Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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148
Chapter
3
5,
Still
not studied ie the problem on the aeymptotic behaviour
of
the
solutione at infinitely growing time
of
epecial importance
would be
to
clarify whether the non-stationary solution reduces to
the stationary one.
6,
Of
great interest from both the theoretical viewpoint and in
the applied
88
re the operation problems for the Mavier-Stokes
equations (eee?%P by J.-L.Lions
1.
New re8
ts
n
this
direction
have been recently obtained by
A,V.
hrsikOV??35f.
7.
In the diffusion
model
from
4
the condition
of
smallness
(4.20)
was imposed on
the
diffusion coefficient Thie limita-
tion can, in
our
opinion, be eliminated.
0.
In the model with the internal degrees
of
freedom from
L4
5,
it
would be desirable
to
study the generalized eolutions,
BE
well
as
to consider the case of
an
inhomogenous medium.
149
CHAPTER
4
CORRECTNESS
OF
THE
PROBLEM
OF
FLOWING
THROUGH
FOR
THE
OF
AN
IDEAL
INCOMPRESSIBLE
LIQUID
This chapter considers a classical hydrodinamical model
-
the
Euler equations for
an
ideal liquid. One of the basic problems
is
that on flowing of a liquid in
a
given domain whose boundaries may
contains the portions where the liquid flowe in
an
out. The inves-
tigation of the initially-boundary problems for the Euler equations
itiated in the papers by N.M.Gunter
[
42
]
and L.Lichtenatein
T8,7l?
These authors obtained the basic results for the caeee when
the liquid
fills
the whole space, or a container with impermiable
walls, and when the vector of mass forces
is
potential.
It
should
be emphasized that since the problem of the correctness of the
problems for the equations of an ideal liquid
is
quite difficult
even when etudied in the
small
with respect to time, the results
obtained in thia field are mostly local
in
time. This chapter also
dwells on the solvability
"in
the mall" of the non-stationary
problem of an ideal
liquid flowing through a bounded domain, when
on the portion of in-flowing the total vector of velocity
is
given,
while on the or ion of out-flowing
-
its
normal component.
It
was
N.E.
Kotchin
a801
who
first
studied the problem of flowing through
in
a
model formulation, wherein the boundary conditiona at the
entrance were formulated for a velocity vortex. The two-dimension-
al non-stationary problem with a given vortex at the inlet part
was studied by
V.I.
Yudovitch
[147].
1.
FORMULATION OF
THE
PROBLEM
AND
THE
BASIC
RESULT
Let
of three
ordinates of the
51
point through
x
=
(x,,
x,,
x3>
E
Q
the time
through
t
E
[O,T]
and
assume
Then, let
?I
be a
unit
vector of the external normal to
I?
On the
l'
surface let us introduce the local curvilinear orthogonal co-
ordinates
q
=
q(x)
=
(q,, q,,
4,)
with the axis
q,
directed to
the external normal. The Lame coefficients of the transition to
the system of coordinatea
Q
are denoted by the letters
52
be a bounded domain in
R3
with the boundary
I?
,
composed
parts: an impermiable wall
I?'
,
the part of flowing in
I?'.
Let
us denote the Cartesian
co-
I"
and that of flowing out
Q,
=
a
x
(0,
TI,
s
=
r
(0,
TI,
si
=
ri
x(~,~~,
i=
O,I,~.
?Ii:
150
Chapter
4
The components of the velocity vector
T‘u
be denoted through ui,
i=
1,
2,
3,
while in the variables
through
W
the velocity vortex vector be denoted through
2,
3,
pressure
-
Zhrough
P,
and the vector of the external
mass
forces
-
through
f
.
Further, let rot, div,
V
and
A
denote the differential opera-
tors rotor, divergence, gradient and Laplacian.
If
these operators
are calculated for the
functions which are given on the surface
F,
then the corresponding symbol
is
supplied
with
an index (for
instance,
divz
,
V,
indicating that differentiation
is
carried
out over the tangential coordinates
9,.
In this case the
vector field considered on
r
,
is
decomposed into the tangential
Let us consider the flow of an ideal incompressible liquid in the
domain
52.
