Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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118
Chapter
3
and then change the order
of
integration
According to Lemma
1.2
of the preceding paragraph, we have the
relation
(Iul(t
+rl)
-
Ul(t"
R
5
cr11/2*
'T
Therefore,
2:
5
Chl/'.Thuo, vie get the estimate
2'
5
C(hl"
+
A').
4
And, if
h
=
O(A
)
then
2
I
C
A.
Hence, in formula
(2.6)
we have
I
'1
1
'
'11
('1Ph)A2A
311
2
,RT
'
(2.8)
The theorems of continuity on average of the functions from
L
and the theorems on the convergence of the averagings
afford th&t
I,+
0
at
A
--t
0,
h
=
O(A4
)
.
te that
Ik
.+
0,
k
=
2,
3.
it,
one can easily verify the property
of
continuity
of
p
with respect to
t
in the norm
L
(Q).
For this purpose let
us
employ the equivalent definition of
a
generalized solution, i.e.
identity
(1.5")
In
a
similar way one can demonstra-
Thus, equality
(2.5)
is
proven.
Using
9
+,
2
-b
t
I
(P
(t2
(I,
(t2
I)
,Q-(P
(t,I t+(t
1,
,Q=
I(P(z>
,4Jz
+(U
-D
(I,)
j2
,Q
d
z
9
which
is
valid
for
any
smooth function
(I,(x,
t).
It
means
that
if
(I,
is
independent
from
t,
then
which
is
valid for any smooth in
Q
functions
+(x).
It
means that
p(t2j
--f
p(t,)
weakly inL,(Q) at
t,
+
t,.
But, in line with
equality
(2.5)
we have
lull
IIP(t2)1I2,Q
=
IlP(t,)l12,Q
t2
jt1
Hence,
p(t2)
converge to
p(t,>
in the norm L,(Q) and, due to
the boundedness
of p(t)
in the norms
been shown in the beginning of paragraph
1,
at
p(t)
of
Lq(Q),
15
q
<
m.
As
has
a weak
limit
t
+
0
is
po,
Now
we have established that the weak
limit
is
in
Initial-Boundary Value Problems
119
fact
a
strong one. Therefore, setting in
(2.5)
t,
=O,
t,
=
t
we
get equality
(2.11,
which
is
the end of the proof of
Lemma
2.1.
A
consequence of this lemma
is
the convergence of the approximations
only
a
weak convergence,
as
has been established earlier. Indeed,
for
any
n
=
1
t
0
at
all
t
E
[O,FJ
the equality
(2.1
)
is
also valid
{P"
)to
P
in
the
norm
Lq(Q)
with any
q,
1
5q
<
m
and not
In particular,
at
sequence
{P
)has the property
q=2
the space
L,
(Ci)
is
a Hilbert one, and the
n
pn+
P
is
we&inL2(q),
II
pn
II
P
II~,~
Hence, true
is
the convergence in the norm L,(k) i.e.
Por an arbitrary
q
>
2
due to the boundedness
of
{p"
}
we have the
inequality
dq
,
I
(181-
m>
1-2'q
II
pn
-
P
I1
2,k
n
II
pn-
P
11
(Iscz
which demonstrates the convergence in
In
its
turn, the property of the approximations
{P
}makes
it
possible to generalize the general theorem
1.1
for the case
of
a
more complete model of a non-homogenous liquid, when the medium
particles
of
various density have, generally speaking, various
viscosity
p= p(
p).The e uations of momentum conservation in
this case are
as
follows
1972,
1951:
I,,&%).
a
ui
at
p[
-
+(u.
V)Ui
The function
P(P
ap
5
a
aui au.
ax.
axi
a5
j=I
ax.
