Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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39
CHAPTER I1
CORRECTNESS "IN THE WHOLE" OF THE BOUNDARY PROBLEMS
FOR EQUATIONS OF ONE-DIMENS IONAL NON-STATIONARY
MOTION
OF
A
VISCOUS
GAS
This Chapter dwells
on
the problems
of
the existence of the
solu-
tions
to
the boundary problems for a system
of
the Navier-Stokes
equations
(1.33)
in Chapter
I.
Besides proving
a
unique solvabi-
lity, the solution behaviour at
an infinitely growing time is
being studied.
1.
FOFMJLATION OF THE BASIC BOUNDARY PROBLEM.
LAGRANGE VARIABLES
Let
us
consider a system
of
differential equations
(1.33)
from
Chapter I, describing the one-dimeneional non-stationary motion of
a complete polytropic viscous gas
ap
a
at
ax
+
-
(p
u)
=
0,
-
(1.1)
au au
a2u
i)p
at
ax
ax' ax
p(
-
+
u-)
=
v-
--
+
pf,
P
=
RPb,
ad
a3
a
'i,
3U au
at
ax ax2
ax
ax
cvp(
-
+
u
-
)
=I4
-
-
p
-
+
v(
-
1'.
Here
p,
a.
v
and
p
are density, velocity, absolute temperature
and pressure, respectively,
V,
R,
cv
and
M
are positive con-
stants.
Let
us
consider the problem of a gas motion in a bounded domain
with impermiable heat-isolated walls. Let
then the boundary conditions on the boundaries
x
=
0
and
x
=
L
are expressed through
Q
=
{XI
0
<
x
<
L}
ao
aLi
ax
X=O
ax
X=L
u(x=o
=
Ulx=L
=
0,
-1
=-I
=o.
(1.2)
At
the initial moment
of
time t
=
Othe distribution of velocity,
density and temperature are assumed known:
UI
t=o
=
U
(x),
=
p
(XI,
=
b0(x),
x
E
[O,
L],
(1.3)
0
0
in which case
p0(x>
and
Oo(x>
are strictly positive and
bounded functions:
o
<
in
5
po(x>
5
M
<
a,
m
5
c0(x>
5
IJI,
x
E
[o,
L].
(1.4)
40
Chapter
2
This formulation will henceforth
be
referred to as a basic boundary
problem. Other possible formulations
of
the problems
of
a viscous
gas motion will
be
given below.
The purpose is
to
prove the olvability of problem
(1.1)
-
(1.4)
at any finite time int rval
(to,
TI. As
there
is
a local theorem
of
existence
[127, 1927,
the main difficulty in studying the
problem in the whole
is related to obtaining a priori estimates,
the constants in which depend only
on
the initial data and on the
value
I
of the time interval, but are independent
of
the length of
the existence
of
the local solution. In this case the
1
cal solu-
tion can be extended onto the whole
of
the length
[O,
'TI.
For the
sake of convenience, the proof of the local theorem of existence
will
be
given in
9
5
in a somewhat different way as in
[127,
1921.
When deducing global eatimates it is convenient to use other inde-
pendent variables, i.e, the Lagrange coordinates.
For
this purpose
let
us
consider The Cauchy problem
ay
(1.5)
where
x
E
[0,
L],
t
E
[0,
TI.
Let
y
=
y(s,
x
t)
be
the solution
to
this problem. Let ue assume
=
y(s,
x,
t)fsZo
and use new va-
riables
5
and
t.
A
Jacobian of the transition
J
=
3
is obtai-
ned
from
(1.5)
by the formula
-
=
u(y,
S),
Y
Is=t
=
x,
as
On the other hand, the equation
of
continuity
aP
au
+u-+
p-=o
dP
at
ax ax
-
can
be
written along the trajectory
y(s>
as
Hence,
Therefore,
in the new variables system
(1.1)
assumes the
form
ap
p'
au
at
po
ag
0,
-+--=
(1.7)
It
is seen from
(1.5)
that in the process
of
transformation
to
the
CO~mtneSS
of
Boundary Problems
41
variables
(c,
t)
the direct lines
x
=
0
and
x
=
L
transform
into the direct lines
6
=
0
and
=
L
respectively. Relation
(1.6) demonstrates that
in
order to prove one-to-one corresponden-
ce of the mapping
(x,
t)
+
(5,
t)
it
is
sufficient to make sure
that density
p(5,
t)
is
a strictly positive and bounded function.
