Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
Подождите немного. Документ загружается.
28
Chap
rer
1
dl
=
Q
x
(0,T)
function with the finite norm
which obeys the inequality
(2.28)
a1
+
X”)
to+i
x
(w(k))2dXdt
+
1-
(
1
<dx)r/’/Q
dt
r
-
‘k,P
where
w‘)
=
max
(iu-
k,
01,
QbT=
K
X(tO,
to+
i)
c
a,y,
q,
r
are fixed numbers,
xo
E
Q
is
an arbftrary point; in which case
Inn 2n
-
+
-2
-
,
q
E
(2,
-
r-2q
4
n-2
,
r
E
C2,
m)
at
nZ
3,
q
E
[2,
a),
r
E
(2,
u3J
at
n
=
2,
k
>M
is
an arbitrary number, and
E(x,
t>,
0
5
5
5
1
is
an
arbitrarily smooth finite in
internal subdomain
Q’
c
(2
the function
u(x,
t)
E
Ca(Q’)
in
which case
Iul,,al
is
estimated through
M,
q,
r,
y and the
distance from
Q’
to the boundary of the domain
Q.
If
(2.28)
is
valid for the cylinders
Q
crossing the boundary
aQ
then
r$,i
function. Then
in
any strictly
P
9-c
u(x,
t)
E
col(s>*
3.
Certain theorems from the analysis
lo.
Conjugate spaces and weak convergence. Let
E
be a normed
space. The
E
mapping onto the set
R
of real numbers which
is
cha-
racterized by linearity and continuity
is
referred
to
as
a linear
continuous functional on
E.
The set of
all
linear continuous
fun-
ctionals on
E
is
called a conjugate space and
is
denoted by the
symbol
E*.
Let the result of the action of a functional
Q
E
E
*
on the element u
E
E
be denoted through
(g,
u)
.
Let us recall some information on conjugate spaces (see, for in-
stance,
[go,
96, 98,
1211).
Conjugate to the space
Lp(Q),
1
5
p
<
m
is
isomorphic
to
the space
Lq(Q)
where q
=
p/(p-I)
if
y
>
1
and q
=
m
if
p
=
1.
It
means that for any
g
E
E
(Lp(Q))*
there exists a unique function
g
E
Lq(Q)
such that
Models
of
Heterogeneous Media
29
(
g,
u>
=
I
u(x1
g(x>
dx
vu
E
LPW,
(2.29)
51
and, vice versa, any function from
tinuous functional on
Lp(Q)
through formula (2.29).
The set
E**
of linear continuous functionals on
E*
is
called a
second conjugate to
E
space. If the element
u
E
P
is
fixed, then
the mapping
g
-t
(g,
u)
,
g
E
h*,
defines the linear continuousfunc-
tional
u**on
P.The mapping
J
,
relating any
u
E
E
with the
called a canonic embedding of the space
E
it$%
the second conju-
gate spfce
E**.
In this case the image
JE
may either coinside
with
Y
or not.
If
for a Banach apace
E
the equality
JE
=
E**
is
valid, then
E
is
referred to
as
reflexive; for instance, ref-
lexive are the spaces
L
($2)
at
1
<
p
<
a,
while spaces
Lq(Q)
defines a linear con-
functional
u**
E
hw*,
such that
(u**,
g
)=
u),vgEC
is
P
L,(Q)
and
L
,(a)
are non-reflexive.
E*
:
sequence {un
}
from
E
weakly converges to
u
E
E
if
for
lim
(g,
un)
=
(g,
u).
The concept of weak convergence
is
defined with the conjugate space
any
g
E
El
n-m
In the case of the space
isomorphous
to
L
(a>
the notion of *-weak convergence
is
intro-
duced: sequence
fun}
from
Lm(Q)
is
*-weakly converges to
u
if
L
,(a>
whose conjugate space
is
not
One more definition related to the notion of weak convergence
should be recalled here. Let
u(t)
be a function on
[O,T]
with
the values in the normed space
E.
