Advances in Nonlinear Partial Differential Equations and Stochastics
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Advances in Nonlinear Partial Differential Equations and Stochastics
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Series on Advances in Mathematics for Applied Sciences - Vo.. 48
Advances in Nonlinear Partial Differential Equations and Stochastics
Editors
S. Kawashima Kyushu University, Japan
T. Yanagisawa Nara Women's University, JapaJ
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-PubUcation Data A catalogue record for this book is available from the British Library.
ADVANCES IN NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND STOCHASTICS Series on Advances in Mathematics for Applied Sciences — Vol. 48 Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved This book, or pans thereof, may not be reproduced in any form or by any means. electronic or mechanical,ncluding photocopying, recording or any information storage and retrieval system now known or to be invented, wiihout 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 981-02-3396-5
This book is printed on acid-free paper.
Printed in Singapore by L)to-Prim.
V
Preface This volume is dedicated to Professor Yasushi Shizuta on the occasion of his sixtieth birthday in 1996. All of the articles presented here by invited speakers at the conference on Nonlinear Partial Differential Equations and Stochastics held at Nara Women's University on September 21 and 22 in 1996, which was organized by Reiko Sakamoto, Sadao Miyatake, Shuichi Kawashima and Taku Yanagisawa. Professor Shizuta recieved his doctoral degree in 1965, under the direction of Professor Hideki Yukawa, from the Department of Physics of Kyoto Univer sity. In the doctoral thesis he studied spectral theory to fundamental equations in neutron thermalization theory. In 1962, he joined the Faculty of Tech nology of Kyoto University, and then in 1965 he transferred to the Research Institute for Mathematical Science of Kyoto University. In 1971 he was appointed Associate Professor of the Department of Mathematics at Nara Women's University and was promoted to Professor in 1976. The scientific work of Professor Shizuta concerns mainly with the existence, uniqueness, and asymptotic behavior of solutions of nonlinear partial differential equations appearing in mathematical physics such as the Boltzmann equation, discrete velocity models of the Boltzmann equation, the Navier-Stokes equation of viscous compressible fluids, and the equation of electromagnetic fluids. Of particular significance is his paper, "On the classical solutions of the Boltzmann equation", of which the basic idea of re duction of the Cauchy problem by perturbation methods to the linearized problem and general theorems about the asymptotic expansions of perturbed semigroups have been applied to various other problems. He has also con tributed to the study of hyperbolic-parabolic coupled systems and the discrete Boltzmann equations by the important work on the characterization of the stability condition. Recently he concentrated his efforts on studying the mixed problems of symmetric hyperbolic systems with characteristic boundary. Through active discussions with Professor Shizuta, many people have benefited much from his strict insight and keen sense to mathematics. He is also an excellent teacher and has passed on his sterling qualities to a lot of students. All of us admire the success he has acheived in mathematical research and education, and sincerely wish that he will develop further his vitality in reseach and in other activities. Skmchi Kawashima Taku Yanagisawa
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VII
Contents Preface
v
Mathematical Aspects of Supersonic Flow Past Wings S. X. Chen
1
The Null Condition and Global Existence of Solutions to Systems of Wave Equations with Different Speeds R. Agemi and K. Yokoyama Scaling Limits for Large Systems of Interacting Particles K. Uchiyama Regularity of Solutions of Initial Boundary Value Problems for Symmetric Hyperbolic Systems with Boundary Characteristic of Constant Multiplicity Y. Yamamoto
43
87
133
On the Half-Space Problem for the Discrete Velocity Model of the Boltzmann Equation S. Ukai
160
Blow-up, Life Span and Large Time Behavior of Solutions to a Weakly Coupled System of Reaction-Dfffusion Equations K. Mochizvki
175
On a Decay Rate of Solutions to One-Dimensional Thermoelastic Equations on a Half Line; Linear Part Y. Shibata
198
Bifurcation Phenomena for the Duffing Equation A. Matsumura
292
Some Remarks on the Compactness Method A. V. Kazhikhov
319
Percolation on Fractal Lattices: Asymptotic Behavior of the Correlation Length M. Shinoda