In equations (1.7) let us make one more substitution, setting
Ob
a
aL
au
au
at
aq
aq
as
aq
The variable
q
is
called a mass Lagrange coordinate.
It
should be noted that
if
the right-hand part
f
initial
Euler
variables
(x,
t)
then
it
will
be the operator over
is
obtained, then the Euler coordinate
x
is
found from the Cauchy problem
CV-=M
-
(p
-->
-p-+up(
-)>'.
is
given in the
U(q,
t>
in the new variables. Indeed,
if
the solution to (1.7)
of
the particle
(5,
t>
dx
t
dt
-
=
u(5,
tl,
Xlt=o
=5,
i.e.
x
=
f
+
u(<,
r)d
T.
The dependence
of
<
on
q
is
obtained from relation
(1.8).
Hence,
0
t
x
=
S(q>
+
I
u(f(q),
TG)dTr
0
It
should be stressed that the presence of
f
in
equations
(1.9)
does not introduce principle difficulties into further investiga-
tion, provided
f(x,
t)
is
a sufficiently smooth function. There-
fore, for simplicity we shall
limit
ourselves to the case when
f
E
0.
Finally,
in
(1.9)
let us
go
over to the dimensionless variables
9
t
U
P
@
,
t'
=
-
,
u'
=
-
,
p'
=
-,
b'
=
-
,
x'
=
-
91
tl
u1
PI
0,
where
v2
41
V
=q1
,ul=-,pl=-,o
=-
L
9,
=I
P0(51d4;,
t,
=
-
0
v
91
L
cv9:
42
ChaDter
2
In this case the domain of
x'
changes
is
a
unit leyth
[0,
I]
and
the
system
of
equations
assume8
the following form
primes
omit-
ted):
a~
au
-+p
-=o,
at
ax
au
ij
au
ap
a6
a
ao
au
au
at
ax
ax
ax
ax
--
--
(p--)--
at
ax
ax
ax
p
=
kpU,
-=A
-
(p
-)
+
p(
-
)2
-
p
-
.
Here only two constants
k=
Wc,,
and
A
=
ti/
vc,,
are
used.
(1.10)
The initial and boundary conditions are written
as
ae
u=
-
=o
at
x
=
Oy
I,
(1.11)
(1.12)
ax
u
=
u0(x>,
P
=
p0(x),
9
=
eo(x>
at
t
=
0,
x
E
[o,
I],
in which case
p0(x>
functionsb Besides,
in
the dimensionless variables the initial
density
p
(XI
has the property
and
Oo(x>
are strictly positive and bounded
1dx
J
-=I.
0
PO(X)
(1.13)
If
the second equation in (1.10)
is
multiplied by
u(x,
t)
and
added
to
the third equation, we get the equality
aw
a
aw
a
ac
a
at
ax
i)x
ax
ax
ax
(1.14)
--
-
-
(p
-
+
(A
-
I)
-
(p
-
)
-
k
-
(p
SU),
where
w
=(1
/2>uz
+
b,
(1.15)
Equation (1.14)
will
be further used together with the first two
equations
of
(1.10) as
a
system
of
equations
which
is
equivalent
to the initial one.
Definition
1.1.
A
generalized solution of problem (1.10)
-
(1.12)
is
a
set of functions
(p,
u,
8),
Correctness
of
Boundary Problems
43
obeying equations (1.10) almost everywhere in
and taking on the given boundary and initial values in the sense of
traces of the functions from the classes in question.
Let us formulate the basic result.
Theorem
1.1.
Let the initial data (1.12) have the following pro-
perties of smoothness
Q
=
(0,
1)
x
(0,
T)
In this case there exists a unique generalized solution to problem
(1.10)
-
(1.12). in which case p(x,
t)
and 6(x,
t)
are strictly
positive and bounded functions.
