Partial Differential Equations And Their Applications - Proceedings Of The Conference 9810240597, 9789810240592

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Partial Differential Equations and Their Applications

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Proceedings of the Conference

Partial Differential Equations and Their Applications

W u h a n , China

5 - 9 April 1 9 9 9

Editors

Chen Hue School of Mathematics, Wuhan University, China

Luigi Rodino Dipartimento di Mathematica, Universita di Torino, Italia

V | f e World Scientific »■

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Proceedings of the Conference on PARTIAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4059-7

Printed in Singapore by Regal Press (S) Pte. Ltd.

V

Preface

The Conference "Partial Differential Equations and Applications" was held in Wuhan, People's Republic of China, from the 5th to the 9th of April 1999. First event for the Conference was the endorsement of the Agreement of Cooperation between the National Natural Science Foundation of China (the NNSFC for short) and the National Council of Researches of Italy (the CNR for short), in the field of the PDE; the Conference in Wuhan was in fact planned as a first act of a four-year program of scientific exchange between the two countries. Other Institutions from Europe, Asia and Australia, also contributed to the success of the meeting, giving an international character to lectures and audiences. On the whole, the Conference was able to record the considerable progress realized recently in the field of the PDE, with particular attention to those themes which got pre-eminence, because of their novelty or importance in the applications. Contributions were presented in the following topics: — Linear and nonlinear microlocal analysis: hypoellipticity, solvability and propagation of singularities for equations with multiple characteristics, hyperfunction theory, Weyl-Hormander calculus, equations on manifolds with edges and corners, Cauchy problem for non-Kovalevskyan equations, Gevrey solutions. — Nonlinear hyperbolic equations and systems: existence of solutions, for­ mation of shocks, blow-up phenomena. — Nonlinear elliptic equations: Hessian measures, monotonicity methods, minimal surfaces, curvature equations, nonlinear Dirichlet problems. — Spectral theory: irregular drums, Schrodinger operators. — PDE and Applied Mathematics: nonlinear parabolic equations (existence of solutions, singular solutions, blow-up), reaction-diffusion equations (existence of wave front, traveling waves), Navier-Stokes equations.

vi The lectures were given by 59 lecturers. The conference was attended by 73 participants: 54 from China, 5 from Italy, 6 from Japan, 8 from other countries (1 each from America, Australia, Belgium, Prance, Germany, P.D.R. Korea, Philippines and Romania). This aiming at a state-of-the-art volume, presents a selection of the aforesaid lectures. Due to the space constraint, not all the contributions, which throughout the meeting testified an impressive research activity in the fields, can be published. Besides thanking the NNSFC and the CNR for financial support, we want to express our gratitude to Wuhan University for the facilities provided for the conference. We also thank Dr. Liu Weian, Dr. Chen Wenyi and Dr. Tian Guji who organized the conference efficiently. Finally, we wish to express our warmest thank to Prof. Zhang Yinglan of the NNSFC for her support.

The Editors CHEN Hua Luigi RODINO

VII

Contents

Preface

v

Bahouri, H., Chemin, J.-Y. and Xu, C.-J. Trace Theorem in Sobolev Spaces Associated with Hormander's Vector Fields

1

Chen, H. and Sleeman, B. D . On the Asymptotics of the Counting Function for Irregular Drums

15

Chen, S. X. Formation-Construction of Shock in Compressive Simple Wave of N x N Hyperbolic System

32

Chen, W . Y . Diophantine Characterization of Hypoelliptic Vector Fields on the Torus T 2

44

Cicognani, M. and Zanghirati, L. Well Posedness of the Cauchy Problem for Nonlinear Weakly Hyperbolic Equations

51

Garello, G. and Rodino, L. Nonlinear Microlocal Analysis

67

Godin, P. Blow-up of Solutions of Semilinear Hyperbolic Equations in One Space Dimension

82

Hong, J. X. Boundary Value Problems for Isometric Embedding in R3 of Surfaces

87

Igari, K. Propagation of Singularities in the Ramified Cauchy Problem

99

VIII

J i a n g , L. and Dai, M . Convergence of Binomial Tree Method for American Options

106

J i n , EL, Li, Z. Y . and Ye, Q. X . Traveling Wave Front Solutions for Reaction-Diffusion Systems