331
1
M A T H E M A T I C A L A S P E C T S OF S U P E R S O N I C FLOW P A S T
WINGS
C H E N SHUXING
Institute of Mathematics, Fudan University Shanghai, 200433, China
§1. Introduction. This paper is devoted to the mathematical theory of supersonic flow past wings. Obviously, the theory has considerable important applications in physical science and engineering science, particularly in aviation and space eight. The first sys tematic discussion on the nonlinear theory of supersonic flow is due to R.Courant and K.O.Friedrichs. Their famous book "Supersonic Flow and Shock Waves" [11] has been highlighting this field for more than half century, and provides basic knowledge and viewpoint for most works in this field, as well as the basis of the discussion in this paper. For readers convenience, in our discussion we will recall some relevant concepts and formula in [11]. While we refer readers to the book or other bibliography for more details. When a supersonic projectile moves in the air, a shock front will appear ahead of it generally. If the projectile has a blunt head, then the shock front is usually dettached. If the head of the projectile has a sharp point head or sharp edge, the shock front may be attached, depending on the shapeness of the head. Such phe nomena are observed in physical experiments for both wing and cone-shape body. According to the principle of relative motion, we may consider the supersonic Sow past a given bodv instead of considering the motion of projectile in stationary air Mathematically,' the first task in studying such problems is to prove the existence of" tlie solution in some neighbourhood of the head. The solution satisfies the system describing the motion of fluid in the whole neighbourhood in classical sense having junips on a surface which is iilso to be determined *md called shock front. Because thT problem often involves m^my troublesome factors such, ™*™^Sfnlta
taSfcSJ^concentrated
in one so ace dimensional rase As for
2 recent results in M-D case obtained by author. Meanwhile, some new ingredient is also included. In this paper we always assume that the coming flow is uniform supersonic and have direction horizontal from left to right, while the flow is assumed to be inviscid, and dominated by Euler system. Moreover, for simplicity we only con sider the motion of perfect gas, satisfying the state equation p = A{S)p1 , where p is pressure, p is density, A{S) is a given function of entropy S and 7 is adiabatic exponent. In what follows our attention is concentrated on the steady inviscid supersonic flow past wings The local existence of solution in some different cases will be carefully discussed.. 1.1 Supersonic Flow Past a Wedge The simplest model of wing is wedge, which is formed by two planes. Assume that the edge of the wedge is perpendicular to the direction of the coming flow, then the picture of the motion is uniform in each plane perpendicular to the edge. Therefore, two independent variables are enough to describe the motion. When the angle between the plane and the coming flow is not large ( more precisely, it is less than a critical value described later ), then the flow on the upper part and on the lower part can be determined independently. Take upper part into account we may determine an plane oblique shock issuing from the edge of the wedge, while behind the shock the flow is still uniform and parallel to the surface of the wedge. The location of the shock and all parameters behind the shock can be determined explicitly by Rankine-Hugoniot relations as follows. Denote by u, v the components of the velocity of flow, by p, p the pressure and density, by a the sound speed, and use the subscripts "0" or " 1 " to represent the states ahead or behind the shock front respectively. Then the Rankine-Hugoniot relation can be written as N0p0 = NlPl (1.1) PO&Q +PO = PIN?+PI LB 1
2
■
= £1 1
2% + ! o = ^ 1
(1.2)
(1.3) + I
■
i
,
(14)
,
where N, L stand for the normal and tangential component of velocity with respect to shock front, 90 is the speed of coming flow ahead the shock, i stands for the enthalpy equal to ^ a 2 . The entropy condition is Pl > p0. If the shock front is a plane with an angle of inclination 0, then (1.1)-{1.4) implies (see [11]) ui = (1 - ^ 2 )<Jo(sin 2 0 - sin 2 Aa)
(1.5)
3 "i = (Qo - Ui)cot/J
(1.6)
with 0 as parameter, A0 is Mach number and fi? = *£. (1.5),(1.6) indicate a relation between ui and v\% which has a graph on u, J plane called shock polar. It means that any state behind shock must locate on the shock polar in order to connect with the state (go,0) by an oblique shock. Eliminating the angle 0 from equation (1.5),(1.6), we find
