Nonlinear wave equations perturbed by the viscous term

341 Pages·2000·10.186 MB·English

by  Maslov, Viktor Pavlovich, Mosolov, Petr Petrovich

de Gruyter Expositions in Mathematics 31
Editors
Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg
V. P. Maslov, Academy of Sciences, Moscow
W. D. Neumann, Columbia University, New York
R.O.Wells, Jr., Rice University, Houston
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30 Relative Homological Algebra, Edgar E. Enochs, Overtoun M. G. Jenda
Nonlinear Wave Equations
Perturbed by Viscous Terms
by
Viktor P. Maslov
Petr P. Mosolov f
Translated from the Russian by M. A. Shishkova
w
DE
_G
Walter de Gruyter · Berlin · New York 2000
Author
Viktor P. Maslov, Kafedra Prikladnoi Matematiki, Dept. Appl. Math., Moscow Insti-
tute of Electronics and Mathematics (MIEM), B. Trekhsvyatitel'skii per. 3/12, Moscow,
109028 Russia
Title of the Russian original edition:
Uravneniya odnomernogo barotropnogo gaza. Publisher: Nauka, Moscow 1990
Mathematics Subject Classification 2000:
74-02; 74H10, 74B20, 76N10, 35Q72
Keywords:
Partial differential equations, asymptotic solution, nonlinear elasticity, existence and
uniqueness theorems
® Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress — Cataloging-in-Publication Data
Maslov, V. Ρ (Viktor Pavlovich)
[Uravneniia odnomernogo barotropnogo gaza. English]
Nonlinear wave equations perturbed by viscous terms / by V. P.
Maslov, P. P. Mosolov ; translated from the Russian by M. A. Shish-
kova.
p. cm. - (De Gruyter expositions in mathematics ; 31)
Includes bibliogaphical references and index.
ISBN 3-11-015282-7
1. Gas dynamics. 2. Navier-Stokes equations. I. Mosolov, Petr
Petrovich. II. Title. III. Series.
QA930 .M2913 2000
533'.01515353-dc21 00-043001
Die Deutsche Bibliothek — Cataloging-in-Publication Data
Maslov, Viktor P.:
Nonlinear wave equations perturbed by viscous terms / by Viktor
P. Maslov ; Petr P. Mosolov. Transl. from Russian by M. A.
Shishkova. - Berlin ; New York : de Gruyter, 2000
(De Gruyter expositions in mathematics ; 31)
ISBN 3-11-015282-7
© Copyright 2000 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced or transmitted in any form or by any means, electronic or mechanical,
including photocopy, recording, or any information storage or retrieval system, without permission
in writing from the publisher.
Typesetting using the translator's TgX files: I. Zimmermann, Freiburg.
Printing: WB-Druck GmbH & Co., Rieden/Allgäu. Binding: Lüderitz & Bauer-GmbH, Berlin.
Cover design: Thomas Bonnie, Hamburg.
Preface to the English Translation
The authors of this book who are specialists in the field of mathematical physics
devoted many years to solving mathematical problems arising in mechanics.
Academician Viktor Pavlovich Maslov hardly needs to be introduced to the reader.
Many of his monographs and papers concerned with urgent problems such as construc-
tion of tsunami wave models or simulation of the causes of the Chernobyl disaster are
well known in the world of science.
V. P. Maslov's friend and colleague, P. P. Mosolov, started working on problems
considered in this book when he had already obtained several fundamental results in
the mathematical theory of flows of viscoplastic media. Unfortunately, Petr Petrovich
Mosolov passed away prematurely.
In the present book the authors studied a wide range of mathematical problems
arising in nonlinear continuum mechanics, such as nonstationary flow of viscous gas
and dynamic deformation of media with different rigidity in tension and compression.
From a mathematical viewpoint, these problems are boundary value problems for
nonlinear hyperbolic equations. It is essentially important to justify their statements, to
study their solvability, to find possible types of solutions, and to construct asymptotics.
For readers who are mathematicians, it is beyond any doubt that these problems
must be studied. At the same time, specialists in mechanics and practical engineers
may think that they can solve specific problems without using any rigorous theory but
only exploiting their experience, intuition, and numerous applied software. However,
this opinion often leads to mistakes.
In this connection, a discussion organized by specialists in mechanics at Μ. V. Lo-
monosov Moscow State University in the early sixties is worth mentioning. It was
discussed whether the tangent component of velocity in a Newtonian viscous fluid
could have discontinuities. Even "solutions" were constructed for flows with this type
of discontinuity.