119

Komatsu, H. Convolutions of Hyperfunctions of One Variable and Laplace Hyperfunctions

130

Lei, Y . T . and W u , Z. Q. Asymptotic Behaviour for Minimizers of a Ginzburg-Landau-Type Functional

146

Liess, O. E . Fourier Transforms in Spaces of Hyperfunctions and Hartog's Type Phenomena

152

Liu, W . A., Li, J . F . and Lu, G. Fujita Phenomenon and Critical Exponents in Nonlinear Parabolic Equations

167

Liu, Y., Li, Y . and D e n g , Y . B . Stability of a Semilinear Cauchy Problem

180

Luo, X. B . Removable Singularities Theorems for Solutions of Quasi-homogeneous Hypoelliptic Equations

200

Ma, L. and A n , Y . L. The Maximum Principle and the Yamabe Flow

211

Miyake, M . Borel Summability of Divergent Solutions of the Cauchy Problem to Non-Kowalevskian Equations

225

Morimoto, Y. and Morioka, T. Hypoellipticity for Elliptic Operators with Infinite Degeneracy

240

Qi, M . Y . Hadamard's Fundamental Solution and Conical Refraction

260

IX

Reissig, M. and Wang, Y . G. Analysis of Weak Singularities of Solutions to 1-D Thermoelasticity

272

Schulze, B.-W. Iterated Asymptotics on Corner Manifolds

283

Tahara, H. Uniqueness of the Solution of Nonlinear Singular Partial Differential Equations

297

Yin, J. X. and Wang, Y . F. Asymptotic Behaviour of Solutions for Nonlinear Diffusion Equation with Periodic Absorption

305

Zhou, M. Quantization on the Symplectic Leave of SU(n) and *-Representations of the Quantum Group SUq(n)

309

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1

Trace theorem in Sobolev spaces associated with Hormander's vector fields H. BAHOURI Universite de Tunis, Departement de Mathematiques 1060 Tunis, Tunisie E-mail: Hajer.BahouriQfst. rnu. tn J.-Y. CHEMIN Analyse numerique, Tour 55-65, 5 s etage, Universite Pierre-et-Marie-Curie, 4, place Jussieu, 75230 Paris Cedex 05, France C.-J. XU Universite de Rouen, UPRES-A6085, Mathematiques 76821 Mont-Saint-Aignan, France E-mail: [email protected] et Institut de Mathematiques, Universite de Wuhan, Wuhan, 430072, R. P. Chine

1

Introduction

Let Q, C R d , d > 2 be an open domain, and P = ( P i , • • •, Pn) a smooth real vector fields system. P satisfies t h e Hormander condition of order 2, i.e. rank{Pi,---,P n ,[Pfc,Pj],fc,.7 = 1, • • • , n } = d. For k G N , we define Hk{Sl, P) = {fe

L2(n);PJf

e L 2 (ft); \J\ < A;},

where P J = Ph •••PjnJ = (ju . •., j E ) , \J\ = l. Denote by X the C°°-module of vector fields spaned by { P i , • • •, P n } . Let E C ft be a smooth surface, E non-characteristic for the vector fields system P , i.e. for all x € E, Xx is not included in T X E. XY, denotes the subspace of T E defined by A^i = T x E f l X \ x . We suppose t h a t there is a generator system p E = { p f , . . . , p£} of XY, , then JVE, P f , • • •, P% satisfy also the Hormander's condition of order 2. We define F f c ( E , Pfc) = {/ e L 2 ( S ) ; P£f e L 2 ( E ) , \J\ < k).

2 We can prove the following trace theorem: T h e o r e m 1 Denote by 7 E the trace operator on E, then ^:Hk(n,P) is continuous

-

and surjective,

[fffc(E,PE)ntffc/2(E),L2(E)]1/(2fc). here Hk'2(E)

is the usual Sobolev

space.