V=
{qo u)
~ \[^
(1.7)
where K = (p2 + (1 - p.2) sin2 A0)q0 U = (l + (1-
(i2)sm2Ao)q0
Moreover, (1.1)-(1.4) yield Pi = Po + PoQo{Qo~ ai) =
^
=
iVi
(1.8)
^ pogoan 0 u - 5o^ J cos J £f
tan/?=^^
(1.10)
The picture of shock polar ( upper half ) is shown in Figure 1. The curve has a unique self-intersection at P0 ! (so00). By using the shock polar the problem of determining a supersonic flow past a wedge can be readily solved. Graphically, drawing a ray I starting from the origin O with slope angle 0, we get three intersections of t with the shock polar, provided 9 is less than a critical value. Among the three intersections the middle one, denoted by P, can be chosen as the state behind shock front. Then connect the points P0P by a straight line, and starting from O draw a straight line perpendicular to P0P, we obtain an intersection D. The angle ZP0OD is just the angle of inclination of the required oblicme shock By this angle we obtain the oblique shock attached on the edge of the wedge. Correspondingly, the state behind the attached shock is also uniform, then ui,wi,Pi,/>i can be obtained from (1.5),(1.6),(1.8) and (1.9). The whole above-mentioned process can be translated to a process of solving a series of algebraic equations. Moreover, make a line starting form the origin and tangential to the shock polar and denote the angle of inclination of this line by 9EXt. Obviously, when 8 < 6elt , the ray ( will only intersect with the shock polar at one point , where the radius is greater than q0. So the above process doesn't
4 work in this case. The angle $ext is called critical angle, which is determined by the parameter of coming flow. Above analysis tells us that an attached oblique shock exists for the problem of supersonic flow past a wedge only in the case that the angle 6 of the surface of the wedge with the coming flow is less than the critical angle. One question is remained. Why we should take the middle one among three intersections of t with the shock polar as a reasonable solution? First, the farthest one P' from the origin corresponds to a speed larger than q0- Therefore, if we took it as a state behind the shock, the entropy condition would be violated. To exclude the nearest one P" we need another physical principle, which says that among two possible states satisfying all Rankine-Hugoniot conditions and entropy condition, only the state corresponding to the weak shock is stable and can actually happen, while the other one corresponding to the strong shock is unstable and cannot haooen ^From mathematical point of view behind the strong shock the system describing the flow would have two complex characteristics. Then the development of small perturbation on the shock can be described by a,n initial boundary value problem of elliptic equation. Such an initial boundary value problem does not have Linmneness and stability neither Meanwhile in the case of weaker (xtz) -f{x,z)' 4>{x,z) -
^,z }{x,z)
=z
(2.1)
By this transformation the boundary y = f(x,z) and y = <j>(x,z) ara transformed to a = 0 and fi = 0 respectively. By direct computation we have
0(x,y) Notice that for frozen problem y = f and y = <j> ara two planes with different slope at the origin, then we have d e t | 5 ^ | / 0
when
( * , ? ) - * (0,0)
Therefore, if 0 ( i , z ) is obtained ds s amall perturbation no fhe eorresponding function for frozen problem, the transform (2.1) is an isomorphism near the origin. After the coordinate transformation (2.1), the system (1.15) takes the form (2.2) where 1 pq- V a
0,P>0
f
(2.6) (2.7)
on a = 0
= a=0 0; on 0 =- 0 (2.8) + hS<j> + mSU = g, 6<j>\ oa oz where the left side of (2.8) is the Prechet derivative of T in (2.5) with respect to its arguments. The operator L is hyperbolic, if u > a. It has five characteristic surfaces issueing from z-axis. Among them three coincide with 6 = 7r/2, one is in the second quadrant and one is in the fourth quadrant. To prove this property we use the original coordinates system Oxyz, which is available due to the invariance of characteristics under coordinates transformation. (2.6)-(2.8) is equivalent to F(SU, H) = p ^ + q^-
1
R+u2 uv
p r + i pu
pv pw
0 -l^-pv pu 0
— , pw 0 pu
ou Sv 6w
d_ Ox u/
0 0
IP
uv
pv _ 2 = i pu
vw
o
w uw vw
pw 0
+w
2
-1
--^pu
pu i+l pv pw 0 0 pw -pv
d_ dy
1=1-pw 1 pv p
5v Sw
KssJ {6p\
\
flU
pv 7+1
(&P5u \
dz
pw
Su Sv 6w
(2.9)
w/ uSu + v6v + w6w +
7 $P _ 7 -6p = 0 7-1 p 7 — 1 p2
fx6u - 5v + fz&w = 0
on the surface of wing
(2.10) (2.11)
12
[H
(
[H
[puw] [p+pu [ ,pvw] &4>x + [puv] [p + pw2] [puw] \[pw(i+±q2)}/ \[pu(i + ^)]/ 2
/
S(pv) S(puv) S(p + pv2} 6(pvw)
S(jm) \ Sip + pu2) 5{puv) Hi + S(puw) \$(pu(i+ %