Table Of Contentde Gruyter Expositions in Mathematics 24
Editors
Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg
V. P. Maslov, Academy of Sciences, Moscow
W. D. Neumann, The University of Melbourne, Parkville
R.O.Wells, Jr., Rice University, Houston
de Gruyter Expositions in Mathematics
1 The Analytical and Topological Theory of Semigroups, K. H. Hofmann,
J. D. Lawson, J. S. Pym (Eds.)
2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues
3 The Stefan Problem, A. M. Meirmanov
4 Finite Soluble Groups, K. Doerk, T. O. Hawkes
5 The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin
6 Contact Geometry and Linear Differential Equations, V. E. Nazaikinskii,
V. E. Shatalov, B. Yu. Sternin
7 Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev,
V. M. Petrogradsky, Μ. V. Zaicev
8 Nilpotent Groups and their Automorphisms, Ε. I. Khukhro
9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug
10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini
11 Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon,
G. Zhao
12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and
Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub
13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S.A.
Nazarov, B. A. Plamenevsky
14 Subgroup Lattices of Groups, R. Schmidt
15 Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, P. H.
Tiep
16 The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti,
A. J. Sommese
17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno
18 Unitary Representation Theory of Exponential Lie Groups, H. Leptin,
J. Ludwig
19 Blow-up in Quasilinear Parabolic Equations, A.A. Samarskii, V.A.
Galaktionov, S. P. Kurdyumov, A. P. Mikhailov
20 Semigroups in Algebra, Geometry and Analysis, Κ. H. Hofmann, J. D.
Lawson, Ε. B. Vinberg (Eds.)
21 Compact Projective Planes, H. Salzmann, D. Betten, Τ. Grundhöf er, Η. Hühl,
R. Löwen, M. Stroppel
22 An Introduction to Lorentz Surfaces, Τ. Weinstein
23 Lectures in Real Geometry. F. Broglia (Ed.)
Evolution Equations
and Lagrangian Coordinates
by
A. M. Meirmanov
V. V. Pukhnachov
S. I. Shmarev
W
DE
_G_
Walter de Gruyter · Berlin · New York 1997
Authors
Α. Μ. Meirmanov V. V. Pukhnachov S. I. Shmarev
Departamento de Matemätica Lavrentyev Institute Lavrentyev Institute
Universidade da Beira Interior of Hydrodynamics of Hydrodynamics
R. Marques d'Avila e Bolama The Russian Academy The Russian Academy of Sciences
6200 Covilhä of Sciences Siberian Division
Portugal Siberian Division 630090 Novosibirsk, Russia
630090 Novosibirsk, and
Russia Departamento de Matematicas
Faculdad de Ciencias
Universidad de Oviedo
33007 Oviedo, Spain
1991 Mathematics Subject Classification: 35-02, 35K55, 80A22, 76S05, 58G35
Keywords: nonlinear parabolic equations, Stefan-type problems, nonlocal equivalence
transformation, asymptotic behaviour of solutions
© 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
Evolution equations and Lagrangian coordinates / by A. M. Meirma-
nov, V. V. Pukhnachev, S. I. Shmarev.
p. cm. - (De Gruyter expositions in mathematics ; 24)
Includes bibliographical references and index.
ISBN 3-11-014875-7 (alk. paper)
1. Evolution equations — Numerical solutions. 2. Lagrangian
equations — Numerical solutions. 3. Mathematical physics.
I. Meirmanov, A. M. (Anvarbek Mukatovich) II. Pukhna-
chev, V. V. III. Shmarev, S. I. (Sergei I.), 1958- . IV. Series.
QC20.7.E88E96 1997
515'.353-dc21 96-54 303
CIP
Die Deutsche Bibliothek — Cataloging-in-Publication Data
Meirmanov, Anvarbek M.:
Evolution equations and Lagrangian coordinates / by A. M.
Meirmanov ; V. V. Pukhnachov ; S. I. Shmarev. — Berlin ; New
York : de Gruyter, 1997
(De Gruyter expositions in mathematics ; 24)
ISBN 3-11-014875-7
NE: Puchnacev, Vladislav, V.:; Smarev, Sergej I.:; GT
© Copyright 1997 by Walter de Gruyter & Co., D-10785 Berlin.
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.
Typeset using the authors' TgX files: I. Zimmermann, Freiburg
Printing: Arthur Collignon GmbH, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin.
Cover design: Thomas Bonnie, Hamburg.
Preface
This book presents recent results of the investigation of evolution equations by means
of the method of Lagrangian coordinates.