To
begin with, let us
limit
ourselves to the case
of
a
homogenious liquid with
its
density
p
constant throughout,so
that
we can assume
p
3
1.
In the coordinates
x
a system of the Euler
equations describing the motion
is
as follows
in the coordinates
X
will
4
-
3
vi,
i
=
1,
2,
3.
Let the corresponding components of
wiand
’
Gi
,i=l,
4,
and
-9
-+
V
=
(Vl,v2)
and normal
v3=
un=
(u.
“n
)
components.
z
at
3 3
3
-9
at
(1.1
1
-
+
(
U.
V)u+
Vp
=
f
,
div
u
=
0,
(x,t)
E
4
.
At
the initial moment of time
t=
0
the field of velocities
is
assumed to be known:
3
-9
3
=
uo
(x),
div
uo
=
0,
x
E
Q
.
Jt=o
(1.2)
On the impermiable part
condition
is
fulfilled
:
r0
of the boundary
r
the unpenetration
u
n
=
G
B
)=
0,
(x,t>
E
so
.
(1.3)
On the part of the in-flowing
I?’
the total velocity vector
is
given
+
u
(x,
t)
=
Z(x,
t),
(x,
t)
E
s’
(1.4)
where
g
is
a vector field on S’such that
-3
g
e
g
n
On the part of the out-flowing
locity
is
known
It
is
also assumed that the boundary values gand
r
of the nor-
mal
component
un
obey the necessary condition of fitting the
equation
divz
=
0,
i.e.
=
(
2
-
n)
5
const
<
0,
(x,t>
E
S‘
.
r2the normal component of the ve-
un(x,t)
=
r(x,
t)
>
0,
(x,t>
E
s2
.
(1.5)
I
g(x,
t)m
+
I
r(x,t)d
r=
o
Y
t
E
[o,
rj.
r’
r2
Finally, to determine the pressure unambiguously, let us add the
re
gu
irement
Correctness
of
the Problem
of
Flow
151
(1.6)
As
far as the known functions
is
concerned,
let
us set that they
have the following properties of smoothness:
-3
-+
f
E
c"
@(Q>,
UO€
GI:
a(
Q),
(1.7)
g
EC'+
r
EC'~
01(s2>,
o
<
0:
<
I
.
It
should be noted that for the
sake
of simplicity the conditions
-9
g.
and
r
are overstated.
Besides, let
us,
impoe%g no limitations on the generality, assume
that the vector field
f
is
a solenoidal one with the nero normal
component on
r:
-+
-3
div
f
=O,(x,tjE
a;
f-
GO,
(x,t)
E
S,
(1.8)
11
since the gradient part
*f
can be referredt to
VP
assumed that there
is
an accordance between the ini
dary data, i.e.
It
is
finally
ial
and boun-
Here
pO(x>=
P(XIt)l
tzo
Newmann problem
are defined from the solution to the
3
a,:
au?
c-
-J,X€Q,
i,j=l
ax. ax.
JJ
(1.11)
-3
-
=
-
((
uo
.v>
uo.
n
>-
-
-+
+
,xcr;
aPo
an
at
t=o
0,
x
E
ro
r,
x
E
r2
The solution of the formulated problem (1.1)
-
(1.6)
is
underetood
as a claesical one, i.e. when all the functions and their deriva-
tives, included
in
(1.1) are continuous.
Our
aim
is
to prove the
correctness
of
this problem on a finite len th
of
time (suffici-
ently small). The boundary problem
(1.1)
-
71.6)
is
characterized
by two important peculiarities. The
first
one
lies
in the fact
that when reducing the problem to an operator equation
it
is
ne-
cessary to ensure the property of the operator continuity. If,for
instance, we shall construct the operator of the problem in a na-
tural way of linearizing equations
(1.1)
152
Chapter
4
(1.12)
-t
+
a%
+
-
+
(wo*
0)
%+
0)
wo+
vp
=
f,,
div
w
=
0
,
at
-
+
++
where we consider
as
a
w0
mapping,
i.e.
w
=
Aw0
then the
property of the operator continuity
is
far from being obvious.
