Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
Подождите немного. Документ загружается.
168
Chapter
4
As
a consequence
of
the equality div
Gk
yields on rthe relation
0
the above formula
As
the
Hence,
of
the
axis
9,
is
directed along the normal to
1'
then
5:
=
("wk
.
n
1,
(rot
F
>,
=
(rot
f n
1.
boundary Elthe equality
+
-b
+-b
herefrom we have for every component
of
connectedness
J.
Prom the conditions
of
agreement
data we conclude:
between the initial and boundary
I
Therefore, the vortex field
dk
satisfies all the requirements ne-
cessary to construct the next approximation
3';'
as a solution
of
problem
(3.1
1.
Now let
UB
consider
in
more detail the procedure
of
finding the
so-
lution to Problem
3.
The problem on constructing the vortex field
at a given velocity field on the basis
of
the Helmholz equations
has been studied in the mentioned above works
by
N.M.Ghter
142
J
yd
the right-hand part
of
f
is
potential. In this case the system
of
equations
(3.5)
appears
to be integrated in a finite
form,
provided we
go
over to the Lag-
range variables.
If,
for a certain period
of
time, we omit the index
k
in
(3.5)
(3.6).
the characteristic curves
of
the system
of
equation
(3.5j
will
be determined as the Cauchy problem solutions
for
ordinary
differential equations
wi
-
ds
L.Lichtenstein
[
1771
for
the case when
ui
(y1, y2,
Y,,
s>
,
yi
Is=t
=
xi,
i
=
1,2,3,
--
(3.8)
x
=
(xl,
xpt
x,)
E
52
,
t
E
[O,
TI,
01
s
St
.
Since the field
is
defined only in the domain
P
there
may
be
only two cases to consider: either the point
with
the kcoordi-
nates
y(s)
=
y(s,x,t),
bounds of the domain
s
E
[
O,ti
on the whole
of
the interval
Q
or at the moment
s
=
so(x,
t)
Let
us
introduce the function
T=
t(x,t)
setting
t=
0
in the
first
case and
T
=
s
(x,t)
in the second one. In other words,
T(x,t>
is
the time
of
the particle
(x,t>
appearance in the
domain
of
the flow.
If
the equation
of
the part
r'
is
Q(x)=
0,
y
=
(y,,
y,,
y,)
remains
within
the
his point gets on the surface
11;
I
Val
#
0,
in the implicit form, then
T(x,t)>
0
can be found
Correctness
of
the Problem
of
Flow
169
from the equation
Herefrom we can find the expressions for the derivatives of the
function
t
(x,
t)
Differentiating
(3.9)
with respect to
t
and
we get
@(Y(
T,
x
I
t))
=
0
.
(3.9)
xi,
3
aip
ay.
as.
at
c
--(L+A-
)
=
0
j=I
ay.
at
as
at
J
In an analogous way
Having determined
z(x,t)
for an arbitrary function f(s,
x,
t)
let
UB
set
f(s,x,t)
(3.12)
if
J
(x,t)
=
f(
t(x,t), x,t)
.
In particular,
[y]
(x,t)
=
y(
t(x,t),
x,t>
particle appearance in the domain Q.In caee when the function
f
is
set on
and on the lateral part
S1
=
1''
x
(OPT
1,
then
it
can be expanded
onto the whole of the domain
41
setting
it
constant on every tra-
ia the spot
of
the
S'
U
Qo,i.e. on the lower bottom of thecylinder
t%
jectory
y
=
y(~,
X,
t)
:
Let, to start withLthe right-hand part in equations (1.1) be
po-
tential, i.e. rot
f
0
In this caee the deduction
of
the
solu-
tion to Problem
3
is
based on the following phenomenon. Equations
(3.5)
for
the vortex
ted as followe
along the trajectories
y(s)
can be presen-
3
aui
--
&i
ds
j=l
j
ayj
-2
w-
,
i
=
1,2,5.
(3.14)
On the other hand,
in
line with
(3.8)
for any derivative
a
yi/a%
and
ay./at
valid are the equations
170
Chapter
4
Here
Bim
is
From (3.151,
=
1,
i
=
m
;
6iIn=0,
i
#
m.