J
f
-
=
c
-
[p
(p)(
-
+
2
1j.t
pfi
,
J
i
=
1,2,j.
then must obey the conditions:
The property of strong convergence
of
the sequence of approxima-
tions
{p
}
allows
one
to
carry out
a
limiting transition in the
non-linear terms of type
P(P
J
including the proof of Lemma
1.2
on compactness
of
{'J
}
in
Lz(d),
are essentially the same.
n
n
a
U:
The remaining considerations,
-+
n
120
Chap
fer
3
2. Estimation of higher derivatives
of
the
velocity vector
Let
us
obtain the a priori estimates for the solution of problem
1,
when the fluid motion
is
plane-paralle, i.e.
2+
x
=
(x,,
x,)
E
Q
C
R
u
=
(U1,
0
I
-i
+
f
E
Lz(q>r
dl
=
Q
X
(O,T),
Uo
Lemma
2.2.
Let
neralized solution of the two-dimensional problem
l
has the fol-
lowing properties:
J1(
Q).
Then
a
ge-
-t
Z(t)
E
L~(O,T;J'(
a))
n
L2(o,T;ivi(
Q>>,
Ut
E
~~(d
Proof. Let
us
demonstrate that for the Galerkin approximations
{@
}the
a
priori estimate
is
valid
(2.10)
with the constant
C2
which
is
independent of
N
.
Then this rela-
tion
will
also hold for the chosen sequence{
u
land for the
limi-
ting function
11-
Therefore, the momentum equation in system (1.1)
is
satisfied not
only
in
the sense of the integral identity
(1.4)
but in a strong sense
as
well, i.e. almost everywhere
in
%.
The
derivation of estimate
(2.10)
is
analogous to that of the estimate
for the equations of
a
homogenous fluid and
is
essentially based
on the known properties
of
the operator
z=
pp
A,P-
is
the opera-
tor
of
mapping of L,(Q)onto J(Q).The difference from the case
whe.
p
in the fact that for the de-
rivativep
~lre
dariori
estimate
(2.703
is
proved not only
by
multi-
plying the equations by
%''
-tn
-+
const
>
0
from
188
J
is
on1
but also by combining the estimates
+.
j
=
1,2,...,
i:
,
let us multiply the j-th equation by dclTj/dt
respect to
J
:
and
sum
up with
+,y
-'IT
(
(
Z*J
V)U"
,
%
)2,Q
Applying the Hhder inequality and using the property
m<
PIi<_
IIs
we
have
Initial- Boundary Value Problems
121
If we multiply the corresponding equation (1.10)
by
A.
c
and also
sum
it
up, we get
J
Nj
+
Here account
is
taken of the fact that
Jlj
are the eigenfunctions
N N+.
3.
of the operator
A: A+J
=
h4
JIJ.
J
According to the Hglder inequality, we again have:
In line with the theory
of
imbedding
(Ilw
with the property
of
the operator
A
(see
[88])
5
C(lw
~[~~~~~~''x~[~~~
))
N
(2.12)
and with the estimate (1.13). we deduce
Let us now apply to the last addent the Cauchy inequality with
E
:
The righ-hand part of relation (2.11)
is
estimated in the same
way
:
Reinforcing the above inequality with (2.13) and applying then
the Cauchy inequality with
E
we finally get
Herefrom, in accord with the Grounwall inequality, and accounting
for the estimate at out disposal
(1.131,
we conclude
Returning
now
to
(2.12) and
(2.13)
and integrating
with
respect to
t
we get inequality (2.10).
Thus,
Lemma
2.2
in proven.
122
Chapter
3
Let
us
now dwell on a general case of the three-dimensional motion.
Here the estimates
of
type (2.10) can be proved only 1fin-3the small':
Lemma
2.3.
Let
three-dimensional problem
1
has the properties:
for sufficiently small values of
T
or
113~11
and
II
f
-3
--t
J1(Q>
f
E
L,(Q) and
uo
6
J'(Q).
Then the solution of the
-3
-3-B
u(t>
Ek
(Ole;
J1(
011,
(ut,
uxx>
E
L,(Q
1,
if
one of the following conditions
is
met:
where
K1(Q)
and
K,(Q)
are the constants of the theorems of
imbedding.
Since this stateme
t
is
prove in the same way as in the case of a
homogeneous fluid
f88,
Ch.
