Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
Подождите немного. Документ загружается.
58
Chapter
2
Therefore,
151a,y5
c
va
E
(0,
1/21.
(4.18)
Let us consider formula (4.3) for an arbitrary ap/ax. The
first
addent
p(uo
-
u)
in
it
is
estimated in the H6lder norm with the
exponent not less than
1/2.
The co-multiplier
pB-lY-'
which
is
contained in the
first
two addents,
is
also continuous by Holder.
This follows from formula (2.7) which can be written as
The boundedness
of
Y(t>
and B(x t) the Hb'lder continuity
bf
Q(x,t)
result in the Holder continuity
of
the right-hand part, as an addi-
tional smoothness over the variable
t
is
ensured by integration with
respect to this argument. Finally,
(4.3)
contains the term
as..well as of B(x,t) with respect to the variable
x
whose Hb'lder co tinuity
is
verified in the same way as for the
function <(x,tr
from
(4.17).
Thus, relation (4.3) allows one to estimate
Now
we can consider the second and third equations
of
(1.10)
as a
parabolic system with respect to u(x, t) and
@(x,
t)
with the
continuous in the Holder sense coefficients:
(4.20)
ao
azo
ap
ao
au
au
at
ax2
ax
ax
ax
ax
-
=
A
p
-
+
A
-
*
-
+
p(
-
)2
-
kpe
-
.
Let
u
=
1/2
min(a,
1/21
where
CI
is
an exponent of the Hb'lder
continuity of the second derivatives of the initial functions
Uo(X)
andQo(x)
(2.42)
from Chapter
I
for linear parabolic equations, we have
From
the
first
equation of (4.201, using estimate
IUL+PV,
l+V,
1;1
C(1
+lo1
1+2v,
v,
Q)'
(4.21)
Here
l.Ly,j+&y
exactly the same
way,
considering the second equation of (4.20)
as
linear parabolic with respect to
addents to be the right-hand part, we conclude from
it:
is
the norm in
C'+y
*j+6(y)(see Chapter
I).
In
0
and assuming the last two
lo
L+2U,1+U1L4
5
c('
+lux(
zu,u,~+~~x~O,~~~x~~V,u,~~*
(4.22)
Correctness
of
Boundary Problems
59
Stregthening (4.21) with (4.22) and using the compactness
of
embed-
ding
c2+",
l+u(sl)
in
cl+2v
'
'(Q1i.e. the relation
IU,I
zv
,v
,Q
~~u~2+Zu,1+V,C,
+
C&WV,V,Q,
we get the inequality
Iu1z+2v,~+v,Q,'
+
IU,I
0,Q
lux
I2v,u,Q
(4.23)
Let us now make use
of
the interpolation inequalities (2.25) from
Chapter
I
i
-a
lUxl
or&'
cIuI~+~~,~+~,Q
IuI
zv,v,Q'
where
a
=
(3
+
4u)-',
b
=
(1
+
4u)(3
+
4v)".
As
far
as
u(x,
t)
is
estimated in
CPU"(Q)
then (4.23) yields
1
I
2
+
PU
,I
+"
,
Q
5s
(1
+
I
UI
:
+2v
,I
+v
,
Q)
where
U
=
a
+
b
in which case, obviously,
0
<
0
<
1.
But
in
this case, applying the Young inequality with
VE
>
0,
'dr
>
0
where
r
=
~u~2+2v,1+v,ii~.
E
=
1/2C-'
preceding relation we get an a priori estmate
I
Er
+
C,
from
the
IUL+2U,l+U,Q
'
N28*
From (4.22) we have
(4.24)
(4.25)
If
in
the conditions of Theorem
1.1
and,
hence, the estimates in the Holder norm have been proved.
If
CL
>
1/2
then
v
<
?/?-a
In this case we have to consider formulas
(2.7) for
p(x,
t)
(4.3) for
ap/ax
as well as equations
(4.20)
for u(x,
t)
and
8(x,
t).
From (2.7) and
(4.3),
allowing for
(4.24) and (4.251,
we
get
1'1
2+2u,i+~,Q
'
Nz9
a
5
1/2,
thenu
=
I/ZCL
lpll+o!,a,~
'
N30'
(4.26)
Then from (4.20). repeating the procedure
of
obtaining
estimates
(4.241, (4.25) withv
=
I/Z
a
we have
lu12+CL,I+~2,Q'
N31'
~'~Z+CL,1+01/2,Q5
'32.