When obtaining estimates for
w
say, in the norm
L,(Sa)
by way of
multiplying (1.12) by
w
the required function
wo
must be
of
greater smoothness,
as
minimum
L,(O,
T;
W'm(Sa))
It
should be
noted studying the correctness of boundary problems for the linear
system (1.12) has been carried out only for the case (see
L45,
48,
120
11,
when the boundary
S
of
the domain of defining the
solution
is
a characteristic surface. In this case
it
is
sufficient
to aet only one boundary condition on
S
while in a more
general case
(Go
$
0)
nome additional boundary conditions must be
sef
on the part of in-flowing
I"
It
should be emphasized that no
general theory of mixed boundary problems for systems of equations
of a composed type, of which (1.12)
is
an example,
is
available at
present.
The second peculiarity of problem (1.1) -(1.6)
is
associated
with
a possible presence of lines of contigency
of
various parts
Po.
The present work
is
mainly devoted to the
first
aspect of the
problem, i.e. to the question
of
reducing the problem to an ope-
rator
equation with a continuous operator. Therefore, we shall
ignore the eecond peculiarity by making a simplifying assumption;
we shall consider the case when
Sa
is
a simply-connected domain,
the parts
PoIrl
and
Pz
do not intersect and each component of
connectivity
C,
of
the boundary
I?
is
a closed surface of class
sed, when the parts of in-flowing
A
+
-3
+
-D
wOE
0
I"
and
I"
of the boundary
I'
.
cz+
a
.
,O
<
0:
<I
(below in
B
6
more general variations are discus-
and out-flowing P'can touch
I'O
1.
Let us formulate the basic result
of
Chapter
IV.
Theorem 1.1. Under the above made suppositions problem (1.1)
-
(1.6) has a unique local in time solution of class
I'u
€Cl-(<)
,
v
p
E
c
"(4)
*
The key moment in provixlg this theorem
is
to transform the formula-
tion of (1.1)
-
(1.6) to an equivalent one, whose variants are the
subject of the next paragraph. The idea of such a transformation
is
a known in the theory of non-linear equations method employing
the so-called extended (broadened) system of equations, obtained
from the initial system by differentiation. This method ha8 been
especially effectively used in the theory of hyperbolic equations
(see, for instance,[ 37, 117
1.
In the problem
we
are going to con-
(1.1) the operations rot and div. Verification of the equivalency
of
the initial and the transformed problems
is,
by opinion
of
the
authors, a key point in proving Theorem 1.1.
eider here, the extended aye
1
em
is
deduced by applying to equations
Correctness
of
the Problem
of
Flow
153
2.
EQUIVALENT FORMULATIONS OF
THE
BOUNDARY PROBLEM
Let us apply the operations rot and div to the pulee equations
(1.1)
and get the system
(2.1
1
a;
3
-t
-t
-t
-
+
(
u
at
-
V>
G
-
(w
V>
u
=
rot
f
,
3
aui
au.
i,j=l
ax.
ax.
J1
Ap=-
C
-
2,
4-t
rot
u
=
w
,
aivii
=
o
.
(2.2)
(2.3)
The boundary conditions on
will
be also transformed with equa-
tions
(1.1)
which are previously written in the curvilinear
coor-
dinates
q
182,
Ch.111
:
avi
2
~
2
a2
v.
v.
aHi
v?
aH.
-
+LA
_._
-J
1
A)=
at
j=I
!ij
aqj
H~
1-1.
as.
H~
:I~
asi
JJ
Here
Fi,
i
=
1
,&2,
3
are the
f
components
in
the coordinates
9-
Let
us
consider the third equation of this system,
i.e.
the pro-
jection
of
the vector equations
(1.1)
on the axis
along
2
excluded through the equation of continuity, which has the follow-
ing
form
in the variables
q
(see
82,
Ch.1
)
q,
directed
to
F*
In this equation the derivative
av,
/
as3
will
be
a
a
a
-
(v~~I,
H,)
+
-
(v2
HI
H,)
+
a%
a
92
(v3
HI
H2)
=
3
.