The function
u(t)
1s
called
weakly continuous with respect to
t
in
E:
if
for any
g
E
h*
and
arbitrary
to
E
10,
T]
lim
(
g,
u(t>)
=
(
g9
u(t,)
)
t +to
2O.
The notion of compactness and criteria of compactness. The set
K
in the Banach space
B
is
called compact
iQ
any sequence of
its
elements contains a converging subsequence.
Various criteria of compactness are known. We
shall
use the com-
pactness criteria
of
the sets in concrete Banach spaces given
beloa.
Lemma
2.8.
A
family
K
of the functions from
C(51)
is
compact
when and only when
all
the functions of this set are uniformly
bounded and equicontinuous, i.e.
’)
The property
of
compactness given in
its
definition
is
often
referred to
as
relative compactness.
30
Chapter
1
VU
E
K4
Iu(x,)
-
u(x,>(
<
E.
For the set
X
to be compact in the space
L
(a),
1
<
p
<
it
is
necessary and sufficient that all the functions of the set
K
be uniformly bounded and equicontinuous
in
the norm of
L
(Q),
m,
P
i.e.
P
where
U,(X)
=
U(x
+
A),
uA(x)
E
0
at
In
the space
C"(Q),
0
5
01
<
1
which are uniformly bounded in the norm
Cp(Q>
It
should be noted that the notion of weak compactness
is
defined
in an analogous way: the set
K
of the Banach space
B
is
called
weakly compact
if
any sequence of
its
elements contains a weakly
converging subsequence.
The
following criteria of weak compactness are most frequently
used.
Lemma
2.9.
A
bounded set of
a
reflexive Banach space
is
weakly
x
+
A
ji
Q.
compact
is
the set
K
of
functions
at
p
>
a.
corn act, and
if
U
is
a weak
limit
of
the sequence
{u,}
then
IIUIP
5
lim
Ilu,ll
n+M
Lemma
2.10.
A
bounded closed sphere
of
the space
I,,@)
is
com-
pact in the sence of *-weak convergence.
It
should be added that sequence convergence by norm results weak
convergence, while the reverse statement
is
invalid. For a
Hil-
bert'~ space the following statement
is
true.
Lemma
2.11.
If
the sequence
{u,}
converges to
u
E
H
and
lim
norm of
H
.
This statement can be easily deduced from the
Riss
theorem on pre-
sentation of the functionals over a Hilbert space: any linear
continuous functional
g
on
His
realized as a scalar product
of a Hilbert's space
H
weakly
Ilu,llH
=
IIuIIH
then
un
+
u
in the
n-+m
<
65,
u
>=
(v,
UIH,
where
v
E
H
is
uniquely defined with the functional
g.
3".
Theorems on the fixed points of operators. While proving
theorems
of
existence one often considers the operator equations
of
type
where the operator
n
is
defined in the Banach space
B
and
u
=
AU,
(2.30)
Models
of
Heterogeneous
Media
31
functions in
€3-
The solution
of
this equation
is
sometimes called
a fixed point
of
the operator
A.
Let
us
recall some most known
theorems which quarantee the existence
of
fixed points
(see
[98l).
The Banach theorem.
If
the operator
A
maps
a closed
set
K
of
a
Banach space
B
onto itself,
A
:
K
4
K
and
is
contractive, i.e.
then there exists
a
single fixed point
u
=
nu
on
H.
The Shauder theorem.
If
A
is
a completly continuous operator and
maps
a limited closed convex set
K
onto itself, then there exists
at least one fixed point
u
E
K.
It
should be recalled that a quite continuous operator
is
a
con-
tinuous operator which maps any bounded closed set into a compact
one.
The Tikhonov
-
Shauder theorem.
If
K
is
a compact convex closed set
of a Banach apace
B
and the operator
A
self-maps
K
continuously
in
the
norm
of
B
then there
is
a fixed point on
K.
4O.
Truncations and averages, their properties. Let u(x)
E
Lp(Q),
p
2
1,
be an arbitrary function. For an arbitrary
k,
I
kl
4
M
let
us
set
u(~)(x)
=
max
(u(x)
-
k,
0).
