* °° ^if * P
(2.8) (2 8)
here the it"' are defined successively by u(°) = w0, ti(' +1 ) = (A ( A - GGf fcc )ti<'> ) t i ( 0 - K J2 (^ «('+!> x
m=0 ^
u (( ' _- mm )) ) + rofllA; r 0 ^/ f c ; e*(« ( m ) . «
(2.9)
7
'
where the G/e are certain singular Green operators stemming from the elimi nation of the pressure g, and
m v] Z[lM]
Mx',t)-v(yW
- /L /Lr L/ a(\*>-y\< d«'-»i-+*)^/^^*■ + t)W * dyda dt
(210) (2.10)
We then define the data norm of $k by ll o||^. /P g . + + II^HB;+».(*) IIV*llBj+>.(*) + II ^ 0 * * ) = (ll/H^-/')^). °HB;,-'-V, >o**) (l|/||r,(«,«/2) -«p
/t)
tJ U
++ 3
,. (S) (S)» >
+^.6)*.
(2.11) (2.11)
where Z5)P)b = 0 if 5 -f 2 — | ^ N, and otherwise equals the possible X term entering in the compatibility condition. The following result on uniqueness and on the existence of solutions on large time-intervals for small enough data, and on small enough time-intervals for large data, is proved in detail in [7].
50 T h e o r e m 2 . 1 Let^ Cl n bbe a smooth bounded open set inn RRnn.. Let fc = 0 , 1 , 2 , 3 ths>s > 2±2 or 4, lett s > $fj- -—i 1 with ^ ^ -—3,3, and let b G R + . Consider <$/- as in (7), . ... — P ,. . satisfying the compatibility condition of order s. 1° There is at most one solution {u, q] with s la /l Tn± 2 K ig, g r ar add?*?} }} G Ih) G H^>°l^ #H^^T * '^\Q 'h)\ Q' ( IQbj JY" xx HJ,'>'M(Q # P ' 'H^'' \QJ
(2.12)
of the Navier-Stokes problem (1.1) for each set of data $k (where q for k = 0, 2 or 4 is subjectct to q(x,t) o r almost all t). J n g(a?,t)cte /o to the the side side condition ■ondition2 J^ J^ qyx, t) dx dx = = 0U //( " ± 2 _- 33 ,i// iL 2° Wften7 s* > n±2 ^ -_ 3 [[«s >> ^±2 ^ _- 3| €€ NN++ , .p ^ 2], there is a constant NSiP£ such that for data $ * wztfi data norm N[\ p\($k) < ^s,p,b there orn?. dp.np.ndinn exists a solution {iz, q) of (1.1) withn (2.12) the norm depending continuously ; [z.LZ), me norm aepenaing ccmntimi.ni on $k- When s > s 0 for some s 0 >.n±2_z ^±^ [-st_± 3 (£[4f -^2]£ ^ N + z/p ^ 2], the norm N+tfp condition
for existence can be replaced by the condition A/^ o ^ b (**) < NNSoSOiPtb ,P,b- . n s > ^ _ 3 I r.hnnsp / > ' < / > 3° When > ^ — 3,j one can for each N > 0 choose b' < b such that there exists a solution {u, q) of (1.1) satisfying (2.12) with b replaced by b', and with norm depending continuously on $k, for any set of data &k with norm ■ ^ C A ' f t ) < N. Forr ss > so, so, Jso as above, the solution can be obtained with b' defined relative to so. The statements hold with Hp replaced by Bp throughout, even without the conditions in [ . . . ]. + M / 2 ,s) ' One concludes furthermore t h a tU q€ G^ Hp ( Q(Qj / 6 , ,)) ^when s > 0 or / is as in (2.4); in some cases q belongs to a better space, see [ 7 ], T h . 3.6. For k = 0, s = 0, the result is consistent with Solonnikov's result [ 18 ], n (2ll) T h . 10.1 for n = 3, showing the existencec of n ^Wp Q / bt ))t ;» >to the U! solutions B u m u u u r , in ill vfp ' ( (Qi V.V/ n Dirichlet problem when / €G L p„((Q = 0, iUo G n ) , >§| . W h e n both / and (pk are 0, one can get solutions with still more general initial d a t a , e.g. in Ln (ft), cf. [18] for the Dirichlet problem (n = 3), GigaMiyakawa [3] and Giga [2] for Dirichlet and intermediate problems, and [8] for N e u m a n n and Dirichlet problems. In [ 8 ], we use the semigroup U(t) associated r C 0 (/{ifbc>;iip[}iy' -;tf - ; j ffpprHp(Q) (fi)") (fi)"),n), = ((—A -A + G k k)T>k to obtain solutions e.g. in spaces C°(/ with AAk Gk)r' C°(Ib'] r n 18 3 2 3r rr > ? p ( Q ,) " , callowed for when u0 is taken ini JH^(U) > 2. £ -- 1. [ ], [ ], [ ] and von Wahl —p [20] also treat problems with / ^ 0, (fo = 0, in related spaces. For the Dirichlet problem, we can include infinite intervals / = R + in certain cases:
T h e o r e m 2.2 Hypotheses as in Theorem 2.1. In the Dirichlet case (k = 0), the existence of solutions with (2.12) for sufficiently small data extends to b —
51 +00 -foo (i (generalizing
Theorem 2.1 2°), when either 1°, 2° or 3° holds in addition to the conditions^ ss>>l p£■ - -—l 7,1,s s— >>^p^^i^ - Z — : 3: 1° n < 4 .
r s TT(*>8M(n
<*n.uni €emw {/,^o,wo} n (0)
initial values, i.e., satisfy n
5
2 5 v p( +2-i,(5+2-i)/2), \\ n v p(5 + -F -ir)/2)^ p>, (v + 2 P
n ( OP. VR ) x x5 B
There is a similar generalization
(0v
\.
/fU
JX x ^ " ^ ((iL F T>^ R. )+ ) xio). i°h v
(2.13)
with Hp replaced by Bp.
T h e m e t h o d of proof of Theorem 2.1 in [7] consists of the following four steps: 1) Reduction of the linearized problem to a truly parabolic but pseudodifferential initial-boundary value problem ([ n ],[ 1 4 ]). 2) Solution of the linear reduced parabolic problem by pseudo-differential machinery (from [ 9 ], [ 6 ]). 3) Solution of t h e corresponding reduced nonlinear pseudodifferential problem, by use of product estimates and iteration. 4) Conclusions for the original non linear problem. For Theorem 2.2 one uses moreover, t h a t t h e resolvent of the linearized stationary problem is really only applied t o t h e solenoidal space, where t h e spectrum for of R + ; thi: this allows sharper r k = 0 is ac closed subset of estimates.
3
A simplified m e t h o d
We shall now explain the m e t h o d of proof in a version where Step 2 is simplified. We first treat t h e associated resolvent problem, where dt is replaced by the complex parameter —A. To be concrete, consider (1.2) in t h e N e u m a n n case k = 1 (which has been studied less t h a n the Dirichlet case k = 0): radgqq— =—f/f grad ((—A -A-— A ) X)u t xif -h+ grad
on on Q, Q,
div w= =0 dHiv i v u7/. = 00
o n O. on Q, on
ft,
(3.1)
U -- 7 o g = ^> XiW ^ on T, with / and (p givenl iin resp. B Bsp+1 lpp{T) ( r )n]n ; Q bounded bo and smooth. n H8(Q)n res 1 1 urtr\ n t i n c r 1 the normal component v ttoothe Applying — div the first firstlin line i nin1.^ (3.1) and f taking of the third line, we find: -Ag = -div/, 7og = 271 u „ - ipu.
(3.2)
52
(-:r
This is a Dirichlet problem for q, so if we denote ( have q = -RDdivf div H;-1^) RD : H;-\Q) y
j
= ■■(ifo (RDK KD),D), we
f + -\- KD{2IIU,, KD(2y1u1/ -- ip (puu),), where s ++1 1 -> m H > - - 1, -* # ;+1 ((Q) f i ) for for s > -+ p {n) f
p
(3.3) (3.3)
+1+ 1 +1 +1 (Di^ :• 5 H ; ' "*(r) _-»■ A H' ' (Q) " (^) tor s K ##!>: '*^ (r) A'c (r) -»■ ( «V /) for R 5; -+ #Hi P/ ";S+1 ^for ^*.« 6€<=e R. R.
Insertion of g into (3.1) gives the equations for u: (-A A - A)u A)M + 2 grad if/>7i /YD71 pr^ titz = AJW-hzgraa A j D 7ipr = /j + -t- grad graa R noD div aiv // + -+- grad g r a aKi \D£<£„, )^, l/ = 70 div u — 0, 70 div w = 0, por_ r r xYi IwU == W ^ rT..
W-4) (3-4)
A solution of u of (3.4) will satisfy (3.1) when q is defined from u, f and == = Vi fa on TI\ fa on T, T/tz ^1 T[u r,
(3.5)
A grad 1 D xr „ A pr„, tfFC = 22grad T T/ Tf XXi, 6 1 Q UK ^^,), -1- = — 771 /ll Hp iy) ^1 = — {pr I P Trr AX Iii,7oc ), To /U div}, /1 + gradif Vi ^1 jD ^ I/ , fi = /f + gradi?£>div/ grad J Rn div f 4-grad KnV>v< fa = — {^r,0}; i
(3.6)
where we have set
n
here / + grad RD div equals projection upeiaiur operator prj m(n) n uctib the ine projection p i j that mapss i7p(ft) n 5 onto the solenoidal spaces J* > I£- -l (1c (cf. ], f . [[5], Jps = = {u {u G G #p(ft) Hsp(Il)n |I divu divu == 00}} for f r «s > _ __ _1 r r t / x.. _ _ _ Example 3.14); K is a Poisson 1, and Ti and T[ are trace Pnissnn operator nnpra.t.nr of order nrHpr 1 operators of order 1. In order to use other known properties of the Laplace operator, we now make a new reduction. Write the problem (3.5) as follows: \)u ~ = = fi - KTu KTu on Sft, ( --Z\ A -- \)u A)iz /1 ATu J i J.i «/ v^j. - Jion w -7i = ?/JI (T! — 7i)tz -vi "\i/ non n TT. fa -— on 7itz?/. = — (T[ {T{ -— 7i)w T. 7l J.
(3.7) ^ ' '
9 { - A -— A,71} The systemn {—A A, 71} isis uniformly parameter-elliptic ^in (in the the sense sense ofof [9[], ], 6 [ ]) for A on rays with argument 6 G]0, 2TT[ ; and it is bijective Lvefor forforAAGGCCC\\ \RHR+. +. ^ By a simple application of [6], the inverse is continuous for each > ^h1 — :h s« > -—1,1,
( A^ k V 7l
A
) /
, ; i +1 nn ,i ++ 2+2 i (flw / {NtX ' " (f rn)-*' -»■ ^(T)ni/' ^^'> = (R ) ) :=(R ^ -ffi'"(fi) "NtX ( fKiN}X ) n" ):H x s5B ; :++1 1- *"nxB; ^^H; ( f(n)n, p-»({l) v> NiX x K (3.8) (3.8)
53
uniformly for A in sets Ve, e > 0, Vee={\£C ={\eC ={\£C
|argAe[£,27r-£],|A|>e}; \arg\€[e,2ir-e],\\\>e};
(3.9) (3.9)
fi = = | |A|a. A | 2 . (The H^ hereS fi and 5* ,M spaces are H* and 5* spaces provided n n )f? H*^(R ff«-"(R withh norms dep< depending on fi, as in the basic case of ), ), which is the ff«."(R»), 22 22 5/2 9 /)u|| 2 space provided with the norm || OP((|£| + |/*| + l) , ]. N Mapping ™ ||\\oi>{(\W OP((|€| ++ |Ai| + 1iy")u\\ ) * H IppP,cf. , cf. cf.'[p]. W + | 15 i • J.T__ literature, I:J..__X _ r e.g. [ M 5 i], except, perhaps properties like (3.8) are well-known in the cf. for the extension to low values of s.) With (3.8), we can write (3.7) as:
«u = iJjv,A(/i + Kff^i KN)X (T{ - 771)11), 7i)« RN,x{fi - KTu) + (^i - (7j {T[ -Riv,A(/i i)«), NtX{4>i
(3.10)
or, if we set T T $ RN,xh R KNi rl>i, ipi, fCx K X) T=( T==T,TL * =K RN vNt,\h ixfi f i ++ AA K fNtX \x1>u © i, A £\,xi = = ==(RN,\K (RN,\K «(RN,\K W I A AKK M I )) ,, , 7 T=(rp, „_. , NIX NI N>x NtX
J)) , V i "" 771 l/ / (3.11)
as:
(J + /C /C AA7> 7> = $. $.
(3.12)
We observe that the operators in (3.11), when considered as depending on the parameter A, have regularity | in the sense of [4], since RN,\, KN,X and T have regularity -foo, and K, being of order 1, counts with regularity | by [4], Prop. 2.3.14. Now we need the elementary Lemma 3.1 Let Let A :: V ::W -f AB : A:V ->W is. If IflI + V ->■ -► W W and and BB :W W-> ->VV be belinear Im mappings. > V is bijective, biiective. with W —> W is niective. bijective, then then I I++BA BA : :VV— -+V 11 l 1l (I + BAT (J BA)= /I-B(I AB)~ A.A. BA)I - B(I + AB)' BA)~l =I-B(I + AB)~ AB)
(3.13)
Proof: One just has to check:
{I+BA)(I B{I {I+BA){I [I+BA)(I - B{I B(I + AB) AB) AB)~__ 11XA) A) 1 11 = /I + + BA BA BABA-B(I BA - B(I + AB)AB)~lAXAB)AA --lA-BAB{l BAB(, BAB(I BAB{I + + Al AB)= B{I AB)~ BAB{I AB)~lA l(I + 1 -= IT + ±RA-R(T + AB){I AR\(T + 4- AB)~ AR\~XlAA = = /I, I., BA-B(I + AB)BA-B(I _
.
_
,
_
.
_
V
,
_
.
V
_
with a similar calculation for the left composition.
1
.
□
((333-' 1144 )
54
The lemma will be applied with 2 A =K Kxx : B 5 s;p++11-*-"(T) ~ ^ " ( rn+1 ) n + 1 -»• ff;+H'+^in)", >"(ft)n, n h,i +l plp n+ +2 n 1 sn+1 n+: B = T:H; T': H+2sp^m '>^B(tt) -» T: H' '^{n) -»• Bn+1 (T)B p , Bp" '~ ^(T) p - -+ I : h°^*'"(uy' (i^(T) \ p
ss>> ± - 1, 1, ^AeeKK,,((33 .. i155)) P and also with the roles of A and B interchanged. The lemma shows that (21) can be uniquely solved (in these spaces) if and only iff /7 + + T)C\ is invertible, in T/CA is which case l1 l1 (I I-ICx(I TICx)(/ + + KxT)~ /lCxT)CAT)"1 = / - K ICx(I TKxY^. T.-l X{I + TKx)~
(3.16)
w I/ + TJC\ TJC\ is is much easier to deal with thann 7I + KxT, Now K\T', isince it is a parameter-dependent ps.d.o. on the boundaryless compact manifold T\ In details, TK\ is an l)-matrix s, TK,\ a i (n + 1) x (n + l)-m TIC
_— (/
TR TRN,\K T.KJV.AK NtXK
TK TI
/ *,x - ^ ( T , _ 1I)RNXK
(T / _ yi)KNtXJ
\\
'
^
/o 1 7 \ (3.17) i 7
J
4
where the entries are of regularity | , in the sense of [ ]. It is parameter-elliptic in the sense of [4], for A on rays in. C C \\ R ++,, since this is a question of bijectiveness of certain model operators at the boundary (and certain matrices), a property that can be traced all the way from (3.4) to (3.12); the parameter-ellipticity of (3.4) was shown in [14], Sect. 6. This implies that for any e > 0, there is an r(s) > 0 so that for any t £ R , n+1
t n+1 n+1 s^(r) ^Bt^s^(r) , I + TKx: TICx B p^{r) p'"(T)
n+1
,
uniformly for>r A e W , Wt>r{t) e>r(t),
= rarg g AAA€ € [€e[e, ,[e, 2 2JT 72TT r --e- ]e], , | A|A| | >> r (r(e) e ) } ;}; W e£Ctr{e) ,.r(e) W = {{\eC {A € G C ||Iaarg el, IAI > r(e] rU) =
^ ^ (3.18)
we denote the inverse 1 1 {I =Qx. (I + TICx)TKx)=Qx.
(3.19)
For such A we also have the inverse, by Lemma 3.1, 11 +2 n n n + 2s +2 (7 + KxTT = I - KxQxT :■ JJp 77; ^(Q) ^77; ^H '"(fi)", ^(n)n, ICxT)7 7 p+s2+^(Q) ^H; ' ; — J — I\-XWXI v ('f i ) ^ 7 7 ; +p^ ( 4
(3.20) (3.20)
uniformly for A 6 We £^ — 1. £,r(£), Altogether, we solve (3.1) by taking u = (I + ICxT)-1® * = (J (7 - ICxQxT)(R /CxQxT)(RN,xh ,xfi + I
(3.21) ( 3 - 21 )
55
and defining q by (3.3). The point is here that all operators except the factor Qx stem from classical resolvent problems for the Laplace operator; and Q\ is a parameter-dependent ps.d.o. on T. For the other boundary conditions (the cases k = 0,2,3,4 in (1.2)), there are similar methods; for k = 0,2,4, the roles of the Dirichlet and Neumann problems are interchanged. ire —A is replaced re Also in the original problem where by dti this approach gives some simplifications. Indeed, as described scribed in [7], the resolvent considerations carry over to solvability of the t-dependent problem with initial data 0, formally by a Laplace transformation. Analogously to the derivation of (3.21) from (3.5) we find that the problem (d - A - g)v + {dtt-A-Q)v t-&-Q)v gi + KTv KTv = = 9l g1 T[v = T!v = Ci Ci
ooonQn, nnQy RR ,, on SR, OT\ST>.
((3.22)
'
'
su with v{x,i) v(x,t),) g\(x,t) gi(x,t) and £i(#,0 £i(a:,2) supported for t > 0, and g < inf{ReA {ReA | A £E a W solution operator de described by Wsr,r(e)}j £ir (j)}, has a solution V
= ( (/ /- -KKQQTT) )( (RRNN N5PPI II ++ K KKNNN C C II ));;
(3.23) (3.23)
with ( R N K N ) = ( ( ** - ^^ - -^ ^) " \
i1
/
K = ( R N tKf K N ) ,
1
>,
(3.24) (3 24) "
oQ — TK T T) -W1 . = rr_L (7 (/ + T
The exists since -f T K is derived from the parameter-dependent op ce 7 + Lhe latter latter exists since erator 7 + TK\+ by replacing —A by IT in the symbol ana and using a pseudoQ xator 7 + T)C\+Q by i tne symDoi us differential definition in one more variable; here 7 -f TK\+ is invertible for A Q lifferential definition ir 3 more vanaDie; nei>re I -f TIC\+e is in 6 6 ], Theorem 3.1 in an obtuse neighborhood of {Re A < 0}, and the calculus of [ of {ReA < 0}, andI the calculus of [ ], 1° is applicable. The resulting estimates for the solution operator are the same as those described in detail in [7]. It is because of the constant Q in (3.22), that we do (s,s/2) not obtain time-global estimates inQ Hp spaces in general. See however the Hp ' J^sp< special considerations for the Dirichlet problem in [7]. TT
4
Exterior problems
Consider now the case where fi is the complement of a compact set in R n , still with smooth boundary T. We can then investigate how the method of Section
56 3 can be used. Applying — div to the first line in (3.1) and taking the normal component of the third line, we again arrive at (3.2), now an exterior Dirichlet problem for q. T h e o r e m 4 . 1 The exterior Dirichlet
problem
—Av —Av = g in Q,ft, has a solution
JQV 70^
— = tp ip on T,
(4.1)
operator (ifo KD) such that g radK Bp;+1+ 1 "'{V) (tynn grad A'p V ) -> Hsp{U) D : 5 n n grad g r a d iRfDo ddiv i v :: Hl(Q,) # p s ( f t ) n -*■ -> H*(Q) HsJQ)n
for s £ R , 1 for 1, / o r ss > > -- -- 1,
y
F
(4.2) (4.2)
p
x
g r a d /KD v o maps into f)q>i ren-Hq{{\ \ > R})> when {\x\ > R} C ft. (More and grad precisely, RD is defined for g with compact support, and grad RD div is extonr\ non nil f inn n i t 11 ) tended by nr^nr\continuity.) r iifere / ! # £> should map Here KD is uniquely determined by the property that grad A KD n n ire 0(\x\ ) for \x\ — > • 00. It is also u into functions that are 0(\x\~ ) /£ uniquely determined by -R}) — or requiring graa grad I\D KD 10 to0 map into p L > i ^ requiring 1 R}), ifP
T h e m a p p i n g KD is constructed as follows: We want to find a solution
-Av
= 0 in
ft,
70U = ip on T, I\
(4.3)
5+1--
P for given V ip €£ BSpP+1 r a d f is 0 ( | x | ~ nn ) for \x\ |x| -» 00. (The (Tl *((rr)),, such t h a t g5radv nn r\ri Tir-i I I r\ \ ns-\ K n t i l derivatives 0 ( | a : | ~ )) , anc and then g r a d u £ #ff£({|x| £ ( { | £ | > R}) stives will also be 0(\x\ R}) for all r, all q >> 1, when {\x\ of (4.3), we can study {\x\> > R} C ft.) Instead Instc
—Av —Avcc = 0 in
ft,
JQVCc = V ip> — 7o^ —cc on on T, T,
(4.4) (4.4)
where c is a constant to be chosen freely; if vc solves (4.4), then v — vc -\- c solves (4.3) (and vice versa), and they have the same gradient. We can assume t h a t 0 is in the complement of ft, so t h a t the inversion x \-+ h->- ;r/|;r| # / | ; r | 22 m n a p s ft iZ onto onto ft* ir \\ {0}, where ft* is a bounded open smooth set with 0 £ ft*, ft*, and 3ft* is the image of T. (In the following, let n > 3; 3ft* == T* r* is there is a. sirr a similar proof for nn = proof for - 2.) Let W(x) be the solution of the special Dirichlet problem ft*, --AW A W = 0 in ft*,
jlQ0W(x) W{x)
= = \x\2~ -nn
on o n TT*; *;
(4.5)
57 since | x | 2 "_ n > 0 on T*, W(0) W(0) > > 0 by the m a x i m u m principle. Now if a function V is h< harmonic in ft* \ {0}, then the function 2 n 2 \x\ ft, | x | 2~_ nV(x/\x\ V r ( ^ / |)a : | 2 ) is harmonic in ft, transformation). in M, and a n a vice vice versa versa (the \i\ Kelvin 111 iraiisiormaiioiij. 2 n 2 n Since -0 on T is carried over to in Bp++ 1 ~ F ( r * ) , to \x\ | x | ~2 -tp(x/\x\ ^ ( ^ / | )x | 2 ) on on T*, lying ngin5; -*(r*), we can find a solution of (4.4) by solving
-AV,V C C = 0 in
n 70Vc = \x\ k | 2--n(^(x/\x\ ( ^ ( ^ 2/)| x | 2 ) - c) on T*. F\
ft*,
(4.6)
T h e problem (4.6) has a unique solution Vc G i ^ + 1 ( f t * ) fl C°°(ft*) for each c; and by the linearity, V Vcc = = Vb — cW, cW, cf. (4.5). Take c* = V VQo(0)/W{0); (0)/W(0);
(4.7)
= \x\2-nVc.(x/\x\2),
(4.8)
then Vc.(O) Vc*{0) = 0. Now let vc.(x) .{x)
it solves olves (4.4). Since Vc. (0) = 0, £> Davacu*(x) 0(\x\1-1n-^) - r *-l Qf l) for \x\ | z | -> 00, oo, any c *(z) is 0(|a?| grad c * and its a (seen seen from the Taylor expansion of Vc* at 0). In particular, gradu derivatives ivatives are 0 ( J| ## | ~ n ) and hence Lq integrable at oo 00 for any q >> 1. On the +1 H*+11 over bounded other H** subsets er hand, Vcc* G i#/pp+ 11 ((fQt ) implies t h a t fvc*cc* is in Hp bounc 1 of ft. Altogether, it is found t h a t v *, and hence v — v * + c*, have gradient cc 1. it is found t h a t Vn*. and hence r; = ^cc- + c*. ha i p as in H*(Q) and in in n<7>i,r€R rP|Lg>>ii , rr €eR R S"J({|ar| ^^ (( ii ll ^^ ll >— ^^ }} )) -- Defining Hp(Q) and — -R}). Defining /KD as the the m maappppiinngg from from ^ to v, we have obtained an operator with the asserted mapping properties. ^ iu c, we iicive u u t a i i i c u dii u p c i d i u i w i t n t n c cissci i c u i i i d p p i n ^ p i u p e i ties.
To show show the the uniqueness, uniqueness, let let uu be be aa solution solution of of (4.3) (4.3) with with grad grad uu = = 0(\x\~nn)) To 0(\x\~ for \x\ \x\ —)■ —>• oo. for oo.Recall Recallt ht ahta tany anyfunction functionu(x) u(x) t ht ahta tisisharmonic harmonicononft fthas hasa aunique unique Laurent expansion expansion Laurent
(4 9)
«(*) ^f (W +) f+E;E]1L Jd ^S^] f.,.,, == f;ffj;(«)+f; Ef ;f^ ; «( «) + «(«) ^^ »W = E k=0 /c=0
k=0
4 -9 (4.9) ((4.9) - )
' '
where the ,ne functions i u n c u o n s H£(x) nk\x) 1 and a n a Hk{x) nk\x) are are homogeneous nomogeneous harmonic narmonic polynomials polynomials of degree k; cf. Brelot [ ], p p . 197-202, where also sets \x\ <^ < R2} ee fe; cf. where aiso also sets {Ri {Rx < < \x\ < ri2j R2} 1. Brelot creiox, [[*], j , p p . 197-202, i y f - z u z , wnere \n\ <^ are considered. T h e first series in (4.9) converges uniformly on bounded sets, sidered.. T h e first series in (4.9) converges uniformly uniformlv on bounded sets, sets. the second ond converges converges uniformly uniformly on on sets sets {\x\ {\x\ > > R} R} contained contained in in ft, ft, and and the the derivatives of u are represented by the termwise differentiated expressions. ives of u are represented by the termwise differentiated expressions. W h e n u has the form (4.9), OO
OO
TT
(
\
ggrad d( z^) ^= / (^kz(x) ^) + ^X g>grad r aa dd -rr^2k> rra da u«(«) =^ z£g3gr grad ra add i^H* |jg -| wp. 2g%^f c,, fe * ==1l
fc=0 k=Q
'' ''
(4.10) (4-10) (4Ai
58 n 2+2k where grad(iJ fkc{x)/\x\ (x)/|x| where gT&d(H - n - 2 +)2 / c ) is 0{\x\1-n~k) for each k > 0. The requirement — n> n rrarl ?/ sshould h o u l d hbe p f0n.7*l n l p s oout u t t.hp n l v n n m i a l s H£ FT? for a s well that that gradu 0(|a:|~ )\ rrules the npolynomials for & k >> 11 as as the term Ho/\x\n~2, so
u(z) = H-0+v{x),
V(x)
=f 2 j £ ^ .
(4.11)
fe=i k= l ' '
Now the Kelvin transform of v> v, V(x) V(x) == |x| |^|2 2- _n nv(x/|x| t;(^/|^|2 )2 ) ha has the representation oo OO
V(x) = ^2Hk(x);
(4.12)
k == l l fc
7 is the unique solution of (33) with it is C°° on ft* with V(0) = 0. Thus V c = #0*. If u = Z+ v(x) also solves (4.3), with v = J2T=i Hk{x)/\x\n~2+2k, then v is the Kelvin transform of the unique solution V of (4.6) with c = q V(0) = = 0. Now V --V V = = (-# {-H* +c)W this also has V(0) c ) ^ (cf. (4.5)), and since 0* + V{0) = = 0 and W^O) W{0) ^ 0, c must equal # * , and V = i'. V. Thus u is 1/(0) = F(0) uniquely determined. uniquely determined. To require require gradtz gradtz 66 L £ 9 ({|a;| > R}) for some q < n/(n — 1) 1) likewise likewise reduces reduces To g ({|x| > R}) for some g < n/(n 1— 1 __ nn £ L9({|a?| > R})\ uu to the form (4.11), since nonzero polynomials and | ^ | to the form (4.11), since nonzero polynomials and | ^ | ^ ^ ( { l ^ l > R})] and and the the analysis analysis goes goes as as above. above. Now consider RDRp. TO solve the problem (4.1) with ip = 0 and a nonzero n g = div/z, div/i, /i h G Hp(Q) , we let / be a continuous linear extension operator s n / : Hp(Q) -> H (R ) such that d i v / / = / d i v / , and search for v in the form v = r^vi rftt>i + ^2, where v\ i;i and ^2 v^ solve ^«*, i n fft, t, ff-Av2 --A M ^„ Avv22 = 00 in ft, -A v i = /flf /# on R nn , resp. < < ' ■Avi _ [[70^2 70^2 = 7 0 ^ 1 oon n T. = --70^1 r.
, (4.13)
(rn denotes restriction to ft.) When g = div/i, we want gradf to depend H*(Q,)nn.. continuously i7*(ft) continuously on on ha in in xip(^j To To begin begin with, with, let let gg have have compact compact support. support. To To get get aa convenient convenient solution solution of the problem for v\, we define R as the operator (for compactly of the problem for v\, we define R as the operator (for compactly supported supported
/)
R:f^wn\x\2-n*f-c(f), where c(/) is the constant / r 7S o vK I^* |f- *^ / )* ^ ^ Jrlda
(4.14) (4.14)
59 and a; n |x| 2 n is the Newton potential (u; n |#| 2 O p (l£l~ 2 )/)- This satisfies -ARf = / , and
n
* / can be viewed as
(4.15)
[/ ioRfd
by el elliptic regularity, so
s +1 +1— -
" p' ( r ) C L p ( r ) , ai Vl G 5 p 7ofi E -Bp (r) C ^p(r), and the expressions are well-defined. Now insert 70^1 in the equ Now insert 70^1 in the equations for v2\ then the operator K& established above gives a solution v2 with gradu2 E HUQ,)12. Altogether, we take Rog Rog = =v v= = rnvi rnvi -\-v + v22 — — rnRlg rnRlg -- K KDDjj00Rlg. Rlg.
(4-16) (4-16)
ben g #= div h is In particular, when = div/i is inserted, then grad -\ grad dR — grad KDJQR div //i. Ih. div 2 = n ggrad grad vv = = TQ TQ grad grad v\ v\ + grad vt>2 = rrn i2 div div Ih Ih — — grad grad KoloR KpyoRdiv Ih. (4.1' (4.17) (4.17) Here we have for gradi>i, by Fourier transformation, gradwi gradfldiv/ft /K| |^|22|)2).),j-i=>j = )lh. gradui gradvi = gradi?div//i gradtfdiv/Zi = O P ( ( ^ i/^/K nn)/A)^ii=,i1. . . , n
(4.18) (4.18)
2 2 )perator OP(( OP ((&£,/|£| ) 1J= i,...,„) extends exte The OP((£i£j/\€\ )ij=i,...,n) to a continuous operator in rhe operator n nnn\n n Kv n r^ciilt. rS (^alrl^rrin nnrl 7,vcn LLp(H ) by a result of C alder on and Zygmund; and it is likewise continu resul p(R ) s n nn ous in H (R ) for all s E R, since : > u s i n # p' ( R ) " f o r
2
s
OP((0 OP((6^/K| „) O P « £ PS) = =O P (((^^//K KI| 22k;=i,..,n). ) Il JJ == Ii „). „). OP ((055)) op((6^/Kr) op((«i/Kr),-2 )liIJ= =iiil...ln) OP((0" op((^7KI IJ= OP«O „)
(4.19) (4.19
Thus l l m gj r a d v i H l ly.(n) ^ ,. ^ < Hgraduillff.^n) < ^i||'"N/f«(R») Ci||//»|| Ci||/ft||/r.( Cyi/iH^,,™NI^mraut7 ff ff..( ( R») R ») i< Cyi/iH^,^)^ HgradviHif.fRn) H8 illff;(R") ^ ^C2 |^| »IM P (4.20) " (4.1 This extends ic to h with arbitrary support, by approximationi by rf(x/N)h r)(x/N)h,) for I lnis exienas r) E Co°(R n ) equal to 1 near 0 and N ->• 00. For v2, we need to know that the mapping from h (or just from grad^i) to 70^1 is continuous from is shown by the help of >m H*(Q,) Hsp(p) to B 5 *p + 1 ~ *p (T). (r). This This \i Lemma 4.2 below. Takei for S H a large ball containing T in i its interior. In view of (4.15), the lemma gives that +1 p +1 ||7oVi|| < CC H7oVi|| ;+i-p CalMljf.+ifs) < (r) < 3| ||gradt;i|| H.(1) ovi\\BsBBp;+i-p - (r) ^ ^C C3||^iHH; (I) ^ CCsllgrad n-p S 33\\vi\\ | | « I | |HHs;p++i I ({E) E) S o o 3CC^WgiadviWjj^ |gi (D
< cn*c!n\\h\\„-.^. < CC < CC3C 3C2\\h\\ H^y 2\\h\\ Hs{^y
^21)
60
Altogether, the mapping grad RD div, as realized in (4.17), extends by closure to a m a p p i n g with the continuity in (4.2). □ In the course of the proof we used the following variant of the Poincare inequality: L esm £ — 1, I, and let S be ai bounded mm m a 4 . 2 Let s > ^ uoujiueu connected cofuiccita n n ofH"R with a sufficiently sufHcientlu smooth boundary boun such \ch that that the the injection injection into i^p(H) Hpi^) is compact. ompact.
open subset uptu suuset +1 of Hp of Hp+1(E.) (E)
Let T be a nonempty smooth closed smooth clos< closed hypersurface hypersurface There is a constant tant C such that for u G #H*+ p + 1 l((liL), H), adw
in in S. S.
(4.22) (4.22)
nt of Bp+1 p {T) (r) C Proof: Note t h a t 70u — u\r is well-defined as an element ot. be b p found, f o u n d there t h p r p is is Lp ( r ) , so t h a t the integral has a sense. If a constant C cannot a n < U A : a n < U A : radw da >> 0. a sequence Uk with ||u/e||#.(=) = l1bbu ut t l||l ggradw/cH^,/^ r a d i*t /l cl ilfl .^(,s/ )^ "^ "~*" 000a l ^^JfJrrjou r^^°k° —± ~^ ^d&— ~* ^*0. ivith||w f c || H . ( 2) ~ +1 Since the sequence is bounded in ^if* (El), it has a ~subsequence u t h a t is con _ _ ! _ . _ „ Kkj . p v - , , *« ~ vergent in Hp(E) to a limit UQ. grad w^H^.px UQ. Since || gradw^H^.px gY&&Uk\\ sub Ukj Hsf5\ —>• 0, the subsequence P P vergent in #Hp is convergent p ++11(E), ( ! E ) , with lim limit UQ, WQ, andi 7 0^ — (y ri j). . Now loUkj ->> 70^o 7o^o Bpp (T). II M U W —r jo UQ in 111 £ ±j so u 1 Le hand h a n d ||i/o||# ||WO||_H-S(=\ on one — 1, 1> so UQ uo o i^ 0; 0; on on the the other otheriv hand, hand, |||| gggraduoll^wE^ rraaddw woollll^^ss//^^ — — 00 s rEF\ = so tio i*o is a constant, and this constant must equal 0 since J fpjouda jouda = 0. This ce rr7owc?cr = Th contradiction proves the statement. □ p
One can also replace the integral in the right hand side of (4.22) by another supplementary term, e.g. the integral of u over some small subdomain. T h e m a p p i n g KD is established in Simader and Sohr [16] for integer s > 0 in a slightly different way, and with lower smoothness assumptions on T; our presentation here was inspired by conversations with B. Fuglede. T h e operator / + grad RD div is the projection onto the solenoidal space Jp = {u { u G Hp(Q)n I div u = 0 } , s > £^ — 1, generalizing the situation where Q is bounded. Insertion 01 of the iformula o r m u l a lfor o r q into (3.1) leads to (3.4) and hence (3.5) and (3.7), /;, with witn the tne same notation as before. Detore. We w e nfind n a by 1use of Theorem 4.1 t h a t n n +1 + 1s s+1 ny n /1 G H;{Q) when / G H^{U) and ip G B -*{T) B ; ( r ) nnn. G H;{U) , when / G H^) and ip G B p~p *~^{T) 9 F or p yt.prinr d omains W P a l s o have h a v p (3.8), (R R\ by K v [T91], so For exterior domains we also so uwe can write the problem in the form (3.12). Again we can apply L e m m a 3.1, using t h a t (3.15) is valid.
61 It is is at at this this point point that that lit is a great advantage that we can reduce to the inversion of 7 / + + TK\. TK\. For For this is a ps.d.o. on the compact manifold T. It on of satisfiess the symbol requirements for being parameter-elliptic of_ _regularity ^ the svmbol reauiren — ^ ^ „ „ - _ - _ „ „ — ^ „ j on the rays in C \ R+, hence is invertible on these rays for sufficiently large on the rays in C \ R+, hence is invertible on these rays for sufficiently large |A|, |A|, i.e. i.e. (3.18) (3.18) holds. holds. The The inverse inverse satisfies satisfies (3.20), (3.20), and and we we get get the the solution solution as as in in (3.21), 0
u = (I - K,xQ\T){RN)\(f
r
1
0
2
x{
which lies in i7 5+2,/i (Q). Also the considerations for the problem with —A replaced by dt go through, and the discussion for nonzero initial values can be completed as in [7]. For the pressure q we use the formula gradg = — gradifo grad i^j) KD{^liu (271^ —<£v), <£v), - * ) //22>(3 ( 5 RR )) -> ^H^ ^( \Q QR K) ") -\ grad*!,
grad RD div : H^s^(QR)n
-> H^'^(QR)n,
(4.24)
for s > ^ — 1. (For grad gradii KD, one uses that the mapping property holds for hnnnHp.H npiichbnrhnnrlfi iof T by [7], and that grad KD maps into bounded neighborhoods P| r G R #p({|:c| > R})n when R is large enough. For gradifo div, the property s, one uses that the operator is straightforward when s > 0; to include lower 5, that Hp H{pSSs'>s/2*(Q)* \ny = is selfadjoint (being an orthogonal projection in L2), and that = s,-ss/2 (/2) I _ I I + i = 1 . ) ff s (fir for s G ] 1 ; [ ; h e r e ffi-:-/*)^. ] I 1,1[, here I + J, = 1.) f o r s £ H^ '- \ny v for s € ]J i - 1, iL[ , here ± + ± = 1.) v
'
y
f
y
p
'
For For the the other other boundary boundary conditions conditions (the (the cases cases k k — — 0,2,3,4), 0,2,3,4), the the above above analysis can be carried through in suitably modified versions. analysis can be carried through in suitably modified versions. The The application application of of the the linear linear result result to to solve solve the the nonlinear nonlinear problem problem goes goes 7 7], so we arrive at the result: mechanically as in the bounded case in [ mechanically as in the bounded case in [ ], so we arrive at the result: Theorem 4.3 Theorem 2.1 generalizes to the case of an exterior smooth do main. (When p > n/(n — 1), gradg is only unique up to addition of functions c(t) grad | x | 2 - n for n > > 2, resp. c(t) grad log \x\ for n — 2.) Acknowledgments We are grateful to B. Fuglede and to M. Yamazaki for useful conversations.
62 References References 1. M. Brelot: Elements de la theorie classique du potentiel. Les cours de Sorbonne, 3e cycle, 3e Edition (Centre de Documentation Universitaire, Paris, 1965). 2. Y. Giga: The nonstationary Navier-Stokes system with some first order boundary conditions, Proc. Jap. Acad. 58, 101 (1982). 3. Y. Giga and T. Miyakawa: Solutions in Lr of the Navier-Stokes initial value problem, Arch. Rat. Mech. Anal. 189, 267 (1985). 4. G. Grubb: Functional Calculus of Pseudo-Differential Boundary Prob lems, Progress in Math. Vol. 65 (Birkhauser, Boston, 1986). 5. G. Grubb: Pseudo-differential boundary problems in Lp spaces, Comm. P. D. E. 15, 289 (1990). 6. G. Grubb: Parameter-elliptic and parabolic pseudodifferential boundary problems spaces, iviuui. Math. ZJCILZCIU Zeitschr. . 218, 43 (1995). pruuieiiiis in i n global giuuai L ±jvpp Sobolev o u u u i c v opcu^co, ^ i o , <±o {JLVVOJ. 7. G. Grubb: Nonhomogeneous time-dependent Navier-Stokes problems 7. G. Grubb: Nonhomogeneous time-dependent Navier-Stokes problems in in LLpp Sobolev Sobolev spaces, spaces, Differential Differential and and Integral Integral Equations Equations 8, 8, 1013 1013 (1995). (1995). 8. G. G. Grubb: Grubb: Initial Initial value value problems problems for for the the Navier-Stokes Navier-Stokes equations equations with with 8. Neumann conditions, conditions, in in The The Navier-Stokes Navier-Stokes Equations Equations II II — — Theory Theory and and Neumann Numerical Methods, Methods, Proceedings Proceedings Oberwolfach Oberwolfach 1991, 1991, p. p. 262 262 (Springer (Springer Numerical Lecture Note no. no. IOOU, 1530, neiueiuerg, Heidelberg, iyyzj 1992) eeditors editors J.. VJ. G. nHeywood Heywood et al. al. Lecture Note no. 1530, Heidelberg, 1992) G. et ljecture iNoie u n o r s JJ. e y w o o u et ai. 9. G. Grubb and N. J. Kokholm: A global calculus of parameter-dependent 9. G. Grubb and N. J. Kokholm: A global calculus of parameter-dependent pseudodifferential pseudodifferential boundary boundary problems problems in in L Lpp Sobolev Sobolev spaces spaces Ada Ada MatheMathematica 171, 165 (1993). matica 171, 165 (1993). 10. 10. G. G. Grubb Grubb and and R. R. Seeley: Seeley: Weakly Weakly parametric parametric pseudodifferential pseudodifferential operators operators and Atiyah-Patodi-Singer Atiyah-Patodi-Singer boundary boundary problems, problems, Inventiones Inventiones Math. Math. 121, 121, and 481 (1995). (1995). 481 11. G. Grubb and V. A. Solonnikov: Reduction of basic initial-boundary value problems for the Stokes equation to initial-boundary value prob lems for systems of pseudodifferential equations Zapiski Nauchn. Sem. L.O.M.I. 163, 37 (1987) = J. Soviet Math. ^ 9 , 1140 (1990). 12. G. Grubb and V. A. Solonnikov: Solution of parabolic pseudo-differential initial-boundary miuai-Dounaary value vaiue problems, proDiems, J. j . Diff. JJijj. Equ. n>qu.87, of,256 zoo(1990). ^lyyuj. 13. G. Grubb and V. A. Solonnikov: Reduction of basic 13. G. Grubb and V. A. Solonnikov: Reduction of basic initial-boundary initial-boundary value value problems problems for for the the Navier-Stokes Navier-Stokes equations equations to to nonlinear nonlinear parabolic parabolic systems of pseudodifferential equations, Zap. Nauchn. systems of pseudodifferential equations, Zap. Nauchn. Sem. Sem. L.O.M.I. L.O.M.I. 171, 171, 36 36 (1989) (1989) = = J. J. Soviet Soviet Math. Math. 56, 56, 2300 2300 (1991). (1991). 14. 14. G. G. Grubb Grubb and and V. V. A. A. Solonnikov: Solonnikov: Boundary Boundary value value problems problems for for the the nonnonstationary Navier-Stokes equations treated by pseudo-differential stationary Navier-Stokes equations treated by pseudo-differential meth meth ods, ods, Math. Math. Scand. Scand. 1991, 1991, 217 217 (69). (69).
63
15. O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralceva: Linear and Quasi-linear Parabolic Equations, AMS Translation Math. Monographs 23 (AMS, Providence, Rhode Island 1968). 16. C. G. Simader and H. Sohr: The Weak Dirichlet Problem for A in L? in Bounded and Exterior Domains (Pitman Research Notes, London), to appear. appear. 17. V. V. A. A. Solonnikov: Solonnikov: Estimates Estimates of of the the solutions solutions of of aa nonstationary nonstationary linearized linearized 17. system of of Navier-Stokes Navier-Stokes equations, equations, Trudy Trudy Mat. Mat. Inst. Inst. Steklov Steklov 70, 70, 213 213 system (1964) = = AMS AMS Translations Translations 75, 75, 11 (1968) (1968) (1964) 18. V. V. A. A. Solonnikov: Solonnikov: Estimates Estimates for for solutions solutions of of nonstationary nonstationary Navier-Stokes Navier-Stokes 18. systems Zap. Zap. Nauchn. Nauchn. Sem. Sem. LOMI LOMI 38, 38, 153 153 (1973) (1973) =J. =J. Soviet Soviet Math. Math. 8, 8, systems 467 467 (1977) (1977) 19. V. A. Solonnikov: Estimates of solutions of an initial- and boundaryvalue problem for the linear nonstationary Navier-Stokes system Zap. Nauchn. Sem. LOMI 59, 172 (1976) =J. Soviet Math. 10, 336 (1978) 20. W. von Wahl: The Equations of Navier-Stokes and Abstract Parabolic Equatios (Vieweg und Sohn, Braunschweig, 1985).
64 L^-DECAY R L^-DECAY RATES ATES F FOR OR S STRONG TRONG S SOLUTIONS OLUTIONS O OF F A PERTURBED P ERTURBED N A V I E R - S T O K E S S Y S T E M I N IR3 HANS-CHRISTOPH GRUNAU Fachgruppe Mathematik, Universitdt Bayreuth, D-95440 Bayreuth, Germany
We want to investigate the asymptotic behaviour in IP (IR 3 ) of sufficiently s m o o t h solutions to the Cauchy problem for the perturbed Navier-Stokes sys tem f u t t - Au+ Au + lv ((v • Vj v) u + + (u • V) (v 4 (u • V) u 4 VTT = 0, i v u = 0, I ddivti IR 3 ,, V7, in 111 (0,oo) ^ U , WJ x /S XIL
, .
J u {t, (t, x)?) -> 0 as |#| \x\ — —¥ 00, for for 2t G G (0, (0, 00) 00), ¥ 00, oc), ((\ T.\ r\ = — 7/.n i,~ (x.\ (r\ I( 11 7/. uw (0 (0, x) — UQ (X) x) = i*o (#)
fnr (Z TR T R33.. ffor or X xr CG IR
(0)
.
.
e
.
Here t> is assumed to have the properties of R. Finn's stationary PR-solution with prescribed constant velocity VQQ G IR 3 \ {0} at infinity: [ div ({i? v (x) (2?) = = 0 for all x G IR 3 , (x) — — v00 foo < ]—1 T—r in IR 3 , \vv (x) I1 \x\ \x\ (( } 00 00 33 q ^ H L Ij ((( vv5} -Voo) -Voo) --TJOO) G eL ^ r\L°° H L (IR for any gg G G (2, (2, 0000)),, UO oo) G G ^ H L 0 0 (IR (IR 3 ))) for for any any gq II
#G L V ((v<E Lrr n L°° (IR 3 ) for any r G ( ^| ,, 00 J) . T h e precise integrability properties of v were pointed out by G. P. G a l d i 6 , they are different in the case of zero boundary conditions at infinity. System arises v; y o b ^ i i i (1) \j-i CLiiD^Q when vv 11^11 studying OLPUVJIJ 111^ the \jii^ stability oIJO.L»IIIhy of \JL the i i i i c stationary o l a i i w i i a i y solution ou u (t) describes the temporal development of the initial disturbance disturbance uo. T h e total >tal flow is given by v 4-f uu (t). It is well known 1 3 , t h a t there exists exist a global weak eak solution u to (1) which satisfies the generalized energy inequa] inequality. (0)
Unconditional nconditional energy stability of v is ensured by a small energetic energetic Reynolds
65 (0) (0)
number number of of vv .. We We assume assume throughout throughout this this note: note: ReE :=
((W ((w VV)w, )w, V vv -VQOJ -VQOJ ((w •• V)w, -Voo) J -^ -* „fj-—5 „ „n ^- < 1, L 1 r——^ '-
sup
(3)
A is the Stokes operator in IR3. We know from 1 2 , 1 3 , 7 , 8 , that any weak solution u to (1) with generalized energy inequality becomes smooth after finite time and decays in L2 according to \\u (t) T - e) ( aa00, , ^-e\ ^-e) {t) ||2 < C (1 + t)~a , a = min (a
(4)
for any e > 0; ac*o 0 is the corresponding decay rate for the semigroup solution of the3 heat heat equation equation with with same same initial initial value. value. See See also also11--44, ,nn, ,1144, ,1155, ,1188. . he time, bevond « Moving wing the beyond which u is regular, into zero or nrascribincr prescribing a. a suffi sufficiently ntly small initial value, we may assume that u is also strong: (ueC (uec111
33 22 ([0,00), ([o, oo), oo), L' L2 (IR n V i / (IR")) (IR (m33)))) n C C° c°u ([0, ([0,00), ([o, oo), oo), H212> (IR3) n H'H2>» *p (IR")) (IR (IR33)))) ,, ([0, (IR°)) D ''- (IB?)
22
3
3
2 3 p 3 eG C° ([0,oo),L CM? G (IR 7n£C° DL (]R \I VTT VTT C° ([0, ([0, oo), oo), L L2 (lR (IR3))) n nV L p (IR (IR 3)) )) ,, 33 3 ™\ r.666 (m \ ) n L°° r.°° CTR ([0, oo), LL (IR (IR 33))^ ((neC nr a.€0—r.° C° rrn L°° . . ~, — , , — —. . . „ _„ v
VL
v
7
v
/ y
for some pp > > 3. More information about the integrability of the pressure 7rn (decay at infinity) can be extracted from its differential equation, see (8) below. r so > ^cording t o 1 2 we may assume that there is some So SQ > 0, sue Moreover, er, according such that for t > 0 there holds \\u(t)\\ + t)-So. (5) \\u{t)\\p l l - ^ ^ M lh increase in precisely that way that can be expected from linear theory. But if v ^ 0, the lower order terms in (1) prevent us from deriving appropri ate energy inequalities for HD^u (t) ||2- It seems that only the L p -decay rates behave as it could be expected with regard to the papers 1 6 , 1 9 . T h e o r e m . Assume that ||V v H3/2 is sufficiently small (in dependence on p). Then we have: IUVC. '^11 ^ r>n a+a+ , ^-r^ \\u(t)\\ \\u(t)\\ ^-W,rt-o+fi-; p
,
(6)
66
Remark, i) By interpolation (6) also holds for p £ [2,3]. ii) The case VQQ t ^ = 0, where in general V v £ L 3 / 2 but only V v £ L
dx= -4- [ \u\pdx, 3 P«WIR
2 p-2u) {-Au) •p(\u\ 33(-Au)-(\u\ - u)
2 pP p2 22 = \u\ Wu\ dx+(r>-2\ -~'\Vu\ = // 3Ju\ M | V2u | 2 dx+{p-2)
= /
luP-'lVur d»+
M j u P 2u{3) uu k | pP"~22)uw{J)( j)) ) ix? r ((W\ 33 x-(\ \ ~
//
dx=Yl dx=Yl
Y" J2 ;
^ .
VluP2
3
U • (|«|pp - 22«) x 3 (Up -V) ' V J «) J ' (i"l ~ u ) ddx=^2 = 2_, /f
/ 3 ( p -V) «) • (|«r «) dx=J2f3 /'
4 j ^ V i 33V ^ jM ^ ! ^ pp"-44dx ^ u^ui }u^u^\u\
f/
/
3
dx d* dx
>(,)%){,))uWuW\ur dx 3 v « «!?> 'l«r * «£ «uW) l«lp-<22*
= - Y [ %){i)W^^W^Y|I«P u^u^\ur2dx = — 2_, / 2J = 1
v
—-
- ■(p-2) (p - 2) Y (p ^
/I
p , 4 4 v tj^'Vl «W«WHP/'«W da: 4*> i4*>
(( )(,)
3s
(p.v) )-(Np-2 )^
= t^ • f| W | p - 2 u) = ---(p-l)J ( (P P -- I )1)J / 3 C ( »' -V)autf$-v)u).(\u\>-*u)dx, = w dx,) hence
( 2 //" 3 ((Y p -#vV ) Uu) -V( | «(l«r | p -2 2 «Wz « ) d x = 0.o.
/" 3 (p-v) u )-(i«r «)dx = o.
The same calculations show that there also holds /I
( u - V ) u ) - ( | u | pp--22uM)) 3 (((u-V)w)-(|w|
= 0. 0. dx =
With help of the embedding
IMI§„ <
2
2
\uf~ |M«pr~2 2 |V«| \vu\ dx, \Vu\22 dx,
3
ueH u e6 1H (m3)),, if*1 *^ (IR
(?) (7)
67
which can e.g. be found in 1 9 , we conclude further: |I\ /
( 2p (« V) U) ^ f({u [{u -"u) dxi <^«.< [f / i t * •■ v V) i v) u Vi j■ (\U\P\• 1(|u| «■* I i* i dx\ i*«L-
pdx IVv Vi0i/v\VI ■ \U\P •i u.|u| dx i KJLUJ
; 3 3 lint 3 V '' VV II " ' | |~" JilR l R 3 Il ,, ,,» . ^ , . 1 1 ™ (((0)11 /" , , ((((0)11 ( p 2o2 < IV \W <<<7„ Co (P()P)IIV IV ^#|| | 3 /3/2 j / ^J^, I7/.IP| «wr 2r \X/u\ |V«| < IIIV v v##II|| 88 // aa •• llull IMfsp Co(nl I v wll \vu\2,92 2d.rdx.dx. Zr < 2 ••
The pressure term has to be treated more carefully. We have 3
r
r\2
( --ATT A T T = - d i v (VTT) = V 2 _- 2 (t? ((V {? --v^ ^ — ~ — -An «V «o o, )) .—, oxidxj I V . ^ oxidxj \ // J
= : —A7Ti — A7T2-1,3 = 1
(i\
WW«(*")««) ^° uMW «tu^(^j)) +
L
"]
JI
/oX
J
v
(8) (8) /
= : —A7Ti — A?T2.
