KOZONO, Hideo

YANAGISAWA, Taku

Kyoto University Research Information Repository

Title Global DIV-CURL Lemma in 3D bounded domains (HarmonicAnalysis and Nonlinear Partial Differential Equations)Author(s)KOZONO, Hideo; YANAGISAWA, TakuCitation数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu(2009), B14: 27-33Issue Date2009-11URL http://hdl.handle.net/2433/176896RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityRIMS Kokyuroku Bessatsu Bl4 (2009), 27-33 Global DIY-CURL Lemma in 3D bounded domains 'J'Ii*a (Hideo KOZONO) Mathematical Institute Tohoku University Sendai 980-8578, Japan 1~Pi1;J~ (Taku YANAGISAWA) Department of Mathematics Nara Women's University Nara 630-8506, Japan 1 Introduction. Let !:1 be an open set in JR3 . It is well-known that if Uj ----' u, Vj ----' v weakly in £ 2 (!:1) and if {div Uj}~1 and {rot Vj}~1 are bounded in £ 2 (!:1), then it holds that Uj ·Vj----' u·v in the sense of distributions in !:1. This is the original Div-Curl lemma. For instance, we refer to Tartar [5]. The purpose of this article is to deal with a similar lemma to bounded domains where the convergence Uj · Vj ---+ u · v holds in the sense that (1.1) k Uj · Vjdx ---+ k u · vdx as j ---+ oo. Our result may be regarded as a global version of the Div-Curl lemma, which includes the previous one. To obtain such a global version, we need to pay an attention to the behaviour of { Uj }~1 and { Vj }~1 on the boundary 8!:1 of !:1. Indeed, an additional bound of { Uj · vlan}~1 , or 1 that of { Vj x vlan}~1 in Hz ( 8!:1) on the boundary 8!:1 plays an essential role for our convergence, where v denotes the unit outward normal to 8!:1. In what follows, we impose the following assumption on the domain !:1: Assumption. !:1 is a bounded domain in JR3 with C00-boundary 8!:1. Before stating our result, we first recall the generalized trace theorem for u · v and u x v on 8!:1 defined on the Banach spaces E~iv(!:l) and E'j.ot(O) for 1 < q < oo, where E~iv(!:l) E'j.ot(!:l) {u E Lq(!:l);div u E Lq(!:l)} with the norm lluiiEq = llullq + lldiv ullq, dw {u E Lq(!:l);rot u E Lq(!:l)} with the norm lluiiEq = llullq + llrot ullq· rot Here and in what follows, II · llq denotes the usual Lq-norm over !:1. It is known that there are bounded operators "fv and Tv on E~iv(!:l) and E'j.0 t(!:1) with properties that "tv: u E E~iv(!:l) f--+ "fvU E W 1- 1/q',q' (8!:1)*, "fvU = u · vlan if u E C1 (D) , Tv: u E E'j.ot(O) f--+ TvU E W 1- 1/q',q' (8!:1)*, TvU = u X vlao if u E C1(D) ' © 2009 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved. 28 l-IIDEO f(OZONO AND T AKU ~{ A_NAGISA_\i\/A -+ = 1. The range ,q1 of l'v and Tv is the dual soace the of the trace on 8!1 of functions in the holds (u, rot¢)= v, -\~vher,e Jo denotes the usual the between to Borchers-Sohr [1], [2], Sirna.der-Sohr [3] g,nd Temam Our result now reE,ds: (L4) DOTne lt E n be as in the and and v E Uj and that c {rot is bmmcled in If eitheT (i) }j~1 is bounded in OT is bounded in then it holds that -----+ In {1. As an in:unediate consequence of our open set in some u E and v E (D), (1.9) ancl {rot Then it holds tl~at Let l < T < oo with 1/r + 1. t!wt Assv,me also that some q > somes> u vrlx as j -j. oo. all = 1, 2, ·, · is we have the open set in _Assurne also that are bounded in -:-tlt,en we have also Div-Cu.rl lemma m an . Let 1 < T < oo. (D\ ' ) (D), (1.10) Uj · Vj ---' u · v iT! the sense distrib•tdions in D. GLOB.AL JJIV-CURL LEMMA TN 3D BOUNDED DOMAINS 29 Remarks, (i) Since n is a bounded we may assume q ;; r and + < s ;; r 1, and hence it holds that Then we have that c (ii) In Theorem it is unnecessary to assume both bounds of and in vvl-l/r',r' what we need is one of these bounds, 2 decon'1position, In this vve recall the for vector fields m L' For a we refer [2], to the tvm types u , ~~ = 0 and tt x conditions we il.rst define harrrwnic vector spaces and i div h = 0, rot h = Q in fl Vvith h · V = Q 011 {hE E div h = 0, rot h = 0 in with h >< v = 0 on Moreover, for 1 < T < oo let us define [2] vector fields ; div tL = 0, r,lvU = E W1''(0); cliv tt = 0, TuU = Let 1 < r < oo. are dimertsion,al vector spaceso and (2) For every u are p E W 1,r(:l1), w