The function
According to the definition
u(")(x)
E
LP(Q)
(u(x),
v(x))
E
Lp(~)
for almost all
x
E
Q
(2.31)
u(~)(x)
is
called a truncation of the function
u(x).
and for any pair
Ju(k)(x)
-
V(")(X>l
5
lu(x)
-
v(x)l.
Lemma
2.12.
Let
u(x)
i
W1(Q),
p
2
I.
Then
u"(x)
E
W'(S2)
in
which case
u(~)(x)
=
u(x7
-
k,
VU'')
=
Vu,
x
E
Ak
=
rxlu(x)
>
k};
u(~)(x)
=
0,
Ou(')
=
0,
x
E
Q
\
Ak.If
ulr
is
limited,
r
=
aQ
i.e.
~~~~~m,r
5
ko
Finally,
if
{u,}
then at
k
L
k,
the function
u(~)
E.
;;(a).
converges to
u(x)
in
Wi&)
then
u,(k)
+
LI(~)
in
wp(~).
i
Alongside with trunsactions the averages
are used, where
w(5)
which equals zero at
151
2
1
and
1
w(g)
dc
=
1.
Lemma
2.13.
Let
U(X)
E
'I$&),
1
5
p
<
m
.
P
internal subdomain
8'
c
Q
is
a smooth function (nucleus of averaging)
15151
Then for
any
strictly
32
Chapter
1
By analogy with averages
(2.32) one can also use the averages with
respect to tlme:
1
t+h
It
Uh(X,t)
=
-
I
u(x,z)dz, uz(x,t>
=
-
I
U(X,T)
dz.
(2.34)
ht
h
t-h
By combining the operations
of
truncation and averaging, one can
introduce the functions
Uhk)(X,t)
(
=
DaX
(Uh(X,t>
-
k,O).
This
definition yields the formula
(2.35)
which
is
valid for any function u(x,t)
6
L,(O,
T;
L2(Q))
at
almost every where
t
E(0,
T).
4. Properties
of
the solutions to differential equations
When studying the bounaa problems for systems of equations
of
hydrodynamics listed in
7
1
one needs information
from
the general
theory
of
elliptical and parabolic equations. This information can
be found
in
a
number
of
monographs
[
1,
92,
93,
1341.
lo.
The Shauder estimates. Let
52
be a bounded domaln in
R"
with
the boundary
I'
of
class
C2+01,
0
<
01
<
I.
In the domain
Q
let
us consider the Dirichlet problem
for
a second-order elliptical
equation
n
n
+
C
ai(xlux
+
a(x)u
=
f,
ulr
=
'p,
a
i
j
(x)~x
ix
i=
1
L
Lu
=
c
i,j=1
(2.36)
the coefficients (aij,
aj,
a)
which belong to the space
CO1(Q>,
'p
E
C2+01(I').
It
is
aesumed
that
in (2.36)
aij
=
aji,
and the right-hand part
f
of
In this case
for
any
solution
u(x)
of
problem (2.36) valid
is
the
a priori estimate
(2.38)
Iul2+a,p
5
C(lfla,a
+
lqlz+
a,r
+
l~10,Q)-
The last addent
in
the right-hand part can be discarded provided
one
knows
that the theorem
of
existence holds,
for
instance
at
a(x)
5
0
according
to
the principle
of
maximum
Iulo,Q
5
C(I'~lo,r
+
lflo,Q)*
If
r
E
C2,
'p
E
W2(r),
T
E
LP(s1),
p
>
1,
the coefficients
of
P
Models
of
Heterogeneous Media
33
equation (2.36) obey the conditions (2.371,
a
ai
E
Lq(a),
a
E
Lql(a),
p
2
n,
q,
=
1/2
q
then valid
is
the estimate
E
C(Q),
ij
at
p
<-
n
and
q
=
p
where
q
>
n
at
in which case the lower norm
((u((
also
discarded,
if
Now
let
us
consider a parabolic equation
in the right-hand part can
be
P
SQ
a(x>
d
0.
au
at
Lu
=
f,
(x,
t>
E
Q
=
n
x(0,
T),
(2.40)
--
with the boundary and initial conditions
0
uJs
=
q(x,
t),
s
=
r
(0,
I!),
r
=
aa,
~l~=~
=
u
(XI.