// V7T VTT •• V (\u\p-2u)wj rfar dx == (p (p -- 2) 2) 2Y^" // ^iruV'uX/uV'W 7rt|W>tiW)ii(0|ti|P-4 da. ywr ^ dx\ 3 3
UlR /
Urn
^ I
l^i^
j
p 2
,
p 4
VTT • (|u| - «) dx = (22p -112 ) V /2 17 r U ( ^ ' V ' > | u | - d x
3
I
.-^Wra,
2 < 5e j/ I\u\r\Vt/| \Vt/I•22Iwl^ dz d^ 2+)-^1 ) ^(iwl^/^-^Vwl) / |l7rl 7 r |22bl^| W r 22 dx dx. dx.
< < 2r»n ii7r7r til
I|2 I|2
II II II
nun| |PP po" ~— —2 2 2
,,, nr»ri ii || II
|I|||I222 MIIii||
ui |it?p| P— - 2 — 2
22 7 r V ■)/(P+D IMIsp +2|NI MMI(3p)/(p+i) II^IISp ||ti||; - 211*1 ll l|l(3p)/(p+ l|l(3p)/( H(3p)/(p+l) llwllsp ++2||T2|| 2ll| M N2|| | pPpINIJ \\u\\ ll«llp IMIp P+11) IMIsp p
) >) <
/
....
V li\
II2
If
\
|
Hl|p-3p-222 N|IHIS;
V / . *,j = l1.2,3||/ W / ' n++1xn) |||| V /^»J ,j = l,,22, 3, 3 | |r3(a3npy/ ) / ( p+1 V = uu -2 2 +C + C ((p) p ) ||w||2pll ||w||2 ll llp llp by +G (p) \\u\\2pP||w||« by the the Calderon-Zygmund Calderon-Zygmund inequality inequality and and (8) (c
0 pP _i_ nirA lUjiP-1 111lUJl3 IK C( pI ? ?' _*, C(p) WuW;WuW;- \\U\\% \\u\\% < C(p) C(p) \\U\\% --vjW2o ooII|f[ IUJIP ||\\u\\ P) fV U ||g3p p3p ++ C(p)
c (v
II
113
{ur2lVul2dx tt P_a Vtt
^^W^| €l ./2/ a{L U"3/2 R . l V 1lH ° l 11
) >a'faJ
p 2 + £e 2 f / ' 3 | u«2| P_2 + - | VVu | 22dx d x > l + C ( p , £ ) | | W | ,f5i +e2' ((/ni j ^ J «lr l | V lU | U2 ld x )) +c(p,e)Nlp +C(p,e)\\u\fp^
by (7) and Sobolev's inequality. So we obtain from (9): by (7) and Sobolev's inequality. So we obtain from (9): (\u\pp-_222u) dx\ w) / 33 VTTVTT- (|t/| (|u| «) dx / 3 VTT- vV(|u| p _ 2 «) UlR }^ dx |
\m
I
(9) (9)
68
\Vuf M3 < ||vv^/2V |)(J )/(J / n(i^. i.«|r«adxM i)v t^t | a d « ) 8 |/ a8 ^^^()p()«( ^++ji^%l P +C(p,e)\\u\\ +c(p, p^. e )|H|^.
Collecting terms we find:
i\~fi (IHIJ) -c.(ri||v VQ} (P) ||V V%'\\ «»|m8/j (p)) («+1 (e*+++II-|||v |V VV<J?L %f ( «ll?)+ W ++{i {l1-C - *» (P) %\ ---CCl (e tfQ} p(p) Cl (l <„) /2)} /2 -c, 0 (p)||v^|| /m | ui«r r p2-|22Viv«i u | 22j£((IHIJ)')I (HI IHJ ;I );^) ". . First we choose e = ^(p) (^ ~~ ») • Now there i s a bound M = M (p), such (0)
that the smallness condition on v that the smallness condition on v
IML^
IML-<«
implies fiitl I y « ' | ^ < 1 ( l - i ) , Co (p) I V ' S ' | w < i ( l - l ) . We com, iup m p lwith i e s SiM
up with
I V ( ; ' | [ / 2 < 1 ( l - l ) , Co (p) ||V ' S ' | s / ! < 1 ( l - l ) . We come
U P_2 V jItt T(IHIp) (IMIP ++(( /4/_. ^. I\ur>wdx^
At last we have II
II
IMIP
^
II
4 | | 3 p —2 |II|
< 1MI2 /
3p-6 | | 3 p —2
INIIsp
,o, N
4 *„c (/ f/ -
5 S
,
pp 22
2 22
x ((33pP - 6 ) / ( 33 p 2 - 2 P ) N
< C(u C^(u l <+) *- S) ~F =' 5* f( / J |«| u | *-- |Vu| | |VVuu| |
P + —(1 + t ) aa 3^ (\\n\\P) &=% < C, i\\nW\ $=* (\\u\\P) (IMIZ) ++O* )" ^ ^(lltill?) J—t (\W\\ (IMIS)^^ <
(■ (10) (10)
Now we follow19. We put y (t) := y(t) := ||u \\u (<) (t) ||P, \\P, (10) is of the the form y 1 v' + ail+tfy a(l+t)l3vl+r
fill (11)
69 > 0, 7 £ where a, 6 > 0, r > 0, /? 0> G IR. If we put
. (0+1 / ? - 7 l
rr -f +1 J
I r L
<—M^reri. —H.(^)".(?)H=mHt (, ( o,,f»a±iiV
fr
,^I"*:
*(i) = d ( l + * ) " ' , we have: 1+r 1+r x1 r x' +- a(l+ (1 ++ 0tf tf" x >b(l+tf>y' b (1 (1+ tf7 > > y■ y'+ ( 1+ + tf tf y1+r * *' + aa (1 ^x1 + r > ++ t)' *) > y' + aa (1 + >> b o(i. H+a(l+tf x{0)>y(0). x{0)>y{0). x(0)>y(Q).
By a standard ODE-comparison result, there follows S + ty tys6. t)~
y(t)<x(t)=d{l y{t)<x(t)=d(l y{t)<x(t)=d{l According to (5), we may start in (11) with So SQ
p- 1 e INo). 7/ck = = -S {k(kee Ik = -&k - 4k ■• pp-^ ^—3r 4 IN, JNo) • 3 (*
= s00p,
By iteration there follows: • f
3
/
ox
4
P
f
p - l 3p-6l
4 + 1=min = m m a ap P _ 2 ) ),a--^ a : + 4 fl 2 ' 3 7 ^+ 4—3-3—^ l P + 4+i((p^3'3^2
r•
^f > Because <J0 > 0, j^^ i • ff^ff >1,1, after after finitely finitely many many steps steps we we obtain: obtain: 1/p a1 ||«(<)|| = l»(<) / ( t1/p ) 1/p << C 7c(al ++ --t)) .. \W(t)\\pJ , = = y(i)
D
References 1. W.7. Borchers, T. Miyakawa, Algebraic I? l? decay for Navier-Stokes Navie flows in exterior domains, Ada Math. 165, 189-227 (1990). R n r r V i p r s , T. T Miyakawa, A/fiva.Va.wa. Algebraic Alcrphrair L T.2 decay r l p r a v for f n r Navier-Stokes Navif 2. W.J Borchers flows in exterior domains II, Hiroshima Math. J. 2 1 , 621-640 (1992).
70
3. W. Borchers, T. Miyakawa, L2-decay for Navier-Stokes flows in unbounded domains, with application to exterior stationary flows, Arch. Rational Mech. Anal. 118, 273-295 (1992). 4. W. Borchers, T. Miyakawa, On stability of exterior stationary NavierStokes flow, Ada Math. 174, 311-382 (1995). 5. R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal. 19, 363-406 (1965). 6. G. P. Galdi, An introduction to the mathematical theory of the NavierStokes equations I. Linearized stationary problems, Springer Tracts in Natural Philosophy 38: New York etc. 1994. 7. H.-Ch. Grunau, The Reynolds number and large time behaviour for weak solutions of the Navier-Stokes equations, Z. angew. Math. Phys. 44, 587-593 (1993). 2 8. H.-Ch. Grunau, L -decay rates for weak solutions of a perturbed NavierStokes system in IR3, J. Math. Anal. Appl. 185, 340-349 (1994). 9. J. Leray, Sur le mouvement d'un liquide visqueux emplissant Tespace, Ada Math. 63, 193-248 (1934). 10. P. Maremonti, Stabilita asintotica in media per moti fluidi viscosi in domini esterni, ester ni, Ann. Mat. Pura Appl. Ser. 4, 142, 57-75 (1985). 11. P. Maremonti, On the asymptotic behaviour of the L 2 -norm of suital suitable weak solutions to the Navier-Stokes equations in three-dimensional ex exterior domains, Commun. Math. Phys. 118, 385-400 (1988). 12. K. Masuda, On the stability of incompressible viscous fluid motions p; past objects, J. Math. Soc. Japan 27, 294-327 (1975). 13. T. Miyakawa, H. Sohr, On energy inequality, smoothness and large ti; time behaviour in L2 for weak solutions of the Navier-Stokes equations in exterior domains, Math. Z. 199, 455-478 (1988). 14. M. E. L2 decay decay tor for weak weak solutions solutions oi of the the JNavier-Stokes equa14. M. h,. Schonbek, schonbek, L* N avier-btokes eqi equaftions, i r m s Arch. Arrh Pntinnnl Merh Annl « « 209-222 90Q-999 M Q«M Rational Mech. Anal. 88, (1985). tions, Arch. Rational Mech. Anal. 88, 209-222 (1985). 15. 15. M. M. E. E. Schonbek, Schonbek, Large Large time time behaviour behaviour of of solutions solutions to to the the Navier-Stokes Navier-Stokes equations, Commun. Partial Differ. Equations 11, 733-763 equations, Commun. Partial Differ. Equations 11, 733-763 (1986). (1986). 16. 16. M. M. E. E. Schonbek, Schonbek, M. M. Wiegner, Wiegner, On On the the decay decay of of higher higher order order norms norms of of the the solutions of Navier-Stokes equations, Proc. Royal Soc. Edinburgh 126A, solutions of Navier-Stokes equations, Proc. Royal Soc. Edinburgh 126A, 677-685 (1996). 17. W. von Wahl, Vorlesung iiber das AuBenraumproblem fur die instationaren Gleichungen von Navier-Stokes, Rudolph-Lipschitz-Vorlesung, Vorlesungsreihe No. 11, Sonderforschungsbereich 256, Universitat Bonn, 1989. 18. M. Wiegner, Decay results for weak solutions of the Navier-Stokes equa-
71
tions on HT, J. London Math. Soc. Ser.2, 35, 303-313 (1987). 19. M. Wiegner, Decay and stability in Lp for strong solutions of the Cauchyproblem for the Navier-Stokes equations, In: J. G. Heywood et al. (Eds.), The Navier-Stokes equations: Theory and numerical methods, Springer Lecture Notes in Mathematics 1431: Berlin etc. 1990, pp. 95-99.
72 ON T W O - D I M E N S I O N A L EQUATIONS OF T H E R M A L CONVECTION IN T H E P R E S E N C E OF T H E DISSIPATION F U N C T I O N Yoshiyuki KAGEI Graduate School of Mathematics, Kyushu University 36 Fukuoka 812, Japan
1
Introduction
This paper is concerned with initial boundary value problem of equations describing t h e two-dimensional Benard convection. We here consider t h e governing equations in which t h e dissipative heating effect is taken into account. T h e non-dimensional form of t h e governing equations for t h e velocity u, t h e pressure p and t h e fluctuation 6 of the t e m p e r a t u r e from t h e static state is written written as as V uu-+h V p = Vu = / ( 0 ) e 22 ,
j^-vAu+u-^--vAu+U' j^-vAu+u-
V - u = 0,
xe x G R xx (0,1), t > 0, xe
e R x (0,1), t > 0, ax G
86 <9# — - ACA0 — ACA(9 + U W-- V 0 - e 2 -u = = rjD(u) ^ ( u ) : D{u),
a?G e RRxx((00 ,, ll ) ,
t* > > 0.
(1.1)
(1.2) (1.2) (1.2)
Here, = is the opposite to n e r e , ee22 = = (0,1) [\j, i) is m e unit unit vector vector opposite 10 the m e direction airecuc of gravity ; v, K and rj are non-dimensional physical parameters to be defined in section 2 ; D{u) : D(u) is t h e dissipation function : rw x ^ / x r>/
l
v^ v-^ (du (dul
k du du \
DW:D(„) = - £ ( - + - )
;
and / is a smooth ;h function on R satisfying f{6) f(9) = = 9, 0,
(1.3) (1.3)
l / U = sup |/(0)| | / ( 0 ) | < oo, |/'|oo l/'loo < oo and l|/"|oo I/loo / " ^ < oo.
(1.4) (1.4)
or eeR
73
Here f and f" denote t h e first and t h e second derivatives of / , respectively. Equations (1.1) a n d (1.2) are supplemented by the boundary condition : u = 0 ; 0 = 0 at
x2 = 0 , 1 ,
and the periodicity condition : U\Xl-0
du I\ —
— U\Xl=ai
_
du_\ - J^_|
59 x i L i = a ' dxi ~ 1==00 " 0*1 L L *l1 L L 11== a a' L=0 " dx
0\Xl=0
— = @\x1=a ;
d0 30 II
d0 80 II ^
*l1 L lL11=0 "" dx ^1 L ^dxi L\Xi=a a ''' =i =00 " 11== a
5dx1 dx
In case 77 = 0 a n d /f(0) ( # ) = 0, equations in f(v) 0, one obtains t h e usual Boussinesq eq which the ;he dissipative heating effect is ignored. For the Boussinesq Boussinesq equations, the global Foias, bal existence and uniqueness of weak solutions are well known. kno 1 Manley' and and TTeem maam m [3] [3] proved proved the the existence existence of of the the associated associated global globa attractor to which for its Haus:h all solutions converge as t —> 00, and derived a bound foi dorff dimension in terms of the significant physical parameters : t h e Rayleigh number Ra, the P r a n d t l number Pr and the Grashof number Gr. Their bound is c\Q\{l + c|J2|(l +P Pr){l r)(l + + G Gr r + + Ra), ifa), where |Q| is is Lebesgue Lebesgue measure measure of Q— = (U,aj (0,a) x constant dehere \\l\ 01 \l x (0,1) (U, 1) a an nd dc c is is a a co pending only on on tt h he e flow flow geometry. geometry. T Th he e purpose purpose of of this this paper paper itis t o study mding only flow initial itial bb oo uu nn dd aa rr yy value value problem problem for for the the system system (1.1)-(1.2) (1.1)-(1.2) with with 7777 > > (0. We first recall the existence results of global weak solutions of the problem call the existence results of global weak solutions of the oroblem aand the associated global attractors A-q obtained in our recent work [5]. We then discuss rrmvprcrenrp thp plnha.l attractors attractors A A„ aO 0. convergence of as v convergence 01 the the global global A v as 77 —> u. T h e paper is organized as follows. In section 2 we deduce the non-dimensional ofthe the governing governing equations equations and and set set uupp the the corresponding corresponding nonlinear no] form Lof evoluiry problem. tionary problem. In In section section 3 3 we we first first present present some some results results in in [5] [5] :: the existence obal weak glol of global weak solutions solutions of of the the problem problem a an nd d tt h he e associated associated global attractors Aq • In [5] the Hausdorff dimension of Avv is bounded by cc|fi|(l | n | ( l + Pr){\ + Gr + + Gr^2Ra
+ 0(77))
with a constant c depending only on t h e flow geometry. Unfortunately, our bound does n o t reduce t o the bound of [3] as 77 —>• 0. However, we here prove the convergence of the global attractors Av t o Ao as 77 —> 0 (Theorem 3.3). T h e o r e m 3.3 is proved in section 4.
74 2
Preliminaries
We consider a a iwo-aimensionai two-dimensional innniie infinite layer layer n Rx x \v,u) (0,d) aand assume tn t h a t the 've consider n a assume layer is occupied occupied by by a a viscous viscous incompressible incompressible fluid. fluid. Suppose Suppose further further tth hat the ayer is t ee m p e r a t u r e a t t h e lower boundary X2 = 0 equals $o, 0o, and t h e t e m p m p e r a t u r e a t t h e lower boundary X2 = 0 equals 0o, and t h e tempee r a t u r e n atit tt h h ee upper upper boundary boundary X2 x^ — —d d equals equals 0\. 0\. Here Here 0Q 0$ aand 0\ are are constants such d 01 constant 1 22 l ,,u t hh aa tt #0 #o > 0iT h e n t h e governing equations for t h e velocity u — (u u ), ) , the 0n > 01. T h e n t h e governing equations for t h e velocitv u — (u . u : pressure p and t h e t e m p e r a t u r e 0 are written as follows (see [1,2]) : ire p and t n e t e m p e r a t u r e v are written as iollows (see L-MJJ On Ql/oAu + QU Ql/oAu — - P POVQAU +P pQu *• Vu °~8t~ °~8t~ Vtz + + + Vp V Vp p= = = pp000g(l g(l g(l -- 70/(6/ 7 70/(0 o / ( 0--6/0o))e 0 o ))e 22,
P
T7
„. _ n
80 80 <9# PoCv K 0OAe Ae v e = D :D u PoClkv— poCcvvK -V0 D(u) PoCy— + P° ppoCyii : D(u). C vuU 0C ~ poC P° QA0 + V '-V6 = D(u) (u) :D{ ( )Here, ee22 = — (0,1) is t h e unit vector opposite t o t h e direction of gravity ; —#e — #e22 is t h e acceleration due to the gravity ; po is t h e constant mean density ; 70 is the volume u m e expansion coefficient ; Cv is t h e specific he heat at constant volume ; VQ is t hlee kinematic viscosity coefficient ; KQ vo Ko is t h e tthermometric conductivity he coefficient; and D(u) : D(u) is t h e dissipation function
JW:D(„) = ^ W M + 0. «.,:i*.)=?!;(£ +^ t,« = l t,« = l
At t h e boundaries a?2 = 0, c?, t h e velocity u is prescribed by uU, — = V07
a t X2 #2 = = 0,d,
C*U * , 2 — W , U,,
and t h e t e m p e r a t u r e 0 is prescribed by 0 = 0O O
a t a?22 = 0,
0 = 0i $i
a t 2:2 = d.
We further require u, p and 0 to be periodic in t h e ^ - d i r e c t i o n with period ao^. o. To obtain t h e non-dimensional form of equations, we introduce t h e following non-dimensional leiisiuiicii variables vdiiduics : .
-~-£ £ - --1 ~d' d' ~~d'
SB-B a-JLiL a°~°
X X
BQ-B^ 0 oo - 0 i '
,1, 1, ii_(l^{0,-0i)\ i_(ntg(0o-8i)\ \\
-
gfi fi--
P P
_(ntg(0o-8i)\ \
{i*9{e,-01)dyi^ P -P PP-P -P Poyog{Bo-Bi)d' PologiOo-O^d'
Po*?o9(0o - 0i)d'
l
- \ d J h *,* f{0 - e0) - HB - Bo) m = W-0o)-W-o°) m = W-0o)-f(d-Qo) n > 00-0, n
>}
0o-0, 0o-0i
75 and non-dimensional parameters : p Ra=
log{Qo-0i)d3 -—, Js 0K0 J^oKo
Pr=
v0 —,
Ra Gr=—,
K K00
Pr Pr
where
§ = !i^°iX2
+
o0.,
F(x22) = j T * 2 f ( ( g^l 2 ~/ 0/ 0) )rr)) dr.
P = P09 Pog (x2 - 1loF(x oF{x2)),
Jo
2d
\
)
The non-dimensional numbers Ra, Pr and Gr are called the Rayleigh, the Prandtl and the Grashof numbers, respectively. Using these new variables, we we obtain, after omitting tildes, -£ - isAu + u-Vu u • Vu + = f{0)e2, -^-vAu + Vp Vp=f(e)e
(2.1)
V - u« = 0, 86 — ACA0 + + uu •• V0 V0 -- ee22 •• uu = = 7/D(ti) 7/D(ti) :: £>(u), £>(u), — -- ACA0 where U
_/Pr\1/2
-\Ra)
'
K
_ /
1
-\PHte)
\1/2
'
(2.2) (2.2) (2.2)
_ 7_ol05gdd
^-"CT
and k r^/ r^/ x \X T^, r^/ r^/ ^ xx vVis IS s~^ ^-^ ^-^ yr~^ (ff 9u' 3U 8U dul du 8d uu \\ D u) : D M = - > -— + -— 1
kk
The boundary conditions at x#2 — 0,1 0,1 are are 2 = u = 0,
0 = 0 at xz22 = 0,1,
and u, p and 0 are required to be periodic in x\ with period a = ao/d. We consider equations (2.1) and (2.2) in the domain Q, — (0,a) x (0,1), together with the above boundary conditions and the initial conditions : u\t=o u\ t=o = uQi
6\ 0\t=o t=o == 000O.
We denote this initial boundary value problem by (BE)^. In case /(0) = 0, we formally obtain the Boussinesq equations from (2.1) and (2.2) by passing to the limit rj —> 0.
76
We now introduce some notation. LPP(Q) (Q) (resp. Hm(Q)) (Q,)) denotes the usual pp 2 -space (resp. the L -Sobolev space of order m) and its norm is denoted by L -space ,2). We define the function spaces C Cl2per(Sl), Hlper || • ||p (resp. || • || m ,2)(£l), L22a, HQ per and IV/ b vy 2 22 C j .er(fi) er® = W n ; V>eC = o , i = 00,, i>(x Coo>P Win; ^ e C 2 ( (R R )),, V ^UUa =o,i tl>{xl+a,x 2) l+a,X2)
iP{x1,x2)}, = 1>{x
22 Ll = 0, «ti22U 2=o,i L^a20 = L22(Q) (Q) •u = ,i — = w, 0, u,u^ 1L|a; ! *^1,=o^u — = > u^ 1U! *^!a^ }} ,. — {u£ l " c -^ V * 6 / 5; V == 0u,l v • u, — w, u, |a?2a 1 22 Hl,per = L 11=a I a = o ,i 1=O = =a} #
Then, the following Helmholtz decomposition holds [6,7] : L2^i) {^)22 where
= L2a®(Ll)L
=
2 LLl®G a®GpeT,
GPer = {Vg; g G # 1 , 2 (ft), ?U 1= o = 9Ui=«}^Ui=a}-
In terms of the associated orthogonal projector P onto L2a, we define the Stokes operator A by
An = -PAii, ue£>(A) = {ue vn# 2 (ft) 2 ; -p-\
= 4H
}.
It is well known that A is positive definite and self-adjoint in the Hilbert space 1 2 L2a satisfying yHA ^^ ^^ = 1^^4i ' / u\\2 = ||Vi/||2, ||Vi/||2, and and there there exists exists aa constant constant C G> > 00 such such that that N| N| 22 >> 22 < < C||Ati|| C||Ati|| 22
for for ueD{A). ueD{A).
(2.3) (2.3) (2.3)
We also define the operator B by BO -A0, B6 = = -A0,
2
nper f 2ntf f(Cl);(Q); 6eD(B) e£(5) = = {0etf {eeHlV £L\ *| per o \nH ' ( Q ) : TET OX
l
1*1=0
= == ^-\ ^^-r| }. lri=a M l
The operator 5B is positive definite and self-adjoint in L2(f2) satisfying y j ? 1 / 2 ^ ^ = ||V0||2, and there exists a constant C > 0 such that \\0\\2,2 < ||0||2,2 < C\\B6\\ C||£0|| 22
for 6eD{B).
(2.4) (2.4)
We sometimes use the following Gagliardo-Nirenberg inequality [4] : ^ / ^ l l . . l l / 22 | | . . | | l //2 ||IL.II. c|Hilf 2 |H|J 22, IM| qitiHlgllull^, U||4 4 < ciMilgiMi^ 1
(2.5) (2.5)
77
where C is a constant depending only on a. In particular, for u E Ho,per w e have /2 /2 C\\Vu\\l/7/2 \\u\\l . . |||M| ||u||J (2.6) U||4 < C||Vti||J IMl4
II
l l ^ i l l l l ^
\
/
Throughout this paper, the letter C denotes a constant which may vary from line to line. Further, C(- • •) denotes a constant depending on quantities appearing in the parentheses. 3
Results
In this section we first recall the existence results of weak solutions of problem (BE) and the associated global attractors A^. We then state our convergence result of An. Arj. We begin with Definition. Given {uo,0o} G L2a x L2(Q), a pair of functions defined for t > 0 is called a weak solution of problem (BE) if TO L\)2 ) nHLL22((0, tui €G£L°°(0, ( 0 , T; T;L 0 , rT;V), ;lO,
{u(t),0(t)}
0 G L 4 // 3 (0, T; L2(n)) (ft)) n L 2 (0, T; L 4 / 3 (Q))
for all T > 0, and the identities - / (u,v')dt + [i/(Vti,Vv) + (w-Vw,v)]dt Jo Jo (uo,t;(0))+ / = (wo,t;(0))+
{f{0)e2,v)dt
Jo
and (0,i/>')dt+ [K{0,Ai;) + {u {u-V4>,9)]dt {u-X7ip,0)]dt eft + [/[ [K(e,Aip) [/c(0, A^) ++ • V ^ , 0)]')dt+ (0,^')cft [K(0,A-^) (u.V^,0)]
/
,,
..
771/ , _ ,
.
_ ,
.
V(0)) u, V-) D(u),i>))dt (ffo, D(u)^)]dt = (0o, (ffo, ^(0)) ^(0)) ++ y| / [(e [(e22 •• ti, ti,V) V) +++ y^(£>(«) y (£>M (£>M :: J3(ti),V)]rf* all vv G £ P^ hold for for_all )HL22(0, T; V) with v(T) = 0 and all $ G (^([O, T]; W 1 ' 22(0, ^ , T; L 2 )HL T\\ f 2 r<2 p err(Ct)) (n\\ with ™,ui C (^)) with V'CO V'CO — — 00- Here Here and and in in what what follows follows (•, (•, •)•) denotes denotes the the scalar scalar CQ
product of L2.
Our existence result is the following T h e o r e m 3.1 Let Jf satisfy satisfy (1.6) (1.3) or or (LA). (1.4). Assume that 5.1 ([5, l|b, Th. TH. 2.1]). *-!JJ- (i) W Let Assun 222 22 7rj7 << 11 ifif ff satisfies satisfies satisfies (1.3). (1.3). (1.3). Then, TVien, TVien, for /or /or each eac/i initial initial value value {UQ,0Q} {UQ,0Q} G G L £ Xx ££ (r2), (r2), a each initial value {t*o 0o} G L x 5 £/iere exists ; aa weak 0} of weafc solution solution {u, {u,0} of problem problem (BE) (BE) defined defined for for all all tt > > 0.
78
(ii) Let f satisfy (1.4) and let {uo,0o} G V x L2(Q). 2
2 ueC([0,T}]V)r)L (0,T;D(A)), ueC([0,T] ]V)nL(0,T;D(A)),
Then 2
2
C C([0,T]; ( [ 0 , T ] ; L2(Q)) ( Q ) )nnLL2 (0,T; ( 0 , T ;Hl 4 %per 9^G p e r ))
> 0, 0 ; and the weak solution {u,9} is unique in the class for all T > 2 2 ((C([0,T\;V)C\L C ( [ 0 ) T ] ; y ) n2L(0,T;D{A))) ( 0 , T ; D ( A ) ) ) x (C([0,T];I (C([0,T]; L2 (f2)) (Q)) n L 2 (0,T; ^ p e rper) )). ). L2(0,T;Hl
(iii) Let f satisfy {1 A) and let {u ,60ii} G {u0)i € VxL 2 (Q), i = 1,2. ForeachT> 0 o ,i,0o,i} i/iere exists a constant C = C(T) such that if {ui(t),0i(t)}, i = 1,2, denote, respectively, the weak solutions corresponding to initial values {wo,f, #o,z}, i = 1, 2 ; £/zen 1,2, then 2 \\vu vvixi^j|| u i ^ n 22 -t-||t;2^J-^UtJ||2 ||022(t) ++11*0,2 ||0o,2 - - MIDHVti (t) + \\e (t)-- e^w 0i(')ll22^<<^c(\\vu c(||vu 2-— - vix v«o,illi Ml!). 0,2 0,i||i 0l0,2 | | v a222(t) ^ j -- vm(t)||l VII vxx vix 0 ,i|| 2 -t- 11*0,2 - co,i
Remark 3.1. The uniqueness problem of weak solutions is unsettled when / satisfies (1.3). Let
H = V xL2(D).
By Theorem 3.1, we can define the semigroup when / satisfies (1.4) : Svvv(t) {u(t),0{i)} ===SSvS(t)U (t)U GJJ.JJ. (t) :H3U :H3U00 0 = = {tx(O),0(O)} MO), 0(0)} 0(0)} *-> ^ {u(t),0{i)} MO^W} JJ. MO), H> vv(t)U 0 00GG In [5] we proved the existence of the global attractor for {Srj(t)}t>o. Then, for Theorem 3.2. (i) ([5, Th. 4.1]) Suppose that f satisfies (1.4). T/zen, every 5ry rj > 0, the semigroup {Srj(t)}t>o associated with problem (BE)^ (BE) has a unique ique global attractor Av which is bounded in D(A) x HQper(Ql), com} compact and nnected in H. Furthermore, Av attracts bounded sets of H. connected (ii) ([5, Th. 5.2]) The Hausdorff dimension of A-q A^ is bounded above by c|JJ|(l + P r ) ( l + Gr+ Gr^Ra Grll2Ra
+ 0{rj)),
with a constant c depending only on a. Remark 3.2. In case rj = 0 and / satisfies (1.3), Foias Manley and Temam [3] proved the existence of the global attractor and derived the following bound for its Hausdorff dimension : c\Q\(l + + Pr}(l Pr)(l + Pr){l + Cr> Gr + c|U|(l + J Ra).
79
Remark 3.3. One can easily deduce from the proof of Theorem 3.2 given in [5] that for any fixed 770 > 0, there exists M 0 = M M00(77o) (r]o) such that Av C £ # ( 0 , Mo) M0) aa,nd n d Srj{t)Uo S ^ ^ o €E55j y^ ( 0 , M M00 ) ffor o r aall* i n >> 0, 0,U U00eB eBHH(0,M (0,M00/2) /2) and and 777 EE[0,770]. [0,770]. Here Here
BH(0,M) = {U£H ; H^II^ < M } .
Remark 3.4. Let 770 > 0. We write Sv{t)U (t)U00 = {wt,W,^W}» { ^ ( 0 A(*)}> ° < 1 ^< < W), and So(t)Uo — {u{t),0(t)}. Then one can also deduce from the proof of Theorem So{t)Uo 3.2 given in [5] that, for every compact interval / C]0, +00[, there exists Co = Coil, r]o,M Co{L 770, M00)) such such that that ifif U U00 E GB BHH(0, (0, M M00),), then then
jf(||i4ti(t)||l + ||Vfl(t)||l)cft
j(\\A + iiv^wiiijtft^co \\^Mi)\\l)dt
-> 0 -*
as 77 ->■ 0,
where dd (( B B 00 ,, B B ii )) = = supv inf | | /\\U-V\\ 7 - y | | ^H.. ueB0 £Bi
4
Proof of Theorem 3.3
In this section we give the proof of Theorem 3.3. For this purpose, we make use of the1following following abstract abstract result, result, which whichisisaaspecial special case caseof of[7, [7,Th.I.1.2] Th.L; : Let H be a metric continuous netric space. Let {S(t)}t>o D e aa nonlinear semigroup of co transformations lations on H. H. For each 77, 0 < 77 < 77 770, 0 we consider a 1nonlinear semigroupP {S on H for t > 0. W We assume {5r?W}t>o, >o, where Sv(t) is continuous or v(t)}t>o, that the following conditions onditions (A1)-(A3) hold : a n attractor at.trart.nr A A which Axrnirn attracts ^ttv^risi :an open neighborhood (^)}t>o neighborl (Al) {S(t)} {S(t)}t>o U. t>o has: an (A2) For every 77, {Sr)(t)}t>o has an attractor A-q Aq which attracts U. (A3) For every compact interval / C]0, +00[, iy uuiiipa^i. m i c i v a i 1 v jw, T ^ [ J sup supd(S v {t)vo, S{t)vo) —>■ 0 as 77 —> 0. v0£U
t£l
80
Then we have T h e o r e m 4 . 1 . ([7, Th 1.1.2]). Under the above hypotheses, Av converges to A as 7rj7 — — >■>■ 00 inin the the sence sence d{A d(Ar,,A)-+0 rnA)^0
(77 -^ 0). (?7->0).
Proof of Theorem 3.3. To prove Theorem 3.3, we appeal to Theorem 4.1. r u u i 01 J_ntJortJiii 0.0. _LU pruve ±iieurem 0.0, we ajjpea-i IU _LIJ Set>t U U — — BH{0, BH{0, M\), where Mi is the the number number MO(T7O) MO(T7O) in Remark 3.3 with BH{0,MI), Mi), where Mi is in Remar 7no 70» = == 1. 1.1. For this £/, we will show that our semigroups For this U, we will show that our semigroups {Sn(t)}t>o, {Sn(t)}t>o,0 < 77 < 1, satisfy ,tisfy the the conditions conditions (A1)-(A3). (A1)-(A3). Then, Then, we we can can obtain obtain the the desired desire result in Theorem 3.3 by applying Theorem 4.1 above. Ineorem 6.6 by applying ineorem 4.1 above. It It is is obvious obvious that that the the conditions conditions (Al) (Al) and and (A2) (A2) are are satisfied satisfied since since A Avv attracts attracts all bounded sets of H. It remains to verify (A3). all bounded sets of H. It remains to verify (A3). We set S^Uo SnftUo = {**„(*)> K(*),M * ) ) (° 0{t)U 0 == LetUo = {uo,600} eU. We M*)} (° << V»7<< *)> 1),SS0{i)U 00 0 {u(t),0(t)} and {u(t),0(t)} and vjt){t) = ur,(t) uJt) - ti(t), u(t)< ^(t) tbjt) = 6Jt) 0,,it) - 0{t). 0(t). Vr] 1>„(t) = 0„(t) - 0{t). Vr]{t) = ur,(t) - u(t), Then the condition (A3) is written as, for every compact interval / G]0, +OO[ Then the condition (A3) is written as, for every compact interval / G]0, +OO[ (4.1) sup supdlV^WHl + |HiMOIl!) | ^ W i l l ) "► 0° as 77 ^ 0. (4.1) u0eu tei We shall prove (4.1). The proof of (4.1) is divided into four steps : (i) For every very t > > 0, supdKWIll + II^WIlD-^O u0eu
as 7r/->0. 7 ->0.
(ii) For ror every every compact compact interval interval I1 G]0,-|-oo[, tju,i-oo[, sup s u p d K (M t )I| l| l! + ll^(t)Hl) -> 0 u0eu tei
as 77 -+ 0.
(iii) For every t > 0, ^O s u p ddl|V V^^W ( *I)I^I + |I |K^W ( 0H^D ) -- > 0 uU00£U eu
as 7 7 ^ 0 .
(iv) For every compact interval /I G]0, +OO[, sup u eu u0Qeu
sup(||V^(t)||i -f- ||V^(<)||1) | | ^ ( 0 l l 2 )-»> ^ 00 sup(||Vt» sup(||Vw,(t)|Jl ||V-,(*)||1) -»■ v (<)||l + tei tei
as 9-*•(). 77^0. ij-»-0.
81
We first show the assertion (i). The functions v^ and 9^ satisfy dv„ -± -£ + vAv„ + + p( P(v +uu■■Vv„) V«„)== P(h{0,9 P{h(0,0„)V>„), Vtfv .• VVu„ u„ + v)i>v), -± + vAVr) vAVr) + p(Vtf . V u „ + u ■ Vv„) = P(h{0,9v)i>v),
(4.2) (4.2) (4.2) (4.2)
dib + nB4> KSV>, + + w„ v„ ■■V0 T]D(u„) :: D(u (4.3) -- ^^n + + KBV>„ V0„ +uu■•VV>„ W , -- ee2 2••v„«„- =t]D(u »?I>K) I>K) (4.3) (4-3) V + v) (4.3) n + v„ ■ V6>„ + u ■ Vipv -e2-v„= v) : D(uv) and w,(0) = 0, MO) = = o, v„(0) 0,
^ (0) = = 0. V>„(o) = o. V,(0) 0.
Here Here h(0,0 f/ f'(0 + Tl{, r4> )dT. )d7 v)= n h(e,e v)= f f(e + Tipnn)dT. Jo
Jo Taking the product of (4.2) with v^, we obtain Taking the L -scalar product of (4.2) with v^, we obtain L 2 -scalar 2
^JT-dKII^ I K I I i + HIV^Ili^lK-Vti^wJI KA^,^^.!;,,)! HIV^U2,^ |(ivVu,,t;)| + |(/»(0,0»M,,«»)| < | K - V u „ t ; ) | + |/'| 0 o||^||2|K||2 '/ '
r
/ I
■
i«f
i"-^-1 M r */ I I - ^ I
< IK-VUn.WjI+l/'loodl^lll+IKHl
\{v \{V VUrf,v)\ v)| <<< ||^||4l|V^|| ||^||4||V^|| CUV^I^H^IbllV^I^ \{VVvV■••Vw^, VUrf,v)\ |K||4l|Vtt„||2 C ,CUV^I^H^IbllV^I^ ||VV I ,||2||^||2||V^||2 2< 2<< "'T/J ^ / l
_:
11^7711411 » ""7711^
_2
^ | |
*
"T/ll^ll "7/ | | Z | |
^^llv^lll + qiv^iiliKHl. ^
^ M ^
n2
.
^»II^-7
n2n
n2
It then follows that yy| | V < Cc(i (l + ++\\Vu \\Vu + \M\l \\l)(\\v,\\l + U.Wl). U lJftlt\\vr,\\l k l l ! +++ yllV^H! liv^lll
(4.4) (4-4) (4.4)
We next take the L 2 -scalar product of (4.3) with tpv to obtain
*I|V^||1< IK •V^,^)|+|(c 2 -t; l , ) ^)H-i ^2^11^112 l l ^ l l i ++«l|V^||i(«,):I>K) ) ^)| J DK): J DK),^)l ||^
■
||
r '/ | 1 4
IV
'/
'I '
' 'I / l
I V •"
• ! ' ' ■ / /
I
■
»i V
V
'# /
v
'/
<■ IK I ^ L•. Vff„ \7fl_ 0,)| ii>-\\4-n\(n<,i-\ »A_i < IK-v^.v,)!+ +«?!(£>(«,) r,\(D(u )■: £>(«„),iMi :nc?;_i £>(«„), t/> )\ + ||V„Hl v0„,0„)| »?|(i>K): ++n||v,||i ++ ihMi *?|(£>K) % ,),W I+IKHlK+ i l lIKHl ||lM v:i>K),^)|
S/1+/2 = h + h++||«,||1+||^,||?. Kill+ 11^,11?.
82 We estimate i\ I\ in in tne the same same way way as as aoove above :: we estimate
h < ciiv^ii^ii^ii^iiv^iiaiiv^ii^ii^H^ IIWl4||v
By the inequalities (2.3) and (2.5), hI2 is estimated as
riWDM Z)K)|| ||^|| <<^C^llV^ll^l^lb CijIlVtgfflliya h■"■A _< r)\\D{u : r>K)|| ' / l I-*-' V * * / /n)*: Z)K)|| V *! /\\4\\Yr)\\z ' /C^||V«,||2||V,||2 ll 22||V,|| 2 _i: 2 ||^||2 1
v
u
< CvllV^llall^Hall^lla 22
where C where C does does not not depend depend on r\. 77. We Wethus thusobtain obtain ^ll^|| 2 2 + + «||V^||| ^lhM2 «llv« 2 2 WVO.WlMWv.Wl \UhA\l\ r7n||v ll\7«JI?||47i. HvtylxiKii! \m\l) +4-CfWVu^WlWAuX CT?\\vu < ^l|Vt;,||l < |liv«,nl1A-C,(\A+ + C(l c(i ++ HV^HlKIKIl! iivtylxiKH! ++4-H^H H^iil) )+
with C independent of rj. r). This, together with (4.4), yields 2 222 rr F r ||2 r ||V^||2)(|| r ^Ill + IIV'nill) < C^7(ll*'-,lli-+-IIV'*,llI>-i-c7»? + C '« » W f«^IIIII-4.«,lli J || ^,lli) ^lli> < ^(i^ll^ C7(lH-||V«^lli-l-||V^-,llI)(ll*'-,lli-*-||V',Hl)H-C7»7 «»,lli-i-ll^ «^lli-i-ll^ ^-,III>(ll»'-U,Hi-t-IIV' ^Hl)H-c=' ?ll^? ll^ ||^'«,|IIII-4.«,lli «,llill-4.«^lli
with C with C independent independent of 77. 77. It then then follows follows that IKWIIl ++ ||V',(*)lll<(IK( constant > 0 there exists a const a 7(Mi t) > 0, 0. independent indenendent of 7 r77 and Tin (=G£/, /V snr.h C(Mi,t) rj and UQ UoG U, such such that that / 9i{T)dr
JO -'O
Jto Jto
> 0, Thus, we see from (4.5) with to = 0 that for every t > ssup u P ((II^COIII | | M 0 l l l + 11^(0111) l l ^ ( * ) | l l ) < + 00
Uo€U C/o€« U0<EU
as r,7 -> , ^ 00..
83
This proves the assertion (i). We next prove the assertion (ii). Let / = [^0,^1] be a compact interval in ]0,+oo[. Due to Remarks 3.3 and 3.4, there exists a constant C ( M i , J ) > 0, independent of rj and UQ Uo G U) such that gi(r)dr
/f |HVti^Hlllyln^llld-r | V ^ | | 2 | llVuJllUuJlUrKCiM.J) |^||2(iT < < C(M C(Afi,/) U I) Jto
for alli tt t£i L. combining Combining tnis this with with (4.0) (4.5) and and the the assertion assertion (ij, (i), we we find find t. that supsup(|KW||i + ||^(t)||l)
as 77 -> 0.
The assertion (ii) is proved. We next prove the assertion (iii). In view of the assertion (i), it suffices to show that for every t > 0, HVt^COIIa2 - * 0 as 77 -» sup HVt^COH -> 0. 0. eu uu00eu 2 Taking Taking the the L L 2 -scalar -scalar product product of of (4.2) (4.2) with with Av Avvv,, we we have have
^^llv^lli H ^ l l ! + HMHIli < IK • vu,,^,)!++Ku-v^.^^i |(u • vvn,Av„)\ + \(h{0,0n)i,v,Av,,)\ Hl^lli^lK-vu^^^l + K/i^^K,^)! Aii; ,f ), )| +| +| (| (««--VVt t»)„„A i 4t i;;l , ) | + |/'| |/ / |oo||^||2||At; < I K - V t«i„„A || 00 ||^||2||At>1| II||2 <\(v ,Avv)\+\(u-Vv )\+ ^{^111 < •■V«,,At) Vtt,, A«,)| + |(u |(uAv-•vVt»„ Vt»„ A«,)| CHVnlll+++|||Aw, |ll^t»,|[| v-V• UllVtc.-Aw,)! n>
|22 + C|K||2||V ||V ||A < lUvnWl \Uv,\\\ ++ C||i;,||2||Vt;,|| C||i;,|| C||«,||2||Vt;,|| ||V«,|| ||A«„|| V l ? ||22 2||V«,|| U ,||22||A«„|| Uf)| 2||Vt;,|| ^I^1|2||VV,1| \\Avv\\l C||^ll2||V«,|| ||V«,|| 22||VU,||2||/1«,||2 2||A«„||22 2
2
2
Similarly Similarly we obtain 2 22 ^2<^||A^|| /2<^||A^|| + q|v^|| q|v^|| ||v«|| ||v, 2 |H| 2 . ^2 < ^11^,111+ q|v«,||i||v«||liHil.
84 It then follows that
^liv^lll c(u,\\l ++ (l|v«,|| (||v«,||2||A«,|| ||Au,||2 ++ ||«||l||v«||i)||v«,||l), Hi!||v«||!)||vi,„||!), jdt" << c{Uv\\l v t\\v*v\\l " ^ ^ which tiich implies implies that that IIV^WHI HV^WHl < |||V^(io)||iexp(/; | V ^'n({to)\\Wp{ft t 0 ) | | i e x p (oo/g«jr >{T)dT) (2 r(r)dr) )dr) t2 t +Cfl\\Ur)hexp(! + C / t t 0 | | ^ ( r ) | | 2 eTx9p2(a)d ( / T tlT)dT.
Here
(4.6)
jfl2(<) ||«(<)||i||v«(t)||i). 92(t) ==c(||v C(||V«,(t)|| ||A ||«(<)||i||V«(t)||i). U | | (t)||22 + u„(t)||22||yK(t)|| c(\\VuJt)\\ ||«m||2|iv«(t)||2 2«)= 2\\AuJt)\\ 2+ +
Due to Remarks 3.3 and 3.4, there exists a constant C(Mi,t) / Jo
> 0 such that
92{r)dr
Since ^ ( 0 ) = 0, we infer from this and (4.6) with to — 0 that \\Vv I \\Ur)\\ldr
||^(r)||i Ct sup sup ||V,(r)||l r£[0,t]
for some = C(Mi,tf) C(Mi,t) uniformly in in UQ U and rj. ome C = Uo E £U 77. Thus, sup H V ^ W U l ^ C s u p sup | | ^ ( r ) | | i u0eu u0euTe[o,t] for some C = C(M\,t)
independent of rj. From the assertion (ii), we see that
sup sup uusup 00eu eu
ass V sup | | ^ ( T r- ) | | | -^ 0u aas 77 —> 0, sup 11^77(^)112 sup ||VM Jll2 — ~~>>•*0 ^ —^ " 0> rG[o,t] rG[o,t]
which implies iplies that that supP |l||V V^^((t0) |l|lll -->>00 u0eu
as iy7|-^0.
This completes the proof of the assertion (iii). We finally prove We hnally prove the the assertion assertion (iv). It suffices to show that for every compact interval I C]0, +oo[, interval I C]0, +oo[, sup sup||V^(t)||2 s u p | | V ^ ( t ) | | l -> as -» ->» 777— >>0. sup sup||V^(rj||2 —►00U as as 77] — u0eu tei
85 We prove this in a way similar to the proof of the assertion (ii). From Remarks 3.3, 3.4 and the inequality (4.6), one can deduce t h a t supp s u pp || || V C ( s u p l||Vv supH ^ ( t ) l lil ) V^^((tt))||||l| < C(sup l V tf^? (*o)||!+ o j l 2 + sup sup hM*) zu tei Uneu Uneu u00eu eu tei u00eu eu u00eu eu tei C{M\, for some C — C ( M i , /I)) uniformly in rj. It then follows from the assertions (ii) and (iii) t h a t sup s u p | | V ^ ( < ) | | 2 —>■ 0 as 77 —>> 0. u0eu tei This proves the assertion (iv), and so (4.1) is proved for every compact interval / C ] 0 , + o o [ . We can now apply Theorem 4.1 to obtain the desired result in Theorem 3.3. This completes the proof. References 1. F. H. Busse, Fundamentals of Thermal Convection, in Mantle Convection : Plate Tectonics and Global Dynamics, ed. W. R. Peletier (Gordon and Breach, New York, 1989, 23-95). 2. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Oxford Uni versity Press, Oxford, 1961). 3. C. Foias, O. Manley and R. T e m a m , Attractors for the Benard problem : existence and physical bounds on their fractal dimension, Nonlinear Anal. 1 1 (1987) 939-967. 4. A. Friedman, Partial Differential Equations (Holt, Rinehart & Winston, IQfiQV 1969). 5. Y. Kagei, Attractors for two-dimensional equations of thermal convection in function, to appear in Hiroshima Math. J. the presence of the dissipation funciton, 6. R. T e m a m , Navier-Stokes Equations (North-Holland Pubi. Co., Amster d a m , 1977). 7. R. T e m a m , Infinite Dimensional Dynamical systems in Mechanics and Physics (Springer-Verlag, Berlin, 1988).
86 EXTERIOR PROBLEM FOR THE NAVIER-STOKES EQUATIONS, EXISTENCE, UNIQUENESS A N D STABILITY OF S T A T I O N A R Y S O L U T I O N S Hideo Kozono Graduate School of Poly mathematics Nagoya University Nagoya 464-01 JAPAN e-mail:[email protected] Masao Yamazaki Department of Mathematics Hitotsubashi University Kunitachi, Tokyo 186 JAPAN e-mail:[email protected]
Introduction Let Q be an exterior domain in Rn{n > 3), i.e., a domain having a compact complement Rn\£l, a n d assume t h a t t h e boundary dCl is of class C 2 + / i ( 0 < /i < 1). T h e motion of the incompressible fluid occupying Q is governed by the equations: L11C Navier-Stokes IXXVlCl-OtUiVCO C4UCLUIUIIO.
{{
—Aw w 4— L±W 4--+-ww• •Vvw -h V7r V7r == div aiv Fr div w = 0 in Q, < div w = =0 in iZ, Q, w on w(x) w — —0 0 on <9fi, dVt, w(x) — —> >
in in Q, \i, 0 0
as as
\x\ oo, \x\ — —> >■ ■ 00, 00.
where w — w(x) = (w1(x), • • •, wn(x)) a n d tr = 7r(x) denote t h e unknown velocity vector a n d t h e unknown pressure of t h e fluid at point x G £2, re spectively, while F = — F(x) = (■Fj(^))»,j=i>.-.,n is the given tensor with div F — AT?1
„
P>T?n
( S ? = i "aF~> *'" > S ? = i ~dF~) denoting the external force. In the series of famous papers(Leray — So) had apers(Leray 2 0 , F u j i t a 6 , F i n n 5 ) , t h e existence of solutions for (N — So) h; been Lvestigated in the class of the finite Dirichlet integral fJan |V oo 00 een investigated \Vw(x)\2dx < < and rid in the t h e class s u p x G n ||a?||iy(a;)| ^ gi j ~LTl V |Vti;(a?)| | Vwwf(az))!| < oo. Most of the these jr1M < 00. x | \w{x)\ M * ) l + s u pP rrccGGnn ^-^\Vw{x)\ 00.Mos results >sults assumed t h a t t h e given iven tensor F(x) decays rapidly at in infinity infinity (see, e.| e.g., tak< a larger clc F u j i t a 6s a n d F i n n 5 ). T h e first purpose of this article is t o take class :rnal forces so t h a t t h e existence of solutions to (N — S off external So) is obtained. obtaine Our investigation yields also t h e asymptotic behaviour of w(x) and Vw(x) as
87 \x\ —> —> oo. Recently, in 3 — D exterior domains, Galdi-Simader Galdi-Simader 11 0Cconstructed onstructed a solution w in the class sup |ar||w(aO| < oo, xeti
Ww Vto G L 27r(Q) (ft)
for all r > 3/2
provided s u p x G n | x | 2 | F ( o : ) | is sufficiently small. Later on, Novotny-Padula 24 a n d Borchers-Miyakawa 3 improved the spacial decay estimate for Vw(x) given by F i n n 5 and found a solution w of (N — So) in the class sup | z | n " 2 | u ; ( z ) | 4- sup l a P ^ l V w O r ) ! < oo xea xen provided s u p ^ ^ | x | n _ 1 | F ( x ) | -+f s u p r^€^n |x| n |Vi ? (a?)| is sufficiently small. They m a d e use of a potential theory of hydrodynamics. It should be noted t h a t any of these results has a great difficulty to find the solution w with Vw G £«/(n-i)(fj). On the other hand, our m e t h o d is based on the functional analysis; the Utheory for the linearized equations to (N — So), i.e., the Stokes equations: + VTT = div F in Q, fi, -Aw + m 12, < div w — 0 in H, == 0 w — on <9Q, w(x) —»)■ 0
{
as
—y oo, \x\ —>
Vww G play an i m p o r t a n t role for our approach. To solve (N — So) in the class V L r ( Q ) , we have to restrict ourselves to the case r = n / 2 which istems from the nonlinear V w . It It should should be be noted noted tthhaatt the the norm norm ||Vu;|| Z/ n/2 ilinear structure wj -•• vw. \7w. norm ||Viu||j is invariant *iant under the change Xw(Xa for eeach lange of scaling such as w\(x) = Xw(Xx) JPP> nCaffarelli-Kohn-Nirenberg » f F a r p l l i - T C n V mri-Nirenberg - l \ r i r p n h p r c r 444^ ). ^ It Tt. is ic known, Lrnrvwrn however, Vmwpw^r ttVi: Caffarelli-Kohn-Nirenberg A > 0(see h a t (So) has a unique solution w with Vu> G Lr(£l) if and only if r satisfies n/(n — 1) < v < n(Borchers-Miyakawa2, Galdi-Simader 9 , Kozono-Sohr 1 7 ). Unfortunately, = 3 is excluded because n/2 n / 2 = n/(n the le most physically relevant case n = n/( — 1)) = 3/2 is the critical power for the unique solvability of (So). TThhuuss the mlethod e t h o di of linearization makes no contribution to solvability of the nonlinear th( nonlii equations quations So). To overcome this difficulty, we introduce a 1larger ccclass ms (N — So). 7 Vw (Q,) w G LP)q md show ttiicii h a t \oo) (So) nets has ct a unique solution w in .L nn> Q ) iwith \ (Q) and P)q n>>o00o00((Q) P)(? unique solution w in 7 Vw w G LIL mal m a L I LJ00 J00)(Q), P)(?(Q) denotes the Lorentz space over Q. By making 0 0 ( Q ) ,, where L,P)(? P)(? (Q) denotes the Lorentz space over i use se of the linearized ized m e t hhod small F G o d in in such such aa Lorentz Lorentz space, space, for for prescribed prescribed small prescril L /2t I L )j00 ( Q ) , we cann construct a solution w of (TV — So) with w G L X7w V G 0 0(f2), n)00(Q,), ict a solution w of (TV — So) with w G L. LJL (0,), scaling / i )OO ( f i ) , the norms lorms of which are also invariant under the change of scs sea )0O described escribed as above. ove. Our possible class for F is larger t h a n t h a t of BorchersBorch Borer Ml iiyy a k a w a ' s 3 and N o v o t n y - P a d u l a 2 4 and we do not need any information informatioi on h^ H p n v a i i v p X7 the derivative V FF1 ooff FF.
