(17) o
seems to be cumbersome and we do not present it.
The similar condition for
The author expresses his thanks to ...
25 downloads
445 Views
3MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
(17) o
seems to be cumbersome and we do not present it.
The similar condition for
The author expresses his thanks to V~ So Buslaev for the discussions of the results. LITERATURE
CITED
J. von Neumann and E. Po Wigner, "Uber merkwiirdige diskrete Eigenwerte," Physik. Z., 50, 465-467 (1929). 2.
T~ Kato, "Growth properties of solutions of the reduced wave equation with a variable coefficient, ~ Commo Pure Appl. Math., 12, No. 3, 403-425 (1959).
3.
A. Poincar~, New Methods of Celestial Mechanics, Selected Works [Russian translation], Vol. 1, Nauka (1971), Chap~ XVIIo
ESTIMATES NAVIER
-
FOR
STOKES
SOLUTIONS
OF NONSTATIONARY
EQUATIONS
V. A. S o l o n n i k o v
1. I N T R O D U C T I O N In the present paper we consider the problem of finding the vector and the function
~0
and for
t=O
~
Some of
these inequalities w e r e established in F17) and used for the proof of the stability of the solutions of the s t a t i o n a r y N a v i e r - S t o k e s s y s t e m in /~(~). Apparently, the r e s u l t s of this p a p e r allow us to c a r r y out s i m i l a r investigations regarding the stability in HSlder n o r m s . In Sees. 2-5 we c a r r y out the a b o v e - d e s c r i b e d investigation of p r o b l e m (1.1), (1.2) in the spaces ~t~(Qr~ ---W~'(0~ , while in Sees. 6-9 in the spaces
~'~((~r) , whose e l e m e n t s have H a d e r continuous
d e r i v a t i v e s occurring in s y s t e m (101). In Sec. 10 we consider nonlinear p r o b l e m (1.3). For this we prove a solvability t h e o r e m of the s a m e c h a r a c t e r as in [3 I, but in the s p a c e s ~ ( Q ~
and ~ " ( ( ~ .
In
addition, we give e s t i m a t e s in different n o r m s for the solutions of p r o b l e m (1.3) when ~=0 in t e r m s of the n o r m s in/,~(~) , ~ 5 2.
and the norm in ~'(s
NOTATIONS
AND
of the vector 1~o~) . AUXILIARY
STATEMENTS
E v e r y w h e r e in the sequel ~ is a bounded or unbounded domain of the t h r e e - d i m e n s i o n a l space ~* with a compact boundary
.~ of c l a s s
whose points a r e denoied by (x,L~; ~
G~ , ( ~ = ~ x [0.T] is a cylindrical domain of the space ~N ,
~ % ~ , re--[0,T], ~T = ~• [0.T] .
We introduce the following Banach s p a c e s . By /..q(~l) we denote the space of functions whose q - t h power ~ t
468
is Lebes~ue integrable and where the norm is defined by
(2 o~) For
~ = ~ we set |t~|.,a:e~ga~h~ I~z)l. By /_~,~(.Q~ for ~ ,
v~
Similarly one defines
Z~(Q~, L~(S) and so on.
we denote the space of functions with the norm T
|~,~,, aT:
,I.
U,(.
,~
(2.2)
9
0
By ~i(fl~ we denote the space of functions having in ~ generalized derivatives up to and includLag the power ~ and belonging to L#(fl) together with all the derivatives. The norm in ~(f~) is given by the f o r m u l a p
where ~:C2,,,3,,jaD, fl~>O Ifll=jlJ,,+3,+N~ ~ f u : By ~ ( { ~
=~r
3'~tt
we denote the space of functions having ~-th power summable generalized
derivatives of the f i r s t and second o r d e r with r e s p e c t to m and of the f i r s t o r d e r with r e s p e c t to 1~;
the norm in ~(,Q~ is
~or
the leadi~ te~ms of ~ e norms in W~Cn~ ~ d
~04~ ~ we introduce special notations
2-
We define now the following spaces of HSlder continuous functions. By
H~)
we mean the
space of bounded and HSlder continuous functions with exponent ~ , defined in ~ (everywhere in the sequel ~ ~ (Off) } . This is a Banach space with the norm
where
a=~p
I ~b
i
I~-.,v,,I It~(~)-uA~c)I.