(2.41)
Here
L
is an elliptical operator of type
n n
Lu
=
C
aij(x,t)ux
+
c
ai(x,t)ux
+
a(x,t)u,
i
j=1
i
j
i=1
1
Let the boundary
I'
E
Cz+
O:,
0
<
CL
<
1,
the coefficients and the
right-hand part possess the following smoothness:
(aij(x,tj, ai(x,t), a(x,t), f(x,t))
E
c
CL'CL/2(~),
and the initial and boundary data
(S)
2+ a,
I+
42
be
in agreement, i.e.
U0b>
=
q(X,t)lt=O,
x
E
r,
uo(x)
E
c2+
"(a),
q(x,t>
E
c
In this case for any solution of problem (2.401, (2.41) valid is
the estimate
112.42)
34
Chapter
1
and the condition of agreement
Uo(x)
=
q(x,t)lt=O,
x
E
r
fulfilled, then valid
is
the estimate
is
(2.43)
2O.
Determination
of
a
vector field by the vortex and divergence.
In Chapter
IV
we shall co5sider
a
classical problem on defining
solenoidal vector field
u(x)
by the vortex and the normal compo-
nent on the boundary
a
+-+
-+
++
rot
u
=
w(x),
div
u
=
0,
x
E
SZ,
(u*n)l,
=
y(x),
x
E
I'
(2.44)
Here
51
c
R3
is
a bounded domain with
a
smooth boundary
p,
$(x)
is
a given vector field
on
R,
y(x
is
a given scalar function on
r
.
The system
of
equations (2.44j
is
elliptical by Douglas-Nie-
renberg
[l].
Let
us
now formulate the basic result referring to
the solvability of problem (2.44)
(for
its
proof see
[l,
32, 46,
1241).
Let
Q
be a single-connected domain with the
-
,
0
<
CL
<I
which consists
and let the given functions
Ck+
ct
C,
y(x)
E
Ck*(I'),
J
ydr
=
0,
(a),
div
w
=
0,
r
-t
k-la
3
w(x)
E
c
of a finite
possess the
in which case when
k
=
1
the equality div
w
the sense
of
the theory
of
distributions:
01
--f
I
(w
vcp)dx
=
0
vq
E
W,(Q>.
a
In this case problem
is
valid. If
y(x>
E
boundary
I?
of
class
number of connected
properties
=
0
VC,,
kl
I,
=
0
is
understood in
(2.44) has
a
unique solution
c(lwlk-i+
CL,Q
+
Iylk+
u,l?)*
-+
and the estimate
(2.45)
I
then
Models
of
Heterogeneous Media
35
3O.
An orthogonal expansion of vector fields. Closely related
to
the above problem is the
operation of expanding an arbitrary vec-
tor field
d(x>
on
Q
c
R'
of
class
L,(o)
orthogonal in
L2(Q?)
addents (see
[
32,
88,
1201
1.
Let
Q
be
a
limited singly-connected domain in
R3
with a smooth boundary. In
this case valid
is the presentation
into a
sum
of the
3
where
v(x)
E
.Yvl(d),
cp(x>
E
Wl(Q)
,
tial components of the vector
Y
on the boundary
r
=
dR
equal to
zero.
The two addents in the right-hand part
rotY
and
Vq
are ortho-
gonal in
L2
(Q)
a$ce, according
to
the Gauss-Ostrogradsky
formula, for smooth
Y
and
cp:
in which case the tangen-
+
.+
(rot
Y,
q'p),,?
=
J
g(rot
Y
-
n)m.
43
r
+
But (rot
Y
rdIr
=
0
if the tangential components
of
Y
on
r
are zero (see formula
(2.49)
below). In this connection let us
cite the formulas of calculating the differential operators
V,
div, rot
in
an arbitrary curvi-linear system of coordinates. Let
q
=
(ql,
q,,
q,)
be
a curvi-lineas system of coordinates in
R3
orthogonal for simplicity, and let Hibe the Lame coefficients
of
transfer from the Cartesian system
x
to
q:
3
ax.