88 T h e second purpose of this article is to show the stability in Lri}O0(Ql) of our sta tionary solution w of (TV — So). If w w is perturbed by a, then the perturbed flow v(x,t) is governed by the following non-stationary Navier-Stokes equations:
_n\ ((NN_n\ ^
( JJI |] I[
v lj
^
| f -- A Vv Vq == div div i*' F iinn ifi, Av H+v v •• Vv Wv ++ V? H 2 , tt >>U0, , div in ^ , ttt > Q, 0,, div ttv> >= = 00 in Q, > 00, vvu = = 00Q on <9ft,*>0, ddQ f tit>0 , * >i 0 , v(x,t)->0 |z| v ( z , * ) - » 0 as \x\ |z -> oo, t;(x, 0) = u>(#) w(x) -f a(#) for # G xEQ. a(x) £1.
We show t h a t if the stationary flow w and the initial disturbance a are both small in the common space Lri)00(Ql), then there is a unique global strong solu tion v of (TV — S S\)i ) such t h a t the integrals /( I
r v x f^\v(x for — 1) 1) fn\ (\v(x,t) 't) ~— ww(x)\ (x)\rdxdx ax for //((nn — :t) J~ — w(x)\' lor nnn/[n— 1J u rr Jn I M M n/(n In l ^ 0 ) ~ Vw(z)| Gte for n / ( n J n | V v ( x , t ) - V w ( z ) | c t e for n / ( n --
oo, <<< rr <<<
converge to zero with definite decay rates as t —> oo. Recently, Kozono-Ogawa 16 2 showed a similar result on the stability of w with small V w G G L nL/ n2l(£2) (Q) 'showed nn for>r small initial disturbance a G L (Q,). (£2). w of (TV — So) with ( l ] ) . T h e solution w — So) Vww G L n ' 2 ( f i ) exists, however, in a special situation. Indeed, such a solution V exists if and only if the drag force exerted to the body is zero:
I/ {T{w,7r) ij / d S (T(u;,7r) ++ F ) -F).vdS=0, 5=0, where T(u>, IT) — ( f j - -f | j ~ - — £J7r) ^J7r)zz-j j ==1) ...))nn denotes the stress tensor and z/ is 1 ) ... Galdi-P the unit: outer normal to dQ,. See G a l d i - P a d u l a 8 and Kozono-Sohr Kozono-Sohr 1 8 . Later 3 on, Borchers-Miyakawa considered the class sup |tf||w(a:)| + sup |x| 2 |Vi(;(a:)| = Cw < oo xen, reri xen and showed the stability of such flows if Cw is small and if a is small in I/ n j 0 0 (fi) • 3 Our class ui of siciuie stable flows larger tmh aa nn tinai h a t of u u r ciciss nuws is is larger oi Borchers-Miyakawa Dorcners-iviiyaKawa ,, and in particular Our us particular we we do do not not need need any any information information on on Vw(x). Vw(x). Our result result enables enal to obtain the coincidence in L (Q) with stable stationary flows and initial U ) 0 0 to obtain the coincidence in L U ) 0 0 (Q) with stable stationary flows and disturbances. stability Dances. As disturbances. As aa result, result, we we see see tt hh aa tt the the existence, existence, uniqueness uniqueness and and st can be unified in the same class L (Q). L ( Q ) . : n)OQ n ) 0 0 n)OQ can be unified in the same class Ln)OQ(Q). Sn) and and (TV (N (N — respectively. the tions of (TV - So) So) -— i i)i),), ,respectively. Let w and v be solutions — So) (TV— —SSSi S respectively. 1TTThhheenn the pair of functions u = :Vv — w,p=q w,p = q — 7r ir satisfies
89
({N-S') N - S " )')
( ^| j - Ait + ID T it> w • vVu u + it • Viu +f uu -- V V uu ++ vVpp = = U 0 in in i n M,t Q,* f t , *> >> 0U, 0,, < i v uu = 0 ^I d iv in ft,/ fi,*>0, ft,/>0, > 0, u(£,t)—>-0 w <9ft,£>0, u(#,t)—)-0 t/ = 0 on 33ft,£ fQt , £ > 00,, [ u\ a. w|t=o t=o = a-
as
\x\ » 00, —> < |a?| — oo,
Thus T h u s our problem on the stability for (N (N — So) So) can now be reduced to inves rice of global strong solutions to (N S' and their asymp tigation into existence (N — S') S') totic behaviour. To solve (N — S') S') globally in time, we need to establish the LpP)OQ oo — .//-estimate . / / - e s t i m a t e for the semigroup e~~t£r , where Crr is an operator defined by' Cr = Viu). = Ar+ + P Pr{™ ' VVuu ++ Uu •• Viu). r{w • Here Prr. is the projection operator from U U(ft) U(Q) (Q) onto ££(£}) Lrraa{£l) and Arr = — —P —PrirA denotes the Stokess operator. If w = 0, we have C £rr = = A Arrir, , and in this case tu = this cas our problem blem coincides usual ides with the initial boundary value problem for the uusui es equations. equations. Many nonstationary ionary lNavier-DtoKes Navier-Stokes obtai obtain ier-Stokes nonstationary Many efforts had been m a d e to ob tAr the L Lpp — — - U U estimates estimates for the the Stokes Stokes semigroup in unbounded do: domains( domaim ates for dome the semigrou] e~ 2 U 1 4 15 1 5 2299 2 j U 5 1 4 j , 1 5>, 2 9 ))-. TTo o[b ttreat rtreat eat treat CC as as such such perturbation of of as the tl LLLP}00 — VULP)OQ Crrr as as such such pperturbation perturbation of AAArrrr as as the the P)OQ P)OQ — , , , , ) . To £>r erturb 33 1 estimate e remains to be Borchers-Miyakawa Borchers-Miyakawa Dno-Ogawa 1 6 a n d Borchers-Miyakawa Borche Dtained, Kozono-Ogawa Borchers-Miyaka Borchers-Miya estimate remains to be obtained, obtained, Kozono-Ogc required ||Vttf||.s. respectively. Making use of the tr Maki requiredI smallness smallness3 of of ||Vtu||.s. ||Vtu||.s. and and of of C CWi re Wi special structure w-Vu + u-Vw of the perturbed convective term together wit with ;• Vu ^ erturbed term special structure w-Vu + u-Vw of the perturbe div u = div it; w = 0, we shall remove such an assumption on the derivatives Vu>. Vi tCr Indeed, we shall establish the LP}00 — U estimates for e~ only by assuming Py00 t h a t w is small in L nL,oon)OQ.
1
Stokes equations
Before stating our results, we introduce some function spaces. < oo, spaces. For 1 < p < Hp{&) denote the closure of CQ°(Q) with respect to thee norm ||V • || p ; || • \\|| p iiis Hp(Q) II i I F ' II \\y ;he norm of the usual IP space. Since Q is an exterior domain, large: the #Hp{&) H 2p{&) (ft) is lomain, n.H^Q) i s llarger ar 11^(0,). By real interpolation, HHp p^qqq(£l' tJian h a n the usual Sobolev space Hp(Q). {Q) (ft) tion, we define define if* np{\L). o y real interpolation, we aenne UpHpqq{i [\i) 1 11 3y H$ H^Q) = ( ^f^fp ^(0(f(f^it )), if,7f^p \ (1 (f^^t ))))M^^gj , where 1 < p0 < p < Plpi < '.1 by = < oo and 0 < 9 < tq(Q) l))e}q, where l < p 0 < p < p i < o oo o and 0 < 9 < 1 >atisfy 1/p — (1 — 9)/po oo satisfy 0)/po + 0/pi 9/pi and 1 < q < oo. Notee t h a ,tt for 1 < q < < oo, o fpi and 1 < q < oo. Note t h a t for 1 < q < oo, HPtq (Q,) Hp (£l) is the closure of C^°(Q) C5°(ft) with respect to the norm (ft); q(£l) tq pL-JQY. > g(Q,)] rm ||V I||V I7-|| V ••- ||\\IPPL) ig9. of of LLp>q (£l Piq f2) with respect to the norm of L (Q); Ptq Piq (Q) denotes the Lozentz space Q, (see Bergh LLP}q BerghP}q(Q) Piq s n a r p over nv&r ft O with AA/it/h the t.Vip norm n o r m ||II| •.• ||p, IL Q P P RprcrVi|| p>ggJ (see Bergl 11 between LPtq(Q) and L pp//)g /(fi 7 .vhere 1/p + 1/p' = 1, 1/q + 1/q' = 1. where I / ? = re l/p+ 1,1/9+1/^
90 It is known t h a t Co°(ft) is not dense in # p ) 0 0 ( f t ) . Inspired by this fact, we introduce further a space H^q(Q); H^q(Q) is the closure of C§°(fi) with respect to t h e norm | | V • || P i 9 . T h e n we have H^q(Q)
= Hp\q(£l)
(1.1) (1.1)
forl
^oc(") C <«,(«),
l H (H^ny,^)^^ l/p (l _ 0)/PO 6>/pi,0 < 6> < 1 H}I00 H} W (ny== (H^(siy,H^{Qy) (H^(siy,H^{Qy) i/p = {i-e)/p {i-e)/p ,o<e< p
H^iay^H^in), ^ ( n r = ^ ii0o 0 o (n),
(1.2) (i.2) (i.2)
where X X** denotes t h e dual space of the Banach space X. < oo, 1 < q < Now let us define t h e generalized Stokes operator SP}q P i ? for 1 < p < i for p follows: oo a n d TPt \ 1 < < oo as Pt SPiq ::{w,7r} {w,ir} eG ^* eH^Cl) f^f ^O^ )x ixx, LL„,,(fl) , Piq , ,(Cl) ^ ) . -.► SS„,, p>, :{«;,*} { A w + V 7 r , d i v w} G H^ ,{Qy xx iLMp ,,(n), eH$, ,{Siy ( !( l ) , M - A w + V7r,div V7r,divw iq(^y •-)■- {-Aw w}} GeH^, H$, ql iqlq(ny TPpp,i,i :: {w,7v} {W.TT} { w>,7r}£H^ ,*}G G ^H^Q) #x(Q)xL ( 0pA))(n)^ (Q)->^4 T (^ Q xxLLpPP|1 (n) >| 11(Q) 1 ( --{-Aw AAw w - -VTT, VTT. i v w} w G# 7? , tO0 Cf -*-* _»* {{-AwVTT,-div VTT,- d-div w}lw} e H$. e iH$. ^.. (fi)* {ay{nyxx LL xppLp,i(n). ,i,i(n). (J2) tO0 Here —Aw V7r 6G Hp, Hp,)9qi(Q,y - A w ++ V7r '(^)* stood as
—Aw — —VTT V7r G G H*, (Q)* should be be under under aanndd —Aw fl"^j00 (fi)* should >00
H ( - A i uW + (Vu>, V<£>) f^ G ^^ H^ (Q) and (-Aw + V ^ ) -- (7r,div lfp# + V7r,^) V7T, y?) = = (Viy, (VW, V(p) V) p')q'ql \S g /(ft)
( -- A u < w; - V T T , ^ ) =
lim [(Vw, V p^ /)) + (7r,div ^ ) ]
for
^ G 6 Hi -ffi ffi «>(«), ^(ft),
respectively, {t ^^jjj}j = ^ !i is Co°(ft) ° ( Q ) such t h a t ||Vy?j— Vy>|| p / j00 —>• iy, where wnere {00 (ft)* (fl)* and H^, ^ (ft). (ft). Clearly t h e operators 5 S P ) 9 and T TPtPi) i are ?p />00 (ft)* and H^,;;00 Clearly t h e operators Sp>q ;00(ft). p>q and TPti a b o t h bounded operators a n d it follows from (1.2) t h a t T p *i(adjoint operator of T p> i) = 5Sp'/i00 Tp^(adjoint ) 0 .0 -
(1.3) (1.3)
Our result on t h e Stokes equations (So) now reads: T h e o r e m 1. ^ (1)j (injectivity) For 1 < p < < n , l < q < c o ; 5 p > g is 25 injectii injective. n, For or I1 < < p < < n, n ; TPt a/so injective. More precisely, let {w,n} G iH iJJp*pg(g(((ffft t)) x P) Pi)\i is also
91 LPtq(Q)(l solutions
< p < n, 1 < q < oo) and {w, TT} G H^Q) of I
v y
- Z A U > + V7r =
X LPti(Q)(l
< p < n) be
/,
[1 azt? div c/zt? w w — — g g
and AZAXX; t ! ) —- VTT -AwVTT = j/ ,, if — v TT = = — div ch't; w w— — gg \ — / o r {/,
x LpA (Q), respecrespecp ,i(ft),
lIIVHU |VHU + + ||Jr|| |M| < OC(||/|| + \\g\\ IMk, Cfll/H^ ||j|| IIvwllp,, ' U I / H HAIl,/ // (( nn )) .. + Hflllp,,), P> p,,), q), p> Pl ,, <
(1.4) (1.4)
IIVtiHp,! UTTIIP,! < CC(((|||||/|/|/|||A||^i^/ l|Vu>|| l|Vt&||p ++ |UTTIIP,! |7r||P,< Pll >i + 1
(1.5) (1-5) (1.5)
p' ,oo
ll^llp,!), .. + ++ \\g\\ ||5|| i), PPl,i),
((v(n nn))). '
where C = C(n,p, q) and C — C(n}p). (2) (surjectivity) Let p and q be as (i) n' n <
for every {f,g} /(Q)* x L PP))g(Q), a^ /ea5£ {/, #} G ^ Hp,/ j9g/(fi)* g ( ^ ) ; £/iere 25 is at least
one p one pair a i r {^, {w, 7r} n} G H^ Hp (Q) X x LP}(} (Q) 5i/c/z such £/*a£ that P) ^(Q) (Vw,V(p)-(n,div
(^1 ggv.
(1.6)
qi(£l).
T h e following corollary is an immediate consequence of Theorem 1: C o r o l l a r y 1.
(unique solvability for (S)) Let p and q be as
1
(i) oo, < oo ;; ( LJ n II < \ p // < v^ n, IL) 1 J. < v^ q if < \ i_^; (ii) p = n'\q — oo. Then for every {f,g} {/, qi{&)* x L P)
92 2
S>tationary t a t i o n a r y INNaavvi ieerr--Ssttoo kk ee ss ee q qu u aa tt ii oo n n ss
n'' < < ^2. < < nn for all n > Since\ n > 3, Corollary 1 guarantees t h e unique solvabi solvability of (S) ) in t h e class Hi
00(^)
x L Z L ) 0 0 ( ^ ) in which we shall solve (N — So) So)by
the ee tt hn oo da oi of perturDauon perturbation oi of xne the nnearizea linearized equations equations (S): tne m mnethod pj: T h e oureiii rem A 2 .. [Lj^exisiencej (1)(existence) Let > 3. S(n) > 0 L>VI n II ^ o. There mere is is a a constant CUWSIUTIL 8 O =. = oyi such that that if if F F G G LIL LIL)00(Q (QI)) satisfies \\F\\JL < 8, then there exists )00 satisfies \\F\\JL)00 < 8, then there exists a pair )00 I {w,7r} G Hi c^(Q) of the solution of (N — So) in the following }00(Q) o ( ^ ) x LIL}00 sense: (Viu, (Viu, V<£>) V<£>) — — (w (w •• V<£>, V<£>, w) w) — — (TT, (TT, V(p) V(p) I = ( F , V y > ) forall(peC§°{S (2.1) = -(F,V ) forall constant constant k = — k(n) k\n) s\ (2)(uniqueness) There is a constant k = k(n) such that any pair {w,n} G Hi ) x LJL (CI) satisfying (2.1) with 00(^ }OQ f2) x LrL)00(£l) satisfying (2.1) with Hi 0 0 ( ^ ) x LJL}OQ(CI) satisfying (2.1) with (2.2) IHkoo < * (2.2)
{
is unique. 25 (3)(regularity) For oo, there £/*ere is ac constant 0 < 8'{n) r) < 8(n) S(n) nYy^ F o r n'{— ^zj) < r < oo, £/zere 25 such that if L InL0) 000(( Q )) nnL n LLr)00 ^ ) 5atefie5 satisfies IIF | | F | | n | 0 0 < 8', £', then £/zen f/ie the solution z/F F GL 0o 0( (Q rr o) (Q) {w, 7r} O of/ (TV (N — 5o) So) gifen given fry by the above (1) has the additional property Vw G La (ft) H I r | 0 0 ( f 2 ) , L±>00 > 0 0(Q)
TT G L ^ ) 0 0 ( Q ) n iLr ,oo(fi). £i,oo(fi) r ,oo(fi).
(2.3) (2.3)
R e m a r k s . (1) By tU11C h e OUUU1CV Sobolev Cembedding and t111C h e lreal L) Uy l l I U C U U l l l ^ , theorem b l l C U l C i l l ClllU C d l interpolation, llltCipUlcltlUIJ n C LL there holds Hi ^(Q) J$l) C £ ,oo(^) with || IHIn.oo << CUVHI^.oo, q | V u ; | | f ) 0 0 , wr H.oo( ) "nn ^ ( ,(S2) ^ ) with IHU.oo where C i ^l|n,oo ^ where ^l|n,oo ^ ^11^^11-^,00) ^11^^11-^,00) C is spending only on n. n . T h u s we ip G HKy ( ^ ) as C tesis x1 where can take (p i^/nx/ a constant depending only on n . T h u s we can take ip G HKy x ((^ ^ ) as test 9 n afunctions constant depending only on n . T h u s we can take ip G HKy ( ^ ) as test x l U l l ^ / U l V J l l O 111 14.11, in (2.1).
(2) For For tt h e on functions (2.1). (2) hin e assertion assertion on uniqueness, uniqueness, we we do do not not have have to to assume assume tt h he e smallness smallness of ||VIU||IL ||VIU||IL )) 00 00 b of bu u tt of of the the weaker weaker norm norm ||w|| ||w|| nn )) 00 oo(3) For For tt h he e assertion assertion on on regularity, (3) regularity, tt h he e smallness smallness for for ||F|| ||F|| rr )) 00 oo is is not not necessary, necessary, which is is closely closely related tI o o tuh h e invariance invariance of ^ I U O C I ^ Irelated CICIIJ^U. t U t i i ce n i v a i i a i i L C \JL which of tt h he e norm norm ||VW;||IL ||VW;||IL )) 00 00 under under such such change of of scaling as as described described in in Introductioi Introduction. >f scaling change Introduction. tion w (N — — So) (4) For For n n = = 3, 3, Galdi-Simader G a l d i - S i m a d e r 1100 constructed constructed solution w of of (TV of (TV {N -— So) with with constructed aa solution (4) 22 ss uu w rovided ss u ppp^,^^,c^^0 |a?| |F(x)| P:ren N M °°5 > 3 3 // 2 2 provided provided IxrlFfa:)! r||w(aO| ° 5 ^w ^ Lrr(Q) for all r :> P:ren ll^^l l^^O) I! < °°°5 |a?| |F(x)| Drchers-Miyakawa 33 found found is sufficiently sufficiently small. Then Then N N PPaaaddduuulllaaa222444 and and Borchers-Miyakawa Borchers-Miyakawa mtly small. Nooovvvoootttnnnyyy---P is the solution w in t h e class the solution w' in h e class in ttne ciass 2 1 sup | zn\| n-~2\w(x)\ | w ( £ ) | + sup |ar| |a:| rln "" |Vti;(x)| ■ < oo - 1 |Vt«(a:)| x£fl xG^
93 for t h e given F with s u p r r 6 a k r _ 1 | ^ ( ^ ) | + sup a 7 G n | z | n | V F ( z ) | small. In The orem 2, we need not impose any assumption on V F , and furthermore our class for F itself is larger t h a n t h a t of 3 , 1 0 , 2 4 .
3
S t a b i l i t y of s t a t i o n a r y s o l u t i o n s
We shall next proceed t o stability of the flow flow w obtained by Theorem Theorem 2. Let us define the t h e Stokes C°°-real operator A in I £ ( f t ) . C™ is the set of all C°°-real vector ;okes operator Arr in L^(i2j. u 0 ( Ta is the set ol functions3 (p — (ip ((p1)1, •• •• •, •, (p (pnn)) with with compact compact support support in in ft, ft, su such t h a t div j r ((ft). f t ) . It It is known t h a t UG = — {u G 17; — 0 on 3 d f t } . Recall the H< Helmoltz Lr; div u ti = 0 in ft, u • jv/ = decomposition: tlUIl. r L L£ e G r (direct s u m ) , Lr r = L a 0
< co, 1 < rr <
where Grr = = { V p £ Lr;p £ L[ o c (ft)}. For the proof, see Fujiwara-Morimoto 7 , M i y ai k a w a 2 3 and Simader-Sohr 2 6 . Pr denotes the projection operator fro: from U r along G G Thhee Stokes Stokes operator operator A Arr on on L L£ Ua0 is is then then defined defined by by A = -—PrA onto Ua along L^ Jr.. T .Arr = 22 rr (ft) # o £ , where ^ # *; ^£ is the closure C^ with domain D{A O i/o'^, ga^a with ){Ar) = ^#tf >> ((ft) #o£> 1 11 r, r — IMIr with t h e ]n o r m ect to the iHf ' -novm - n o r m |||<£>||#i>r H| ^^| H llv^llr H- ||Vy>|| r . Equipped wil respect ||^>||r + | HH1 . '^- = H^r^llr, ^ interpolation, n)^Arr^) = D(( ^Arr)r ) is a Banach space. By real in interpols IM|.D(A — ||w|| \\u\\r || r + -h H-Ar-w||y-, | | A r i i | | r , D(A r qq r r L^q = (L£°, iefine L a> = (L^°, (L we define ' by U« U^ L ^}q1, ) ^ , where 1 l << r r00 << r r< L 9q = r j {u t{ue i EG L r >,g(ft);div ix = 0 g (ft);div u
in ft, u • i/ = 0 on 5ft} <9ft}
and t h a t the Stokes operator A is also well-defined on Lra>q with t h e domain D{Ar>q)
= {ue
r L™; L «\V'uV ' t i G L r > gg(fi), ( f i ) , ij - l , 2 , i i | a n = 0 } .
On t h e stationary flow, we impose the following assumption: A s s u m p t i o n 1. w is a solenoidal vector field on ft with w\dn — 0 in t h e class
weL^DL00,
Vw<ELr*
for some r* with rz < r* < oo. R e m a r kk.. Take so tt nh aa tt r* rr* < rr 00 < < oo. oo. If (ft) + < l a k e rr 00 so it F r G G L«. ^ ^ ) j00o0 (ft) ("J n n L ^rr0)OO 0 ,ooV^J H^llf.oo < < ^^ (( rwi j, rro0 )) ,, then Theorem 2 (3) and t h e Sobolev embedding then Theorem 2 (3) and t h e Sobolev embedding L°° yields solution w w of of (TV (TV — — So) So) satisfying satisfying Assumption Assumption 1. 1. i s aa solution
satisfies satis 1,r C HH1^*
94 Now we ve define aenne an an operator operator B arr on on U .L^a for tor 1t < < rr < <. r* r* by by BrTu = Pr(w •- Vu Vu -f it u • Vtu) Vw)
with the domain
D(Br) = # QH^ '£.
It should be noted that if Vw G £ r * and if 1 < r < r* then we have uit • Viu Vw G L r an nence rr for u E well-defined on on H^ H^ aa.. Then Then ££rr isis introduced introduced by by G HQ'I i? 0 a and d hence 55rr isis well-defined Cr = Ar + Br,
1< r < < r*
with domain
D ( £ r ) = D(A r ).
On the initial disturbance a in (N — S\) we impose the following assumption. Assumption 2. The initial disturbance a is in L"'°°. Our results on stability of w tu now read: T h e o r e m 33.. (1) (1) Let Let w and and a satisfy satisfy Assumption Assumption 1 and and Assumption Assumption respectively. There respectively. There is a positive positive number number K = K(n,r+) K(n,r+) such such that that if IMIn.oo < K,
2,
(3.1) (3.1)
||a||n,oo < « «, ,
1 then there exists a strong solution u of (N — SS') ) with the following properties.
(i) W i iG G55CC((((00,,ccoo));;LL^^--))nnC C((((00,,ccoo));;D D((A Arr JJ )) nn C C 11 (( (( 00 )) oo o ) ; ^ ) ; [ti/d^ + Prr.r^, (u Vu) ==0, in m LJL-^ LLrarar*,a*,*, t>>0; 0; (it) du/dt + £Crr#.uw ++ P ti> in u. -f JT ^(u a •.••Vu) vVu) uj = — u0, ,0, m t/t , ti> iY/^ — __v> a /7 Czn' t/(£) (Hi)^ u(t)
inpnhhl* in in weakly*
(iv) (uniform
estimate)
/", n >°° nc J_D as tf II -j-0; +0;• L™'°°
\\u(t)\\
< r < n
r*
(3.2)
for all t > /or > 0 with wi£/i a constant C depending only on n, r and r*. (BC(I',X):the (BC(T,X):the set ^\i\j\):uie set of oj bounded uounaea and ana continuous continuous functions junctions values in X) lues in X) (2)(uniqueness) There is constant = k(n,r*) such )(uniqueness)) There There is is aaa constant constant kkk = = k(n,r*) k(n,r*) si such (N — S') satisfying fying the above properties (i)-(iv) witl with
lim m ss uu pp ^^i-i-"-^^i l!ml! ^* ) ! ! ^ < k n+o is unique.
on on the trie iinterval I with that any solution u of that any
(3.3)
95 T h e more rapidly spacial decay at infinity of the initial disturbance a(#) is assumed, cissumeu, the uie sharper snarper asymptotic asymptotic behaviour oenaviour for lor u(t) u(l) as as tI -> —> oo oo isis obtained: obtained: T h e o r e m 4. For n'(= ^ y ) < p < n, there is a constant £ ( n , r * , p ) < « ( n , r * ) (/c(n,r*); the constant in (3.1)) such that if a E £"'°° C\ H LLJpa and and ifif HH ««,, H kk o o < K,
£/zen the £/ze solution i uu, given then yiucu by uy Theorem ±iicuicm 3o has nan the aic following juiiuwniy «(•) tx(-) «(•)
and and
(3.4) (3.4)
||a||„ ||a|| ||a||n,oo i 0 0 < k, n ,oo additional uuuuzoTiai
p
< t*Vu{.)eBC{[0,oo)-L Vw w (( -- )) €€ 55 C C (( [[ 00 ,, oo)]oo )) ;; L LP P )) ;; < 55 V
((
\\u(t)\\i -= 0 ( rr*^ pp~"T^) ))
f/oro r
pp
as
Moreover, for every q with p < q < n, there is a constant such that if we assume in addition to (3.4) that
HHkoo < «, HHkoo < «>■ then i there mere
properties: properties: (3.5) (3.5)
t --+ + ooo. o. k —
(3.6) k(n,r*,pyq)
(3.7) (3.7) (3.7)
a IMkoo < «, ll l|n,oo < «,
holds holds n. / 1
IN
1
||Vti(t)||/ = 0{t~^p"^ 0 ( t " ^ ( p}~^) -T)-i) 7u(t)\\i =
for
p
as
t -> oo. o
(3.8)
R e m a r k s . (1) Theorem 3 shows t h a t L™,co is the class of stable stationary flows and t h a t it is the same class as t h a t of initial disturbances. BorchersM i y a k a w a 3 obtained, among others, similar results to ours including r = oo in (3.6) and q = n in (3.8), respectively. They impose, however, such a stronger assumption as sup |x||u>(x)| + sup |#| 2 |Vu>(x)| xGH
x££l
is small On other iidiiu, hand, uour h a t the assump cxii eenough. iiuugii. w n the tile ULIICI u i rresults e s u l t s nhave a v e eclarified i a i i n c u tuiidi; biiv <x»s r r,n tion on V w w is superfluous for the stability in L < r < oo. M Moreover, on Vu; is superfluous for the stability in L ,n < r < oo. Moreovei the 50 space, L£'°° \x ^n,oo is 'ls larger } a r g e r tt hh aa nn the the class class of of functions functions such such tt hh aa tt ss uu pp ^^ ^^ |a?||it;(aj)| |a?||it;(aj)| < oo. • j _ i _ _ ,i. _ i ^ _ _._ T12 _ x _ i _ : i : i . _ . 2 -stability for weak so Hence:e Theorem Theorem 44A ihas has improved improved also the results on L also the results on L -stability for wea lutions utions with homogeneous boundary condition at infinity, i.e., w(x) —> 0 as \x\ - » oo obtained by H e y w o o d 1 2 , 1 3 and M a s u d a 2 1 . x\ -> (2) is not strongly continuous in L"'°°, we [2) Since the semigroup {e~tC}t>o cannot verify whether our solution u satisfies lim^-'^llumiL U+o
=0.
This is the reason why we impose the condition (3.3) on the uniqueness. regularit on u>, Kozono(3) W h e n Q Q, = — Rn{n (n > > 3), without assuming any regularity 19 >tained its stability with sharper decay t h a n (i(3.6) and (3.8). Y a m a z a k i obtained
96 References 1. Bergh,J. and Lofstrom, J., Interpolation Spaces. Berlin-Heidelberg-New York: Springer-Verlag 1976. 2. Borchers,W. and Miyakawa, T., Algebraic L2 decay for Navier-Stokes flows in exterior domains. Acta Math. 1 6 5 , 189-227 (1990). 3. Borchers,W. and Miyakawa, T., On stability of exterior stationary Navier-Stokes flows. Acta Math. 1 7 4 , 311-382(1995). 4. Caffarelli,L., Kohn, R. and Nirenberg, L., Partial regularity of suitable week solutions for the Navier-Stokes equations. C o m m . Pure Appl. M a t h . 3 5 , 771-831 (1982). 5. Finn, R., On exterior stationary problem for the Navier-Stokes equations and associated perturbation problems. Arch. Rational Mech. Anal. 1 9 , 363-406 (1965). 6. Fujita, H., On the existence and regularity of steady state solutions of the Navier-Stokes equations. J. Fac. Sci. Univ. Tokyo, Sec IA M a t h . 9, 59-102 (1961). 7. Fujiwara, D. and Morimoto, H., An Lr theorem of the Helmholtz decomposition of vector fields. J. Fac Sci. Univ. Tokyo, Sec.IA 2 4 , 685-700 (1977). 8. Galdi, G. P. and Padula, M. Existence of steady incompressible flows past an obstacle, preprint 9. Galdi, G. and Simader C.G., Existence, uniqueness and Lq-estimates for the Stokes problem in exterior domains. Arch. Rational Mech. Anal. 1 1 2 , 291-318 (1990). 10. Galdi, G. and Simader C.G., New estimates for the steady-state Stokes problem in exterior domains with application to the Navier-Stokes problem. Diff. Integ. Equations 7, 847-861 (1994). 11. Giga, Y., and Sohr, H., On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo, Sec IA 3 6 , 103-130 (1989). 12. Heywood, J. G., On stationary solutions of the Navier-Stokes equations as limits of nonstationary solutions. Arch.Rational Mech. Anal. 3 7 , 48-60 (1970). 13. Heywood, J. G., The Navier-Stokes equations :On the existence, regularity and decay of solutions. Indiana Univ. M a t h . J. 2 9 , 639-681 (1980). 14. Iwashita, H., Lq — Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces. M a t h . Ann. 2 8 5 , 265-288 (1989). 15. K a t o , T., Strong Lp-solution of the Navier-Stokes equation in Rm, with
97 applications to weak solutions. Math. Z. 1 8 7 , 471-480 (1984). 16. Kozono, H. and Ogawa, T., On stability of the Navier-Stokes flows in exterior domains. Arch. Rational Mech. Anal. 1 2 8 , 1-31 (1994). 17. Kozono, H. and Sohr, H., On a new class of generalized solutions for the Stokes equations in exterior domains. Ann. Scuola Norm. Sup. Pisa 19, 155-181 (1992). X<7, 1 U U l O l y±OC7±). 18. Kozono, H. and Sohr, H., On stationary Navier-Stokes equations in unbounded domains. Ricerche M a t . 4 2 , 69-86 (1993). 19. Kozono, H. and Yamazaki, M., The stability of small stationary solutions in Morrey spaces of the Navier-Stokes equation. Indiana Univ. M a t h . J. 4 4 , 1307-1336 (1995) 20. Leray, J., Etude de diverses equations integrales non lineaires et de quelques problemes que pose VHydrodynamique. J. M a t h . Pures Appl. 9, 1-82 (1933). 21. Masuda, K., On the stability of incompressible viscous fluid motions past object. J. Math. Soc. J a p a n 2 7 , 294-327 (1975). 22. Masuda, K., Weak solutions of the Navier-Stokes equations. Tohoku M a t h . J. 3 6 , 623-646 (1984). 23. Miyakawa, T., On nonstationary solutions of the Navier-Stokes equations in an exterior domain. Hiroshima Math. J. 12, 115-140 (1982). 24. Novotny, A. and Padula, M., Note on decay of solutions of steady NavierStokes equations in 3-D exterior domains. Diff. Integ. Equations 8, 1833-1842 (1995). 25. Serrin, J., The initial value problem for the Navier-Stokes equations. Nonlinear Problems, R. E. Langer ed., Madison: University of Wisconsin Press, 69-98 (1963). 26. Simader, C.G., and Sohr, H., A new approach to the Helmholtz decomposition and the Neumann problem in Lq-spaces for bounded and exterior domains. " M a t h e m a t i c a l Problems relating to the Navier-Stokes Equa tions" Series on Advanced in Mathematics for Applied Sciences, G.P. Galdi ed., Singapore-New Jersey-London-Hong Kong: World Scientific, 1-35 (1992) . 27. Sohr, H., and Wahl, W . von., On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations. Manuscripta M a t h . 4 9 , 27-59 (1984). 28. Solonnikov, V. A. Estimates for solutions of nonstationary Navier-Stokes equations. J. Soviet M a t h . 8, 467-529 (1977). 29. Ukai, S. A solution formula for the Stokes equation in R^_. C o m m . Pure Appl. M a t h . 4 0 , 611-621 (1987). 30. Wahl, W . von, The equations of Navier-Stokes and abstract parabolic
98 equations. Braunschweig-Wiessbaden: Vieweg 1985 31. Wiegner, M., Decay results for weak solutions of the Navier-Stokes tions in Rn. J. London Math. Soc. 3 5 , 303-313 (1987).
equa-
99 ON DECAY PROPERTIES OF SOLUTIONS TO STOKES SYSTEM IN EXTERIOR DOMAINS P. MAREMONTI Mathematical Department, Universita della Basilicata Via N.Sauro,85, Potenza, Italia V.A. SOLONNIKOV Mathematical Department, Steklov Institute Fontanka, 27, St. Peterburg, Russia In this note we report the statements of some results concerning the nonstationary Stokes problem in exterior domains, which are contained in a forthcoming paper [13]. In particular we want to communicate new results about the asymptotic behaviour (in time) of the solutions. Here for the sake of to brevity we furnish no proof of our theorems, as well as we limit them just to the well posedness of quoted results. Actually the aims of [13] are larger than the ones showed here.
1
Introduction
In [13] we study the solvability of the nonstationary Stokes system with Dirichlet b o u n d a r y conditions. T h e initial boundary value problem is considered in an exterior domain Q C IR n (n (n > > 2). As is known, t h e results concerning the Stokes system are preliminary t o investigate t h e nonlinear Navier-Stokes equations. In this connection, we recall some recent papers [1,3,4,6,7,10,17]. However in [1,3,4,6,7,10,17] some questions are not solved about the asymptotic behaviour with respect t o time of the solutions. T h e aim of [13] is two-fold. On the one hand we want t o propose the solvability of t h e initial boundary value problem by t h e well known Schauder's method n n 2 1(in 1 1 1 this U 1 1 1 Q way VV fJOJ we » » V essentially V _ . U t J V l l U l U . l l J extend VAUV/11U t Uo W the U i l V case V / U U V / of v y i exterior V / A U V 1 1 U I domains U.U111UP111U Q UU C v_- IR JLL\j IV > ^__ J-l the results obtained in [17] in t h e case of three-dimensional exterior domains), then t o improve the estimates concerning t h e resolving operator associated t o the system. U11C Stokes U UUJ\.CO CiJ Oli^LlL. Our results may be considered sufficiently complete in t h e sense t h a t they are complementary with respect t o the ones already known in literature, either for the results or t h e technique employed. Finally we are able t o prove t h a t our estimates concerning t h e resolving operator are sharp ones. 2
N o t a t i o n s a n d s t a t e m e n t of t h e chief results
T h r o u g h o u t this paper t h e symbol Q C IR n , n > 2, means an unbounded domain exterior t o a finite number of compact regions. Boundary <9Q is as-
100
sumed of class C m , m siimed m an even positive integer such that 2m 2ra > n. n. By Co(£l) we denote the set of all solenoidal vector fields (*) €e L £f {Q)} o c (0)} • f x p VVO (f, W ) = / n ( ) * V^(«)dx = 0, V(f(a), V^(x)) ViKx)) G J (ft) ( ^ ) x G*(fi). By the (Q,) into J P (Q). J 1 , p (fi) is the symbol P- we indicate the projector from LP(Q) 1,P closure of C0(fi) in W (Q). The norm of a function in Wm>p{Q) is denoted by | • \m,p- By the symbol P 0 ' P (Q) we denote the set of functions (vector field) p from Besov space Bli,p>(Q,) (P) vanishing on the boundary, if £ > 1/p; for £ < 1/p l p l p we set B 0' (Q) = B > {Q)\ finally in the limiting case £ = 1/p we define Bl0lP(Q) as the space of all functions from B1^P,P(Q) whose zero extension into IRn — Q belong to Bllp*p{Q). The norm in BQ,P (Q) may be given by the formula llUU\\r,p \\ llr,p = IHlBr-.P(n) +
7
r,
[dist(x,dQ)]~1dxl)dx, otherwise Irr = 0. For rr < 1 + 1/p, where Ii/p = fn \u(x)\p[dist(x,dQ)] this norm is equivalent to |u°|j9r,P(n) where u°(x) = u(x) for x G 0, u°(x) u°(#) = 0 for x G nr - a In Q we consider the initial boundary value problem vvttt(x,t)-Av(x,i) (a?,*)-Av(<M) = V7r(a?,<)+f(a;,t), V7r(a?,t)+f(a;,t), T7ir(x,t)+f(x,t), on on JJ Q xx (0,T), (a?,t) - Av(x,t) = (0/ onQ x (0,T), V-v(a?,t) = 0, onQ vfa?,^i«o. = 00> vfx,tl—>■ for b l --> > oo v(a?,t)|an 0,5 v(a:,t) v(a?,t) \x\ V^ > 0, v(a:,t)|an ->- 0 /for o r |x| oo, V* vv(a,0) ( a , 0 ) = vvo0((a?). x).
(2.1)
In this paper following theorems. ;er firstly urstiy we we state state the tne loiiowingp Lneorems. p T h e o r e m 11 -- Let f(x,t) G L {{0,T); L {n)) f{x,t) #>(((), T); #>(£})), vv00(x) PP0022"-22//Pp)' Pp (fi)n^(fi), Letf{x,t) G isLP((0,T);LP(JJ)), (x) G y*,i) KZ \\y,± ),IS\IL)), ) VQ\X) tG J^o V(LfPi ) 'n ^ ( f i ) , p > 1. T/ien Then there e^'s^s solution {v{x t),p(x to system (1) iii)^p{x iit)) ?n ^/zere ^/zere exists e^'s^s {v{x t),p(x t)) (1) such 725^5 aa unique unique solution [\{x t)) to system (l such ) ) 1 Lp{{0 T)2 J^p{Q)nW2'p{n)),v {x,t),Vn{x t)eLp{{0,T);Lp{Q)) tfza£v(x,*)Gi:M(0,T);^^ thatv{x t) G y | G L M ( 0 , T ) ; J ^ ( ^ ) nJ^ ]^ ( Q ) ) , v t ( x , t ) ,t V ^ , t ) G ^1 ( ( 0 , T ) ; ^ ( Q ) ) and I
(Mt)|£
+|O 2 v(0l^ + |V7r(f)pdi
J 0 «/
< Co j ^T ((|f(t)|> (|f(t)|P t))£^^F( (nnniiJ)) )d A ||vo||5. ,,
(2.2)
w;/iere Q' where Clf CC ^^ isis aa s£np s£np of of dQ, dQ, CC isis independent independent ofT. ofT. Let us consider A = —PA. —PA. It is known that A can be considered as an operator defined on the set J 1 ' ^ ) n W2>P(Q), which is dense in Jp(ty. Then system
101 (1) can be considered as a Cauchy problem in t h e space J p ( f i ) : v t (a:,t) (a;,i) + A v ( xi , t ) = f(<M), Av(x,t)
(z), v(a?,0) = v 00(a:),
f(z,*) G £ P ( ( 0 , T ) ; J'(fi)), v0(x) G 6 JJPp{Q), (ft), p p> > 1. 1.
(2.3) (2.3)
Let U(s,t) be t h e resolving operator of equation (3). For f ( # , t ) = 0, operator U(t,s) associates t o any v 0 ( z ) G J p ( ^ ) t h e solution v(x,t) = U(t,s)vo(x) of the equation vt{x,t)-\(x,t) + Av(x,t) Av{x,t)
= 0, 0,
G JJ p (pQ)) ( (p 1). vv(a?,«) ( z , * ) ==vvo0((sx)) E p >> 1).
(2.4) (2.4)
22 2 p2 p
Actually, by virtue of Theorem 1 U(t s) is defined on set 5 ( Q ) n J PJ(p{tt) Q) U(t,y s) B - "' >/ ^(Sl) p which is denses in (Ct). Operator semigroup: in Jjr(\i). u p e r a t o r U(t,s) u\t,s) has nas tt h n ee property property of ot se = W (r)t , r ) , V* =U(t V* > > ss > > r, r, U(t,t) W(M) = = J. J. i i
U(t,s)U{s,r) U(t,s)U{s,r)
We prove t h e following properties of operator U(tys). T h e o r e m 2 - Let v(a?,t) 6e a solution of the initial value problem (4). there exists a constant C such that (p (p > > 1 for n > 3, p p> > 1 for /or n n = 2) V--*sa >>>(0)0, ,, | v ((t**) )|),| |<, ,<
Then (2.5) (2.5)
_ nn // l l __] l. \
fo,co] 3, j!p,oo] ' /n> n >>3, ItP. °°] ifz*/» 3, 2 2\p V p q) J *' ' ff*%>,«>) pp,,oooo ) it if/ nn = 2; = 2; ^|l^vWI^CIv^Ut-*)^, ^ -f S- )S -) '^i , Vv v( < ( <) )| g| ,<
P _--(--V\, ee9
'*-
tui£n with (( _ II ^^ _ _ // ii | / i ~ ~~ 1 1 ~ 1 {II
(2.6) (2.6)
1; |h+pif + / i « 7 gqe\p,n],t-s>0 -5s >>ip> 2 , p > 1; f-h/i2/ tGL [Pp, ,nnj ], ,r t-in>2 u0 , n > ^ z i; !| + + / i 2 7 ? e [ p , o o ) , * s e ( 0 , l ] , n > 2 ; / i z 7 g G [ p t s G + /^i2z/7 ggG G [Lpp,,oooo )J ,, tt -- ssG |+ Gl(U0,,llJ], ,nn > 2 ; % if ^ *7 9gg> >>>PPv.a »P,9,g g> >>>n,t w > > 333;;; if % nq>P,q>n,t-s>l,n>3; ,, tt— > >l,n > ^ *i /7a M ---s*«s > lll ,,, nnn > & 2 ff Q > P i 0 € ( 2 , o o ) , * s > £ * / 0 > P i 0 € ( 2 , o o ) , * s > l )2, h ~ &f Q > PiQ€ (2, oo), t — s >l,l,n,nn=== 2,
where S > > 0 can 6e chosen arbitrarily small. Moreover, estimate (6) for q > p,g > n,p > nn/2,n / 2 , n > 3, z's sharp, in the sense that it is not possible to improve the exponent ft in fi -f e for any e > 0. In i n particular it is not possible to have ft — 1/2 + -f p. p,. Finally |\Mt)\ v t ( qi
> 2,q 2,q eG [p, [p, oo],< oo],t— ifif nn > — 55 > > 00,,pp > > 1; 1; if if nn = — 2,q G [p, > 00,,pp > 1; 2,q E [p, oo), oo), 11 > > tt — — ss > if 2,q 2,g i7 / nn = = 2,q=:2,t-s> 2, g= = 2,t-s> 2 ,, *t -- ss>> 0 , p G (1,2]; 00 1 if27 2,<7 G \v.oo)D oo) o,oo^)),),tt^-— [p, H 2i /7nn = == 2.a 22,g , g GG G [p,oo) [[p, p ^oo) n )H(2.oo).t ((2, ^(22!,,o2oo),t - ss s> >>l 1 , pp > 1; i n = 2 / = ' ^ b 5 2 ] ^2-iqe\p,2],t-s>l,p>l, 5>i,p>i,
(2.7) (2.7)
102
where S > 0 can can be 6e chosen arbitrarily small. Corollary 1 - Le£ Let (v(£,2),p(#,£)) be the solution of system (1) determined in Theorem 1. Then there exists a constant C(T) such that aap p P p a c T n\MW {{\Mt)\ MtO IP? P+++Ht)\i Wm, \vm)iPlPlPp+\v*(t)\ + |V7r(t)|J} \V*(t)\pPp}}dt dt \f(t)\$dt\ tap £[£ " ttap M |V7r(i)|j;} eft < C( ('<*) 7 a) » hMvol£ £t\f{t) r\f(t)\rdt\ 0|j; + {\v (t)\ < '\f(t)\$dA
where a is a suitable exponent. Moreover, assume Vo(x) = 0 and f(x,t) G p L g9((0,T); with the following restrictions: ( ( 0 , T ) ;L ^ ( (Q)), f i ) ) , iwz£h if pp > n/2, then z'/ then qg G (1, 2p/(2p - n)) z
/ P P £€ (1 (1,3 nrc/2], / 2 ] , then tAen g > 1.
Then, there exists a constant C such that ( rT T rr
\
1 / r
\1'<>
( ,T T T
[[j[ j /[ |v(t)|;
< f ||f(t)|9d«J )(<)| | 9 d«dt < jj
where s > p, r > g, ^ — ^ > ^- (£ ——7i )—— i . 1-me^ T^fl-\ r
q ~ 2 \p
(2.8)
constantLy Cz* isniucpcjiueiii independentujof cun&iuiii
b
= ifk-l-n.(l-L\ %{l-i)-l,otherwtseC
-1 i lJ r
e
,,VT>0, VTTT>>>00o,,, ,V V
= C1T ,b=]:-\-%(l-\)
s )
+
l,
p PP
C\ is independent ofT. Finally, assume f(x,t)- ) = 0 and vVQ(X) vo(x) (Q). sume ffxj n d l £G JJ (Q). fQ). Then lim|vW|p t lim|v(<)|„ tt—>-00 —>-00
= 0;
(2.9)
(5)-(6)-(7) become \v(t)\q = = 0o{t-»), o(r»), |v(*)|, = |v(i)|, (r"),
\Vv(t)\ = o(r*), o(t->), ((rr""' ')),, |Vv(<)|, o(f-A), |v | Vvtt|,| , = 00o(r"'), |Vv(t)|, q =
(2.10)
where //, //, p, //' p! —are are defined in (5)-(6). — . _ „ ._ vo(x) G T h e o r e m 3 - For arbitrary f(x,t) G Lq>r (QT) problem (1.1) with vo(tf) q)r(QT) —
/
2-
_.j
/
?
l—
—
> /
-. —
—
\
/
\
/
-
5 00 q9 (Q) nas B /las a unique solution with v(x,t) G ^W^^(' ^QQTT)) ^^ V7r(a?,t) V7r(#,tf) G Lq^r(QT) and the following estimate holds r £\f(t)\ |r; . (2.11) ^/Tr ((|v«(0l$ | (iv.Wla v . ( < ) i ; + |v(*) l|v(<)vK| v W r^^(„)+|V r n(n)+|V7rW|;)df i + l Vff (t)|5)(ft T r W I ^ d
Moreover, £/iere Moreofer, there /io/c/s £he ^he inequality 2 r r r [fTT(\v (\Mt)\ + \D\(t)\ \D v(t)\ +\y^t)\rqq)dt )dt qq+\V7r(t)\ q)dt t(t)\qqt)\ q+\V7r(t)\ q q + Jo Jo
T r
(2.12) (2.12)
103 if q G ( l , f ) , (n > 3) then C 2 is independent of T. Otherwise for n > 22 C{ = 1 + C T 6 withb= «;rtA6= 1l +17 + r j- - ^i L , V77 ^ X>) 0. . For For 9q > > f andVre andVrE (l,oo) C C22 can can no£ be not 6e constant with respect to T. However instead of (1.11) we can prove the following r rr jf {\Mt)\ \P^{t%)dt
Jo)
r r |fW|^+|v(x) ff I nt)\ \f(t)\ + | v q(,x ) o0||J, J: gdt+\Hx)ol\ qdt Jo
(2.13) (2.13) (2.13)
for any r,q G ( l , o o ) . Rl e m a r k 1 - Our Our chief results are contained in Theorem 2. It gives the asymp asymp totic otic behaviour with respect to time of the solutions, in particular the orde order of )f decay. In this connection, apart from partial results obtained obtained i\in [2,9,3], 2,9,3], we stress that in the two-dimensional case our results are quite neu new, as is well as, the estimates concerning \| • |oo and |V • \Lp of the solutions. Here w we give a scheme of solved questions in this respect. We can resume the results already known in literature as follows: known
results
:
(2-14) (2.14)
f a M > 3 3| v\v(t)\ ( t )rc>3|v(f)| ,qE\p,oo),>> 000;; q| ML2 ::■ ||Vv(<)|r | Vv(<)| V vv (( tt )) || r < < C\v C ||vv000\pt~°~ -- 1 / 2,, rr G (p,n],Vt > 0; 0; r
. _ r, A A33L:: 3 : nnn = = 22
0 p
i /.\i ul/a-l/v\ _ ^ /i o\ . ^ ro l |v(<)|, oo); |v(*)|, o{t^-^),p €G (1,2), (1, 2),qqq G [2,oo); \v(t)\q = = ooo
A44:
n>2
l Nv(t)\ | V v ( t ) |22 = = o (o(tt - 1 / ''n,pe(i,2]\ )lpG(l,2];
\v (t)\88kkdtdt < oo, \v(t)\ 00,
n - l = n/k + 2/s, Jo Jo 5 // oo rr ff (( xx ,, tt )) G GL L 5 (( (( 00 ,, T T )) ;; ^^ (( Q Q )) )) .. open (2.15) open questions questions solved solved by by Theorem Theorem 22 :: (2.15) Si :: it it extends extends estimate estimate A\ Ai to to qq — — 00; oo; Si 5522 •: it extends estimatelie A2tor> > n; to r > 53 : it gives for n = an estimate of the same type of Ai, 5 22 ; 2 = 54 :: it it gives gives directly directly on v(x,t) estimate A4 increasing the range of k.s k,s 54 k,s > 1; 55 :: it vvtt(x,t). 5s it gives gives an an estimate estimate for for (x,t).
In the seems interesting to stress that Liie light iiyiiL of uj results results stated stuicu, in in Theorem ± IICUI cm 2, &, it it seems uitci for - nn > 3, q > n, estimates (6) give a result different from Cauchy problem. > 3, q > n, estimates (6) give a result different fro However, as stated in Theorem 2 the result is sharp, in the sense that it is ywever, as stated in Theorem 2 the result is sharp, in not possible to improve the exponent ft in fi + £,V£ > 0. In particular it is
104
not possible to have ft = 1/2 + p,. fi. This is strictly connected to the fact that Q, is an exterior domain. Not only (11) is different from the analogous result of but of uj Cauchy ^/uuu^iiy problem, yi UVI^III, i/ut the n t c proof yiwj uj "sharpness" oiiuti JJIIK^OO fails junto in in the LIIO case c u o c of uj the vnc whole uunuic space. Finally, Finally, we we want want point point out out that that our our considerations considerations also work work in in twotwospace. also dimensional case. case. Therefore, Therefore, if if one one proves proves that that for for nn = = 2, 2, qq > > p, p, || V V vv (( tt )) || gg < < dimensional v _1 C | v o | p ^ _ 1 ) P > 1> then this estimate is certanely sharp. Actually, on the base of (6), we are able to assert that it is not possible to improve the exponent fi (i ~r in fi + ce JUI for tiny any ce > in (6) IUJ in in fj, / 0. u. Of course, about estimates (5)-(6)-(7) an interesting consequence is the pos sibility to analyse the asymptotic behavior of the solutions to Navier-Stokes system [5,8,9,11,12,14-16,18]. Here, for the sake of brevity, we omit the devolopment of these results. However they can be obtained, for example, following the argument lines proposed in [11,12]. Finally, the first statement in Corollary 1 proves that problem (4) is well posed. Apart from its own interest, inequality (8) is important as application to the weak solutions of the Navier-Stokes equations see [6,11]. We conclude with a Remark on the results of Theorem 4- More precisely we point out that estimate (12) is not true for q > ^ (Vr € (1, oo)) with a constant C independent of T. However we are able to give an estimate of C2(T) for Some words about the employed technique. Theorem 1 is obtained with the well known Schauder method, in the way proposed in [17]. Making use of the idea of V.I. Yudovich [19] we obtain estimate (12) for q £ (1, |§)) from (2). The "sharpness" of estimate (12) is obtained making reference to the solutions of the me steady sieaay Stokes DtoKes problem. prooiem. As far far as as Theorem Theorem 22 is is concerned, concerned, the the method method of of the the proof proof of of (5)-(7) (5)-(7) relies relies As on energy energy estimates, estimates, imbedding imbedding theorems, theorems, L Lpp — —L Lqq estimates estimates for for the the Cauchy Cauchy on problem. problem.
References 1. W . Borchers and T . Miyakawa, Algebraic L2 decay for Navier-Stokes flows in exterior erior domains Acta Mathem., 1 6 5 (1990) 189. 2 Flows in un 2.I. W . Borchers orchers and T . Miyakawa, L -decay for Navier-Stokes Fi bounded ed domains, with application to exterior stationary flows Arch. Arc] Rational Mech. Anal. 1 1 8 (1992) 273. 3.}. W . Borchers archers and W . Varnhorn, On the boundedness of the Stokes Stoke, semigroup in two-dimensional 275. >-dimensional exterior domains Mathem. Z., 2 1 3 (1993) 271 4.1. R. Farwig rwig and H. Sohr, The stationary and non-stationary Stokes Stoke system in exterior domains with non-zero divergence and non-zero boundary values Math.
