Free Boundary Problems in Continuum Mechanics: International Conference on Free Boundary Problems in Continuum Mechanics, Novosibirsk, July 15–19,1991

348 Pages·1992·7.712 MB·English

by  G. V. Alekseyev, A. Yu. Chebotarev (auth.), Prof. S. N. Antontsev, Prof. A. M. Khludnev, Prof. K.-H. Hoffmann (eds.)

ISNM106:
International Series of Numerical Mathematics
Internationale Schriftenreihe zur Numerischen Mathematik
Serie Internationale d'Analyse Numerique
Vol. 106
Edited by
K.-H. Hoffmann, München; H. D. Mittelmann, Tempe;
J. Todd, Pasadena
Springer Basel AG
Free Boundary Problems
in Continuum Mechanics
International Conference on Free Boundary Problems
in Continuum Mechanics, Novosibirsk, July 15-19,1991
Edited by
S. N. Antontsev
K.-H. Hoffmann
A. M. Khludnev
Springer Basel AG
Editors
Prof. S. N. Antontsev Prof. K.-H. Hoffmann
Lavrentyev Institute Institut für Angewandte
of Hydrodynamics Mathematik und Statistik
Novosibirsk 630090 Dachauer Str. 9 a
Russia D-W-8000 München
Germany
Prof. A. M. Khludnev
Lavrentyev Institute
of Hydrodynamics
Novosibirsk 630090
Russia
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Free boundary problems in continuum mechanics /
International Conference on Free Boundary Problems in
Continuum Mechanics, Novosibirsk, July 15-19,1991. Ed. by
S. N. Antontsev ... - Basel ; Boston ; Berlin : Birkhäuser, 1992
(International series of numerical mathematics ; Vol. 106)
ISBN 978-3-0348-9705-1 ISBN 978-3-0348-8627-7 (eBook)
DOI 10.1007/978-3-0348-8627-7
NE: Antoncev, Stanislav N.; International Conference on Free Boundary
Problems in Continuum Mechanics <1991, Novosibirsk>; GT
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically those of translation, reprinting, re-use of
illustrations, broadcasting, reproduction by photocopying machine or similar means,
and storage in data banks. Under § 54 of the German Copyright Law where copies are
made for other than private use a fee is payable to »Verwertungsgesellschaft Wort«, Munich.
© 1992 Springer Basel AG
Originally published by Birkhäuser Verlag Basel in 1992
Softcover reprint of the hardcover 1st edition 1992
ISBN 978-3-0348-9705-1
Contents
Preface IX
0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 000 0 0 0 0 0
Some extremum and unilateral boundary value problems in viscous hydrodynamics
Go V. Alekseyev andA. YUo Chebotarev I
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
On axisymmetric motion of the fluid with a free surface
V. Ko Andreev 13
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
On the occurrence of singularities in axisymmetrical problems of hele-shaw type
Do Andreucci, A. Fasano and Mo Primicerio 23
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
New asymptotic method for solving of mixed boundary value problems
l. V. Andrianov andAo Ivankov 39
00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Some results on the thermistor problem
So No Antontsev and Mo Chipot 47
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
New applications of energy methods to parabolic and elliptic free boundary problems
So No Antontsev, Jo l. Diaz and So l. Shmarev 59
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A localized finite element method for nonlinear water wave problems
Ko Jo Bai and Jo W. Kim 67'
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Approximate method of investigation of normal oscillations of viscous incompressible liquid
in container
Mo Tho Bamyak 75
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
The classical Stefan problem as the limit case of the Stefan problem with a kinetic condition
at the free boundary
Bo V. Bazaliy and So P. Degtyarev 83
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A mathematical model of oscillations energy dissipation of viscous liquid in a tank
l. Bo Bogoryad 91
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Existence of the classical solution of a two-phase multidimensional Stefan problem on any
finite time interval
Mo Ao Borodin 97
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Asymptotic theory of propagation of nonstationary surface and internal waves over
uneven bottom
So YUo Dobrokhotov, P. N. Zhevandrov, Ao A. Korobkin and l. V. Sturova 105
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Multiparametric problems of two-dimensional free boundary seepage
V. No Emikh 113
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Nonisothermal two-phase filtration in porous media
Ro Eo Ewing and V. No Monakhov 121
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Explicit solution of time-dependent free boundary problems
YUo Eo Hohlov 131
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
VI
Nonequilibrium phase transitions in frozen grounds
1. A. Kaliev ........................................................ 141
System of variational inequalities arising in nonlinear diffusion with phase change
N. Kenmochi and M. Niezgodka .......................................... 149
Contact viscoelastoplastic problem for a beam
A. M. Khludnev ..................................................... 159
Application of a finite-element method to two-dimensional contact problems
S. N. Korobeinikov and V. V. Alyokhin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167
Computations of a gas bubble motion in liquid
V. A. Korobitsyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 179
Waves on the liquid-gas free surface in the presence of the acoustic field in gas
1. A. Lukowsky andA. N. TImoha ......................................... 187
Smooth bore in a two-layer fluid
N. 1. Makarenko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 195
Numerical calculation of movable free and contact boundaries in problems of dynamic
deformation of viscoelastic bodies
L. A. Merzhievsky andA. D. Resnyansky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 205
On the canonical variables for two-dimensional vortex hydrodynamics of incompressible
