The Stefan Problem

256 Pages·1992·32.09 MB·English

by  A.M. Meirmanov, Marek Niezgodka, Anna Crowley

de Gruyter Expositions in Mathematics 3
Editors
O. H. Kegel, Albert-Ludwigs-Universität, Freiburg
V. P. Maslov, Academy of Sciences, Moscow
W. D. Neumann, Ohio State University, Columbus
R. O. Wells, Jr., Rice University, Houston

The Stefan Problem
by
Anvarbek M. Meirmanov
Translated from the Russian
by
Marek Niezgodka and Anna Crowley
W
DE
G
Walter de Gruyter · Berlin · New York 1992
Author
Anvarbek M. Meirmanov, Institut für Angewandte Mathematik, Universität Bonn,
Wegeier Str. 6, D-5300 Bonn l, Germany
Translators
Marek Niezgodka, Institute of Applied Mathematics and Mechanics, Warsaw University,
Banacha 2,00-913 Warsaw, Poland
Anna Crowley, Department of Mathematics, Royal Military College of Sciences,
Shrivenham, Swindon Wilts SN6 8LA, England
Title of the Russian original edition: Zadacha Stefana. Publisher: Nauka, Novosibirsk 1986
7997 Mathematics Subject Classification: 35-02; 35R35,35K20. 80-02; 80A22.
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
Meirmanov, A. M. (Anvarbek Mukatovich)
[Zadacha Stefana, English]
The Stefan problem / by Anvarbek M. Meirmanov ; translated
from the Russian by Marek Niezgodka and Anna Crowley.
p. cm. (De Gruyter expositions in mathematics,
ISSN 0938-6572 ; 3)
Translation of: Zadacha Stefana.
Includes bibliographical references and index.
ISBN 3-11-011479-8 (alk. paper)
1. Heat—Transmission. 2. Boundary value problerns. I.Title.
II. Series.
QC321.M45B 1992 91-39310
536'.2-dc20 CIP
Die Deutsche Bibliothek - Cataloging-in-Publication Data
Meirmanov, Anvarbek M.:
The Stefan problem / by Anvarbek M. Meirmanov. Transl. from
the Russian by Marek Niezgodka and Anna Crowley. -
Berlin ; New York : de Gruyter, 1992
(De Gruyter expositions in mathematics ; 3)
Einheitssacht.: Zadaca Stefana <engl.>
ISBN 3-11-011479-8
NE:GT
© Copyright 1992 by Walter de Gruyter & Co., D-1000 Berlin 30. -
All rights reserved, including those of translation into foreign languages. No part of this book may be
reproduced in any form or by any means, electronic or mechanical, including photocopy, recording,
or any information storage and retrieval system, without permission in writing from the publisher.
Printed in Germany.
Printing: Gerike GmbH, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin.
Cover design: Thomas Bonnie, Hamburg.
Preface to the English edition
This English edition is the exact translation of the Russian original publication except
for two points: Theorem 11 on pages 32-36 of Chapter 1 and the Appendix.
Very few new results on the Stefan problem have appeared since the publication of
this book in 1986. The most important results in my opinion concern the behaviour of
the mushy region and are due to Berger & Rogers and Götz & Zaltsman. I have included
the results of Götz & Zaltsman for the reason that their proof is simpler and clearer for
the reader.
Another point concerning the Appendix. The Russian original publication of the book
presents a very short version of the model which describes phase transitions in a binary
alloy. Since this book appeared we have published a full version of the Appendix in
"Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki", No. 4 (1989), pp. 39-45 (English
translation: LG. Götz, A.M. Meirmanov, Modelling crystallization of a binary alloy,
J. Appl. Mech. Tech. Phys., No. 4 (1989), pp. 545-550). The Appendix in the English
edition is based on this paper. In my opinion it is more understandable for the reader to
have this full version of the mathematical model of phase transitions in a binary alloy.
Bonn, November 1991 A.M. Meirmanov
Preface
More than twenty years have passed since the appearance of the first monograph on the
Stefan problem. In the meantime, many new ideas have been discovered and research
techniques set up that have contributed to solving numerous complex problems. To some
extent, the Stefan problem in its primary setting has settled itself, and time has come to
consolidate the material collected.
In this book, the primary concerns are existence and uniqueness questions for the
Stefan problem and a study of its structure. The choice of material presented was at first
dictated by the author's interests, for a couple of years focussed on the Stefan problem.
A large part of the book presents results obtained by the author and his colleagues. Many
important aspects remain beyond the scope of the book, including numerical methods