However, an analysis showed that the tangent component of velocity in a Newtonian
viscous fluid could not possess discontinuities [3]. This result was first proved and
explained on the basis of a refined mechanical study of the conditions arising on
discontinuity surfaces in dissipative media. Then the authors of [3] developed an
instructive mathematical interpretation. This result immediately followed from the
properties (well-known at that time) of generalized solutions to systems of partial
differential equations.
Any complications of phenomena and processes that can be described within the
framework of continuum mechanics give rise to development of new models with
internal variables. For such models, it is sometimes not sufficient to prove the existence
vi Preface to the English Translation
of discontinuities of some specific type, but it is necessary to study the structure of
these discontinuities.
The methods developed in this book can be used for this purpose. These methods
show how to regularize the problem by introducing a small parameter chosen in a
reasonable way and how to match the solutions that hold on different sides of the
discontinuity surface. Such methods, as well as those for constructing asymptotic
solutions, can be used for obtaining justified numerical methods and algorithms for
solving a wide class of nonlinear problems in mechanics and technology.
The logical structure of the book is described in Introduction.
It should be noted that this monograph does not make easy reading. However,
the study of this book will undoubtedly help the reader in his independent fruitful
research.
Editor of the English translation, R. V. Goldstein
Preface
The present book deals with the classical system of equations describing the motion
of a one-dimensional compressible gas. This system consists of the one-dimensional
Navier-Stokes equation, the continuity equation, and the equation of state relating
pressure and density. It is assumed that the viscosity parameter is small, i.e., the coef-
ficient of the second-order derivative in the Navier-Stokes equation is small. Usually,
energy estimates for equations of such type are obtained under the assumption that
viscosity is constant and the constants involved increase exponentially as viscosity
tends to zero.
In this book significantly sharper estimates are obtained, which have involved
considerable efforts in overcoming substantial difficulties. In particular, using these
estimates, the authors succeeded in obtaining necessary and sufficient conditions for
the existence of smooth (for vanishing viscosity) solutions of the classical system.
This theorem, like the KAM method, is proved only under the assumption that the
parameter (viscosity) is sufficiently small.
Chapters 11 and 12 were added after the untimely death of P. P. Mosolov. These
chapters contain asymptotic expansions for vanishing viscosity. They were obtained
in the multidimensional case in [48, 54] but not justified rigorously, i.e., the existence
of solutions close to the asymptotics was not proved, but only formal asymptotic
expansions were constructed.
This book is an extended and revised version of the Russian original publication.
The appendix is based on the preprint "Elasticity theory for media with different
moduli of elasticity" by V. P. Maslov and P. P. Mosolov, which was published after
P. P. Mosolov had passed away.
In preparing this edition, great help was given by Α. V. Babin, V. A. Tsupin,
G. A. Omelyanov, V. M. Kuz'mina, A. B. Solov'eva, and I. V. Knyazeva. I wish to
thank them all.
V. P. Maslov

Contents
Preface to the English Translation ν
Preface vii
Introduction 1
Chapter 1
The Cauchy problem for an infinite one-dimensional system of particles
with nonlinear viscoelastic ties 7
Chapter 2
Main estimates for the solution of the discrete problem 13
Chapter 3
Interpolation of grid functions 57
Chapter 4
Existence, uniqueness, and smoothness theorems for the solution of the
Cauchy problem for a partial differential equation that is the
limit equation for a nonlinear viscoelastic system 72
Chapter 5
Estimates for differences between solutions of the Cauchy problem for
the basic equation (4.17) 105
Chapter 6
The Cauchy problem for an equation in general form 138
Chapter 7
The Cauchy problem for a second-order hyperbolic equation with
small third-order viscous terms 147
Chapter 8
Solvability of the Cauchy problem 155
Chapter 9
Solvability of the Cauchy problem for a system of equations 192
Chapter 10
Solution behavior in the case of vanishing viscosity 203