H~
=
(
c
(
-11
>2>1/2,
i
=
I,
2,
3.
j=i
asi
Let
q,
be
a scalar function, in which case its vector-gradient
vcp
has
Components
aq/
axi,
i
=
1,
2,
3,
in
the
coordinates x and
,
i
=
1, 2,
3,
in the coordinates
q.
Then, let
a
be
a
1
acP
--t
--
H~
aqi
3
vector field in
R'
the components of vector
a
in the coordinates
q
denoted through
ai,
i
=
1, 2,
3.
In this case
3
I
a
a
a
div
a
=
-
[
-
(a,H,H3)
+
-
(a,HlH3)
+
-
(a311,Hz)].
H1H2H3
3%
a
9,
(2.48)
3
I
a
a
a
div
a
=
-
[
-
(a,H,H3)
+
-
(a,HlH3)
+
-
(a311,Hz)].
H1H2H3
a
42
a
9,
(2.48)
-+
The vecor rot
a
in the coordinages
q
is expressed in the
36
Chapter
1
following way:
-i
Here
curvi-linear coordinates. In particular,
if
j,
certain surface
r
and
jl,
j,
are the tangential to
I?
vectors,
then
for
the vector field
2
with the zero tangential components
j,,
k
=
1,
2,
3,
are the orths taken a?pg the axes
of
the
is
orthogonal to
a
+-+
99
a,
=
a2
=
0
we have (rot
38),
=
(rot
a
n)
=
0.
Formulas
(2.48). (2.49) are used in Chapter
IV,
for
its
proof see
4O. The Stokes stationary system and the Bubnov-Galerkin method.
Let us consider the system
of
equations
[81
I.
-i
.*
p
A
-
Vp
=
f(x),
div
u
=
0,
x
E
2,
p
=
const
>
0,
-3
ulr
=
0,
1'
=
ad
.
in a boundsd domain
sk
c
H',
It
is
known
[
88
]
that for any right-
hand part
f
€
L2(8)
E
wp>,
VP
E
Lp).
Closely related to this problem
is
the
problem on the eigenvalues
the problem has a unique solution
?(XI
E
Lemma
2.14.
There exists a countablenumber
of
negative eigennum-
&re
hk
which are corresponded
to
by eigenvector-functions
+k(d
are obtained by a closure of finite functions in the
norms
L2(s1)
and
Wi(Q).
tely differentiable, and
if
r
E
CZfCL,
0
<
u:
<
7,
then
+,
E
The system of eigenfunctions
of
problem (2.50)
is
used as a basis
for
constructing solutions to the Navier-Stokes equations by the
Bubnov-Galerkin method. !@he essence of the method lie8 in the fact
that the solution of a certain arbitrary equation
Lu
=
0
(2.51)
is
constructed as a limit
of
ttapproximatett solutions
{un].
Each
of the approximations
un,
n
=
1,
2,
.
.
.
,
is
sought as a finite
linear combination
of
the basic elements
forming a basis in the spaces of solenoidal vectors which
Inside the domain
9
the functions
Gk(d
%re infini-
E
C'+"(Q).
+k,
Models
of
Heterogeneous Media
nk
u,=
c
Ck'p
9
k=
1
37
(2.52)
with the unknown coefficients
In order to define the coefficients
ci
equation (2.51) must
be
fulfilled for un
orthogonal to
the
n
elements
of
the basia
c:
,
k
=
1,
2,
...,
n.
in an "approximste" way, i.e. Lun
must
be
(Lu,,
cpj)
=
0,
j
=
1,
2,
...,
n,
(2.53)
where
{qj}
is
a certain complete system, in particular,
cpj
=
Jlj.
Substantiation
of
the method
is
reduced to,
first,
proving
solv-
avility of problem (2.53)
and,
second,
to
carrying out a limiting
transition
at
n-t
m.This
procedure
is
carried out for
any
par-
ticular problem.