fluid
O. 1. Mokhov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 215
About the method with regularization for solving the contact problem in elasticity
R. V. Namm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 223
Space evolution of tornado-like vortex core
V. V. Nikulin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 229
Optimal shape design for parabolic system and two-phase Stefan problem
S. P. Ohezin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 239
Incompressible fluid flows with free boundary and the methods for their research
A. G. Petrov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 245
On the Stefan problems for the system of equations arising in the modelling of
liquid-phase epitaxy processes
A. G. Petrova .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 253
Stefan problem with surface tension as a limit of the phase field model
p. 1. Plotnikov and V. N. Starovoitov ........................................ 263
The modelization of transformation phase via the resolution of an inclusion problem with
moving boundary
H. Sabar, M. Buisson and M. Berveiller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 271
To the problem of constructing weak solutions in dynamic elastoplasticity
V. M. Sadovskii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 283
VII
The justification of the conjugate conditions for the Euler's and Darcy's equations
v. v.
Shelukhin ...................................................... 293
On an evolution problem of thermo-capillary convection
V. A. Solonnikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 301
Front tracking methods for one-dimensional moving boundary problems
U. Streit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 319
On Cauchy problem for long wave equations
V. M. Teshukov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 331
On fixed point (trial) methods for free boundary problems
T. Tiihonen and J. Jarvinen .............................................. 339
Nonlinear theory of dynamics of a viscous fluid with a free boundary in the process of
a solid body wetting
O. V. Voinov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 351
Preface
Progress in different fields of mechanics, such as filtra
tion theory, elastic-plastic problems, crystallization pro
cesses, internal and surface waves, etc., is governed to a
great extent by the advances in the study of free boundary
problems for nonlinear partial differential equations.
Free boundary problems form a scientific area which
attracts attention of many specialists in mathematics and
mechanics. Increasing interest in the field has given rise
to the "International Conferences on Free Boundary Problems
and Their Applications" which have convened, since the
1980s, in such countries as England, the United states,
Italy, France and Germany.
This book comprises the papers presented at the Interna
tional Conference "Free Boundary Problems in Continuum
Mechanics", organized by the Lavrentyev Institute of
Hydrodynamics, Russian Academy of Sciences, July 15-19,
1991, Novosibirsk, Russia.
The scientific committee consisted of:
Co-chairmen: K.-H. Hoffmann, L.V. Ovsiannikov
S. Antontsev (Russia) J. Ockendon (UK)
M. Fremond (France) L. Ovsiannikov (Russia)
A. Friedman (USA) S. Pokhozhaev (Russia)
K.-H. Hoffmann (Germany) M. Primicerio (Italy)
A. Khludnev (Russia) V. Pukhnachov (Russia)
V. Monakhov (Russia) Yu. Shokin (Russia)
V. Teshukov (Russia)
Our thanks are due to the members of the Scientific Com
mittee, all authors, and participants for contributing to
the success of the Conference. We would like to express
special appreciation to N. Makarenko, J. Mal'tseva and T.
Savelieva, Lavrentyev Institute of Hydrodynamics, for their
help in preparing this book for publication.
July 1992 S. Antontsev, K.-H. Hoffmann, A. Khludnev
International Series of Numerical Mathematics, Vol. 106, © 1992 Birkhiiuser Verlag Basel 1
SOME EXTREMUM AND UNILATERAL BOUNDARY VALUE
PROBLEMS IN VISCOUS HYDRODYNAMICS
G.V.Alekseyev, A.Yu.Chebotarev
Institute of Applied Mathematics,
Vladivostok 690068, RUSSIA.
This paper is concerned with investigation of direct and
inverse problems for the stationary Stokes system. At first we
prove the unique solvability of a direct unilateral boundary va
lue problem and establish some properties of the solution. Then
we formulate problems which are inverse to the direct problem and
investigate the solvability of one inverse extremum problem.
Key words: Stokes system, unilateral boundary value problem,
inverse extremum problem, variational inequality.
1.INTRODUCTION.
Simulation of some new processes in fluid mechanics causes
the necessity of solving the new boundary value problems for the
Navier-Stokes and Euler equations. As an examples of such prob
lems are so-called unilateral boundary value problems and extre
mum problems arising in viscous and ideal hydrodynamics. Various
authors have considered the extremum problems in hydrodynamics.
The works most closely related to ours can be found in Lions
[ 1,2), Fursikov [3,4), Chebotarev [5,6). In all of them some
standard linear boundary conditions were posed on the boundaries
of domains considered. Unilateral boundary value problems were
studied in Lions [7), Kazhikhov [8-9) and others(cf. references
to [7), [9)) for the nonstationary Navier-Stokes equations and in
Chebotarev [10) for the Euler equations. For an ideal fluid uni
lateral boundary conditions arise in a natural way in the prob
lems of liquid flowing through a 'tube'. For the first time these
problems were considered in stationary case (wi th classical
boundary conditions) in Alekseyev [11-13J. For the Stokes system
unilateral boundary value problems connected with the steady vis
cous fluid flowing through a bounded domain and corresponding
extremum problems were studied in Chebotarev [14), where some
properties (existence, uniqueness, estimates) of the solutions