacement
Thrn t , h c boundary v a l u e p r o b l e m ( 1 . 1 6 ) i n
( i . 3 ) o f S c h r a d i n g e r t y p e has a u n i q u e g l o b a l s o l u t i o n
wh2)(o, - ) ) nwL1)((o,
Q" T f o r t h e system
u ( x , t,) € Lm((O,
T);
T); ~ ~ ( m0) )..
By t h e similar way i t i s n o t d i f f i c u l t t o o b t a i n t h e c a l s s i c a l g l o b a l solut i o n s and t h e smo0t.h g l o b a l s o l u t i o n s o f the boundary v a l u e p r o b l r m and
52.
(1.16) i n
(1.4) in
QT
f o r t h e n o n l i n e a r systPms o f t h e g e n e r a l i z e d S c h r 6 d i n g e r t y p e .
S;istems o f P s e u d o - p a r a b o l i c Type. H e c e n t l y many a u t h o r s havp p a i d g r e a t a t t e n t , i o n t o t h e s t u d y o f t h e l i n e a r
and n o n l i n e a r p s e u d o - p a r a b o l i c e q u a t i o n s .
The n o n l i n e a r p s e u d o - p a r a b o l i c
equa-
t i o n s o f t e n o c c u r i n p r a c t i c a l r e s e a r c h , s u c h as t h e s o - c a l l e d BBM e q u a t i o n s
ZHOUYu-Lin
442
f(dX = Uxxt
u + t
(2.1)
f o r long waves i n n o n l i n e a r d i s p e r s i o n s y s t e m s [11-131,t h e e q u a t i o n s i n t h e c o o l i n g p r o c e s s a c c o r d i n g t o t w o - t e m p e r a t u r e of h e a t c o n d u c t i o n , t h e e q u a t i o n s f o r f i l t r a t i o n o f f l u i d s i n t h e broken r o c k s , t h e e q u a t i o n s o f Sobolev-Galpern t y p e a n d s o f o r t h [14-z"1. These e q u a t i o n s c o n t a i n t h e d i f f e r e n t i a l o p e r a t o r
u
t
-
u
XXt
as t h e i r mran p a r t .
Some f a i r l y g e n e r a l f a m i l y o f n o n l i n e a r pseudo-
p a r a b o l i c s y s t e m s [21y 22', which c o n t a i n a l l above m e n t i o n e d e q u a t i o n s a s s i m p l e s p e c i a l c a s e s a r e c o nsid e r e d , such a s t h e systems ut + (-l)'Au
= f ( u , ux, X2Mt
..., u 2M )
(2.2)
w i t , t i t h e s p e c i a l form o f thr r i g h t hand p a r t
M
m m+l f. = (-1) Dx m=l
M
aF
+
J
whcre
u
u;,,-~)
and
and
h(u)
j = 1, 2
h.(u), J
a r e N-dimensional v e c t o r v a l u e d f u n c t i o n s ,
G(u, ux ,..., u
X
)
X
M- 1
m = 0 , l,..,, M - l), and
A
a r e smooth f u n c t i o n s , is a
N x N
p. = u. J,m ,Ix'"
,...,
N,
F(u, u x . .
(2.3)
. .,
( j = 1 ,..., N ;
p o s i t i v e l y d e f i n i t e constant matrix.
The g e n e r a l i z e d global 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 o f t h e p e r i o d -
i c boundary problems and Cauchy problems f o r t h e s y s t e m s (2.2), ( 2 . 3 ) a r e o b t a i n e d ir1
[el]. I n t h i s s e c t i o n , f i r s t l y we a r e g o i n g t o c o n s i d e r t h e q u e s t i o n s o f a p r i o r i
e s t i m a t i o n f o r t h e l i n e a r p s e u d o - p a r a b o l i c s y s t e m s , t h e n we t u r n t o s t u d y t h e problems f o r n o n l i n e a r pseudo-parabolic systems. For t h e l i n e a r pseudo-paraboli c systems
l e t u s c o n s i d e r t h e boundary v a l u e problem
k(O, t ) = u k ( k , t )
U
X
= 0
( k = 0 , 1,..., M
-
l),
X
(2.5)
Nonlinear Evolutional Systems u
where
443
i s a N-dimensional unknown v e c t o r v a l u e d f u n c t i o n ,
N-dimensional q u a d r a t i c i n t e g r a b l e i n
9,
f(x, t)
vector valued function,
is a
Cp(x)
is a
N-dimensional i n i t i a l v e c t o r v a l u e d f u n c t i o n , s a t i s f y i n g t h e homogeneous b o u n d a r y conditions,
A(x, t )
is a
...,
B k ( x , t ) (k = 0 , 1, matrices.
