Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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228
Chapter
5
and finally
--t
IIvph
[[Al,
Ql
<-’
{aKh
llpl,Q
(
+
+
(Ifh
Ilk
,
Q
}
*
1
Now for the proof to be completed
/h
I=(h:
+
hi
+
h:)’/‘
tend to zero on the
basis
of
the statements
1
s:
2,
Ch
1.
Consequently,
(4.23)
can be written
in
the
form
must
J
For the proof of
r
with
the help
the estimate up to
rl
c
ri
we straighten locally
of transformation
y=
y(x),
y,(I”)=o,
y3(~>
>
0.
where an increment
h
is
only
gi-
and continue
it
by
r:
with
the method described
Deduce a function
Zh=
ph- poh
ven for
y,,y,
in
theorem
4.1.
In the same way we are led to the case already con-
sidered
for
pyl,py2
and
py3
is
evaluated from the theorem.
Si-
milarly, we consider
1
=
2
.
“
Remark
4.
Theorem
4.3
will
essentially be used for investigating
smooth solutions
of
the joint filtration problem in a 9simplifiedff
formulation:
1-
-3
Let
j:
EL
,,(%),
(ox
f,
p
E
~~(4~)
(or
E
L
(1
1).
qP
for any solution p(x,t)
crease infinitely with
under corresponding conditions on boundary data.
5.
Smooth solutions
on
t
parameter
Let
and we
will
evaluate
Pt
by corresponding increments
of
the given
functions
satisfies the
fol-
lowing
similar
(4.11,
(4.2)
problem:
from
Lm[u,~z;,ji(
Q)
derivatives
lIx
p
E
E
L1(,‘)
(v
1+1
P
E$,m(k))
respectively), where
h=
h(q)
in-
q
to grow.
It
is
also true
for
regions
Q6
P
.(x,t>
=
p(x,
t+
TI-
p(s,t)
*
li,
f,
po,
R.
It
is
evident
that
pT
-t
div(K
V
p
=+
Q
)
=
0,
x
E
Q
9
-+.a
(4.24)
..
P
lr2
=
po,=,
(K
VpT
+
G)
n
lr,
=
-
I$
,
where
G
=
f
+
K
vp.
One can apdy theorems
4.1
-
4.3
to evaluate
ded
in
Q,
which satisfy Helder conditions over
t
integrally.
These saturation
s(x,t)
properties correspond to the case
of
united singularity problem (see the compactness principle
$
2).
44
z
p<Let
K,f
(
s(x,t)
respectively) be mesurable functions boun-
Filtration
of
Immiscible
Liquids
229
where
u
<
n
%
/
(n
-
g>,
n
2%;
u=m
,~1,>n.
The proof immediately follows from the inequality
of
theorem
4.2
(4.26)
%,Q
+
+
115
1Iq,r1
9
9,
=
so
(1
/(Y
-
9,
1,
which
s
to rise to a power
of
A
and to integrate over
tE
LO,
ding theorem
of
JJ1
in
4,
.mote that for
n=2,s,>
n,
A=m
l‘-
T
f*
The
first
estimate
(4.25)
is
the corollary
of
the imbed-
and with the theorem
4.1
taken into account
it
follows
that
YO
P(x,~>
E
c
’(w,>,
0
<
P
<
min
(
a,
(%D
/
yo
>.
5
IIJt2
,
q<w
.
Theorem
4.5.
Let
tion
of
the problem
(4.11, i4.2)
Then
a0
In addition,
if
I?
E
H1Z1
,O
51
5
2,
p(x,t>
EL
,tO,T;
‘/J:(Q)]-
be a generalized
solu-
(K(x,t), ?(x,
t))
E
C
-
v
p(x,
t>
EC
(k’)
,
a.
>
0,
c
%
.
and
We begin the proof with
inequality
(4.26)
exists for any sufficiently large
‘I,
>
q
and
consequently
for
>
n
1
=
0
Under the theorem conditiona the
N
i.e.
p(x,
t)
satisfies the Helder condition over
x.
