Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids

88

Chap

ref

2

where

integral

f

V(t)dt

5

li

and in existence of finite

limit

lim

U(t>=

5:

U

m

which

is

likely not to be equal to zero. The followi

is

to

prove a uniform over

t

bound from below for density

2x"lt%

Lemma 8.1. There exists a constant

nil

>

0

p(x,

t)

2

n1

>

0

v(x,

t)

=

p-'(x,

t),

fi

=

U(O).It results

in

convergence of

M

0

t+m

such that

vx

E

Lo,

11,

t

2

0.

(8.8)

Proof. Use auxiliary correlations from

9:

2

and equality (7.39) from

lemma 7.1. ;ie would remind these formulas.

A

strain function

au

o=p--

PO

ax

possess the

following

property:

xo

=

xo(t)

E

LO,

I]

such

that

for eech

t

there exists a point

(u2

+

B)dxdT

+

Taking equality

(8.5)

into account we can write that

1

I

d5

xO

0

x

PO(<)

0

+

J

uo

I

-

dx

-

I

u0(x)dX.

(8.9)

Now

momentum equation

au ao

at

ax

---

-

is

first integrated over

t:

Correctness of Boundary Problems

89

then over

x

from point

xo(t)

to an arbitrary

x

E

[o,

11

X

t

I

(U

-

uo)dg=

I

G(x,

t)dt-

I

U(xo(t),

z)dt.

(8.10)

0

xO

By the definition

of

0

and due to equation

of

continuity we have

t t

J’

u(x,

t)dt=

-

In

p(x,

t>

-I

p(x,

t)B(x,r)dr+

In

po(x).

0 0

Therefore,(8.9), (8.10)

result

in

equality

t

p(x,

t)eq

{J’

pe

dt}

=

pO(x)exp

{t+

1

It~~~(r>~~~dr

+

A(x,

t)}

20

(8.11

Here

dx.

(8.12)

A(X,

t)

=

J’

u0(5.)d5-

I

u(<,

t)d<-

J’

u0(x)

I

-

X

1

d5

0

P

(5)

xO

0

Note that a uniform over

t

estimate results from (8.6)

IA(X,

t>l

5

c

Yx

E

LO,

I],

t

L

3.

(8.13)

Kultiplying

(8.11)

by

8

and integrating over

t

we come to cor-

relation

-

It

p

‘(x,

t)

=

E’(x,

t)eq

{-t

-

I

Ilu(r)I12dr

-

A(x,

t)}

.

(8.14)

20

Here

1

t

1r

F(x, t)

=

-

[I

+

po(x)

I

@(x,

z)exp

{T

+

-

J’

Ilu(s>l12ds

+

Pow

0

20

Hence due

to

(8.13) we have

t

m-’(t)

5

Ce-t

[I

+

J’

hi

,(-c>e

tdr

J.

P

0

Introduce again an auxiliary function

(8.16)

90

Chapter

2

as In

0

6 when the Cauchy problem were being studied. Note, that

+(Q)

=

o(E")

for

B

+

m

It

is

evident from

(8.5)

that for each

t

function

@(x,

t) takes

the values less than

1.

If

O(xl,

t)

>

I

in

some

point

xz

and

so,

+(B(xz,

t)>=

0.

But then

for

all

x

there

will

be O(xz, t)=l

Therefore, for all cases inequality

is

valid

in which due to (8.6)

Lip

5

C(1

+

m-;(t)G2(t)),

Strengthen (8.16) with the help of (8.17):

(8.17)

-1

Hence easily obtain

rn

(t)

5

C

VJ!

2

0.

Lemma

8.1

has been proved.

Remark. In the proof

of

lemma a condition has been used, that a

constant

k

is

equal to

I

in the equation of state

E

=

k

p

0.

The same considerations are applied when

0

<

k5

2.

In

case

k

>

2

IlU(~)ll

and

118

(x

)I1

L,

(Q

)

must be exchanged when deducing formu-

las (8.11)

-

(8.15).