2M),
N x N
symmetric p o s i t i v e l y d e f i n i t e m a t r i c e s ,
A(x, t ) a n d A,(x,
t)
a r e m e a s u r a b l e and bounded
.
Taking t h e s c a l a r p r o d u c t o f
u ?M
w i t h t h e l i n e a r system ( 2 . 4 ) and i n t e -
X
g r a t i n g t h e result,ing r e l a t i o n with respect t o
Since
A(x, t )
x
i n interval
[O,
a],
we g e t
i s a symmetric m a t r i x ,
From t h e i n t e r p o l a t i o n f o r m u l a e
(k = 0, 1,..., 2M)
and t h e r e l a t i o n
o b t a i n e d d i r e c t l y from t h e homogeneous boundary c o n d i t i o n ( 2 . 5 ) , t h e above e q u a l i t y becomes
444
ZHOUYu-Lin
On a c c o u n t o f t h e p o s i t i v e d e f i n i t e n e s s o f t h e m a t r i x
A ( x , t ) . we g e t t h e e s t i -
mation r e l a t i o n
Again making t h e s c a l a r p r o d u c t of v e c t o r
u
with t h e l i n e a r system xZMt
(2.4), we g e t t h e r e l a t i o n
Then i n t e g r a t i n g t h i s e q u a l i t y i n t h e r e c t a n g u l a r domain
Q,,
it f o l l o w s t h a t
where i n t h e d e r i v a t i o n , t h e boundary r e l a t i o n s
uk(O, t ) = u (a, t ) = O k x t x t have b e e n u s e d and
A(x, t).
a
( k = O , l , ..., M - 1 )
(2.9)
is t h e l e a s t eigenvalue of t h e p o s i t i v e l y d e f i n i t e m a t r i x
Hence t h e r e i s t h e e s t i m a t i o n r e l a t i o n
(2.10)
By means o f t h e method o f c o n t i n u a t i o n of p a r a m e t e r , it f o l l o w s from t h e o b t a i n e d a p r i o r i e s t i m a t i o n s ( 2 . 8 ) and ( 2 . 1 0 ) , t h e boundary v a l u e p r o b l e m ( 2 . 5 )
of t h e l i n e a r p s e u d o - p a r a b o l i c s y s t e m ( 2 . 4 ) h a s t h e s o l u t i o n i n t h e f u n c t i o n a l space
Wk”((0,
T); Wh2M)(0, a ) ) .
S i n c e t h e problem ( 2 . 4 ) , ( 2 . 5 ) i s l i n e a r , t h e
u n i q u e n e s s o f t h e s o l u t i o n i s a n immediate c o n s e q u e n c e o f t h e e s t i m a t i o n r e l a t i o n s .
445
Nonlinear Evolutional Systems Theorem 2.1.
Suppose that the linear pseudo-parabolic system (2.4) and the
boundary value condition ( 2 . 5 ) satisfy the following conditions. (1) A(x, t)
is a
N
X
N
symmetric positively definite matrix and is differen-
tiable with respect to t. (2) Bk(x. t)
(k = 0 , 1,..., PM), A(x, t)
and At(x, t)
are N
x
N measurable
and bounded matrices. (3) The N-dimensional vector valued function f ( x , t)
is quadratic integrable
in QT.
(4) The N-dimensional initial vector valued function y(x) 6 Wh2")(0, L) equals to zero together with the derivatives of order up to of segment
M
-
1 at the ends
[o, L].
Then the boundary value problem (2.4), (2.5) has a unique solution
L))
6 Z(QT) = Wbl)((O, T); W;2M)(0,
(X, t)
and the estimation relation (2.11)
holds.