(2.4)
By ~(QT ) we mean the space of functions defined in QT and H51der continuous with exponent ~, with respect to the variable ~ and with the exponent ~ with respect to ~ . The norm in ~ (
is
(2.5)
t
~,~,t
l~-~'i~
--
~.t.t'
It-t'l "~
(2.6)
469
THEOREM
2.1. Let
tl,<e:.'~.l,(.i~ ~ . If p,~.>q/>t
are such that we have
jo : 1 - 2ul,-z - & p +•c,,,v or
:~--N ~+-c~o or v =~ we assume a s t r i c t inequality in these relations), then
(for p = ~
.P
p-i
(2.7)
or ~~- il__p,~,~ [~] T~o(~ .~ [~1~0,~Z-t ~#,~T), . r e s p e c t i v e l y , with any 5 ~-C0.5.) ; 5. depends on ~ .
LtHQ ~0(.5 and under the condition
(2.8)
Moreover, if 0-~ ~ - ~ < t , then
~.~llq,Q~t~
Ilullr
(2.n)
O0
, 0~>1 in the case of a bounded
Proof. Let
V
I~ and ~>5 in the case of an unbounded
P be the solution of the problem (2.29) with j3=O and with a r b i t r a r y finite ~ .
We have
f tr'l&~'lt~'vP&=-I ~Pg:~"-I ~'p&+ I ~ vP&-I R.,'.,PJS, fl
fl
~1
fl
~I
8
and, consequently,
where
~ =~4_ ~
If we take
Cng]~,,~ by virtue of Theorem
< 9 4 4 , then all the norms 2.3 and we obtain (2~176
of P in the right-hand side are bounded by
Since the vector ~ must be arbitrary, in the
case of an unbounded domain 1~ one has to require that ~'4~ , i.e., ~>5 9 Theorems 2.3 and 2.4 hold also for ~=~§ and in this case they can be obtained with the aid of the explicit representation of the solution of problem (2.29) in t e r m s of Green's matrix [24], which for ~=0 has the f o r m
g+
. GREEN'S
R,
MATRIX
FOR
BOUNDARY-VALUE
A
(2.38)
SEMISPATIAL
PROBLEM
We consider in the domain ~ ( ~ > 0 , t > 0 ) of the space ~ the problem
4
, V.g=O, ~/~=o=0,
~/1~=o=0
(3.1)
with a smooth finite (or well decreasing for ~ I - * ~ ) right-hand side ~<m,D , satisfying the conditions V'~ = 0 , (in the notations of See. 2, ~ d ~ (~) with any
~3 %'~ =0
(3.21
~>~ ). Our aim is to construct Green's matrix for
problem (3.1). F i r s t we find the solution of the homogeneous system with nonzero condition on the boundary ~3 = 0
477
-•-VA• + Yp :0,
ut:o:O '
tt.
Y t~:0,
:acz,,x,,t,),
After applying a F o u r i e r t r a n s f o r m with r e s p e c t to
(3.3)
~,~T~O
~c,,x~ and a Laplace t r a n s f o r m with r e s p e c t to
with the aid of the f o r m u l a ao
(3.4) we convert (3,3) into a boundary-value problem for the s y s t e m of o r d i n a r y differential equations
+ Lp :o,
(3.5)
1~0~:0
~'~+~
We shall seek %t, in the f o r m of the sum w=~+9~, where q
(3.6)
is a harmonic function in ~§ and U~ is a solenoidal vector satisfying the heat-conduction
equation. Since t r a n s f o r m (3.4) of e v e r y function satisfying the heat-conduction equation and vanishing for ~ ---~r
is equal to 0,(~,S)e" ~
, while that of a harmonic function is equal to
0~C~,s)dm~ , it
follows that (3.6) is equivalent to
where
The functions
~ , ~, ~ are easily determined f r o m the boundary conditions. We have
Solving this s y s t e m , we obtain the f o r m u l a established in [12]
478
(3.7) o
~
We make use of (3~
:-
,
,
=-v
(J~I~l)~
'~'~'.
