Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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68
Chapter
2
-
+
p;
-
+
P(P,
+
P,)
=
0,
at
ax ax
au
a
au
au,
~
a
at
ax
('1
ax ax ax
-+p
-
J
-
k
-
(PI@
+
pe,)~
-=-
a8
a
a0 a02 au, au, au
-
=
h
-
fp,
-
+
p
-
)
+
p,(
-
+
-
)
-
+
at
ax
ax
ax ax ax ax
+
p(
au2
-
)z
-
kp,B,
au
-
-
k(p,8
+
pe,)
au2
-a
(5.24)
ax ax ax
with zero Fnitial
and
boundary values. Nultiplying by
p(x,
t)
and
integrating from the first equation we deduce:
+
On
account of this relation, the second
equation
of
the system
(5.24)
being multiplied by
U(X,
t)
and
integrating gives:
Likewise, from the third equation we obtain (with the use
of
(5.25)
and
(5.26)):
Hence by the Grounwall lemma it follows:
C3
z
0.
Then from
(5.25)
and
(5.26)
it fol1ows:u
0
and
p
P
0
this is what is required
for the proof
of
uniqueness. The theorem
1.1
has been completely
proven.
6.
THE CAUCHY
PROBLEM
The method of investigating the global solvability
of
problem
(1.10)
-
(
1.12)
presented in the previous paragraphs essentially
applies the property
of
boundedness
of
flow domain.
In
conaidera-
tion
of
problems in unbounded domains there are two peculiarities
impeding the application
of
the given method. The first one is
connected with the fact that physically the absolute temperature
must
be
highly positive, therefore the total energy
of
the whole
gas
mass
(integral over the domain
of
the function
w
=I
/2
u2
+
@>
cannot be finite. Consequently, when studing, let
us
say, the
Cauchy problem it is impossible to use the first a priori evalua-
tion
(2.9).
The second distinction
lies
in the method
of
obtaining
the relations
(2.111, (2.12)
between the maximum and minimum va-
lues of the density and the temperature at sections
t
=
const.
Correctness
of
Boundary
Problems
69
Namely, imbedding inequality
L,(Q)
in
L,(Q)
has been used in the
formula (2.7) from which the relations were deduced. Such inequa-
lity doesn't occur for the unbounded domain
Q*
The key oint of the
investigation
is
the proof
of
inequalities of the kind
f2.111,
(2.12) in the case when the flow domain
is
bounded. To be precise
consider the Cauchy problem. The local theorem of the amooth
solu-
tion existence
of the Cauchy problem has
been set up in
[181].
In
the paper
[TO]
the Cauchy problem for the system
(1.10)
has been
studied under the additional smallness condition, namely under the
condition of priximity of initial data
(PO,
uo,
63')
to rest state.
The methods of obtaining evaluations in
1701
are not connected
with relations of the kind (2.111, (2.12). We demonstrate that the
application
of
connections similar to (2,111, (2.12) enables
us
to
establish the global solvability of the Cauchy problem without any
smallness conditions.
So,
consider the Cauchy problem for the equations
(1.10)
of uni-
form
movement
of
viscous gas, written in Lagrange variable8 (for
simplicity take
all
physical constants equal to unit):
ap
-
+
p2
a"
=
0,
at ax
au
a
au
a
(PS
1
,
-
@--I--
at
ax ax
ax
--_
(6.1)
ae
a
ao
au au
at ax ax
ax
ax
---
-
(p
-
+
p(
-
I,
-po
-.
At
an initial moment
t
=
0
function values
p,
u
and
0
are
assumed
to
be known:
(pot
uo,
Oo!
bounded
are continuous,
(PO,
So)
are highly positive and
o
<
m
5
po(x)
I
ivi
c
m,
m
I
e0(x)
5
M,
(6.3)
and have finite
limits
on infinity:
(without
loss
of
generality these
limits
can be assumed
to
be equ-
al to unit
on
infinity).
Theorem 6.1. Let the initial data (6.2) satisfy the conditions
(6.31, (6.4) and
(Po
-
1,
uo,
Qo
-
1)
E
VJ:(R).
Then in
a
strip
0
<
T
<
m,
ll
=
R
x
(0,
T)
with an arbitrary finite height
T,
there exists a unique solution of the problem
70
Chapter
2
(6.1), (6.2)
where
au
a0
Give a detailed proof of estimates of high positiveness
and
boun-
dednees of density
p(x,
t).