8,the nonlinear
Let us turn to condiser in the rectangular domain
pseudo-parabolic system of partial differential equations
+
where u N x
N
and
g
are two N-dimensional vector valued function, A(x, t)
(2.12) is a
sysmmetric positively definite matrix with bounded derivative At(x, t)
with respect to t. Matrix for
g(x, t, u, ux ,..., u 2M-1) '
(x, t)
€
Q,
B(x, t, u , ux ,... , ux2M-1) is semibounded, i.e.,
and for any
uv
ux)''.'
such that for any N-dimensional vector
5
x2M-l 6 IRN E M
N
,
there is a constant b,
446
ZHOU Yu-Lin
Assume t h a t
g
i s a term of lower d e g r e e , which means t n a t
f O ( x , t ) E L2(QT) and
where
K4
i s a constant.
In order t o prove t h e e x i s t c n c e of t h e g l o b a l s o l u t i o n o f t h e homogeneous
boundary value problem ( 2 . 5 ) f o r t h e system (2.121, we t a k e t h e f u n c t i o n a l space
( 2M-1)
T); W,
G = L,((O,
(0,
L))
a s t,he base space f o r t h e f i x e d p o i n t theorem
treatment. For every
v & C,
u
w e c o n s t r u c t a N-dimensional v e c t o r v a l u e d f u n c t i o n
d e f i n e d as t h e unique s o l u t i o n o f t h e boundary v a l u e prohlem ( 2 . 5 ) f o r t h e l i n e a r pseudo-paraboli c system
+ Xg(x, t,, v ,
0
with a parameter
<
< 1.
Theorem 2 . 1 a r e a v a i l a b l e , so
The correspondence of t h e b a s e space v
c
G,
Z GG
G
t h e image
v
x
u ( x , t)
to
u
v 2M-1)
(2.15)
u(x, t,)
i s uniquely d e f i n e d and
d e f i n e s a f u n c t i o i i a l mapping
<
0
belongs t o
i s compact, f o r every
,...,
It can he e a s i l y seen t h a t a l l c o n d i t i o n s of
i n t o i t s e l f , where T v = u
vx
6 l
Z C G.
i s a parameter.
TA : C,
-+
G
of
For e v e r y
Since t h e imbedding mapping
Th : C
0 d h 6 1, t h e mapping
-+
2 GG
i s com-
p l e t e l y continuous. Let there are
be a bounded s e t of
M
T v = uA and
A
T-v = uA h
For any
G.
.
+ Aw
2 M t = XLl(x,t,v
,..., v
2 M - 1 ) ~ ~M X
M C
G
and any
The d i f f e r e n c e v e c t o r
f i e s t h e l i n e a r pseudo-parabolic systems (-l)Mwt
v g
X
0
<
-
A , A 6 1,
w = ux - ux satis-
441
Nonlinear Evolutional Systems w k(O, t) = w k ( k , t,) = 0 X
(k = 0, 1 , . . . , M - l),
X
w(x, 0 ) = 0. It f o l l o w s immediately that thc ei,t,imation (2.16)
holds,which means that for any bounded subset is uniformly continuous f o r When
X
= 0, for any
0 6
M
G, the mapping
of
Tx
: M
-f
C
X 6 1.
v E. G ,
T0v
= uo
is a fixed vector.
Now we turn to consider the a priori estimations of the solution of the boundary value problem (2.5) for the nonlinear pseudo-parabolic system
+ A(x,t)u x2Mt = X5(x,t,u,ux ,...)u 2M-1 )Ux2M
(-:)Mut
+
with parameter
0
<
Xdx,t,u,ux
,...,
(2.17)
2M-1)
6 1.
Taking the scalar product of the vector
u 2M
with the above system (2.17)
X
and integrating the resulting rclation in the rectangular domain we have
Since
5 is semibounded, there is
For the last term of the above equality, we have
Qt(O
<
t
<
T),
ZHOUYu-Lin
448
By t h e procedure similar t o t h a t used i n p r e v i o u s s e c t i o n , w e have t h e e s t i m a t i o n relation
where
K
5
i s a c o n s t a n t independent of t h r pnramPtPr (1s
X g 1.
It follow.: t h a t all
p o s s i b l e s o l u t i o n s of t h e n o n l i e a r problem ( 2 . 1 7 ) , ( 2 . 5 ) are uniformly bounded f o r 0 <
A <
1 i n t h e base space
( 2M-1) G = L m ( ( O , T); W, (0,
a)).