(4.27)
Finally,
from the
first
equation
of
(1.10)
we obtain an estimate
for
Thus, we have obtained
all
the estimates necessary to substantiate
Theorem 1.1.
60
Chapter
2
5.
PROOF OF
THE
THEOREM
OF EXISTENCE
AND
UNIQUENESS
1.
The existence of generalized solutions as limits of smooth
As
has already been pointed out
ins1,
the existence
of
solutions
results from the local theorem and the global
a
priori estimates.
The theorem of existence of the solution in the small with respect
to time has been proved (see
[
127,
1921)
for a classical solution
under the
following
conditions for the initial data
solut ions
(uo,
Qo)
E
C*+"(Q),
po
E
C1+"(R)
(5.1
and when the agreement conditions
u0
=
(So>'
=
0, (pouo')'
-
k(po@O>'
=
0
at
x
=
0,
1.
(5.2)
are fulfilled.
Let
us
now state the existence
of
a generalized solution when the
initial data have the properties
(5.3)
(PO,
uo,
0°)
E
w;(a>,
u
0
(0)
=
uO(l)
=
0.
Function uo(x>
is
approximated
in
the norm
W:(Q)
differentiable finite
in
Q
functions u:(x). For
po
let
11s
take
such a sequence of smooth functions pE(x> converging to
po(x>
in
hV;(Q>
that
by infinitely
(P;)'(o)
=
(Pi")
=
0.
(5.4)
It
can be carried out in such a way. Let
w(x>
be a certain smooth
function for which
w(0)
=
p0(0>,
w(l>
=
pO(l>,
w'(0)
=
w'(1)
=
0.
It
is
obvious that there
is
an
infinite lot functions
w(x>
which
obey these conditions. For instance,
w(x)
=
(po(~~
-
p0(0)>(3x2
-
d)
+
~~(0).
In
this case the difference
cP(X>
the boundaries x=
0
and
x
=
'1
and, hence
it
can be approxima-
ted
in
the norm
W:(Q>
with finite functions Cp,(x>. Hence, we can
assume
In an analogous way, we can approximate Oo(x> in the norm
W:(Q)
with the smooth functions Oi(x)
Po(x>
-
w(x)
equals zero at
P;(x>
=
w(x)
+
cp,(x>.
(ei>t(o>
=
(eO,)I(i>
=
0.
such that
(5.5)
It
should be noted that convergence in
W:(Q>
also ensures a uniform
convergence, therefore, we can consider
P:(x)
and e;(x> to be
Correctness
of
Boundary
Problems
61
strictly positive and bounded:
-0
0
<
m
-
.sn
5
(pn(x),
an(x>>
5
IVI
+
.sn,
E~
0
0
For every n
=
1,
2,
3,
...
due to the finiteness
of
ui(x)
and pro-
peties
(5.41,
(5.5)
of
the sequences
p:(~)
ment conditions (5.2)
will
be fulfilled. Hence, there exists a
sequence of smooth solutions
(pn,
un,
en>
of the problems with
the initial data@:,
u:,
f3:).
Let
us
now recall that estimates
(3.l),
(3.3),
(3.4) as well as (4.6)
-
(4.9) depend only on the
norms of the initial data
in
N:(Q)
and,
hence, are uniform with
respect to
n.
Singling out a converging subsequence and going
over
to
the
limit
n
+
m
we understand that the limit
(p,
u,
0)
is
a generalized solution. The uniqueness of a generalized solu-
tion which
will
be proved below demonstrates
that
the whole
(pn,
un,
0,)
converges.
2.
Proof
of the
local
theorem
of
existence
The presented above method of proving solvability in a whole
is
based on the local theorem from [l27, 1921 for classical solutions
which, in
its
turn,
is
proved in quite a cumbersome way. Hence,
it
is
more expedient to ive a simpler method of proving the solv-
ability
of
problem
(
1.107
-
(1.12) in the small with respect to
time.