Such estimates being obtained, the
further consideration are similar to corresponding calculations
in
theorem
1.1.
To
prove estimates
0
<
rill
i
p(x, t)
5
K1
<
w,
(x,
t)
E
rI,
(6.5
1
relations similar to
(2.11). (2.12)
for maximum and minimum values
of
the density and the temperature at sections
t
=
const are re-
quired.
blultiply the first equation
of
the system
(6.1)
by
p-'(l
-
p-l),
the second by
u
the third by
I
-
0-',
add
and
integrate over
R
:
d
1
-
dt
2
[
-
u'
+
p-I(p
ln
p
-
p
+
11
+
(Q
-
In
8
-
I>]&
+
P
au
P
a@
Q
ax
e2
ax
+I
[-(-I2+
-(-)2]dx=O.
Integrals over
X
the conditions of
1
2
EE
/[
-
(u')'
+
(6.6)
are taken within limits from
-
m
to
+
m.
From
theorem
6.1
it follovrs that nonpositive quantity
Correctness of Boundary Problems
71
Break up the axis
R
and the band
Il
into finite segments and
rectangle
s
m- m-
R=
U
QN,
n=
U
(=+q
1
N=-
m
N=
-
m
QN
=
{x~N
<
x
<
N
+
I},
QN
=
QN
~(0,
T),
N
=
0,
i
1,
i
29
Obtain
relations
of
the kind
(2.111,
(2.12)
in
each of the rectang-
les $lX.Take arbitrarily
one
of these rectangles. Since in
(6.6)
functions p-'(pln
p
-
p
+
I),
e
-
In
BI
are non-negative for
p
>
0,
k3
>
0
the estimate
(6.10)
is
valid, where integrals
in
the defhition
UN
and
VN
are taken
over
QN-
From
(6.10)
it
follows
that for any
t
E
[0,
T]
in each
QN
there
is
at least one point
a(t>
=
aN(t>
E
[N,
N
+
I]
such that
p-'(plnP-
P
+
l>lx=a(t)
5
E,
(0
-1n0
-1
)Ix=a(t)
S
E.
(6.11)
Let
v1
and
v,
be roots of the equation
It
is
evident from
(6.11
1
that
v
-1nv
-1
=E,
O<v1
C
1
Iu,
<m.
-1
v,
5
p
(a(t>,
t)
5
v,,
v1
5
Q(a(t>,
t)
5
v,
(6.12)
This property
is
an analogy of the relation
(2.2).
Continue com-
paring with the auxiliary constructions from
92,
and further de-
duce relations, which role
is
similar to the property
(2.1
1,
used
in
theorem
1.1.
Lemma
6.1.
Let
functiEn and
v(x,
t)
be non-negative
in
uN
=
(N,
N
Then there exist constants n(E) and
M(E)
such that
N
+
0
<
n(E)
5
I
V(X,
t)dx
5
M(X)
<
00
N
Vt
E
[0,
T],
(6.14)
n(E)
=I/,
exp
{-2E;
-
I},
M(E)
=
2[1
+
E/(I
-
In
2)].
(6.15)
Proof. The estimate from above
is
deduced rather simply: break up
QIV
into two eets
A-(t)
=
{xIO
<
V(X,
t)
<
2,
x
E
QN},
A+(t)
=
QN\A-(t).
72
Chapter
2
Then
N+1
I
v(x, t)dxS
2
+
1
v(x,
tldx.
N
A+
On
the set
A+
where v(x,
t)
2
2
we have
2
1-In2
v(x, t)
5
[~(x, t)
-
In v(x, t)
-
I],
and under the condition
(6.13)
conclude:
N+l
N
I
v(x, t)dx6
ME).
Go
on
to the proof
of
the second inequality
out by the method from the contrary: assume
M+
1
I
N
v(x, to)&
<
n
E
n(E)
(6.14).
It is carried
that
(6.16)
for somet,
E
[o,
TI.
Introduce the set
B-
=
{XI
v(x,
to>
<
2n,
x
E
QN}.
Then
mes
B-
>
1/2.
Actually, if mes
B-
5
1/2
the value of its
addition
I)+
=
ON
\B-
on which v(x, to)dx
1
2n,
1/2.
But then
is not less than
f
v(x, to)dx
2
I
v(x, t,)dx
1.
n,
B+
ri
which contradicts
(6.16).