Therefore t h e boundary v a l u e problem ( 2 . 5 ) f o r t h e n o n l i n e a r pseudo-parabolic system ( 2 . 1 2 ) has at l e a s t one g l o b a l s o l u t i o n
u(x, t ) E W L 1 ) ( ( O ,
T I , WL2M)(0,k)).
Theorem 2.2. Suppose t h a t t h e n o n l i n e a r pseudo-parabolic system (2.12) and t h e boundary c o n d i t i o n s (2.5) s a t i s f y t h e f o l l o w i n g assumptions.
is a
(1) A(x, t )
derivative
A
t
N
X
symmetric p o s i t i v e l y d e f i n i t e m a t r i x and has bounded
N
( x , t ) w i t h r e s p e c t to t.
(2) B ( x , t , u , ux, ..., u 2M-1)
i s a semibounded m a t r i x valued continuous func-
X
t i o n of v a r i a b l e s
( x , t ) € QT
( 3 ) g ( x , t, u , ux,
..., u
and
2M-l)
u , ux,.
. ., u 2M-1 ' IRN
'
i s a N-dimensional v e c t o r valued continuous func
X
tion satisfying the relation
where
(4)
f o ( x , t ) E L2(&r) u,(x)
€
Wa2M1(0, e )
and
K,, i s a c o n s t a n t .
s a t i s f i e s t h e homogeneous boundary c o n d i t i o n s (2.5).
Then t h e problem (2.12),(2.5) has a unique g l o b a l v e c t o r valued s o l u t i o n
"WL1)((O, T ) ; WLzM)(O,
a)).
u(x, t )
Nonlinear Evolutional Systems
449
The uniqueness of t h e s o l u t i o n can b e o b t a i n e d by u s u a l e s t i m a t i o n o f t h e d i f f e r e n c e v e c t o r valued f u n c t i o n o f two g i v e n 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 . The r e s u l t s of 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 and t h e smooth g l o b a l solut i o n s o f t h e boundary v a l u e problems ( 2 . 5 ) f o r t h e n o n l i n e a r pseudo-parabolic system ( 2 . 1 2 ) can be o b t a i n e d by t h e similar way. I n t h e c a s e of t h e system
with s p e c i a l r i g h t hand s i d e (2.3), t h e boundary v a l u e problem ( 2 . 5 ) can b e d i s c u s s e d by t h e method used above.
S i m i l a r l y w e have t h e f o l l o w i n g r e s u l t .
Suppose t h a t t h e system (2.19), ( 2 . 3 ) and t h e boundary c o n d i t i o n s
Theorem 2.3.
(2.5) s a t i s f y t h e f o l l o w i n g c o n d i t i o n s . (1) A(x, t )
derivative (2)
is a
N
X
symmetric p o s i t i v e l y d e f i n i t e m a t r i x and has bounded
N
A (x, t ) with r e s p e c t t o
t
F ( p O , pl,
...,
pM-l)
is
spect t o a l l its variables
t.
M + 1 times c o n t i n u o u s l y d i f f e r e n t i a b l e w i t h repo, p1
times continuously d i f f e r e n t i a b l e .
,...,
pM-l
E
?!I
N
.
The Hessian m a t r i x
5
semibounded, i . e . , f o r any NM-dimensional v e c t o r
(m = 1,..., M; k = l,..,, N )
where
b
(3) h(u)
i s a c o n s t a n t for
G(pO, p1 H
6 RMN
,...,
pMbl)
is
of t h e f u n c t i o n
5
=
M
F
(cmk)
such t h a t
PO’ P1””.
PM-l€
N *
i s a N-dimensional continuous v e c t o r valued f u n c t i o n s a t i s f y i n g t h e
relation
where
C
(h) y(x)
and
d
a r e constants.
Wh2M)(0,
a)
s a t i s f i e s t h e homogeneous boundary c o n d i t i o n s ( 2 . 5 ) .
is
ZHOUYu-Lin
450
Then the problem (2.l9), (2,3), (2.5) has a unique global vector valued solution U(X, t)
43.
d
WL1)!(O,
k)).
TI; WL2"'(O,
Systems of Pseudo-hyperbolic Type. In the study of the practical problems in physics, mechanics, biology etc.,
such as the forced vibration of plane boundary layer, the transfer of the bioelectric signal in animal nervous systems, the linear and nonlinear equations with the principal part
A
utt - uxxt of pseudo-hyperbolic type often appear.
lot of authors have paid much attention to consider various problems for the linear and nonlinear pseudo-hyperbolic equations[23-281.