{3.~}
and (3,8) to solve problem (3.1), We shall seek the solution in the form O' = 'u,
+
~,
(3.9)
where
If
''
FC~,t) is the function (2.11)o Obviously,
while
where
t
~,~=0. Vector
~(~,~) m u s t be the solution of p r o b l e m (3.3) in which 0~ is d e t e r m i n e d by f o r m u l a (3.10).
Now we compute ~ , we i n s e r t ~ into (3.7), (3.8), andwe p e r f o r m the i n v e r s e F o u r i e r - L a u l a c e transform.
In this connection we shall make use s y s t e m a t i c a l l y of the convolution t h e o r e m and of the
f a c t that t r a n s f o r m (3.4) e s t a b l i s h e s a c o r r e s p o n d e n c e between the following functions:
~(t).~i~cP =~
e
'I 8~c3
Therefore,
t o
~§ 479
~
t
":
u,c'j
aJ.
8
O~C
"
r
o
O
a J
J
]
Y'- , /
(3.11)
t
p:,
F o r the solution of problem (3.1) we obtain the e x p r e s s i o n s t
%
t
(3.12)
I v &c=-~)~,~Ii,Ircy.', t-'~)I,c~,'~)i,-Ul, p(~;,t) = ~
These formulas hold under conditions (3.2). In the general case we make use of the d e c o m position ~ = ~ ~p . Since ~
+~
and we leave in the right-hand side the vector ~
while adjoining
~t:~
to
satisfies the conditions (3o2), we have
(3.13} The solution of the Cauchy problem
-vA~-vp=~., ,7.~':o, c~,b eFIT=P,,x [O,T] (3.14)
b0=0, is expressed by similarf o r m u l a s : for ~_JpC~T)
t
p=o; if, however,
(3.15)
~eLpCi]T), then r
(3.15,)
THEOREM 3.1. For any ~L$~,~(~>4, ~C "f
and estimating the norms of the derivatives of ~ 2~
with the aid of Theorems 3.1, 3.2 and Lemma
we obtain
#S~.tlq,,~cf(); T +l T)ll~4tq,,Q++.m~mma,mtvF1(~7".11~,,~+, '~,:' u I/,~d. ).,'t ";-
++ (z.z+,,+:,,+. z+,C;+j :+++(+,++,+o! We c o n s i d e r the o p e r a t o r
We v e r i f y that the finite function Taking into account that
~)t~+ o The v e c t o r
++'+)++,+.o+
uJ':+t~,+ is the solution of the p r o b l e m
] - V~t+2Vo~ s a t i s f i e s condition (2.36). L e t ~ =~J]+2{o~ , Vo=V .
V~. u%0 (k~0) , we obtain
+2Wv)+.~.+Z+~,~/Z'j:'c+:_t)vy~j+L ~+o
We trmsform
~+, (.v- -vy-t;~ ++. +--[-
~+,
the r i g h t - h a n d side by making u s e of the identity
(4.12) which is valid f o r any v e c t o r functions
k
4 ~
4
k
k
a(~s) and ~Cm) . We have
-i
-+
k
Since (?k-~)~=%~-~+(V~vF~'~)=-v'n(+)~(V~vF~'9) , from here we obtain condition (2~
where p,
r e p r e s e n t s the l a s t t h r e e t e r m s and
485
+cJ, I
~r
~r
' ~' Ur
(5,=S~,I:o.1]),
,,o,1~o ~: (~k % ll~%~To,) ,c ~,~,~,~,--c~ ~:h '/') I%.~ ~,t~7 ,o
'
F r o m these estimates, by T h e o r e m 2.4, there follows that for any ~>~ in the case of a bounded domain ~ , and for any ~ >5 in the case of an unbounded domain l~, we have
I~'~,11,~.~,.~.f +cd') q-k~ , . Thus, for the indicated ~ we have
.,~.~,,~,C,~s~%~,It%r p~ --.(J,~,~-r '~+a:'T) ' ~l.k~, l and t h e r e f o r e one can fix ~--~o and for O- ~
t-'l
the solution exists. We extended it in an e~en manner
with r e s p e c t to the plane t =T. into the domain Q~, and we denote the extended functions by tl~ We define now g' and p' for T. ~ ~
as the solution of the problem
fT, -t) v.~=o,
f g' and for %~.To we set lem (4.1) in
Q~.