It meanames B->I/il.Since
2n
<
1
,
on
B-
we have the relation
v(x,
to>
-
In v(x,
to>
-
1
>
-
ln2n
-
I,
x
E
B-,
because
v
-
In
V-1
is
a_
decreasing function for
0
<
v
<
1.
tegrating over
x
on B using nonnegativenessv
-
lnv-
1
and
condition
(6.13)
find:E
>
1/2(-ln
2n
-
1)
definition
(6.15)
of
a constant n
=
n(X).
Thus, estimates
(6.14)
are proven.
Applying
(6.12)
and
(6.14)
one can obtain formulas similar
(2.7)
in
each of the rectangle
yN.
(6.1)
in the
form
In-
which cotradicts the
Putting
down
the second equation
au a21n
p
a
at axat ax
+
-
+
-
(p@)
=
0,
-
firat integrate over
t
,
then over x from point
a(t)
=
aN(t> to
random
x
E
[N,
N
+
I].
equality, multiply by
@
and again integrate over t.
Aa
a result
After that,.potentiate the received
Correctness
of
Boundarv Problems
73
obtain the relation similar to (2.7):
(6.18)
Since points a(t> and x are from one section
zIi
by the Cauchy
inequality
Therefore for the function
B
evaluations of the form (2.8) are
valid independent of the choice of the rectangle
kN
:
0
<
NY1
5
B(x
,
t)IN,
<m.
(6.19)
ositiveness and boundednese of the function Y(t)
=
Now prove the
=
Y,(t>.lJVrite p6.17) in the form
Y(t)p-l(x,t)
=
B-’(x, t)[(po(x))-’
+
i
Y(-c)B(x,z)O(x,z)dz
1
0
and integrate over x from
N
to
N
+
I.
Applying the
low
evalua-
tion (6.14) from lemma 6.1 for v(x,
t)
=
p-’(x, t) and the upper
evaluation for 0(x, t) and also evaluations (6.19) for B(x, t)and
(6.10) obtain:
t
Y(t)
<-
N,(1
+
I
Y(t)dT).
-0
Hence
by
the Grounwall lemma conclude
:
Y(t)
5
N2
exp
{N2T}
.
(6.20)
If
on the contrary we use the upper bound
of
lemma 6.1
for v(x,t)=
=
p-’(X,
6) then obtain:
Y(t)
Z
N,
>
0
Bt
E
LO,
TI.
(6.21)
74
Chapter
2
The inequalities
(6.19)
-
(6.21)
being
(6.17)
produces the relation
t
hi
(t)
5
N,(7
+
I
m
Q(z)d-c
)'I
,
P
-0
obtained, the formula
(6.22)
for
maximum
and minimum values of density and temperature
In
particular,
from the first inequality
(6.22)
we have the uni-
form estimate from above for the density:
5d
(t)
5
N,
5
hi,
<
m
Vt
E
LO,
TI.
(6.23)
P
In
order to find the bound from below for
m
(t)
introduce the
auxiliary function
P
(6.24)
From the evident correlation
by the Cauchy inequality conclude:
(6.25)
The behaviour
of
the function
+(Q)
for8
+
m
as seen from
(6.24)
is characterized by the order
Q112
therefore
(6.25)
results
in
inequality
P
ae
Q'
ax
,(t>
5
N,[1
+
m-i(t)
J
-
(
-
)'dx].
(6.26)
Owing
to
(6.6)
for the function
Correctness
of
Boundary
Problems
75
the estimate
Strengthening
we find:
T
I
A(t)dt
5
E
is
valid.
(6.26) with the help of the second inequality (6.22)
0
t
5
N,[1
+
A(t>N,’(I
+
Ul0(~)d-c)~.
0
Using the Grounwall lemma again, we have
rn
I
Iyi
@(t)dt
5
N,
.
0
(6.27)
Then from (6.22) we obtain the required evaluation from below for
mp
(6.28)
la
(t>
L
N5(1
+
IT7)-’
2
IL~,
>
0
Bt
E
LO,
TI.
P
Inequalities (6.23) and (6.28) provide positiveness and bounded-
ness of the density
P(x,
t>.In this way non-degeneralit
of
pa-
rabolic equations (6.1) are provided for velocity
u(x,
tg
and
temperature
problem become standard and we omit them.
B(x,
t).All further considerations in the Cauchy
7.
OTHER
BOUNDARY
VALUE
PROBLEMS
Give
a
number of results, close to the basic theorem 1.1, concer-
ning other problem on viscous gas flows.
1.