105 Methods Appl. Sciences, 1 7 (1994) 269. 5. G.P. Galdi and P.Maremonti, Monotonic decreasing and asymptotic behav ior of the kinetic energy for weak solutions of the Navier-Stokes equations in exterior domains, Arch. Rational Mech. Anal., 9 4 (1986) 253. 6. Y. Giga and H. Sohr, Abstract LP Lp estimates for the Cauchy problem with ap plications to the Navier-Stokes equations in exterior domains J. Func. Analysis 1 0 2 (1991) 72. 7. Y. Giga and H. Sohr, LP estimates for the Stokes system Func. Analysis and Related Topics,1991 1 0 2 (1991) 54. 8. R. Kajhikiva and T . Miyacawa, On L2 decay of weak solutions of the Navier-Stokes equations in E T , Math. Z. 1 9 2 (1986) 135. 9. H. Kozono and T . Ogawa, Decay properties of strong solutions for the Navier-Stokes equations in two-dimensional unbounded domains, Arch. Ra tional Mech. Anal., 1 2 2 (1993) 1. 10. H. Iwashita, Lq — Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces, Math. Annalen, 2 8 5 (1989) 265. 11. P. Maremonti, On the asymptotic behaviour of the L -norm of suitable weak solutions to the Navier-Stokes equations in three-dimensional exterior domains, Comm. Math. Physics 1 9 8 8 (1988) 385. 12. P.Maremonti, Some results on the asymptotic behaviour of Hopf weak so lutions to the Navier-Stokes equations in unbounded domains, Math. Z. 2 1 0 (1992) 1. 13. P. Maremonti and V.A. Solonnikov, On nonstationary Stokes problem in exterior domains submitted for publication. 14. R. Salvi, The nonstationary problem for the Navier-Stokes equations in regions with moving bounadaries, J.Math. Soc. Japan, 4 2 (1990) 495. 15. M.E. Shonbek, L -decay for weak solutions of the Navier-Stokes equations, Arch.Rational Mech. /\nai., Anal., ou Anal.. 89 yiyov) (1985) 209. /\rcn. national iviecn. zuy. 16. H. Sohr, W . von Wahl and M.Wiegner, 16. H. Sohr, W . von Wahl and M.Wiegner, Zur Zur Asymptotik Asymptotik der der Gleichungen Gleichungen von Navier-Stokes, Nachr.Akad. Wiss. Gottingen, 3 (1986) von Navier-Stokes, Nachr.Akad. Wiss. Gottingen, 3 (1986) 45. 45. 17. Navier-Stokes 17. V.A. V.A. Solonnikov, Solonnikov, Estimates Estimates for for solutions solutions of of nonstationary nonstationary Navier-Stokes equations, J. Sov Math., 8 (1977) 467. equations, J. Sov Math., 8 (1977) 467. 18. M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations 18. M. nWiegner, Decay results for weak solutions of the Navier-Stokes equations on Rn, J. London Math. Soc, 3 5 (1987) 303. on R , J. London Math. Soc, 3 5 (1987) 303. 19. 19. V.I. V.I. Yudivich, Yudivich, Method Method of of linearization linearization in in hydrodynamical hydrodynamical stability stability theory, theory,
(Rostov (Rostov Univ. Univ. 1984). 1984).
106 COM MP PA AC CT TN NE ES S O OF F S T E A D Y COMPRESSIBLE C O M P R E S S I B L E ISENTROPIC NAVIER-STOKES N A V I E R - S T O K E S EQUATIONS E Q U A T I O N S VIA V I A THE T H E DECOMPOSITION M E T H O D (THE M ( T H E WHOLE W H O L E 3-D 3 - D SPACE) SPACE) A n t o n i n Novotny Department of Mathematics, ETMA, University of Toulon and Var, BP 132, 83957 La Garde, France We investigate steady compressible isentropic (Il(p) = p 7 , 7 > 2 where II is the pressure and p is the density) Navier-Stokes equations with arbitrary large external forces, in the whole space in three dimensions, when the prescribed mass is finite. We prove a general compactness of sequences of solutions to these equations. We also study the summability, the regularity and the compactness of different quanti ties in the flow field. We thus get similar results as P.L. Lions in 5 - 6 , by a different method. The main tool in the proofs is the "supersonic" method of decomposition which is a global version of the similar method proposed by Padula and myself in 10 for studying compressible flows near the equilibrium states.
AMS classification: 76N, 35Q Keywords: Steady compressible Navier-Stokes equations, Navier-Poisson equations, weak solutions, global existence of weak solutions, Helmholtz decomposition 1
Introduction
The present paper is devoted to the study of the weak solutions to steady com pressible isentropic Navier-Stokes equations. It has two sources: 1) The works 5 6 - , where P.L. Lions proved, among others, the compactness of the steady compressible Navier-Stokes equations in two and threedimensional bounded domains and in the whole IR3, provided the mass of the gas is finite. 2) The paper 10 where M. Padula and myself have introduced the method of decom position for solving the steady compressible Navier-Stokes equations. It is closely related to our paper 9 , where we have applied the method of decom position to steady compressible isentropic flows in two and threedimensional bounded domains. Here we provide the details of the similar approach in the 3 whole IR .. wnoie in The The equations equations describing describing the the motion motion read: read: -HxAv - (//i + /i 2 )Vdivv + Vp1 — pf -HiAv pi - pv • Vv,
in
IR3,
107 div(pv) / pdx = m, iJ nR 3
= 0,
in
IR 3 ,
p>0
in
IR 3 .
(1.1)
Here p and v = (t>i, v2, V3) are unknown functions (p is the density and v is the are velocity ) while p\,p2 (p\ > 0, p2 > > — -§A*i) §A*i) are given given (constant) (constant) viscosities, viscosities, m > 0 is the given total mass and / = ( / 1 , / 2 , / 3 ) is the prescribed density of m external forces. We give the alternative proofs to the following results of P.L. Lions 5 - 6 : 1) Esse] -0 For 7 > 2 : Essential compactness properties of sequences of solutions to the system (1.1) which wh: are sufficient to pass to the limit in all nonlinear t e r m s (the most difficult term in this context is the term p1). This achievement is supposed to be the main ingredient in the proof of the existence of weak solutions* Moreover: □ T h e incompressible part u of the velocity is more regular then the velocity itself (the first gradient of u belongs to certain Sobolev space with positive fractional derivatives, i.e. it is compact in a convenient Lebesgue space). In particular, Vw does not allow the j u m p s on twodimensional surfaces of IR 3 . As a consequence, the vorticity rotv = = Totu rotu has the same prope properties. ;icity rotv □ T h e effective pressure V) V := p11 — (2//i (2pi + ^ 2d)divt;, i v v , is imore regular then re V, p1 and diwv themselves3s (V belongs to certain Sobolev SD space with positive fractional derivatives; i.e. it is compact in a convenient Lebesgue space). In particular, it does not have j u m p s on twodimensional surfaces of IR 3 . See bee Theorem I n e o r e m 5.1 o.l (a), (aj, (b) (bj for lor details. details. p 2) For 7 > 3 : Further regularity and summability of density up to p E L (£l), v e HQ,P,P{JR33)
, 2 (IR33 n #i70011,2 (IR ), 1 < p < 00. See Theorem 5.2 (a), (b) for details.
All these results were already proved in 5 - 6 , by a different m e t h o d . T h e m a i n goal of this paper is the application of the m e t h o d of decomposition to problem (1.1). It gives another look at the equations. According to this approach, the compressible Navier-Stokes system is split onto the three simpler equations: 1) A Stokes-like system for the incompressible (solenoidal) part u of the ve locity field and for the "effective pressure" "P. 2) A Neumann-like problem for the compressible (potential) part V<£> of the velocity field. 3) A (nonlinear) transport-like equation with r.h.s. V, for the density p.
"However, we do not prove the existence of weak solutions. In this respect, we leave open ""However, only one thing: the existence of appropriate approximations to system (1.1). This problem is easy to solve for the nonstationary version of system (1.1), cf. 6 , n o r 8 . For the steady case, this problem is more difficult and as far as we know, it is still an open question.
108 2
T h e m e t h o d of d e c o m p o s i t i o n
Let us look for t h e solution (p, v) in the form /?, where
(2.1) (2-1)
v — u -f+ V<£>
3 divu divu = =0 0 in in IR IR 3 ..
(2.2) (2.2) (2.2)
T h e n system (1.1) reads -pxAu -pixAu
+ W = pf - pv -Vv
in
IR 3 ,
divw = 0 in E
3
(2.3) (2.3)
where (2.4) (2.4)
V = p1 - (2pi + Aiajdivw /i 2 )divi; which is equivalent transport equation equation for for p p which is equivalent tt o oa a nonlinear nonlinear transport
(2.5) (2.5) (2.5)
pi = *P. V. p* + (2/ii (2AXi + fi *2i)v 2 K•VVlnp lnp = Finally, (p is governed by uy A
in IR 33 .
(2.6) (2.6)
T h e use of t h e above decomposition goes through the entire work; more pre cisely, we employ only a part of it, in particular equations (2.3)-(2.4): 1) t h e elliptic regularity of (2.3) is used for obtaining the compactness properties of Vw and V (in this point, our approach differs from 5 - 6 ); 2) t h e identity (2.4) is used for transferring the compactness properties of V to the strong convergence of p (in this point, we essentially use t h e ideas of 5 - 6 ). 3
F u n c t i o n a l s p a c e s ( P r e l i m i n a r y r e s u l t s I)
Here we give a list of functional spaces employed, recall their definitions and basic properties needed in the sequel. ( I ) Spaces of distributions We denote by V the space of infinitely differentiate functions with com pact support in IR 3 and by V its dual, t h e usual space of distributions. £S\M\^\J
KJ «_«.J-r^ v-r J- v>
XXX
JXU
U 1 1 U
J
XUU
\XUUIXi
U11V
U.UUUIJ.
Ul/UIVV^
\S J.
UlUUllL/UVlUliOi
( I I ) Lebesgue spaces, Sobolev spaces a n d H o m o g e n e o u s Sobolev spaces j y 0°*^ ^ (3I)R 3 ) t h e Let k = 1, 2 , . . . , 1 < p < oo. We denote by L^(IR 3 ) = W
109 usual Lebesgue space equipped with norm II • ||o lP and by Wk>p(JR3) the usual Sobolev space with norm ||-||fcjP = Ylm=o ll^ m , llo,p- Further define, p for 1 < p < oo, the homogeneous Sobolev spaces H^tf^IR (JR33)) := p l | V ' 1 | o ' p (here the superposed bar with the norm denotes the completion in this n o r m ) . This is a Banach space with norm | • |i >) Pp = ||V • ||o, ||o >pp . Recall several well known properties (see x , 4 ): □ Homogeneous Sobolev spaces. We have H1,P(JR3) = {u : u G L3-p (IR 3 ), 3 P Vu G G L LPP(JR (JR33)} )} (1 (1 < < pp < < 3), 3), H^ H^{JR = (IR 33 ), Vti G Vu {JR)3) = {u {u :: uu G G Lf Lf 0C 0 C (lR ), Vti G PP 33 L ( I R ) } | ] R I (3 < p < oo) where l^i denotes the factorisation with re spect to the addition of a real constant. P Dlmbeddings. Let 1 < p < 3, then H^ff^Ht (JR3) P
3
ding H^ (JR ) k
p
C L ^ ( I R 3 ) ; the imbed
33
C L? (1R ), ), s G [ 1 , ^ ) is compact. If {k - r)p > 3, Lf 00CC (lR 3
then W > (JR ) C Crr < |||| •• || || ^^ ^^ || ll '' IIS^, 118^ = ^^ ^yy >r < R3)3 )) H R 33 )) C R 3 3))) aa and 118^ aa = L*(JR kk p§ p§ p§ 3 333))* to W ^ ' ^ I R 33 ) k p DDual spaces. Let < < oo. Dual space (W > (JR DDual,1 spaces. spaces. Let Let 111 < < ppp < < oo. Dual space (W > (JR ))* to W ^ ' ^ < oo. Dual space (W * {IR (IR))*))* to ^ ^ ' ( II R ) k,p f 7 k,p 33 k,p ((l/p' l / p + l / p = 1) is denoted by W~ (JR ) and the du (l/p di: + l/p l / p = 1) is denoted by W~ (JR ) and the corresponding correspc P 33 0 0 p 3 33 is |||| •• ||_* ||_* Obviously ((W°> = L^(IR L (IR ). space ality norm ||_ (W P . Obviously fc|P norm is W ^(JR ^ ('(JR 1 ))* R ))* ))* = ). ]Dual spac ii P 1 P3 lp ' (JR333)) corre {l/p' to ^H^*'^ ^'-(( I'(IR I)R R 333)) ((l/p' l / p ; H-l/p = 1) is denoted by H~tf-^IR and the c om and - i ) Pp . For 3/2 < p << oo, we have imbeddin imbedding sponding duality norm is | • ||_i
L^(JR3)
C
P H-^P{]R CH'^ (JR33).
( I Il Il ) R e m a r k s for n o t a t i o n of f u n c t i o n a l s p a c e s n o n in a Banach □H If not introduced in items (I)-(II) of this section, a norm D space ipace X is denoted || • ||x • do □H All norms refer to IR 03; if a norm refers to another D domain (say Q), ar hen we indicate itt as another index at the norm; then e.g. || • \\k ||fc,)Pp means noi k p 3 ||* >Pjn is a norm >pp(Q). ( norm in Wkk> ai norm in W > (JRI 3)) while || • \\k,p,n Similarly, 3 3mit systematically the symbol symbc "IR "" i] inn the sequel, we omit in the notation kp liP ss; e.g. Wk,p ' )f functional spaces; sta of or H1,p mean, iif not stated explicitely 33 1 P 3 333 \ \ ™ n >p(m \ otherwise, otherwise, W*'"(IR ) or J JH^ ^tf^IR f f(JR i )).. distinguis between the □H If not stated explicitely otherwise, we do not distinguish D k,p spaces of vector and scalar valued functions; e.g. both both Wk}P spaces (Q) (Q) and k p m k p W > (Sl]JR )} m G IN are denoted W > (Sl). T h e difference is always clear from the context.
110
4 4.1
Auxiliary linear problems and elements of convex analysis (Pre liminary results II) Auxiliary linear problems
In the proofs, we often use various properties of Dirichlet problem for the Stokes operator, of the Dirichlet problem for divergence operator and of the Helmholtz decomposition. These results are nowadays considered as mathe matical folklore. We recall here their statements. We start with the Stokes problem: 3 -/iiAu + W — -T j - in —p\i\u-\-yy in IR itt 33 divu divu — in IR IR (4.1) (4.1) Hiv?/. = = gng in in TR ... 3 4 The following two theorems are due to Cattabrigga (see also Galdi for different variants of it): Lemma 4.1 (Stokes problem in Sobolev spaces, weak solutions) Let T E H~1}P, g E Lp, 1 < p < 00. Then the problem (4-1) possesses just one solution (u,V), u E H1,p, V E Lp which satisfies estimate
(4.2) (4.2)
l«|
Lemma 4.2 (Stokes problem in Sobolev spaces, regularity) p k k+1 t Letkk = 0,1,..., 0 , 1 , . . . , l' , the problem (4-1) possesses just one solution {u,V), '*, Vu k+1 pp E L\ V P € EW >' which satisfies estimate V £ Wk+1 2 2
2 ||Vt/||o,t •\\o,t++||V \\V u^ |U,pPp-f||7>|| \\V\\ ,t ||W|| ||V^|U + \\VV\ < ||V«||o,, ++|p>|| W 00,t 0+ U |U, f c , p)P< ».(\T\ 1 .-4II^IL_-4ll/ilL. J-ll/ill, c(|.F|_ M + \\r\\kiP + \\g\\0,t + ||j||*+i,p).
(4.3) (4.3)
Next we investigate the divergence equation: diva; in IR divw — = gg in IR33..
(4.4) (4.4) 2
The following theorem, as it states, can be found in Bogovskij . Lemma 4.3 (Div equation in Sobolev spaces) Let 1 < p < 00 co and g E Lp. Then there exists at least one solution u E HliP of problem (4-4) which is such that M i l P < cIMkp. c||flf||fc>P. MI,P
(4.5) (4.5)
Last problem of this subsection is the Helmholtz decomposition, i.e. the prob lem to find find u (a vector field) and , divu diwu = 0. 0. =u + (4.6) (4.6) 4 The following results follow from e.g. Galdi .
111 L e m m a 4 . 4 Let 1 < p < 3 and v G # 1 , p . Then T/ien there tfiere exists just one (u,(p), u G#1,p, (p G Hl)*-p , V2<£> G L £ pp satisfying problem (4.6). Moreover we have estimate IIV^IO.JE. clvli^ MI,P + livdlo, .E. + + l|VVl|o, l|vVllolPP < < c|«|i lP
(4.7) (4.7)
' 3 —p
,p L e m m a 4.5 Let Le£ 1 < p < oo and and vv GG W W1liP .. Tften Then tfiere there exists exists just just one one(u, (u, (p), (p), 1,p u G WliP ,i (p G H1,p, V 2 ^ G L £ pp satisfying problem (4-6). Moreover we have estimate
IMIi,p + |||vVlkpp „/.£ 4.2
(4.8) (4.8)
S o m e elements of convex analysis
Here we recall two theorems: 1) about the weak lower semicontinuity of convex functionals with respect t o the *weak convergence, 2) about the close range of the monotone operators. Denote Cg(IR 3 ) a Banach space of continuous functions with compact support in Q (equipped with norm m a x ^ ^ s |^(^)|) and denote by M its dual. We have the following statement (cf. e.g. S v e r a k 1 3 ) : L e m m a 4.6 Let m G IN and Q, be a domain ofJR . Let G : IR m —>■ IR 1 be a lower continuous convex function. Let the sequence of functions un G L 1 (IR ; I R m ) converges weakly * in M to a function u G L]oc (IR 3 ; I R m ) . Then < liminf / G{u G(u G(u) < G(u n)dx. n)di n)dx.
(4.9) (4.9)
T h e second statement reads (see e.g. 7 ) ) : L e m m a 4.7 Let AX. oe be a a rejiexive, reflexive, separaoie separable nanacn Banach space ||\\'\\xj •• \\x) \\x) wun with with auai dual dual J\ X* X*(now (norm (norm space (norm (norm \\ an || • \\x*) d M : X — > X* be a bounded, monotone operator (recall that \\x*) and M : X —> X* be a bounded, monotone operator (recall that M A is monotone iff < Mu Mu -— — Mv,u Mv,uu -— —v v >x*> >x*> 0 Vu, Vu, v E GA X,, wnere where < < •, •,• •, •• >x* >x* J < vu, v monotone iff Mv, >x* > U 0 G X, where < >x denotes X* lality between between X, X,X* such that that (i) (i) for for any any u,v,w u,v,w G wtes the the duality duality between X, X* ))) ,,, such such that (i) for any u,v,w G X, X, *< 11 11 <M } X <M >x * ~> M(uu + w >x* V \\^ >x* is Z; + -v), -v),w >x* is a a continuous continuous function function from from IR IR 1 — —> >•• IR IR 1 ;; (ii) (ii) <M\\ \\'j\\ * 0 as as \\v\\x \\v\\x — —> >■ ■ 00. oo. Then 00. Then M M is is a a surjective surjective operator, operator, i.e. i.e. to to any any zz G G X* X* thei there exists v £ X such that z = Mv. 5
Main Theorems
Here and in the sequel, we denote by K a generic positive constant, which is, in particular dependent on m , ||/||o,oo-
112
T h e o r e m 5.1 Let Let 7 > > 2 and m > 0, f / £G L°°. Put q = 3(7 - 1) (7 < 3), 7 + 1 < q < 2 7 (7 7 > 3). Let {(/>n, tt>>nn)}£^_i ) } ^ i be 6e a sequence of weak solutions^- to problem (1.1)] (l.l)i_3 £ Lq, vn G £ H1*2. such that pn G there exists a subsequence ( a ) Then Tnen ^nere p £G Lq, v G #H1l)2, 2 5wc/i such i/zai that p)nn'-> ' —t -»/> pp
weakly wecuviy
vt>n'n' ->^ vv
Moreover
(p,v)
6iN //ciN {{pn ,vn )}n/GiN
q q in 111 L IJ' and ciiiu
weakly weakly weakly
satisfies
aann f l 7 a
d
a couple couple
(p,v), (p,v),
33 flfl in 111 ^ i ^ (o c H ^irt ) ) D 1fl I L L (1 {l2 3), if H L ^ ( I R in H and strongly i n LJ*^ (IR/ c r in ^ and strongly i n L ^ c ( I R ) , 1 < r 2 < 6.(5.1)
strongly stiun^iy
( l . l ) i _ 3 and we have
estimate
\\p\\o,g + IM|o,6 ++ \\v\\o,e | |v|i,2 u | i | 2 <S ifdl/llo.oo, #fL(\\f\\o ( | | / | | 0 |}ooo,rri) o , m) << oo. 00. \\P\\o,q \\v\\o,e ++ |v|i,2 oo.
(5.2) (5.2)
iflt ( b ) Put = the quantities quantities V, u, (p
11^110^ + + I11^110^+11^110,6 l|P|lo + 1^^110^ l|V^||o,« + H|o,g+IN|ol6 + + l' -fe g+3 | | V U« ||||00|)i33^^. + ||V ||o,2 ||||VVUUH | ||oo,,22+++||||V ||V V222«uu|||||0|00,,f)el. + ' O + Qf
' 3+g
||Vy>||o,6 < K(\\f\\o,oo,m). l|V^||o, ||VVl|o,2 X(||/||o,oo, m). 6 +L l||VV||o,2 \\T7,n\\ l\7 /nlL « ^ A' ril f I L rr^ 2
/n particular,
r
(5.3) (5.3)
rott> G £L ,*+ rott; '+ 33, Vroti; G L Ly
and and ||Vrotw|| K{\\f\\ ||roti;|| 0 Jj^L^ + ||Vrotw||o,« ||Vrott;||o^ ^ ( | | / | |0tOO o , o,m). o , "»)■ ||rot«|| 0 ,« < A"(||/||o,oo,
(5.4) (5.4)
' 3+g
T h e o r e m 5.2 00 Le£ 7 >> 3, m > Le^ > 0 and ana1 f / G L L°°. . Then there exists €0 €o €Q such that if 2^ 2y — eo < n n q < < 27, we have: Let { ( p , v )}^L1 be a sequence of weak solutions to problem q (\ l . l ) i -—«-» 3 such that p1 n G U> H L L*, ,7 vunn ^GH # 1*, p 0■ H1'2, 7 max(6, 2I7/) <. Jrp <^ 00. — — \ > /-L
*We say that the couple (p, v) is a weak solution to problem (1.1) 1-3 if it satisfies integral ^We ;ies m W : V rf:r 2 identities ^1 JJRR33 VVv : V£dx (^1++^ ^2) (m J R3 3 divvdiv^da; divvdiv^a: divvdiv£dx --J JR R3 3 p^div^da? p^div^c/x p^div£cte == J R 3 p / • ^rfx £cte + ^ ++ (^1 ) JR JJD3 pv ® v'-V^da;, V£d:r, V£ V^ G V, j 3 pi» • Vipdx = 0, Vi/> G 2) and conditions (1.1)3.
113 (a)
n n Then the v) (a {(p uiz couple cuupie (p, \p,v) (a weak weax limit iimu of oj a a chosen cnosen subsequence suosequence \(p {{p ,1\v ') p In'eiN'ciN see (5.1)-(5.4) (5.1)-(5.4) in Theorem Theorem 5.1) 5.1) is is such such that that p p G GL Lp,, 'v G N'ciN -- see in W1* n C ° ( I R 3 ) , m a x ( 6 , 2 7 ) < p < oo and satisfies, besides (5.2), (54), (5.4), estimate
IMIO,P I M| |iI,J PP <<# (A| r| (/ l|l|/o| | ,oo, oco, ,m m)). IMIo>P + + N
(5.5) (5.5)
p (2.4)). men V G GW W ( b )) Kjonsiaer Consider uie the quanzuies quantities r,(p,u V,
WP\\I,P K(\\f\\0tOOm ,m). \\ni,P + + l|w||o,r I||«l|o,r M k r + ||Vti| V « |1>p HV^IIij, | | V ^ | | i > p < K(\\f\\o,oo,m). K(\\f\\o,oo, ). 1l > i Pp + ||V^||i 6 6.1
(5.6) (5.6)
P r o o f of T h e o r e m 5.1 A n equivalent formulation of m o m e n t u m a n d continuity equa tions
It is possible t o verify t h a t eq. ( l . l ) i implies the identity pi p i / 3 Vv : V£dx + (^1 ( p i + P2) /
3 Ju JJR33 JTR
3 Ju Ju Jn3
-1— —1—
/I
1 p~ div{pt)dx= >71_-f7-11-1div(p£)
7 -- 1 1 JTR3
pf-£dx-\pf-£dx+ pf^dx+f
Jn3
I
divudiv^Gta + divi;div^da: divvdiv£dx
(pv®v):\7£dx (pv®v) :V£dx (pv®v):V£dx
(6.1)
Jn3
, L ([it), ft), P Hfy (1 < valid any p,v,c,, p , v , f , sucn such tinai hat v v tG #^n 1 , 'r ([\i), 9), £ < E #n £ 0Q1 , r '(^)> 11a for lor any c,? G ^ p € *z L>*\I&) \± r 11 r < + ^ div(p£) L**^"*, p 0 v q,rr < o o , i h ^7 = 1) such t h a t div(pf) G L^T^, p / G P ^ " pv 1 1- '^r ' , . -*. !) t y (8 V < o o , i - h ^7 = 1) such t h a t div(pf) G L^T ^, p / G W ' \ pi; 0 ?; G GL q ,r and eq. (1.1)2 implies, forp£L q (1 <
pdivi; = —div(plnpi;), pdivu — div(plnpv),
o5ddivv divv = = -^—A\v{p - ^ - d i-dW(p vsv), ( p d^v),) , pp'divt; 0o — — l1
rr >>
--,,
ag q--— 11
l <S ^^"r~— 0 < \s^^ L,,S^ 1,, 0 < <S 5<
(6.2) (6-2) (6.3) (6.3)
see35 --6 .. (Here a n d in the sequel, for (a,j), l<7 ), b = {bij), lor two tensors a = (a (aij), [bij)} a : b means meai thee sum aijbij a-iibii - s u m m a t i o n over repeated indices.) 6.2
A priori e s t i m a t e s ( 1 )
Here we prove t h e following lemma.
114 n n nn L e m m a 6.1. Let 7 > 2, m > 0, / G L°° and let {(p ,,t; v)}£° )}™ {(pnn,u , 1 , {pn,vn) G )}~ =1 =1= q9 1,2 LL x i 7 , fce be>aasequence sequenceofofweak weaksolutions solutions totoproblem problem(l.l)i_2 (l.l)i_2satisfying satisfying (1.1)3. (1.1)3. Then n (6.4) \\p I|K||o,6 K I||o,6 Io.6 + + K K ||hli , 2 < < A d(H/lkoo, l / l l o . o o , m). m). \\pnn\\0,a l|p \\o, ||o,g + ||« If'K(\\f\\o,oo,m). (6.4)
Proof of Lemma 6.1 For the sake of simplicity, we omit indexes n at (p, v)\ we write simply (p, v) instead of (pn,vn). Putting in equation (6.1) £ = v, we get, after the use of continuity eq. (1.1)2(6.5) (6.5)
l|Vv||o2 pf-vdx\ ll^llo Z9/ '*v v"^x lx l llv^llo 2 < cl\ / P/ JR3
The r.h.s. can be estimated by using Hoelder inequality, Sobolev imbedding and interpolation of Lebesgue spaces | J^pf • vdx\ < ||/||o,oo|M|o,6||p||o,6/5 < / C | | / | | O , O O | | V H | O , 2 | | H I M A HPII^ C||/||O,OO||VH|O,2||HIM | I P I I ^ < c H/llo 1 eo||V V ||o 1 2|H|^ where q > 6/5, A = Hence g^TT(6.6) ||Vw||o, (6.6) l|V«||o,22 < c||/||o,oo||p||^. C||/|| 0| oo||p||^. Suppose q > 7 and consider it in the form q 0. Let 1 < 5s < < 00. 00. Acording to Lemma 4.3, there exists a solution u G H1*8 of problem IIT-7..II
^
- I I JMI
II _ l l A
divuj = pa in IR3 divw which satisfies estimate (6.7) (6.7)
|\\Vu\\o,s
I I " ) "
I l»
I I V,«J13
Multiplying equation (l.l)i scalarly by u>, we get IMR+a =- ""^1 " ^ i // IMR+o
JR
3
v : (/L/i + + A*2) ^ 2 ) // ^^ v : ^Vu;cfo ^ ^ -- (^1
JR3
/
.-/R /R3
pfudx+ pf>wdx +
JR3
divvdivwdx + + divvdivwdx
(pv®v)
:Vudx.
(6.8)
Now, estimate each term at the r.h.s., essentially, by using the Holder inequal 1 ity, estimate (6.7) Lq. The and yyj.ij and a-ina the ijiic interpolation liiucipuiauiun between uciwccn L u and diiu ±J*. ± lie first iiisu emu. the second terms at the r.h.s. furnish, for 7 > a: p\ f Vt; : \?u)dx + (pi u3 arms at the r.h.s. furnish, for 7 > a: p\ fu3 Vt; : \?u)dx + (/ii 4-4a A pL diwdivwdx < cc||V«||o, | | V « | | 0 , 2 l| 2| p|H|^ | | S «a < rif^divvdivudx livwdx < | | V « | | o , 22||V | | VW ||0,2 < < ||V«||o ||V«||o,2||p||2* < c||/|| c | | / | |0o,oo|Hlo , o o | | H l o ,t , t A2)fn3 W ||o,2 z3g \\w\\ , / ?? < << ||/||o,oo The third term ■wdx\ wdx\ <<\\fp\\ \\fp\\ av/a /» \\u\\ \\u\\^j. ||/||o,oo ;rm gives |\f/R3Rpf-udx\ q/a 3 a,/ 3 pf • U
' 44g/or q / a -—3 3
U
'' 3—q/tt 3—3-q/a q/ct
< cc||/||o,oo c||/|| I I P I I J ^a and and the the last last term term yields, yields, for for 7 7> > 2±«: 2±«: 0,oo \\p\\\+ \\p\\ IMI0t0>s gl* ?f% l|Vw|| H^V|a|, l00k>,A *i < ll/llo,oo IHIo^" 2±^: q \f | /M3 (3^® ® Vv)) : Vudx\ Vwc/xl < < |H|o, ||p||o,7 a||^||g, ||w||g,66||Vu;|| ||Vu;||00i2^± i . < c||p|| c|H|0|9 ||V V ||g, R 3(pv 7+ +a 0 ) g||Vt;||g |2 ||p||? 2 |H|^ >g +2A / < cC'll/llo,ooll/'llo^ ||/IIO,OOIIPIIO^ + 2 A --
These
estimates and (6.5) furnish
H k , << ^(||/||o,oo,m), IIMIo.g M # (X(||/||o,oo,m), ||/||o,oo,ro),
||Vt;||o ||V«||o,, /f(||/||o,oo,nx). ||V»||o,,
(6.9) (6.9)
115
provided q = 3(7 - 1) (2 < 7 < 3), 7 + 1 < q < 2j (3 < 7 < 00). Lemma 6.1 is thus proved. 6.3
A priori estimates (2)
We continue the calculus of a priori estimates by using the decomposition. We prove Lemma 6.2 Let 7 > 2; define 6 as in Theorem 5.1 (b). Let f E L°°, m > 0 n {(pnniv, nt;)}^L )}£° and let {{p {pnn,vnn) G Lq x if # 1 ',22,, be 6e a sequence of weak solutions li =1 , (p to problem ( l . l ) i _ 2 satisfying (1.1) 3 . Le* Let un,
iimk#m 0k, ^3, ++ iivnk* inks+n«io,,+nv«"ii^. #. ++ ir'|| ||W>"|k* + + inks+IKIIO,,+nv«i «" o,6 + «" o,, + l|V«"|U ' 3+g
3+g
n nn ||VM H Wnl||o,6 o . e ++ ||VV l|VV /^(H/lkoo, ||V« ||o,2 << K(\\f\\o,oo,m). V« n ||||00,2 + II||||W W || || oo ,. «« ++ ||V^ HV^Ho.6 l|VVllo,2 K(\\f\\o.oo, m). m). (6.10) (6.10)
Proof of Lemma 6.2 For the sake of simplicity, we again omit indices n at (p,v,V,
C c
9+6g »» 9+6g
\\p\\o>q l | V ^ | | 0 ) 2 |Mlo,6 ||v||o,6 < < A"(||/||o,oo,m), A ( | | / | | o , o o , ™ ) , ||pHlo \\PVV\\Q _3g_ _3g_ < < c||p|| C | | / 9 | | 0g|M|§ g||v||g>6 \\p\\o >q l|V^||o,2 0| | >6 ' 3+g r < A (||/||o,cxD, (||/||o,oo, Tn). ^ ) - Lemma 4.2 applied to equation (2.3) thus furnishes
n ll^n|lo ||o,« ++ IKIk,H iinio,^ + nvnio,* iKiio,gg+ 0 IA.+ll^
iini ,^ + iivpio,* + iKiio, + ' 3+g
| | V7uunn\\j*_ "||J| > | ^SL. + + ||VV>|| ||VV*|| , 9
(6.11)
n |||Vv» | vy"||o,6 V ^ |B|||o,6 + ||VV ||o,2
(6.12)
3+g
Inequalities
follow directly from Lemmas 4.4, 6.1 and Sobolev imbedding. Lemma 6.2 is thus prooved. 6.4
T h e limit process
By virtue of Lemmas 6.1-6.2, we can choose a subsequence of {pn, vn} (denoted lnpn —> again {p n',, vn}) with the following properties: pn —¥ p weakly in Lq, ppnn\np n1+1 7+1 q q n nn 7 1 q n74 ) ->-» weaklyininLL^ ^, ^, ,(p") (/9) )" ->-^ ->/ /p^ ^ +11 kly ininL L, , Kq^ ~pweakly {p 1 2 3 nn 3 weakly inLiL^*q/^+ _> L[ //(^^l\+)1,) ,^n ->vvvwweakly weakly in ^0 0C(2C(IR H )),,11< 1< FF1 1- -2a2and in L[ (IR3 ), e a y y iin nin#1,2 nand d s tstrongly rstrongly 0 ngly in
116
6; VVn -> V P weakly in Le and Vn -+V -> V strongly in £[03C(IR3), 1 < r 3 < ^ ; nn 1 22 n n u —>> t* ■w u weakly — >> Vu weakly in U iiv/rA44,,, min(2, ^^- ) < r 4 < tkly in H '*,,, Vu Vu ->L v u —r v a wcciKiy i n iniii^ 222 nnn 22l u weakly in L^, 9 Vw nn — 3 (IR max(2, ^ ) , V u u -* -> V Z/, > y ■ Vti Vti strongly strongly in in L[^ (IR33)) and and u weakly in L , Vu ->■ —> Vti Vtistrongly strongly in L[^ V i/ -> V w L[ o ccc( n 5 3 n 1 6 5 3 n u ->» -+ u strongly in L[ (IR ), 1 < r < max(6,g); (p -> -> ^2 2 weakly nn n 2 2 2 (IR 3),), V 2ip n -^ 2y> 2 . .Moreover, V(p -► Vv? -^ Vy> V9?strongly strongly in in L^ L[^ (IR V ip ->■ V V ^? weakly weakly in in L L Moreover, it it is is i t r r m d v in ^ C C^TR V X7 LD -+ X7 CD w e a k l v in L . M o r e o n nn nn n n nn n n n n n nn nn easily seen that V pp -» -+ P/?, Vp, /? p vt; ®t) v -> -+/9v pv0v,v,p pv v ->->/ov, pv,p pln/9 \np ?; v ->-> plnpv at least in the sense of distributions. plnpt; The only thing which remains open is the strong convergence of p. To prove it, we closely follow5-6. The identity (2.4) with pn,Vn,un yields, when n -> oo, 71 V — ~(F p7" ——(2//i (2//i++//2)divt>, //2)divt>,hence henceafter aftermultiplication multiplicationby byp,p,Vp Vp==p p/? p——(2//i (2p,\++ 7 ^2)/>divt; )/>divi; and finally by (6.2) Vp — /> />+(2/ii -h/X2)div(/>lny9v). /^/>+(2/ii + /Z2)div(/?ln/?t>). Integrating the last equation over IR3 we get, after the integration by parts in the second term at the r.h.s.: p~* pdx. / Vpdx — - \ ~fppdx. (6.13) JJn R33 JJu R 33 n nn n On multiplying (2.4) p On the the other other hand, hand, multiplying \^.^±) (2.4) ^(written (written with ppn,v ,v,v)) ) by by p n ,,, then then using u , xiiuibipiyin^ w r i t i e i i with witn p uy p m e n using us nn nn n n nn 7+1 n n 7+1 the we (2/ii the identity identity (6.2) (6.2)i (written (written with with ppn,v ,vn), ), we get get V Vnp pn = = (/? (/? n)) 7+1 -- (2/i (2^ : + + n ^2)div(p ln/9nnt> f nn)) which ^2)div(/9nln/9 which gives gives after after the the integration integration over over IR IR and and the the passage passage to to the the limit limit n n— —> >■ ■ oo: oo: / Vpdx = I p~i+ p^+lxdx. dx. p7+idx. (6.14) 33 3 Ju Ju JiR Jn Estimates and Estimates (6.13) (6.13) and especially especially (6.14) (6.14) require require some some explication. explication. We We easy easy ver ver nn nn ify that \\V p \\o,s < #(||/|| oo,m), S G [1, m i ^ ^ , ? ^ = ^ ] . Morover, since 0 | ify3 that \\V p \\o,s < #(||/|| G [1, m i n ^ , ^ = % Morover, since 0 | oo,m), S ^ P \\o,6 ^ ^ u i ^ lio.oo,"*;, « t [ i , m m ^ , ' \. iv 3q Vn nn 1+1 1+1) • oo (eventually with a chosen subsequence): he limit n —>• oo (eventually wii ter the passage to the with a chosen subsequence): tssage 1+1 77 + 1+11)(pdx + 11 = div(plnp^) ffn3 (Vp — p — f div(plnpv)(pdx. From here Vp — u3 (Vp — p )(pdx — f div(plnpv)(pdx. From here Vp — — ppp1+1 = div(plnpv) p^+ )(pdx — J ^ div(plnp?;)v?^. From here Vp n3 u3 3 3 a.e. a.e. in in IR IR3;; the the integration integration of of the the last last inequality inequality over over IR IR3 furnishes furnishes (6.14). (6.14). By By (6.13), (6.14), we finally get the identity (6.13), (6.14), we finally get the identity /
J R 33
(p^7TT+ 1-7-rP)dx (P P71p)c?x p)dx = = I0.
ffi.lM (6.15)
Next we use Lemma 4.6 with G : IR2 -+ -> IR1, G{t, G(t, z) = = \t\\z\ |t||z| and with tn = n 7 nn (p ) , znn = = p .. We thus get , for any open set K, C IR3, Xc(p Xc(^ 7 + 1 -~^p)dx > > 0. Hence pip = ~p+i pi+i a.e. in IR JTp IR33.. (6.16) Now, define Mz = = z1 (z > 0), Mz Mz = - | z | 7 (z < < 0). and apply to M Lemma 7+1 pi = —p 4.7 with X = L ; the surjectivity thus yields ^Jp = pi1 a.e. iin IR3. The last n 77 33 —±ppininLL(IR (IR) ) and andconsequently in equality yields strong convergence of p -> 3 l Z/ 1l(IR q,l 11<<<^i 1.Theorem Theorem5.1 5.1isisthus tfocOR3) ) nOLfl ^ 3 ) ), ' 11< «i nri<<<777+++1. 1. Theorem 5.1 thus proved. proved.
117 7 7.1 7.i
P r o o f of T h e o r e m 5.2 Bootstrapping from p E e Lq,v ?>37
E e H1,2
t o v E Lpp,, 6 < pp < oo, p E L*, Lg,
L e m m a 7.1 Lee Let 7 >> 3 3, , 6 < p < oo, m ra >> 0 0, , / E L°°. T/ien Then there exists eo swc/i £/ia£ that to < < q < 27, then £/ien we have: /icme; Let Lei { ( p n , uvnn)}^ )}£° 6e a sequence lat iif/ 22*y 7 — eo vn)}^ =i ==i fre tn problem nmhlp.m. (*\ farina (\ l^o and surh that (pn ,i> of weakb solutions solutions to ( l . l^\^ ) i _ 2n satis satisfying (1.1)3 and such , i>nn) E q pp 1 ,22 1,p L HL C\L x H # ' fl # . T/zen tfzere ariste exists 9q > > 37 3~f such that n |l okO,?^?++ IIlKlK^II|l|oko,p,PPH+- ||V«"|| | | V ^ | | 0ol 2, 2 < AAr(||/||o,oo,m). '(||/||o,oo,m lI I|PPpWn| H A'(||/||o,oo,m). IIV^IIo.2
(7.1)
Proof of Lemma 7.1 For the sake i>); (p,v) >axe n of 01n simplicity, simplicity, we we omit omit indexes inaexes nn at ai (p, [p, v); v)\ we we write write simply sn n,vn). instead of (p Clearly, (p,v) satisfy estimate (5.2). Equation (6.1) with )f (p ,v ). Clearly, (p, v) satisfy estimate (5.2). Equation £ = t;|t;| 5 _ 2 , 5 > 2 furnishes
^il|v +/ii||v(H*)||2 d™ i vM,. H | |^,|^22 <s /'iH Vv H^||g, H t ^ 2l 0l )k222 + M >2 |v.M^II A«il|v(|«|*)|li p±gi^^| Id| I|i\\aivviv\ ^ ^| ^*||gH l2 + + ^i||v(H*)||§ /ii||vuwMj||0>2 + ^ iY 0i2 2j „ | , I / „ I J : „ / . . I . . I Js-5222\2J „ I - f l / ~t <s c{\ c{\ pf ■■„.l»,l«-2 v\v\s-~s2-2dx\ dx\ ++ II /[ p^div(t;|t;| pidiv(v\v\ - -)dx\. < dx\ )dx|. [I pf v\v\ pMiv(t;|i;|*)da:|. Jn Jn
in
o\ (7.2) (7.2)
T h e Sobolev imbedding gives 2
2 dx\ ll«llo,3. < C{\ Pf dx\ ll«llo,3.
./O
s 2 +++\ || / // />^^7ddiv(v|i;| d ii v H ( Vt| fV--r ,22dx\). ^|}. ./O
Jn
(7.3) (7.3)
Jn
For s G E (2,oo), 27+T 2^fT < ^27, 7 , first first term at the r.h.s. yields estimate \f^pf {f^pf s 22
•
1X
v H *~- dx\ c t e | < H/llo.oollHlo.^ll^llo^i H / H o . o o l H l o . ^ l l ^ l l o ^ l < c^11/||o,oo||^| v\v\ l l / l k o,ooll^l|q, o l M l1^7 s r -*• Second term ffurnishes i
ii2
2 1 s 2 2 \f \\0^,^2\\p'\v\ /M xiii|H ||V|VV t ;v|^l«t||/^£^i|H l|WVvlv^WvMlP^vl^Wo.i M '>' M l k, 22 <
«-.
O-.
1 hese tllP IM P IlIJJ2 '7W estimates A »iI|l|||V V««^|bt«;|||£1^^T2|2|||§|g§>,I22 ++ II HMI SI^^^IIIM ^ -. TThese - 1 1 ^ 1 ^\v\^Ho.2 1 ^ 121|!o 0 ,>22 < M V W ^estimates t i m a t e s yield in
particular 74
MIIO o ,, 33 ..<<|I|H p "|If';SW II H V+ 1-.
(7-4) (-)
U
>' 8 + 1
Take q0 = = q\ we realize t h a t | | : F | | _ M o < K(||/||o,oo,
m),to
= £3 ^^ . . L e m m a 4.1
applied to eq. (2.3) then gives H^Ho.to ^< ^ (Il/||o,oo, ^ ) - Equation (2.4), when qo(
'
o
f and then then integrated integrated over over IR IR gives gives (after (after employing in multiplied by po f*o °o and q
ii II ),qi IMIo,*! ||p||o, gi
/^ < < <
turn /ii/n n\f\\ c(||/||o,oo c(||/||o,oo
. ^\ m + + m), m )) ,,
1) 2 g0 0+4 -^377-- 3 .i q99o(*o o ( * o~- - l**) ) °"° 2g ^1 jrj = 3o5 •• gi = = 77 ++ = 10
O
(7.5) (7.5)
118
We see t h a t qx > q0, but still gi < 5 7 - 1 < 37. Now, for s G (2, 3 ^
], we
nave
have | +j yT < 9i; gi; therefore, (7.4) yields IM|o,3 < c(||/||o,oo c(||/|| 0> oo + m), \\v\\o,3 ^||o,3Sl55ll <
si =3 sx = Sl
-—-— ^ 3^ — _ qgi 37 x
(7.6) (7.6)
3 After this; we find find H 4.1 ap | | ^^|H| _-ii>, ^t l < c(||/||o,oo c(||/|| 0 ,oo + rn), ti tx = 66 77__ 22^^ii ++ 33 . L e m m a < plied to eq. (2.3) then gives H^Ho,^ < -Kfll/Ho.oo, if (||/||o ) 0 o, rn) and finally (7.5) becomes
I H I| oo^^cdl/lkoo ,,9922< c ( | | / | | o , o c + m), m),
=^77 7++ g 2a 2=
3 7 3 55 < 9 ?l 1 « =H = ^" --33 f7 -" -J3 .-. +g l^i i g^^ p=
(7.7) (7.7 (7-7)
We find t h a t qi < qq22 < 37, q#22 — Qi #1 > 7 ~ 3 ~ e ° - We are thus forced to repeat the procedure several times getting finally after a finite number of steps (say io) q = qio > 37. At the end, we obtain IMIo,3. < ^ ( | | / | | 0 , o o , m ) ,
|H|o,,- < X ( | | / | | 0 ) o o , m ) ,
s G (2, (2,00). 00).
(7.8) (7.8)
E s t i m a t e (7.8) completes the proof of L e m m a 7.1. 7.2
B o o t s t r a p p i n g f r o m peL*,ve max(6, 27) < p < 00
Hl>2 n Lp t o p G Lp, v G H1*,
P u t q\ = q. In this subsection we conclude the proof of Theorem 5.3 (a). We find \\pf\\o,qi < K{\\f\\0too,m), pv <S> v G L*{Q) < # ( | | / | | 0 > o o , m ) , i.e. Il^ll-i.gi < K(\\f \\o,00, rn). L e m m a 4.1 applied to eq. (2.3) yields in particular ll^llo.gi < ^ ( | | / | | o , o o ) ^ ) « Equation (2.4) thus furnishes (after the multiplica tion by pqi~x,) l|p||o,g32 < \\p\\o,q < c(||/||o,oo + m ) ,
q2 = 7 + 9i - 11.
(7.9) (7.9)
We repeat the whole procedure several times. For any p, max(6, 27) < p < 00, we find, after a finite number of steps: I pMl k ,P <<^ ^( |( |H tf(||/||o,oo,m), / |/kl ko o c, m ) ,
\\V\\ A'(||/|| \\V\\o, | | P | | 00}P ,PP <
(7.10)
Now, equation (2.4) yields ||divi;||o,p < c(||/|| 0 ,oo + rn) and (7.1) furnishes c r ll-^ll-i.P (ll/llo,oo+ra). Applying L e m m a 4.1 to the nonhomogeneous | | - i , P < (ll/llo,ooH-^i)(ll/llo,oo+ra)nonhomogeneous Stokes r»Klpm problem --pxAv /iiAv + V = JF -/iiZ\t; + VV V Pf = = divt> divt; divu divt> = divu divt;
3 in 1H IR IR3 in 33 IR in IR IR 3
(7.11)
119 where V — — (pi + ^ 2 ) d i v u + -|- p1 we obtain, in particular H i , P < ^ ( l l| / | | o , o o , m ) .
(7.12)
Hence, due to the interpolation, v G H , r G [2,p]; which yields, by the Sobolev imbedding, v G C°(IR C°(1R ) fl U, L r , 6 < r < oo. This completes the proof of estimate (5.5) and of Theorem 5.2 (a). l,r
7.3
P r o o f of T h e o r e m 5.2 ( b )
We come back to the Stokes problem (2.3). Due to estimate (5.5), we have T G LP(Q). This implies, by L e m m a 4.2, V, Vw G W1*. T h e rest of the proof is an easy application of the Sobolev imbeddings and L e m m a 4.5. T h e proof of T h e o r e m 5.2 is thus complete. References 1. A d a m s R.A.: S o b o l e v s p a c e s , Academic Press, 1975 2. Bogovskij M.E.: Solutions of some problems of vector analysis with the operators )erators div div and and grad, grad, Trudy I r u d y Sem. bem. S.L. b.L. Soboleva Soboleva (1980), (19oU), 5-41 5-41 3. C aEtttabriga t t a b r i g a L.: L.: Su Su un un problema problema al al contorno contorno relativo relativo al al sistema sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31 (1961), uazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31 (1961), 308340 to 4. Galdi aldi G.P.: A n i nntt r o d u c t i o n t o t h e m a t h e m a t i c a l tthheeoorryy of of t h e N aavviie r - S t o k e s e q u a t i o n s , Vol. I, Springer 1994 icite des solutions des equations de Navier-Stokes Navier-Stol c com 5. Lions comons P.L.: Compacite iques, C.R. Acad. Sci. Paris 317 (Serie I) (1993), (199 pressibles 115-essibles isentropiques, 120 >0 nee globale de solutions pour les equations de Na 6. Lions Navierons P.L.: Existence Stokes \okes compressiblesles isentropiques, C.R. Acad. Sci. Paris 316 I(Serie (Ser I) (1993), 993), 1335-1340 7. Lions nmeess a u x ons J.L.: Q u e l qm u e s m e t h o d e s d e r e s o l u t i o n d e s pprroobblleem l i m i t e s n o n l i n e a lires, i r e s , Mir (1972) (in Russian), french original Di Dunod (1969) 969) 8. Neustupa Dtny A.: Global weak solvability to the regularized regular eustupa J., Novotny regularized vis cous flow, Appl. Math. 36(6) (1991), 417-431 ms compressible flow, 9. Novotny mpactness of steady compressible isentropic isentropic Na Navierovotny A.: Compactness Stokes Toulon tokes equations via the decomposition method, Preprint Univ. To (1994) .994) 10. Novotny . : Lp-approach -approach to steady flows of viscous comp compress ovotny A., P a d u l aa M M.: ible >lefluids fluids in exterior domains, Preprint Univ. Ferrara (1992), Arch. R a t . Mech. tech. Anal. 126 (1994), (1994). 243-297
120
11. Padula M.: Existence of global solutions for 2-dimensional viscous com pressible flow, J. Funct. Anal. 69 (1986) 12. Padula M.: Erratum , J. Funct. Anal. 76 (1988), J. Funct. Anal. 77 (1988) 13. Sverak V.: Nonlinear equations and weak convergence, Proc; 14th. Conference on PDEs, Hrensko (1989), 103-146 (in Czech)
121 A NOTE ON A VECTOR TRANSPORT EQUATION WITH APPLICATIONS TO N O N - N E W T O N I A N FLUIDS MARIAROSARIA PADULA Dipartimento di Matematica, Universita degli Studi della Basilicata via N. Sauro 85, 1-85100 Potenza, Italy ADELIA SEQUEIRA Instituto Superior Tecnico, Departamento de Matematica Institute* Av. Rovisco Pais 1, P-1096 Lisboa, Portugal We prove, in appropriate general spaces, the well-posedness of a vector transport equation arising in the study of certain non-Newtonian fluids. The method consists of showing the equivalence between the given problem with a scalar transport equation and a Neumann problem for the Poisson equation.
1
Introduction
Recently, there has been a great interest in the study of the well-posedness of the boundary- and initial-boundary-value problem for the equations governing the motion of second and third grade fluids, s e e 6 , 8 , 2 , 1 6 , 3 , 1 0 , 9 . T h e appealing features of these contributions are due to the fact t h a t the approach they use leads to existence of classical solutions, assuming only the no-slip boundary condition, usually adopted in the Navier-Stokes theory. This approach consists essentially of splitting the given problem into two auxiliary problems, the first of which is a Stokes-like system and the second is a transport-like problem of the following type wJ +-h vV •• V w == V VW v pp- t+- F r 1 } inQ (i.i) V7- ww = 00 JJ (1.1) w • n = 0 on dQ. Here, Q, is the region of motion, dCl its boundary, n the unit outer normal vector to dQ, F is the (given) body force acting on the fluid, p is the pressure field and v denotes the velocity of the particles of the fluid satisfying the kinematical conditions V7 •■vv == 00 inin fifi (1.2) v = 0 on dQ. Moreover, in problem (1) w , p are unknown functions related to v . Let us explicitly note t h a t the Stokes problem has been extensively studied for different types of domains (bounded, exterior, with non-compact boundary,
122 etc.), s e e 5 and the references quoted there. On the other hand, we observe t h a t also the "scalar version" of problem (1), namely an equation of the form w + — G in Q, + vv •• Vw =
(1.3)
with G given, has been investigated by several authors, under very general assumptions on Gr, G, vv ,, \l, Q, see see *,, 1 3 ,, 1 2 ,, 1 0 ,,1 4 , , n and Dtions on and the the literature literatui cited therein. Precisely, (bounded, :ly, such such equation equation has has been been solved solved for for very very general general domains don exterior, (1) is quite r, with with non-compact non-compact boundary, boundary, etc.). etc.). Despite Despite this, this, problem pro! new from the m m aa tt hh ee m m aa tt ii cc aa ll point point of of view view and and by by no no way way can om the ca it be reduced to the scalar equation (3). As a m a t t e r of fact, the only contributions to the cont solution of problem (1) t h a t we are aware of, are those of 2 , 3 for bounded domains. On the other hand, is <±iso also cm an increasing uomaiiis. \JLI me oilier iiciiiu, there mere is inei easing motivation muiivcitiuii for lor investiinvestigating the motion of fluids of higher order, either in domains with corners, or in gating the motion of fluids of higher order, either in domains with corners, or in unbounded regions. We quote, for instance, the study in exterior domains, due unbounded regions. We quote, for instance, the study in exterior domains, due to) the importance of calculating the "drag" exerted by these fluids on b o d i e s 44 . to the importance of calculating the "drag" exerted by these fluids on b o d i e s . T hhe e objective of this note is to furnish a general m e t h o d for proving proving existence T h e objective of this note is to furnish a general m e t h o d for proving existence and elementary ad uniqueness of solutions to problem (1) reducing it to two and uniqueness of solutions to problem (1) reducing it to two more more el< elementary 17 problems, roblems, namely (see 1 7 )) and problems, namely aa N N ee uu m m aa nn nn problem problem for for the the Poisson Poisson equation equation (see (see and aa scalar two auxiliary :alar transport equation of the type (3). In particular, since the scalar transport -equation of the type (3). In particular, since the two auxiliary r n K l p m c nr<=> solvable C/-^1 VAQK1P> i n a a l«rcr<=» class n n c c AT n nmmnc f h p cnm<=> holds n n l n c Tr»r problem problems problems are are solvable in in a large large class of of domains, domains, the the same same holds for for problem
m.