~'=0 ,
p'=O.
t~'lS=0,
=0 b--To
'
It is easy to verify that
O=~+g', p =p§
is the solution of p r o b -
. Repeating this argument, one can construct the solution of problem (4.1) in (~
Now for the case of a bounded domain ~ one can prove completely the t h e o r e m . F i r s t of all, f r o m the solvability of the conjugate problem (see T h e o r e m 3.1) there follows the uniqueness of the constructed solution. Estimate (4.2) is established in the following manner~ Since for %~T~ and let
I
=%~,(.b) be a smooth partition of unity on
[0,T] such that ~ p
[z,,j~,] , j~k=~+T . The functions t~k=~ , zk=p~ f o r m the solution of the problem k
uJk
486
=0,
u~kI$=0
~,~
on the interval [z~ ,fi~] o Applying to t0k the estimate we have just obtained and summing with respect to all k, we prove (4.2). For an unbounded ~ the solvability of problem (4.1) in the class I~CQ~, ~>~, is proved~ If we would have:established a priori estimate (4,1) with any ~>t , then,.approximating 1r Lr vectors, we would have established the solvability of the problem under consideration for any
by finite ~>~ ,
and the theorem would have been completely proved. A deficient a priori estimate will be obtained later, at the end of Sec. 8o We proceed to problem (1.1), (1.2). We assume that
do(X) belongs to ~'~(~) , the closure of
the set of smooth finite solenoidal vectors equal to zero on 5 , in the norm of ~'~r (/xl,b)} ~1§ [l'l,b'~ fl
for
#
where p(~) is the distance from ~ to 5 and
I
for ~4J~, for ~>~;
if, however, 0v=~ , then the seminorm \,,w,,,,,~,~=( =0, at,
u,c~,o)=t~(~:).
Therefore, an equivalent norm in J:~(~) is
where the in~ is taken over all the extensions gC~,5) of the vector v,(~) in (~( from the indicated class. THEOREM 4.2. Let
5r
and
~OAs,~,~.t.ZT j ]O.,j~s,,~,.o.,~' where
Qa.ls,~,a , =~,~ IIC~ ',~li,,~,Q , w h i l e the n u m b e r s
s,d, ~, ,d,
s a t i s f y the c o n d i t i o n s
3 t~-< 4 I , !~s, ,,. ~. ~ . ~-~
Then, for any ~ ~.~CO.~ and 1/.r
vp
(4.13)
~q) problem (1.1), (1.2) has a unique solution t~ r
(4.14) ,
and (4.15)
487
where the constants ~ and C~ do not depend on Proof. F i r s t we consider the case
T~
~ =0 and we write the problem in the form
(4.16) where
is operator (4.3) and ~g = ~ 0, g=~+0~ . We have
Here
and, consequently, 3. ~ 5" 5
t . E #
~,~ _ 5
I
By virtue of Theorem 2.1, 3
[g
.Z.~,II@lse~. +e ll~llssa) ~,=i
w
tl
Y
,
'
*
(C ~
(4o17)
Ua,llsf~,0T+C~llccllsea~t/t~ll~Q--@ ":
9
,
'
"
,
l
where
is a quantity which can be made a r b i t r a r i l y small by selecting 5 and T small. We fix these numbers in such a manner that
[~'~#,~
~/~[r
; then Eq. (4.16) will have a unique solution q~r
~
Reasoning now in the same manner as in Theorem 4.1, one can prove the solvability of the problem under consideration in the cylinder (~T of a r b i t r a r y height T o Estimate (4.15) for
~0=0 follows
from (4.2)and (4.17). Let tI0(~0 and let ff'(Z,~) be a vector from
~(QT~ such that tI~,0)=tI0(:~), il5 =0, IIt~71t~((~
g III~oII1%,n ~ We write ~J= ~J'+~ +t6, p=v*S, where -vhu.?~,=O, 7.u=-7.g ', ~JS=0,
(4.18)
(4.19)
~t,:o =0, u;is=O. According to Theorem 2.4, for any 0vH in the case of a bounded domain and for any ~>5 the case of an unbounded domain, we have
488
in
,
,
%,QT+ ll~'~
E~35,Q~, II~ p lira ~-~c, 01~ li~,Q+ ill t/o Ill~,a + II~II$,QO + O~ll~ II~,Qr. F r o m the l a s t i n e q u a l i t y and f r o m (4.17) we obtain (4.15). Thus, the t h e o r e m is proved f o r the case of a bounded ~_ w h i l e f o r r unbounded ~
for all ~>~
also f o r the case of an
~ At the end of See. 8, the estimate (4.15) w i l l be obtained as an a p r i o r i estimate, but
Approximating 'Jo~ ~~(Gh by smooth finite solenoidal vectors, equal to zero on 5
one
can easily prove the solvability of problem (4.1) for any ~/>t . COROLLARY 1. For any real T r 0
-~t -~t z6=~ , v =pe .