In [7,?9], the quasi-
linear systems of pseudo-hyperbolic type of higher order M utt + (-1) Au = f(x, t, u , ux )..., u M' Ut) x2Mt
(3.1)
with the special right hand side
[;IM f. =
m=O
are considered, where u
(-l)m+' :D
and
aF
(-)
aPmj
g(x, t, u , . . . ,
+ gj,
j = 1,
u M, u,)
2,..., N
(3.2)
are the N-dimensional
X
is a N
vector valued functions, A matrix, F(u, ux,.
~,
. ., u
)
N symmetric positively definete constant
is a smooth non-negative function and
g
is the
[ 21 X
term of lower degree.
The generalized global solutions and the classical global
solutions of the periodic boundary problems and the initial value problems for the systems (3.1) with special right hand side (3.2) are obtained.
In this section we are going to consider the boundary value problems for the nonlinear systems of pseudo-hyperbolic type of higher order.
First of all we
will talk about the linear case for the use of further investigation. Suppose that in the rectangular domain hyperbolic systems
&r,
the general linear pseudo-
45 1
Nonlinear Evolutional Systems
and t h e boundary v a l u e c o n d i t i o n s
(k = 0, 1,..., M - l),
u k(O, t) = u k ( L , t) = 0 X
X
A (x, t )
Here we assume t h a t t h e c o e f f i c i e n t m a t r i c e s
%.
l,.,., 2M) a r e measurable and bounded i n d e f i n i t e matrix and bounded d e r i v a t i v e
B (x, t)
BOt(x, t)
with r e s p e c t t o
f(x, t)
v e c t o r valued f u n c t i o n s
[O,
is a N
AO(x, t )
t.
N
X
(k = 0 , positively
The N-dimensional f r e e term
%.
i s quadratic integrable i n
L)
"(x) € WL2")(O,
and
$(x)
€
M
v a n i s h t o g e t h e r w i t h t h e i r d e r i v a t i v e s of o r d e r up t o interval
k
i s a symmetric p o s i t i v e l y d e f i n i t e m a t r i x having
0
v e c t o r valued f u n c t i o n
B (x, t )
and
k
The i n i t i a l
W P ' ( 0 , !L)
-
and t h e y
1 at t h e ends of t h e
~1 .
Taking t h e s c a l a r product o f t h e v e c t o r and i n t e g r a t i n g i n
Qt(O 6 t
<
u and t h e l i n e a r system ( 3 . 3 ) x2Mt
T), we g e t
''
2M
(-l)'Ij
)dxdt
Qt(ux2Mt, u t t ) d x d t + k=O"'4 t (' x 2Mt ' 'irux2M-kt
(3.5)
Using t h e boundary r e l a t i o n s
u
x t
2(0, t) =
u X
2(L, t) = 0
,...,
(k = 0, 1
M
-
1)
t
o b t a i n e d d i r e c t l y from t h e homogeneous boundary c o n d i t i o n s ( 3 . 4 ) , we have
(3.6)
ZHOUYu-Lin
452 Since AO(x, t)
is positively definite, there is . a
Because B (x, t) 0
0, such that
is symmetric positively definite and BOt(x, t)
is bounded,
there is
If BOt (x, t) is a non-positively definite matrix, B0(x, t) can be non-negatively definite. Hence it can be proved that the equality (3.5) may be replaced by the inequality
By the similar way, we can obtain the following theorem. Theorem 3.1.
Suppose that the linear pseudo-hyperbolic system (3.3) and the
boundary value problem (3.4) satisfy the following assumptions. (1) AO(x, t)
is a
N
x
N positively definite matrix.
(2) BO(x, t)
is a
N
X
N
symmetric positively definite matrix. When
Bo (x, t)
t is non-positively definite, B (x, t) is symmetric non-negatively definite. 0
(3) Matrices %(x,
t), Bx(x, t )
measurable and bounded in
,..., 2M)
(k = 0, 1
and
Bo (x, t) are ali t
”r.
(4) f(x, t) is quadratic integrable in Q,. (5)
( p ( x ) € Wh2M)(0,
a ) , $(x)
6 Wy)(O,
a ) and they vanish together with all
453
Nonlinear Evolutional Systems of their derivatives of order up to M
-
1 at the ends of the interval
[0, k . ] .