Let us confitrUfit %local generalped solution as
a
limit
of
appro-
ximated
(p
,
u
,
0
)
where
u
and
en
expressed
as
the finite
and
0i(x)
the agree-
sums
n
i=
i
un(x,
t)
=
c
u:(t)
sin
(nix),
n (5.6)
QYX,
t>
=
.C
O:(t)
cos
(njx),
n
=
I,
2,
...,
J
=O
with the unknown coefficients
uq(t>,
i
=
1,
2,
...,
n,
OS(t),
j
=
0,
1,
2,
...,
n.
In order
to
define these coefficients let us
demand that the second and third equations of system (1.10) be
fulfilled
in
an approximate way:
~~
aun
a
aun
a
(
p
-)
+
k
-
(
p%n)
sin
(aix)dx
=
0,
ax
ax
-
.-
o
tat
ax
ax
ax
ax
J
x
cos
(njx)dx
=
O,
i
=
I,
2,
...,
n,
j
=
0,
I,
...,
n.
(5.7)
62
Chapter
2
Let
us
seek function
pn
as a solution
of
the
first
equation of
(1.10)
aun
+
(p"f
-
=
0
(5.8)
aPn
at
ax
-
with
the initial condition
n
Having defined
pn(x,
t)
P
Itzo
=
P0(X)*
from
(5.8)
and
(5.9)
by the
formula
(5.9)
let
us
substitute them into
(5.71,
setting
t
zy(t)
=
I
u:
(t)dz
,
i
=
I,
2,
...,
n.
0
In this case relations
(5.7)
and the
given
above definition of
the
values of
z:(t>
will
be written as a system of ordinary differen-
tial equations
._-
dzk-~i,
i=l,
2,
...,
n,
j=O,
I,
...,
n,
k=l,
2,
...,
n.
dt
It
is
designated here
A_
=
I,
A;
=
2,
j
=
I,
2,
...,
n,
naun
"
a
-
)
-
k
-
;P%~)]
sin
(nix)dx,
ax
ax
n
un,
On,
are defined
b
formulas
(5.61,
(5.10).
Take initial
data for tie system
(5.113
from the expansion of initial functions
uo(x),
Oo(x)
in Fourier series
by
sines
and
cosines respectively:
63
Correctness
of
Boundary Problems
n
'0
ui(0)
=
up
3
2
I
u
(XI
sin
(nix)&,
i
=
I,
2,
...,
n,
0
1
en(0)
=
00
=
2
I
eo(x)
cos
(njx)dx,
j
=
1,
2,
...,
n,
(5.13)
J
J-
0
1
0
Qt(0)
=
J
Oo(x)dx,
zC(0)
=
0,
k
=
1,
2,
...,
n.
Local solvability of the Cauchy problem (5.11)
-
(5.13) for each
fixed
n
=
1,
2,
.
.
.
results from the Cauchy theorem for the system
of
Drlinary differential equations (see
[I101
1.
Now show that such time interval
is
available
[0,
to],
0
5
to
5
T,
on which there exist solutions (5.11)
-
(5.13) for all
n.
For
it,
it
is
sufficient to obtain estimates
for
u:
and
Sn
uniform over
That
is,
verify that on a sufficiently small interval
[0,
to]
a
priori estimates of the type
(4.4)
-
(4.9) are valid independent
of
n. i.e.
J
One more condition from which the value of an interval
further chosen
is
connected with the positiveness
pn(x,
t).
0
<
m
5
p
(x>
5
k
<
m
tions are satisfied
to
is
Since
0
demain that in formula (5.10) the rela-
1/2
m
<-
pn(x,
t)
5
2.1u1
for all
n
under
x
E
[O,
I],
t
E
[0,
to].
Begin evaluating (5.14). Multiply the
first
group of equations
(5.7) with numbers
over
i
from
I
to
n
,then add to the equation of the second
group with number
J
=
0.
Then obtain equality
(5.15)
i
=
1,
2,
...,
n
respectively by
u:(t)
sum
(5.16)
64
Chapter
2
Then multiply the equations (5.7) with numbers
i
=
1,
2,
...)
n
by
(ni)2u:(t)
(nj)2ey(t).