In

formulas (8.14)

and

(8.15) indices of an

exponent

will

have an expression

P

t;

+

A(x,

t).

0

hen inequalities (8.16)

is

deduced an exponent with a negative

index

Correctness

of

Boundary Problems

91

is

bounded from above by

unit.

That’s why

in

this

case the correla-

ion (8.16)

is

also valid,

so

is

a

resulting estimate

(8.8)

(see

I

591).

Now

obtain

an

evaluation from above for density uniform

over

t.

Lemma

8.2.

Inequality

is

valid

p(x,

t>

5

Iull

<

rn

vx

E

Lo,

I],

t

2

0.

(8.18)

Proof. First, from (8.141, (8.15) we have

It t

-2

0

i;lp(t>

5

C

exp

{t

+

-

1

Jlu(z>IILdr}[l

+

1

m,(z)exp

{z

+

Second, from evident relations

owing

to

the previous estimates

(p-’

5

C),

clude

:

(8.5) and (8.6) con-

Consequently, inequality

is

valid

t

ir

(t)

5

Cet[l

+

2

I

mQ(r>ezdz

1’’.

(8.20)

P

-0

In

analogy with the previous lemnia introduce another auxiliary

function

r

Q

J,

?(O,

r)

=

J

(s

-

In

s)’/‘s-‘ds,

r(t)

1

-

?/zllu(t>ll‘.

(8.21)

By

the property (8.5)

ub(t>

5

r(t>.

Therefore,

92

Chapter

2

where

2

%(t)

L

1

-

min(1, fo(t))

(8.22)

Strengthen (8.20) with

where fo(t>

3

Ilu(t>l12+ CQ2(t)

E

L,(O,

m).

the help of (8.22):

*

t

ldp(t>

5

C[1

-

e-t

f

eTuin(l, fo(r))dt

]-I

.

(8.23)

0

Function f(t) rein(7,

fo(t>>

belongs to

L,(O,

m)

therefore

t

t+m

0

lim

e-t

1

eTf

(T)d-c

=

0.

(8.24)

m

Actually, since

1

f(z)d-c

<

w

then for any

E

>

0

there exists

0

t

to

>

0

such that for all

t

>

to

inequality f(z)d'l;

<

E

is

valid. But then fort

>

to

t t

-t

e

f

eTf(r)dT

=

e-t

I

ezf(t)d

+

e-t

1

e'f(T)dT

5

to

f(l;)dz

5

Ce-(t'tO)

+

E.

0

t

5

,+-to>

1

f(t>dt

+

-

to

0

Choosing

t

follows

that

a

function

cp(t>

=

e-t

f

ezf(z)d-G non-negative

and

continuous over

(0,

m)

reaches

it

maximum value at

a

finite point

to

E

(0,

m).

Besides this maximum value

is

less than unit. Actually,

since

f

(T)

I

1,

large enough come to correlation

(8.24).

From

which

it

t

0

cp(to)

=

sup cp(t>

=

eeto ezf(z)dz

5

I

-

<

1.

t>O

0

Thus, from (8.23) conclude:

61

(t)

5

c[

1

-

sup cp(t1l-l

=

MI

00.

P

t

>o

Lemma

8.2

has

been proved.

Introduce some more estimates from which particularly follows a

uniform over

t

boundedness of temperature

8.

As

in

Tii

3

introduce a total energy

w

=

1/2

u2+

0

and

sum

the third

Correctness

of

Boundary Problems

93

equation

(8.1)

and the second, multiplied by

u:

aw

a

a8

au

a

--

--(p-+

pu-)--

at

ax ax ax ax

(PU

0).

Nultiplying by

w

and integrating over x we find:

(8.25)

Deducing this inequality we didn't use the fact that coefficients

for the second derivatives

0,

and us

the same.

In

a general case when they are different a similar re-

lation

is

valid. Further,

if

momentum equation

in equations (8.1) are

au

a

au

a

(P8

1

-

(p-)--

at

ax ax ax

---

is

multiplied by 4u3 then after integrating we have

5

3

I

1

pu2uz;dx

+

12

I

1

pu2e2dX.