We study now the characteristic case. We consider only the problems on the Heisenberg groups H ^ , which means t h a t we study the vector fields system P

Hd

=

iXJ

== d

*j + yjds>

Y

J

= d

Vj ~ xids>

3=

h'-,d}

on R 2 d + 1 . We take now E 0 = {(x,y,s) € R 2 d + 1 ; s = 0}, then, for the system of vector fields P J J , (0,0,0) is the only characteristic point, and (x,2/,0) is not characteristic if (x, y) ^ (0, 0). We have a family of radial vector fields on So K> = {xkdxj

+ Vjdyk; xkdVj

- xjdyk; ykdXj - yjdXk; j , k = 1, • • •, d } ,

which is the projection of P J J onto T*(Eo fl {|(z,y)| > e > 0}) parallel to a transversal vector fields. We define the function space on Eo: T 4 ( E 0 ) = {u € L 2 ( E 0 ) ; lllu

6 L 2 , / < 4; |(x, y) such that, for any X G R 2 n , / Y# H ) with Supp u C {p(x,j/,a-) < 1}, and consider a function V? e £({1/2 < |tf< 2}) satisfying

V 0 < | t | < l , £>(2*t) = l. p=0

We define the family (^ P ) p 6 N

and

( w p) P eN b ^

^ P (^,2/^) = ^ ( 2 M and up(x,y, s) = {pii) = uPy>P + p = (p'(2pp)2pPp. Since |Pp| < C, we have \\P( — 0, if \p—p'\ is big enough. Then almost orthogonality implies that oo

p=0

and the same argument as before gives oo

II«II^(H 1/2. T h e o r e m 4 Let m ( x i , x ' , ^ ) ™(*,£') < (£')•

Then

is also a gi i-weight S

M

be a gi i-weight

defined

on T * R d satisfying

(£')2


1/2, where m ( x ' , f ' ) = m(0, * ' , £ ' ) • First we have t h a t the function M is a p-weight and m is a g-weight. Proof of the continuity of trace operators for non-integer case is more complicated than integer case, we omit this proof here, and go ahead directly to the more interesting part, i.e. surjectivity of trace operators. T h e idea of the proof is to interpret the following classical formula in the frame of Weyl-Hormander's calculus. Let v 6 i J 5 ~ 2 ( R d - 1 ) j set

then

\\u\\Hs < C\\v\\Ha_i

and j(u)

Take a function x € ^ 0 ^ )

w

= v.

^h xM

=

1 n e a r 0>

we set

f°r

v

£ «S(E), (4)

JT*E

with /iy(^l) = C71m(y)25-1X(x1(77)2) /

cte^M-2'(0,y,ei,»7)^i.

7R Since M 2 ( 0 , y, f i , 77) = £ 2 + m2(y,rj), and x(0) = 1, then /x(0) = 1 and 7 o Rx = I d . We have to prove t h a t Rx is a continuous linear m a p from H{ms~i) S to H(M ). T h e key lemma is the following L e m m a 4 For all integers N, there exists C such that for all function f £ L 2 ( E ) , we have

II^MO^/Jlli.jR-)

< CA(Y,n(Y)) -N

m

2s-l

00

MS{Y)

11

where Jy is a function of real variable such that I Jy(r)dr yy > R JF

< - C

m2S-i{Y)

From this lemma, we can obtain immediately the continuity of Rx. fact, take / = 0yV, we have «X«IIL*(R-)

*Y

^ CM-*{Y)

f J l

(,>"i(i +

(v^r'JYim)

2J

x\\0y-v\\L^)A(Y^(Y))-Nm2s-1(Y)dY. Apply Cauchy-Schwarz inequality with measure A(Y,7r(Y))-Nm2s-1(Y)dY,

< CM~2°(Y) f

\\e^v\\l^)AiXMY))-Nm2s-l{Y)dY

JT*T.

x

/

(v)-Hl

(r,)ii21)-2J^m)A(Y,n(Y))-Nm2s-1(Y)dY.

+

JT*T,

By the definition of norm in H(MS)1 ll^xHllf(M-) < /

[

(ff\)'i

JT*E

f

with

\\0y-v\\l2mA(Y,Z)-Nm2s-\Y)dY,

F1(Z)= I F2(Z)=

F1(Z)F2(Z)dZ

(l +

(V)t2)-2j}(r)A(YiZ)-Nm2s-1(Y)dtdrdY.

JR.*

Since

/

(i + (n)t*)-*j*{T)dtdr