Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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BOUNDARY VALUE PROBLEMS
IN
MECHANICS OF
NONHOMOGENEOUS FLUIDS
STUDIES IN MATHEMATICS
AND
ITS APPLICATIONS
VOLUME
22
Editors:
J.L.
LIONS,
Paris
G.
PAPANICOLAOU,
New
York
H.
FUJITA,
Tokyo
H.B.
KELLER,
Pasadena
NORTH-HOLLAND
AMSTERDAM
NEW
YORK OXFORD TOKYO
BOUNDARY VALUE PROBLEMS
IN MECHANICS
OF
NONHOMOGENEOUS FLUIDS
S.N. ANTONTSEV
A.V. KAZHIKHOV
and
V.N. MONAKHOV
Lavrentyev Institute
of
Hydrodynamics
U.S.S.R.
Academy
of
Sciences
Siberian Division
Novosibirsk,
U.S.S.R.
I990
NORTH-HOLLAND
AMSTERDAM
NEW
YORK
OXFORD
TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V.
Sara Burgerhartstraat
25
P.O.
Box
21
I,
1000
AE Amsterdam, The Netherlands
Distributors for the United States and Cunudu:
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of
the Americas
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Library
of
Congress
Cataloging-in-Publication Data
Antont^sev,
S.
N.
(Stanislav Nikolaevich)
[Kraevye zadachi mekhaniki neodnorodnykh zhidkostei. English]
Boundary value problems
in
mechanisms
of
nOnhOmOgeneOuS fluids
S.N. Antontsev. A.V. Kazhikhov.
and
V.N. Monakhov.
22
)
p. cm.
--
(Studies in mathematics and its applications
;
v.
Translation
of:
Kraevye zadachi mekhaniki neodnorodnykh
Includes bibliographical references.
zhidkostei.
ISBN 0-444-88382-7
1.
Boundary value problems.
2.
Fluid dynamics.
I.
Kazhikhov. A.
V.
(Aleksandr Vasil 'evich)
11.
Monakhov.
V.
N. (Valentin
Nikolaevich)
111.
Title.
IV.
Series.
OA379.A5813 1990
532'.05'0151535--dc20
Translation
of:
89-25572
CIP
Kraevye Zadachi Mekhaniki Neodnorodnykh Zhidkostei
0
Nauka Publishers, Siberian Division of the U.S.S.R.
Academy
of
Science, Novosibirsk
1983
ISBN:
0
444
88382
7
0
ELSEVIER SCIENCE PUBLISHERS B.V.,
1990
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PRINTED IN THE NETHERLANDS
PRETACE
The objective in
writing
of
this book
is
to
investieate the
correctness
of
the initial boundary value problems for systems of
partial differential equations describing the
flows
of
viscous gas
mid
density-inhomogeneous fluid
as
well
as
the filtration
of
a
mul-
tiphase mixture in
a
porous medium. Qualitative properties of the
solution are under investigation,
such
as
the fulfilment
of
appro-
priate physical requirements,
for
instance, nonnegativeness
of
den-
sity or inhomogeneous-liquid component concentration,
an
asympto-
tical behaviour
of
the solutions wi.th
an
infinitely increasing time,
uniqueness
&?d
stability of the solutions, This book
sms
up the
results of the investigations made by the authors in the
701s
at
the
Institute of IIydrodpamics
of
the Siberian Division
of
the
US73
Academy of Sciences.
It
involves the results concerning both
to
classical 3uler equations for
a
ideal incompressible liquid and to
the problems of justificating the models
of
liquid
and
gas with
complicated physical properties which recently appeared.
IL
characteristic mathematjcal peculiarity of
all
the systems
of
equations under consideration, apart from their nonlinearity,
is
rel9ted to the fact thet they are of combined type. This dictates
the necessity to develop suitable method
of
investisation
for
each
specific systen since
a
general theory
of
conbinec equations, even
linear,
is
far from beinz fully developed. The peculiarities
of
each
model
show
up
when obtaining the so-called a priori estimates md
a General scheme
of
proof
of the existence theorems
is
rather
stan-
dard.
some data from continuum mechanics. There the closed models are
formulated, which are used in hydrodynamics and the stntements of
b8sic boundary value problems are presented. First
a
general model
of
liquids
rnd gases
is
suggested, which takes into eccount com-
pressibility, viscosj.ty 'and thermal conductivity. This model called
the system
of
Xavier-Stokes equations
is
rather complicated
for
both
theoreticel investigation and practical
application when solving
specific problems of mechaxics. Therefore simplified models are more
widely used;
for
example, in fluid dynamics as
a
rule, compressibi-
lity
is
not taken into account.