0

(8.27)

Adding to (8.26) find:

I

15

max

u2(x,

t)blp(t)(lw

i)iX<-l

Hence applying boundedness

hi

(t)

and evaluation

(8.19)

conclude:

P

Consider again momentum equation. Write

it

with the

first

equation

(8.1)

in the form

a

In

p

ap

au

ao

at

ax ax

at

ax

--

+e-=---p

-.

P:ultiply this equality by alnp/&x

and

integrate over

Q

:

(8.29)

94

Chapter

2

The last addent at the right-hand part

we

estimate by Cauchy

inequality:

ae

a

In

p

1

alnp

1

I

aa

ax

2

ax

4

G2

ax

1'

+

-p(l

+

-

I(

-

)'.

PI

-y--

Is

-Po(

-

Therefore by

(8.61,

(8.28)

we have

t

IlP,(t)lI'

+

0

II/@'/'px(T)JJ2d

T

C

C[l

+

Il~u,(~)J~~d

0

T].

(8.30)

because

of

0

<

m.,

I

p

5

PI

<

m.

From the momentum equation one has

Id

1

-

IIU

II'

+

I

Pu:k

=

-

I

(p

Bxu

+

bPXU)dx.

2

dt

no

0

For

the

first

addent at the right part

we

have

1

11

0

20

I

poXucix

I

5

c~~Q~~~

Jjuli

I

-

J'

pu;d~

+

~110~11'.

And for the second one

I

;

opxudx

1

I

Ell0'/'PXI1P+

CE

max

u2

.

0

Ix

51

Therefore the relation

(8.31)

leads

to

inequality

t

:/z

t

I

lIux(t)Il'd

t

5

C

+

E

I

(10

pxllLd

T

+

CE

I

max

u'dz

(8.32)

0 0

0

O'-XIl

And together with

(8.30)

it

gives

(8.33)

Now

let us consider the equations for velocity and temperature

rewriting them in the next form

Ut

=

PUXX

+

PXUX

-

PB,

-

OP,'

Ot

=

pe,

+

pxUx

+

p(u,l

-

pollx.

2

(8.34)

The

first

equation we multiply by

U~

and integrate over

C2

:

Correctness

of

Boundary Problems

95

Evaluate each

of

three addends in the right side:

I,

I(PXUX'

uxx)l

5

llPxll

llUxxll*

luxl

6

05x51

1/2

5

ClIPxl/

'

IbxXIl

3/2*

IIux

II

5

E,

Ib.J

+

CE1

bXll

2*

Id

=

I(POx'

U.J

5

EYIIUXXIIL

+

CErllq2*

Here inequality

(8.33)

is

used. Further, in view of boundedness of

density

p

a relation

is

valid

Beeides

I,

(8.5)

I

(Opx,

u,)l

5

~,(Iu~ll'

+

GE3/IC3px(('

and due

to

BZ(X,

t)

5

2[8(x,

t)

+

IlOXll21.

Therefore, choosing

Ei

0,

i

=

1,

2,

3

so

small that

E,+

cZ

+

E~

<

'/z

m,

se

of

(8.28)

and

(8.33):

and integrating

(8.35)

over

t

conclude on the ba-

Likewise,

if

the second equation

(8.34)

is

multiplied by

8,

and

integrated over

x

we obtain:

The

first

two

addends in the right side are estimated in the way

similar

to

(8.35):

Finally, in the last term we have

I

(PUi,

@,,)I

5

E311@XXl12

+

C,,lluxlr::,Q

'

With the help

of

inequality

from

(8.37)

choosing

E~

small

enough we conclude:

d

dt

-

II@,1l2

+

mlll~xxl12

5

C[IlOxll2

+

lIuxl12(1

+

11Q,112

+

IIUxxl12)l*

96

Chapter

2

Integrating over

t,

with the help

of

(8.28)

and

(8.56)

we have

Add

that from

(8.35)

and

(8.37)

follows evaluation

Together with

(8.28),

(8.33)

it

gives

Ilu,(t)ll

--t

0,

Il@,(t>(l

--$

O

for

t

+

m

.

