Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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48
Chapter
2
Let
us
raise both parts to the power
4/3,
followed by streng-
thening the inequality with the relation (2.12) for
m
As
a re-
sult,
P'
It
means that
Let
us
apply theXolderinequality to the second addent, and then
Young inequality with
E
:
By using the Grounewall lemma, we get (2.16).
3. ESTIMATSS
FOR
DENSITY
AND
TEI'PERATURE
FROM
ABOVE
AND
BELO\'/
Using the obtained in the preceding paragraph additional relations
(2.111, (2.12) and (2.16) one can get the estimates
of
strict po-
sitiveness for temperature and density.
Lemma
3.1.
There exists a constant
initial data (1.12) and
T,
such that
mo
>
0
which depends on the
m
8
(t)
c
m
0
Yt
c
[O,
TI.
(3.1)
Proof. Let us discuss the third equation of system (1.10) for tem-
perature transforming the last addents by adding and subtracting
(1/4k200'
:
..
I
,-
ao
a
Eie
au
I
I
-
=
h
-
(
p
-
)
+
p(
-
-
-
E)2
-
-
k2pG2.
at
ax
ax
ax
2
4
Dividing into
e2
we get the equation for the function
z
6-l:
a
(0
a~
at0
a
to
au
k
k2
at
ax ax ax
ax
2
4
-
=
A
-
(p
-
)
-
[2hpb(
-
)2
+
PO2(
-
-
-
0)21
+
-p
Let us multiply this equality by 2p
w
where p
>
I
is
an
arbibrary number, and then integrate with respect to
Q.
Since the
expression in the square brackets
is
non-negative, we get the
in-
e qua1
i
t
y
.
d
k2
1
-
Ilw
(t)ll
5
-
&
pw2P-ldX.
L,p
4
llw(t)ll
2p-1
LZp(Q
1
dt
Let
UB
apply the Hb'lder inequality to the right-hand part:
Correctness
of
Boundary
Problems
49
In this case after contraction by
a
general positive co-multiplier
2p-1
llwllL2p(Q)
we get the formula
d
kZ
-
IILJIIL
(Q)
<-
-
IIPllL
(0)'
dt
2P
4
2P
Integrating with respect to
t
and going over to the
limit
we find:
p
4
rn
In
the denotations of
(2.10)
this relation can be written
as
follows:
k2
t
Q
40
-1
-1
m
(t)5
m
+-
I
Mp(2)dz.
Using inequality
(2.11)
for
lii
we obtain the relations featuring
only
one function
mg:
P
(3.2)
Thus, the Lemma
is
proved. Indeed, let us set
In this case
y"
Nk2
The second differentiation gives
UI
e(t)
=
-
-
(-)
and formula
(Y')
4n
(3.2)
assumes the form
of
a
second-order differential inequality
-
YY"
2%-
It
is
evident that
if
we add unity to the left-hand
$yl
),2
Nk2
par
,
it
will
become
a
differential
dY
4n
-(
-12
1
+-.
dt
y'
Nk2
Hence
Y
4n
4
Y'
-
or, which
is
the
same,
50
Chapter
2
Y'
mNk'
y
5
After the second integration
we
conclude:
-
4
+
m(4n
+
m2)t'
Nk2
4
y(t)
5
rn-'[l
+
m(n
+
-
It]
Vt
E
[0,
TI,
where
y
=
1Tk2/(4n
+
Nk2)
<
I.
Returning
to
(3.21,
we get an estimate from below
for
%(t)
of
type
(3.1),
in which case
m
,(t)
Z
mo
=
m[l
+
m(n
+
Nk2/4>T]'.
An estimate for
M
('6)
from above is obtained as a result of
Lemma
3.1
and
relation (2.11):
P
Mp(t)5
IT
Vt
E
[0,
T].
(3.3)
$(t>
2
no
Vt
E
[0,
TI.
(3.4)
Let us now prove the density estimate from below.
Lemma
3.2.
There exists a constant
no
>
0
such that
Proof. Due to inequality
(2.12)
it is sufficient to obtain an esti-
T
mate from above for the integral
1
hl
,(t)dt.In
its turn, it re-
sults from
(2.16) that the summability of this integral is quaran-
teed
by the bounded
of
the function J,(t).Let us prove this pro-
perty, taking equation
(1.141, multiplying it by
w(~,
tj
and inte-
grating it with respect to
51
0
1
0
=
k
I
pOuwxdx.
(3.5)
By the Cauchy inequality the right-hand part does not exceed the
value
1
1
1
0
4
0
6
I
pwkd~
+
-
k;""
p0'u2dx,
where
6
>
0
is an arbitrary number. Then'a',
6<A/;!
in line with the
definition
(1.15)
of
the function w we have
(I
-
6)w;
+
(A
-
1)0
w
2
(h
-
26)O:
-
[
4
S"(1
+
A)*
+
1
xx
+
26
+
h
+
21uzI.l;.