In this note we give the proof of the above statement in general abstract spaces. Precisely, y, we we first nrst introduce introduce aa general general functional mnctionai setting setting and a n a prove prove t h a t the resolution and m of of problem problem (1) (1) can can be be reduced reduced to to tt hh aa tt of of the the N N ee uu m m aa nn nn problem pro! of the scalar equation. Then, using the contraction argument we calar transport transport equation. Then, using the contraction argu: isport equation, m e n , tne contract] show t h aatt this this coupled coupled system admits admits aa unique unique solution. solution. Finally, Finally, we we observe >upled system Fi t h a t thiss general result can can be be applied applied to to several several concrete concrete cases. general result cases. This T h i will be ca the object aa pp ee rr 111 55 ,, where are :ct of of the the pppaper where existence existence and and uniqueness uniqueness theorems theorems ar proven the in Q exterior tube, :erior to to aai compact compact (also (also empty) empty) region region of of the the space, space, an an infinite infin an half-space, a domain with corners, or in an aperture domain. Results will space, a domain with corners, or in an aperture domain. Res also be deduced in weighted spaces for two different reasons: on one hand, we deduced in weighted spaces for two different reasons: on one ] want to obtain the decay rate of our solutions at infinity; on the other hand, obtain the decay rate of our solutions at infinity; on the oth we want to control the singularity in the corners. For the sake of generality, we present our results in the n-dimensional case. 2
A General Theorem
In this Section we state, in a convenient general functional setting, the existence of a unique solution to the problem (1), by showing t h a t a certain m a p C admits a fixed point. To this end, let X, V be two Banach spaces, of sufficiently regular
123
functions locally integrable in a domain Q of R n (n > 2), with norms denoted by II * II*, || • ||v respectively. Assume that it is possible to define, in a suitable sense, the trace of the elements of V on the boundary 3Q and similarly for the elements of X and their gradients. We assume that the following subspaces X Xo 0 :=
{UJ
e X : V • w = 0,
u • n | a a = 0}
V0 : = { v G V : v - n | a n = 0, v • Vu • n\dn = 0, for all u <= X}, where n is the unit outer normal to <9£2, are closed. Consider the composite affine map C : u —> p —>• w
defined in Ao, AQ, where pp solves the Neumann problem, for v G6 VVo, o , F GE X^ 1
..
r
—
.
—
—
_ .
„—
r~.„
3
—
.
^.
Ap = V uu;V V• •vv— - t) F) in ft, Ap = fi, i\p — v -•( (w w -• vVv v - - wv <9p
„
§^| ^^ = - F - n on On, onSfi, on = - F - n on On,
(2.4) (2-4)
and w is a solution to the scalar transport equation (2.5) wv i^+vv V w w1 ^* ^=r --^fr1 F+1 F' 1, ', t = l , . . . , n infi. infi. ox ox1 It is easy to check that, if C admits a fixed point w in A'o, then it is a solution to problem (1). In fact we have Lemma 2.1. Let v E Vo, F G X.The operator C has aa fixed point w in Xo if and only ily if 11ww isisaasolution solution totoproblem problem (1). (1). Proof:F: It suffices to prove the equivalence between problems (1) and (4)-(5), with to0 replaced by w in the Neumann problem (4). First we note that (5) t gives the of (l)i. Moreover, taking the divergence on both sides ;he scalar version bo rsion 01 \\)\. Moreover, taking the div< of (l)i and using (1)2 the scalar product produ with 1)2 we we get get (4)i. (4)i. Now, Now, taking taking in in (l)i (l)i th< n on the Neumann ;he boundary, we obtain the boundary condition (4)2 for the N< y, we obtain the boundary condition (problem, recalling the defining properties of the space Vo:m, from (1)33 and V and recalling the defining properties < a Conversely, (5) gives just (l)i, it remains to prove (1)2,3The :rsely, since equation (1)2 rma.t.irm (M CHVPS i n s t (l)-i it rpma.ins solenoidality idality condition (1)2 follows by taking the divergence on both sides of (5) andd using (4)i. Finally, the boundary condition (1)3 is recovered cor computing the scalar condition alar product of (1) with n and taking into account the boundary cc for the Neumann problem (4)2. This completes the proof of the lemma. Next step is to prove our main theorem.
124
Theorem 2.1. Let v G Vo, F G X. Assume that the Neumann problem (4) admits a solution p, with Vp G X, which in addition satisfies the estimate lU i vI lMHI A U' ++ ||F|U) HFiU) IIIIVplU |IVVPp| | > . <<^CdcxdMlvll^lU I (d| |iVv|i| V IIF
(2.6)
Moreover, assume that the scalar transport equation (5) has a solution w G X such that |l | w | U < c 2 ( | | V p | U + ||F|U) ||F|U) (2.7) Then, if ||v||v is sufficiently small, the map C has a fixed point in XQ and for the solution (w,p) of problem (1) we have l|w||* + ||||V * < c |l||FF|||U *, l|w|U Vpp|||U
(2.8)
with the constants c\, C2 ine positive posiuve constant consiani cc depending, depending, in in particular, particular, on on the me cc and on ||v||v>n ||v|| v . Proof: f: Let us consider again the affine operator C C. In view of ithese assumptions, using in (7) the estimate (6) we get, for w := \— £(CJ), £(CJ), l|w|U < c(||v||v|M|Ar + ||F||;r), ||F||*), with c depending on c\ and C2- It follows that if ||v||v < 8, and ||w||^ \\w\\x < R with, say, cSR < 1/2, we obtain ii
..
c
F\\x
w\\ x < < JjiUf^5 B < < r, ||w|U 1 — con cdR r, with r < R. Then it is easily seen (using the fact that C is affine), that £ is a contraction in the ball BR, provided 8 is sufficiently small. Finally we remark that the statement (8) is straightforward. Acknowledgements Research supported by GNFM of the italian CNR, 60% and 40% of the MURST and by JNICT and FEDER, contract STRDA/C/CEN/531/92. The authors wish to thank Prof. G. P. Galdi for invaluable discussions. References 1. H. BEIRAO DA VEIGA, On an Euler Type Equation in Hydrodynamics, Ann. Mat. Pura App., 125, (1980) p. 279.
125
2. V. COSCIA and G. P. GALDI, Existence, Uniqueness and Stability of Regular Steady Motions of a Second-Grade Fluid, Int. J. Non-Linear Mechanics, 29, (1994) p. 493. 3. V. COSCIA, A. SEQUEIRA and J. VIDEMAN, Existence and Uniqueness of Classical Solutions for a Class of Complexity 2 Fluids, Int. J. NonLinear Mechanics, 30, (1995) p. 531. 4. R. L. FOSDICK and K. R. RAJAGOPAL, Uniqueness and Drag for Fluids of Second Grade in Steady Motion, Int. J. Non-Linear Mechanics, 13, (1978) p. 131. 5. G. P. GALDI, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Linearized Steady Problems, Springer Tracts in Natural Philosophy (Springer-Verlag, Vol.38, 1994). 6. G. P. GALDI, M. GROBBELAAR-VAN DALSEN and N. SAUER, Existence and Uniqueness of Classical Solutions of the Equations of Motion for Second-Grade Fluids, Arch. Rational Mech. Anal., 124, (1993) p. 221 . 7. G. P. GALDI and K. R. RAJAGOPAL, Slow Motion of a Body in a Fluid of Second-Grade, Int. J. Engng. Sci., 35, (1997) p. 33. 8. G. P. GALDI and A. SEQUEIRA, Further Existence Results for Classical Solutions of the Equations of a Second Grade Fluid, Arch. Rational Mech. Anal, 128, (1994) p. 297. 9. G. P. GALDI, A. SEQUEIRA and J. VIDEMAN, Steady Motions of a Second-Grade Fluid in an Exterior Domain,Adv. Appl. Math.Sci., (to appear). 10. S. A. NAZAROV, A. NOVOTNY and K. PILECKAS, On Steady Compressible Navier-Stokes Equations in Plane Domains with Corners, Math. Annalen, (to appear). 11. A. NOVOTNY, About the Steady Transport Equation I - Lp Approach in Domains with Smooth Boundaries, Comm. Mat. Univ. Carolinae, 37, (1996) p. 43. 12. A. NOVOTNY and M. PADULA, LP Approach to Steady Flows of Viscous Compressible Fluids in Exterior Domains, Arch. Rat. Mech. Anal., 126, (1994) p.243. 13. M. PADULA, A Representation Formula for Steady Solutions of a Compressible Fluid Moving at Low Speed, Transp. Th. Stat. Phys., 21, (1993) p. 593. 14. M. PADULA and K. PILECKAS, On the Existence of Steady Motions of a Viscous Isothermal Fluid in a Pipe, Navier-Stokes Equations and Related Non-Linear Problems, A. Sequeira (ed.), Plenum Press, (1995) p. 171.
126
15. M. PADULA, K. PILECKAS and A. SEQUEIRA, On a Vector Transport Equation with Applications to Non-Newtonian Fluids, (in preparation). 16. A. SEQUEIRA and J. VIDEMAN, Global Existence of Classical Solutions for the Equations of Third Grade Fluids, J. Math. Phys. Sci., 29, (1995) p. 47. 17. C.G. SIMADER and H. SOHR, The Weak and Strong Dirichlet Prob lem for the Laplacian in Lq in Bounded and Exterior Domains, Pitman Research Notes in Mathematics, (1996).
127 CO N V E R G G E N C E R A T E S IINN H22>>rr O F R O OT TH HE E' S M ME ET H O OD TO T HE N A AV VI E R - STOKE S EQUATIONS R. Rautmann Rautmann Fachbereich Mathematik-Informatik, Fachbereich Mathematik-Informatik, Universitat-GH Universitat-GH 33098 Paderborn, 33098 Paderborn, Germany Germany
Paderborn Paderborn
Rothe's semidiscrete approximation scheme to the Navier-Stokes initial-boundary value problem has a remarkable smoothing property which implies that the H2bounded and L 2 -convergent approximations are convergent even in H2>r, if the starting value is smooth enough, for any r G [2,oo). In the present note we will use this result of [16] in order to derive explicit convergence rates in If and H2'r for all r G [2, oo). After recalling some notations and recent results in section 1 we will formulate the error estimates in Theorem 1.3. Its short proof will be given in section 2.
1
Notations and results
Let Q denote a bounded open and pathwise connected set in the three-dimensional Euclidean space IR 3 . In order t o avoid technical difficulties we assume t h a t the b o u n d a r y dCl is smooth of class C°°. T h e velocity u(i,x) = (^1,^2, {ui,u2) ^3) and the kinematic pressure p(t, x) > 0 of a viscous incompressible flow in £1 a t t i m e t > 0 solve the Navier-Stokes initial boundary value problem — u — Au ++ u-Vu u ■ Vu++Vp=f, Vp = / , —u-Au
V -•uu = = 00
inftfor*>0, in Q, for t > 0,
(1.1) (1.1)
u\dn = 0 on dQ, for t > 0, u\t=o = u UQ. 0. Here we set the viscosity constant and the mass density equal t o 1. function / denotes the prescribed density of the outer forces.
The
We will formulate (1.1) as an evolution problem in the space Xq — closure of D in the Lebesgue space
Lq(Q),
where D denotes the linear space of vector functions
128
by a well known result of Fuji war a and Morimoto [5] the space Lq{Q) =
Xq®Gq
is a direct sum for 2 < q < oo. Let Pq:Lq
-> Xq
denote the projection along the space Gq of generalized gradients. Since Q is bounded, for the Sobolev spaces Hm,q(£l) with norms \\9\\H~.* \\9\\H-.* IMI*"M, == {{
qq a 2£J /[ \d \da9{*)\ \d9(x)\ 9(x)\ dx}K dx}<, E dx}±,
a
g
\a\<mJfl
where aa = — (c*i, (ai, a 2 , a 3 ), c*j = 0, ...,m, |a| = ]Cj=i a?, rn G IN, 1 < g < oo, the imbedding ^ ^ (1.2)
m m,q jHjm,r * n , r ^c_). H^/" >
holds if q < r. We will write tfm-2 = H2,H°>q = Lq = Lq{Q). Therefore the construction of Pru — u — V<£>
in [5 p.694] shows that we have Pr = — Pq\L
if 2 < q < r, Pi\Lr denoting the restriction of P 2 to Lr. In the following we will restrict us to spaces Lq with q > 2. Therefore we can write P = P2. The projection P is bounded on each Sobolev space Hm,q, m G IN, q G [2, oo), [20 p.XXIII]. As usual for s G (0,ra) we denote by Hs,q the complex interpolation space 5 9 between Hm'q and L 9 . if i l 5,(? ' is independent of m > s, for details see [18]. In 0
addition Hs'q stands for the closure in Hs>q of the subspace C£° of C°° -vector A
-
C
- -
—
0 0
of X or i/ Hx1>' 9q are functions which have compact support in Q. The elements v of divergence-free or fulfil the condition of adherence VQCL = 0 in the generalized sense, respectively. q
129 A formal application of the projection P on the first equation in (1.1) leads us to the Navier-Stokes evolution equation d%u + Aqu + Pu-Vu PwVu = Pf, tu + u(0) ti(0) = - u no 0
t:> 0, tt> 0,
(1.3) S}
(
for a strong solution u(t) G D -DA,, the closure Aq, where the Stokes operator Aq is the of —PA in the space Xq. Its domain domainis is 2
1
q
9 l q q q DAq4_ = ' '"n n 1H = #Hff22*nH * C\X nx ., >q'nX i
(1.4) (1.4)
[17, 6, 7]. The operator Aq fulfils the apriori estimate IM|tf2,g < < C H AqJu\\ U HLq L* IMIffa.g c||A,t/||r,g \\U\\H^ < c\\A
(1.5) (1.5)
IK^ A))---11|1|!i!!g!< ^^^^ (^ + +A A
(1.6) (i.6)
and the resolvent estimate
with some constant ce>q > 0 for all complex numbers \e^2 Xe^2 E > = = {z \z e G C\\z\ > 0, U, \argz\ \argz\ < 0, acting on X 9 ,[17, 9, 6, 7, 20]. Moreover, on the right halfplane ReX > 0 (and even on a neighborhood of A = 0) the estimate
iiK+AniL^j-^ IK^ + A IIK ArMlL^T^TT rMlL^Y^j
(1.7) (i.7)
holds with some 0, powers A^ &U111C constant ^UIIOIJCXIIL c i, q > ^ U , [20,p.79]. [ . £ U , J J . J tfj . Thus 1 I 1 U S the U11C fractional I1CIU Q C q with dense: domains DAQ C-> Xq are defined for all real a > 0. In the case a < (3 the domain DAA? is dense in DA«, the imbedding D DA? $ c-> DA« being compact [4,p.l58, 10, p*.69-74]. p*. 69-74]. q
Let JJ be a real interval, X a Banach space. Then by C°{JtX) we denote the set of all uniformly niformly bounded and continuous functions / defined define^ on J with values in X.".. In the following we will always assume Pf = 0. Q. 0. This implies that the density msity of the outer forces in (1.1) is a gradient field. We use this unessential restriction only in order to simplify the notations below.
130
For sufficiently small T > 0, the existence of a unique solution u £ G C o ([0,T], DAq) of (1.3) for given UQ G £ DA P>A wellknown. known. IfIfthe the initial initialvalue valueuu00has has q q isiswell some more regularity, i.e. if we assume u0 G UQ G
DDAAI+< I+< Aq
for some (£ G (0, l/2g), then the unique local solution u of (1.2) fulfils u G C°([0,T], 23.i+0 D,n-<) for some T > 0, [20 p.127 - 128],[12], without any nonrealistic compatibility condition concerning u0. In the following we will assume that a solution u G C°([0,T], D^i+<) of (1.2) is given for some q G [2,oo), £C G £ (0,1/2?). (0,1/2*). Starting from the given initial value VQ = u0 £ D4i+<:,C i+<;,£ G (0, l/2#), we ap(0,1/2*), proximate a solution u G C°([0, T],DAq) of (1.3) by means of Rothe's scheme A ? ^ — ~rvk_x vvk, "ll^Ll +I~A.v^-Pv^-Vvt,
(1.8) (1.8)
...,K,where u£ stands for some approximate value of t/(t) u(t) in the grid fc = 1, ...,-K",where point t = tk tk of a suitable time grid
tk = k.h, A = - L * = o,..,#, if = 1,2,.... In [15] we have proved In [15] we have proved
A A
T h e o r e m 1. 1 Let u G C°([0,T], C°([0}T],DA2D^ ) 2 ) denote denote aa solution solution of of the the Navier-Stokes Navier-Stokes initial value problem (1.3) with right hand side Pf = 0, vk the approximations from (1.8). Assume the initial values VQ G DA2 are uniformly bounded in H2 and satisfy |\\v | ^ho-uv\\ 0. Then for any h G (0, ho], ho being sufficiently small, all vk GGDA approximations v% DA2 2 exist, exist, and andthe theerror error estimates estimates
\\v£-u(t h, \\vhk-u(tk)\\
(1.10)
b«h
((1.11) l.ii)
for (3 £[0,1] £[0 j^-.The constants z[0,l] constai bo,bp depend }1] hold uniformly in k = 1,..., K < -£-.The on supp ||A2v(t)||, c,T, the constant bp additionally onfi, too. fi, toe t€[0,T]
131 For Navier-Stokes solutions u E C([0,T],£> £), even convergence C ( [ 0 , T ] , £>AAi+<,C i+<,C E € (0, (0>i), rates in H2 have been established in [13, Theorem II]. In in in of
order to prove t h a t the scheme (1.8) even converges strongly and uniformly H2> 'q q for each q E [2,oo), in [16] we started with a linearized Rothe scheme 2 q H > ,q > 2: W i t h VQ = u 0 , the assumptions of Theorem 1.1 hold because the imbeddings Lq <-» ^ L2 and
-> DAq = H2>qn H^q nxq ^ H2r\ n H1 nx2 = DDA2M.
DAI+<
Let~j (v%), 1, 2,...., be the sequence ol of Rothe (f£), \vk)i" h = — ^,k jc>k = — uQ,...,K,K= , •••> A , A = 1,2,...., Rotf approxi 2 t i o n s vkk E A D \\v^\\ < M 1.1 and m a itions DA ^ 222 , from (1.8) which exist by Theorem A2) 42)2 , |H^llif | ^ | | iHf22 < Theory 2 which ich are of first order convergent to u(kh) in L 2,, uniformly with with respect to k. Starting with (vk) we have found a sequence (uk) of approximations to u approxim; i-i-O „ r. . i i in H2,q from the linearized scheme .
^^ —~h - k^^~l
I
T
+ AqU 1 hVt; = /fc + Aqu^h"k == + A = ~-Pv - PPVh^".V k__x.Vv k=f^* k, = / ,', 1 •
* fc=1 = ! '-' , . .A.'-, # ,
Muh
= vj. txg 0 = o=^S-
(1.12) (1-12)
Namely, after having established suitable bounds for //fk in X Xq9, , inductively we c istence of uk E DA*+< -> A F>A uk = = vvkk from the see the existence D,4 4 qgq C DA2 and conclude u£ uniqueness 3S of the linear Stokes resolvent boundary value problem in D L A2 T h e n using Ashyralyev [1, D. p . 93. 93, I(2.11)],in ivralvev and Sobolevskii's coercivity coercivitv inequality ineaualitv fl. [16, p.386, Remark 4.1] we have proved
C o r o l l a r y 1. 1 For each q E [2,oo), the Rothe approximations vk = vk E DAi+< from (1.8), (1-12) (1.12) exist for all k = — 1,..., A' and are uniformly jioin (l.oj, [1.16J exist JUI ua K — i , . . . , i \ tnia u / c UIUJUI uuy bounded 2 q in D . i + c <-+ H > . q )y Next by means of aa compactness shown compactness and and uniqueness uniqueness argument argument we have sh ( E (0, T h e o rr e m 1. 2 [16, ( 0 , 1l/2q) /l/2q) 2 ? ) let u E 16, p.380] p.380] :: For For some some qq E E [2,oo) [2,oo) and and C T],D < ) denote denote aa solution C f 0 ([0,T], £>4 iA+i+<) UQ E DDAAi+c I+C solution of of (1.3) (1.3) with with initial initial value uo Then the ^/ze approximations nations vk in Rothe's scheme (1.8) with iinitial value value VQ = uo converge -» ^ge in H2>q (and even in DAi+n for all n E [0,C)) to u(k • h) with wi K -> co, h == j£ —>• 0, uniformly
in k — 1,..., AK..
In this3 note, we will prove T h e o r e m 1. 3 F For o r some some q E [2,oo) [2, oo) and and C ( G E ((0, 0 , 1l/2q) / 2 ? ) /e* letuu E C°([0, C ° ( [ 0 , TT]] , DD A i+<) Ai+<) denote a solution
of (1.3) with initial value UQ E ^DAI^+< +<
an
d Pf = 0. Then T/zen the
132
approximations v1^ in Rothe's scheme (1.8) with initial value VQ — UQ fulfil the error estimates \\v£ -k)\\ u(t )\\Lq0hKo,< b0hK\ H-u(t (1.13) {1.1.0) Lqk.. M1 "6' 1 ,, ! - « ( t(<*)IIH».« * ) | | HH»».. . . <^<^M M l|t>£-«(t*)||
(1.14)
K
\\v£-u(t foranype[2,q], k)\\L,
(1.15)
where "•0 — 7 + 4 f _ 6 / g , K
l — K 0Y+£,
«2 = i T ^ U / P - 1/9 + (1/2 -
1/P)K0}.
+ TAee constants T/ze ^w(t)||x,p,T,
t€[0,T] t€[0,T]
additionally on p, too. 2
P rr oo oo ff of of T T hh ee oo rr ee m m ]1.3 P
From Corollary 1.1 and our assumption u E C°([0,T], D 4 i+<) on the Navierq Stokes solution u we conclude the uniform boundedness \\A\«{vhk-u{t -u(tkk)\\LL,<M
(2.1) (2.1)
with a suitable constant M > 0, therefore I | V £ -u(t*)ILff - U ( * * ) | | I P3.'« « < < cc00M ||v£ M
(2.2) (2.2)
22 , 9
because of of -D £>AAi+< i+< «->• ^ £>A„ £>A, ^^ i? # ' . Recalling Recalling the the first first order order convergence convergence H
- u(tk)\\LL,2 < Clh
(2.3) (2.3)
(£l) stated in Theorem 1.1, by interpolation of the approximations v% in L2(Q) 0 22 between H #°> ' = L2(Q) and H H22>« * we wefind find IK"
(2.4) (2.4)
from the multiplicative inequality [4,p.27], where I = (
A l u 7 K-i 2 - p ^ 6 " ^
(2.5)
133 in the case 2 < p < q. Together with (2.2) and (2.3), (2.4) with p — q implies the error estimate Together with (2.2) and (2.3), (2.4) with p = q implies the error estimate \\vhk - u(tk)\\Lq H \\vhk - u(tk)\\Lq
(2.6)
l lQ
(2.6)
T h e m o m e n t u m inequality for the operator Aq reads T h e m o m e n t u m inequality for the operator Aq reads h1 ,.U,.\\\\T„ \\A {y k))\\ L, \\AqqJv {yhkk} -- u(t u(t k))\\L,
1h + h A\Ax+^(,b . II-,,! +<(vhk < u(t ■ \\v u(tk)\\^. < c\\A] c\\A] <(v k --_7/f*,.'rtll!+< u(tkk))\\tf ))\\]£ ■ \\vhkk -- -,,(i,.\\\l* u(tk)\\^.
(2.7) (2.7)
Therefore Therefore from from (2.1) (2.1) and and (2.6) (2.6) we we get get the the error error estimate estimate h \\A u(tLgk))\\L, \\A,(vl-u(h))\\T., q(v k-u(t-k))\\
)) a << cc,3//ii((11_c"3aah^e<.. T' i ?f>^.
(2.8)
But then from (2.3) and (2.4) with p — q recalling (1.5) we find the sharpened error estimate a a 1+( 1 o ) ( 1 + h H-u(tk)\\ h^\\v | | ^ k--u(tk)\\ « ( i fL
(2.9)
T h e last two steps in (2.8), (2.9) show: For n — 0 , 1 , . . . having established the convergence rate fi finn in in L,^ Lq, , fio /io — = 1i — — a, a, irom from [z.i) (2.7) we we get get m thee convergence rate •gence rate converg q q fin - J^T in DA and then the better convergence rate fio + L*>nY+7 finj+7 in in L . L . q hr in DA9 and then the better convergence rate fio + H>n Y& in from (2.4) with p = q because of (1.5). T h u s by induction from (2.4) with p = q, (2.7) we find (l-a)
\\v£\\v^ - u{tk)\\Lq
<
1^l-z-
^'"IT? rfc
(2.10)
and h
\\A h^^H (vhk-u{t u{t
(2.11)
Ti nhee nn recalling recalling ^I.OJ (1.5) by estimates Dy aa short snort calculation calculation we we get get the m e first nrst two LWO error error es (1.13), (1-14) with A C O and K\ in Theorem 1.3. T h e third error estimate (1.13), (1-14) with ACO and K\ in Theorem 1.3. T h e third error estimat (1.14) in Lp with p E [2, q] follows from (2.3) and (1.13) by the interpolation Z\W\\)i-\\u\\\-\ \\f\\L*<\\u\\h-\\u\\l-\ ll/lkp
A A=£—fA == ^i ^4 i 2
2 between L Lq in in case case 2i <^ < pp <5 q.q. L" aand n a is* Supported by Deutsche Forschungsgemeinschaft.
qa
134
References R eferences 1. Ashyralyev, A., Sobolevskii, P.E.: Well-posedness of parabolic difference equations. Operator Theory Advances and Applications 69; Birkhauser, Basel, Boston, Berlin 1994. 2. Beale, J . T . , Greengard, C : Convergence of Euler-Stokes splitting of the Navier-Stokes equations. IBM Research Report RC 18072 (1992) C o m m . Pure a t h . XLVII 1083-1115. i u i c Appl. zippi. M x\xa,\jii. Y Y X J Y A X (1994) y±oo-rj J.UU«_J-x A A U . 3. Chorin, A.J.: Numerical study of slightly viscous flow. J. Fluid Mechan ics 57 (1977), 785-796. 4. Friedman, A., Partial differential equations, Holt, Rinehart and Winston, New York 1969. 5. Fujiwara, A.J., Morimoto, H.: An L r -theorem of the Helmholtz decom position of vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA M a t h . 24 (1977), 685-700. 6. Giga, Y.: T h e Stokes operator in Lr spaces. Proc. J a p a n Acad. 57 Ser. A (1981a), (1981al. 85-89. 7. Giga, Y.: Analyticity of the semigroup generated by the Stokes operator in Lr spaces. Math. Z. 178 (1981b), 297-329. 8. Heywood, J . G . , Rannacher, R.: Finite approximation of the nonstationary Navier-Stokes problem I. Siam J. N u m . Anal. 19 (1982), 275-311. 9. Miyakawa, T.: On the initial value problem for the Navier-Stokes equa tions in Lp spaces. Hiroshima M a t h . J. 11 (1981), 9-20 10. 10. Pazy, Pazy, A.: A.: Semigroups Semigroups of of linear linear operators operators and and applications applications to to partial partial dif dif ferential equations, Springer Berlin 1983. ferential equations, Springer Berlin 1983. 11. Pironneau, O.: On the transport diffusion algorithmen and its applica tions tions to to the the Navier-Stokes Navier-Stokes equations. equations. N Nu um m .. Math. Math. 38 38 (1982), (1982), 309-332. 309-332. 12. R a u t m a n n , R.: On o p t i m u m regularity of Navier-Stokes 12. R a u t m a n n , R.: On o p t i m u m regularity of Navier-Stokes solutions solutions at at time time tt = 0. Math. Z. 184 (1983), 141-149. = 0. Math. Z. 184 (1983), 141-149. 2 13. 13. R R aa uu tt m m aa nn nn ,, R.: R.: H H2 convergence convergence of of Rothe's Rothe's scheme scheme to to the the Navier-Stokes Navier-Stokes equations. Journal of Nonlinear Analysis 24 (1995), 1081-1102. equations. Journal of Nonlinear Analysis 24 (1995), 1081-1102. 14. R a u t m a n n , R., Masuda, K.: iif 2 -convergent approximation schemes to the Navier-Stokes equations. C o m m . Math. Univ. Sancti Pauli 43 (1994), 55-108. 15. R a u t m a n n , R.: A remark on the convergence of Rothe's scheme to the Navier-Stokes equations, Stability and Applied Analysis of Continuous Media 3 (1993), 229-246. 16. R a u t m a n n , R.: A regularizing property of Rothe's m e t h o d to t h e NavierStokes equations, in: Navier-Stokes Equations and Related Nonlinear Problems, A. Sequeira (ed), Plenum Press New York (1995) 377-391. —
—
_
x
_
—
—
_ -
—
^ —
—
j
}
_
—
135
17. Solonnikov, V.A.: Estimates for the solutions of nonstationary NavierStokes equations. Zap. Nauch. Sem. Leningrad Math. Steklova 38 (1973), 467-529. 73), 153-231. English transl.: J. Sov. Math. 8 (1977), (1 18. Triebel, jbel, H.: Interpolation theory, function spaces, differential d operators. (1978), IS), North-Holland, Amsterdam. 19. Varnhorn, nhorn, W.: Time stepping procedures for the nnonstationary Stokes equations, ations. preprint 1353 (1991), Technische Hochschule Hochsd Darmstadt. 20. von Wahl, W.: The equations of Navier-Stokes and abstract parabolic equations. (1985), Vieweg Braunschweig.
136 O N EQUILIBRIA IN T H E I N T E R A C T I O N OF F L U I D S A N D ELASTIC SOLIDS
M. Rumpf Institute for Applied Mathematics, Freiburg University Hermann-Herderstr. 10 79104 Freiburg, Germany The mutual deformation Ucll influence l l l l i u c m ^ c of U l the Llic u c i u i i i i d i i u i i of u i an d i i elastic c l a s t i c obstacle u u a i a t i c and auva a a, viscous V W L U U B flow U U Y Y is studied. The interaction of the liquid and elastic material at the common boundai boundary is analyzed ;ed and existence results in terms of a small d a t a analysis are obtained f(for equilibria, a. where the normal stresses balance at the interface. A peculiarity of this th problem is that it involves both Eulerian and Lagrangian coordinates.
1
Introduction
We will study the interaction of an elastic solid and a fluid flow. T h e elastic body is influenced by the boundary stresses of the liquid and conversely in fluences the fluid as the domain of the flow changes when the body deforms elastic. T h e two m a i n m a t h e m a t i c a l results are given in Theorem 1, 2, below. T h e firstt result establishes existence for small d a t a in suitable Holder spaces spaces and t h e ; second theorem studies a free elastic body under gravity with a velocity velc prescribed force. ;scribed at the container wall in the counter direction of the gravity fc We: will prove t h a t under supplementary symmetry conditions and physically physic nder supplementary symmetry conditu meaningful aningful assumptions of the <elas ons the the gravity gravity and and the the stress stress at at the the boundary bo tic body will balance: in a stationary equilibrium. in a stationary equilibrium. T h ee introduction to m m aa tt hh ee m m aa tt ii cc aa ll elasticity elasticity and and the the approach to it by by the appi 5 implicit plicit function theorem, as they were explained in the book of Ciarlet Ciarl , >orem, as they were explained in the motivated ►tivated this work.. There There is is aa lot lot of of literature literature discussing discussing viscous flow -with 18 surface face tension. We mention in particular the work of Solonnikov and BeSole 2 11 4 m eIl m a n n s . IInn Lundgren, Sethna and Bajaj describe the behavior behavior and instability tability of a m e m b r a n e induced by a flow. flow. For example focusing on rr medi cal modeling Peskin and McQueen study elastic fibers in a fluid16. But But they inquire [uire lower dimensional elastic structures interacting with a higher dii dimen sional nal flow phenomena. In comparison with these contributions our int< interest ffinding m r h n c r stable st.a.blp configurations r o n f i cm r a t i o n s of o f liquids l i r m i r l e and a n r l thick thir«Lr elastic «=>lactir' solids, cr-klirlc wVi<=>r<: is inn finding where the balance law at the common boundary leads to the coupling of the two different problems.
137
Figure 1: The different sets and boundaries
2
T h e Problem Setting
A first we have t o derive and discuss the equilibrium equations for an interacting elastic - fluid system. Before we summarize and define t h e problem in stationary form we should mention one serious problem. Imagine a free elastic body in a liquid container. For t h e existence theory as well as for t h e mechanical understanding t h e elastic body has t o be stabilized in some way. Otherwise a t i m e independent setting is meaningless. In t h e second part of this paper we will t u r n t o a pure traction problem for t h e elastic solid under reasonable s y m m e t r y assumptions. B u t at first adding a mixed boundary condition we Figure 1: The different sets and boundaries will simplify the configuration. We do this by gluing t h e elastic body QE onto a fixed inelastic material tip (Fig. 1) in such way t h a t (3QE n<9ftc) DdQp — 0 where Qc is the reference domain for the fluid. This zero intersection condition 2is crucial T h e in P r the o b l eapplication m S e t t i n gof strong regularity results for t h e displacement. Even t hhave e linear case there a breakdown our a r g u for m e nan t s 2 0inter. To A first inwe t o derive andwould discussbethe equilibriuminequations clarify elastic notation we denote 0Q,EN we — summarize d&c H d&Eand t h edefine traction N e u m a in nn acting - fluid system.byBefore t h eorproblem b o u n d a r y and by 8QED — d^F t h e zero displacement or Dirichlet boundary stationary form we should mention one serious problem. Imagine a free elastic and summarize h e assumptions t h e domains: body in a liquid tcontainer. For t hon e existence theory as well as for t h e mechaniA 1 (Domains) cal understanding t h e elastic body has t o be stabilized in some way. Otherwise a t i m e independent setting is meaningless. In t h e second part of this paper £IE,QC,&F bounded domains in Rn , n — 2 , 3 , VLE is connected we will t u r n t o a pure traction problem for t h e elastic solid under reasonable Ct C F U ^ ) CC B Qu t atQfirst = Qc U {QEa U SlF) boundary &E{
138 in t h e deformed configuration will be indicated by a superscript (p. T h e n t h e Navier Stokes equations hold in £]). Q,c{
+ v* • V vv v* + -\- W vy ■= — 0u divv^ = \J \J divv^ = \j0 0
in £lc(fp) bib Q,c(ip) Mb(jyyj in £lc(fp) on on dSlENif) on dSlENif) dSlENif)
v^ v^ = = 0 0 v* = v° II JticM
ondtt on 8Q
If v° • n = 0 ./an Jdn
p* p* =p =p
To simplify t h e exposition we introduce with w* = v^ — v° and q^ = p^ — p new variables iables on £lc{v) where we regard v° as a divergence free extension extensic of v° with compact C\<9£2 dQ,supplied supplied with with an anappropriate appropriatee$estimate m p a c t support c(^) H pport on ^ Qciv) dQ for t h e extension extension operator depending on the spaces H1, ,C C2'a where we actually a settle our nr problem. This implies t h e solving of the standard Stokes problem prol in a fixed domain. In terms of w^ and a^ q^ the Navier Stokes ecmations equations an appear as u v°) ( u ^ + v°) + ({w* --vc&w* i / c A^w; w 1*^ + ur + v )•V V(w;^ vu) + + V g ^ = //**
f* =■■i/nAv vcAv° 0
,
t^£ = ^ = 00
o n 5SQQc^( ^M)
,
//
divw* iivvu ^^ = 0 d
^) in O c ( p
a^ g^ = 0
Now we will focus on t h e elastic solid and its governing equations. T h e fact t h a t tthe h e material in CIE behaves elastic leads us to a stress tensor T^ = Tg depending V<£>. introduce t ht he e nding solely ?1-1_T1T^(V<£>)CofV<£>. T£ |((VVV^ ) C o f V yv ? . For i t e 5 . Note Note t h a t E is For aa reference reference we we ccite symmetric, assume t h a t metric, because T is symmetric. In addition we will further on as ;cause furt E is differentiate natural state and in Vit and t h a t t h e elastic material is in natural able ii isotropic. opic. Therefore this operator has the well known Navier Lame Lame equations as itss linearization. Defining an operator C(u) = ( i d -+hVV«u))EE((V Viui ) we formulate the equilibrium equations for t h e elastic solid A 3 (Quasilinear Elasticity Equations) — divC(u) —0 yyu) — u u = 0
C(u)n
in ILE Q,E in on
8CIED
= T(v < ^,p < / ? ,^) n n
on
8CLEN
139 Heree T(^,p^,<£>) is t h e pull back of t h e flow flow stress te tensor influencing t h e formation at at tt h he e interface. interface. Taking Takin into account elastic itic deformation accoun t h e transformation da
C(u)n U(u)nE
= --T*(f, D(v*))CoiV^n T £ ( p * , -TZ(prnivr))CofV(pn L>(t/"))Cof Vpncc
where Cof indicates the cofactor \{vitj+Vj T^(p T ^ ( pip^i , £)(i;^)) D^)) :ofactor matrix, D(v) = %{vi i)ij and Tg(pV, D{v^)) tj+vtjti)ij the s t a n ddard a r d stress stress tensor tensorr oi of a fluid. We finally achieve a Newtonian Newtonian fluid. We finally achieve for the stress ference configuration: at t h e interface iterface in t h e reference A 4 (The Boundary Condition) T{y*,p*,
= ({-p*id+2v - p v , i d +c2D(v*))(
on
dnEN d£l
T h e Equilibrium State of t h e Coupled P r o b l e m
At present we will formulate our m a i n theorem concerned with t h e existence of an equilibrium state. T h e appropriate setting is in strong Holder space; spaces, because Lse we search for solutions of t h e Navier Stokes equations in a deforme deformed d o m a i n . Even a result in weak spaces is only known if t h e deformation deformation is ), 1 in C 01,1 function. ,, in t h e sense t h a t d£lE is locally t h e graph of a Lipschitz functio] This forces us t o look for Lipschitz elastic LiDschitz continuous displacements disDlacements in t h e elast material. B u t in nonlinear elasticity case we should then settle our problem in the context of classical solutions. T h e o r e m 1 (Existence of an Equilibrium State) Let Q, Q,E, Q &F E be domains fulfilling (A 1) with boundaries of Class C3 and let \\v0\\c*.<* and p be suf ficiently small depending on the geometry, the material properties exprt expressed )rm of the second in the form :ond Piola Kirchhoff stress tensor near the ;natural . state deformation )) (ft )) ,> and and pressure p* p*in G G CC^ *"^ C2>a(Q~^)') , a velocity v* (ftc(v?)) aridI aa pressure C^ (n (ncc((p)) ((p)) such thatt (A 2), (A 3), (A 4) hold. This theorem will be proved in a separate section below. Here we only sketch an outlinei aa n hat a nda introduce introduce aa frame irame for ior what wnai follows. ionows. We vve2will win suppose suppose tin C s t a n d a r d extension extension estimate estimate u < C ||u°||c 2 >«(dn) holds. estimate || || ff °°°||||||ccc22a)) ,tt tt«// (nn n ^,, j )u W^Wc^(dCt) < C ||u°||c >«(dn) holds. T h e proof will consist in demonstrating a contraction property in an approp appropriate consist in demonstrating a contraction property in an appropi set of deformations. Let Let us call 3> this subsequent contraction m maappppiinngg. We will show t h a t $(
140 we first should focus our attention to some regularity considerations. If dQ,EN 2,OC 2,OC is s m o)oth o t h then
(p EECC C (QE) leadsusustotoa adeformed deformedfluid fluid domainHc() , the boundary stress pulled back to the rreference i o u s t h a t T(t>^,p^, T ( f ^ , p ^ , (p) 1,a
,2 a 2 a $:C ( Q ^ ) - >2>CQ{nE) ' (n^) :C2>'a{QE)->C ip* $ ( yo)=id+u(T(v* O=id+u(T ( i^(p)) p^y>))
as the convolution of the following three operators • T h e5 solution operator for the Navier Stokes equations equations on the deformed d o mnain ain • T h ei evaluation T(wip, g^, (p) <£>) of stresses on the reference reference boundary • T h e2 solution operator for the deformation with uuppddaatteed boundary condition To justify the contraction property
r) - nr)\\c^(^) ^ W "v - vllc-^)
cc*(e) ||v> | | *^(1^)) - *(¥> < ^ 2 )2|)llc».-(fiF) | c 2 , < . ( n ^ < c*(e) | | ^x - V 2 | l c ^A\( cn^(^) ^) H
for all (px,(p2 E Be(id) in C2)OC(HE) (£IE) and a constant c*(e) < | , where e is a ! 7 30U||I I _ „ a(f^"), ,——, Pr> We W P will x v i l l discuss H i f i r n s f i these t V i p « p operators r»r>p>rntr»rc separately and at b o u n d for ||^ c2) the same time discuss in detail the existence and regularity results sketched above. N o t a t i o n a l P r e l i m i n a r i e s : In the following we will utilize the same names for dependent variables on the deformed as well as on the undeformed domain and we often skip thej cp (p From p in a superscript to abbreviate the notation. notal the context it will be clear what is meant. If any confusion should shou occur in a particular case we supplementary upplementary note the point of evaluation. Finally we denote by C a universal constant and by c(e) a universal small constant of the order of e.
141 4
The Navier Stokes Equations
We search for a velocity w^ and a pressure q^ and we supplementary have t o estimate tthe h e difference of two solutions (w* , q^ ), (w* , q^ ) on two diffe: different 1 ( 2 < 22 domains Q,c{
,
<
C
Wu\\ci,"(^)
supp E(u) E(u) C\ n supp i>° v° = —0
For 6 small dl enough enough the t h e extended deformation id-\-E(u) id-\-E(u) on on Q is in invertible. Denoting this extension again by by 'a{U) («)
(fic)
,
ip =
(id)mC >«(n) v^eB ^^eB (id)mC (id)mC>«(n) ^(U) ^eB eE c{e) c(e) c{e)
Now we are able t o transform t h e Navier Stokes equations on flc(^) onto t h e fixed domain Qc • Introducing the following notations A(
i=i,...,n = (vc^ij^kjmj ijipkjWi,i k = l,--,n
detVy?)i=i,-,n detV<£
k = l,- • ■ ,n
\{(p)(i,i ,,, , B Bii((p) ==tl> ipi !{
W=([^K],,). , / i = l,--,n \ ' / i = l,--,nd e \ ' / i = l,--,n N{
i=lj
n
for t h e deformed Laplacian A(
142
where the left hand side defines the deformed Navier Stokes Operator S(a O F 1 ) x (C1'* H L L22( C°>a x C 1 ' " 2 2 L22(^) L () (Q CC))CH
gM M 01 1 g?M * )**))<))^)^^===001
A difficulty for a functional analytic discussion is the dependency of the spaces on the unknown nknown (p. Therefore let us denote by S( = ((Cof^7(p)u,i) {{CofVc = [f
dfi
((CofV
t
dfi -- ((CofV
JClc
JClcc JCl
= / (CofV
da = 0 da = 0
JdClc
C22^{U^)UH^(n q* (fi^)nLg(nc) we obtain that for functions w* E C ' a ($}c)U#£(fic), ^ G C 1 >Ca ltOC {nc)nLl(Qc) c), 0,oc 0,a L22 ). Furthermc C the imagee of the operators is contained in C x (C1,a H L ). Furthermore l locally Lipschitz continuous ina the argument (p. t 5' ,5 are locally the y?. S 5' (id) is obviously t] 4 22 pp liP proved H > 1 standard Stokes Operator. I n Cattabriga H ' x H regularity ibriga ffor this problem. lem. Therefore the complementing boundary condition is fulfilled and a anting ai 11 so also the (tic) (Qc) Le strong strong regularity regularity holds holds11 and and for// EEC°> C°>aa(Q^) (Tl^) and and ggEEC1,aCC *"^] id for '"^) H LQ(QQ) C22',aa(Qc) #o(£2c) and a pressure qg E Lg(Qc) there ( n c ) H HQ(£IC) there exist a velocity w E C l 1 = (f,g) . C ^ ^ ) flLg(JJc) such that ^S {id){w,q) (id)^,^ = To deal with the original operator S() (#(¥>) 5'M = S'(0) S*(0)
5'l (0) _ 1 is bounded and Sl is Lipschitz continuous in (p. Therefore if (p is closed S [entity in C 2 ' a , S 5'l (tp) is invertible and the inverse is uniformly bou to the Identity bounded schitz with respect to (p near the identity. Now let us now turn tto the and Lipschitz nonlinear problem. We search for a solution of the nonlinear problem with a
143 given right hand side e)(w*,0) 5S(
= 0Owe w e immediately get for v° = 0 Dw*tq*S(
l = = SSl(
T h e n due to the implicit function theorem f theorem 15.2) the m a p p i n g S((p) is locally invertible in a neighborhood of the zero solution and the inverse is Lipschitz continuous in , because because an inverse, exists, depends on parameters (here the build in | |c| c° > o ,«- < C C||u°||ca.I I ^ H c ^ < c(e) we obtain lHl it ^ l l c^^. a + llo^llci.or l l ^ l l c i , * < c(c) c(e) Now let us inspect the relation of the solution (w^.q^) with respect to the operator S(
w^ =■ w^ *" = «"
?TTT measS2cc(<£>)
/
F
JaccM Ja
Obviously the same smallness conditions as above also holds for w^, u>^, q^. Now ble to prove the following l e m m a estimating the cdifference of two we are able stationaryy fluid flow C22'a, neighborhood. ow solutions on two distinct domains in in a C 111 111 222 222 L e m m a 1 Let ),), (u> (u> be two solutions of the Let (w (w (u> ,,, qqq ))) be be two two solutions solutions of of the the Navier i; ,,, qqq ), NNavier Stokes equations S((fi)(wi,qi) , ql) = ( f/ ' det det Vy>*', V ^ ' , 00) ) for for iii== = 1, 1,22 and and *', 0) Beec(id) 2,a Be(0) in C (£lc)Then their difference can be estimated estimated on the reference domain Q,c •' 11 e 2 ) 2 2w \\W \\w ||||Cc 2,, ,^(?^) IkIk2"2-9- ^llci.-(^) ^llci.-(^) ^(Olb )lb2-V - - ^Wc*'*^) ~- - ^^H ^ ( n(? )^)+++ II? Wc^i^^^nW <<^( l^Wc*'*^) lc.«(nl) c2>
P r o o f : At first we have to estimate the correction t e r m in the pressure adjusting the S solution
144
-I
lc{
\\measQc(
_
/[
2 Jn 7Jcto(^) ncc(f( v)2 )
fdfi22
2 eet V
^-IN_ "M ^ _ 22 measQc(<^ m e a s fIci^ li c ^2)2 ))
11/ r 11/ lMn lUncc
lL__
—r-^ I/ 11
A M Y1* 1 )
1 tf1^fdA
m e a s Qftcl^ c b )) Jcidf meas \\|]|| measilc('p) Jcic(^)) J1 22 d e t v 2 V^> det V Vy? Vyp detV Vyy11 \\\ -i^„t f/ det yy^ det 222 1 measQc(<j V m eeaassQQccl^^ )) measQc(<^ m e a s ^ c)/^ 1 )/ )/ Vmeasfic^ measficC^ VmeasQcCv measQcif
IIII \\||
\\
2 ^ C I I S - 1 ^ 22 ) ^ / 2 d e t V V y^ >. O ^ 1K) C ^ / 1 d e ttV ^CWS-'i^Kif , )0 )) -) ^- S- -1 V V^^11) )00)) )| ||(|c( c» ^. -xxcC1i,.«- ) +
c (W 0 lHl v^2-- ^ ll llc^i Recalling the Lipschitz continuity of 5 with respect to -«; ||c».<. + l l « - ? | | c i . « 2 2 2 1 l / 111 d e t V ^111 0 ) ) | | ^ < CC|| | 5S~--11V c« x Cc111 ,,«) ) « )) + (( ^)2 () ( (// 22 d ee tttV 0 ))^))- -5S^--11 V (^ ^1M VV^^^.,O ) )!(!( // detV^ t V ^ ),0))|| 05 ) ) | | ( Cc 22( ,,« )x
c{e)\\
° | ||Cc 22 ), a« c(e)||y> c{€)\\
/ n ^1., O / ^^11 1^) )!Z(!/( /f/111 d r le^ tf \V7 /^ .~ |( |C((2xccci.~) ^^xxCC11.«) < C | | 5lQ{v - -12)((f (/ ^/ o2ll )^(/ (/ /f1 dr Wee t^ttV O detV^ ,«xc + & VV ^ O^^l Hl/ detV
2
r
1
1
1
1
cc(e)\\
T h e Boundary Stress
Here we will briefly consider the behavior of the boundary stress influencing the deformation of the elastic material. We observe t h a t though Tcip^, -D(f^)) on thee deformed boundary d^d^) dticitp) is linear in p^^v^ the transformation to transforma the reference sference boundary dtic involving the Cof Vc;rhit.7. Kprfl.nsp n f aactor r t . n r is i« obviously n l ^ v i o n s l v aa smooth «mr»r»tn function. fnnr'tini remains because t.hp the rcof The following l e m m a expresses this property in terms of the appropriate function spaces. 1 2 2 22 , a 1 1 2 2 Lemm Suppose ip (p ^>
\<£> e (id) C i naa 2^ ouppose
145 6
T h e Elasticity System
Now we will focus our attention to the nonlinear elasticity problem and the essential estimate, which will constitute the final step in our fixed point discussion. To begin with we recall (A 3),(A 4) and rewrite it in operator notation r K{u) i(u) = ( - d i v C{u),C{u)n)
= (( 00, ,Tr((^^, ^, ^, v^?)) nn )
where K is a m a p p i n g a xx ({C C22>'' aa(tt ({ILE) HE)^ ) ICiV)-> HV)-> {STE) K(u)l) :. ^O C°'a' {Q I V ) —r C°> O \*&E) E) 1
V = H {QE)C){u
= 0
on
O
'
lta CC^idQEN) {dQEN) \UILE.
dnED}
Our goal is to prove t h a t the Inverse J i " 1 exists and is uniformly Lipschitz in the neighborhood of the origin. As in the case of the Navier Stokes operator let us first inspect the linear operator K'{0) u = ( - d i v C"(0)ti, C'{Q)u n) Q# is a domain with smooth boundary 8£IE and let us assume t h a t Q,E which consists of two separate p a r t s 8QED,
9Q,EN
with DCLED C\0QEN
E C3
— 0 and
a positive cap(<9Q#£)). Then especially due to Korn's inequality the Navier Lame operator K'(0) is invertible and for a solution of K'(0)u — (g,r) the regularity estimate . — < cr, (\\n\\... . — . j - M rT I U , ~ . ^ ~ IMIc2,«(n^) < (ll#llc°.«(?^) + ll llci.«(an B A r ))
holds. For the proof we refer to f Theorem A.3). A crucial loss of regularity takes place if the two boundaries d&ED and OQEN intersect. In general 2 0 even the H2,2(Q,E) regularity fails and there is no hope to demonstrate the existence of classical solutions u p to the singular intersection set. T h e following l e m m a deals with the dependency of the solution of the boundary stress. L e m m a 3 Under the above assumptions there exists a neighborhood Be(0) of the origin in C2,CC(CIE) H V such that the nonlinear operator K(u) is invertible for u E Be(0). For two solutions v},u u1^22 of K(u K{u{l) — (0, (0,T(w T(wi,qi,{,^')n) q\ y>')n) the following estimate holds ll« H«22 -
ul ul
2 22 Wc>,°(T^) 7) < < C \\T(v\p \\T{v2,p2,^) ,9 )) IU«(nT ,?