Then the boundary value problem (3.4) for the linear pseudo-hyperbolic system
(3.3) has a unique generalized global vector valued solution u(x, t)
-
wm ("((0,
a))
T); W r ' ( 0 ,
Wil)((O, T); W h 2 M ) ( 0 ,
€
Z
a)).
k . ) ) n W h 2 ) ( ( o , TI; L2(0,
There is estimation
(3.7)
N o w we turn to consider a nonlinear pseudo-hyperbolic system
(-l)Mutt + A(x,t,u,ux ,..., u 2M-l 'Ut ,Uxt9 . X
. - ,u
M-1 )' 2 M t x t
x
,...,
Suppose that A :A(x, t, P. q ) 5 A(x, t, p0, P~,..., P ~ ~ qo, - ~q1, is a positively definite matrix valued function of variables 3M vector variables dimension N.
...,
pk (k = 0, 1,
2M
-
1) and
(x, t)
'M-1
&r
qh (h = 0, 1,..., M
)
and
-
I) of
For the sake of brevity we assume that N-dimensional vector valued
function f(x, t, p. q) means that for (x, t)
€
is the term of lower degree of the system (3.8), it and
p 6 IR2MN,
MN q E. IR ,
(3.9) where
fo(x, t)
€
L2(&r).
We construct the solution of the boundary value problem (3.4) for the nonlinear pseuto-hyperbolic system (3.8)also by the method of the fixed point technique. Let us take G = L,((O,
T); Wm
454
ZHOUYu-Lin
as the base space. Now we define the functional mapping space into itself with parameter
0
of the base
G
+
as follows. For every
A 6 1
Q
TA : G
v
E
G, we
take u = TAV satisfying the following linear pseudo-hyperbolic system (-l)Mutt + A(x, t, v, vx ,..., v 2M-1’
Vt’ Vxt,.
X
= Af(x, t, v, vx,..., v *M-l’
.
v M-1 ‘ ) 2M x t x t
*,
(3.10)
Vt’ VXt’..., v M-lt )
X
and the boundary conditions (3.4). By the compactness of imbedding Z L it can easily seen that for every
0
<
A
continuous and for any bounded subset M uniformly continuous with respect to 0
G
<
and the estimation relation ( 3 . 7 ) ,
TA is completely
1, the mapping
of
TA
G, the mapping
:
M
+
G
is
6 1.
In order to prove the existence of the solution for the problem (3.8),(3.4), it remains to get the uniform boundedness for
A in G
of all possible solu-
tions of the boundary value problems for the nonlinear pseudo-hyperbolic system with parameter
0
< A
d 1
and the system (3.11) and then intex2Mt grating the resulting reaction in Qt(O < t 6 T), we can obtain that all possi-
Making the scalar product of vector
ble solutions of problem (3.11), (3.4) are uniformly bounded in G
for
O Q A 6 l .
Theorem 3.2. Suppose that the nonlinear pseudo-hyperbolic system (3.8)and the boundary value prcblem (3.4) satisfy the following assumptions. (1) A(x, t, po, pl, ..., p2M-l, qo, ql, ..., qM-l)
is a
nite matrix valued continuous function of variables vector variables p k ( k = 0, 1,..., 2M
-
1) and
N
X
(x, t)
N
Q,
6
...,
q (h = 0, 1,
h
positively defiand
3M
M - 1) of
dimension N. (2)
f(x, t, PO’ Pl, ..., p2M-1, qo, ql, ..., qM-l)
is a N-dimensional vector
Nonlinear Evoiutional Systems
455
valued continuous function, having the properties (3.9).
q(x)6
(3)
Wb2M)(0, 8 ) and
$(x)
€ WP'(0,
e)
are two N-dimensional initial
vector valued functions, vanishing together with their derivatives of order up to M
-
1 at the ends of the interval [O, II].
Then the problem ( 3 . 8 ) , (3.4) has a unique generalized global solution u ( x , t) E
z.
Similarly we can obtain the results for the boundary value problems of the pseudo-hyperbolic systems (3.1) with special right term (3.2). Also by the similar methods, we may obtain the results for the classical and smooth global solutions for the above mentioned problems.
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