If, in addition,
po
E
GItcL
(Q),
(UO,
so>
E
C'+"(Q),
o<
a
<
1,
and the initial data agree with the boundary ones, the solution
is
classical
Let
us clarify the scheme
of
further considerations.
As
has been
mentioned above, the main role belongs to the global a priori
estimates; the estimates of strict positiveness and boundedness of
density and temperature being central among them. The
first
stage
is
to
check positiveness of the temperature and boundedness of the
density. Then the density
is
estimated from below. The deduction
of these estimates
is
based on some additional relations and lem-
mas
which
will
be the subject of the next paragraph. Finally, the
derivatives
of
the sought functions are estimated, and the diffe-
rential properties
of
the solutions
(6
4)
are investigated, fol-
lowed by verifying the statements of the basic theorem
1.1
on the
bases of the obtained a priori estimates in
65.
2.
AUXILIARY
CONSTRUCTIONS
Let us assume that the initial data (1.12) are sufficiently smooth
functions and problem (1.10)
-
(1.12) has a classical solution,
p(x,
t>
>
0
and
b(x,
t)
>
0
(in the smallwith respect to the
it
is
guaranteed by the local theorem of existence). Let
us
state a
number of additional properties
of
the solution.
Lemma
2.1.
p
x,
t
is
a positive and continuous
in
function,
then at
E
f
0,
T
the equality
holds and there exists
a
bounded measurable function
a(t),
0
5
5
a(t)
5
1,
such that
p(a(t>,
t>
=
1
vt
E
[o,
TI.
(2.2)
Proof. Let us write the
first
equation
of
(1.10) as
a1
3U
-(-)=-
at
P
ax
44
Chapter
2
and integrate
it
with respect to
x
from
0
to 1. Allowing for con-
ditions (1.11) and property (1.13) of the initial density
p0(x>
one gets
(2.1).
Prom (2.1) and continuity
of
p
xy
t
we
get that
at any
t
there exists at least one point
a
E
[O,
13
where the
p
value equals unity.
In
case when there are more than one points,
one can choose, for instance, the least one.
As
a result, we get
the curve
x
=
a(t>
to
continuity
of
the function
~(XY
t>
The
Lemma
is
proved.
Let
us
now deduce another additional relation between the sought
functions which results from equations (1.10).
Let
us
write
the
first
equation in (1.10) as
which
is
obviously limited and measurable due
au
i3
p-=--
In
P
at
and exclude
P
-
from the second eauation of the system:
au
ax
ax
au
a21np
a
-=---k-
(PO).
at
ax
at
ax
Now let us integrate the above equality with respect to
t:
+
U0(X)
-
u(x,
t).
The second integration with respect to
X
at
a
fixed
t
from the
point
a(t>
where
p(a(t>, t)
=
1
to
an arbitrary point x
E[O,
I]
followed by taking the exponent, yields the equality
where
If
both parts of
(2.3)
are multiplied by b(x,
t)
then the left-
hand part
will
contain a derivative with respect to
t
of the ex-
ponent. Integration from
0
to an arbitrary
t
results
in
the rela-
tion
Correctness
of
Boundary Problems
45
Returning now to
(2.31,
one obtains
t
p(x,
t>
=
po(x>Y(t>B(x,
t>[l
+
kpo(x>
J’
Y(-c)B(x,
-c>G(x,
T)d7
]-I.
(2.7)
0
This formula makes
it
possible to reveal important relations bet-
ween density end temperature, or, more exactly, between their maxi-
mum and minimum values on the truncations
t
=
const.
Lemma 2.2. Under the conditions
of
Theorem
1.1
valid are the
e
s
t
imat es
0
<
N;’
i
B(x,
t)
I
N,
<
G-,
IVY’
I
Y(t>
i
N,
<
w,
where
B(x,
t)
and
Y(t)
are from (2.5) and
(2.41,
while the let-
ters
N
with indices
will
henceforce denote the positive constants
which depend on the initial data (1.121, the parameters
A,
k,
T,
m,
hl
and the conatanta from the theorems of embedding.