There are two basic methods for describing the motions of continua, those of Euler
and Lagrange. In the first method all characteristics of the motion are considered as
functions of time and independent space variables in a coordinate system not connected
with the medium itself. In the approach of Lagrange, the observer follows the particles'
paths, e.g. all parameters of the motion are assumed to be functions of time and the
initial positions of the particles. Each of the two approaches has its own merits. In
particular, the method of Lagrange is sometimes preferable to deal with the motions of
continua which involve free boundaries, a priori unknown.
As was noticed some ten years ago, one may profit from this mechanical ideology
to study abstract evolution equations. Given an evolution equation of divergent form,
whatever its physical applications may be, it may always be viewed as the mass balance
law of some motion of a continuous medium written in Euler variables. This fictitious
motion may then be given the Lagrangian description which is of different mathematical
nature than the original equation. The study of the Lagrangian counterpart proves very
informative and if the formal correspondence between the solutions of the starting
equation and its "twin" has been somehow established, we advance in the study of both.
These ideas are developed in several directions: we outline their algebraic and
numerical aspects, and use them to study free boundary problems. The contents are
structured into four chapters.
Chapter 1 is devoted to the study of Verigin's problem. This problem occurs in the
mathematical modeling of joint motions of two immiscible liquids through a porous
medium. The process gives rise to the free boundary that separates the regions occupied
by the liquids. The complete investigation of this problem in the one-dimensional
formulation is presented, the description of the physical process and the governing
mathematical models included. For the deduced boundary value problems the existence
and uniqueness theorems are proven. The self-similar solutions are constructed and it is
shown that the self-similar solutions of Stefan's problem yield the similarity solutions
of the limiting Verigin problem.
In Chapter 2 we discuss algebraic properties of the coordinate transformation em-
ployed. Special symmetry properties of second and higher-order parabolic equations
are derived. In particular, we describe a class of equations possessing the nonlocal sym-
metries which turn out to be the classical Lie symmetries after the passage to Lagrangian
coordinates. The classes of equations are stated 1) which are invariant with respect to
vi Preface
Lagrangian coordinate transformation; 2) that can be linearized or simplified via this
change of variables. In the latter case we construct special exact solutions generating
new explicit solutions of the original equations.
Lagrangian coordinates prove to be a favorable framework to obtain new classes of
exact solutions for certain boundary value problems. We present exact solutions to the
one-phase Stefan problem and to the Cauchy-Verigin problem. Of special interest are
those exact solutions of Stefan's problem which describe the process of phase disap-
pearance. The method is also extended to some systems of equations and to equations
with complex variables.
Chapter 3 contains a systematic study of the Cauchy problem for a number of equa-
tions contained in the class of equations usually referred to as the generalized porous
medium equation or the advection-diffusion equation. Here we are interested princi-
pally in the regularity and propagation properties of the interfaces separating the regions
where solutions are strictly positive from those where they are identically zero. Al-
though an appropriate mathematical theory of such problems already exists, we employ
the Lagrangian coordinate transformation to get detailed information on the interface
properties. We establish conditions on data providing C1,a and C°°-smoothness of the
occurring interfaces, the asymptotic representations for the interfaces are derived, and
it is shown that, under certain restrictions, these asymptotics are differentiate. For a
subclass of the generalized porous medium equation the effect of instantaneous loss of
initial smoothness is discovered: whatever the input data, the space derivative of the
solution instantly becomes discontinuous and, moreover, the parameters of this "jump
of discontinuity" do not depend on the nature of the data.
In the concluding sections of this chapter we demonstrate an application of this
approach to the study of unbounded solutions for parabolic equations. A special non-
classical variant of Stefan's problem with degeneracy at the free boundary is discussed
in the final section.
Chapter 4 collects results of the investigation of multi-dimensional free boundary
problems for parabolic equations. The first of the problems under consideration is the
classical one-phase Stefan problem. The Lagrangian formulation of this problem is
presented. After the passage to Lagrangian coordinates we obtain, instead of a single
parabolic equation, a system of evolution equations (which need not be parabolic). The
dimension of this system equals the number of independent spatial variables in the
original equation. The well-posedness of the linear model of this problem is studied.
For the complete problem, the existence of self-similar solutions is proven. These self-
similar solutions can be interpreted as describing the process of ice melting. The initial
temperature is assumed to be constant along each ray exiting from the top of some cone,
the temperature is positive inside the cone and zero otherwise.
By their very nature, Lagrangian coordinates can always be introduced in a number
of ways and we benefit from this arbitrariness to treat the equation of polytropic gas
filtration (the porous medium equation). For this equation an analog of the system of
mass Lagrangian coordinates is proposed in which the equivalent system of equations
adopts a canonical form. Next, we study Boussinesq's equation of filtration theory. The
group analysis of Boussinesq's equation is given and two types of self-similar solutions
Preface vii
are constructed and investigated. This study serves the groundwork for the construction
of new nonlocal invariant solutions to Boussinesq's equation. We present an example of
the numerical treatment of these self-similar solutions and propose a method to linearize
Boussinesq's equation at its solution.