Correctness
of
Boundary Problems
51
Therefore, assuming
6
=
l/amin(l,
h)
from (3.5) we get
If
the second equation of system (1.10)
is
multiplied by
and integrated with respect to
Q,
then we get the relation
u3(x, t)
31
31
-
$
p~'u;dx
+
-
k2
IpQ2u2dx.
20
20
5
hhltiplying the above equation by
2/3N,
conclude
and
where
Let
us
now estimate the right-hand part from
the fact that
N8
=
N6
+
k2K,,
y
=
1
/6N7.
(3.7)
adding with
(3.61,
we
above, making
use
of
p(x,
t)
5
IT,
02(x,
t)
5
Iid2@(t),
1(u112
5
2No.
Thus,
it
does not exceed the value 2N,NN,Ikg(t).Let
us
make
use
of
inequality (2.16) and note also that the-left-hand part of
(3.8)
contains the term
J,(t).Therefore, we get the inequality
d
3h
dt
)
+
-
J,
5
2NofTN,(~J1
+
GGJ2
+
KE).
(3.9)
Let
us
choose
E
>
0
in such a way that
4N0NN,~=
h
and allow
for the fact that according to definition (2.15)
Then in
(3.9)
there
is
a
differential inequality
d
5
N,z
+
N,,
J,(t)
=
dz
J2(t).
4
for
the
positive
function
z(t> EllW(t)l12+ ~Ilull~,~
+
hJ2(t)
Integrating this inequality we deduce that z(t)is bounded, which
means
J2(t)
I
Nll
Vt
E
[0,
T],
(3.10)
Ilw(t)ll
5
N,2,
llu(t)l14,Q
5
N12
vt
[o,
TI.
(3.11)
and, besides,
The boundedness of function J2(t) by formula (2.16) results in
summability
on
[0,
T]
of the function ivlZ0(t)and, moreover, of
52
Chapter
2
Now equality
(2.12)
gives an estimate from below for
mp(t>
m
(t>
L
n(l
+
NN~~)-'
=
no
>
o
vt
E
[o,
TI.
(3.13)
P
Thus, Lemma
3.2
is
proved.
Let
us
cite the two direct corollaries from
it.
First according to definition (1.15) of function
w(x,
t)
from
(3.11
)
we conclude
0
5t
nax
5T
118(t)l)
5
N,,,*
(3.14)
Second, inequality (3.10) obtained when proving the Lemma together
with
(3.13)
results in the estimate
(3.15)
Therefore, we have proved strict positiveness of the density
p(x,
t>
and temperature
p(x,
t).
For the temperature
0
an estimate from above by maximum
has not yet been obtained, but we have demonstrated the summability
on
[0,
T]
(3.14) and
(3.15).
O(x,
t)
as well as the limitedness
of
of the function
Id'@(t)
in
(3.121,
as well as estimates
4.
A
PRIOR1
ESTIlilATES
FOR
DERIVATIVES
Having obtained the estimates from above
and below
for
the density
scheme. Equations (1.10) for velocity and temperature are conside-
red as strictly parabolic with res ect to
u(x,
t)
and
8
x,
t-)
and on the basis of known methods !see, for instance,
[92))
the
corresponding a priori estimates are deduced from them. Then from
the equation of continuity for
p
the density smoothness
is
in-
creased and, hence, estimates for
u
and
0
are improved. The
process can be repeated,
if
necessary.
Let us describe
the above method in more detail, beginning at the
first
step with
the use of
only
those estimates that were obtained
p(x,
t)
let us carry
on
our considerations by the following
in
0
3.
The momentum equation
au
a
au
at
ax
ax
(P-1-k
is
multiplied by
u(x,
t)
and integrated
-=-
(4.1)
a
-
(PO)
ax
with respect to
51
:
pQuxdx.
Correctness
of
Boundary Problems
The right-hand part
is
estimated by the Cauchy
value
53
ine qua1
it
y with the
11
11
-
J
puidx
+
-
k2
20
20
pQ2dx.
The boundedness and strict positiveness of
p(x,
t)
as
well
as
re-
lation
(3.12)
finally yield
rn
Let
us
again make use of formula (2.7). Differentiating
equality with respect to
x
we get:
(4.2)
this
(4.3)
where
A(x,
t>
mates, we have:
u0(x)
-
u<x,
t)
Herefrom, due to the above esti-
ao
Let
us
rewrite
(4.1)
in
a
more detailed form:
au a2u
ap
au
ap
a@
at
ax2
ax ax
ax
ax
-=p
-
+---k-Q-kp-
and, multiplying by
uXx
integrate
it
with respect to
51:
(4.4)
The
fi.rst
addent in the right-hand part
is
estimated by the
k[l
pxll
uxxl[
-
i,i
and the second one
-
through
kN
11
@,I1
11
L&)
.
For the third addent we have an inequality from above through the
quantity
max
I
uxI
.llPxlI
'IlUXJ
0
5x
51
As
far
as
u,
turns to zero
at
every
t
E
[0,
T]
at
least at one
point
xo
E
LO,
I]
then
'Jib,
t)
5
211Ux(t)ll'IIUXx(t)ll
II
P,ll
*
II
uxJl
3'211
UAI
1'2
Therefore, the third addent in the right-hand part of
(4.5)
is
not
greater than
Using the Young inequality with
E
and allowing for the obtained
earlier estimates
(4.4).