Similarly
having theorem
4.3
in view the case
12
1
is
considered.
230
Chapter
5
5.
ON
DISTRIBUTION
OF
PHASE SATURATION
ON A
GIVEN
FIELD
OF
A
SUMblED FILTRATION
VELOCITY
1.
Problem design
The present paragraph studies the properties
of
problem solutions
on determining a reduced saturation
s(x,t)
for a given summed
filtration velocity
?(x,t>
Possible case8
of
independent determi-
nation
of
the latter have been described in
3,
$1. The existence of
a generalized solution
of
a general problem was established in
Si2,3. We have
0
5
6,
5
S(X,t)
51
-
6,
5
I,
(x,t)
E
&
(5.1)
And now let
us
consider the problem for
as
++
a
-
=
div
(K0a
vs
-
b
v
+
P
),
(x,t)
E
L+,,
s
ls2=
so,
(KO
a
vs
-
b
v
+
I!)
n
Is,
=
li
,
s(x,
0)
=
so(x,ii)
(v
n
Is,
=
div?
1
at
--t
+-t
++
=
0
),
(5.2)
(5.3)
(5.4)
Here we suppose
e
summed filtration velosity
$(x,t)
be given. The
exietense
of
solution
s(x,
t)
E?lz(-)
n
L
,(%)
(5.4)
is
proved.
Now
over purpose
is
to investigate
the
uniqueness
and
the smooth properties v(x, t) with respect to the given functions
of
problem (5.2)
-
+
u(X),
Ko(X>,
a(x,S),
b(S),
F(x,s),
s~(x,
t).
2.
The solution properties for parabolic equations
We'll give some facts from the theory
of
parabolic equations.
Theorem
1.
Let
Q
be bounded domain with the boundary
r
c
H2.
There
exists the generalized solution
of
the problem
(5.5)
au
-9
--
at
-
div
(
v
u
+
GI,
(x,t)
E
Q
;
u
Is
=
u
it=o
-
-0,
for which estimates are valid
T+
It
IIlJ
(%)
5
By
It
II
q,u
9
(I
(5.6)
where
.%I
cc
and
A'
=
A'(q,q)-
ie a continuous function over
q
and
hT
=
1
.
The pdof
of
the theorem
for
q
>;!results directly from the pro-
perties
of
thermal potentials
of
the corresponding problem, and
for
y
=
2may be established (assuming the existence
of
the
solu-
tion with the necessary smoothness) when integrating by parts.
Actually, multiplying (4.5) by
u(x,t)
we have inequality
cl
Filtration
of
Immiscible
Liquids
23
1
hence after the integration and evaluation of the right side
by
the
Young
inequality
and in the left side (5.8) one may take vrai max over
Multipluing (5.5) by
after common operations we are led to the inequality
t
E
[0,T].
is
a smooth finite function),
5'
u(
E(x,t>
+
1105
It
I
11;.
Q
}
9
from which the internal evaluation
(5.7)
results.
Now consider a generalized solution
of
the equation
au
u(x,t)
E
V,(%)
n
La(%)
+
(5.9)
--
-
div
(K
V
u
+
),
at
satisfying
it
in the sense of the integral identity
Theorem
5.2.
Let
of the equation (5.9), and
u(x,t)
E
V2(&)
n
La(,)-
be a generalized eolution
(5.11
and
q,r
are arbitrary positive numbers such that
Then
.i/
1:
+
n
/
q
=I
-
w,
q
E
[n/
(I-
M),
a],
(5.12)
II
*I1
01
5
all
llq,r,k
+
IIU
l12,Q
1
9
(5.13)
+
-
c
(a')
where
a=et
(q,
r,n,M)
>
0,
u1
c
.
In addition
if
u
=
0
on
Soc
S
then (5.13) occurs for
=I
so.
The proof
is
based on the lemma
2.7
y2
ch.
1
keeping the admitted
symbols.
Assume
cp
=E2
u$l)
,
w(l)
r
[niax(i
II
6-1,0)]
6
smooth function
5
(5
E
[O,l)]to posses the properties mentioned
in lemma 2.7
y
2
ch.