Summing over
i
and those with numbers
j
=
1,
2,
...,
n
by
and
j,
accounting that
finally obtain the relation
+
In
in
(5.17)
conformity
with
the assumption (5.15) estimate
the left side
pn
2
1/2
m
pn
from below
and from above
pn
5
2ld
in
the right
side. For each
n
inequalities
(5.15)
will
be satisfied
on
a cer-
tain sufficiently mall time interval
[0,
tn]
actions
will
be performed
on
[0,
tn]
luated from below
tn
Z
to
>
0.
right side (5.17), considering all addends
Ik(t), k
=
1,
2,
...,
6.
in turn.
In
I,
so
that
all
further
then show
all
t,
to be eva-
For
this
purpose estimate the
first,
on (5.10) and (5.15) we have
t
0
(5.18)
n
lpxl
5
C(1
+
I
lu&(x,
~lld~),
and for
u:
by imbedding theorem
Consequently,
Correctness of Boundary Problems
65
Here
a
constant
C
depends on
m,
bl
and
T.
Applying the
Young
inequality with
E
we conclude
t
I,
I
E,IIU&II'
+
CE,[l
+
IIu214
+
(
I
IIUaIzd.C)41*
0
For the second addend
I2
e
s
t
hat
e
in
the right side
(5.17)
we have the
1
0
Ip
5
IrJ
I
p%$&dxl
5
2Uc
lp;l[.l/UgJ
I
In the third term
first
account the inequality
1
0
)On)
5
I
I
Qn&l
+
110:11,
and then use quality
(5.16)
and imbedding inequalities
I
C(1
+
llUnl12
+
II0,nll)
5
C(1
+
IIugIz
+
p",l).
I,
5
C(1
+
11.;
(I2
+
IlOn,ll)U
+
I
11.;
112~~~~11~~11
*
Then due to
(5.18)
obtain
t
0
From which with the help
of
the
Young
inequality
66
Chapter
2
Now
choosing
sic
i
=
1,
...,
6
come to the inequality
sufficiently mall from
(5.17)
we
This relation can
be
written in the
form
of differential inequa-
lity
for a non-negative function
Since a constant
c,
in
(5.19)
is
independent
of
n
and initial
data
yn(0)
are bounded uniformly overn,y,(O)
5
C,
an estimate
uniform over
n
is
valid on a sufficiently small time interval
LO,
to]
YnW
5
Y(t>,
t
E
Lo,
to]
(5.21)
Correctness
of
Boundary Problems
67
Here
y(t)
us a solution of the Cauchy problem
(5.22)
and
to
>
0
is
time
of
y(t)
existence.
It
is
evident that from
(5.21) comes the relation (5.14) from which comes the uniform over
n estimate
of
the value
on an interval
[O,
to>.
In
this
way continuity
of
Cauchy problems (5.11)
-
(5.13) over
[0,
to)
des, there
is
also
a
condition
of
smallness on
7
5.14) we easily obtain
aolutions of the
is
provided. Besi-
to
which would
rovide the conditions (5.16) to be satisfied. From (5.10) and
'
-
Lv'Ly
34
llo
Here
N3,+
is
a constant from (5.16). Therefore,
if
one chooses
(5.23)
the inequality
n
P
(x,
t)
5
a;,
x
E[O,
I],
t
E
[o,
to].
holds.
Similarly
it
is
verified that
in
view
of
this condition the second
relation (5.16)
is
also true, i.e.
The estimates (5.14) allow us
to
choose converging subsequences
from sequences
{Un(x,
t)},
{On(x,
t)}
!'lith
the help
of a limiting
transition in equalities (5.7)
-
(5.10)
it
is
easy to show that
limiting functions
u(x,
t
,
O(x
t),
p(x,
t)
give a generalized
solution on the interval
[O,
t
-
(1.12).
By the estimates of
9
3,
4
it
pollows
that a local solution may be
continued onto the whole interval
LO,
TI.
So
the theorem of solu-
tion existence has been proven.
3. Uniqueness of a generalized silution
For
the proof
of
uniqueness assume that there exist two different
solutions
(p,,
u,,
S,)
and(p2,
up,
02)
of
the problem
(1.10)
-
(1.12).
Their difference
p
=
p,
-
p2,
u
=
u,
-
up,
8
=
8,
-
B,
is
a solution
of
a 1Fnear uniform system
pn(x,
t)
2(1/z)m
of
the problem
(1.10)