E and hE can be as +rot w -:-Such a w, h} is sttbonlinate to the estimate with the constant C' = C'(ll, if u has another j5E (3) Fo·r every tt E can be as u. The above u = h +rot w + and hE the11 we have h = ft, rot w =rot , there are p E W~''(fl), w E u = h + wt w + Such a {p,w, is subordinate to the'- estimate IIPIIw' r + llwllwl,r + llhllr ;; Cll·ullr and E 1) ~s such that to In such t.hat u 30 HID EO KOZONO AND T,AKU Y Al\fAGISA WA will& tl&e constant C = of u. The ab01je if ti haS another r"t:'n'IF.,S'U u = h + rot 'iiJ + p'C aTtd .hE ther& VJe have h = h, rot w = rot .A.n immediate consequence of the above theore111 is 1ue have two k,inds 7) 1 < r < oo. r:then ·we .have all u E L./ theTe .holds and where C' = denote the If t& has '111 additional EfJ rot EfJ rot =rot w, is the constant such may choose the scalar and the ve·dor More we have ( i) in 1) can be choserl', as ?JJ E (ii) di\T 1-t E wiih seal aT zn 1) can be chose·n of .) esti1nate on In < r· < oo. ]Vf oTeover, and for son1e 1 < q :£ r, in the class ~ r) then the 'With_, the estirnate GLOBAL DIV-CURL LEMMA IN 3D BOUNDED DON!An,;s 31 Let us consider the (i) in div "u (2,4) can be chosen asp E with the estinwte (ii) vectoT J then the with the estimate, r, q) is the constant 3 Proof of Tlu:~oren1 1. (i) Let us first consider the case when we make use of the v) on r and q, In such a case, operators from Notice that the holds for all u E (L3), we have and all v E Stokes formula and h)==-w, h) = for all p E (:11), wE (fl) and hE 'LU, = 0 for all wE pE Thus >Ne obtain 1), vve ses that the convergence (L'T) can be reduced to (3A) the ranges of are finite rank operators, therefore compact, frorn vii1ich it that to r 3r max') 1,-- < q ;= T, 3+r-and Since !1 is max ~, 1, 3r' .t 'l( < s ;= rJ , 3 + r ,, we 1nay assun1e c<· ,)J.l1.C8 I-iiDEO I\OZONO /\J,TD TAEU Y AJ'JAGISi~~ \~l A and T 1, we see that with the estirnate Vj lis+ llvj llr1 ) :;:; AI, for all j = 1, 2, · · · of j. and hence we see with the estimate (3.3) follows. and that <::: lVI for all j = 1, 2, · · folluvls. =rot V'1e next conside:r the case when , In this case, \Ne the former case ( i) make use of the above. !Then the argv.n1ent is This proves Theorem L 1.1. \Ne rnay prove that for every (p E Let us cake bounded dormdn Q c Then it suffices prove that it holds that Since div with t ---> j zpu · vdx . .. D vvith the smooth -> (L8) and EJQ so that supp y:; c Q c D. (Q). is bounded is also bounded in (n), q = T and s = T 1 in the ::onvergence 'I) follows from and Theorem GLOBAL DIV-CURL LEMMA IF 3D BOUNDED DOMAINS Heferences (1] Borchers) --:\lv .... , Sohr1 I-I.; On the equations rot v = .9' and div tt = f 11.1ith ;:;ero boundary condition.;s. =-:Iokkaido IVIath . .J., 19, 67-87 [2] Kozono)-L, Yanagh~;a ... va 1T., I./'-variat'iona~~ ineqv.a,lity faT vector and the H-elrnl&oltz- VVeyl ae-cornpo8ition, bounded dornains. to appear in Indiana tJniv. ldatb, J" [3] Sirnad.er1 COC~.i Sohr) I-I.j A neu) apJlTOcch, to the Ilel?n.l-~oltz decon~posit·ion and the l\Teu1nann )JTOb-lent in Lq-sp,_:tces fat' bounded a·nd e;;;terior dorn,ains. 1'l\1athernatical Problerns relating to the I\Tavier-Stokes Equation_s);· 011 .Ptdvanced in lviathsrnatics for J\_pplied Sciences1 {J.P. Cialdi eeL, Singapore-1\J e·\v Jersey-London-I-Iong I\~ong: V\Torld Scientific~ 1-35 [4] Silnader, Sohr, I-L, The Dirichlet problen~ the Laplacian bounded an,d unbo,un.ded do<-rnains. PitrDa:n Research r\f otes in lVIathe1natics Series 300: Logn18Xl l 996. [6] Tart.ar)L., con&pactness ancl apphcations to po/tiial eJ}V,o~tions, 1\Tunlinear a.nd Mechanic.s: I-Ieriot-V/att Symposium 4, R.J. Knops, ed., Research Note:l in Math., Pi',man, 1979. [6] Tlerna.rn, Pt,., l+lav-ier-Stokes equations. TlteoTy and J+lurn,erical _,4nalysis ]\Jortl1-I-Iolland Pub. Co., PLm.sterdam-Ne'N Y-ork-O~xford 1'979 

Global DIV-CURL Lemma in 3D bounded domains (Harmonic Analysis and Nonlinear Partial Differential Equations)

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Global DIV-CURL Lemma in 3D bounded domains (Harmonic Analysis and Nonlinear Partial Differential Equations)

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