The estimate follows from (4.15) for
COROLLARY 2. For any ~ >C~> 0 we have the inequality
r
Q
~
T
II v" U~$(&~t tlv II where C, and r Indeed, for
"
(4.21 ')
do not depend on T . ~>C2 we have t
0r
-!
~ ~($ IItql$,Qt + 65~r~ (~).
&IIl/,llq,,a+q,llIIIlo,,O.t(6,~,(t')+r Since I!VI*{,o --- ~ IItr ~'q,,~t , we have t
and, consequently, for
~'~(C,,~) we have T
T
0
0
t 0
$
,
T 0
C~ $ _. ~T
IIO q,,Qr,~W 'ra, (T)e .
489
F r o m these e s t i m a t e s and f r o m (4.15) and (4.20) we obtain inequalities (4.21) and (4.21'). These inequalities may hold also for s m a l l e r , sometimes even negative, values of ~ (see Sec.
5). In conclusion, we mention that the boundary conditions in problem (1.1), (1.2) can be taken to be nonhomogeneous:
where ~ ~_W,
( 9 (for the definition o~ this space, see, e.g., [26]) and ~.~=0
the right-hand side of (4.15) the norm
]]~llW[m,-'&Cs ) occurs.
. In this ca.,e, ~n
In the proof of the theorem the v e c t o r
g'(~,~) has to be chosen so that
II{II~/,c~,~r
IIr ~ ~' ,-~(s,)")
The remaining arguments are left unchanged. One can show that estimate (4.15) does not hold always if q . ~ r
.
Condition (4.13) in the case of an unbounded domain ~ c a n be relaxed, requiring instead the boundedness of the local n o r m s
' ~ ~ ~t 5'~ %,(T)=Z_~CI,a.,
t)~
where oJ is the intersection of ~ and the unit sphere with the center at an a r b i t r a r y point ~E.~ . by a countable number of such domains c0k , we obtain an inequality s i m i l a r to
Indeed, covering (4.17)
r,-T
~,
a=4 Jo
k 0
"b
K
K
~
J P"mk
r, k
'~' ~
% c0 II~ U~,a,,
which, just as (4.17), is used for the derivation of the estimate (4.15). 5.
ESTIMATES
FOR
THE
RESOLVING
OPERATOR
In this section we shall consider problem (1.1), (1.2) as a Cauchy problem in the space j~(l]) ,
(5.1) where
Act)v = A~ + Bcbtr,
490
(5.2)
are o p e r a t o r s defined on the set, dense in J~(~), of the solenoidal vectors f r o m ~r on
, which vanish
~ o
We denote by 1)vtt,%) the resolving o p e r a t o r of Eq. (5.1), which a s s o c i a t e s to the solenoidal vector the vector t~(t)=~Ct,%)9, solution of the problem
&~ &t + Act)~~0, ~ t,Cq By virtue of T h e o r e m 4.2, the o p e r a t o r s
(5.3)
~(~,~) a r e defined on the set of the solenoidal vectors f r o m
J : ~ ( ~ ) , dense in j~(O) ~ They p o s s e s s the semigroup p r o p e r t y
~r
=~(t.s~,
t >.v~s, 11 Q,b -- I.