Proof. Integrating
(1.14)
with respect
to
52
the boundary condi-
tions taken into account, we get
In line with
that
l/2
IIU(t)ll
where
/I
*
11
According
to
we have
di
dt
0
-
I
w(x,
t)dx
=
0.
definition (1.15)
of
the function w(x,
t)
it
means
+
llw~l(~)
=1/2
IIuo;12
+
llho
l[L,(2)
5
No
VtE[O)TI,
(2.9)
is
the
norm
in
L2(Q)
the Holder inequality, in formula (2.5) for
B(x,
t)
A
Hence, the first relation in
(2.8)
is
fulfilled with the constant
11,
=
exp
IIIuoIIL,(:l)
+
(2N0)l/Z}
.
To
prove the second
part
of the Lemma, let us write
(2.7)
in the
form:
and then integrate
it
with respect to
B
allowing for the property
(2.1
1:
46
Chapter
2
I
dx
t
1
Y(t)
=
I
+
k
f
Y(T)
I
B-l(x, t)B(x,
‘I;)~(x,
7)dxd-c.
0
0
As
far as for
B(x, t)
estimates
(2.8)
have already been proved,
from this fact and from (1.13) we conclude:
B(x, t>po(x>
t
1
Y(t)
1
IVY’
+
kNY2
J’
Y(T)
f
O(x,
-c)dxdz,
Y(t>
5
It,
+
!XI$:
J’
Y(z)
1
b(x,
-c)dxd-c,
0
0
t
1
0
0
Since
the second, with allowance made for (2.9)
(
,/
L(x,
-c)dx5
K
)
6(x,
t>
20
the
first
relation yields:
Y(t)
2
Nil
while
1
0
gives
0
t
Y(t)
5
N1
+
kNoN:
Y(-c)dt
.
0
Using the Grounewall lemma, let ua finish proving Lemma 2.2.
Let
us
introduce abridged notations for the maximum and minimum
values of the density and temperature on the truncations
t
=
const:
m
(t>
=
min
p(x,
t),
%(t)
=
min
C(X,
t),
P
0
5x
51
0
Ix
51
(2.10)
Lemma
2.3.
The following relations are valid for the introduced
quat
it
ie
s
t
Bi
(t)
5
N
[I
+
n
J’
m,(-c)d
‘I;
]‘I,
(2.11)
P
0
t
m
(t>
2
n[l
+
N
J’
NI,(T)~IT
I-’,
(2.12)
0
P
where
N
=
NlN2max
{
lvi,
km}
,
n
=
NY‘min
{
kM,
m)
.
(2.13)
The roof of the above inequalities results directly from formula
(2.77
provided we use estimates (2.8) for
B(x,
t)
and
Y(t>.
Let
us
obtain one more additional relation for the temperature.
Assume
Correctness
of
Boundary
Problems
47
t
Jz(t)
=
I
J,(t)d~.
0
Lemma
2.4.
At
any
E
>
0
the inequality
$(t>
5
EJl(t)
+
CEJ2(t)
+
I$
(2.16)
holds, with the constants
C,
and
KE
depending on the data
of
the problem,
T
and
E.
Proof. Let
us
introduce the function
1
0
+(x,
t)
=
G(X,
t)
-
.f
S(<,
t)d<,
for which the next equality
is
valid
1
0
I
&(x,
t)dx
=
0
Yt
E
[o,
TJ.
Hence, at any
t
there
is
a
point
Therefore, the presentation
is
valid
xa
x1
E
[O,
13
where
&(x,,
t)
=
0.
1/2
.
~+(x,
t>l3/2
=
I
-
I+(E,
t>13/zd5
=
I+(<,
t>l
sign
+(g,t)x
x,
at;
a+(<,
t>
a<
x
d5.
Let us estimate the
integral in the right-hand part
by
the Cauchy
inequality, taking the
first
co-multiplier with the weight
p-1’2(<,
t)
:
Since
then we have (using denotation
(2.14))
1+(x,
t>l
’/‘
5
-
3
(2No)’/2mp-1’2(t)J~’2(t).
2