(4.21,
(3.121,
we conclude
54
Chapter
2
Directly from equations
(1.10)
we get:
(4.7)
Besides, differentiating the
first
equation of
(1.10)
with respect
to
x
and allowing for (4.3) and (4.61, we have
By
multiplying the third equation by
0
analogously with pro-
ving (4.61, one can easily get the esti%te
(4.9)
which,
in
particular, demonstrates the boundedness of temperature
By
way of concluding the deduction of the estimates for generalized
solutions
it
should be noted that when obtaining them use was made
only
of
those properties
of
smoothness
of
the initial data which
were given in Theorem
1.1,
i.e.
(p
,
u
,
0')
E
Wl(L2).
Now let
us
go
over to the estimates in the Hb;lder norms. First
of
all, account should be taken of the fact that..inequalities (4.41,
(4.7) and (4.8) for the de-naity guarantee a Holder continuity
of
the function
p(x,
t>
in
Q.
Indeed, if x, and x,are two arbitrary points from
[0,
I]
then
@(Xt
t)
from above.
00
From this relation, due to (4.41, we conclude:
The Hb'lder constant over
t
i3
estimated in the following way. For
the two values
t,
and
t,
from
LO,
T]
we have
Correctness
of
Boundary
Problems
55
Then, by the embedding inequality
In line with
(4.7)
and (4.8) we get
Therefore,
i.e. we have estimated
a
Hglder modulus of continuity even
a
greater power than 1/2.
For
our further considerations
it
is
im-
portant to have
an
estimate
of
the Holder norm of density
with
some positive exponent, and therefore
it
has been proved that
JPI
5.
Nz3
v
a
E
[O,
1/21.
(4.10)
a
,G
Let
us
now demonstrate the Hb'lder continuity of the velocity
temperature
0
and derivative
ap/ax.
For
this purpose let us
first
differentiate the second and third equations of system
(1.10) with respect to
t:
u,
au
a
au,
a
au
a
('t ax ax
at
ax ax ax
-
)
-
k
-
(ptO
+
pOt),
A=-
(p-)+-
a aot
a
ao
au
au,
--
-
A
-
(p
-
)
+A
-
(p,
-)
+
2p--
at
ax
ax ax ax ax ax
t
au au, au
ax
ax ax
+
pt(
-
)2-
kpO
-
-
k
-
(PtO
+
PO,).
(4.11)
The first of these relations
is
multiplied by
grated with respect to
x
from
0
to 1:
ut(x,
t)
and inte-
2
Id
--
IIu~II'
+
I
P(
-
ax
=
2
dt
o
ax
The first addent in the right-hand part
is
not greater than the
value of
kMollptll
-
/Iudcll
positiveness
of
p
and inequality (4.71,is not greater than the
and, hence, due to the boundedness and
SUm
56
Chapter
2
1
I
pu:,dx
+
CE
Mk(t).
0
1
Directly from the Cauchy inequality we get an estimate from above
for the second addent
'2
1
P@,ux,dx
I
E,
I
puXtdx
+
CE
JIQtll
2o
0
2
Finally, the last addent does not exceed
since
max
luxI
5
l(uxx((.
Choosing
Ei
>
0,
i =
1,
2,
3,to be
sufficiently mall, by integrating (4.12) we get
05x51
m
The second
of
relations (4.11)
is
multiplied by
Q,(x,
t) and
integrated with respect to
Q
=
(0,
1).
1
6
2
dt
0
(4.14)
where
1
1
I1
=
-
I,
PtOxQxt&,
1'
=
2
I
0
PUXUX,~,dx,
Let us estimate each
of
the expressions
Ij
one by one. First,
OSXXl
Then, in an analogous way, we find
1121
I
~pll~xtll*ll~tll*
ml3.x
IU,I
I
CIIUxtll.IIUxxll
*lPtIl
i
05
x51
Correctness of Boundary Problems
57
Finally,
From (4.14), using the obtained estimates, we conclude:
(4.15)
The additional integral estimates (4.13) and (4.15) give
first,
an estimate
of
Holder continuity for
u(x,
t)
the same as for density
p(x,
t)
i.e.
and
Q(x,
t)
which
is
IuIa,U
I
N2,9
I@Ia,U
5
N,,
V
a
E
(0,
1/21.
(4.16)
Second, they show
a
Hzlder continuity of the function
t
acl,
o
ax
S(x,
t>
3
J
-
(x,
TIdT,
(4.17)
which, in
its
turn,
will
be used below to estimate the Hslder con-
stant .of the derivative
ap/ax.
Indeed, for arbitrary
x1
and
x,
from
LO,
I]
we have:
t
t
x2
I
1OX(x1,
T)
-
Ox(x2,
T)ldT
5
I
1
I
Oxx(X,
T)dX
I
dt
C
O
0
And for any
tl
and
t2
from
[0,
T]