1.
And symbol
6
means averaging over t.Sub-
stituting
by parts and common limitting transition
for
6
+
O(see
y
3) we
come to the inequality
-
where a
q(x,
t)
of the given type in (5.10) after integrating
232
Chapter
5
to+
z
+
c
{
I
1
[lw(1)12
(
I
V5I2
+
I
55,
I>
+
to
%,p
-9
+
Id
1'
5'
]
dx
dt
}
9
U:
=Up
x
(toy
to
+
t)
.
The
last
addend in
it
is
evaluated in the following way:
Consequently, inequality
(2.38)
b
2
ch. 1 holds with
2q(
n+Zx)
2r
(
n+dx)
+
9,
=
n(--
;
r
=
;
H
=
2%;
y=
I+
lid
11
2.
n(r-
2)
On
q9rY%
0
Thus,
in
accordance with lemma
2.7
u(xyk)
E
C
"(%'>
and evaluation (5.13)
is
a
corollary of the problem linearity.
2
ch. 1
5
on
so
is
sufficient to prove (5.13) in
%'
3
So
.
Theorem 5.3. Let
K(xy
t)
satisfy (5.11) and additionally
)
5
6i
1 1"
(
11':
IlC(,)Y
IIDx
'19
Dx
II%yy
where
%>
2
are numbers large enough. Then for the solution of
the equation (5.9)
u(x,t)
E
V,(k)
estimate
is
valid
1+
1
l+
(5.14)
IID,
UII
5
c
(I1
ux
dllo,,$
IIU
II
1
Y
4YY'
CIS4
where
C=
C(q,L,
%)
and
q
5
s,
for
l=Oy
q
<
s,
and
1
21
.
Besides
if
u
=
0
on
So€
S
correapondance conditions are
satis-
fied fied up to the order
[1/2]
(see [92, page
96.21
then (5.14)
occurs also for
3
SO
.
The proof
is
similar
to the theorem 4.1. We
begin
directly from
the problem
-9
ult
=
div
(K
vul
+
d1
)
(x,t)
E
a
-
1
(5.15)
-f
andv
is
a
solution
of
the problem
(4.5)
with
f
=
K?'u
5,-dv5,-
-
UXlt*
Consider that
lc..
(x
)
=
6..
lJ,
x1
E
Qi
and with
q
already
fixed diameter has been chosen from the condition
=J
1
Filtration
of
Immiscible Liquids
233
(5.16)
Considering
u1
as
a finite solution of the problem
-+
2
=
div
[
v
"1
+
G1
+
(K-
i?;)
vull
,
due to (5.6) and (5.15) we have
Sununation
of
these equalities over and evaluation
of
V
V
with
the help
of
(4.5*)
-
(4.7*)
similar
to the theorem 4.2 yields
(5.14) with
1
=
0
can be taken from half-interval
(19
s,
1.
Now let
=
1.
Consider functions
satisfying an equation
Uh
(x,t>
=
Ih
I-'[u(x+h,t)-
u(x,t)],
+
--
:2
-
div
(Kvuh
+
I(
vu
+
Gh
.
12
According to the previous
where
(1
<
ao=
Y'io
A%-
q),
.X'e
yll
a
*
here
-i
(5.18)
II
vu
llA,~*l
'
C(ll
Ilk,%
+
IIU
I1
1,
E
(0,
w).
1
I+
tending
to
(5.17) due to
1
Y
2
ch. 1 yields
or
using (5.18) and
a
final evaluation (5.14).
If
u
=
0
on
it
Is
sufficient to take
be repeated up to any finite
3.
A
regular problem
of
a two-phase filtration
Consider the flow of-inhomogenious liquid, in which conditions
(2.7) and
sO(x,t)c
6,
1,
bi
>
0.
It
provides the zones
with residual saturations to be absent the equations (5.2) to be
uniformly parabollic respectively. Thus
for
these solutions for
almost
all
(x,
t)
E
q,
Soc
S
II
5,
)60=1
.The described process can
1.