'iru
the Kronecker symbol,
in particular, we get the relations
a
yi
3
aYi
at
j=1
J
a
xi
+
C
u.(x,t>
-
=
0,
i=
1,2,3,
-
(3.16)
and from (3.10)
As
is
seen from
d
and
(3.11)
az
3
az
+
c
Uj(X,
t)
-
=
0
.
-
ax.
J
at
j=l
(3.141, (3.151, the function
wi
(3.17)
and ayi/axm obey
the same quations along the trajectories
y(s)
the initial data
for
ayi
/%xh,however, are given at
s
=
t
while for
wi
at
s
=
T
If
we introduce the Lagrange coordinates
(y,
t)
where
Y=Y(
T(xtt>j
X,
t>
then the Ehler variables (x,t) are obtained from the Ca-
uchy problem
=
ui
(x,,
xgt
x3,
s),
xi
Is===
yi,
i
=
1,
2,
3,
-
&i
ds
with
the initial data
at
S=
2.
Therefore, the solution
w
of the
Helmholz equation (3.5) along every trajectory can be expressed as
a linear combination of the derivatives
axi
/ay.the clefficients
in this combination being defined from the initial data at
s
=
z
This representation, as can be easily seen, hae
the
following form:
J
Since the Jacobian of the mapping
(x,
t>
+
(x,
t>
after goin over to the variables (x,
t)
the solution of
Helmholz
equations 73.5) with the zero right-hand
part
assumes the form
equals unity,
Here
(i,
I11,
1)
is
a cyclic recombination from
(1,
2,
3). the va-
lues of
m
+
j
-
i
and
1
f
j
-
i
are taken by the modulus 3, and
for the minors
of
the transition matrix the following denotation
is
used
a(~u,
370)
=
'3
'YB
ayu ayg
,
a,
p,
m,
1
=
I,
2,
3.
a(xm,
xl>
ax,
ax, ax, ax,
-3
The index
1
at
d
for the case
rot
f
P
0.
denotes that
it
is
the solution
of
Problem 3
Correctness
of
the Problem
of
Flow
171
-I
In the case
of
non-uniform equations (rot
f
$
0)there arises an
additional problem
on
the non-zero right-hand part and the zero
conditions at
solve$ by the Dugaumel method, i.e. solving the uniform system
t
=
0
and
on
r’.This problem can be, for instance,
a\y
+
-I+
-B
-
+
(u
V)Y
-
(Y
V)U
=
0,
(XI
t)
E
Q5=
{
(X) t)lx
E
n
,
at
t
>
5
L
O})
with the conditions
c
c
4-
‘Ylt,5
=
rot
f(x,
51,
Ylrl
=
0.
In
line with formula
(3.18)
we have
where
-3
The sought solution of the non-uniform system with the zero data
on
and at
t
=
0
is
set by the integral
+
t-b
w2(x, t)
=
Y
(5,
x,
t)d5.
0
For the complete problem
(3.5)
we have the formula
+-b
-B
+
t-b
w
=
w1
+
wz
=
w1
+
I
Y
(5,
x,
t)dg,
-0
(3.20)
+
-b
where
u1
and
Y
are obtained with formulas
(3.18)
and
(3.19).
We
can add that the expression for
3‘
can be transformed to the form
3t
+
wi(x,
t)
=
c
(rot
fIj(y(s,
x,
t),
s>
x
(3.21)
j=i
-c(x,t)
172
Chapter
4
4.
A
PRIOR1 ESTIMATES
Let
us
get down
to
deducing the a priori estimates, on the bade of
which the solvability of the problem
is
proved.
As
a basic func-
tional space, the vortex
w
belong to, let
us
take
Ca(Q,)
with the
norm
In the considerations to follow, the
a
priori esti tea
will
be ob-
tained on a sufficiently
small
time interval
[O,
Ty
fore, while proving them, we should be careful about the dependence
of the constants on the parameter
T
and
(Zk-’l
a,~9
bearing in
mind that the final aim
is
to
find the value of T(sufficient1y
small), such, that
a
certain sphere
C
tinuous mapping
A
*
.
zk-
+
zk.