VII,
we are not going to discuss
it
in
detail. The only difference
is
related to the prespce of the
variable coefficient
is
deduced by multiplying by
p-
nu.
In this case,
however, an additional difficulty arises, since for the Galerkin
approximations no additional multiplying by
made, as such a function
is
not allowed in equalities (1.10).
Therefore, as was the case in Lemma
2.2,
the eq$imate
is
obtainef
through a combined multiplication
-
first by
ui
,
then by
.
3.
Density continuity by Hb'lder
Let
us
again consider the case of a plane-parallel fluid. (For the
three-dimensional problem the further results are also valid, but
onlyf5n the small", since local are the estimates
of
the hie;her
velocity derivatives.
)
Lemma 2.4.
At
almost all
t
E
LO
,
TJthe velocity vector
u(x,
t),
x
E
(1
c
R'
the variable
X:
p
at the higher derivative
A.
The estimate
"LI
Au,
rather than
(p")-%
T'"
can be
-3
obeys the Lipschitz's quasi-condition with respect to
(2.14)
where xl,xp are arbitrary points from
Q,hIl
is
the Q-dependent
constant.
The
p
oof
of this lemma results from the known theorem of imbed-
ding
f5d.
Lemma
2.5.
Let in the two-dimensional problemIpo(x)
6
C
(
Q),
1
T'uo
6
J'(Q>.Then the density p(x,t)
is
continuous, by Hb'lder,
Initial-Boundary Value Problems
123
-
in
Q,
=
Qx
[O,Tl
with the index
@,o
<
p
<
q
which
is
determi-
ned by the quantities
11
po
Ilci(a),
Proof. Set (2.14) of th velocity to the Osgood theo-
,T
and
Q
.
rem
of
uniqueness (see
PllO,
Ch.
a unique
sol-
vability of the Cauchy problem on finding the trajectories
of
liquid particles
In
this
case the solution
Y=Y(
T,
X,
t)
is
a continuous, by
Hglder, function of the arguments
X
and
t
is
uniform over
-c
E
1
0,rJ
:
I
Y(
where
and the
Such an
ma
2.61
constants
~~i~
and are obtained through
C,
and
C?
estimate
was
obtained in the paper by
T.
Kato
under the supposition that the coeffi ient
shitz -quasi-condition (2.14)
is
bounded on
70,
!?]
only the property of summation
was
made use of. Let us make sure
in
it,
verifying the Hb'lder continuity
first
with respect to the
variables
X*
Let
XI
and
X2
be two arbitrary points from
9
and
let
I
xl-
x2
15
d,where
'i'*
d
=
eq
{I-
exp
{B.,
(1
u(t>i12
dt
}}
<
1.
$12(
sl)
If
'we assume
rence
Z(
T)=Y~-
y,
yl=
y(
t,xl,t),
y,=
y(
z,x2
,
t
then the diffe-
is
the solution to the Cauchy problem
-P
$2
d-c
-
=
u
(Y,,
T>-
U(Y,,
TI,
=
XI-
y2
At
IZI
<I
we have from (2.14)
L
-
.I(
t)
(Z
I(i-ln
(Z
I
>,
o
<:
z
I
I;
,
cl
Izl
dz
d
-
dt
In
(I-
In
lZ(
T>
I>
5
A
(T
>.
Integrating this inequality from
z
to
t
we have
1
Z(
T>
I
5
d-
I
x,
-
x,
IcL
<
1,
t
E
[
0,
'llJ
,
124
Chapter
3
T
where
Let
us
now estimate the continuity modulus with respect to time.
Let
t,
and
t2
y,
=
y(t, x,t,).
For
the sake of definiteness, let
t,
also assume that
XI
=
~~l~=~~.
Then, due to uniqueness, the in-
tegral curve
problem
CL
=
exp
{
-
I
h(t)dt
}
<
1.