-- T(o i V .W pW ) | j| Ic ll c, „i .( aa ^n nE i T(t,
id+K~1(glJ(g,r) r) for (g,r) (g^r) E K(B€€(0)) A solution (p — i d + u = id+K preserving and injective.
is
v )
orientation
146
Proof : We again apply the implicit function theorem f Theorem 15.2) to ; ge C°'a(^), (nE),
K(u) = (g,r)
rr € C ^ '1", 0([ (flJW) cKW)
11 2,a 2,a and obtain K~ K (g,r) {g,r) Furthermore aue due to the Diam aa solution solution uu — — i\ (a(QE) H V) and therefore we finally the differentiability of A'" A " fferentiability of A'" in B€{0) C {C > {QE)C)V) and therefore we obtain the following Lipschitz estimate l \\u2 -- «^ l lllcc ,^,n^j )) = .T^.p1,^Jn)!^,^^ = \\K\\K {Q,T{v\p\S)n) Wl\v',p',
We will examine the last statement of the lemma not only for u but for an extensionI of u in C1. . Therefore we suppose e to be small enough such that an extensionl E(u) of uuu onto ||i£(u)||(7i iis suffionto aa convex convex neighborhood neignbornooa of ol £l \lEEE exists exists and ana ||E(w)||(7i || ciently = id+E(u) is orientation preserving because id+t E(u) ltly small.. Then
0. >bviouslv invertible for t G TO, and det(id) detfid) == 11 => => det detfid+E'fuV It is also injective; otherwise there would exist two points x1 ^ x2 with (p(xl) = -x ^ ii 2 9 n ^ N mi 29 x > \\x - xx1\\n -- sup||V£(ti)||||z || > > sup||V£(u)||||a: - xx1]] >0 T-,/
which contradicts our assumption. 7
The Fixed Point Property
In the preceding sections we have separately inspected the three different operators the mapping $ consists of. Collecting these results we are now able to prove our first theorem. Proof of Theorem 1 : We choose e small enough such that c*(e) c*(e) *(e) = C(Cc(£) C ( C c ( £ ) + cc (( cc )) ))) < << -ii Then for two deformations (px,
II^V^IU-^)
= IKr(«2,p2,^2)) - txcrc^.p1, ^1))llC3,-(n^) '"T*
C
\\T(v3,p\
147 Lemma 2 _
^
/
\
cc [(ll^-^MU-^ W - V ^ ^ + WP2-- ^ i u . ^ ) < < cc CC ( | | , 2 - ^ | | c 2 ( Q ( - ) ++ |lb^ 2_pi|Ua(_) 2
2 » lb +Cc(c) | b -^IU«(n - ^ H ^ ^ j +Cc(c) | b 2 - ^ H ^ ^ j LCm al < C (Cc(e) + 6(e)) \\
Finally we obtain that 22 B£e{id)) • $(B (id)) C CB B€c{id) (id) in CCC '«(n^) ^(Q^)
• $ is a contraction map iin
C2)CX(QE)
Therefore a fixed point ) in the boundary condition for the elasticity problem is the cofactor of the elasticity solution id -ft/* = tp* . 8
The Case of a Non Fixed Elastic Body
Up to now we have inspected the interaction of an elastic body with its liquid environment iment where the body is fixed on a subset of its boundary. Let us now turn to another physically interesting question: Under what circumstances would an nonfixed elastic body under gravity forces rest in a deformed stable state in a viscous incompressible fluid? From now on we will assume that the prescribed velocity v° on 80, is a constant. The underlying to those lerlyine; equations and assumptions now look slightly different difl originally stated in (A 1), (A 2), (A 3). Let us denote the differences: A 5 Changes of the Domains (A 1) &E, ^Cj &> mnded boundeddomains domainsin inR?R3 &E(
=
Q#Q#CC CCQ ^2 &cQc= =^ ^——&E &E
fic(^)
= £2 -
fifi(^)
A 6 Changes in the Stationary Navier Stokes Equations (A 2) - i / c A / + v* • vVv'^ -f+ vVp^ = yg y — v^ = v°
constant on dd£l
148 We assume without h a t the v i i n u u i any G'lLj irestriction c o i i H / U i u u tLiia/i) 1/i.ic ldensity a c n o i t j uofi the d u e nliquid ^ u i u 1is equal to 1. Differing from Theorem 1 now a constant gravity force g affects rom Theorem 1 now a constant gravity force g affects the liquid. lp(p Let us introduce f££ = = 2v 2ucD(v p^ with Luce a stress tensor T -— p*> with p^p* == p^p — g • x*. cD(v )) T h e n we can rewrite the Navier Stokes equations in terms of TQ -divf£ + -divTg + v* .- VVv^* = 0 which will1 subsequently subsequently be be of of importance. importance. A 7 Changes nges in the Quasilinear ^ ,
— divC(u)
x
= pog
Elasticity
Equations .
.
constant
.
in
(A 3)
^
Q#
. i n
#11^°
Here g is the gravity force and po the density of the isotropic elastic body which we suppose to be larger t h a n 1 . T h e above question not only demands for a scale of the prescribed velocity v° in the counter direction of the gravity force, but also some other conditions have to be fulfilled. Thinking of a propeller shaped body, which would never behave static, we recognize t h a t there have to be restrictions on the geometry of the elastic solid as well as on the shape of the container walls. Furthermore two additional assumptions will be necessary, one on the mean value of the hydrodynamical pressure influencing the stability of a solution and the other on the smallness of the gravity keeping our considerations in the general framework for small d a t a . At first we will focus on the elastic component of our system and we have to recognize significant changes in both steps of the solution procedure, in the considerations for the linearized model as well as in the application of the implicit function theorem. To demonstrate t h a t the linearized m a p p i n g is an isomorphism we have to reformulate the image and pre image spaces of this m a p p i n g . From the classical work of Stopelli 1 9 we know t h a t for the traction problem bifurcation might occur at the zero solution even if the linearized equation is uniquely solvable. But under certain conditions on the angular m o m e n t u m of the volume and on the boundary forces these difficulty can be ruled out. Stopelli 1 9 , Chillingworth, Marsden, W a n 8 and Lanza de Cristiforis, Valent 6 have studied these conditions. T h e presentation in 8 is based on geometric concepts and we will here sketch the result. Let us first restrict to an admissible set of deformations D = {(p £ C2,CX(QE) \I ^ ( 0 ) = 0} were the translational part is ruled out. Above we have introduced the operator K(u). Now we define an operator E on D and observe t h a t its image is contained in the space L : E : D - > L ; E{
149 where L is a restricted space of loads a 1a f GiT)eC^ ^ j {Q e EC)xG ^E )'x{dn CEl> )\^a{dn E EJ) l fI /f L =^ {{G {(G,r)GC°'"(ft xC
GDX+ GDX+ /
JQ,E
TDA = 0}
JdttE
Here we remark t h a t a deformation
I x
vanishes. In these spaces the following existence result can be formulated. L e mi m (Existence of of an an Elastic Elastic Deformation Deformation under under Fixed Fixed Traction) Traction) Let m aa 44 (Existence Lei us 33 assume , IQ E L EE QQ = { L G L | A(L) = 0} and thattr a{lo) is not an \me that^E E- C a(/o) not o , to t i^EQ — 1> e Li I A^L; = u / ana inaiira eigenvalue sufficiently nvalue ofa(lo). of a(Jo). Then for loads XI with I close to /o in L EQ and X sufficie) there exist exist aa.deformation deform.ation
{Xg,Xr) {Xg.Xr)
and
Coonnssii adeerraa tt ii oo nn ss ifor Deeifoo rr m m ee u d r>ouy Body ^ o r aa iNxoonnfni xx ee d u u
In the following paragraphs we deal with a deformed body where the deforma tion is fixed and observe in detail the circumstances which keep it in a stable state. Furthermore we will verify the assumptions of L e m m a 4 in this context. We have Lave already mentioned t h a t symmetry conditions would be essential essentia to obtainn an equilibrium state. T h e principles of m o m e n t u m and angular mo angular ate. ±ne principles 01 m o m e n t u m a n a an; m e n t uurn m conservation have to be be satisfied. satisfied. T T hh ee propeller propeller geometry geometry rmentic mentioned re to abovee is an example with. point gra point symmetry symmetry in in planes planes orthogonal orthogonal to to the t] gravity force which would only guarantee the conservation of m o m e n t u m perpendicular perpendic arantee the conservation of m o m e n t u m Derr to thee gravity direction, but we will see t h a t the following two planar symme sym tries, provided with a smaller symmetry group t h a n the rotational symme symmetry, whichh would also fit, is sufficient to conserve enough m o m e n t u m quantities. quantitie Let us choose an orthogonal coordinate system {ei, e2,63} e2, es} in our domain space
150
Figure 2: The symmetry of the elastic body
such that ei ei is parallel to the gravity force and the prescribed velocity v° (g - v° < 0) and introduce a notation for reflected points and vectors
(6,*'6,i6) * y = (6,*'6,J6)
; for £ = ( 6 , 6 , 6 )
; i,j
e{-i,+i]
Now we formulate the symmetry assumption A 8 (Symmetry of the Elastic Body and the Container) Let Cl and QE have the two symmetry planes e i 0 e 2 and ei^e^ , where g , v° \\ t\ and g -v° < 0 or £ HE) QE) the points x^ for ij E {—1, -f 1} in other words : for a point x £ Q, (x E QE) Furthermore let 0 be the center of gravity. are also contained in Q, (in Q>E) (Fig. 2). This domain symmetry enables us to define symmetry groups acting on the spaces of the velocity, the pressure, the displacement and the deformation a a ii J V {v£ C C22'> (n^p)) ||\v{x' v{xiJii)=v S)=v *{x);iJ {-1,+1}} V={vE (Q^p)) v{x V = = iv€V~(Uc({x);i,j v' (x);i,je €E {-1,+1}} {-!,+!}} 2 a 22a a ij 1 Jij U (Sh^p)) u(x«) -l.+l}} u = {u {ue cC >«(p^p)) <* )===uu^{x);i,j u (x);i {x)-i,j £6e {{-i,+i}} E{>p)) |I {uE£ C^ <{Q(Sh^pj) | u(x'J) «(*«) u^(x);i,j 22l 22aa aa > (n (
T\ i
\ /If
JClc(
/ i P p;p(xii)=p(x);i,je{-1,+1}} ; p(s") =p{xy,i,j E pp = =p p;p(x i)=p(x);i,j € {{-1,+1}} -!,+!}}
Finally this kind of symmetry is preserved while solving an elliptic problem where the operators parameters, the domain and the right hand side have the
151
appropriate s y m m e t r y and the solution is unique. Therefore if (v,p) E (V x V) then solution m aa solution solution uu of oi (A (A 7) () is is always always in in U U and and vice vice versa versa \iu it u £U £ u then then aa soluti (v,p)p) of (V of (A (A 6) 6) is is always always in in (V (V xxx V). V). Under der the the introduced introduced symmetry symmetry assumptions assumptions the the conservation conservation of of angular angular mo n m e nn tt uu m turns out out to to be be always always fulfilled. fulfilled. T T hh ee conservation conservation of of momentl momentum m turns reduces from the vector valued equation to a scalar identity in the direction of the gravity force. Taking into account (nc = — TIE)
/If
onadx= p0gdx = fI
T*(v*.r)*.iD\ntd T*(v*,p*,ip)n*da*
At present we will have a closer look at the astatic load in the reference con figuration. figuration. which has shown to be of significant importance for the stability and the existence of a unique elastic deformation. a(pog (pog}}Tn) Tn)
— \ J£l J£IEE 5
xx < <& pogdx dx-\+ < 6g>) pog pog ax -\- /I /
JJJ d^lE d£l 8£IEE
xxx0w®®-L T T( \v (/ ,/ p, p*,p*, < , <,p(pP)n ) )nnddaaa a T(v^,p*,
x iTn( r Tn n) i / ( E/ n B *i)/>o<7i * i W i ++ / a ffen 0°0 ndniE !E *i(* ((L i)i( ) i + 00 U*2(Tn)
V V V
k^(rn)2
00 0
000
0° 0
2
\
o
/8* / 3(x*{Tn)* (TTn« )) 33 // U f l3 f_
T h e integral Jn x\ vanishes because 0 is the center of Q,E- If we assume smallness of the pair of unknowns v,p — p , which will subsequently be verified in our small d a t a analysis, we further evaluate a(p0>og,Tn) g,Tn) = diag
( / \\JdtoE
za(Tn)acta) / «=!,...,3/ a=if-,3/
diag I I( / -px ] +0(\\Dv\\ + 0 ( | | D v | | + \\p \\p-p\\) - p\\) = diag -p anxa dx) na dx ) \\\Jda \Jda ...3/ //„=■. a = l,:;3/ E E and on account of fda
xana
a(p0g, Tn)
dx = J n
= -p\SlE\
daxa
dx —\ QE | we end up with
id +0{\\Dv\\
+ \\p - p\\)
This leads us to an additional stability assumption A 9 (Positive Hydrodynamical Pressure) The mean value of the pressure small positive constant. ip p= / p* dx* 0 p= p*dx> >0 Jtoc(
is a
152
In the following L o w i n g , let iv.u us n o always a i n a j o understand u i i u ^ i o u a n u the UXAV. smallness o i i i c * i i n v , o a of wx v,p v, y — p y in m the sense t h a t 00((||||JD ODv\\ iV; | | + \\p — p\\) < cp |\ QE I\ f°r for a sufficiently small constant constanl c . This ensures the hypothesis of L e m m a 4 t h a t tra(pog,Tn) is not an eigenvalue of a(p0g,Tn) . In our current model we have left one free parameter, the scale of the prescribed velocity dd which naturally has to have the opposite directic direction city at the boundary oil of the le gravity force to stay in the space of symmetric velocities V . We have 1to prove the coi con here is a velocity v° v constant on the container walls such si re t h a t there servation woul release aan ation of m o m e n t u m is satisfied. In the dynamic model we would elastic stress therel the strei tic body in a large container. Then it will accelerate and thereby at its the st is b o u n d aary r y will increase. We suppose t h a t after a while the stress and tl gravity the asvmptoticlv balance. This limit behavior should be re dty will asymptoticly related to tl static ic modeling. Therefore we n a m e v° the limit velocity. Subsequently vwe will prove existence of a small limit velocity. Therefore we multiply the Navi< Navier Stokes equations by v* and integrate over Q c (<£>)• 0 == / K • V)u div ('Zi/cD(v*) ( 2 i / c £ > K ) -p*id)V' - P* i d K dx* (v* V j i rVv ^ -- div dx^ Jnc(tp) p D{vtptp)D(vipip)-ff )-pPdivv dxtptp-2vcD(v divvipipdx
== 0 + /
- [f =
o2vc^D(v*)D(v*)dx* n ^ m ^ u ^ . ^-v° o
JCLc(
[/
II /f
fgn^i fii T^n^da*
T*n*cdaA
UdO,
J
T h e nonlinear t e r m vanishes because of the constant boundary conditions. Us ing this as a representation for the second integral in the m o m e n t u m equation we obtain we oDtain / p0gdx= Tgn*da*=[ f^n^da^-f pogdx= [/f T*n* da? = jf fgn*da* P* — -L U cU,U, — J. ll U,U, Jdtp(n Jdip(n JCtE E Jdtp(nEE)) Jdip(a Jn Jdtp(a Jdip(n E)E) C
= jf = /
C
divfgdx*div fgdx*divT%dxv-
C
/f f
C
-~ f Jd(p(CtEB) Jd
v v g-x^n^da^ 9-x*n cda
f£n*da*+g\ f£n*da*+g\
K-V K dK a cfe* : ^ - - / / f*n*dav f£n*da*- + g\(p{SlE)\ = /( (v* ■V = Jn / c(
-g -g
ff T£n* f£n*da* cda* Jon Jan
+ gg\
•v°(p \n
=>
I
vvv D(v*)D{v )) D{v*)D{v D(v*)D(v*)
v dx dx* dx*
153
Let us denote the left hand side of the last equation by a\((p)v° and the right ie by a2((p, v°). Here we look at v^ as a function of v° ;and take into hand side account that v* is a solution of the Navier Stokes equations ggained by the implicit function theorem. This ensures first order linear growth depending on v°. c\v°
| < I K | | J J I
\
Whence we obtain the estimates 2 c | v° | 2 < c\\v*\\ cll^H^! clKUlp H1 < 2i/ c /
ip D{v )D(vip)dx^ D{v*)D{v*)dx*
< CH^H^i C\\v*\\2H1
where C, c can be chosen independent of the deformation (p if
such such that that aa2(t>°) < a2(<£>,^°) < a2{v°) 2(v°) < a2(^p,v°) < a2(v°) Furthermore we know that a,2(v) a^-, •) is a continuous function. We recognize that for small ip (p the term ai((p) 1 t>o vo are lall ip the term ai[(p) is is positive, positive, because because po po > > 1 1 and and g g and and ^o of opposite directions. For small \g\ there exist small bounds v°,v° for >osite directions. v°,v° ^ciioiis. For ror small sman \g\ \g\ there mere exist exisi small siiian bounds uounus y_ , v0 for the velocity defined by the following relations ai((p)y° = 0,2(21°), a,i(ip)v° ^ i ( ^ ) = a (v°) :y defined[ by ^^ii((0^^))^^^00 = == o^2(^°) 2{2 by the the following following relations relations ai((p)y° ai((/?)v° = = 0,2(21°), 0,2(21°), 0 ] 00) > and v° < < v°. v°. Therefore Therefore we know that a (v_°) < < ai( a>i( aa-i((p Then by by continuity continuity small limit limit velocity velocity v°((p) v°((p) exists exists in in the the interval interval [v°. [v 0 ,^ 0 ] mity aa small 0 with ai((p)v° d2((p,v ). We We formulate formulate this this result result in in the the following following lemma lemma. ii((p)v° = = a2(tp,v°). a2(^p,v°). Lemma configuration na 5 5 (Existence (Existence of of the the Limit Limit Velocity) Velocity) Assume Assume that that in in the the configura described by (A 5), (A 6), (A 7), (A 8), (A 9) the gravity g is sufficiently small. Then there exists for a fixed deformation
The Fully Coupled Traction Problem
Above we have discussed the fixed point argument for the mixed boundary oblem. Following these lines we will now prove existence for th< problem. the pure traction problem of an elastic deformation coupled with the flow in a surrounding fluid container.
154
Figure 3: The cone of possible equilibrium points
Theorem 2 (Existence of an Equilibrium in the Nonfixed Case) Let Q,,Q,E be domains fulfilling (A 5), (A 8) with boundaries of Class C3, assume that (A 9) holds for p and let po > I and g be sufficiently small. Then there exist a constant limit velocity v°, an injective deformation a velocity v e C2>a{ttc( V (
u(T((v^p«)(v0(v)))
Now we can start with our analysis and derive the appropriate estimates similar to those in the mixed boundary case. At first we recall that the constants
155
c(e), c(e) c(e) can in some sense more precisely be written as c(e),c(e) =
0(v°,p)
Lemma 3 and Lemma 2 carry over with no change. Only the result of Lemma 1 has to be reformulated. We know that for fixed v°
\\(v, r2(vo)-(v,pr,(vo)\\ ^ , .
\\(v, | | ( «PP,yp ) *(v»)-(v,p)* V ) - (v,P)vl{v°)\\c'.°xCi.~ < 0{v\p)\\^ - ^ |l\\c |C2,„ c xCl c (v»)\\c>.*ipxci.°<0(v\p)\\P. y(v ,« , , ^ , .
ol
2
01
which is an obvious consequence of the properties of the mapping S((p)(.) discussed above, we obtain
o
2 \{v,Py\v\v -- KprVwiic^xci.{V,PY\VQ(^))\\C^XC^ \\{V,PY\V\^)) \\{v, Pr\v)) {^))-{v,pY\v°{^))\\c^c^ 2 < 0(V°,P)\\
Due to Lemma 5 0(v°) < 0(\ g |) . Finally it remains to prove that
\v\ |l«V)-«V)l zV,>V2)-v\^)\<0{\g\)y ) - * V ) l < 02(-vUl\\c^ | ) l |2^-^||c>.-^||c^ Therefore we have to observe the momentum conservation even more carefully. We define a new function F(
a2(
such that the zero set of F mVx{R+ -ei} consists of the set of pairs (c • This is possible due to the differentiability of differential the mapping S(
+ O{(v0)2)
(p ) ) ( 0 ) ; ||C(^)|| with C{>cc> > 00. . Inserting this into the JD t,oDK voD{v /p p f i n a t i n n W P n b t . a i n above equation we obtain
F(
(Po0\Q \-\ I - I
C(
3 0((v°) ) 0((v°f)
156
and end up with 2 2 2 DvoF(p{SlB)\) E | - | ^ 2 £ ) ^ ++ G
<0
>c>0
Taking into account the general quadratic behavior of the function F((p,.) it follows from F(>clip) 2W c(>cc>>00 v o( p andJ. v°( c i)ii^ n i q u e o solution u i u i i w n u of i z F((p,.) ^y> «y —= w0i nin o the n e i iinterval iici vai 0 [v0 ,>iv^°]Now we can estimate the value of F for the point v°((p) with small ]. Now we can estimate the value of F for the point v°((p) with a asmall offset S o > 0 in the second argument vo with v set S(Lo with t\ t\i ■ •< 5 „ o > 0 in the second argument ovo u tne F(c() -c((p) -c((p) U N\g\\S \g\\S o\ F f t t . v°(
/
<
22 2 00 A(ip i ( ^ 2 )(v^(v°), ) ( ^ 2 {v°), v*\v v^iv0))
2 1 \g-v°\\\ |5-w° - -1E1)\\ ^^( (^^) )1111 + | 11(Q 111E^)\-\
22 2 22 2 1 C U I A2 I2 llv. - v-^ H !!^.< C(\ v° | 2 + |I ga || ««° v°0 Dllv \)\\
with A((p)(v,w) ^-{vijipij A(
+ Vj + Wj,i4>i,i)det'Viipiii)(wiiiipitj +
1 1 1 i ( ¥^>2 2))(( X v^)2 ) - Afa ( y 22)(v ) ( vX',v*') , vv') -- A{y ^ 2,, v^ i ^ 1)^ ), ^ , vf v^1)1) == iA(ip i(v?l)(v* )(v2,v ' 2 , v^) X ) ++
2
1
2
l 2 1 >* 2 ,v^) Aivvl ,a)^^* Aip1)^'^*') )(v i(vp )(v ¥ ' 2 ,^ ) -- - A{ip Mv1)^.«* ),v* ) + 2 1 1 1 i(v? 1l){v' )(wplv,' 1 , w^ A{ip vv') ) - iA( y^ 1 ))(^^ 1 , v*1)
Summarizing all the estimates we obtain for a small 85^V F(
c(>
\ gI H \| ^\\6 U | | VcV\\»\\CC.*.?\ -,,.\ ) \\6
(J„o I -I——I a I \\ ll<J,„IUa.« % +h(L, ^ , v° v °-8„o) - M <<^c(u>) ) | < /I |a I I - I (-\tvo\+£r\g\\\S (-\& v0\+-^-\g\\\8 v\\c^\ v c>.<.) Therefore know that outside the LC we we Know unau ouusiue wie cone cone \(
^ + |^o |<-£- \g\ \\8J\c>.\\6 v0\<—\g\ + 6v,v°±6 ^^.^i^o) , t ; 0 v±o)\M | \5 \&vo\<£-r\g\ l l ^vl\\c>A l c ^ }\
157 the function F does not vanish. Then the set {(
i
2 0 \v {^)-v\^)\
158 12. J o h n G. Hey wood: The Navier Stokes Equations: On the Existence, Regularity and Decay of Solutions, Indiana Univ. Math. J. 2 9 , 639 - 681 (1980) 1980) 13. Trhomas h o m a s J.R. Hughes, Jerrold E. Marsden: A Short Course Course in Fluid Mechanics, Mechanics, Publish or Perish, Berkeley (1976) Boundaries for Flow 14. Tr.S. . S . Lundgren, ren, P.R. Sethna, A.K. Bajaj: Stability Boundaries Boi nduced Motions Nozzle, Journal J >tions of Tubes with an Inclided Terminal ft Induced of Jounds and Vibration 6 4 ( 4 ) , 553 - 571 (1979) Sounds Jerrold E. Marsden, Marsden, T h o m a s J.R. Hughes: A Mathemati Four 15. Jerrold Mathematical Foundation of rentice-Hall,Englewood Elasticity Prentice-Hall,Englewood all,Englewood Cliffs, Ne^ Elasticity New Jersey ((1983) i, D.M. McQueen: A Three-Dimensional Three-Dimensionc Com} U.S. Peskin, Three-D 16. C.S. Computational El Wethod for Blood Flow low in the Heart, I. Immersed Imi F\ Method Elastic Fibers in a ompressiblehie Fluid, Journal of C Coom mppuuttaattiioonnasl Physics Viscous Incompressible Com Physi( 8 1 , 372 Viscous - 405 (1989)) / . A . Solonnikov, proble for the 17. V.A. nikov, V.E. V.E. Scadilov: Scadilov: On On aa boundary boundary value valut problem stationary system In stationary of Navier Navier Stokes Stokes equations, equations, Proc. Proc. St< Steklov Inst. Math. system of 1L25, 2 5 , 186-199 (1973) )9 (1973) 18. V.A. / . A . Solonnikov: motion of a heavy i i k o v : Solvability Snluahilitii of nf a problem nmhlp.m. on nn the t.hp plane nlnrie. \motion viscous nscous incommpressible capillary liquid partially filling a container cont Izv. Akad. Vkad. Nauk. SSSR Ser. Mat. 4 3 , 203 - 236 (1979); English English transl. in M Math. a t h . USSR Izv. 1 4 (1980) 19. F. ?. Stoppelli: SulV existenza di soluzioni delle equazioni delV elcelastostatica isoterma Isoterma nel caso di sollecitazioni dotate di assi di equilibrio, equilibrio, Ricerche M a t . 6, 244 - 282 and 7, 71 - 101 , 138 - 152 (1957) 20. N . M . Wigley: Mixed Boundary Value Problems in Plane Domains with Corners, M a t h . Z.115,33-52 (1970)
159
Regularity for Steady Solutions of the Navier-Stokes Equations MICHAEL RUZICKA, JENS FREHSE
Institute of Applied Mathematics Beringstrafte 4-6 53115 Bonn, Germany rose @ fraise.iam.uni-bonn.de We show the existence of a regular solution to the steady Navier-Stokes equa tions in a bounded five-dimensional domain. Further we give an overview of re cent results to the question of regularity of weak solutions to the steady NavierStokes equations.
1 Introduction. One of the most challenging open problems in the m a t h e m a t i c a l theory of the Navier-Stokes equations is the question of regularity of weak solutions of the t i m e - d e p e n d e n t three-dimensional Navier-Stokes equations <9u dt dt
. „ -_ « A = A uu + + uu •• V V uu + + Vp Vp = ff F F
div u = 0
iiinnn(((000,,,TT T)))xxxOO O...
(1.1) (1.1)
T h e existence of weak solutions u <E L ° ° ( 0 , T ; L 2 (ft)), V u G L 2 ( 0 , T ; L 22(Q)) (ft)) was already proved by Leray [19] and Hopf [17]. Even today, sixty years later, Lave only partial answers, (1.1 can we have mswers, to the question whether isolutions of (1.1) lop singularities orr not. They are not at all satisfactory. Basically there develop satisfac Basically wo types of results related to this question. are two irstly, Serrin [22], using ideas of O h y a m a [20], showed t h a t a weak so Firstly, solution which locally satisfies the additional condition u €ELLPp((00, T, T ; L; 9L( J^ l)) ))
(1.2)
for 3 q q
22
> < 11,1,,, --- + -- < p p
1
(1.3) (1-3)
160
is locally in space regular. Afterwards many authors generalized this result in several directions ections allowing equality sign in (1.3) see e.g. Fabes, Jones, Riviere [4], Sohr [23], 23], von Wahl [28], Galdi, Maremonti [12], Struwe [24] and ; Takahashi [26]. Secondly, mdly, Caffarelli, CafTarelli, Kohn, Nirenberg [2] using ideas of Scheffer [21], proved that "suitable" weak solutions for which 1
/8 rt+r2/*
lim sup - / r-+0
r Jt-r2/8
n,
\Vu\2 dyds < e
/ J\x-y\
are regular in a neighbourhood of (tf, x) (They also prove that the singular set of a "suitable" weak solution solution has has one-dimensional one-dimensional parabolic parabolic Hausdorff Hausdorff rr measure utable weak zero).. In In this this connection connection we we also also refer refer to to Struwe Struwe [24] [24] who who studied studied the th fivedimensional steady problem. problem. This This is is motivated motivated by by the the scaling scaling properties of nsional steady propei (1.1) (see Caffarelli, Kohn, Nirenberg [2]). Indeed, if u(t, x), p(t, x) is a solution (see Caffarelli, Kohn, Nirenberg [21). Indeed, if uft, x), v(t< x) is a s< to (1.1) then also 22 u\(t,x) X\i(X u A (t, x) = Au(A *,t,Xx) Ax)
2 z (t,yx)x) = = XX2p(X t, Xx) pxx{t p{XH,Xx)
(1.4)
3 2 is a=t solution /, Xx). solution to to (1.1) (i.i) with witn force iorce f\(t, ixyi, x) xj — — AATf (A (A~I, AX). From rrom this tnis point point of oi view view thei time variable has scaling dimension two, which motivates the study time variable has scaling dimension two, which motivates the study of of the the steady five-dimensional problem
- A ,u u + u Vu V u + Vp = f div u = 0
i] in
fi Q
(1.5)
instead of the time-dependent three-dimensional problem (1.1). Also the integrability integr ability properties of the solutions of both problems are sim similiar. In r deed, from the E L°°(0, £°°(0,T; L2(Q)), the properties of a weak solution to (1.1) u G J 2 22 10 3 10 3 V u G L {0,T;L E L / (0,T; LL 10/ (Q)). T;L(Q)) (Q)) one easily deduces u (G (Q)) On the 2 other hand, in the five-dimensional which ( s-dimensional steady case we hhave Vu G L (Q.) 10 3 5 3 gives u G L °// (fi). Similiarly, for both cases. miliarly, one obtains p G £ / fo These arei the reasons attention on the question Dns why we here focus our atl of regularityr of solutions of the steady problem (1.5) in higher di] dimensional NN domains Q, CC M M . . ItIt isis known known that that aa weak weak solution solution to to (1.5) (1.5) whic which belongs N to the space W^2{Q) n [11], von Wahl fl L (fl) {Q) is regular (see Sohr [23], Galdi [11], [27]). Thus for 2 < N < 4 every weak solution is regular, a result, A which was also proved by different methods by Ladyzhenskaya (N — 2,3), Ger Gerhardt [13] (N = 4), Giaquinta, Modica [15] (2 < N N < 4). Recently, there was some progress in the studies of the problem (1.5), also for N > > 5. Frehse, Ruzicka [5], [6], [7], [8], [9], [10] and Struwe [25] showed the
161
existence of 3i regular regular solutions solutions of oi (1.5) (1.5) in in several several situations. situations. The The purpose purpose of oftthis note is to survey the main results in this direction and then to explain the survey the main results in this direction and then to explain the basic ba ideas of the le proof proof in in the the case case of of aa five-dimensional five-dimensional bounded bounded domain, domain, which whicl is also instructive for the other situations. Let us start with the main results. 1.6 Theorem. Let Q, C RN, N = 5,6, be a smooth bounded domain and let f G L°°(Q,). Then there exists a weak solution u , p to problem (1.5) with Dirichlet boundary conditions which is regular, i.e. for all r G [l,oo)
uew&rr($2), (J2), ocV
(1.7) (1.7)
'
pe<;r(fi). ni PROOF : Frehse, Ruzicka [8], r-i [10] [10]
■
1.8 Theorem. Let Q = M5 and let f G C£°(IR5). Then there exists a smooth solution dution u,p to problem (1.5). P RROOF O O F : Struwe T251 [25]
■
1.9 Theorem. Let Q = (0, L)N, 5 < N < 15 and let f G L°°(Q). Then there exists a smooth solution u , p to problem (1.5) with space periodic boundary period conditions. is. PROOF : Frehse, Ruzicka [7],[9] and Struwe [25] (for N = b). 5).
m
Note that all proofs are quite different and use different properties of the Theorems 1.8 and solution. Theorem rheorem 1.6 is an interior regularity result, while The 1.9 are global worth mentioning bal existence results of smooth solutions. It is also wo: here the following new regularity criterion, which holds in all dimensions. 1.10 Theorem. Let Q C 1 ^ , 5 < N < oo, be a smooth bounded domain and let f G L°°(ii). }). Then every weak solution u , p to problem probk (1.5) with Dirichlet boundary conditions nditions which satisfies
( y ++ p +) P) W(Q) ) (y+p) eC(fi) + e£L% (\
+
C
(1.11) (l.n)
is regular. PROOF : Frehse, Ruzicka [8] [8]
■
162 2 M a x i m u m P r o p e r t y for t h e H e a d P r e s s u r e . From now on we restrict ourselves equations - Au + + -t- uu •• V v uu -h
to the study of the steady Navier-Stokes Vp = fi vp =
in Q Q in
ddiv i v uu = div = 00 u = r\0
(2.1)
on <9Q, £>
5
for a bounded smooth domain Q, C M . We assume t h a t f G L°°(Q,) and consider weak solutions u , p of (2.1), i.e. u G Wli2(Q), p G Wl^lA{Q) (mean value of p is zero) satisfy for all
JJ^^^ dt+S Jd 1tlx ^+Pidx+ J UJwi ^iVij d x + j ^ i --—dx+
n nn
/ Ui-r-^-Vi dx+
n nn
= fi9idx fitpidx
JI -
I -—<£>i = / fi
n nn
a na
(2(2.2) 2)
'
2
We will firstly show t h a t the head pressure ^ - + p has a m a x i m u m property. is was was observed observed by by Gilbarg, Gilbarg, Weinberger Weinberger [16] [16] and and Amick, Amick, Fraenkel Fraenkel [1 This [1] in a pprpnt. rnnt.Pivt, F o r m a l l v . m n l t i n l v i n P ' ( 9 T l hv 11 and a d d i n c t o t.Vip rpsii different context. Formally, multiplying (2.1) by u and adding to the result the pressure equation -Aprr
dui duj __-^L_dlvf
OXj OXi OXi OXi
(2.3)
we get, denoting V u o V u = §^-§^f, |^-|^f, 2
2
-- AA ( ^ -( y + p) + u - V v ((^y- ++ pp ) = V u o V u - | V u | 2 + f - u - d i v f . (2.4) - A ( y + p) + u - V ( y + p ) = V u o V u - | V u | 2 - f f - u - d i v f . (2.4) Now one easily realizes t h a t Now one easily realizes t h a t V u o V u - |V Vuu| 2| 2 << 00,,
(2.5)
which implies by the m a x i m u m property of the elliptic equation (2.4) t h a t u2
y+P
c.
(2.6)
cannot cannot be De m m aa dd ee rigorously rigorously for tor all a weak reason why Theorem 1.6 does not reason why Theorem 1.6 does n< hold the properties of a weak solution one
163
easily sees that u • Vu G L5/4(Q.) and thus we cannot use u G L10/3(12) as a test function in (2.2). The situation changes if we consider the following approximation of (2.1) for s > 0 ££ = f - A u ££ + u££ • Vu £e + £ | u ^£ |uV£ f+ vVp // — i
Q in Q
£££ div div uuu = = 00 ££ uu == 0o
(2.7) (2.7)
on <9Q . on an
It is easy to show the following (cf. Frehse, Ruzicka [5]) 2.8 L e m m a , Let f G L°°(Q). Then there exists a weaAweaA' solution u£, p£ to (2.7) satisfying for all
y 4^+u^+£|u£|2u^dx+1 ^ d x = J f i r i
n
such that
a
uu e| | |ll , 2 < t#f , Illl|K 111,2 i ^ , e\\u% i4
-
n
, N (2.10)
i4
2.10 (2.11)
£ £ £ 2 1/4 | | £ iu£\u | u\%| A/3 |o,4/3 | |
K /f | | e u £||p*Hl,5/4 K | 2 | ,5/4 | o , 4< < /, ,^ 1 / 4 , ||P*Hl,5/4
dx (2
(2.11) (2.12)
w
l
' *
Note5 that that by by continuity continuity the the weak weak formulation iormulation (2.9) (z.9) holds holds for all
in Q
111 a£
oonSfl on n c<9ft, to,
(2.13) (2 13) '
Note t h a t here one should also introduce a further smooth divergence—free approximation of u e , such that uke —>• u e e W*'*(Q). However, since all estimates are independent on A;, we will skip this approximation here. The interested reader can find all details in Frehse, Ruzicka [5].
164
dist(xo,d£l), where u = \i£ is a solution of (2.7) and &h{xo), 0 < h < dist(x 0 ,9^), is a smooth non-negative approximation of the Dirac distribution satisfying supp 8h{x (x0) C Bh{xo) ,
/ S8h(xo)dx = 1. h(x0) dx = n
2 ((2.14) -14)
We now have 2.15 L e m m a . For all e > 0, h > > 0 there tiiere exists exists aa solution G = = G\ G£h G e C°°(fi) G°°(Q) H D 1,2 2 iy0 {Q) (Q) to to (2.13) ( 2 . 1 ^ satisfying WQ' [vGV
lu-VG
such that
1,22 Vy> V
G G >> 00 ,, l|G||o,oo C(h) IIGIkoo < C(h) c(&),,,
(2.17) (2,17)
2 2 | 2 ),
(2.18)]
l
"
where the constant c(h) is independent ofe and u. where 1,2 PROOF : The existence of a solution G G E C°°(Q) n P7 W001,2 (Q) follows imme ^— ^ ~— „.— ^ v_ ^ v-w7 ■ ■ " u v-v diately from the Lax-Milgram elliptic :-Milgram theorem and the regularity of linear lin from the Lax-Milgram theorem and the regularity of linear e equations. The non-negativity of G G follows from the non-negativity of the -negativity non-negati LS. The non-negativity of G follows from the non-negativity right-hand side of (2.13)i. (2.13) 1. In order to show (2.17)2 we use in the weak formu nd side of (2.13)i. In order to show (2.17)2 we use in the weak i lation (2.16) the test function (p = Gss. After some rearrangements (using the !.16) the test function (p = G . After some rearrangements (usi structure of the equation and Sobolev's embedding theorem) we arrive at
+1 S 5+1 (j |G|( s+1 >f >t da;) < c||«5 c\\S ||0t00 ,oo(s (s + + 1)J 1) da . (J |G|(' )J dx)* dx)1t < c||^||hh\\0> l)JJ \G\ dx 0oo(«
Denoting
.0 So
= 2,
«+!=(§) «+!=(§)
«o *o + | ( | ))
we get Moser's recurrence formula 1 s 1 s IIG||n. llGllo^^c'^'isiY^lMl^llGllo^, s_
-1
(2.19)
165
which immediately gives (2.17)2 (see e.g. [14]). Finally, if we use 2,IIVGII2 ||VG||2 .i4 . O + UVulln II ' — I I U , i^2 -
But one easily checks that 2
2 rflL« IIW2II?2 )44.<||G||o,oo||V < II/7IL._ll\7 l|VG||^ G||o, l|VG|| <||G||o,oo||V G|| 0 ,2, 2) which immmediately implies (2.18).
■
2
2.20 R e m a r k . In fact, we want to use \+p as test function in (2.16). This is not possible mainly due to the boundary values of the pressure p. Naturally we only have j £ = f • n + Au • n and not p = 0 on <9ft as required by
Now ow we can derive from (2.9) the weak formulation of the pressure e< equation he approximative equation (2.7) using , where %/> somesmooth for the \j) is is some ,4 ir function. By Rv continuity continnitv we we get cret for all all ip IIJ GGL°°(ft) L°°(Q) ODW0W} (Q) scalar £°°(ft) ' (ft) I Vp Vip dx = / Vu o Vu ipdx — e / ||u| u |2' u .-Vr/>dx+ • Vrf>dx+ V ^ dx + /I/ ff •• Vi/> VV> Vipdx. dx.. VpVipdx VuoVuipdx-e dx JJ J J J (*) *)f)\ oo\ J J J (*) oo\ z,ZiS aa a a a l a a a \L.LL)i 22 From ip = GC ,0 < G is is aa possible Froma (2.17) (2.17) and (2.18) follows that C£°(ft) a possible [z.i.i)222 and ana (2.18) y^.io)222 follows ionows that LnaL V \p = = GC crc, ,0 ,u < ^ ((s 2G t: G£°(ft) 0 5 [*<•)is 2 test G test function function in in (2.22). (2.22). On On the the other other hand hand
y +p) GC2dxdx++£ ej\u\\4uGC dx V(Y+P)V(UC)+U>V{Y+P) \ uc> dx a a 22 22 22 22 = // ((Vu V u oo V u -- I|Vu| GC dx-e |u| V{GC dx )) GC uu ••• V{GC = Vu I Vu| Vu|*) G'C dx-e dx-e ff |u| |u|*u V (G'C))Jdx dx a a
+ // ft -• VV(tfC ( G C 2 J) ++f * --uGC uG uGC C 2 2dddx. xx.. + [f a
( 2 .23) (2.23)
166
Using now (2.5) and (2.17)i we obtain
/y VV((y^ + )) V V V(G( p) )G p)G( GCC222dcadx br - +ppp) (( G2C) 2 ))+++uu -•VV( (yyy++p+ n ft ft
(2.24) 2 2 l« . // \ 2)_L+ ff • dx + V(GC |u||2„ u .• Y7^/0/-2>i V(GC2) J^ dx+ f ff . •T7(nr
< --e £ j/f
2
u GC dx d ,
ft ft
which together with (2.21) (note that ^ + p E W 1 ' 4 / 3 (Q)) implies f u /* u22 Jy SMh{x P)C22dz dx *0o)(Y )(y + + p)C
ft ft
2 < /f uu .- V (y + + pp))Gdx-2JGV(\+p)ve ) GGd^x-- 22 j/ G V ((yy++pp) )VVCC2 d2 x^ V C?(^ GV
ft n ft
ft nft
2
2 22 - jG{^+p)&(; A dddxxz-- e / |u| V(GC V(G(222)dx )dx / ^ ( ^y - ++ppp)))A ACC C2dx-ej\u\' |u|22uu-V(GC ft n ft ft a. u
(2.25) (2.25)
2 2 ) 2 ) ++ f - uuf-uGC + ff-V(GC / f -V(GC GGCC2 d2dx, ^x , a ft
where of course u = uue£ and G = G£h. Let us first pass to the limit in (2.25) as e tends to 0 for fixed but arbitrary h > 0. This is possible due to the estimates in Lemma 2.8 and Lemma 2.15. We get u2 Sh{x0)(Y+P)C2dx
f J ft
v 2( 2 ■2(Y+P) V C 2 (Gdx y - + pGdx ) G~ d x -//jsGV "V< (3(y+p) < Ju-VC yV+( yp )+Vp )
-
/G(y+p)AC2dx+
(2.26)
O
/ f .V(GC 2 ) + f - u G C 2 d x . ft
ft ft
In order to handle also the limiting process h —> 0 we need estimates indepen dent of h. In the same way as for the Laplace operator one can show
167
2.27 Lemma. L e m m a . For the solution Gh of (2.IS) (2.13) (now with u solving (2.2)) we have q VgE J\G f\Gh\ \qhqdx
(2.28) V r66E[1,5/4), [1,J Vr Vr [1,5/4),
where K is independent of h. This enables us to prove the following L°° -estimate of Gh, Gh away from the sin gularity X X00.. 2.29 Lemma. Let B2R C ft be be a bail such that Bh{x0) 0 BR = 0 for 0 < h < ho. Then we have for S > 0 C \\Gh\\Q ||G*|| 0> 5/3-*• • HGTJIO.OO.BH ^ ) l|G/i||o,5/3-(5 too,BR^ < (C{R)
PROOF :
(2.30) (2.30)
Let BR+P R, rp = 0 in ft \ 5 H + /,9 , 0 < p < R be L.ei rrpp E t G£°(ft), u 0 " ^ i j , rrpp = —1 i in in sB OR, rp = u m it \L>R+ U <^ P,
s given cut-off the off functions. functions. Using Using (p >0,0,inin(2.16) (2.16)we weget get(using ( structure of the "convective" term u • VG and Bh(xo) C\ BR = 0)
3 3/5 5 3/5^ Co J_ ( j ( S ++1 l)5/3 rf;c) +11)10/7^)7/10^ 10 7 ( yfG G^ dxf J/ £(. G(+s+1 dx)V1°, ^+1^3^dx) W '7cte; 0^{0\( (
BR
(2.31) (2.31)
BR+P
which is the starting inequality for the local Moser iteration technique (see e.g. [14]), which immediately gives (2.30). ■ Now we can handle the limiting process in (2.26) as h —>• 0. Indeed, all terms on the to Lemma he right-hand side of (2.26) involving V£ can be handled due to 2.29.. The remaining integrals on the right-hand side of (2.26) converges converge; due to (2.28) considerations 8) and (2.10)i.. Note, 1NUUC, that LllClb the bllC last I d S l term LCI111 in 111 (2.26) ^Zr.ZrU^ restricts I C O b l l U l S our U U i consid< UUlli to the ,use already for N = 6 we have G EE£L9q(ft), tie case TV = 5,, because <3/2 and (ft), qq < x1 u E L 3 (ft) and hence G does not belong to L x (ft). Also ]Lemma 22.29 cannot nee u G be proved the e: exponent >roved in this way, because on both sides of inequality (2.31) th< would Id be 3/2 and hence the Moser iteration technique does not woi work. Jut for N = 5 we proved a local L°°~ estimate for ( ~ -f p) . • But
168
L°°(tt). Then the weak solution of (2.1) constructed 2.32 T h e o r e m . Let f G L°°(J2). in this section satisfies for all compact subdomains £lo QQ ofQ u2 sup —c(f,fi sup — +p + p <
(2.33)
(2-33) 2
This This one-sided one-sided estimate estimate enables enables us us to to get get aa weighted weighted estimate estimate for for \\ -\-\- p p using in fact only the pressure equation. Due to the local character of ict only the pressure equation. Due to the local character charac using in fact of the the estimate now localization. us therefore 2.33) we we will will from from now now on localization. Let Let us estimate (2.33) (2.33) on often often use use aa locahz< therefore "^00 i_ _ _ J. J.: r..„ D /r\\ fix some notation. Let r G CQ D(QI) be a truncation function and denote Q\ = Dtation. Let r G CQ (QI) be a truncation function and dei fix some notation. denote Q\ = {x ((x) 1). Let CS°(Bi(0)) be {x G G Q;T(X) Q;T(X)v) = = 1}. 1}. Let C C ee Cg°{Bi(0)) Cg°{Bi(0)) be aa given given function function with with C(x ((x) = = C(kl), C(kl), aa n n (( l C = 1 o on n ^1/2(0)- For given x0 G ^1 and 0 < R < dist(5^i, XQ) 00 < C < 1 < C< 1 l C= 1 ^1/2(0)- For given x0 G ^1 and 0 < R < dist(d£2i, £0) we we define define CR,X =C C(( Ol^oO*) C*,*o(z) = C( 0{X)
°) p °) °) •••
fl fl
We shall omit the subscripts R and xo when there is no danger of confusion confusion. Note that |V*C*.*ol<|j *=1>2> suppVCn,ro SUppVCil,ro iUppVCil,ro C T f^T i / 2R/2( i{xo), ro), where T XR(x ^ o0)) = =
B2R(X0)
\
BR(x0).
2.34 T h e o r e m . Let f G L°°(Q) L°°(0) and let iet u , p be the the weaA' weak solution of (2.1) constructed before. Then /|U2 , CR)XO [ (U'(x-Xo))2 2 dx / h r + P 1 —i? + / ^ ^ R^-CRrn^
(2.35)
where cc — wnere = c(f, CR) does not depend on xQ G ^2 = {^ G £V, ^ i ; dist(x 0 , <9Qi) > 2i?}. 2i^}. of ut :: Choose ^noose in 111 the me2 weak weaK formulation iormuiation 01 the me pressure pressure 1 (2.22) with e = 0) ip = = C 2 | ^ - ^ o | ~ 1, where £ = CR,X0- After I) with 6 = 0) ip = C |a? —icol" , where £ = CiJ,ar0- After
PROOF
equation equation (compare some computations some com]
169
we end up with
3■ / ■
( u ^ o ) ) !
^
b \x-xXn\ 0\
J
b
X -
a"2
=: 22 / (ir ++p)pI UU i UJ U -+P)\~ )l^F^-. „A3[3dx-dx- / Ui ' ^^fci^ J ' « - ^ - T 7 — — r dx i(Y J
v
^X-XQ]6
2
J
J
dxidxj
\X - X0\
a /• (X - X0)j dC2 , a f . . 22 1 / t/, i/j^ j j - — eta - / pAC -j r da: J \x-x0\3 dxj J " \x-x0\ o
o ft
da; 3. id 3 - 5 r - J// /'«dxi ^ -ldxi\x-x Ta;I-\x-x —ar0T\\ ^ ^ 7J P I\x-x k„ c-x -0a\0-3\rdxi ,c/a:,dxt 0 5x 0
a 7J
a
2 \x | £-- zx \| ^. *f^c 00
d 6
wce get Using the identity (*£ ))+ -- | |!i r£ + ( 1 T++ p) P/)-= =' 22\ ((2T^ + + FpP>+ + pnP| |"we J + ^ i2 ^
(u-(*-z0))2,2j„
J aa
55
10
/>
2
, _,
c2
,_
J7 ' 22 ' |' a|z: -—z 0#| 03 | 3 a a C2 dx / w <9 u 2C 1i /f uu 2 c /* C2 da: g - J T+ + Ppy+u . 1 \5 // u^ » ^Jio—5—1 rT d d+ /) + /+ /J v(vl-^ „ . 3 + ^ r s r T i i — T ^ '+ \x■— ——a?or XQ\ JJJ dxidxj \x XQ\ 22I — a?o| OXiOXj \x x \ \X|ar £ J OXiOXj \X — X 0| 0 OXiOXj \x — xo\ a ana {x-x0)i dC , a f AA2 1 , a / - 20 / muj) '-j—dx;dx Uj[ 3 ^ — dx- / pAC] pAC 1 ;dx J 'Ix-xo^dxj \x-x \z dxj J \x-x \ Q \X — X0n + 2 UiU \X — Xn\ dxOXA pA aI dx #0| p£ — - £o|
J
J
6
^^^^ -J ^\x^\
2[0
J
|a?-a?o| \x — XQ\3 c/a:,dxj
JJ
dxi \ dxi \x\x -—x0XQ\
|z-£o| 3x,-
7
* 3z,- | s - a ? o |
(2.36)
(2-36)
+ 2 a/ J « - « o ) ^ & + aa/ / i ^ _ L _ d , 3 J
a
h^^Cdx \X — X n P
The first integral on the right-hand side is bounded due to (2.33). All integrals containing VC2 or V 2 C 2 are bounded using the properties of the weak solution and the fact that \x — #o|~ 7 € LOO(QI\BR(X0)) for 7 > 0. The last two integrals are finite due to f G L°°(Q). Thus we get (2.35). ■
170
3 Hole-Filling Technique. In order to get further information, which are better than (2.35) we will use the so-called hole-filling technique (see Widman [29]), which is widely used in the regularity theory of elliptic equations. We have 3.1 Lemma. Let z G Lloc(Q) satisfy for allx0 G £1, R > 0 such that
/I
\z\dx
BR(X0)
B2R(XO)
C
|z| + cRa \z\dx + cjHr \z\dx \z\ dx dx-\-cK +
//I TR(XO)
with constants c and a > 0. Then there exists (3 G (0,1) and CQ = co(Oo) such that I
13 \z\dx
V V Xx0 Q0tEeG^QQ Q Obzbz CCCCCQ Vx ^Q VX CC i^^" . VX'o 0 0C
BR(X0)
In principle one first had to show that the quantities for which we will apply Lemma 3.1 belong to the space LlQC(Q). But basically this follows from the way we derive the hole-filling inequalities and thus we will not stress this point here. Also we will use singular test functions of the type (?\x — #o| _(5 , S > 0, which are not allowed at first sight, but they can easily be replaced by C2 (jx — xo\2 + h2) , h > 0, which enables us to make all operations rigorous. But in order to make the idea clear we will use these singular test functions. Let us first derive a useful inequality. Using in (2.9)
•((u')2 £€ £ h =- ff-u ~fdx+ (^-+p)u fu i --/dx+ u 7 aa; -t- /f/ ^—9 i — {^—t-+p)u h"p/j uu vV~fdx. vV'ydx. laxaa a a But the second term on the left-hand side is non-negative and thus the limiting process as e —> 0 gives VuV(u7) c?x u(—+ u VL (vu^117^ 7 ) c^ tux e <<\ // i uu( (u\^_—-++p+ p))p)Vydx VVVjdx 77V7 cc tt ux e ++ f if-wydx r // VVuV(u7) dx p) +-r / f-wydx. f-wydx. ——rjf^j o o n n a a
(3.2) (3.2)
171
Note again that for a general weak solution of (2.1) this procedure is impossible. In the inequality (3.2) appears the term \+p, for which we have already more information. Let us therefore first investigate the following auxiliary problem u2 Ag = T +
infl
P
g=0
(33)
on dQ,.
3.4 L e m m a . Let f G L°°(Q) and let u,p be the weak solution of (2.1) constructed in Section 2. Then there exists a weak solution g G W2}5/3(Q) n WQ'*,2(SI) to (3.3) such that Il#||o,oo,loc < C y| BRFt
I x -- x«or ol3
(3.5)
(3.6)
F| ' J o|3 ./ '' 2 2, ^ '| zx --z xnr
-
B-2R E>1 R B2R
W'^v^ 4- JK dr. 1 1i / \X7n\* 1 / iv^l \x \Z— XQ\~ Ili"* j TR
where c = c(f) and K\ is independent of xo G £2i and 0 < R < dist(a?o, d£l\). : The right-hand side of (3.3)i belongs to L 5 / 3 (Q) and we immediately get the existence of a weak solution g G W2>^3(Q) H W 0 1,5/2 (fi). From (2.35) follows that we can use the fundamental solution of the Laplace operator C2\x~ x0\~3 as a test function for (3.3)i. This immediately gives (3.5). Using now the test function ( t f - ^ K ^ - ^ o l " 3 , where C = CiMo a n d # = I 7 *!" 1 ITR(X0) 9(X) dx> in the weak formulation of (3.3)i we get PROOF
r
"2
Vo^
-
77;dx+ /
U_77l2 V
^
~' C V .
^rfa
ft ft /•
„2
A2
-oh n - 22
"
'
r l^, _ 77|2
r:;dx+ n
^—r^-VC\
^-
d
X
Xn\a
_
/ Va(a-~a\.
n
and therefore using Young's and Poincare's inequalities and the definition of
172
the fundamental solution of the Laplace operator we arrive at 2 I IY7„|2
_ i
/„ , | „ / ^ \ -^12
\Vg\ [3 dx + \g(x °)-<» /l^irr^r^ + Wl^)-^
J ^9l\ir^3aX
+
X0
BR
< ,oo,loc // |hI TTrT r ++ ++ pP I| lI i T ^#| 221-. - d.* < C||^|| c||^||o,oo,ioc ' 0 0o,oo,ioc i7»dx 333t o^ ++cccc ///| vVv\Vg\ 1 n l■ s0 | 3 JJ ''' 2 2 | 5 c — £ 7 F —a?o| ~, ^ o | 33 7 2 '|a: —J?O| J |x \xx \ J \X■ | 5 c — £ | 7 F ~ ^ o | 0 0 BoR TR
;clWlo.oo.ioc
l T ^ u r T j 3 ^ + / l ^ 77—;^<
which is nothing else but (3.6).
■
Based on (3.2) and Lemma 3.4 we can now show 3.7 Lemma. Let f G L°°(Q) and let u , p be the weak solution of (2.1) constructed in Section 2. Then we have for some (3 > 0 2
2
/|Vu| U l l h—?—dx+ & U^ // iVV U I \f aX ^ + /// u I—L1 3rgdx 3^ J \X XQ\ J \X - \x-x XQ\^ \x - x 0\ 0\ BR
BR /
•
22
I
J7JI TR TR
/*
1
2 l -—dx js dx + 2K K22 j// \Wg\ |V V ^$ | 2-] h— x —r -dx Sdx $dx+K dx 6 6d6 6 | z— - x#o| 7 | j ?\x-xo\ -\X-XQ\ ao\ rE \X-XQ\ \x 7 F 0 r 0| d \x-xo\ J \X — Xnr / \X — Xn\ B B2R R
l
(3.8)
B2R
l + K V ^ | 22-dx + K2BP K2R?, K22 /I ||V<7| —$dx , d3 JJ! 7 k -—-^\X-XQ\ |\xx\X-Xo\ &o ol r6 XQ\" TR T R TR
where K2 is independent of XQ G &I and 0 < R < dist(#o, dQ\). PROOF : Using in (3.2) 7 = £2\x — # o | - 1 , where ( — (R)XO we obtain 2 2 2 C -dx /|V | Vuu| | — -£ ^ - d s + [u / u 2 ., ._ ds n3dx \x-x J \x -— 0\\o F | 7 F-a?or F-a?o| F xxx*o| o00\l\33 J7 \x-x ^ -—z XQ\ J j \x \x-x 0 .0.
