A Local Strong Solution of the Navier-Stokes Problem in L 2 (Ω)
11 A Strong Solution of the Navier-Stokes Equation on )( p L Maoting TongDepartment of Mathematical Science, Xi’anJiaotong-Liverpool University, Suzhou, 215123,P.R.China, E-mail address: [email protected] Ton*Hohai University, Nanjing, 210098, P.R.China, E-mail address: [email protected], Currentaddress : 1-3306 Moonlight Square, Nanjing, 210036, P.R.China. *- The corresponding author
Abstract
In this paper we prove that the Navier-Stokes initial value problem (1.1)-(1.4) has a uniquesmooth local strong solution ),( xtu and ),( xtp if p DLtf )( is H ö lder continuousabout t on T ,0 and the initial value ))(( Du C satisfies ).(0 jixu ji Mathematics Subject Classification (2010). Primary 35Q30, 76N10; Secondary 47D06.Keywords. Navier-Stokes equation, Existence and uniqueness, Local solution, Semigroup ofoperators, Invariance, Fractional powers. ( ) Introduction
The Navier-Stokes initial value problem can be written in the following form as )4.1..(.............................................................................................................., )3.1.........(....................................................................................................,0,0 )2.1.(..............................................................................................................0 )1.1.....(............................................................,,,)( xuu tu divuu ttxfuuputu tt where is a bounded domain in R with smooth boundary of class , C u )),(),,(),,((),( xtuxtuxtuxtu is the velocity field, )( xuu is the initial velocity, ),( xtpp is the pressure, ),( xtff is the external force. In [17] we proved that theNavier-Stokes initial value problem (1.1)-(1.4) has a unique smooth local strong solution on )( L if the initial velocity u and the external force f satisfy some conditions. In this paperwe will extend this result to ).( p L Let )( p L ( p ˂ ) be the Banach space of real vector functions in )( p L with theinner product defined in the usual way. That is )3,2,1)((),,,(,:)( iLuuuuuRuL pip .For ),(),,( p Luuuu we define the norm .)( )(3 1)( ppp pLi iL uu The set of all real vector functions u such that div u =0 and )( Cu is denoted by ).( ,0 C Let )( p DL be the closure of )( ,0 C in ).( p L If )( Cu then )( DLu implies div u =0. (see p.270 in [4] ). We have )( ,0 C )( C )( p L and ),()()( LLCDL pp ),()()( ,0 ppp WLDL , )()( pp LDL and .)()()( ppp DLDLL (2)From [4] and [5] we see that .;)( ,1 pp WhhDL
Let P be the orthogonalprojection from )( p L onto ).( p DL By applying P to (1.1) and taking account of theother equations , we are let the following abstract initial value problem, .Pr II )2.3.....(...................................................................................................., )1.3.(................................................................................,,
00 0 xuu ttPfFuuPdtdu t where .)( uuPFu We consider equation (3.1) in integral form Pr.III .))(()( )(0 dsPfsFueuetu tt PsttP (4)For )(),,( p Luuuu we define ),,( uuuu and ),,( xuxuxuu where is considered as an operator operating on vector functions. Since the operator i i x is strongly elliptic of order 2. From Theorem7.3.6 in [12] is theinfinitesimal generator of an analytic semigroup of contractions on )( p L with )()()( ,10,2 pp WWD . Hence is also the infinitesimal generator of an analyticsemigroup of contraction on )( p L with )()()( pp WWD , where )( ,2 p W and )( p W are the Sobolev spaces of vector value in )( ,2 p W and )( ,10 P W respectively. Wewill prove that is also the infinitesimal generator of an analytic semigroup of contraction on ).( p DL This is a key point for our purpose. (2)Some lemmasLemma 1. If )( Cu then )( p DLu implies div .0 u Proof. Suppose that ).()(,, p DLCuuuu
Then there exists a sequence .,..2,1:)( ,0 nCu n such that , nn uu lim that is )( RDLnn p uu uniformly on and )( p Lniin uu uniformly on for .3,2,1 i And so .. eaniin xuxu and )()(lim .. xuxu ieanin uniformly on for .3,2,1 i (see Theorem 1.39 in [15] ) It from theproof of Theorem 7.11 in [14] follows that .0limlim)()(limlim )(lim)(limlim)lim( i inini inini ii iniinixxn ii ininininxxii i nini ii xuxuxx xuxu xx xuxux uxudivu ioi ii This is to say div .0 u Lemma 2.