'Phis chapter also contains the de-
rivation of the equations
of
Lncompressible liquid, both viscoii9 and
ideal, from the general system
of
Kavier-Stokes equations; the state-
ments
of
initial boundary value problems are formulated; the mathe-
matical investigations are produced which are related to
the ahove-
mentioned equations; and the problems are emphasized which are in
the focus
of
this book. Besides, some models me presented which may
be used when describing the motion
of
continrun with more complex
physical properties, such
as
nulticomponent mixtures,
a
medium havine
internal degrees of freedom
as
well
as
the model of filtration
of
two non-mixing liquids in
a
porous medium. In
2
of Chapter
1
the
necessary information belonging to functional analysis and theory
of
differential equations
is
presented.
The second Chapter
is
devoted to the investigation
of
the
Na-
vier-Stokes model for
a
compressible viscous heat-conducting gas.
The theory
of
boundary value problems for this system of equations
in general case
of
R
three-dimensional motion at present has
bees
The
book
consists of five chapters.
The first one, being of
a
supplementary character, contains
vi
developed only in
a
local statement. i.e.
or
within
a
rath
r
small
time interval
[I811
or
at
rnther limited initial data
L1791.
A
glob-
al
behaviour of the solution ha.a been investigated only for the case
of one-dimensional motion. The results for one-dimensional lJavier-
Stokes equations being of
a
global character represent
a
main con-
tent
of
Chapter
2.
The third chapter considers the model of
a
nonhomogeneous vis-
cous incompressible liquid. Different Aspects of the theory of equa-
tions of homogeneous viscous incompressible liquid have been pre-
sented in the book by 0.A.Ladyzhenskaya
L88J.
Interest to the model
of inhomogeneous liquid
is
conditioned by
its
si,mificance for ap-
plied fields of hydrodynamics, such
as
oceanology and hydrology.
It
should be noted that
teking
into account heterogeneity brings
some peculierities to
a
theoretical analysis, which are connected
with
an
additional nonlinearity of equations. Besides generalization
of the well-known theorems from
88
to
the case of nonhomogeneous
liquid, the pew statements
of
boundary value problems with one-side
restrictions are analyzed. In addition, the results
of
studying the
model of inbomogeneous two-component liquid taking into account dif-
fusion
and
the models
of
a
medium having internal
degrees of freedon
Eire considered.
an ideal incompressible liquid.
It
is
necessary
to
note that the
problem of correctness of boundary vc?lue problems for u%ler equa-
tions
is
nontrivial even the studies in local statement. The funda-
mental results.in the theory of equations of ideal liquid belong
to N.M.Gunter
1421
and
L.Lichtenstein
11771.
In these works the
CRBCS
have been considered when the
flow
region boundaries are
ab-
sent
or
are characteristic surfaces for Euler equations, i.e. vhen
the liquid occupies the whole space
or
when the vessel has imperme-
able
walln.
Nore general boundarj-valae problems which a2pear when
considering the liquid motion in
a
given region are cliffwent in
principle: in that section of the
flow
regior? boundary where the
liquid flow-in takes place,
it
is
necessary
to
prescribe
the addi-
tional boundary conditions. 'I'he problem of flowine through
a
given
region
'FIBS
first considered by N.E.:Cochin
30
in
a
model statement
when the values of velocity vortex were prescribed
as
additional
boundary conditions in th.e section
of
flowing-in. The problem for
which the whole velocity vector
is
prescribcd in the floming-in sec-
tion
is
nore natural. from the standpoint of physics. However, the
attempts to justify the correctness
of
such
n
statement have been
unsuccessful
for
a
long
time. The main result of Chapter
IV
consists
in proovinz the solvability
of
the problem
of
flowing-in
jvst
in
the above-xentioned physical statement. Eesides, other variants of
correct statements of the flowing-through problem are presented.