Together with

(8.291,

(8.30)

it

is

easy to demonstrate that

m

d

alnp

i

1-11

-

11'

ldt

5

C

.

o

dt

ax

is

also valid.

Consequently,

II~,(t)ll

+

0

for

t

+

m

Owing to boundedness conditions for

U

and equalities

(8.5),

it

is

evident that convergence occurs to a stationary solution

u

E

0,

prl,

0

E

1.

It

can be demonstrated that convergence velocity

is

exponential

i

analogy with the case of barotropic gas movement considered in

~561.

Stabilization in stronger ranges than

rVi(0,

1)

is

verified in the

same way. Thus, the following generalization of the basic theorem

1.1

is

valid.

Theorem

8.1.

The solution of a non-stationary problem (8.1)-(8.3)

occurs for all

t

>

0

and converges

to

stationary under infinite

1

time increase in the norms of

rN,(O,

I),

1

=

1,

2,

... .

In conclusion we add that stabilization of solutions in more gene-

ralized designs considered in

96,7

is

similarly demonstrated

if

boundary values and right parts have limits

for

t

+

9.

UNSOLVED

PROBLEMS

This paragraph formulates a number of problems unsolved or partly

solved which in our opinion are

of

great interest and significance

for theory and application.

1.

The problem of existence of Wavier

-

Stokes equation solutions

Itas a wholef1 for multi-dimensional case

is

the most interesting

and complicated. Yet, all obtained resulto, except one-dimensional

flows are local,

i.

.

either a time int rval

LO,

is

considered

to be small enough f127, 170, 181, 191 for initial values are close

to the state of rest

[179J.

At

the

first

stage

it

is

advisable to consider the simplies modal

-

the Burgers system

-

in which

it

is

admitted that

P

const.

Correctness

of

Boundary Problems

97

Then system (1.25) from ch.

1

has the form

aP

-,

-

+

div

(p

u)

=

0,

+

at

(9.1

au

-,

-3

p[

-

+

(u

v)uJ =VAU

+

(A

+

p>V(divb),

at

p> 0,

3A

+

2p

10.

The energy equation

is

separated from

the system and temperature

is

determined after velocity and density

p

being found.

2.

Investigation of one-dimensional flows with cylindrical or

spherical symmetry gives rise another class

of

problems. The equa-

ti0

system in Euler variables has the form (see L117 ch.

11,

Q

2%

a

at

ax

-

(X"p)

+

-

(xm

pu)

=

0,

p

=

R

p8,

am

a

ad"

at

ax ax

-

(x

p

u)

+

-

[xm(p

+

pu

)

-

v

-

J

=

(9.2)

Here an independent variable

corresponds to cylindrical symmetry, and

III

=

2

to spherical, and

R,

V,

cv

and

H

-

positive constants,

c

=

I'i

+

cv.

A

characteristic peculiarity of system (9.2) in comparison with

(1.1)

is

that

it

is

degenerative

if

a symmetry axis (a symmetry

center for

m

=

2

lies in the

flow

domain. In

a

simplier eitua-

tion when

x

Z

a

>

0

and system (9.2)

is

non-degenerative, the in-

vestigation of validity of boundary value problems began in

[106].

It

is

important to carry out a complete analysis for a general

case.

3.

Investigation of a

so

called

flow

problem when there are areas

of in-flow and out-flow on the boundary of flow domain gives rise

to another interesting problem. For example, in the case of one-

dimensional movement when the equation system has the forn

(1.1

),

a flow problem can be set up in the following way. Let the fluid

flow in through the boundary

x

=

0,

i.e.

Then

it

is

necessary to set a boundary condition for density

x

-

a

polar or spherical radiusp

1

P

uIxE0

=

qo(t)

>

0.

Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids

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