0
be
two
arbitrary points
from
[0,
TI,
y,
=
y(t,
x,
t,),
>
t,.Let
us
y,(t) can be considered as a solution
to
the Cauchy
with the initial data at
T
=
t
Hence, as in the preceding case,
the difference
Z(
z)
=y,- y,
3s
the solution of the Cauchy
problem
4
LIZ
=
u
(Y,,
T)
-
U(Y,
,
T),
z
IT=t,
=
x
-
x'.
dT
By definition of x'
,
we have
t2
-3
tl
t2
t,
YEQ
X'
=
x
+
f
U(
yp(s,
X,
t,),s)dS.
Therefore,
/x-
x'I
6
J
m_ax
Ry,s)l(&'Jsing the imbedding
in
d:(
Qhn
C(Q)
and the Cauchy inequality, we get
Hence,
if
we assume
Itl-t2
I
5(
dCil
relation
*
then the inequality
b-
X'
15
d
holds. But in this case for
(z(Z>
I
valid
is
the
IZ((t)
15
d-
CLIx-
x1IU
5
C
(tl-t2
Is,
@
=
I/
2a:
,
wherefrom results the Ii&der continuity with respect to
t
.
In
view
of
all the above notes, the roof of Lemma
2.5
on the
Hb'lder continuity
of
density
p(x,
t?
results from the presenta-
tion
(2.15)
p(x,t)= P0(Y(9,X,t)>, PO(Y)E
Cl(Q1,
Y(O,X,t)
6
ce
(L),
-t
Indeed,
if
the velocity
u
is
a smooth function, then the above
formula gives a classical solution of the problem
Initial- Boundary Value
Problems
125
+
Through a limiting transition,
by
approximatin
U
with a sequence
of smooth functions, one can verify that (2.157
is
a generalized
solution in the sense of identity (1.5). Lemma 2.1 yields that at a
given
3
the generalized solution
is
unique and, hence,
is
defined
by formula (2.15). Lemma 2.5
is
proven.
4.
The existence of a classical solution
Theorem 2.1. In the two-dimensional problem
1
let
I'=
aR
E
C'?
(ql,
o<
a
<
I,
PO(X>EC1(Q),
2+
0:
+
CL
,a/2
uO(x)
EC
.
(a),
f(X,t)E
c
--t
and the following congruence conditions are met
+
0
div-)uo
=
0
,
u
1.
=
0,
+
+
p0(x>
f(x,
0)
+
PA
uo
-
V
Po
=
0
,
x
~r,t=~,
where po(x>
=
p(x,O)
is
the solution
to
the Neumann problem
1
P++
+ +
div(
-
V
pol=
div
[
a
A
uo+
f(x,
0)-
(uo
.V
)uo],
x
E
R
,
pow
P
dPO
+--t
+
+
_.
(in
lr
=
p(
A
uo
*
n
>+PO(
f
(x,
01
-
n
>,
x
E
r,
--f
n
is
a unit vector of the external normal to
I'
.
Then the generalized solution
is
a
classical one.
Proof. Let
us
turn
our
attention to the momentum equations and con-
sider them,
at
the given function
P(X,t>
E
C
(h)
as a linear
system with respect to
B
(
3,
p
1:
--f
-b
--t
-b
+
put-
pAu+vp=FSpf-p(u-V)~,
+
div
u
=
0,
(x,t)
E
'u,
,
+
-t
-b
I*?=
0
9
t
E
[
0,
'I3
i
u
=
U0(X),
x
E
sz
.
Denoting
y
=
1/2
min(a,B)let us demonstrate that the estimate
(d)
Czk+2
y,k+y
is
valid, where
I
*)
2k+2
y,k+
For
the case when
p
zconst
>0
this inequality was stated by
V.A.
Solonnikov [123, 1251.
For
the general case, when
p
const
denotes the norm in
126
Chapter
3
it
is
deduced with a standard method
of
dividing the domain
Q
into
a finite number
of
subdomains, in each of which the function
p(x,
t>
oscillations are sufficiently
small,
and of applying the
estimates from 823, 1251 with the Ilfrozent1 coefficients.