In
order to make the cumbersome ex-
pressions more presentable,
let
us denote through
C
the constants
which depend only on the data of the problem (on the domain
Q
ini-
tial and boundary data, the right-hand part). Through
N
we shall
denote the constants depending on
T
and
1.
%,Q
P
but such that
at
T
-D
0
the
N
value has a
limit
which
is
independent of
lzk-’l
a,Li
}).
In other
(usually
N
is
an expression
of
type exp{Tl
cases
we
shall follow the behaviour of the constants at
T
+
0.
1.
Velocity estimates
As
before, for the sake of simplicity, let us not use the index
k
which
is
the iteration number.
Let
us
set
and, there-
CL
(dis self-mapped by a con-
’k-1
I
Zk-I
la,*
-D
lil
=
1.1
(4.1)
and
get an estimate for %using problems 1-3 and assuming that
for
a,a
the preceding approximation
zk-’
the inequality
has already b88n deduced. First, from Problem
1
we get (see
[I,
1241):
-b
-B
5
c[l~(t)l,,Q
+
I~(t)l
q+
CL,r
]
<-
C(1
+
M),
l‘(t)l
I+
a,Q
Correctness
of
the Problem
of
Flow
173
Let
us
take two arbitrary values of
t,
and
t,
from
the difference
u(t,>
-
u(t,)
and deduce therefrom:
[0,
T]
compose
+
-3
Since the norm in
L
is
estimated through that in
C
and
y(t)
is
a
differentiable function
[O,
T]
-3
Wi(r)
9
we conclude:
Further, taking into account
that
div
3
=
0
from (3.1) and (3.5)
we have
-3
au
-3
-9-3
-3-3
{rot
-
+
(u
-
V)U
-
u
v
.
u
-
f}
=
0.
(4.4)
at
-3
+
Here the vector
u
V
has the components
3
au
.
j=
i
J
axi
c
u.
2,
i
=
I,
2,
3,
In
the exact solution
of
the initial Euler equations (1.11, thia
vector
is
potential and equals
I/,
V
Iu1
At
each iteration
its
components are written by the formulas
+2
and, generally speaking, do not
form
a scalar function gradient.
Therefore, in (4.4) the potentiality of thef field
u
v
-
u
is
not used. From (4.4) we have
+
-9
-3
(4.5
1
au
-9
-3-9
+ +
-+(u*v)u-u~v
'U+
vx=f.
at
Besides,
it
should be emphasized that the estimates are deduced for
the approximate
solutions
of
the problem in the formulation (2.1)
-
(2.31, (2.24)
-
(2.311, where the
nents
of
the velocity on
ri
are not given.
Let
us
introduce an additional vector field
Ocp
as
a solution of
the Droblem
values of the tangential compo-
acp
Acp
=
0,
x
E
Q
,
-
1
=
y,
I
q(x,
t)dx
=
0.
an
r
a
(4.6)
174
Chapter
4
For
Vcp
valid are the relations
(4.7)
10
dtll
1+
5
c,
lo
-
v
Cp(t,)la,Q
5
Clt,
-
tJ.
-b
Multiplying equations (4.5) in
a
scalar way in
and using
the
equalities
L,(Q)
by
u
-
v
cp
-9
-b
-9
(
n,
u
-
09)
=
0,
(f,
v
Cp>
=.o,
hi,
v
Cp)
=
(
qt,v
Cp),
whose validity results from the properties
-9
-b
-9
div
(u
-
vcp)
=
0,
((t
-
vCp).n)lr
=
0,
div
f
=
0,
f,lr
=
0.
we get the formula
Id
-b-9
-b -b
Id
2
dt
llv
911
+
(f,
u)
+
(u
*
T7
u
-
-
(t
o);,
"u
-
v
Cp)
(remember,
that
(*
,
-1
and
11
*
11
is
a
scalar product and the
norm in
L2
(Q):
Having modified this formula using integration by
parts, the Cauchy inequality, and relations (4.21, we get
a
diffe-
rential equation for
ll<(t)ll
which affords after integration:
-b
Ilu(t)ll
5
C(l
+
M'T).