(5.4)
The solution of problem (5.1) is e x p r e s s e d by the f o r m u l a t 0
Let us show that
~(t~5) is bounded, while for t>s it is a smoothing operator in J~(~) .
THEOREM 5. 1. Assume that the conditions of T h e o r e m 4.2. hold. There exists a number such that for all
~>~o , ~,~>~ ~ 5
3~
and if
~,=oo, then ~>-~ ,
(5o6')
> ~~> ~ 5 -3-
andif
~ =oo, then ~ > 5 ,
(5.7')
0C,
for any
o
Let
~(~,~) =(~-~(~)~ d(%), where ~ is the function defined in T h e o r e m 2.1. We have
Therefore t 0
0
Let us p r o v e now (5.6), for example. Under condition (5.6'), for any v e c t o r for
t=O ,
t~(~,t)~p((~,) , vanishing
we have the multiplicative e s t i m a t e
(5o11) where the constant ~ does not depend on Oo This e s t i m a t e is equivalent to inequality (2.7) with v = ~ . Setting U : t(C "~: V(~ - ~ (~))e "~ we obtain
and for
~ >d
IIt~ II~,a ~0r>] , one constructs the numbers ~ o , . . . , ~ , possessing the following properties: 0Vo=#, #~=~,~.> ~>~-~k.~3- 9 By virtue of (5.4),
492
and, according to what has been proved,
-~t ; ' )
i
Consequently,
~C$~)
~
lleg(t)qli~.a.0
Let
~
, 0 , . ( ~ ) ~ L s ( ~ ) , acg0)~-L~C~), where
~>~
, ~5
o Then t h e r e
~ >0, depending on the constants f r o m inequality (4.15), such that the domain
~-2,~ =
~lC~ and if we s e l e c t 4
We take ~=~ +1s
(5.18)
. If ~c~>0, then
T
so that (5.18) can be satisfied by the selection of a sufficiently l a r g e
T.
The s a m e holds for [~c~ =0.
If, however, ~.~0 is an integer,
0 ~ , ~ ) we denote the Bananh space of functions f r o m
~(,~)
whose derivatives satisfy the Hblder condition with the exponent ~ , In this space one can introduce the norm
where
is the norm in the space
C"(~) , while 0 we have (6o14) moreover, r
c.L)
twl, .-eta> s
(6.15)
P r o o f . By v i r t u e of S t o k e s ' s f o r m u l a we have
$
and t h e r e f o r e (6.15) follows f r o m (6.6).
Then,
(6.16)
499
where
~(:~) is the intersection of ~ and the sphere ~,t~:l~-t~l.,.-~,],while ~ is the point of ~ n e a r e s t
to ~ o The f i r s t two t e r m s in (6.16) are bounded by
~(v'Z~/>~'~+,~a~lWI) while the third one is t r a n s -
f o r m e d with the aid of the Stokes formula in the following manner:
(6.17)
Obviously, this t e r m does not exceed c
,b4+$ _
~s
.(~
" The l e m m a is proved.
LEMMA 6.3~ For the solution of problem (2.18) which d e c r e a s e s at infinity (in the case of an unbounded ~ ) we have the estimate C~'~,)
I~)ln
C.O
~-~)~C~
9 for the potential
with the aid of (6.22):
=
~(~-~36{~)~ they are established 5
(6.23) ~c~a~ 161 -~ c s. The l e m m a is proved~ THEOREM 6.1. E v e r y vector ~ C~) ~ H~(~) satisfying the condition o
(6,24)
500
o
where ~ - L~(fl), while ~@~) is a function with finite norm . LEMMA 7:2. If K(~,~) satisfies the conditions of Theorem 7.1, while
where I ~ I ~ < ~ and if for any fixed D
~T
~ J ~,
sccpQf.c~0),then
~t.~
.... it_t'i@
....