Problem on gas flow into vacuum
Ve set up the problem
first
in Euler variables, which are designa-
ted by (y,
t).
Let
at
an initial moment
a bounded domain,
let’s
say a unit interval. Let the right boundary
be
a
boundary, where gas contacts with vacuum, and be unknown in
advance. In domain
““y
=
{
(y,
t)l yo(t)
<
y
<
1,
0
<
t
<
T
}the
so-
lution of the equation system
is
required to be found
t
=
0
gas
occupy
0
<y
<I
y
=
1
be an impermeable solid
wall,
and the left one
y
=
y (t)
0
aP
a
-
+
-
(pu
)
=
0,
at
-
ay
au
au a‘u
ap
at
-
ay
aY2
ay
p(-+u-)=v---
9
P=RP@,
ae
ao
a
*e
au au
at
-
ay ay ay
.
ay
cvp(
-
+
u
-
)
=
H
7
-
p
-
+
v(
-
>2?
satisfying initial data
76
Chapter
2
and boundary conditions
au
22
=
u(yo(t), t),
(v
-
-
P)
dt
ax
I
Y=Yo(t)
=
0,
ae
ae
The condition on a free boundary
y
=
yo(t)
au
ay
u--
p=o
means that the strain equals zero. An additional condition
dt
means that the velocity of boundary movement
yo(t)
equals the ve-
locity of a material particle with a coordinate y,(t).This requi-
rement is a condition
for
the unknown boundary
yo(t).
Note that
under the transition
to
the
La
range coordinates
(x,
t)
main determining the solution
$Pc
U,
0)
becomes known, and the
problem on gas flow into vacuum is formulated
in
the following
way: in the domain
q
=
{(x,t)(O
<
x
<
1,
0
<
t
<
T
}
one must
find the solution
of
equation system
(1.10)
the
do-
aP
,au
at ax
+
p-=
0,
-
au
a
au
at
ax
ax
(P
-
-
PI,
E
=
lspe
,
--
--
ae
a
ae
au au
=A
-
(p-
)
+
p(
-
l2
-
p
-
at ax ax
-
ax ax
-
satisfying boundary conditions
(7.1
)
Boundary value problem (l.lO), (1.121,
(7.1)
has been studied with
the correctness
in
work
[54]
in view. The theorem of existence of
this problem solution is proved
in
the same way as
1.1.
The main
difference is the absence of lemma 2.1 in this problem. But if we
Correctness of
Boundary
Problems
77
take the boundary
x
=
0
where strain
is
known, as a curve
a(t)
all constructions are kept. Deduce the additional correlations
(2.111, (2.12)
between values
np,
%,
illp
way. Integrate equation
and
Idg
in the following
au
a
au
at
ax ax
(P
-
-
P)
-=-
over x from
0
to an arbitrary point from
[0,
I]
using
the
first
condition
(7.1)
and obtain the series of formulas
of
the
form
(2.3)
-
(2.7)
in which auxiliar function
Y(t>
from
(2.4)
is
identical constant.
So,
formula
(2.7q
in'
which inequalities
(2.1
I),
(2.12)
are deduced have the form
(7.2)
where
Thus, the proof of estimates
is
simplified a little and chronolo-
gically the problem on gas flow into vacuum was
investigated be-
fore
L54J
the problem with adhesion conditions
E6].
Present the basic result of the given problem investigation.
Theorem
7.1.
If
initial data
of
problem
(1.101, (1.121, (7.1)
possess the same smoothness as in theorem
1.1,
the problem
is
sol-
vable over tine
as
a whole.
2.
Inhomogeneous boundary value problems. Problem on piston
In problem designs considered above there has been essentially
used the condition under which velocity and
strain
on boundaries
x
=
0
and
x
=
1
are equal to zero.
It
is
easy to verify that
a
number of functions
u
=
ax,
p
=
(1
+
at)-',
0
=
(1
-
dk)(l
+
at)-k
+
a/k
(7.4)
is
a solution of equation
(1.10)
for any a=const. However for
a
<
0
at finite time
to
=
-
a
density and temperature become
unbounded. This example demonstrates that the solution of equation
system
(1.10)
with random smooth inhomogeneous boundary values
can, generally speaking, be destroyed
at
finite time.
Consider a possible version of inhomogeneous boundary value prob-
lem for the system
(1.10):
-1
(7.5)
Leave boundary conditions for temperature and initial data un-
changed:
a8
(7.6)
--
-
0
for
x
=
0,
I
ax