(4.8)
-b
From (4.5), (4.2) and (4.81,
for
the difference
"u(t,)
-
u(t,)
we deduce:
-9
lu(tl)
-
u(t2)1j
5
clt,
-
t,l(i
+
M'T)
5
NTI-O"I~,
-
t,)
(4.9)
2.
Pressure estimates
Let
us
now consider the Neumann problem (3.2) for the pressure.
Bearing
in
mind that in the boundary problem (3.5) for
the
vortex,
the pressure
is
allowed for only through the boundary conditions
on
r1
(see formulas
(3.611,
it
is
the pressure estimates on
I?'
that we should seek.
Let
51'
be an arbitrary subdomain whose boundary contains
r1
and
is
located
at
a
finite positive diatance from
r2*
Such
a
domain
8'
does exist, since
I?
and
P2
do not touch one another. For
p
in
9'
valid
are
the local estimates:
1
These relations
are the known Shauder's estimates for elli tical
equations (see,
for instance, the
way
they are deduced in
81,
931).
Correctness
of
the Problem
of
Flow
175
From equation
(3.2)
we have
-b
-b
IIA
Pllq,Q
<-
IUxlo,Q
lluJq,Q*
Allowing for the kind of boundary conditions for
ap/an
and for the
imbedding theorem
Wi(Q)
in
C1+
',
a
=
(q
-
3)/qand using (4.2),we
conclude
:
+
I
vP(t)l(Y,Q'
5
c[1
+
M
Ilu(t)ll
+
IIP(t)ll
1
<-
Wq(Q
1
(4.10)
C
C(1
+
IN2
+
IIP(t>Jl),
g
=
3/(1
-a>.
In an analogous way, but this time allowing for (4.31, for the dif-
ference
p(tl)
-
p(t2)
we get
-b
Let
us
also get the estimates for
purpose let us introduce one more
A+
=
P(x,
t),
-
1
=
0,
a+
an
r
p
in the norm L,(Q).For this
auxilary function
+(x,
t):
I
9(x,
t)dx
=
0.
Q
As
far
as
P(X,
t)dx=0
the problem for
$
is
unambiguously
SOlV-
able and
Q
(4.12)
Then by the Green formula,
we
deduce:
From equation
(3.2)
we
have
-D
-+
vp
=
-div(
(u
a
V
)u)
and, hence,
-3
-4
--t
--t+
(AP,
$1
=
((u
V)u,
V
9)
-
I
Q((u
.
Vh
n)d?
r
On the parts r'and
rz
and
(We
out
ap ay
-b
++
((u
V)U
-
n)
=
-
-
-
-,
an
at
1
on
r
((u
-+
+-+
a~
ag
v)u
n)
=
-
-
-
-
+
diVT[d%
-
V,>l*
an
at
-
should not forget that all the
above operations are carried
for the approximate solutions,
where, generally speaking,
176
Chaprer
4
9
vt
p
on
I?'
.
Then, integratirg by parts,
we
get
9
9
-t
-b
--t
((u
vb,
$1
=
-
((u
v>
4,
U>
+
I
Y(U
-v
+Id
r,
r
Using
the Cauchy inequality, the theorem
of
imbedding
\'/:(a)
in
L4(l?)
as
well as (4,l2), we
get the estimate
(4.13)
In an analogous way, for the differences
at
the p
we deduce:
nts
t,
and
t2
--f
--t
IIP(~,)
-
p(t,)II
5
CLlt,
-
t21
+
max
11u(t)l14,Q Ilu(t,)
-
0
It
IT
(4.14)
Using the inequalities from the theorems of imbedding, we get
(i
=
3/(1
-
a),
a
=
5/(3
+
2
a),
b
=
5/(3
+
a),
--t+
+
for the function
w
=
u(t,)
-
u(t,)
estimates (4.3) and (4.91, we obtain:
and the, allowing
for
the
IIP(t,)
-
p(t,)ll
5
NT
a(1-
a)/(3+
a)l
t,
-
t,I"
(4.15)
Returning to inequality (4.11) to reinforce
it,
we conclude:
a(?