Similar results hold for a potential with a singular or absolutely integrable kernel in the space or in the plane [~' ( ~ : 0 ) , e.g., for
R'
ill,:~ 503
We consider now the thermal volume potential t
2,
l+
t
LEMMA 7.3. If
j and l(=,,X,,O,O}=O, then
~r
(.=*,0
~)
,
(=.~')
(~)
(7.8)
Proof. The function ~C~,t):~x,b-~'(x,t) is the solution of the problem
~t'v~=~, t~=o=O,
~1
(7.9)
=0,
and the function t#(0~,[)=~(=,[)-~'(x,t) can be written in the form
~(=,t)= ~ rcm-~,t-$)J(~,~)&~J+=LI(=.b for =,,0. It is a solution of the Cauchyproblem , =
t.0
=0,
(z,[)E[1,=gx [0,T]
(7.10)
Inequalities (7.8) follow f r o m the known estimates of the solutions of problems (7.9) and (7.10) (see, e.g., [16, 26])
ruin, ,.etJ]~,,
l:Valn,.5 and then
F r o m T h e o r e m 3.1 it follows
, ~=0, and this concludes the proof~
P r o b l e m (3.1) is solvable in the c l a s s
~'"(s
Then in the right-hand side of (7.13) one adds
also in the c a s e when
~ -"" J (~,] but
%~:rc~o.
[Pal]~) and the consistency condition b e c o m e s
~It=o,~=o:0 8.
THE
SOLVABILITY (4.1)
IN THE
OF
THE
CLASSES
Ps R O B L E M ~
((~T]
This section is devoted to p r o b l e m (4ol)o F i r s t of all we p r o v e an a p r i o r i e s t i m a t e for its solution in the n o r m s of ~ " ( Q r ) 9 2*~
THEOREM 8010 We a s s u m e that p r o b l e m (4.1), in which the v e c t o r
~cH
~cj'(Q~
0 Let lJ~ ~'((~T)
' v P ~ ( Q T ") be the solution of
s a t i s f i e s the consistency condition
(8.1)
~ (~c,O) L~s:O. F o r any ~ r
we have the e s t i m a t e
L ~'' § IV],," + IpJQ,'~"'-'- q (,El], +~ l~C~:.U)' where
/
I
(~T=.O x [0,'[] and
_QI
c, ~,~r
(8.2)
is an arbitrary bounded subdomain of ~ such that
S N(~\fl')--0. The
constants 6, and C, do not depend on T .
Proof~ It is s~ficient to estimate L~:,~' since . t e r that Evp3~% is e s t i m a t e d f r o m the s y s t e m , and since ~p=v~o"
is the solution of the p r o b l e m
IS
15
(8.3)
$
(the l a s t condition is posed for an unbounded ~'l), by virtue of L e m m a 6.2 we have
lal ~c(.l~ol r QT
The n o r m 506
Eg]aT
:~
+s~p.It-tl t.(
_sSol~Lec~,t)-~,gc~,bl)-' ~ o W e shall seek the solution in the form ~=~+u~, ~=~,,~, where t~,~, v, and S are the solutions of the problems
u'/t:o=~
(8.22)
~'Is=o,
~ -VhU~+V5 =l(~.0),
q~=O, (8.23)
l~lt=o=O, u~l~=0 The v e c t o r s g ( ~ , ~ ) - ; ( ~ ) ~ r
, ~ ( ~ ) = ~ o t , ~ ( ~ ) , ( = ) , where :~,e-r
and
3 fi
a r e finite and p o s s e s s the following p r o p e r t i e s : for v . - ~
uniformly
in any bounded subdomain of fl, P~ (a,C) and ~
~ --~,
(~) =g~'(~) satisfy condition (8oi) and
I in (8.22) and (8.23) we r e p l a c e ~' by
and ~(~.o) by ~
proved, these problems will have the solutions w~)
,
(:r,)
, then, according to what has been
~'~ and for the sum
t ~ ' ) + -~- ~
= 0~) we have the
estimate
JqT~O,t~lqT+C~,~l~ 1-C,I~.IQ +C,T [o ]Q, ensuring for small T the boundedness of
Q,~ [0{~)]{='
and, consequently, the solvability of problem (4.1).