.0.
=: / ((\+p)u-V ^ p ) JU^dx+ d * + / f •• fUi U- r -u ^j ^fC V .• V V-jr ^ -dC x ^ 7^ ^22+ +■pjU i |X ^-—x X jo00l|fr| ^d X + J ^ j \x 7 -- za?o| XQ\j |a: a? kF ^ F 7 o l \X-Xo\ \x-Xo\ ft ft
(3.9)
o
X
2
Xo
* *!%H - )* dx. dx dx - [Uu I a:—- a? xo\ oxi \x \x -- x§\ \x7 dxi 2 |a? JJ + J7 ^^u—zi ^rt—da 2 \x - x-0\| J dx{ \x - xx0\\3 + / ^!AC n
XQ\ 0
36
n
3 XQ\ 0
Using equation (3.3)i in the first integral on the right-hand side of equation
173
(3.9) we get after partial integration /-2
,2
22
7
I s—- a^ol \x XQ\
2 2 < IVU|2T-^ r i\X-XQ\i J |ar — is ar 0 |
i/i^'c^i^'/^irrrp ^
B2R
c
2 l' V^V^l |^TT—ris + Ce / ^ _ i -s F' ^ + u^T iR^^^F ^ * L / /u ^ i]-] .^. .^. i 3
+ / TR
TR
\h2v
+ ^ / u 2 r - ^ — 3- d x , *o| which together with (3.9) and the assumptions on f imply (3.8).
■
The only term without hole on the right-hand side of (3.8) is fB 1V DR
1^+^1.7-^^+ 2
. : : r «*« (3.11)
R„
<
A
l
dx
u2
; r 3rdx a x ++^A 3 i^ 1 ^—J3
P l j o f l X t A 3 3 // U"-j 3 // |h'T2rT t+ - ^HiI x - —ti x o l 3 "+^*"3 ; " l I x - x o ld ' T7R
' 2
' |x - z
0
r
7
TR
'
P ~ zo|
r^ is * independent .• J i J. of ~r XQ „ ^Gr* i n0 ^ * i./~ a5 n where K3 ^1 and < r> R^ < jdist(#o, ^ i\) .
But in the inequality (3.11) we have only terms with a hole which also appear on the left-hand side. Thus the cycle is closed and we have 3.12 Theorem. Let f G L°° (Q.) and let u,p be the weak solution of (2.1) constructed in Section 2. Then we have for some a G (0,1)
If\Vuf\^\dx^IURa'
BR
(313)
174
where K4 is independent of xQ G &I and 0 < R < dist(# 0) dQi). : Add to (3.8) {K2 + 1) times (3.6) and then ({K2 + l)I 0
PROOF
/
| V u |12 1
j '1
:dx+
|a: —s oxn arl0| '' lhe a -—
BR i
/ U21
JJ/
B BR R
f (u- {x-xo)) iz. --x£o| J 55
, /* , u 2 + *+ J lT 2
d
R
+
r,dx
x nr \xx~—~xxo\ o \he
BH
d 77 3 ^ ^ ' />l Vv^*''"l F1Ii^W ^-*
p
i 1 l' i|zI 3-^aT 5
5R BR
2
D _
BR
2 2 22 tf //Ui—1—dx u ^dx K CA' ++ IU ||V V
TR
TR
pP : di T ++Hir-^ l ] ^ r ^ i rf 3r^cte+/< ■ ^ / /l T— «+ ^ TR
which is the hole-filling inequality. Now Lemma 3.1 immediately gives (3.13).
■ From (3.13) we easily get that 2 Vu€^+a(fi),
(3.14)
which will be the starting point for the iteration procedure in the next section. 4 Full Regularity. Let us start this section with two useful lemmas, which can be easily deduced from the results in Chiarenza, Frasca [3] by a usual localization argument. 4.1 Lemma. Let v G WQ,r(Q), 1 < r < 5, satisfy VvG^(O), 2
0< A < 5 - r .
(4.2)
B y Lq'X(£l) we denote the Morrey space consisting of functions from Lq(£l) for which R X sup ~ fn„(xn) \f\qdx < °°> where SlR(xo) = Q,C\BR(x0).
x o efi,0
175
Then we have v€i^(Q),
(4.3)
where 1 q
1 r
1 5— A
(4.4)
4.5 Lemma. Let f 6 L r (0) n L\£(£l), 1 < r < oo, 0 < A < 5, and let v, IT be a soiLition to the Stokes system -Av +
VTT
= f
in
div v = 0 V
Then we have
e
VTT
S
(4.6)
on <9fi.
= 0
V2v
n
^ l o c (J2).> h
\oc
(4.7)
(0)
Note that Lemma 4.1 and Lemma 4.5 hold also in arbitrary dimension N. Based on these two results and (3.14) we can increase the integrability of our solution of (2.1). 4.8 Lemma. Let u , p be the solution of (2.1) constructed in Section 2. Then there exists a / > 2 such that /,l+a
V u € i ;-"loc 7"(Q) r
(4.9)
peC"(fi). where a is determined from (3.14). PROOF
: Let us denote
IQ
= 2. From (3.14) and Lemma 4.1 follows
ueig' 1+a (n),
(4.io)
1 1 1 - = - - . q0 l0 4- a
,
where 4.11
176
From Holder's inequality and (3.14), (4.10) we get u-VueC1+(,((l),
(4.12)
where r0
go
«o
«o
(4.13)
4- a
Now Lemma 4.5 implies V2u,VPGC1+a(fl),
(4.14)
which due to Lemma 4.1 gives Vn,peLl^1+a(Q),
(4.15)
where 1 _ 1 /i r0
1 _ 2_ 4— a /0
2 4 — a'
(4.16)
Clearly, for a > 0 and /o = 2 we get *i > Jo , which concludes the proof.
■
4.17 T h e o r e m . Let u , p be the maximum solution of (2.1) constructed in Section 2. Then u , p is regular, i.e. for all gG [1, oo) C uew?; uJ G W" "^ci *(fi), i )" ; o( cfV
„ P W j^(fi). wi-'ffi) pP ee€ ^iccl")-
(4-18)
PROOF : An iteration of the above procedure gives recursive formulae for /„, rn given by 1 2n 2 4 - = — - ~r—{T - 1) for i- > a , /n
/0
1 2n+1 — = —: r„ /0
4
~
a
2U
2n+2-3 4- a
3 for — > a . 2n
M1QY l
j
177
Notice, that both — and ^- tend to —oo, but formulae (4.19) make sense only if the right-hand sides are positive. Notice further, that the condition in (4.19)2 is violated at first, but this means that (4.14) holds for all r < 4 — a and therefore (4.15) holds for all/ < 00. This and once more an application of Lemma 4.5 imply
Vu, P eC a (fl) v2u)vPGir-1c+a(n) which is more than stated in (4.18).
/e[i,oo), re [1,00), ■
A cknowledgement We would like to thank our colleague M. Steinhauer for careful reading of a previous version. REFERENCES 1. C.J. Amick, L.E. Fraenkel, Steady solutions of the Navier-Stokes equations representing plane flow in channels of various types, Acta Math. 144 (1980), 83-152. 2. L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier- Stokes equations, Comm. on Pure and Appl. Math. 35 (1985), 771-831. 3. F . Chiarenza, M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl. 7(7) (1987), 273-279. 4. E.B. Fabes, B.F. Jones, N.M. Riviere, The initial value problem for the Navier- Stokes equations with data in Lp, Arch. Rat. Mech. Anal. 45 (1972), 222-240. 5. J. Frehse, M. Ruzicka, On the Regularity of the Stationary Navier-Stokes Equations, Ann. Scu. Norm. Pisa 21 (1994), 63-95. 6. J. Frehse, M. Ruzicka, Weighted Estimates for the Stationary Navier-Stokes Equations, Acta Appl. Mathematicae 37 (1994), 53-66. 7. J. Frehse, M. Ruzicka, Existence of Regular Solutions to the Stationary Navier-Stokes Equations, Math. Ann. 302 (1995), 699-717. 8. J. Frehse, M. Ruzicka, Regularity for the Stationary Navier-Stokes Equations in Bounded Domains, Arch. Rat. Mech. Anal. 128 (1994), 361-381. 9. J. Frehse, M. Ruzicka, Regular Solutions to the Steady Navier-Stokes Equations, Pro ceedings of the 3. International Conference on Navier-Stokes Equations and Related Nonlinear Problems, Madeira 1994 (ed. A. Sequeira), 1995. 10. J. Frehse, M. Ruzicka, Existence of Regular Solutions to the Steady Navier-Stokes Equations in Bounded Six-Dimensional Domains, (to appear), Ann. Scu. Norm. Pisa. 11. G.P. Galdi, An Introduction to the mathematical theory of Navier-Stokes equations, Vol.2 Nonlinear stationary problems, Springer, New York, 1994. 12. G.P. Galdi, P. Maremonti, Sulla regolarita delle soluzioni deboli al sistema di NavierStokes in domini arbitrari, Ann. Uni. Ferrara 34 (1988), 59-73. 13. C. Gerhardt, Stationary solutions to the Navier-Stokes equations in dimension four, Math. Zeit. 165 (1979), 193-197. 14. M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Birkhauser, Basel, 1993.
178 15. M. Giaquinta, G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math. 330 (1982), 173-214. 16. D. Gilbarg, H.F. Weinberger, Asymptotic properties of Leray's solution of the stationary two-dimensional Navier-Stokes equations, Russ. Math. Surv. 29 (1974), 109-123. 17. E. Hopf, Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen, Math. Nach. 4 (1951), 213-231. 18. O. A. Ladyzhenskaya, Investigation of the Navier-Stokes equation for a stationary flow of an incompressible fluid, Uspechi Mat. Nauk 14 (3) (1959), 75-97. 19. J. Leray, Sur le movement d'un liquide visqueux emplissant Vespace, Acta Math. 63 (1934), 193-248. 20. T. Ohyama, Interior regularity of weak solutions to the Navier-Stokes equation, Proc. Japan Acad. 36 (1960), 273-277. 21. V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math. 66 (1976), 535-552. 22. J. Serrin, On the interior regularity of weak solutions to the Navier-Stokes equations, Arch. Rat. Mech. Anal. 9 (1962), 187-195. 23. H. Sohr, Zur Regularitatstheorie der instationaren Gleichungen von Navier-Stokes, Math. Zeit. 184 (1983), 359-376. 24. M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. on Pure and Appl. Math. 41 (1988), 437-458. 25. M. Struwe, Regular Solutions of the Stationary Navier-Stokes equations on R5, Math. Ann. 302 (1995), 719-741. 26. S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manus. Math. 67 (1990), 237-254. 27. W. von Wahl, The continuity or stability method for nonlinear elliptic and parabolic equations and systems, (to appear), Rend. Sem. Mat. Milano. 28. W. von Wahl, The equations of Navier-Stokes and abstract parabolic equations, Vieweg, Braunschweig, 1985. 29. K. O. Widman, Holder Continuity of Solutions of Elliptic Systems, Manuscripta Math. 5 (1971), 299-308.
179
DECAY OF NON-OSCILLATING SOLUTIONS TO THE M A G N E T O - H Y D R O D Y N A M I C EQUATIONS M.E. SCHONBEK University of California, Santa Cruz, California, CA 95060 E-mail: [email protected] In this paper we study the decay of solutions to the Magneto-Hydrodynamic equa tions. We show t h a t if the magnetic energy decays to a limit L; that is, if the energy of the magnetic field is non-oscillating, then the energy of the velocity decays to
1
Introduction
In this paper we study the large time behavior of solutions to the MagnetoHydrodynamic equations in all of R 3 -space ut + u • Vw - B • V 5 + Vp = Au Bt + u-VB-B
• Vu = 0
(1)
div u = div B = 0 with the initial conditions (ti(s, 0), B(x, 0)) = (tio(s), B0(x)) e X where X will be described below. We show that if the magnetic energy J R 3 \B(x,t)\2dx tends to L as time goes to infinity then the energy of the velocity J R 3 \u(x,t)\2 dx tends to zero at the same rate. This establishes a conjecture described in Moffat's paper 3 . We will present the proof for smooth solutions. The proof is also valid for weak solutions for which a sequence of smooth approximating solutions can be constructed which converge weakly in L2. In the second case one applies the proof we give to the approximating solutions and then passes to the limit. The proof is based on the Fourier splitting method 2 ' 4 . Here we use the modified version which permits to treat the integrated equations x . We will use the notation
\\u\\2 = /3
|u|2rfx,
JR
\W\\H-=
E / jR JR \\«\<m a\<m **
\Dau\2dx.
The main theorem we establish in this paper is the following
180
Theorem 1 Let (u(xyt),B(x^t)) (L2{K3)nL1{K3)nH2{Il3))2. \\u{-,t)\\2 ^0 fort-* oo. 2
be a smooth solution to (1) with data (ito, BQ) G Suppose that \\B{.,t)\\2 -> L fort -> oo. Then
The decay
In this section we establish Theorem 1. As was said in the introduction, the main tool is the Fourier splitting method. The proof we present can be used for solutions in R n , n > 3. In what follows we suppose the existence of smooth solutions. Such solu tions should exist at least for small enough data. Proof of Theorem 1. The first step is to derive an energy inequality which involves the energy of the magnetic field, the velocity and the gradient of the velocity. Multiplying the first equation in (1) by u and the second by £ , integrating in space and adding both equations yields after some integration by parts:
±■1
(\u\2 + \B\2)dx = - /
iVufd*.
Hence integrating in time over [s,t] we have (\u(x,t)\2+\B(x,t)\2)dx
/ JR.*
(2) = - 2 f/ \Vu\2dxds+ dxds + [ Js Js
(\u(x,s)\2+\B{x,s)\2)
dx.
JR3 JR3
We first need to estimate the Fourier transform of the solution in a neighbor hood of the origin in frequency space. Lemma 2 Let u(x,t) be the first component of a solution to the MHD equa tions (1) satisfying the conditions of the Theorem. Then
N*,<)l ■<(! + £)■ Proof. Taking the Fourier transform of the equation we derive an ordinary differential equation which yields for u(£,t) u{£yt) = u0e-W2t-
f Hfos^-M'V-Us Jo Jo
where H(S, s) = uVu(£, s) - BVB(t, s) + Vp(£, s).
(3)
181
Noting that the pressure p satisfies the elliptic equation A
d2
"=-Eft^j(«*«i-^j).
it follows that
Vfcp V*p = ZkP - i^tkZiZjiuiUj i^£kZ£j{uiUj
- BiBj)
and hence
\Vkp\
\H\
so that
m, t)\ < c+Co f \mn\i+wBtfje-w-') dS Jo
(4)
*c+w Here we used that UQ £ L1 and hence UQ G L°°. D Returning to the proof of Theorem 1, we use the Fourier splitting method with the modification introduced by Wiegner x . Let v(s) = \\u(s)\\l -f ||5(5)||1 5 t h e n ( 2 ) r e a d s
v(t) -v(s) < Let S(t) = {£ : |£| <
r
■\t\<[f]
I
\Vu\2dxds
mil/2
},}, where g(t)
will be specified below. Hence,
by the Fourier splitting method, vwe have v(t)-v(s)+
f
g{rf\\u{r)\\ldr
JS
(5) 2 < < f/ 9(r) 9{ry I/
Js Js
JS{r) JS{r)
2 2 2 m, r)\2 didr < f/ g(r) [C(g(r)nn«'2 + " )] dr \*{Z,r)\'dtdr< g{ry[C(g{r) + 5 "/g»«-')]dr
Js Js
182
where we need (3) for the last inequality. Now following Wiegner let e(t) = * / > > ' * , hence = e(t - h) f g(r)2 dr + hc(h) Jt-h where e(h) —> 0 as h —> 0. Now write
(6)
e(t) - e{t -h)
e{t)(v(t) -L)e(t - h)(v(t - h) - L) = {e(t) - e{t - h))(v(t) -L)+ e(t - h){(v{t - h) - L) - {v{t + h) - L)) so that by (6) e(t)(v(t) - L) - e(t - h){v(t - h) - L) = e(t - h) [ g{r)2 dr(v(t) - L) + e(t - h){v(t) - v(t - h))hc{h)(v(t) - L) = Jt-h g(r)2[v(t) - v(r)] dr + v(t) - v(t - h)
e(t -h)\ Ut-h
g(r)2(v(r)-L)dr +
/ + Jt-h
he(h)(v{t)-L).
Writing v(r) --L L = = \\u(r)\\$ \\u{r)\\2 + + (||B(r)|^ (\\B(r)\\22 - L), we get Siting v{r) - L) - e(t e(t -- h){v{t h){v{t -- h) h) -- L) L) = = e(t){v{t)t) -L)r /■*
e{t -h)\
I
g{r)22[v{t) - v(r)] dr + v(t) - v{t - h)
(7) V + /'
Jt-h
\\u(r)\\l h) //f 9(r)J2\\u(r)\\ldr]+e(t-h) \\uv)\\2UI dr\ f+" ce(t v t ' _- '*; J
Jt-h
{r)22{\\B{r)\\\ (\\B(r)\\2 9g(r)
- L) dr
i)-L). +he{h){v(t)-L). +he(h)(v(t) - L). _1 ) , with a sufficiently lai — a(t + + l1))-_11 , with a sufficiently large. Then Let g(t)2 = a e{t) eft) = =eeaL So #&r = =z(t (< + + !)«. U<*.
Let T0 be such that for t > T0 we have ||||5(*)||1 | | 5 ( t ) | | | - L\ < c. From (5) and (7) it e. F follows that, for t > To, e(t)[v(t) -L]-
e{t - h)[v(t - h) - L)
183
< e{t - h) [ g(r)2[v{t) - v(r)] dr + C [ e(r)(r + l ) ~ n / 2 dr Jt-h Jt-h + ee{t -h)
[ g{r)2 dr + he(h)(v{t) - L). Jt-h
(8)
Note that \v{t)-v(r)\
if
<0{h)
|*-r|
since, by (5),
\v{t) - v{t - h)\ < [
(r+l)-n/2
Jt-h Hence summing (8) over intervals of length h it follows that for t > T e(t)[v(t) - L) - e(T)[v(T) - L] <0{h)
[ e(r)g{rf
dr + [ e(r)(r + l ) ~ n / 2 d r + e /
JT
JT
g{r)2e(r)dr
JT
+ c(h)[v(0) - L] Let h —»■ 0, then e(i)[v{t) -L}<
e(T)[v(T) - L] + C(t + l ) « " n / 2 + 1 + eC0e(t).
(9)
Here we used that / g{r)2e{r) dr = C0 [ {r + l ) " " 1 dr < C{t + 1)*.
JT
JT
Dividing by e(t) yields / \u(t)\2 dx + j \B(t)\2 dx-L<
e
-^(v(0)
+ L) + C(t + l ) - " / ^ 1 + eC0
concluding the proof of Theorem 1. A cknowledgement s Research partially supported by NSF Grant No. DMS-9020941. References 1. M. Wiener, Decay results for weak solutions to the Navier-Stokes equa tions in R 3 , J. London Math. Soc.(2) 35, 303-313 (1987).
184
2. M. E. Schonbek, Decay of parabolic conservation laws, Comm. in P.D.E. 7,449-473 (1980). 3. H.K. Moifatt, Magnetostatic equilibria and analogous Euler flows of ar bitrarily complex topology,J. Fluid Mech. 159, 359-378 (1985). 4. M. E. Schonbek, L2 decay of weak solutions of the Navier-Stokes equa tions, Arch. Rational Mech. Anal. 88, 209-222 (1985).
185 T H E STOKES P R O B L E M FOR E X T E R I O R D O M A I N S IN HOMOGENEOUS SOBOLEV SPACES * HERMANN SOHR Fachbereich Mathematik-Informatik, Universitat-GH 33098 Paderborn, Germany
Paderborn
MARIA SPECOVIUS-NEUGEBAUER Fachbereich Mathematik-Informatik, Universitat-GH Paderborn 33098 Paderborn, Germany, e-mail [email protected] We prove the Lr — solution theory, 1 < r < oo, of the Stokes equations — A u + V g = / , div u — g, U\Q$I — h for Ftn and for exterior domains Q, C Fln, n > 2, in homogeneous Sobolev spaces characterized by | | V m n | | / / r < oo, m > 0. Further we prove the embedding properties for such spaces.
1
Introduction
It is well known t h a t the t r e a t m e n t of elliptic boundary value problems causes m a n y difficulties in unbounded domains. In particular the usual Sobolev spaces Hm,r with m — 0 , 1 , . . . and 1 < r < oo are not adequate in this case. There are several possibilities to overcome these difficulties. One way is to use weighted Sobolev spaces H™,r with weights of the form (1 + l^l)^, S £ M] this was done by several authors 2 ' 1 0 , 1 2 ' 1 6 . T h e theory of elliptic operators in these weighted spaces has been intensively studied and leads to a rather complete theory. However, there exists another way to obtain appropriate Sobolev spaces for elliptic operators in unbounded domains. These are the homogeneous Sobolev spaces or Beppo-Levi spaces L m ' r , which consist of all L\oc-functions with finite ||V m w||L r where V m w means the system of all derivatives of order m5'11. Let Q be an exterior domain in ]Rn, n > 2, with sufficiently smooth b o u n d a r y d£l. Our purpose is to investigate the Stokes system - A u + Vtf = / ,
divu = g
u\dn = h
in Q
(1.1) (1.2)
in these spaces L m , r , m — 0 , 1 , . . . . T h e force / = ( / i , . . . , / n ) with | | V m / | | L r < oo, the divergence g with IIV™"4"^!!.^ < oo and the boundary values h G Hm+2~1/r,r(dQ.) are prescribed (see Section 2 for exact notations). *This research was supported by the DFG research group " Equations of Hydrodynamics", Universities of Bayreuth and Paderborn.
186
We also consider (1.1) for the entire space Mn where the condition (1.2) is omitted. We show the existence of some velocity field u = ( w i , . . . , un) with ||V m + 2 w||L r < oo and some pressure q with IIV™*1?!!^ < oo, which solve the system (1.1), (1.2). Observe that the prescribed f and g may increase polynomially. Further we will characterize the solutions of the homogeneous system where / = 0, g = 0 and h — 0, they constitute the null space of the operator (u,q) »-> S(u, q) = (—Au -f Vg, div u, t/|an) defined in the next section. In particular the case m > 1 has not been considered systematically up to now. Some results 17,6 are known for m = 0. The theory of weak solutions (u, q) of (1.1) within the class defined by ||VU||L^ < oo and \\q\\Lr < oo is omitted here. This problem requires to introduce the case m = — 1 and has been solved rather completely by Galdi and Simader7 and by Kozono and Sohr8. Related problems for the Laplacian were examined by Simader15. The key for solving (1.1) in homogeneous spaces is the structure result on the space Lm,r in Theorem 1; for any / G Lm,r we obtain a decomposition / = p + / , where p is a polynomial and / can be approximated by smooth func tions with compact support. In Mn this enables us to apply the convolution procedure and the Calderon-Zygmund theorem to get approximating solutions converging to a solution (u, q) of (1.1). The case of an exterior domain Q, can be reduced to the case Q = Mn by an extension argument. Furthermore we need a fundamental potential theoretic result for exte rior domains which has been proved for n = 3 essentially by Ladyzenskaja9; a complete proof for all n > 2 has been given by Varnhorn17. The homogeneous Sobolev spaces considered here have been frequently used in several fields and investigated mainly for special cases 5 ' 6 ' 7,8,11 ' 15 . How ever, up to now it seems to exist no systematic treatment of the spaces including embedding properties.
2
Notations
Let G C Mn, n > 2 be a domain with boundary dG and closure G, 1 < r < oo and m E W0 = { 0 , 1 , . . . } . We recall the notations of some usual function spaces. For 1 < r < oo, m E W, Hm>r(G) consists of all distributions f with finite norm x
r
\\f\\Hm,r{G) = I J2 J \D"f\ dx
/'
187
where A =
fai
= l , . . . , n , £>« = ( ^ ) " \ .. ( ^ ) " n , | t t | = £ ? = i a,- for
a = ( a i , . . . , a n ) G W0n. For m = 0 we get H°>r(G) = Lr(G) where Lr(G) denotes the space of all Lebesgue measurable functions with finite norm
ii/iM G) = ii/ikG = (/ G i/r^
±/r
If dG is bounded and of class Cm, we write dG E Cm. There exists a contin uous linear trace operator 7 : Hm>r(G) -» # m _ 1 / r ' r (<9G), see e.g. 1, so that 71/ = U\QQ is the restriction of u to the boundary if u is sufficiently smooth. Let C°°(G) denote the space of all smooth functions in G, G Q ° ( G ) is the subspace of all / E C°°(G) with compact support in G. If G is unbounded, we set C$°{G) = {f\G, / E G£°(JR n )}, i.e. G£°(G) is the space of all restrictions / | G to G of functions / E G£°(JRn), here we have to observe that / E C$°{G) need not vanish on the boundary dG. For an unbounded domain G we also use the local spaces H%?{G)\ f E Hj%r{G) means that / E # m ' r (G n 5 ) for each open ball B so that G Pi B ^ 0. By £ r ( G ) n , tfm>r(G)n,... we denote the corresponding spaces of vector fields / = ( / 1 , . . . , / n ) ; the norm is again ex pressed by ||/||i, r (G) n = ||/||r,G etc. Sometimes it is convenient to write simply L r (G)for Lr{G)n etc. In the following we are mainly interested in the cases G = Mn and G = fi where Q always means an exterior domain in Mn, i.e. a domain having a compact complement Mn\Q,. To introduce the definition of the homogeneous Sobolev spaces for such a domain G we abbreviate by V m / = ( D a / ) | a | _ m the system of all derivatives of order m. In particular, V / = (D\f,..., Dnf). We define the space L".'(G) = {/ E L[ oc (G), | | V 7 | | r | G = ( £
||D"/H;.G
\\a\=m
I
< 00}.
(2.1)
)
Then L°>r{G) = L r (G) for m = 0. For m > 1 the expression | | V m / | | r ) G does not define a norm on L m , r (G), but only a seminorm. We have ||V m /|| r ) G = 0 if and only if / is a polynomial of degree less than m. To avoid this difficulty we fix the ball K = {x E Mn, \x\ < 1} if G = Rn and # = {z E JRn, | s | < R} if G = Q where i? > 0 is chosen so that dQ C K. Then we define the norm ||/||L».r(G) = H/Hir(GnJO + l | V m / | | L , ( G ) .
(2.2)
Lm>r(G) becomes a Banach space under the norm (2.2). To prove this and some other elementary properties we use the following
188 well known facts. First of all we need the generalized Poincare estimate 1 3 . m IMI^G)/^., < Ml^ayiP^ < C\\V C\\Vmv\\ v\\r>d ri6
(2.3)
where G C G is any bounded subdomain with s m o o t h boundary;
IHL-fG)/^-, =pjPl_1\\V
+ P
Wr,G
m e a n s the n o r m of the quotient space Lr {G)/Pm-i and F m _ i , m E N, the space of all polynomials with degree d(p) < m — 1. JPo means the space of constants. Lr(G)/Pm-i is a Banach space. Further we need the equiva lence of some norms. T h e equivalence of the norms |M|#m,r/ G \ and \\u\\r Q + | | V m u | | r G with G as above is well known 1 3 . Further, if K' ^ K is an other open ball with d£l C K1 in the case G = fi, we can easily show t h a t the norms HullL^Gntf 1 ) + | | V m u | | L r ( G ) and ||u||i,r( G n tf) + | | V m u | | i , r ( G ) are equivalent. Indeed, let (uk)kelN be a sequence with Hm/c_).00(||iZfc||Lr(Gni<:) + ||V m t// c ||L'-(G)) = 0- We choose a bounded subdomain G C G so t h a t K C\ G C G, K' H G C G and use (2.3) which yields polynomials pk E P m - i with lim*_>oo 11Wife + P ^ I I L ^ G ) = °- F r o m l i m *-»oo ||^/c||L-(GnX) = 0 we see lim^-^oo IbfcllL'-(G) ~ ^ a n c * therefore l i m ^ o o | | ^ | U r ( G n A ' / ) = 0. This leads to lim/c_foo(||u/c||L'-(GnK'/) + ||V m w/ c ||i / ^(G)) = 0 and the assertion is proved. To prove the completeness of Lm,r(G) we consider a Cauchy sequence {uk)k6N m Lm>r(G). We choose bounded subdomains Gj C G , j E Wo, such t h a t G D K = Go C G i C G 2 C • • • and G = (J^Lo G J - T h e n f o r a n Y n x e d J> (uk)keiN is a Cauchy sequence concerning the norm || • H L ^ G J ) + | | V m • ||L^(G) which is equivalent to the norm defined in (2.2). Further, (uk) is a Cauchy sequence in F m > r ( G j ) , j ' G W o , hence we get some u^ E Hm>r(Gj) with ^ ( l l t i W - ti f c || r | G . + | | V m ( t ^ ) - uk)\\rtGj) for all j E W 0 .
From G 0 C G i C • • • it follows u^\Gjl
Therefore we get some u E ISloc(G) fclim
with U\GJ = u^
= 0 = u^
for /
< j.
for all j E W 0 and with
(\\u - ii fc || r>Gj . + | | V m ( u - Uk)\\r}Gj)
= 0.
Now we conclude t h a t | | V m u | | r ) G < oo, l i m ^ o o | | V m ( u - uk)\\r,G lim (||ii - ti A || r > G o + | | V m ( u - uk)\\rtG)
= 0 and
= 0.
AC — f OO
This shows t h a t L m , r ( G ) is complete. Consider again the space Pm-i of all polynomials p of degree d(p) < m —
189 1, m E N. Further we define P _ i = {0}. Since i P m - i is a finite dimensional subspace of Lm>r(G) there exists a (topological and algebraical) complement of Pm-\. We fix such a complement Qm,r (G) by the following construction: m r Let Q > (G) = Lm'r{G) if m = 0 and let r i , T 2 , . . . , Td, d E W , be a basis of i P m _ i if 77i > 1. By an elementary consideration we find functions ^ i , . . . , ^ G L r ' ( G fl K), r ' = r / ( r - 1) such t h a t
(Tjyjy7p 7pkk))GnK GnK
= /
Tjip Tjipkkdx dx
= 8jkjk,,
j,k=l,... j,k=l,...}}d,
JGC\K
where Sjk means the usual Kronecker symbol. To each ipj we can assign a continuous linear functional on the space Lm,r(G) defined by (ipj,u) = IGHK ^3U
dx
f o r a11 u e
Lm,r{G).
By d
Pm-lU -
^2(lpj,u)Tj 3=1
we define a continuous linear projection Pm-\ from Lm,r(G) onto the subspace JP m -i. Indeed it holds Pm-\p — p for all p E iPm--i a n d -Pm-i is obviously continuous. Then /— P m _ i , where J means the identity , defines a projection onto the subspace m r m r r Q - (G) = (/ (I- - P Pm-^L^iG) Lm'r(G),(^,u) , . . .,<*}. == 0, 0,jj == 11,..., d}. m-i)L > (G) = {ue L™'(G),
This leads to the direct decomposition L m > r (G) = FPmm__ie<2 ! e' rr(G) (G)
(2.4)
in the algebraic and topologic sense. We next define the subspace
Lm>r(G) =
C?(G)MLm'r{G\
i.e. L m > r (G) is the closure of C§°(G) in the space L m > r (G). 3
Main Theorems
Our first theorem yields some information on the structure of the space L m , r (G) for G = FT and G — Vt. An important fact is that each u E L m , r (G) can be decomposed in the form u — u0 + p where uo E Lm'r(G) and p E iP m _i. This means that each u E Lm>r(G) can be approximated modulo Pm-i by functions from G Q ° ( G ) .
190
Theorem 1 Let m G N, 1 < r < oo, n > 2 and let G = Mn or let G =ft6e an exterior domain with 3ft G Cm. (i) Then for all ueLm>r(G), m m C l | | V mmu«|| | | LL ,r( (GG) ) < < |||tl Pm-lU\\Lmtr{G) < C 2 |C| V | L rLr{G) Ci||V | u-- Pm-i«||L».'(G) < (G) 2\\Vt l |u\\
(3.1)
where Ci, C2 are constants not depending on u; thus \\^7m-\\Lr(G) and \\-\\Lm>r(G) are equivalent norms on the subspace Qm>r (G) C L m , r (G). (ii) /£ no/c/s the decomposition r L Lmm> -r(G) (G)=P =m-i+LrnP>mr(G), -1+Lm'(G),
(3.2)
z.e. for each u G Lm,r(G) there exist a polynomial p G IPm-i and Uk G C Q ° ( G ) , k E N, so that lim^oo \\u — p — Ufc||z,m,r(<3) = 0. (iii) If f3 e No, p > r and m - n/r = j3 - n/p, then Lm>r(G) C # ' ' ( G ) t* a continuous embedding, i.e. (3.3)
U \\\\U\\L0>P(G) \\L».P(G) < C\\u\\ C||«||L™.'-(G) Lmtr^G)
for all u G Lm,r(G) with C > 0 no^ depending on u. (iv) Le£ /? G Wo 6e the smallest number such that m — n/r < /3 < m and let p > r be such that m — n/r = (3 — n/p. Then mm l|V'ti|| C\\V u\\L rLr(G) l|Vti|| LL,,((G) «|| {G) G ) < C||V
for all u G Lm,r(G)
(3.4)
with C not depending on u, and F ^ . i C Lm'r(G). Fu-iCL^tG).
In particular it follows IPm-i C Lm,r(G)
(3.5)
and hence
r Lmm>' rr(G) = LmL>mr'{G) (G) L (G) =
(3.6)
if r > n. Remark. (3.6) shows that (3.2) is not a direct sum in general. Setting xa = x"1 • . . . • x%n with a — ( o n , . . . , an) G Ng, we can conclude from (3.2), (3.4) and (3.5) that r L Lmm> 'r{G)=l {G) = I
Y, ]T
\P
a mm rr aaX aaxa,a ,aa€c\®L (G) aec\®L >' {G)
J J
is a direct decomposition if f3 < m and Lm>r(G) = Lm,r(G)
if /? = m.
191
To describe the properties of the Stokes system (1.1), (1.2) we use the operator formulation. For this purpose we introduce the operator S : (u, q) H-> S(u, q) = (-At/ + V#, div u) if G = Mn and the operator S : (w, q) *-> 5(«, g) = ( - A n + Vg, div u, u|$n) if G = Q. Our aim is to show that S and 5 are surjective operators with finite dimensional null spaces, this means that 5 and S are Fredholm operators. As usual the null space of S or S is the set of all (u, q) in the corresponding domain of definition such that S(u,q) = (0,0) or 5(ix,g) = (0,0,0), respectively. To characterize the null space for G = Q, we need some well known potential theoretic facts which are listed in the following lemma: L e m m a 1 Let 1 < r < oo, n > 2, m G N0, h G Hm+2-1'r>r(dQ)n. (i) For n > 3 the system - A u + Vg = 0,
divu = 0,
u\dn = h
(3.7)
has a unique solution pair 1(h) = {u,q) with the following properties: u G H%+2'r{G)n H C°°(Q) n , |V*tx(s)| < C ( k | 2 " n _ ; c ) ,
9
G J C + 1 ' r ( J l ) n C°°(«)> |V*$(s)| < C ( H 1 " " " * )
(3.8) (3.9)
/or a// ft G Wo and sufficiently large \x\, where C > 0 is a constant not de pending on x and k. (ii) For n = 2 and each additionally given constant vector a^ G C2 there exists a unique solution pair X(h, aoo) = {u,q) of (3.7) with the properties (3.8) and (3.9) where for k=0 , \Vku(x)\ < C{\x\2-n~k) must be replaced by u{x)-aoo\n\x\
= 0(l).
(3.10)
In both cases, from (3.8) and (3.9) it follows uE L m + 2 , r (ft) n and q G L m+1 > r (ft). Remarks on the proof. For n = 3 and continuous boundary data h one obtains a unique solution9 u G C x ( ^ ) n nC°°(fi) , g G C°(H) nC°°(Q) of (3.7) with the properties (3.9) as a sum of boundary layer potentials. If h G ffm+2-1/r'r(0fi)n is continuous we consider this pair X{h) = (u,q) and apply Cattabriga's lemma 4,7 to the system (3.7) for the bounded subdomain QC\K. Then we obtain the regularity properties
ueHZt2'r{Ti)\
«e/C +1,r (n)
192 which lead t o (3.8). If h E # m + 2 - 1 / r ' r ( < 9 f t ) n is not continuous, which is possible if ra + 2 < n/r, we m a y approximate t h e given h E i J m + 2 _ 1 / r , r ( 9 Q ) n by continuous functions and then we find t h e desired solution pair X(h) = (u,q) by some approximation argument. This proves the l e m m a for n = 3. A detailed presentation of these arguments and t h e complete proof for n = 2 were given by Varnhorn 1 7 . Our results on t h e Stokes system read as follows: T h e o r e m 2 (Stokes system in Mn). Let m E Wo, n > 2, l < r < o o . Then the operator m+2 rr n n >' (M {lRn))n 5S : LLm+2
m+1 r
n
n x ) x LLm+1'r'{M {M
m+1 r n n r r n nn n (R{R ) ) xx L mL+ 1 ' r ('J(M R )) ) -> -»■ LLmm><
defined by S(u, q) = (—Au-\- Vg, div u) is bounded and surjective dimensional null space
with the finite
r
m nn M {R) ) Mm(M
= = {{u,q)eP^ {(u,q)elP2l++l1 xPm, x P
m
, Aq Ag = = 0, -Au -Au+ + Vq Vg = = 00}. }. Let m E Wo, n > 2, with dQ E Cm+2. Then
T h e o r e m 3 (Stokes system in exterior domains). 1 < r < oo, and let Q C Mn be an exterior domain the operator
m+1 r m r n m+1 r m + 21 r 1 r r r n m + 2rr n n S : Lm+2 ' ' (ty xL //m+2 ( Q ) x Lm + 1 ''r ((fi) f i ) -*• -> L m '' r (fi) (Q)n x L m + 1 ''r ((ft) ft) x H - '-> / > (5Q) (dtl)n
defined by S(u,q) — (—Au + V g , d i v u , w|an) ^ bounded and surjective finite dimensional null space m
with the
n
m Af JVT (Q) (J2) = = {(u,q)\ {(«, g)|n -- Il{u\m), ( « | a n ) , («,
/or n > 3 and m +1 m+1 m
m
2
A^ (Q) = { ( i x , g ) | a - X ( T i | a n , a 0 ) , ( i i , g ) E A r ( i R ) , t / =
J
aax« + a0}
l « |==i M
for n — 2. T h e following corollaries are immediate consequences of these theorems. C o r o l l a r y 1 Let m E Wo, n > 2, 1 < r < oo. T/ierc / o r eac/j ( / , #) E Lm>r(lRn)n x L m + 1 ' r ( i R n ) ttere ariste a *oZtj*ion ( u , g ) E £ m + 2 ' r ( i R n ) n x m+1 r n L - ( J J ) of -Au + + Vg Vg = / , div -Au = /, div w w ==#;#; ( u ^ 25 unique modulo Afm(Mn) ^
and it holds the
estimate:
w
,m x (ll + ^o||L-+2,r ( i R n ) + \\q +
Tn{Mn n) {u qo)eNTn {u0i 0iqo)eN {M )
< C (||/||L~.'-(«») +
with C > 0 no£ depending •^
i
?
7 •
r
on fg.
\\g\\Lm+^r(]Rn))
qo\\L™+i,r(]Rn))
193 C o r o l l a r y 2 Let m G W 0 ; n > 2, 1 < r < oo one? let Q C f f 6e an exterior domain with dQ G C m + 2 . Then for each {f,g,h) G Lm>r(Q)n x Lm+1>r(Q) x m+2 1/r r n # - ' ( < 9 f t ) there exists a solution {u,q) G L m + 2 > r ( Q ) n x L m + 1 > r ( Q ) o / + Vq + Vg = = / , div w u=g, = #, w|aa u\dn = = /*; h\
-Au
(u,q) is unique modulo J\fm(Q) ( ^ k m , o x ^U ("o,
+
and it holds the w
estimate
<>IU m + 2 ' r (n) + Ik + 0o||L™+i.r ( n ) ) v y/
<< Cr(\\f\\ -4- |i ki /|i|iLi ,^- .+^l , r. ^( ax) +-L ii/,n„_ ^) 1„ ( | | rm / | | L « . ' -w^x (n) + |N|iJ™+ 2 - l /m+2-l/r,r( r,r \\h\\ fy) H
wi£/z C > 0 no£ depending 4
on
( < 9 ad )
f,g,h.
P r o o f s of t h e t h e o r e m s
Each proof is carried out in several steps. W i t h i n the proofs we use positive constants C, C i , C 2 , . . . which m a y have different values from line to line. Proof of Theorem
1. a) To prove (i) we use (2.3) with G = G D K and obtain inf
C\\Vmu\\r}GnK
||u + p\\r,GnK <
for all u G Lr(G n K) with | V m u | G Lr(G fl i f ) . T h u s we get inf
||tl+p||L».r(G)
for all u G Lm>r(G) and therefore, on t h e quotient space L m > r (G)/JP m -i t h e norms | | V m • || r) G and || • ||L^. r (G)/JP m _ 1 are equivalent. Since I - Pm-\ is continuous we conclude t h a t ||(7 -
Pm-l)u\\Lm,r{G)
and since {I - P m _ ! ) L m > r ( G ) = Q(G), m a p p i n g theorem yields
<
Cf||l/||Lm)r(G)/Jpm_1
(I - P m _ i ) J P m _ i = { 0 } , the open
l h l | L ~ . r ( G ) / J Pm lhl|L~.r(G)/JP -i < < C\\(I C\\(I -m -i
Pm-l)u\\ Pm-l)u\\ ,r(GG) Lrn,r( Lm
which proves (3.1). This leads t o
m
m
\\^ M|L^(G) < l|vmu\\ u||rr),GG < < Ihh-r(G) < CHV^tlHr.G c||v ii||riG
(4.1)
194
for all u G
<
C\\Vmu\\r}GnK
for all u eQm,r(G). Since the norms ||V m -|| r) GnK + |H|r,Gntf are equivalent, we get the inequality | | V m u | | r , G n x < \\u\\Hm,r{GnK)
and
IHIiJ^^Gn*:)
< C\\Wmu\\r)GnK
(4.2)
foralltiG(? m » r (G). b) To prove Assertion (ii) for G — Mn, wefixu G Lm,r(Mn) first that there exists a sequence Vk G Co°(iR n ),feG W, so that
and show
lim ||V m w - V m ^ | | r ) i R n = 0.
(4.3)
/e->oo
For this purpose we use a scaling procedure. We set Kk = kK = {x G Mn,\x\ r{Mn)withpk = Pm_iuk. Let pk be defined by pk(x) =pk{k~1x). We use the notations x =fey,£>£ = ( a ^ ) " 1 • • • ( s ^ ) " * (^T)
' ' ' ( d^w
' Then fr°m (4-2) w e S e t
tne
and
&% = &" =
estimate
11^(^-^)11^
11^(^-^)11.,^ = ( ^ = ^ - l « l + n / r (J
\D"x(u(x)-pk{k-'x))\
\D^(u(ky)-pk(y))\r
dy
k-^+n^\\D^(uk-pk)\\riK
=
< Jfe- |a l +n / r C||V™u*||r,jr = fc-l-l+n/'C ( 7 |V™«(fcy)|rd») =
andi similarly similarly
Ckm-\a\ff
l y m ^ ^ r ^
a k \\D )\\kr)\\ ,K,k,K, | | £ >(u-p > -P r k
_ £ ^m-M || V m u|| r i J f t
m m < Ckm-W\\V < Ckm-M\\V u\\rtKu\\ ,k r,K2k
(4.4)
195 for all k G IN and \a\ < m. Now we need the following cutoff procedure. Consider a function G C§°{Mn) so that 0 < <j> < 1, (x) = 0 for \x\ > 2, and set k(x) = <j)(k-1x). Then we define wk = ( ^ ( tlc/{»-P - ^V)))||r,JR >iRn = r)iRn \\V""ttffcllr.Hwk\\rr,Mi~ '))\\r,R*
kk a a k \\D\\D (u-p(u-p )D^<j> )D^ \\riK2k \\D«(u-p )D^kkk\\r,K 2k
^ E £ \<x\ + \l\=m \a\ + |or| +\-y\=m |7|=m
£ E £
a kk \\Daa(u-p (u-p )\\ \\D (u-p )\\rtK2k \\D^ ^KK2k) \\D )\\rk,rtK2k \\D^ ,k) K2) k\\ k\\ LL^ IP>-P
|a| + |7|=m |a| +h|7|=rn |7|=m \a\ + \-y\=m m m m m
||I> 0, 1«*||r,«- < C | | V m « | | r , R . ||^^||
r ) i R
n
< C||Vm|i||r)iRn
for | a | =ra.Therefore, the sequence (Dawk)kej^ is bounded in U(Mn) for each fixed a with \a\ = m. An easy calculation shows that the sequence (Dawk)kejN converges weakly in Lr(Mn) to Dau for | a | = m. Since there is only a finite number of such a we can use Mazur's theorem 18 , and find a sequence (wk)k£]N where each wk is a finite convex combination of suitable elements wk such that even a lim \\Dawk-D-au\\ Dr)Mn u\\Ti]R« = 0
/c-*oo
holds for all a with \a\ = m. This yields lim ||V m iu* - Vmu\\r)]Rn = 0.
fc—>-oo k—voo
Now we can apply Friedrich's mollification procedure to find a sequence K ) f c e j v in C^(Rn) with m = 0. lim | | V m ^ - V r ou\\ u||ri]R «. = 0. r>n
(4.5)
A:—>-oo
c) Since (4.5) does not imply lim/^oo \\vk - ii||r,/r = 0 in general, we will modify each vk constructed above to obtain uk G C™(Mn), k G N such that lim \\U-P~ lim \\u-p-
k-+oo /c-*oo
U \\ m,rf n) = 00 U*||L»WJI») = k
L
M
(4.6)
196 with some p £ F m _ i . Then we obtain u = u — p £ Lm,r(Mn) which leads t o the desired decomposition u = p + u for G = JR n . For this purpose we choose t h e smallest number (3 £ No and p > r such t h a t m — n/r < j3 < m and m — n/r = f3—n/p. Then Sobolev's embedding theorem 1 leads t o t h e estimate m I| |IV VV V ** "- vi)|| Vl)\\p,*< C\\V C\\Vm(v (vkk P f «» <
- i;0|| t^Hr,*r>J Rn
(4.7)
for all k,l E N. Using (3.1) with m,r replaced by /?,p we obtain polynomials pk — Pp_iVk, k £ N, so t h a t (vk ~Pk)keJN is a Cauchy sequence in L^,p(Mn). p n T h u s we get some v £ lf> (R ) with lim ||r7 \\v-— (v = 00. (vkk -Pk)\\Lfi>p(iR») - Pk)\\Lfi'P(JR») = Since p > r we obtain I K --Pk~ P/c ~ (V/ {Vl -- Pl)\\r,K P/JHr.A- < < C\\v C||Vfc -Pk~ p* - (V/ (Vl ~ ~ Pl)\\p,K Pl)\\p,K k -
(4.8)
for all k,l £ N. Using (4.3), (4.7) and (4.8) we conclude t h a t (vk — Pk)keiN is a Cauchy sequence in Lm>r (Mn) with a suitable limit v. Now we show v = v. From (4.8) we get V\K = V\K> B u t t h e same conclusion holds if we replace K by any larger ball K D K\ it follows v\g = v\^ for all open balls K D K and therefore v = v. In particular we get l i m ^ o o ||V m t> — VmVk\\r,]Rn = 0. W i t h (4.3) this leads t o Wmu = Vmv and hence there exists some p £ i P m _ i with u = p + v and lim \\u-p|TX — J> —(vk(V-k -Pk\\L'».'-(Mi») Pk\\L'».'-(Ml») = 0.= 0. It remains t o show t h a t each polynomial pk £ IPp-i can be approximated in Lm>r(]Rn) by C^°(Mn) -functions. We m a y suppose /3 > 1 since pk = 0 if f3 = 0. For this purpose we fix any k £ N and show t h a t m l|V m(^p (^P*)||r,R» f c )|| r | J Rn < C
(4.9)
for all j £ W , where j(x) = <j>{j~lx) is defined as in Step b) and C is a constant independent of j . Let a £ W " with | a | = m. Using (3 > 1 and /? — 1 < m — n / r which follows from the construction of /?, we get for | # | > 1 that m n rn/r |<71 \Dapk{x)\ C i l ^ l1^-^- 1 - ^ < -r '--W (4.10) \D"p < Cilxf' < Ci\x\ Ci|« k{x)\ for all <J = ( < n , . . . , crn) £ N£ with cr < a , i.e. Oj < a j for j = 1 , . . . , n . If we keep in m i n d t h a t D°pk — 0 for a = a , (4.10) leads to ||i5 a (^-p*)||r,R-
<
C i £
/
?< k <2.? °--aai<\*\
\Da-°4>i\r\DaPk\rdx
197
<< Cc2J2 2 ^
m n/r W[ T r {j-\ {\x\m--nlr-W) dx (rlaa-°\) -aly{\x\ - ydx
f/
a aa
°^ ^ j<\x\<2j j<\x\<2j
- °312
I (i" |a|r+|cr|r )(i mr ~"~ k|r )rfx
a a
°± ± j<\x\<2j j<\x\<2j
< C4j~"
I
dx
i<\x\<1j j<\x\<2j
which proves (4.9). Now we have l i m b e c \\pk — <j>jpk\\r,K = 0, Da(4>jpk) con verges weakly to Dapk = 0 and using Mazur's theorem in the same way as in Step b) we obtain the existence of functions Wjk G C^(Rn), j E IN, with lim ||£>>fc - Dawjk\\ri]Rn
= 0.