I
In
u5,
5)
/
15
l2
I
5
l.i(
so
96,
).
(5.19)
Theorem
5.4.
Let conditions (i)
-
(iv)
2
be satisfied (existence
of
a
(5.197
--t
eneralized solution
in
a
united problem) and in addition to
;
Il~m
a)
5
I$!,
s,
>
n.
(5.20)
234
Chapter
5
Then
for
any solution
s(x,t>
E
vZ(
P~),
o
<
6
5
s(x,t)
51-
tjl
<
I
(5.2)
-
(5.4) the estimation
is
satisfied
where
a=
a(%
,
k,
Ivl
)
7
0.
In addition
if
r
E
H',
14,
E
II
Sot
i
v
S"
II-
+
IIR
llq,
",S1
5
bl
9
991.1
-
(5.22)
q
>n
t
2,
n
y
>
(n-1)
s,
,
(5.23)
w
then
II
s(x
,t
5
c
(N"
+
lJq
)
.
(8)
We begin
with
a
remark
that
under the theorem condition (5.19)
holds and
I
b;
3
11"
5
C(,,~).I.ntroduce functions
9%
U'
(x,t)
=
us
(x,.l;)
-
hrnso
(x,t),
A=
0,1
,
G
'=
AKa
V
so
i
F
+
Ka
Vl/ms
+
hVvr
-
b
v
,
-9
-3
-9
wherew(.f
wdx
=
0)
is
a
solution of the problem
'DH,
(x,
t)
E
s',
1
-
Is,]
i(%sot)"(b,
J,
I
s=
S1
(x,
L)
E
s2
.
(5.24)
!i
Q
-B
AW
-
wet,
(x,t)
€%;
Vw-n
Then uA(x,t) satisfies the problem
au
9
at
--
-
div
(~a
vu
'+
G
'1,
(x,t;>
E
,
in the sense of a corresponding integral equality. Since
theorem
In
~(s)
E
c
Q),
to prove (5.21)
it
is
enough to
verify that
?'EL
(,)
with
q,r
satisfyi?% (5.12).
It
i8
sufficient
to
assume here
q
=
%,
r=
2%
/
(%(I-
~)-n),
1-
H
>
from
the
>
1.3,
q,r
>
n
/
and
to use an inequality
N
-9
-+
0
-B
>,
II'
Ilqo,p,
is
C(ibl)
(
112'
IIm,?<
+
IIV
is
v
II
9,s
"9%
true for any
(5.23)
is
made
for
u'(X,t)
because
from
the condition
of
the
theorem
accounting (4.5*) we8ave
I'
>
'I.
Thus (5.21)
has
been proved. The proof of
so(x,t)
E#'(~)cc
"(c),
CL
"(4
-n
+I)
/
5
.
firther,
Filtration
of
Immiscible Liquids
235
and coneequently (5.23) exept vicinities
S1
has been proved
(u'=
o
on
s2
and for
t
=
0
).In the vicinity of
S1
(including
s1
0
S2
we'll act in the following way. Strai hten
S'
locally
with the help
of
transformation
y
=
y(x).Here 75.24)
is
written
this way:
au(y,t)
-
--t
J
=
div
(Ka
vu
+
G
?.)
,
at
where
J=
#
and
is
yacobian
of
the corresponding transformation,
-
-'A
-
au.
au.
K
={
I<.
.}
=
{
K
12
J},
t&
=
{G~
13
"11
axv ax,
a
xi
Extend
u(y,
t)
throughd by the method described in-theorem 4.1
and introducing a function
u
=
Ju
(J(y)
E
C'(
;)),we come for
it
in the vicinity of
S1
to the case already mentioned.
The theorem has been proved.
Then we have
In addition
if
r
c
H;''
,
and conditions
of
agreement up to the order
then
[1/
z],are satisfied
We begin the proof with the estimate
(5.26). In agreement with the
theorem
5.4
in
our
case
s(x,t)
E
c
CL(:Lf)and correspondingly
1C
E
C
a(u').Therefore from theorem 5.3
J
u0(
VS
respectively)
236
Chapter
5
-D
belong to
Lq(“’)
since
Go
q
sr
<
CQ
.