In particular,
it
has been shown that in the problem studied
by
B.E.Kochin the whole velocity vortex vector ccvlnot be arbitrarily
prescribed,
it
can be done
for
only
its
tangential components.
tions for two unmixing incompressible liquids in
a
porous medium.
The appearance of this model and the interest to be genernted
anions
matheme.tici,uls In
it
are conditioned by elaborating tine method of
oil forcing out
by
pumping water
or
special solutions into tlie oil
layer. The main mathematica,l peculiarity
of
the above
model
is
that
it
is
reduced to the system of
two
second-order equations, one being
elliptic and the second being parabolic. The lat,ler may degenerata
to
the first-order e uation at definite values of the required
30-
lution. The correctness of
bounda;y
value problems for equations of
of multi-phase liquid filtration has been analyzed recently,
and
the
most
complete results in this area have been obta.ined by V.N.Xona-
The fourth chapter investigates the classical Ihler model for
Chapter
V
is
devoted
to
the investigation of filtration equa-
vii
khov and S.I\J.Antoncev
[
16
-
221.
These results comprise the content
of Chapter
V.
There the solvability Oheorems have been proved, dif-
ferential properties
of
the solutions, their uniqueness, stability,
asynptotical behaviour with an infinitely increasing time have been
studied, and approxiaate methods of the solution have been justi-
fied.
It
should be noted that in ,Chapters
I1
-
V,
along with the new
results, in the
last
paragraphs some new unsolved problems have
been formulated, which on the authors
f
opinion, are significmt
scientific interest.
All
the basic results included
in
this book were discussed
ct
the seminar guided by Prof. L.V.Ovsyannikov, Zlavrentiev Institute
of
!-Iydrodynmics.
A
number of useful notes
and
advices were given
to the authors
by
N.N.Yanenko, V.V.kl;hnatchov, S.A.Tersenov,
V.11.
Vragov, P.I.Plotnikov, V.P!I.Teshukov am3 V.V.Chelukhin. Ve
also
wish
to
express
our
thanks to A.A.Papin. V.E.Mikoleev and 5.Ya.Relov for
their help given in the preparation
of
this book.
This Page Intentionally Left Blank
ix
CONTENTS
Preface
V
CHAPTER
1
MODELS
OF
THE DYNAMICS OF HETEROGENEOUS MEDIA AND THE BODY OF
MATHEMATICS
......................................................
1
1
.
Models of mechanics of heterogeneous fluids
...................
1
1
. General equations of motion of continuum and laws of
thermodynamics
........................................
1
2
.
Model of viscous perfect polytropic gas
...................
7
3
. Heterogeneous viscous incompressible fluid
...............
10
4
. Ideal incompressible fluid
...............................
12
5
. Models of multicomponent and multiphase mixtures
.........
13
6
. Model of a medium with intrinsic degrees of freedom
......
14
7
. Equations
of
filtration of two immiscible incompressible
fluids
...............................................
15
2
.
Auxiliary information from analysis and differential
equations
................................................
20
1
. Functional spaces
........................................
20
2
.
Special inequalities and embedding theorems
..............
24
3
. Certain theorems from the analysis
.......................
28
4
. Properties of the solutions to differential equations
....
32
CHAPTER
2
CORRECTNESS "IN THE WHOLE"
OF
THE BOUNDARY PROBLEMS FOR EQUATIONS
OF ONE-DIMENSIONAL NON-STATIONARY MOTION OF
A
VISCOUS GAS
.......
39
1
.
Formulation of the basic boundary problem
.
Lagrange
variables
................................................
39
2
. Auxiliary constructions
......................................
43
3
. Estimates for density and temperature from above and below
...
48
4
. A priori estimates for derivatives
...........................
52
5
.
Proof of the theorem of existence and uniqueness
.............
60
1
. The existence of generalized solution
as
limits of
smooth solutions
.....................................
60
2
. Proof of the local theorem of existence
..................
61
3
. Uniqueness of a generalized solution
.....................
67
6
. The Cauchy problem
...........................................
68
7
.
Other boundary value problems
................................
75
1
. Problem on gas flow into vacuum
..........................
75
2
.
Inhomogeneous boundary value problems
.
Problem on
piston
...............................................
77
3
. Inhomogeneous problems for temperature
...................
79