This construction yields a sequence w^ G Co°(]Rn)1 k G W, so that lim ||V m p fc - Vmu»fc||r,R- = lim \\Vmwk\\ = 0. k—voo
/c->oo
Moreover due to the properties of j we can construct wk in such a way that PU\K — ^U\K for all k G IN. This leads to lim \\pk - wk\\L™.r(m») = 0. Setting Uk = Vk — Wk we obtain the desired assertion (4.6) which completes the proof of (ii) for G = Mn. d) Next we prove (iii) and (iv) for G = Rn. Let u G Lm>r(Mn) and let uk G C£°(iR n ), k G N, so that l i m * . ^ ||u - u*||i,m,r(1i») = 0 . If /? G W0 and p > r are chosen so that m — n / r = /? — n / p we obtain from Sobolev's embedding theorem in the same way as in (4.7) that
IIV(ti* - uOILjR- < C | | V m K - ti/)||r>jRn and
(4.11)
||V' J ti f c ||p.«n
(4.12) are
Further we observe that ||V • \\ryK + || • \\r,K and || • \\H™>r(K) norms in Hm,r(K)\ therefore, Sobolev's embedding theorem yields I K - Ui\\PtK < C (||Vm(tlfc - tiOlkx + I K - ti|||r,tf) and
m
I K I U K
equivalent
198
Hence we obtain ll^fc Ihfc
n Ul\\LP'P(IR W / I I L ^ )^ * )
< C\\uk C\\Uk <
UlWLm.r^JRn) W/||L»».'-(JR»)
and < C\\u m,^ r(<]R C | | ^ k/ \\ c |L| L - ( i Rny ^)-
|IM|L0>P(JR») |W/C||L^'P(JRW) <
r
(4.13)
Now we; may pass to the limit in (4.12) and (4.13) which yields u E L(3^p(Mn) liicty pct»s I U u n c 1111111 111 ^ t . x ^ ^ a n i a ^-±.±oy w n i ^ i i ^ I C I U O u. ^z J n and proves »ves (3.3) (3.3) and and (3.4) (3.4) for for G G= = M Mn.. The remaining part part of of Assertion Assertion (iv) (iv) for for G G— — JR JRnn was was already already ]proved in fie remaining e Step c). Indeed, if /?, p are chosen as in (iv) and if p E JPp-i, then Step c) we Indeed, if /?, p are chosen as nin (iv) and if p E iP/3-1, thenk by by S kk) eJN 1 in Co°(JR n ) so that lir C%°(M find a sequence (w lim^oo ||p — w \\ m,r^j k L equence (w )keJN in Co°(]R ) so that lim^oo ||p— U^HL™.^ Rn^ = 0. This shows (3.5) for G = = M Rnnn... The proof of for ows (3.5) for G = JR The proof of of the the theorem theorem is is complete com G = Mn. In the last step we prove the remaining Assertions (ii), (iii) an e) In and (iv) for G = Q by by using an extension procedure which reduces this case to G = Mn. ,hat K = {x \x E G JR Hn,. \x\ < R} R\ is chosen with R > 0 so that 80, Recall that dQ, C K and r m that ||w||L^. (a) = ||V ti|| r) n + IMIr.tfnn- For L m , r (Mn) we will use now the same K as for L m , r (Q). Since <9Q E Cm we may apply the well known extension theorem x>13 and find for each u E Lm>r(Q) an extension ix E Lm'r (]Rn) such that £t|n = w and ll^llif-^(K) < C\\u\\Hm,r{Knn) (4.14) with C independent of u. This yields I NI |lLL™^>. r^( 2HR"«)) P
<
||w||Lm,r(n)
+
| | f i | || jj yj m , r ( t| f )
<
| h | |UL~~.. '' -- (( n )
+
< < < <
m ||ti|U«.r(n) + C C22(\\V (||Vmu\\ ti|| IMknnx) r)nnK r | nnx + IMIr.nmr) m C 3 | | tm,r i|| m,r+ ( ) .C (\\V u\\ anK \\u\\ + IMIr.nmr) L {n) 2 ri
C , l | | t i | | ^ Cl\\u\\ m , r (Hm,r^ n n / nK f ))
f
jL
n
Hence we get the estimate Hence we get the estimate II^IU^.^iR^) < Cff||t/||jLm,r(a) (4.15) INlL-.^lR'*) < C ||t/||Lm,r(a) (4.15) m r for all wE L > (Q). rove (ii) for G = ft (Q), u E Lm>r(Mn). T\ To prove Q we fix u u EL £ mm>' rr(ft), Then we get some 0 n p E FPm _ i and u wfck E C C%°(lR g ^ " )),, k E W, l , with lim/^oo lim*_>TO ||up -- pp -- u*|| u Lm,r(jen) = 0. m _i w h \\u p -~ wUfcllL™-'-^*), ufe||L-.-(iR-), k\\Lm,r(<]Rn), we Since5 ||u e have li lim^oo \\u - p ||w - p - Wfc||L~.»-(n) UfcllL^n) I < ||# IIs -~ P u r mm k\\Lmm,r(n) >r(to) == 00, E LL ', r'(Q). (Q). too, which whichh means means uu — — pE p£ 3 too, m r m,r To prove (iii) for G = ft we pick up u E I/ ' (Q), a sequence sequen u/c E Co°(r2), Q if L (^), Ar E W, W, with lim/c^oo \\u ||u -— i«fc||i,m,r(n) Uk\\L^>r(n)= = 00and andthe thecorresponding correspon extensions
199 u,Uk G Lm>r(Mn). Then (4.15) leads to l i m ^ o o \\u - Uk\\L™>r(iRn) = 0. Since the support of each uk is compact we can apply Friedrich's mollification proce dure to find a sequence uk G CQ°(Mn), k G N, withlim^^oo ||€t—Uk\\Lm>r(mn) — 0. This shows u G Lm'r(Mn). Using (3.3) for G = Mn and (4.15) we obtain |M|L£'P(ft) ^ H ^ I I L ^ . P ^ " ) 5: Cl|H|l, m » r (tfi*) < C^ \ \ U \ \ L ™> r(ft) for all u G Lm>r (Q) which proves (3.3). To prove (iv) for G = Q we first observe t h a t the validity of (3.5) for G = JRn immediately implies the validity of (3.5) for G = Q,. To show (3.4) for G — Q we argue by contradiction. Let us assume the existence of uk G L m , r ( Q ) , k G N, such t h a t lim^^oo ||V m u f c || r > n = 0 and H V ^ I ^ n = 1 for all k G N. We set uk = u k - Pp-iuki then Pp-iuk G Lm>r(Q) by (3.5) and uk G £ ^ ( $ 2 ) by (iii). Using (4.1) with ra, r replaced by /?,p yields l|fi*l|L/».p(n) < C f i||V /3 ti /c || P) n = d
(4.16)
with G\ independent of k G W . Then (3.3) leads to the estimate
HVtifclLn < C2||tifc||L-.r(n) = C2(||Vmtx*||r|n + IM|r,nnx)-
(4.17)
Using p > r, (4.17) and lim/^oo ||V m t/A;|| r) nnK = 0 we obtain: ||£fc||tfm,r(nnK)
<
C 3 (||V m ii/ c || r ) nnK + l l ^ H ^ n n x )
<
C 4 (||V m fi f c || r i nn* + ||ti*||L/>.p(n)) <
ft
with C5 independent of k G W . Therefore we may assume t h a t i//- is strongly convergent in U (Q fl A') because i J m , r ( Q fl A') is compactly embedded in L r ( Q fl K). But this implies t h a t (uk)k£]N is a Cauchy sequence in Lm,r (Q) since lim | | V m % | | r , n = lim | | V m U f c | | r , n = 0. (4.18) k—>-oo
Ac—)-oo
m,r
Hence there exists u\ G L ( ^ ) with l i m ^ o o ||wi—^A;||L m ' r (n) — 0 and by (3.3) lim/c-^oo \\u\ - Wfc||L/j,P(n) = 0. From (4.18) we have V m t / i = lim/^oo Vmuk = 0; thus u\ G JPm-i. Moreover, | | W | U
= lim HVtifcllp.n = lim | | V u f c | | , , n = 1. Ac—Yoo
k—¥oo
Therefore u\ is a polynomial with V ^ u i G LP(Q). This is only possible if V ^ i = 0 contradicting ||V^i/i|| p > n = 1. This proves the estimate (3.4) and the proof of Theorem 1 is complete.
200
Proof of Theorem 2. a) First we recall the well known fact that Au = 0 and u G Lr(JRn) implies u = 0. To prove this we apply WeyPs lemma and conclude that u is harmonic in Mn. Then we use the mean value property for harmonic functions to balls with center x and radius \x\ and Holder's inequality. This immmediately yields the estimate \u{x)\
xeMn.
It follows u — 0 by Liouville's theorem. b) Now we consider the null space ker S = {(ti, q) G L m + 2 > r (iR n ) n x L m + 1 ' r (Mn), -Au + Vg = 0, div u = 0} of the operator S in Theorem 2. Let (u,q) G ker 5. Then Aq = 0 and V m + 1 A ? = A V m + 1 g = 0. Since V m + 1 g G Lr{Mn) we get Vm+1q = 0 by Step a) and therefore q G Pm. Furthermore we calculate 0 = V m + 2 ( - A u + Vg) = - A ( V m + 2 u ) + V V m + 2 g = - A ( V m + 2 u ) . Again by Step a) we conclude V m + 2 it = 0 and u G Afm(Mn)\ the other inclusion is obvious. c) In the next step we solve the Stokes system -Au + Vq = / ,
divu = #
1P£+I-
Hence ker 5 C
(4.19)
in Mn for given polynomials / G ^P^_i and # £ Pm. We look for a solution (u,q) G i P £ + 1 x JP m , where m G N0. If m = 0, then / G J F ^ = {0} and g — go is a constant. In this case we set u = (<7o#i,0,..., 0) and q = 0. If m > 1 we apply div to (4.19) and obtain the equation Aq = f
(4.20)
where / = div / + Ag is a polynomial with degree dg(f) < m — 2. To solve the Laplace equation (4.20) for given / G iP/c, k G W0 we use an elementary argument 14 : Let iff/ be the space of all homogeneous polynomials
p = J2 a<*x"
( 4 - 21 )
where aa e C, xa = x%1 • • -x%*, a = (au...,an) G Ng, I G W, H0 means the space of constants. Let k > I and b = ^ | a | = / ^«^ a G iff/ with 6 ^ 0 . Then the operator b(D) = J2\a\=i baDa defines a surjective mapping q \-± b(D)q from Mk onto JHk-i . To show this we observe that (p,g) = p(D)q, where g
201 is t h e complex conjugate of q, defines a scalar product in Mi for each / G IV. Assume t h a t b(D) : JHk —> Mi-k is not surjective. Then t h e range V — b(D)Hk is a proper subspace of Mk-i and there exists a nonzero s G Mk-i with (s,b(D)q) = s(D)b(D)q = 0 for all q G Hk- Choosing q — sb we have 0 = s(D)b(D)sb = s(D)b(D)sb = (sb,sb), hence s6 = 0. This is only possible if either s = 0 or 6 = 0 which leads to a contradiction. Now we can solve t h e equation (4.19). T h e case m = 1 is trivial since then / = 0. In t h e case m > 2 we know from above t h a t A : JHm —>• i ^ m - 2 is surjective. Therefore (4.20) possesses a solution q\ G j F m so t h a t A g i = div / + A#. T h e same argument yields some u\ G i P ^ + i w ^ n —A^i = f — V g i and some u> G lPm+2 with A w = div u i — g. We set ii = u\ — V w , g = q\ — Aw a n d obtain div u = div u i - A u ; = gy — A u + Vg = — A u i + V A w + Vgi — V A w = / . Hence (u, g) G i P ^ + i x P m is a solution of (4.19). d) To prove t h e surjectivity of S we pick u p some / G L m + 2 > r ( j R n ) n and # G L m + 1 , r ( i R n ) arbitrarily. Then we use (3.2) and decompose f,g in t h e form / = / + / ,
g = g+g
with / G F » _ ! , / G L m ' p ( « n ) , geJPmJe # G Co°(Mn), i G IV be sequences with Hm | | / - /i||L-.-(iR-) = 0, V
I—)-00
'
L m + 1 > r (iR n ). Let /,- G C H K " ) * ,
Km ||'-(2R'M = 0. V
X—)-00
We use t h e fundamental solution 7 , 9 E — (Ejk)j,k=i,...,n+i given by
E
y
of t h e Stokes system
= ^(|^F
+
^ ( l ^ l („>3)))'
En+l,k{x)
=
Ek,n+l{x)
=
, . , u; UJnn|a;| \X\
En+1>n+1(x) £Vi+i,n+i(z)
= =
8{x) S(x)
»M
(4.22)
where ion is the surface of t h e unit sphere in Mn and S(x) means Dirac's delta distribution. For any h = ( / i l 5 . . . / i n + i ) G C £ ° ( i R n ) n + 1 we define t h e usual convolution n+l \\ « n+l «+ + ll
(
by jg?ifc * A Ejk Afcfc = /
y ^ Ejk * /*&)
JJRn
£j£j/c(z fc(jc - y)hk(y)
dy.
202
Setting (ui,qi) = E * (/»,£,•) we get -Aui + Vqi = /,, divu,- = git i E N. The Calderon-Zygmund theorem yields m E L m + 2 ' r ( J R n ) n , q{ E L m+1 > r (iR n ) and the estimate
||vro+2«j||ri«» + ||vm+1?i||r,ffin < c(||v m /i||,,«. + ||vm+1ft||r,R«) for all f E W. Applying this estimate with w»,g», /i,<7» replaced by w2- — tz/, 0i - 9/, /." - //, 9i ~ 9i with ij E N we get: lim ||V m+2 (ii,- - «/)|| r ,«» = 0,
lim ||V m+1 (g t - - gi)\\ = 0.
By Theorem 1, (i) we find polynomials ui E F ^ + 1 , g» E JPm> « G W, so that (ui - €ti),- is a Cauchy sequence in Lm+2>r{Rn)n with limit u E Lm+2>r(mn)n m+1 r n and (qi-qi)i is a Cauchy sequence in L » (jR ) with limit q E L m + 1 ' r ( i R n ) . Then we obtain m V (-Au + = Vm(-Au + Vg) V?) = = M-ZXUHvgj = =
m lim - A ( u -j - «-,itji -Hi) ) + V f af-j--- fg.ft)) t)) lim V V m ((-A(«,++ vig,V(g nm v"^-z\iu t m m lim lim Aw u ii + lim V V m (( ---A A w + Vg,-) Vg,-) = = lim lim V V" / f
f—)-00
i—)-co m »—)-oo »—)-oo
vm 7/ = v"7, Vmm /, m m m = vvlimQ///VV= v+ (div(w, /, =m+1 mv 1 /, = (div(«j-ui)) m+1
i—foo i—foo i—foo
t—>co V mm ++ 11 (div (div «) .lira V m + 1 (div(«,(div(«< - «<)) fi<)) V w) = = I—)-CO lim V I—)-CO lim Vmm++11 d i v u i I—)-CO = lim lim V rmo ++ 11 5 i = *—yco lim V V m + 1divwi divui = = i—>-oo lim V 5i *—)-oo
+ 1g = Vmm++ 11 # . = V v mm+1 # = v #.
t—foo
Setting / = — Au + Vg — / and g = div {/ —
AU + Vg|| L m,r ( i R n ) n + ||dlV u\\Lm+l,r{JRn)
=
m
||V (-Ati + Vq)\\r)Mn + || - An + Vq\\r,K + ||V m+1 div ti|| r>jR . + ||div u\\r,K < d (||V- + 2 l/|| r ) i R , + ||V- + 1 9 || r)JR n + \\u\\Hm+>,r{K)n + \\q\\Hm + 1,r{K)) < C2 (||V m + 2 ti|| r | J l . + ||V r o + 1 g|| r | «. + \\u\\rtK + \\q\\r>K) = ^2 (|MUm+3'r(Jln) + lklU«+i.'-(JR*)) • This completes the proof of Theorem 2.
203 Proof of Theorem 3: First we prove the surjectivity of the operator S. Let / E L m > r ( f t ) n , g E L m + 1 ' r ( f t ) , and let h E Hm+2-1tr>r(dtt)n. We ex tend / , # to / E Lm>r{Mn)n, g e Lm+1^{Mn) and choose u E Z / n + 2 > r ( i R n ) r \ g G L m + 1 - r ( i R n ) using Theorem 2. Then A = u\dfl E . f f m + 2 - 1 / r ' r ( d f t ) n . By L e m m a 1 for n > 3 we get a unique solution pair (tz, g) = X(h — h) of the system - A u + V ? = 0, d i v u = 0, w|an = ft-ft (4.23) with the properties (3.8) and (3.9) while for n — 2 we fix a solution (u^q) = X(h — h, 0) of (4.23) by choosing a^ = 0 in (3.10). We set u — u\n-\-u, q — q + q and obtain S(u, q) — (f,g, h) which proves the surjectivity of 5 . Next we consider the null space of S. It is clear t h a t Afm(Cl) is contained in the null space. Now let u E L m + 2 ' r ( f t ) n , q E L m + 1 > r ( Q ) with S(u, q) = 0. We extend (ti,g) to (u,q) E L m + 2 ' r ( i R n ) n x L m + 1 > r ( i R n ) . Since S(u,q) = 0, / = - A w + Vg and g = div iz have a compact support in Mn . Therefore (uyq) = (u,q) - E * (f,g) is well defined and we get u E L m + 2 ' r ( U n ) n , q E L m + 1 - r ( l n ) , and 5(ti,g) = 0. Hence (u,q) E Mm{Mn). On the other hand (u, q) — (u, q) in Q and 0 = u\da
= u\dn = u\dn + E*(f,g)\dn
(4.24)
This means (u, q) — — E * ( / , # ) solves the system —Aw-fV(7 = 0,
div w = 0 in Q,
w|aa = u\dn-
For n > 3 the decay properties of E give (3.8), (3.9) and hence (tz, q) = X(w|an), which means (it, g) = (u, o)- So we need a slight modification. For any b E C 2 , x E Q we have (w,g)
=
(ti,$)|n-(£,£)
=
(ii + 6 - a 0 , g)|n - (ii + 6 - a 0 , g) m+ l
= ( J2 a^a + M)|n - (« + * ~ a0iq)
(4.25)
|a| = l
We abbreviate ( X ^ f i i aaxa + M ) = (v,g). Then we still have (v,£) E NTn(]R2). Furthermore it holds S(u + b-a0,q) = 0 in Q since S(uyq) — 0, and i>| a n = (^ + 6 - a0)\da due to (4.24). For any b E C2 the pair (u + 6 - a 0 , tf)
204
also obeys the regularity properties (3.8) in Lemma 1 and the decay properties (3.9) for k > 1. So with (3.10) we conclude (u -f b — ao, q) = X(v|an, b) if we find 6 G C 2 in such a way that u(x) + b-a0-b
ln\x\ = 0(1)
(4.26)
as \x\ tends to infinity. For this purpose we observe that supp(/,#) C M2\Q^ hence we obtain from (4.22) for j = 1, 2 that UJ %
=
~y^ jk{x-y)dy- 2 ^ / 2 fk(y)E my)&jk(x -y)ayjjrj J]R \ci
-- y)dy j3(x / 2 g(y)E g(y)&j3{x Jm \n
" ^ /' ( A ( xli / ) (-yi) - y-r + / 2 x(2! / -)- (2/2) *yi)2 - ^ ) ) ^ l (fi{y){xi -* lyi) +i )J2\y)v h{y)(x2 \jiyy)V 27r v y JiR2\n F ~ y\
~2TT JiR2\ / n S ^I fa; -* y\^2 ^ ^ 2TT - 7iR2\n / £(y)ln|*-y
-1 = : / 1 + / 2 + /3. //3. 3.
\x-V?
?■* Jm?\n
(4.2' (4.27)
It follows Ii -f ^2 — 0(1) as I a: I tends to infinity. To estimate ^3 we expand : \x — v\ \n\x — y\ = ln|x — y\ — \n\x\ + ln|a?| = In—7—. \- ln|x|. \n\x — y\— In I a: — t/| — ln|ar| + ln|#| = In—r-j h \n\x\. \x\ \x\ If y G JR2\ft the first term remains bounded as |a?| tends to infinity. Substi tuting this into the integral J3, we get :
W&
h
d +++h ^ii [ dy fj(y)dy^\*\ fjivW-^-dy Mv)dM*\ fav^Kzrdy === hkT«\iJm.*\n LJw **w * LJ^nir ^ Jm?\n \\ J]R*\n *>* Jm?\n \\ 27r 27r
= =
x
x
bjln\x\ 6 i ln|s| + + 0(l) 0(l)
(■(4.28)
with bj = ^ fm2 fj{y) dy. We substitute 6 = (61,62) into (4.25). Then (4.27) and (4.28) lead to the decay (4.26), which shows the decomposition (u,q) = ivi 9) In - Z(v\dn, b). Hence the null space of 5 has the form asserted in Theorem 3. The boundedness of the operator S follows in a similar way as for 5 in the previous proof if we additionally use the following trace estimate I M d n | | t f m + 3-l/r,r (flfl) < C\ \ \ U \K \\H ™ +2>r (K) < ^ 2 | M | L m + 2,r( a ) .
The proof of Theorem 3 is complete.
205 Acknowledgement T h e authors are very grateful to Prof. G.P. Galdi, Prof. C.G. Simader and Prof. W . Varnhorn for helpful discussions. References 1. R. A. A d a m s , Sobolev spaces (Academic press, New York 1975). 2. V. Benci and D. Fortunato, Weighted Sobolev spaces and the nonlinear Dirichlet problem. Ann. Mat. Pur. Appl. 1 2 1 (1979) 319-336. 3. A. P. Calderon and A. Zygmund, On singular integrals. Amer. J. Math. 7 8 (1956) 269-309. 4. L. C a t t a b r i g a , Su un problema al contorno relativo al sistema di equazioni di Stokes. Sem. Mat. Umv. Padova 3 1 (1964) 308-340. 5. J. Deny and J. L. Lions, Les espaces du type de Beppo Levi. Ann. Inst. Fourier 5 (1954) 305-370. 6. G. P. Galdi, An introduction to the mathematical theory of the NavierStokes equations Vol. I, Linearized problems. (Springer tracts in n a t u r a l philosophy, New York, 1994). 7. G . P . Galdi and C. G. Simader, Existence, uniqueness and Lq- estimates for the Stokes problem in an exterior domain. Arch. Rat. Mech. Anal. 1 1 2 (1990) 291-318. 8. H. Kozono and H. Sohr, On a new class of generalized solutions for the Stokes equations in exterior domains. Sc. Norm. Sup. Pisa 19 (1992) 155-181. 9. O. A. Ladyzenskaja, The mathematical theory of viscous incompressible flow. (Gordon & Breach, New York, 1966). 10. R. Lockhart and R. C. McOwen, On elliptic system in Mn. Acta math. 1 5 0 (1983) 125-135. 11. V. G. Maz'ya, Sobolev spaces. (Springer, New York, 1985) 12. R. C. McOwen, Boundary value problems fot the Laplacian in an exterior domain. Comm. Part. Diff. Equ. 6 (1981) 183-198. 13. J. Necas, Les methodes directes en theorie des equations elliptiques. (Masson, Paris, 1967). 14. U. Neri, Singular integrals. (Springer, New York, 1977). 15. C. G. Simader, T h e weak Dirichlet and N e u m a n n problem for the Lapla cian in Lq for bounded and exterior domains. Applications, in Nonlinear Analysis, function spaces and applications 4 4 (Teubner, Leipzig, 1990). 16. M. Specovius-Neugebauer, Exterior Stokes problems and decay at infin ity. Math. Math, in the Appl. Set. 8 (1986) 351-167. 17. W . Varnhorn, The Stokes Equation (Akademie Verlag, Berlin, 1994). 18. K Yoshida, Functional Analysis (Springer, New York, 1965).
206
B O U N D A R Y VALUE PROBLEMS A N D I N T E G R A L EQUATIONS FOR THE STOKES RESOLVENT IN B O U N D E D A N D EXTERIOR D O M A I N S OF R n WERNER VARNHORN Institute of Numerical Mathematics Technical University of Dresden D-01062 Dresden, Germany
1
Introduction
In the present paper we construct a solution of the Stokes resolvent equations \u-Au
+ Vp = f
in
G,
V - u = 0 in
G,
u = 0
on
T (1.1)
using methods of hydrodynamical potential theory. Here G C Mn (n > 2) is a bounded or an exterior domain with boundary T = dG of class G 2 , / i s some given external force density, and A ^ 0 is some complex parameter with |arg A | < 7T. The construction is based on a new explicit representation of the fundamental tensor of (1.1). It includes a detailed study of the corresponding boundary layer potentials and ends up with a boundary integral equations' method for the spatially homogeneous (/ = 0) boundary value problems with prescribed Dirichlet and Neumann boundary data. We solve these problems in bounded domains as well as in exterior domains by reducing each of it to a system of second kind Fredholm boundary integral equations. The resolvent equations (1.1) have been studied with methods of hydrody namical potential theory for the first time by McCracken 13 in the case of the half space (n = 3). In 13 also the corresponding fundamental tensor has been computed. Another representation of this tensor, which can also be used for numerical purposes, was presented in 1 6 , 1 7 in the framework of a boundary ele ment method for the solution of the interior Stokes Dirichlet problem (n = 3). Deuring 5 , 6 , 7 considered (1.1) in the case of an exterior domain (n = 3) and established U-estimates for the solution. A potential theory for the equations (1.1) in the case n = 2 has recently been developed by Borchers, Varnhorn 2 3 , . The general case n > 2 is considered for the first time with potential theoretical methods in the following. As in the classical potential theory we require Green formulas in bounded domains as a starting point. To derive these formulas we define the formal
207
differential operators S\, S'^ by
* =
(1.2)i2
*:©-*?=((X-_T:")-
(13)
<»
These operators are adjoint to each other (A defines the complex conjugate of A). The corresponding stress tensors adjoint to each other are denoted by T
: ( " ) — ► T " :=-2Du
+ pIn, (1.4)
r
: (;)->T>$
:=
-2Du-pIn,
where the deformation tensor is given by Du := | ( V u + ( V u ) T )
(1.5)
with (Vu) T as the matrix transposed to Vu := {diUk)k,i=i,...,n- Here and in the following, In is always the nxn identity matrix and N = N(y) the exterior (with respect to the bounded domain A) unit surface normal vector in y G dA. For vectors a, 6 G C n and complex nxn matrices C = (C,j), D = ( A j ) we set n
n
a -b \— 2_.aibi,
C : D := \_] CijDij-
With these notations, for sufficiently smooth, solenoidal vector functions u,v and scalar functions p,g in a bounded domain A C ^ n (n > 2) with the boundary dA G C1 we have Green's first and second formula S
A /j{(Sx{5p) 4)P )
A
d y = T N -■ Du.Dvdy, ) -vdo++ f I\u-vdy Xu-vdy ++22j2j J Dw.Dvdy, Du.Dvdy, ' 00( dP y ^z z = f(f(T(TpN)-vdo+ pNP)'vdo
dA BA
A
(1.6) (1.6)
A
do(1.7) /j{{{S4)[Sip) N O -■ ■0 0g - -- (;)-{s' 0W ■ {SxVq)}dy=J{(T%N).v •xv)}dy=J{(T%N).v-u.(T'V {SYqJJW = J {Vp")qN)}do. ■ v -- uu-(T'" ■ l JqN)} qN)] do. - (1.7) UA
8A
With help of Green's second formula (1.7) a representation to the solutions u,p of the resolvent equations
S$ = (i) in G
208 can be developed, if the corresponding fundamental tensor E\ = — (\E^ F k^ )
V J^ / j,/e = l,...,n + l V
is determined as the solution of the equations is determined as the solution of the equations S\E\ = 5In+i S = SIn+1 (1.8) XEX in the space S'(Mn) of tempered distributions. Here in (1.8) the term in the space S'(Mn) of tempered distributions. Here in (1.8) the term S\E\ = (S\E1,..., S\En+1) S\E\ = (S\E1,..., S\En+1) means the application of S\ to the column vectors Ek := (■E'j)fe)i7=i,...,n+i f° r each fc = l , . . . , n + l, and 8 is Dirac's distribution in Mn. We determine E — E\hy Fourier transformation of (1.8). Setting E = {FE E7jki f7{£) ){£) -cfc(0 O = (FEjkjk)(0
jf/ = c^ n cKtn
(1.9)
exp{-i£.x)E exp{-it'x)E e x p - i ^jkjk{x)dx -{x)dx ^^W^
with c^)n — (27r) - n ' 2 , for each k = 1 , . . . , n + 1 we obtain the distributional vector identity 2 Ejk + *E ZjEnn +i + xfi) (/ (\Z\ {\i?Ejk \Ejhjh + iiZjE tk )
.
+1
.
Jj
\ = h n
>
=
CWtn (*i*)i = l l ... l n + l •
Here c ^ is the Kronecker symbol. Now for the Fourier transform E of E for all j , k = 1,.. .,n we find
*»
£n #n + l,*(f) = E ^fc,n +1l(t) ( 0 == k)n +
-jjjf ^
,,
En + 1 ,n + l ( 0 = C^n CWin( (l l ++ — — jJ.. In order to calculate F from F by means of inverse Fourier transformation we use F = (F^EJMX) ( F " 1 ^ ) ^ ) = cr.n / Ejkj /(x) c (z) =
e x p ( iexp(i£.x)E ^ * ) £ i f c jk(($)dt. 0#.
(1.10)
For all n > 2 and j , * = 1 , . . . , n this leads to c?r Cn nn C7r>nn Eik = Ssjkjk (F~1 Cn ' >' "] + a?. f( VF"- 1 Jk Jk 2 +0 k 2 2 \ \t\ + \ ) > V KI (KI +
^ A)J
209
1 ltk = , n + l = —9k -dk ((FEn + l,k = ^£/c,n F
£ nn+i,n+i + l,n + l — = (F ^~ C CTT^ E T r J^ J4"- fA A (l FF~
ZLA ) J, -r-jy
((ll. .lH i ))
-j-j^- j '. 1712")
Due to
eeie c e ++A)v
2
A\\e Ve
ee++ Ay'\)'
for j , k = 1,. . ., n it follows E = (n{x) e „ ( x ) - efaj) £,-*(*) - S*jki je^(x) \d%{e e*(*)), jk(x) k k e*(x) + \8] x Enn+ lik(x)) +i,k(
En+i, +iinn+i(x) +i{x)
= Ek,n+i{x) Ek}n+l(x)
(1.12)
= -dken-d {x), = ken(x),
= <&(z) S(x) + AeXen (x). n(x).
Here e n (n > 2) is the fundamental solution of the Laplace equation in JRn, i.e.
e2W ln e (x) := := ±-\n±2
h w\' Z7r
|x|
en{x) en(x)
:=
|a| ' ' " (n -— 2)cj 2)unn
(n ( n>> 33))
(cj n stands for the surface area of the (n — 1) -dimensional unit sphere in i R n ) , and e^ is the fundamental solution of the Laplace resolvent equation, defined by (see 4 ) n- 2 n
*v:=h(m)'
"v^™-
(113) (1.13)
Here and in the following, VA G C stands for t h a t particular square root of A G C\{z < 0} which has a positive real part, i.e. Re y/X > 0. For 0 < v G JR the function Ku is the modified Bessel function of the order vx , which we will only need for integer and half integer orders. Now by an elementary but lengthy calculation the equations (1.12) lead to the following representation of the tensor E [Ef( k*(x)) i ?— = £E\A — =(£& )) \
J
J j,fc = l,...,n + l
210
n>2 n >2
(j, kfc= = l,...,n): 1 / <^i
^(«) = i{]#b«^W) + ^ e ^i«i)}. _ (tr- 1 /^-!^)
«) = e ii(W
v f
~
c 2 («) =
+
r(m)
2(frgro(/c) 2
r(m)«
4(fr+1/^m+1(K)
n
,
*n\i,*(«) = ^ , » + i ( « ) =
i
«2'
m:=f,
(1.14)
^ F
ln ^n + l,n + l(«) = *(*) + — {
R' \2-n
I n-2
n= 2
>
, n > 3.
With help of this fundamental tensor E\ now we derive a column representation of a solution u,p to the resolvent equations
Sx? = (i) in G in the form 18
J E^(x - y) T%N(y) doy - J Dx(x, y) u(y) doy - J Ex(x - y) (f(^\ dG
8G
dy
G
_ | f-r)(*),*€G, -(p)w^eG, 0
(1.15)
, x $ G.
Here the (n + 1) x n double layer tensor D\(x, y) is denned by Dx(x,y)
:=
((-T.^^-y^yJV^y))
using the column vectors E$ := (1^)^=1,..., n +i for fc = 1 , . . , , n + 1. This tensor has the following form (z := z — y, AT := N(y)).
211 n > 2
(Ar, i — 1 , . . . , n ) :
—{^^^'-(^^"^^W'+TF^^wt}' dl(K) == rfl(K)
d2[K} da(,e) =
8(fr+1Jgro+1(/c)
2n In
^^++1'1'
FR^ FR^ 8(%r+lKm+1(K)
-
d3{K) =
T(mp FR^ FR^
++
2(z)mKm(K)
2n
FR -fM +
^'^ '
—^—•
(L16)
m:=~,
n
^^ = ~^^""^-^}The representations of the fundamental tensor and the double layer tensor given in (1.14) and (1.16), respectively, are valid for all n > 2. These tensors have a much more complicated form as the corresponding tensors in the case A = 0 for the stationary Stokes equations. We will see, however, that the singular behaviour of the fundamental tensors at x = 0 coincides in both cases (in contrast to their behaviour at infinity). The same is true for the singularities of the corresponding double layer tensors at x — y. To see this we need additional statements regarding the asymptotic behaviour of the modified Bessel functions Kv which will be derived below. 2
The Surface Potentials
With help of the tensors E\ and D\ calculated above now the surface potentials with complex vector-valued source densities ^ G C°(T) are constructed for all n > 2. We need the single layer potential (£*,„*)(*) (£*,»*)(*) = = J E[ E{c\x \x-y)*(y)do - y) 9(y) do y , y, r
x^T,
(2.1)
212
and the double layer potential (£>A,n*)(*) = j Dx{x, y) 9(y) do y , x £ T. r The n-componental velocity parts are supported with a dot, i. e.
(£*,„*)(*) = / 4r'C)(* - V) *(V) y) *(y) d°y> r
x£ T, * $ r.
Moreover, we require the normal stresses of the single layer potential denned in a neigbourhood U C Mn of the surface T by (Hln9)(x)
= JTx(E<{\x-y)y(y)) r =:
Hx{x,y)V{y)
(2.2)
(2.3) (2.4) E\^
N(x)doy (2.5)
doy,
x£T.
Here x G T is the uniquely determined projection of x G U onto T. For the n x n kernel matrix H\(x, y) now we find the following representation. 2.1 L e m m a For x,y G T we have ffA(z,2,) = ( ^ ( j / . o ; ) ) 7 ' .
(2.6)
(r)
Here D\ ) is the n x n kernel matrix which results from cancelling the last row inDx. Proof: In the following computation we use the summation convention. For ky z, j — 1 , . . . , n we have (Tx(E{c\x-yMy))N(x))k = -(dXjE^(x
- y)%(y)Nj(x))k
- (d^E^x
+ (^^n+i,,-(*-y)*i(y)^(*))fc = (#*,-(*, y)<Mi,))
-
y)*{(y)Nj(*))^
213
with H&x, y) =
(-dXjE^i(x-y)Nj(x)-dXkE^(x-y)Nj(x)+6kjE^+hi(x-y)Nj(x))ki.
Using the properties of E\ this implies Hx(y,xf - (dyjEfoy = -(dyjE^y x
x)Nj(y) SijE^^y - x)Nj(y) + dy,E$k(y - x)N X + i , * ( y "j{y) - *
x)Nj(y)) *)Nj(v))
= -{d ~(dyEik{* }E^k(x = \ Vi
y)Nj(y) + + dy.E^(x dy,E$k(x -- y)Nj(y) y)Nj(y) + + <5 )j ^ +1>fc (x -- v)Nj(v) fcX+i,*(* -
y)Nj(y)) y)N j(y)) / kl
x
= =
-(-d y)Nj(y)-d x,E^k(x-y)Nj(y)+6ijE^+lik(x-y)Nj(y)) -(-dXjXjE^(xE^(x-y)N j(y)-dx,E^(x-y)Nj(y)+6ijE^+lik(x-y)Nj(y)) J ki
\ \
J ki
x +((-TxE E£(x-y)) k(x-y))ijNj(y))
=
/ ki
\
=
/ ki
D
as asserted. We need further statements regarding the continuity behaviour of certain surface potentials with special densities. Here and in the following, Ge C ]Rn (n > 2) denotes an exterior domain with boundary T := dGe of class C2 such that Gi := Mn\Ge is a bounded domain. 2.2 L e m m a 1. For the double layer potential D\ nb (see (2.4)) with some constant den sity b G Cn we have
[ 6, c)
(Dlnb)(x) + xJE^ (x-y)bdy
= J \b, 2"> o,
G,
x e Gi, xeT,
(2.7)
x e Ge-
2. For the single layer potential E\N (see (2.1)) with the exterior (regarding Gi) unit normal field N as density we have f {Exx,nnN){x) N)(x) (E
E{'\x-y) N(y) do, = I = J E^(x-y)N(y)do y r
-®,xeG -®,x€Gitit -±§,xeT,
(2.8)
214
Consequently, E\nN
= 0 in Mn.
Proof: 1. Let us first assume x E Ge. Then it follows from Green's second formula (1.7) with
0 =0.
Q) = !#(*-•),
* = 1,...,«
due to
SX? = 0 ,
S'xEt(x-.) = Q,
TgN = 0
the relation
J((X/3, E$'e\x - y))) dy=- J«J3, T'XE[C\X - y)N{y))) doy Gi
V
=-((Dx/3)(x))T. Analogously, for x £ G{ we obtain
J((\/3,Exr^(x-y)))dy-(f3u...Jn)
= -{(DxP)(x))T
Gi
by cutting off a sufficiently small ball centered at the singularity in x £ Gi. Now let x £ T, let K£(x) C Mn denote some open ball centered at x with radius £, and define G] := Gi\K£(x). Then it follows
J(W, Exr'c\x - y))) dy = - J « P,T^E{c\x - y)N(y) » doy
-
j
((P,T^E{c\x-y)N(y)))doy.
dKenG~
From the case just considered it follows that we would obtain the contribution —(3 from the last integrand, if the integration would be carried out over the entire 8Ke instead of dKe Pi Gu, (the normal N changes its direction), plus an additional volume integral, which disappears for e —> 0. That means, integration over dK£ 0 Gi with e -> 0 contributes — |/?. This proves (2.7). 2. For # £ G e we use Green's second formula (1.7) with
(P = ©>
Q)=^(*--),
* = !,...,»+!.
215
From S*; , SA? = Q ©>
S ( z - -• •) )==©©>, S'x^E£(x
T°AT ==Ar T°JV iV
we obtain J((N(y),E<(\x-y)))doy, J{{N{y),E^(x-y)))doy,
(0,0) = = rr
i.e. the last equation in (2.8). For x E G» the representation formula (1.15) implies r) J({N(y),E J({N(y),E^(x-y)))do x (x-y)))doyy,,
= -(0,1) = r
and the case x E T is treated as in the first part above. The continuity and jump relations of the Stokes surface potentials are described in the next proposition. 2.3 P r o p o s i t i o n Let tf E C° (T) and let E*Xn^, D^W, H^V denote the surface potentials defined in (2.3), (2.4), (2.5), respectively. Then we have ((E^y J^*)'' = = E £ x>n 5 i n9* = ( ^ , „(El^r, *)a, (£>;,„*)<-£>*,„* (£>!,„*)' - !>*,„* = + |§** = =Dl D* nV-(Dl Xin9 -nVy, (D\in9)°,
(Ht,nn hence
HI„*
(2.9) (2.9)
= - i * = ^ > n * - (#*,„*)»,
fl (£>$,„*)< -- (D\ == ** == (Hi - (Hini vy. (£>$,„*)< (xn,„*) (^n*y in*y i n *)°-(^ n *)'.
(2.10)
A detailed proof of the relations (2.9) in the case n = 3, A = 0 can be found i n 8 (compare also n for n = 2, A = 0). In the resolvent case A E C \ {z < 0} we can apply the decomposition Efjkk{x (x --y)y) = E y) + y), •£&(* 2/) = #?*(* b - #jfc(* hjjkk{x {x --y)2/) +R Kjkjk{x (x -- \2/), r
216
2.4 Lemma Let n > 2 and ^ G C°(T). Then for the single layer potential E\ n \P and the double layer potential D\ n\I> (see (2.3) and (2.4)) we have the following decay behaviour for \x\ —> oo: (£$,„*)(*) = 0 ( 1 * 1 - " ) ,
(2.11)
= O(\x\-n+1).
(2.12)
(Dln0!)(x)
Proof: The asserted asymptotical behaviour of the single layer potential fol lows directly from the representation (1.14) of the fundamental tensor. Since the modified Bessel functions Kv {y > 0) decay exponentially 10 we find ei(V\
\x\) = C(\x\-2),
e2(V\ \x\) = 0(\x\~2),
\x\ -> oo,
(2.13)
and thus the assertion. Similarity we obtain the decay behaviour (9(|ic|~ n+1 ) for the double layer po tential from the representation (1.16) of the stress tensor, since for \z\ := \x — y\ —)• oo we have d1(^|z|) = l + 0(|2|-2),
3
d2(V\\z\)
=
0(\z\-*),
d3(V\\z\)
=
0(\z\->).
The M e t h o d of Integral Equations
In this section G2 C Mn (n > 2) is some bounded domain with smooth bound ary T := dGi of class C2 such that Ge :— Mn \ Gi is the complementary exterior domain. We consider the following boundary value problems for the Stokes resolvent equations (X E C \ {z < 0}): S
*p = 0
in Gi,
u= $
on T,
(3.1)
s
^p = (o)
in Gi,
T^N = $
on T,
(3.2)
SxUp = Q
in G e ,
u — 3> o n T ,
(3.3)
s
in G e ,
on T.
(3.4)
^p = (o)
T^N = $
In the next lemma the uniqueness of classical solutions u,p (i.e. u G C2(G) O Cl(G), p G Cl(G) O C°(G)) to these problems is studied.
217
3.1 L e m m a Let n > 2 and A G €\{z < 0}. Then the solution u of the interior Dirichlet problem (3.1) is determined uniquely. The pressure p is unique up to an additive constant. The solution u,p of the interior Neumann problem (3.2) is unique. For the exterior Dirichlet problem (3.3) and the exterior Neumann problem (3.4) the solution u,p is unique, if (\u\\Vu\)(xU (Miv«l)(«)\ (|«| |V«|)(x) 1
+1
/i .. ix/ A r = C J ( l a ; l
)> k l - * - o o .
(MIPIK*)/ (MH)(*)J
(3.5)
[
'
Proof: At first let u,p denote the difference of two solutions to some of the interior problems. Then we obtain from Green's first formula (1.6) with v = u (u defines the complex conjugate to u) the identity 0 =
f{\\u\2
+
2\Du\2)dy.
Due to |arg A| < tr it follows u — 0 in G,-, and the differential equations S\p = (°) in Gi provide for p = c, whereas the constant c has to vanish in the case of the interior Neumann problem (3.2) due to TpN = 0 on T. For the exterior problems we apply Green's first formula in GR \— GeC\ KR(0) and obtain 0 = f (\\u\2 + 2\Du\2) dy + f T^N -Udo. GR
dKR
Here the last boundary integral vanishes for R —> oo due to the decay condi tions (3.5). This proves the lemma. In order to show the existence of solutions u,p to the boundary value problems (3.1) - (3.4) in the following we will use the method of boundary integral equations in C°(r), as it has been carried out completely for the case n = 3 in 18 (see also Borchers, Varnhorn 2 , 3 and Hsiao, Kress f1, for A = 0) regarding the exterior Dirichlet problem for n = 2 as well as Deuring 6 and McCracken 13 regarding the exterior Dirichlet problem and the half space problem, respectively, in the case n = 3). Let us consider firstly the interior Dirichlet problem (3.1) and assume that the boundary value $ G C°(T) satisfies the compatibility condition $N
:=
f- Ndo
= 0.
(3.6)
xed,
(3.7)
For the solution of (3.1) we choose an ansatz (£)(*) = {Dx,n9)(x),
218
in form of a pure double layer potential. With help of the jump relations (2.9) we obtain the weakly singular boundary integral equations' system {\l + J£>X,„)* D ^ ) * = $* (!ln n +
on
T.
(3.8)
As (\ln + D\ n) : C°(T) -> C°(r) is a Fredholm operator, we consider the homogeneous equations on
( § ' » + #*,„)* #*,„)* = 0 (K»+ =0
on
r
(3-9)
r>,
which are adjoint to (3.8) with respect to ( * , * > : = /[ V$ t •f * (*,*>:= d odo== / f^ ^$ 2, *$,*, 7 d odo. . r
(3.10)
ii=1 =1
r
By means of the Fredholm alternative the next lemma shows that condition (3.6) is not only necessary but also sufficient for the existence of a solution \£ G C°(r) to the system (3.8). So here we have exactly the same situation as in the case n = 3, A = 0 1 2 . 3.2 L e m m a Let n > 2 and A G C\{z < 0}. Then the null spaces of the operators |ll7n+n Dl +n:C°(r)->C ^ , n : C 0 0((T) r)->C°(r) \ln + H*x
= {/3N\pe {(3N\0eC}. C}.
(3.11)
Here N is the exterior (with respect to Gi) unit normal field on T. Proof: From the properties (2.8) of the single layer potential with the source density N it follows a a (H*nXtn {Hl N)N)
= 0,
(H*XtnNY =
hence H*xnN =
on
Tr
due to the jump relations (2.9). This implies NeM
:=N(\in
+
Hln).
-N,
219 In the following we will show t h a t every solution \£ E C°(T) to (3.9) can be represented in the form \I> = @N with /3 E C. For this purpose let us consider the single layer potential ( u ) = E\ n^ where ^ E C°(T) denotes a solution of (3.9). Since this potential solves the exterior N e u m a n n problem with homogeneous boundary values and satisfies the decay conditions (3.5) (see L e m m a 2.4) we obtain in
E-Xtn9=Q
Ge
and, due to the continuity of the velocity part, £$,„* = 0
in
Gl.
Moreover, ^ n f also represents a solution of the interior Dirichlet problem with homogeneous boundary d a t a , and by means of L e m m a 3.1 (uniqueness) it follows £Sl„*=(°) with some constant c £ C
in
Gi
Due to
(H-Kn*y=o,
(H^y=cN
we find ^ = — cN from (2.10). This proves the lemma. Next we consider the i n t e r i o r N e u m a n n p r o b l e m (3.2). Here a pure single layer potential ansatz ( p ( z ) = (Ex,n*)(x),
(3.12)
x e d
is sufficient. From the j u m p relations (2.9) we obtain the boundary integral equations' system (-i/n +
ffj[|n)tf
= $
on
T,
(3.13)
and we will show t h a t for each $ E C ° ( r ) it has exactly one solution \& E C°(T). Note t h a t for A = 0 the system (3.13) cannot be solved uniquely, since the homogeneous version (3> = 0) has n ^ n 2 + ^ non-trivial solutions (see 1 5 for n = 2 a n d 1 2 for n = 3). In order to prove the unique solvability of (3.13) we consider the adjoint homogeneous b o u n d a r y integral equations ("!'»+£$,„)* = 0 Here we find
on
T.
(3.14)
220
3.3 L e m m a operators
Let n > 2 and A G C \ {z < 0}. Then the null spaces of the
-lln
+
-lln +
Dln:C°(T)^C°(T), Hln:C°(T)^C°(T)
are trivial, i.e. tf(-lln
+ Dltn) =
tf^In
= {0}.
+ HQ
(3.15)
Proof: Let tf 6 C°(T) be a solution of {-\ln + Hx,n)^ = ° o n r - T h e n t h e single layer potential Ex}n^ with source density \£ solves the interior Neumann problem with homogeneous boundary data. Due to the uniqueness of such solutions (see Lemma 3.1) we have in Gt,
Ex,n*=Q
EXin9 = 0
in G~.
As Ex,n^ represents also a solution to the homogeneous exterior Dirichlet problem and, moreover, satisfies the decay conditions (3.5) (see Lemma 2.4 above) it follows Ex,n^ — (0) m Ge. Consequently,
(#;,„*)" = (HI^)' = o, and (2.10) provides for
* = (ffX,n*)° - (#;,„*)' = o, as asserted. The existence of solutions to the exterior Dirichlet problem now follows directly from Lemma 3.3. We choose a pure double layer potential ansatz (,)(*) = (Dx,n*)(x),
xeGe,
(3.16)
and from the jump relations (2.9) we obtain the uniquely solvable integral equations' system ( - § / „ + I ^ n ) * = $ on T, (3.17) according to Lemma 3.3.
221
Finally we will consider the exterior Neumann problem (3.4). Here a pure single layer potential ansatz (;)(*) = ( £ * , „ * ) ( * ) ,
xeGe
(3.18)
leads to the boundary integral equations' system ( | I „ + #*,„)* = *
on
T.
(3.19)
Now it follows from Lemma 3.2 and the Fredholm alternative that (3.19) can only be solved for functions $ E C°(T) which satisfy the orthogonality condi tion
*oGAf(Vn + Dl)n) ^ y V t f o d o = 0.
(3.20)
r This strange condition is caused by the fact that we are looking for solutions to the exterior Neumann problem in the form of a pure single layer potential. The decay for \x\ —>■ oo of such a potential in general is too strong (in the resolvent case the single layer potential generally decays faster than the double layer potential, as shown in Lemma 2.4 above). In the following we will show that the condition (3.20) can be avoided, if for the solution of (3.4) a combined ansatz of a single and a double layer potential is used instead. For this purpose we need the next lemma (see 14 in the case n = 3, A = 0). 3.4 L e m m a Let n > 2 and A E C\{z < 0}. ThenforO ^ tf0 eN(\ln + D\n) we have f N . ^ O d o £ 0.
(3.21)
r Proof: We use the identity — | ^ o = D * n ^ o and the regularizing properties of the double layer operator
Dlin:C°(T)^C°(T) to swing up the regularity and finally obtain \I>o E C2(T) (note that the bound ary T is of class C2 according to the assumption). Next we use the fact (see 9 18 , for n = 3) that the normal stresses of the double layer potential pass continuously through the boundary from the interior G,- as well as from the exterior Ge, if only the tangential derivatives of the source density ^ are Holder continuous, i.e. we have (T{Dx,n9)Ny
= (T(DAin¥)tf)°
, *6Cll0,(r).
(3.22)
222
Since D\)n^o solves the interior Dirichlet problem with homogeneous boundary data, due to Lemma 3.1 (uniqueness) we obtain the representation
Dx>nVo = 0
in d
(c£C).
From the above mentioned continuity statement (3.22) it follows that D^.n^o solves the interior and exterior Neumann problem with the prescribed bound ary value <£ := cN. Here the case c = 0 is excluded. Otherwise D^n^o would be a solution to the interior and exterior Neumann problem with homogeneous boundary data and consequently, due to the uniqueness of such solutions, in particular D\ n\£o = 0 in G, and in Ge. This would imply ^0 = 0 due to (2.10), a contradiction to the assumption ^ o / 0 . Hence we have shown that (r{DX)n^0)Ny
( O ^ c G C).
= cN
(3.23)
Moreover, due to (D;>B*O)'
= 0,
(3.24)
from (2.10) it follows (l>5,„*o) a = - * o -
(3-25)
Now applying Green's first formula (1.6) to
(-):=(5I^),
0 =0
in GR := G e 0 KR(Q) we find
0 = - /(T£7v)a.iZado+
/ {T*N).udo+
[(\\u\2 + 2\Du\2)dy,
where the normal N on T has changed its direction. Letting R -+ oo the integral over 9-fiT.R vanishes and we obtain / ( A M 2 + 2|£>u|2)cfy = - / c N GC r
Vo do.
If here the volume integral would vanish, so would (u)a = (D* n ^ o ) a = —^o due to (3.25), in contradiction to the assumption. This proves the lemma. For the solution of the exterior Neumann problem we will now choose a combined ansatz of the form (p)(*) = (£*,»*)(*) + P(Dx,n90)(x),
x € Ge.
(3.26)
223
Here we use P := I / $ •tfo do G C r with the complex constant c from (3.23) and we assume that the function 0 ^ ^o G N(\ln + £>A>n) i s n o r m e d such that / r
AT •tfo do = 1
(see (3.21)). By means of the jump relations (2.9) and using (3.23), from (3.26) we obtain the boundary integral equations ( | / n + JTJt f J* = $-N
/ V t f o d o =: $
on
I\
(3.27)
r the right hand side <£ of which satisfies the required compatibility condition (3.20), i.e. / $ • # 0 do = r
r
/ $ •tfo do - ( I N • ^o d o V / $ • \P0 do) = 0. r r
Consequently, for all $ G C°(r) there exists a solution ^ G C°(r) to (3.27). 3.5 T h e o r e m Let $ G C°(r) be given. Then the boundary value problems (3.1)-(3.4) in bounded and exterior domains of Mn (n > 2) have a unique solution for all A G C \ {z < 0} provided that we require the decay condition (3.5) in case of the exterior problems (3.3) and (3.4) and the compatibility condition (3.6) in case of the interior Dirichlet problem (3.1). The solution of both Dirichlet problems can be represented by pure double layer potentials, the solution of the interior and exterior Neumann problem by a pure single layer and by a combination of a single layer and a weighted double layer potential, respectively. In all cases, the source densities of the potentials are solutions of second kind Fredholm boundary integral equations3 systems on C°(T).
References 1. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover Publications Inc., New York, 1970. 2. W. Borchers and W. Varnhorn, On the boundedness of the Stokes semi group in two-dimensional exterior domains, Math. Z. 213 (1993), 275300.
224
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225 LIST OF C O N T R I B U T O R S Doz.-Dr. P. Deuring
Fachbereich Mathematik und Informatik Universitat Halle 06099 Halle, Germany e-mail: [email protected]
Prof.-Dr. R. Farwig
Fachbereich Mathematik TU Darmstadt SchloBgartenstr. 7 64289 Darmstadt e-mail: [email protected]
Prof.Dr. J. Frehse
Universitat Bonn Insitut f. Angewandte Mathematik Beringstr. 4-6 D-53115 Bonn T e l : 0228-733142 Fax: 0228-733141
Prof.Dr. A. Fursikov
Department of Mechanics and Mathematics Moscow University Lenin Hills Moscow 119899, Russia e-mail: [email protected]
Prof.Dr. G. Grubb
Department of Mathematics University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen, Denmark Tel.: 0045-35320743 Fax: 0045-35320704 e-mail: [email protected]
Dr. Ch. Grunau
Universitat Bayreuth Institut f. Mathematik 95440 Bayreuth Tel.: 0921-553292 Fax: 0921-553293
226 Prof.Dr. J.G. Heywood
Department of Mathematics University of British Columbia Mathematics Road Vancouver B.C. Canada V6T 1Z2 Tel.: 001-604-822-0569 Fax: 001-604-822-6074 e-mail: [email protected]
Dr. Y. Kagei
Department of Applied Science Faculty of Engeneering Kyushu University Hakosaki, Fukuoka, 812 Japan Tel.: 0081-92-641-1101 Fax: 0081-92-651-6024
Prof.Dr. H. Kozono
Graduate School of Polymathematics Nagoya University Nagoya, 464-01 Japan Fax: 0081-52-789-3724 e-mail: [email protected]
Prof.Dr. P. Maremonti
Dipartimento di Mathematica Universita della Basilicata Via N.Sauro,85 85100 Potenza, Italy e-mail: [email protected]
Prof.Dr. K. Masuda
Tohoku University Mathematical Institute Sendai 980, Japan Fax: (+81)-22-2151-6400 [email protected]
Prof.Dr. A. Novotny
Universite de Toulon et du Var Faculte des Sciences et Techniques B.P. 132-83957 La Grade Cedex, France Tel.: 0033 94 142122 Fax: 0033 94 142168
227
Prof.Dr. M. Padula
Dipartimento di Matematica Universita di Ferrara Via Machiavelli 35 44100 Ferrara, Italy e-mail: [email protected]
Prof.Dr. R. Rautmann
Universitat-GH Paderborn Warburger Str. 100 33098 Paderborn, Germany Tel.: 05251-602649 Fax: 05251-603836 e-mail: [email protected]
Dr. M. Rumpf
Universitat Freiburg Insitut f. Angewandte Mathematik Hermann-Herder-Str. 10 79104 Freiburg Tel.: 0761-2035638 Fax: 0761-2035632
Dr. M. Ruzicka
Insitut fur Angewandte Mathematik Universitat Bonn Beringstr. 4-6 D-53115 Bonn Tel.: 0228-733142 Fax: 0228-737864 e-mail: [email protected]
Prof.Dr. M.E. Schonbeck
University of California Department of Mathematics Santa Cruz CA 95064, USA Tel.: 408-4594657/4592085 Fax: 409-459-3620 schonbek@math. ucsc.edu
Prof.Dr. A. Sequeira
Instituto Superior Tecnico Departamento de Matematica Avenida Rovisco Pais, 1 Lisbon Codex, Portugal Tel.: 351 1 8417073 Fax: 351 1 8417048 e-mail: [email protected]
228 Prof. Dr. H. Sohr
Universitat-GH Paderborn War burger Str. 100 33098 Paderborn, Germany Tel.: 05251-602648 Fax: 05251-603836 e-mail: [email protected]
Prof.Dr. V.A. Solonnikov
St. Petersburg Branch of Steklov Mathematical Institute Fontanka 27 191011 St. Petersburg, Russia Tel.: 007-812-3124058 Fax: 007-812-3105377 e-mail: [email protected]
Dr. M. Specovius-Neugebauer
Universitat-GH Paderborn Warburger Str. 100 33098 Paderborn, Germany Tel.: 05251-602641 Fax: 05251-603836 e-mail: [email protected]
Prof.Dr. W. Varnhorn
Universitat-GH Kassel Fachbereich Mathematik Heinrich-Plett-Str. 40 34132 Kassel, Germany Tel.: 49 561 8044614 Fax: 49 561 8044318 e-mail: [email protected]
Prof.Dr. M. Yamazaki
Department of Mathematics Hitotsubashi University Kunitachi, Tokyo 186 Japan