For every )( p Lu , div u if and only if div uI for : < arg < r , where ˂ ˂ . Proof. It is similar to the proof of lemma 1 in [17].
Lemma 3. (1.5.12 in[8]) Let ttT be a C -semigroup on a Banach space X . If Y is a closed subspace of X such that YYtT )( for all t , i.e., if Y is )( t tT -invariant, then the restrictions Y tTtT )(:)( form a C -semigroup ttT , called the subspace semigroup on the Banach space . Y Lemma 4. (Proposition 2.2.3 in [8]) Let ))(,(
ADA be the generator of a C -semigroup ttT on a Banach space X and assume that the restricted semigroup (subspacesemigroup) ttT is a C -semigroup on some ))(( t tT invariant Banach space XY . Then the generator of ttT is the part ))(,( ADA of A in . Y Lemma 5 . The operator )( p DL is the infinitesimal generator of an analytic semigroup ofcontractions on ).( p DL Proof. We have know that is a infinitesimal generator of an analytic semigoup ofcontraction on ).( p L Let ttT be the restriction of the analytic semigroup generated by on )( p L to the real axis. Then ttT is a C semigroup of contractions . Wehave already noted that )( p DL is a closed subspace of )( p L . We want to show that )( p DL is )( t tT invariant.For every )( p Lu with div u and :)( Lemma 6. ( Lemma 2.2 in [5]) Let < .2/)1( rn Then rrr vAuAMvuPA ,0,0,0 ),( with some constant ),,,,( rMM provided ,2/12/ rn ˃0, ˃ >1/2.From the Lemma 6 and the formula (5) we see that if take rn , and ,4/3 then )()(20 )( )()(( )()( )() pp pp ppp DLDL DLDL WLDL vuML vuM vuvuvu )() )(( with some constant M for any ).(, p DLvu Hence we have Lemma 7. Suppose that )( p DLvu , are velocity fields and ),()( p DLvu then .)( )()(20)( ppp DLDLDL vuMLvu Now we introduce the main lemmas of this paper. Assumption (F). Let )( p DLX and U be an open subset in )10( XR The function XUf : satisfies the assumption (F) if for every Uut ),( there is aneighborhood UV and constants ,0 L < such that for all )2,1(),( iVut ii )(),(),( uuttLutfutf X . (7) Lemma 8. ( Theorem 6.3.1 in [12]) Let A is the infinitesimal generator of an analyticsemigroup )( tT on the Banach space )( p DLX satisfying tT and assume furtherthat ).(0 A If, < <1 and f satisfies the assumption ),( F then for every initial data Uut ),( the initial value problem )2.8.(..............................................................................................................)( )1.8.........(............................................................,)),(,()()( 00 0 utu tttutftAudttdu has a unique local solution ))(:),(())(:,( pp DLTtCDLTtCu where ).( uTT In what follows we will need Banach lattice (see [3] ). A real vector space G which isordered by some order relation is called a vector lattice (or Riesz space) if any two elements Ggf , have a least upper bound, denoted by gf , and a greatest lower bound , denoted by gf , and the following properties are satisfied:(i) If , gf then hghf for all ,,, Ghgf (ii) If f , then tf for all Gf and Rt .A Banach lattice is a real Banach space G endowed with an ordering such that , G isa vector lattice and the norm is a lattice norm, that is gf implies gf for Ggf , , where )( fff is the absolute value of f and is the norm in . G In a Banach lattice G we define for Gf ,0: ff ff .The absolute value of f is fff and . fff gfgfgf .In what follows one will need the above formula .In )( p L we define the order for )(, p Lgf )()( xgxfgf for .. ea x and )(),(max:))(( xgxfxgf , )(),(min:))(( xgxfxgf for .. ea x . Then ),(),( ,1 pp LW )( p L and )( p DL are all Banach lattices.