If
we apply the inequalities for the norm of the product,
and
allow for the kind of
F
and the bounded
lpl
from (2.161, we
obtain
2Y
9Y
(2.17)
Let us now make use of some interpolation inequalities and of the
properties of the generalized solution for estimating the right-
-hand part of (2.17) in euch a way that the formula
(2.18)
u
-b
-+
14
2+2y,
l+y
+
IVPIzy
y
5
L',
+
b1,Iul
2+2y,,+Y
*
is
valid, whherein
0
<
n
<
1.In this case, with the
Young
inequality
we get the estimate
IU~,+~~,,+~
+
IVPlzy,y
the second and third addents in the right-hand part of (2.17) sepa-
rately. According to the theorem
of
imbedding
-b
5
E,.
Let us estimate
3
-b
-b
a
IUl0,Q
5
~I1Uxllp,Q
lbll;;;
at any p
>
2
where
a
=
1/2
p(p
-I>-'.
In this case, in line with
the Nirenberg
-
Gagliardo inequality (2.25) from Chapter
I,
we
have
Combining these two inequalities and employing the-property of the
generalized solution
4
Inax
IlUxl12,Q
5
CY
0
<t
5T
+
In
a
similar way one can get the inequality for
the following form
luxlo
which has
-b
62
1
+
6,(1
+
2y)
luxlo
5
cIul,+2y,l+y'
6*
=
(2.20)
2+2y
-b
-9
and
lu
I
2y,y
through the
norm
2Y
,Y
Finally, the estimates for
lul
in
C2+2y'1+y(~,,)
result in the relatione
Initial-
Boundary Value Problems
127
'
1
+y
cy
,
lfy
With the above relations and formulas
(2.191, (2.20)
we can rein-
force inequality
(2.17)
and,
as
a
result, get relation
(2.18).
In
this case,
as
can be easily calculated,
n
=
6,
+
6,
=
6,
+
6,
and
As
has already been noted, from
(2.18)
we have
Iu(~+~~,~+
and, hence, the density smoothness
is,
in fact, higher, Le.
p(x,
t>
E
C1(u>.
It
means that one can set
p
=
1,
y
=
1/2
a
the smoothness of the solution
is
just
as
it
is
given in the
theorem.
Notes.
1.
A
further improvement
of
the differential properties
of
the solution with growing smoothness of the initial data of the
problem
is
verified in the same way.
2.
In
the three-dimensional problem the existence
of
a
smooth
so-
lution has been established only "in the
small"
[
911.
3.
In
the class
of
smooth solutions the theorem of uniqueness
is
valid (see
L
166J,
as
well as
[
91,
Theorem
3.1J
1.
0
<
n
<
*;
at any
p
>
2
and
y,
0
<
y
<
1/2.
-t
5
Id9
i.e.
3.
UNILATERAL
BOUNDARY
PROBLEKS
1.
Formulation of the problem
Let
us
consider the flow of an inhomogeneous viscoug liquid
through a bounded domain
52
of
a
Euklid space
R
.
Let
l"
deno-
te a part
of
the boundary
I'
=
as1
where the inflowing of the
li-
quid into the domain
51
takes place,
r2
be the portion of the bo-
undary where out-flowing
of
the liquid occurs, andrO be an imper-
miable part of the boundary,
=
52
x
(0,
T),
S
=
l?
x
(0,
T),
Sk
=
I?"
x
(0,
Y),
k
=
0,
I,
2.
Let us
also
assume that
on
r'the
tangential components of the velocity vector
3
are known (without
violating generality they can be considered equal to zero),
as
well
as the fluid density and the
total
pressurep
+
1/z
plsl
'.
On
the
portion
r2
are given and again one can assume these to be zero.
On
r0
we
set
the lbo-eliptl conditions, and at
t
=
0
we give the initial data.
Thus, our problem
is
to find
a
solution to the system
the velocity tangential components- and the pressure
a<
+
-#
-t
-t
p[
-
+
(u
*V
h]
=
p
A
-
vp
+
pf,
at
aP
+
+
-
+
(u
*v)p
=
0,
div
u
=
o
at
(3.1