Table Of ContentAdvances 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
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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 1
S. X. Chen
The Null Condition and Global Existence of Solutions
to Systems of Wave Equations with Different Speeds 43
R. Agemi and K. Yokoyama
Scaling Limits for Large Systems of Interacting Particles 87
K. Uchiyama
Regularity of Solutions of Initial Boundary Value Problems for
Symmetric Hyperbolic Systems with Boundary Characteristic
of Constant Multiplicity 133
Y. Yamamoto
On the Half-Space Problem for the Discrete Velocity
Model of the Boltzmann Equation 160
S. Ukai
Blow-up, Life Span and Large Time Behavior of Solutions to a
Weakly Coupled System of Reaction-Dfffusion Equations 175
K. Mochizvki
On a Decay Rate of Solutions to One-Dimensional Thermoelastic
Equations on a Half Line; Linear Part 198
Y. Shibata
Bifurcation Phenomena for the Duffing Equation 292
A. Matsumura
Some Remarks on the Compactness Method 319
A. V. Kazhikhov
Percolation on Fractal Lattices: Asymptotic Behavior
of the Correlation Length 331
M. Shinoda
1
MATHEMATICAL ASPECTS OF
SUPERSONIC FLOW PAST WINGS
CHEN 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
Np = N (1.1)
0 0 lPl
PO&Q +PO = PIN?+PI (1.2)
LB = £1 (1.3)
1 2 ■ 1 ■ , ,
2% + !o = ^1 +Ii (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 > p. If the shock front is
Pl 0
a plane with an angle of inclination 0, then (1.1)-{1.4) implies (see [11])
ui = (1 - ^2)<Jo(sin20 - sin2 A) (1.5)
a
3
"i = (Qo - Ui)cot/J (1.6)
with 0 as parameter, A is Mach number and fi? = *£. (1.5),(1.6) indicate a
0
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 A)q
0 0
U = (l + (1- (i2)sm2Ao)q
0
Moreover, (1.1)-(1.4) yield
Pi = Po + PoQo{Qo~ ai) (1.8)
^ ^ pogoan 0
= =
iVi u - 5o^J cosJ £f
t a n / ? = ^^ (1.10)
The picture of shock polar ( upper half ) is shown in Figure 1. The curve has
a unique self-intersection at P ! (so00). By using the shock polar the problem of
0
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 P P by a straight line,
0
and starting from O draw a straight line perpendicular to P P, we obtain an
0
intersection D. The angle ZP OD is just the angle of inclination of the required
0
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
9 . Obviously, when 8 < 6 , the ray ( will only intersect with the shock polar
EXt elt
at one point , where the radius is greater than q . So the above process doesn't
0