(see [1] p.148) Lemma9. Suppose that )(, p DLvu are divergence free satisfying jiji xvxu ).( ji Then ,)( vu ).()( p DLuv Proof. If vu , )( p DL are divergence free satisfying jiji xvxu ,0 ).(0 ji From 【 】 and 【 】 we have hdxvhdxu for all ).( ,1 p Wh That is .0 dxxhu iii Since )( p L is a Banach lattice and )( p DLh for ),( ,1 p Wh then .)()(),(,, hhhDLuuuuu p From proposition 10.8in [3] the lattice operations and are continuous, we have ),(),,(),,(lim ),,(),,(lim ),,(),,(lim )),,(),,(lim( )),,(),,(lim( )),,(),,(lim()( )( )()( 303 202 101 303 202 101 hxhxhxhxx xxxhxxxh xx xxxhxxxh xx xxxhxxxh xx xxxhxxxh xx xxxhxxxh xx xxxhxxxhxhxhxhxhxhxhh xx xx xx xx xx xx Similarly, ).()( hh ) ()()( ,0,1 ppp LWW are all Banach spaces. )( ,1 p Wh implies ),(, RWhh p .0)( dxhu Since v is divergence free, ,0 i i xv so i xv are all bounded, ( Lxv i for some constant L ˃0. We have dxxhuLdxxhxvu dxhvudxxhuL ii iiiii i ii i Hence .0)()( dxhvu Similarly, .0)()( dxhvu So we have .0)()()()( dxhhvuhdxvu Similarly, .0)( hdxvu Therefore hdxvuhdxvuhdxvuuhdxvu and so ).()( p DLvu Similarly ).() p DLuv ( (3)Main result Now we study the Navier-Stokes initial value problem (1.1)-(1.4).A function u which is differentiable almost everywhere on T ,0 such that , u )(:,0 DLTL is called a strong solution of the initial value problem (1.1)=(1.4) if )0( uu and u satisfies (1.1)-(1.4) a.e. on T ,0 . A function )(,0: DLTu is aclassical solution of (3.1)-(3.2) on T ,0 if u is continuous on T ,0 ,continuous differentiable on T ,0 , )( Du for ),0( Tt and (3.1)-(3.2) are satisfied on T ,0 .Let )(;,0 p DLTH denote the space of all Hölder continuous functions ),( xtu on T ,0 with different exponents in and with divergence free functions valuessatisfying ji xu )( ji in the Banach space .)( p DL Then from lemma 9 for any Hu )(;,0 p DLT and any Ttt ,0, , );()()( p DLtutu and for any , uu )(;,0 p DLTH and any Tt ,0 , ).()()( p DLtutu In the following wewill use these facts. Let .,0,)(;,0:)( TtDLTHutuH p H is a subset of )( p DL which consists of function values of all functions in ;,0 TH p DL .)( Let ),,(),( kkkxtu k ,,0( Tt , x )).3,2,1, iRk i Then Hu k ;,0 T )( p DL and Htu k )( for all Rkkk ,, and all .,0 Tt Suppose that )( ii xu )3,2,1)(( iDL p satisfying .0)( i i ii xxu Let ))(),(),(()( xuxuxuxu and )(),( xuxtu for .,0 Tt Then )(;,0),( p DLTHxtu and .)( Hxu Hence H is not empty. Take the open kernel H of H in .)( p DL H is also not empty. It isclear that H is a open subset in .)( p DL The bilinear form uv )( on H takes value in )( p DL . Let .,0 HTU Then U is an open subset of .)(,0 p DLT If , Hu that is, there exist u )(;,0 p DLTH and Tt ,0 such that ).( tuu Let )( utu for all t .,0 T It iseasy to see that .)(;,0)( p DLTHtu Hence u is also a value of another function.That is to say that a function )( Htu can be value of different functions in .)(;,0 p DLTH Theorem . The Navier-Stokes initial value problem (1.1)-(1.4) has a unique smooth localstrong solution ),( xtu and ),( xtp if the following condition are satisfied(1) p DLf is H ö lder continuous about t on T ,0( ,(2) The initial value ))(( Du C satisfies ).