As
a result, we get the relation
It
should be recalled that the left-hand part contains the velo-
city derivatives only along the
coordinates
9,
and
9,
which are
tangential to Therefore, considering this relation on the sur-
face
1’
the
tangential components
V. =(V,,
v,)
and the normal component
V3
a8 well as the operators
div
the form
and uskg the expansion
of
the velocity vector into
--t
and
v,
we can transform
it
to
154
Chapter
4
2
1
1
ap
-
c
(v;+v;)-=--
-
i=l
H3
aq3
+
F3
Since everywhere on
r
the normal component
Un
=
V3
is
known and
F3=
fn
=
0
formulas: on
then on some parts
fi,i
ro
where
v3
=
0,
we have
=
0,
1,
2, valid are the
2
vfl
aHi
(2.4)
Here we take into account that
a/
as3
=H,a/
an*It should be noted
that the right-hand part contains no velocity derivatives.Together
with equation (2.2), relation
(2.4)
makes
it
possible to conclude
that the smoothness
of
pressure P
inside the domain
Q
up to the
boundary
is
one unit higher than the smoothness of the velocity
This fact was for the
first
time accounted for in the paper by
R.Temam
L193]
who studied the problem with the condition
of
-flowing through the whole of the boundary
I'
i.e. when
1''
=I?.
A
formula
of
type (2.4) was also obtained in [129] devoted to
Navier-Stokes equations.
Let us now consider the part of in-flowing, where the total velo-
city vector
5
=
"g
is
set.
Let
(g,,g,,g,
)
be
the
g
components
in the coordinates
9
with the normal component
g
denoted
without an index:
pressure derivative with respect to the normal
on
l?!
--
-
c
---
,
(x,
t)
E
so.
an
i=l
13,
Hi
as,
aP
non-
the
3
+-P
g
=
g3E(g
.
n
).Then we get an expression for
2
1
aHi
I-I
3Hi
93
+
c
(gi
+
g2)
-
-
,
(x,t)
E
s'
.
i=l
(2.5)
On the part of out-flowing
is
set, therefore
1''
only the normal component
v
,
=
r,
ap
ar
-t
+
an
at
--
_--
+
divz
(rvt
)
-
2(vz
-
Oz
r)+
Relations (2.4)
-
(2.6)
will
be used as boundary conditions for
equation (2.2). Besides, we add the condition
Correctness
of
the
Problem
of
Flow
155
I
P(X,
t>
dx
=
0
.
(2.7)
a
For
the elliptical system (2.31, considered with respect to
u,
to
set the boundary conditions
is
to set the normal component
-#
0,
x
E
I'O
i
r,
x
E
r2
g,
x
E
r1
(2.8)
For the hyperbolic system (2.1) there
is
no need to
set
any condi-
tions with respect to the vector-function
3
on the
parts
ro
andr2
ded in the domain
Q
we must additionally set the
IJJ
values,which
are obtained
in the following way.
First of all, by the two tangen-
tial components of the velocity we can find the normal component
of
the vortex. Indeed,
if
5,s
5,s
5,
are the components in the
coordinates
q
then on
r
where
vi
=
gis
i=1,2,
for the normal
component
<,=
(G
*%>
we have the formula (see
(2.49)
Ch.1)
un
=y
(x,tl, (x,t>
E
s;
y
I
On
where the characteristics
of
equations (2.1) are inclu-
-f
1
Besides, equations (1.1) allow one to find two relations on
r1
which relate the tangential components
GI
and
5,
with the pree-
sure derivatives with respect to the tangential coordinates
q1,q2
For
this purpose
let
make use
of
the Euler equations presented in
the Gromeka-Lamb form (see [82, Ch.IIIJ
1:
art
1
+2
+
-P
-+v(p+-l
uI)-[uxrs]
=f.
at
-
2
In the projections onto the axes
q,
and
q,
on the surface
where2
=
3
,
these equations result in the equalities
I'1
P
+
g,
5,
-
F,,
a
1-
ag2
H2
3%
2
at
g
5,
=
1
-
(P+
-
IB
12)+
-
(2.10)
The obtained formulas (2.91, (2.10) together with the initial data
-3
--*
*
(2.11)
w
=
wo(x)
3
rot
uo(x),
x
E
,
are the boundary conditions for the system of equations (2.1).