But then
-D
-D
-D
-D-D
Dx
Go
=
(Dx
8,
+
li’
*s
Dx
s
-
b
v
Ux
s
-
b
D,
V)
E
Ls/,
(Q),
-3-8
E’*
=
P
+
K
il~
v
(I
/
I,I)
2
and consequently
(
v
uo,
v
‘s)
E
L
i.e. satizfies (5.12) then
have
rentiation
1-times
yields the needed result for any finite
1.
Con-
sider the function
u”(x,t)
satisfying (5.24), (5.25). Evaluation
in
~12
SO
is
completely
similar
to
the previous, because
(
vl+’
be locally straightened
(y,(rl)=
0)
and the function
=
J(Y)U
satisfies equalities (5.241, 5.25). Then continue
for
1
=
1
through
S’
using the method described in theorem
4.1 and obtain
(
VU”,
0s)
E
L
(c61),
Then differentiate (5.24) over
y,,
i=
I,
2
and extending
u1
through
sl
we
find that
(VU’
)
E
Ld2(%,j consequently
tuii,
S,
)
EC~
which means
v
”yie
(C1)
and
as
q
>2(n+
2)
,
vs
Ef
(,‘).Returning
to
(5.24) we
da
D,
do
E
L
(J,v
2s
E
L
(%),
respectively. Sequential diffe-
cl
9
-
so
’
k~
).
Let the piece
6:
c
S1
being considered
1+
1
v
Sob
9
u
’
(y,
t
)
‘1
Y
y:
v
Yi
Ji
E
Lq
.
Low
a
one-dimensional equation
is
left to be consi-
dered
au”
a
au”
at
ay,
ay3
(K33
a
-
y1
)
f
1,
,
--
--
Yi
from
which
it
follows that
uy
Theorem
5.6
(uniqueness of solution). Let
r
ell1,
and
s(x,t>
be
a
generalized (in the sense of the definition 2.1) solution of
problem (5.2)
-
(5.4) for a given
v(x,i)
E
L
(,>
.
This
solu-
tion
is
unique if
E
I,
s)
E
Lq(dC)-The case1
2
2l
’is
considered in the
similar
way.
(.,I),
i.e.
(V’U
,u;,
st,
2
9
V
Lll
’
z
The proof
is
made from the opposite. Let two solutions exist
(2.2) one from another we are led to the followim:
si,
i
=
1,2
and
s
=
s
-s2
.
Subtracting the respective identities
Filtration
of
Immiscible Liquids
237
Substitute in (5.29) the function
cp
of
the
type
t
t
1
6
t-6
d
t-
6
CP
='pa
=
-
J'
(p
dt
5
f
sd
(x,z
)
xr(
T)dz,
co
t
5
0,
t
L
t,.
05
t
5
1
/
r,
l/r
st-
5
t,
-
I/
r,
t-,
-I/r
5
t-
5
T-
6
.
:t:
r(tl-
t
X,(t>
=
,
Due to averaging properties (see
39
2
ch.
1)
(
Ins
,
qt
=
-
(m,
s6
xr
1%
,
)%
z
or
turning
r
+
m
(x,
+I),
we
have
(I1
66
(7;)
I1
).
-(Illsbt
,
s6
)<
=
-
-
1
%2
2
951
Passing further
on
to
a
limit
for
6
--t
ij
the equality
from (5.29)
we
obtain
--t
1
-
II
6
IT
+
v
s,
v
s)
=
(6
s,v
s)
+(BFrS,S)
,
.
st-
2
2
.Q
%
"t
Hence the inequality follows
in which the fact
V
ELql,rl
;x),V
s
EL
is
used. But due to the inequality
(2.23?
,r2(<G),
R
E
Ly,,ro
(S')
2
ch.
1
hence the integration over
inequalities yield
t
and the use
if
Young and Hglder