(0 jixu ji Proof. ( Step1 ) First, ))(,( tutF )())(( tutu is a function : )( p DLU because )()())(( p DLtutu according to the definition of . U We will find that by incorporating the divergence-free condition we can remove the pressureterm from our equation. (see p. in 【 】 , p. and p. in 【 】 ) In fact, from )(;)( ,1 pp WhhDL we see that )( p DLp and so .0 pP For u )( p DL we have )( p DLu because to Lemma 5. Hence by applying P to the equation(1.1) we have . uuP It follows from uu )( )( p DL that .)()( uuuuP Therefore we can first rewrite (1) into an abstract initial value problem on )( p DL xuu TtttutFudtdu tt , ,)),(,( (9) where uututF )())(,( . From Lemma 4 )( p DL is the generator of an analyticsemigroup )( tT of contraction on ).( p DL So .1)( tT From Theorem 2.5.2(c) in 【 】 ).(0 (Step2) If tu is Hölder continuous about t on T ,0 in )( p DL , so there is aconstant C and 0˂ such that ),(),( ttCxtuxtu RDL p for .,0, Ttt (10)For any Ututtut ))(,()),(,( we have ),())(( tutu )()())(( p DLtutu )(2211)(22)(1120 )(22)(1211 )(12111120 )(121211 )(121111 )(12121211 )(12111111 )(12121111 )()()()( ))()()( ))()(()(( )()))()(( ))()()()(( )())(()())(( )())(()())(( )())(()())(( ppp pp ppp ppp DLDLDL DLDL DLDL DLDL DLDLDL tutututuML tututu tututuML tututu tututu tutututu tutututu tutututu )( .)()( )()( )()()()( )()( 121 2121 2121 2121 ttCtutu tutuML tutututu tutuML p pp pp pp DL DLDL DLDL DLDL (11)We used lemma 6 in the above third step . For any )()(),( p DLtutu we have .))()(( )()())()(( )()( tttutuCML tutututuML tutututututuML tutututututu tutututututututu tutututu pp ppp pppp pp ppp DLDL DLDLDL DLDLDLDL DLDL DLDLDL (12)We used the Lemma 6 in the above third step and the formula (10) in fifth step.( Step 3) Suppose that )( p DLu is smooth satisfying ji xu i ( ). j Then u isdivergence free from lemma 1 and .),( Uut Set .,1:))(,(),( )(0000 p DL uuttUtututBV Then for Vtut ))(,( , . )(0)(0)(0)(00)( ppppp DLDLDLDLDL uuuuuuuu Let , )(0 RDL p uL ),(22 CCLMLLLMLL , ),( LLMaxL and ),,( Min then from (10),(11) and (12) for all Vut ii ),( we have ).)()(( )()( 22)()(2 )())(()())(()())(()())(( ))(,())(,())(,())(,( ))(,())(,( 212 221 121 )(2211213 212)(22111 212021120)(221120 )(22221212)(12121111 )(222121)(121111 )(222111 pp p pp ppp DLDL DL DLDL DLDLDL tututtL ttLtutuL ttLCMLttLCMLtutuLML tutututututututu tutFtutFtutFtutF tutFtutF Hence ))(,( tutF satisfies the assumption ).( F Therefore from lemma 8 for every initial data ),( Uut the initial value problem (9) has unique local solution )( tu ))(:,())(:,( pp DLttCDLttC ) (13 )where ).( utt The solution (13) of (9) is also the solution of (4). The Theorem 3.4 in [5] mean that as longas the solution of (4) exists , this solution is smooth. From Theorem 3.4 in [5]we have the solution .,0),( tCxtu Substituting ),( xtu into (1.1) we get the solution ),( xtp . We alsohave ),( xtp .,0 tC It follows from the lemma 1 and )( p DLu that the solution ),( xtu is divergence-free. Changing the value of u on to zero we get a unique smoothlocal strong solution of the Navier-Stokes initial value problem (1.1)-(1.4). If is a bounded open subset of R then the solution (13) is a classical solution of (9) and(3.1)-(3.2). Acknowlements The first author is grateful to her supervisors Chi-Kun Lin and Xinyao Yang for their teachingand cultivation.