Let
us
prove that the initial-boundary problem (2.1)
-
(2.11
1
is
equivalent
to
the problem (1.1)
-
(1.6).
Lemma 2.1.
If
(
j,p,u)-
is
the aolution
of
problem (2.1)
-
(2.11)
and such that
($,PI
EC'(U>
then
P>
is
the solution to prob-
lem (1.1)
-
(1.6).
156
Chapter
4
Proof. From equations (2.1) and (2.3)
it
follows that
Hence, the equations of type (1.1)
al'u
+
++
at
+
(u.0)
u-f
=-
vx
.
(2.12)
-
are valid, but, generally speaking, with a different "pressureff
x*
To
prove the equivalency
is,
fFrst,
to
prove the coincidence of
the functions
p
and
x
and, second, to check
if
the tangentzal
components
of
the velocity
(v1,v2)
obtained from problem
u
(2.1)
-
(2.111,
are equal to the
I"
tangential components of the
vector
7f
on the part of in-flowin (Since
in
the boundary con-
ditions (2.4)
-
(2,6) and
(2.8)
-
72.10) the values
of
the tangen-
tial components
3t
on
17'
are not given in
an
explicit
form),
Let
us
ap ly the operation div to (2.121, followed by subtraction
from (2.27. In this case we get that the difference
cp=
p-
x
is
a
harmonic in
Q
function at all
t
E
LO,T],
A9
=O,
x€Q.
(2.13)
On the parts P'and
r2
repeating the derivation
of
formulas (2.4)
and (2.61, we find
acp
--
(2.14)
-
0,
XE
ro
u
r2
.
an
On the boundary
I'1
we shall seek the value
of
an/ an
in
the same
way
as
equality
(2.5) was deduced, only instead of the tangential
componente
of
the vector
3
we shall use the components of
vector
the
-+
vt
Hence, on
r1
for
cp
valid
is
the relation
acp
--t+
+--t
-
=
divT
Cfi
(e+
-
vT
>I
-
2((%
-
vT).vT
g
)
+
an
2
I
aHi
,
XE~'
(2.15)
Besides, for the tangential derivatives from
IT
on the equali-
ties of type (2.10) hold, wherein
g,
and
fi,
are substituted for
2
+
C(gi_?'Vi)--
-
i=l
H
,Hi
a%
v
and
v
Hence, on
r1
1
(2.16)
Correctness
of
the
Problem
of
Flow
157
The value
of
the normal oomponent
of
the vortex
and, hence, together with (2.15). (2.16) the following equality
ia
valid on
1”:
5,
is
given on
I”
Finally,
as
far.as
in (2.12)
an arbitrary additive
time
function,
let
us
assume
TL
is
determined to the accuracy
of
I
cp(xY
t)
dx
=
0
,
t
E
[OyT].
(2.18)
Q
From (2.13) and (2.141, integrating by parts, we get
acp
I1
v
cp
112=
I
cp
-
IT
(2.19)
rz
an
where
11.
I(
is
the norm in
L
(
Q)
Let
us
exprese
acp/
an
from
(2.151, and then in the integ6al
+-+
I
2
i
cp
div,
LG
-
v,)jdr
rl
make use
of
the Gauss-Oatrogradeky
formula.
As
a
result,
we
get
+*
I=-
I
f;(
g
-V
1.
v,
cp
)dr
.
,
r’
Then, expressi.ng
D,cp
lation
from equalities (2.161, we obtain the re-
-++
I+
I
+
3+
3
(2.20)
*V,(l
%
1-
IV,
I
))dr+
I
~p
((h
-
V,
)*
w)dr
rl
-+
Here
W
is
a
vector with the components
(si
-
vi
asi
H,
fli
as,
i
=
1,2
.
(2.21
wi=
-
--
Let
us
estimate the terms
in
the right-hand part
of
(2.20)
one
by
one. Using the property
of
the derivative
gi
boundednese and the
conditiong
5
conet
<
0,
we
get