-a)
(2
+a>
I
P(t,>
-
P(t,)la,Q,
5
C(l
+
Id2
+
N'P
>x
x
It,
-
tJ.
(4.16)
3.
Estimates for particle trajectories
Let
us
now turn to the problem
(3.5)
for
a
vortex.
Aa
inted out in the introductio
a
boundary problem of this type
hs
been studied by N.E.Kotchin
Tho],
and, while obtaining
our
eati-
mates for
a
vortex, we shall mainly follow this work. There are
two peculiarities differing our approach from that employed in the
above cited work. Pirlst, the boundary conditions for
3
on
r
in-
clude the pressure and, therefore, account should be taken of the
dependence
of
vortex estimates
on
the pressure,
in
order to finish
the procedure
of
proving the
a
priori estimates, using the inequa-
lities obtained in
2.
Another peculiarity
is
related to the pre-
sence
of
the tangential velocity components in the boundary condi-
has been po-
1
Correctness
of
the Problem
of
Flow
177
tions
on
l?'
(see formula (3.611, this difference hampering but
insi nificantly, the procedure due to estimates (4.21, (4.3j, (4.8)
So,
let us consider the solution
w
of problem (3.51, (3.61, which
is
given by formula
(3.20)
and consists
of
two addents:
u'
and;'
The vector
2'
is
an integral with respect to
t
from function
Y
the type
of
which
is
analogous to that
of
vector
3'.
Estimates "in
the
smallqt
are meant here, i.e.
T
is
a
small parameter,
so
that
if
the
estimate for
w"'
will
be obtained in some norm, then for
z2
the same norm
will
be estimated, in which case the constants in
some estimates
will
contain the
small
parameter
as
a
co-multiplier.
In
this sense the addent
2'
in (3.20)
is
a
basic addent,while
2'
is
a
subordinate one. In order to make the presentation
less
cm-
bersome, and to emphasize the basic points
of
the procedure without
paying much attention to unimportant
details, in formula (3.20) we
shall distinguish the basic and subordinate quantities. Namely,let
us assume that out
of
the two quantities
A
dependent,
B
is
subordinate
as
compared to
A,
if the following con-
dition
is
met:
as
soon
as
a certain norm of the function
A
is
esti-
mated through
Id
Z
Iw
la,S2
the same norm of the function
B
is
also estimated, having a
small
parameter
as
a co-multiplier. In
other words,
if
the norm of
A
is
bounded, then the norm of
B
can
be made
small
to any extent, provided the value of the parameter
T
is
chosen sufficiently
small.
It
kshould be recalled that through
C
we denote the absolute constants, through
N
-
those depending on
M
and
T
but having M-independent finite
limits
at
T
+O.
Besides,
let us denote through Nothe estimates
of
the functions norms which
contain the subordinate addents, i.e. Noturns to
N
as soon
as
the
estimate for the basic quantities
is
obtained.
Let
us
now
go
over to considering solutions (3.20) of problem
(3.5).
We shall begin with the estimates for the integral curves
y(sl
x,
t)
of the Ceachy problem (3.8), since
it
is
these func-
tions and their derivatives are included in presentations (3.19)
and (3.18)
for
w
*First, from (3.8) we have
(the definition of
[
f]
is
given in (3.121, (3.13)). From equa-
tion (3.15) for the derivatives ayi/ax, we, analogously, get
and 74.9).
-D
+
-+
and
B,
which are
3
-'k-
1
-f
]y(s,
x,
t)
-
XI
5
This
I
[Y](x,
t)
-
XI
5
TIU
uniformly
with
respect to
s
E
[t,
tJ.
Hence, the
same
estimates
are also valid for
the
functions
[ay1]
(x,
t)
=
-
(T(x,
t),
a
yi
XI
t):
axla
axm