One can get rid of the assumption on the smallness of T by the aid of the method p r e s e n t e d in the proof of T h e o r e m 4.2. The t h e o r e m is proved. At the conclusion of this section we prove for any solution of problems (4.1) and (I.I), (1.2) e s t i m a t e s (4.2), (4.15) and t h e r e f o r e we conclude the proof of T h e o r e m s 4.1 and 4.2. Let ~ % ( Q T ) ,
vpeLr
be the solution of problem (4.1), while m=~ , ~=p~ - t h a t of problem (8.6). The estimation
of t~ , ~ reduces to the estimation of the solutions of problems (8.7)-(8.9). We have
513
Then, from formula (8. 1 2 ) it follows that the derivative tegrafs ~(%,t):]
K(z,~)
-~,~ where i ~
~e~
~Ve@CA)
This inequality, together with (9o16) gives the estimate (9,14) for any ~r
For
~r
and, consequently, for
, we obtain (9,15) by combining the e s t i m a t e s (9.14) and (9.17), where in the l a t t e r
one has to take 6 = ~
:
~0, so that problem (10.1)is solvable in ~(QT~ s m a l l e r than ~. 522
for a r b i t r a r y
~s
qo~j~{~(~l) whose norms are
If, however, one can take in (4.21) Y-~O , then the number P~, as can be seen f r o m
(10o7), does not depend on T and, consequently, the solution of problem (10.1) will be determined for any t ~ 0
only ff II~-II~,Q+lllt/fl~.n ~R,~
We denote by Vet,5)
the resolving o p e r a t o r of the nonlinear problem
~--T +
,
t:s :~~
F r o m T h e o r e m 10.1 it follows that it is defined in the sphere
(10.10) K :[ll~lll~,~-r of the space ~:~(1~)
for 0~t-s ~T, provided T and ~ a r e connected by the relation
In this c a s e for
~ =1~ we have estimate (10.8) with ~=0 and
ft, =q~ . Since for any fixed ~ we
have ff(~,t)~{~(I1) , f r o m this estimate we obtain the boundedness of the operator 1/ :
In addition, for any ~
~'~. K~ the vector
t~:Vct, )r -Vcf.,9~ - U-~] is a solution of the linear problem
&~'~ § ~,ch ~ + K(.',)',w)+ K(~,r
-&t
wtoo=q'-q/.
(lO.11)
T h e r e f o r e , f r o m inequality (4.21') it follows that
IIIVct, 9 ~ -V(t, 9 ~' III~,~ -~c, (t -~, ~) IIIq - q'lll~m, i.e., the o p e r a t o r Vct,9 is continuous. w e show that the operator 1](t,9 can be extended by continuity to all of ~(1~) , ~/~5 . It is connected with the resolving o p e r a t o r U(t,s) of the linearized problem (5.3) by the equation
(1 O.12) which we shall solve by the method of successive approximations, by setting !](~,S)? --~(t), ~(~)--0 ,
s We estimate
s
~
. We fix the indices
d v>{ so that
From the inequality
and from (5.6) and (5.7) it follows that
=C4 ') ~Ct-s~ S
523
a.~!
i
/__llII (t)ll~n~O.,Ct-$)
e
~%n +C,qICt-~)
e
LlltI~ I~.~n ~'
(10.14')
T h e r e f o r e , for the functions
~=~
we have the r e l a t i o n s
$
~O,~(b~(~)~,~C~) we obtain
f r o m which f o r
~. ct) ,.o, [il ~ ~.n+c, r where ,
~t[
(~-~)
-
",~
~t-~)
-~,,-~
+(t-9
j(~-g
e
(~-s)'"'d~1.
(10.15)
g
F r o m L e m m a (10,2) it follows that if
t- ~,d, r
(10.16)
then " (~ +/Z) 2
and so f o r
~;~-(5,t) we have
T h e r e f o r e , f o r any p satisfying the eondition Og~p.-~ liRil~,~
"
('I + ~ ) '
~-
4
if q,-< p -1~"~1~; ~, for
u+: ~
~Q+ we obtain the estimate
Uo.r~c,(~++ +~1+1~++~.;/+mgguy'c~p ~,+h?++!_~ .