The final section of the chapter discusses local smoothness properties of the free
boundaries in solutions of the N-dimensional porous medium equation. We prove that
the interfaces preserve the initial regularity in the space variables and become infinitely
differentiable in time.
The results and presentation of Chapter 1 are due to A. M. Meirmanov. The material
of Chapter 2 was prepared by V. V. Pukhnachov. Chapter 3 was written by S. I. Shmarev,
except for Sections 26-27 presented by V. V. Pukhnachov. Sections 28,33-34 of Chapter
4 were written by V. V. Pukhnachov, Sections 31-32 and 35 by S. I. Shmarev, Sections
29-30 are due to A. M. Meirmanov and V. V. Pukhnachov.
The idea of writing a book was born when all three authors were collaborators of
the Lavrentyev Institute of Hydrodynamics, Siberian Division of Russian Academy of
Sciences, Novosibirsk. The original motivation was to survey the results which had
already been obtained up to that time by the Lagrangian coordinates method. However,
the selection of material and preparation of the text have taken much more time than was
expected and the final version of the book contains a number of new results obtained
while this monograph was in progress.
At different stages of work on this book the authors were supported by different
institutions. Primarily, in 1991 S. I. Shmarev was supported with a Research Grant
by Lavrentyev Institute of Hydrodynamics, during 1993-1994 he was given a scholar-
ship of FICYT (Principado de Asturias, Spain) at the Mathematical Department of the
University of Oviedo. The work of V. V. Pukhnachov was partially supported with the
grant EP1/93IS by the Japan Society for the Promotion of Science, and with the grant
NR 6000 by the International Science Foundation. The authors would like to express
here their deepest gratitude to these institutions and their members for encourage and
support.
Our pleasant obligation is to thank here the colleages whose aid, discussions, and
criticism were invaluable in making this book a reality. Especially, we would like to
express our sincere gratitude to V. S. Belonosov, A. P. Chupakhin, J. I. Diaz, T. Nagai,
L. V. Ovsyannikov, C. Rogers, J. L. Vazquez.
Contents
Chapter I
The Verigin problem
1 Review of results 1
2 Filtration in a porous soil 2
3 Formulation of the problem 4
4 Self-similar solutions; Stefan's problem as a limit case of Verigin's problem . . 6
5 One-dimensional problem: main statements and formulation of results 9
6 Proofs of Theorems 5.1 and 5.2: Verigin's problem with given mass flux
on the known boundaries and the Cauchy-Verigin problem 12
6.1 Equivalent problem in a fixed domain 12
6.2 Solutions to approximate problems 17
6.3 Uniform estimates for solutions of approximation problems 21
6.4 Passage to the limit 26
6.5 Existence of classical solutions to original problems 27
6.6 Uniqueness of solution to problem (Ej) 28
6.7 Global in time solvability of problem (Ei) 30
7 Proof of Theorem 5.3: Verigin's problem with the given pressure
on the known boundaries 33
7.1 Equivalent problem (L3). Existence of solution 33
7.2 Uniqueness of solution to problem (L3) 37
Chapter II
Equivalence transformations of evolution equations
8 Main ideas. A historical survey 42
9 Reciprocity transformations of second-order equations 47
10 Hidden symmetry of evolution equations 51
11 Linearization by means of Lagrangian coordinates 62
12 Lagrange-invariant equations 65
χ
13 Equations with spherical and cylindrical symmetries 69
13.1 Gradient-nonlinear equations 69
13.2 Equation of heat conduction in solid hydrogen 73
13.3 The nonlinear Schrödinger equation 75
14 Equivalence transformations for higher-order equations and systems
of equations 79
14.1 The Kingston-Rogers theorem 79
14.2 Third-order L-invariant equations 82
14.3 Reciprocity transformations for systems of one-dimensional
evolution equations 84
14.4 Evolution equations in complex domains 87
15 A remarkable equation of nonlinear heat conduction 89
16 The one-phase Stefan problem: explicit solutions with functional
arbitrariness 94
16.1 Statement of the problem 94
16.2 Solvability conditions 96
16.3 Global in time explicit solutions 98
16.4 Explicit solutions describing the process of phase disappearance . . . 100
16.5 Explicit solutions of the Cauchy-Verigin problem 101
Chapter HI
One-dimensional parabolic equations
17 Introduction 104
17.1 An example 104
17.2 Finite speed of propagation. Interfaces 106
17.3 Preliminaries and results 108
18 Lagrangian coordinates in one-dimensional evolution equations 121
18.1 Equations in divergence form 122
18.2 Some equations not in divergence form 126
18.3 A commentary 129
19 Analysis of the problem in Lagrangian terminology 130