for solving the Stefan problem, questions of optimal control for phase change problems
and the multidimensional quasi-steady Stefan problem.
A historical overview and bibliography of the Stefan problem up to 1967 can be found
in [198]. At the end of this book, a current bibliography on the presented material is
given. The bibliography does not pretend to any completeness, for this we refer the
reader to the comprehensive review papers by Magenes [144] and Tarzia [216].
The author wishes to take advantage of this opportunity to express his gratitude to
B.M. Anisyutin who has offered the material for Section III.l, to I.A. Kaliev who has
written Sections V.5, VI.4 and VII.4, as well as to N.A. Kulagina and A.G. Petrova for
assistance in collecting the list of references.
Table of Contents
Preface to the English edition v
Preface vi
Introduction 1
Chapter I
Preliminaries
1. Problem statement 8
2. Assumed notation. Auxiliary notation 18
2.1. Notation 18
2.2. Basic function spaces 18
2.3. Auxiliary inequalities and embedding theorems 20
2.4. Auxiliary facts from analysis 22
2.5. Properties of solutions of differential equations 23
2.6. The Cauchy problem for the heat equation over smooth
unbounded manifolds in the classes Hl+2-(l+2^2(S ) 25
T
3. Existence and uniqueness of the generalized solution
to the Stefan problem 26
Chapter II
Classical solution of the multidimensional Stefan problem
1. The one-phase Stefan problem. Main result 37
2. The simplest problem setting 39
3. Construction of approximate solutions to the one-phase Stefan
problem over a small time interval 47
4. A lower bound on the existence interval of the solution. Passage to
the limit 50
5. The two-phase Stefan problem 60
Chapter III
Existence of the classical solution to the multidimensional
Stefan problem on an arbitrary time interval
1. The one-phase Stefan problem 65
2. The two-phase Stefan problem. Stability of the stationary solution . . .. 79
2.1. Problem statement. Main result 79
2.2. Formulation of the equivalent boundary value problem 80
viii Table of Contents
2.3. Construction of approximate solutions 81
2.4. A lower bound for the constant 63 84
2.5. Proof of the main result 87
Chapter IV
Lagrange variables in the multidimensional one-phase Stefan problem
1. Formulation of the problem in Lagrange variables 90
2. Linearization 91
3. Correctness of the linear model 94
Chapter V
Classical solution of the one-dimensional Stefan
problem for the homogeneous heat equation
1. The one-phase Stefan problem. Existence of the solution 99
2. Asymptotic behaviour of the solution of the one-phase Stefan
problem 107
3. The two-phase Stefan problem 112
4. Special cases: one-phase initial state, violation of compatibility
conditions, unbounded domains 122
5. The two-phase multi-front Stefan problem 127
6. Filtration of a viscid compressible liquid in a vertical porous layer . . .. 130
6.1. Problem statement. The main result 130
6.2. An equivalent boundary value problem in a fixed domain 132
6.3. A comparison lemma 133
,00 j
6.4. The case / --— dp = oo 134
J\ f(P)
6.5. The case f(p) = exp(p — 1) 135
6.6. The case f(p) =ρΊ, 7 > 1 136
6.7. Asymptotic behaviour of the solution, as t —> oo 139
Chapter VI
Structure of the generalized solution to the one-phase
Stefan problem. Existence of a mushy region
1. The inhomogeneous heat equation. Formation of the mushy region . . .. 142
2. The homogeneous heat equation. Dynamic interactions between the
mushy phase and the solid/liquid phases 149
3. The homogeneous heat equation. Coexistence of different phases 158
4. The case of an arbitrary initial distribution of specific internal
energy 162
Table of Contents ix
Chapter VII
Time-periodic solutions of the one-dimensional Stefan problem
1. Construction of the generalized solution 174
2. Structure of the mushy phase for temperature on the boundary
of ΩΟΟ with constant sign 177
3. The case of $+(t) with variable sign 181
Chapter VIII
Approximate approaches to the two-phase Stefan problem
1. Problem statement. Formulation of the results 191
2. Existence and uniqueness of the generalized solution to Problem (A^) . .. 199
3. Existence of the classical solution to Problem (Ao) 203
3.1. Auxiliary Problem (C) 205
r
3.2. Differential properties of the solutions to Problem (C) 208
r
3.3. Proof of Theorem 2 211
3.4. Proof of Lemma 5 213
4. The quasi-steady one-dimensional Stefan Problem (C) 215
Appendix
I.G. G tz, A.M. Meirmanov: Modelling of binary alloy crystallization . . . 222
References 231
Supplementary references 243
Index 245