Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L (0,T; L n (Ω))

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1 Aca Mahemaica Sinica, Englih Serie May, 29, Vol. 25, No. 5, pp Publihed online: April 25, 29 DOI: 1.17/ Hp:// Aca Mahemaica Sinica, Englih Serie The Ediorial Office of AMS & Springer-Verlag 29 Energy Equaliy and Uniquene of Weak Soluion o MHD Equaion in L (,T; L n (Ω)) Yan YONG Iniue of Applied Mahemaic, Academy of Mahemaic and Syem Science, Chinee Academy of Science, Beijing 119, P. R. China yongyan@am.ac.cn Quan Sen JIU School of Mahemaical Science, Capial Normal Univeriy, Beijing 137, P. R. China jiuq@mail.cnu.edu.cn Abrac In hi paper, we udy he energy equaliy and he uniquene of weak oluion o he MHD equaion in he criical pace L (,T; L n (Ω)). We prove ha if he velociy u belong o he criical pace L (,T; L n (Ω)), he energy equaliy hold. On he bai of he energy equaliy, we furher prove ha he weak oluion o he MHD equaion i unique. Keyword MHD equaion, weak oluion, energy equaliy, uniquene MR(2) Subjec Claificaion 35Q35, 35Q99, 76B3 1 Inroducion Conider he following iniial boundary value problem of he MHD equaion: u u +(u )u (B )B + ( 1 2 B 2 )+ p f, (x, ) Ω (,T), B B +(u )B (B )u, (x, ) Ω (,T), (MHD) u, B, (x, ) Ω (,T), u(x, ),B(x, ), u(x, ) a, B(x, ) b, (x, ) Ω (,T), (x, ) Ω (,T), where Ω R n (n 2) i a domain, uu(x, )(u 1 (x, ),...,u n (x, )), BB(x, )(B 1 (x, ),...,B n (x, )) and p(x, ) denoe he unknown velociy field, magneic field, he calar funcion of preure, repecively. f f(x, ) (f 1 (x, ),...,f n (x, )) i a given exernal force. a, b denoe he given iniial daa aifying a, b. The fir goal of hi udy i o inveigae he energy equaliy of he MHD equaion in he criical pace L (,T; L n (Ω)). Energy equaliy i a very imporan propery of he Navier Soke equaion and he MHD equaion. However, up o now, he obained weak oluion o he MHD equaion only aify he energy inequaliy, wheher hey aify he energy equaliy i ill an open problem. For he hree-dimenional Navier Soke equaion, he energy equaliy hold if u L (,T; L q (Ω)) wih q 1, 2 <<, 3<q<, which i called he Serrin cla (ee Sohr [1]). Acually, o ge he equaliy of he weak oluion o Received April 3, 27, Acceped May 23, 28 The reearch i parially uppored by NSF of China (Gran No & )

2 84 Yong Y. and Jiu Q. S. he hree-dimenional Navier Soke equaion, i i no neceary o aume u o belong o he Serrin cla. I ha been proved ha weak oluion o he Navier Soke equaion aify he energy equaliy only if uu L 2 (,T; L 2 (Ω)), which i aified if u L 4 (,T; L 4 (Ω)) (ee [1]). In ligh of Serrin [2], we prove in hi paper ha he energy equaliy hold for he weak oluion o he MHD equaion if he velociy belong o he criical pace L (,T; L 3 (Ω)). I i worhy o emphaize ha here are no aumpion on he magneic field here. In oher word, our reul demonrae ha he velociy field play a dominan role in eablihing he energy equaliy for he MHD equaion. Wheher he condiion can be relaxed o uu L 2 (,T; L 2 (Ω)) remain unclear. Though he MHD equaion have many properie imilar o hoe of he Navier Soke equaion, here are more nonlinear erm and rongly coupled erm in he MHD equaion which are more difficul o deal wih. The econd goal of hi udy i o prove he uniquene of he weak oluion o he MHD equaion. There have been a number of lieraure on he uniquene of he weak oluion o he Navier Soke equaion. Foia [3] howed ha if Ω R n and if u L (,T; L q (Ω)) wih 2 + n q 1, q>n,henu i a unique oluion o he Navier Soke equaion. Serrin [2] proved he ame reul in arbirary domain Ω R n wih 2 n 4 under he aumpion ha q>nand 2 + n q 1. Since Fioa Serrin work, many effor had been focued on proving he uniquene of Leray Hopf weak oluion in uch a criical cae a and q n. Sohr Von Wahl [4] howed ha if Ω i a bounded domain and if he iniial value a and he exernal force f are boh regular enough, hen he weak oluion u in L (,T; L n (Ω)) wih ome addiional condiion i unique. Mauda [5] conidered arbirary domain and removed he aumpion on he regulariy of a and f impoed by Sohr Von Wahl [4]. More preciely, he howed ha if u L (,T; L n (Ω)) and if u() i righ coninuou on [,T) in he norm of L n,henui he only weak oluion o he Navier Soke equaion. Kozono Sohr [6] howed ha he condiion on he righ coninuiy of u() on[,t) in he norm of L n can be removed in domain uch a he whole pace R n, he half pace R n +, bounded and exerior domain. In fac, Ecauriaza Seregin Sverák [7] proved he regulariy of he oluion o he hree-dimenional Navier Soke equaion in L (,T; L 3 (R 3 )), which implie he uniquene of he oluion o he Navier Soke equaion. He Xin [8] obained ome ufficien condiion for he regulariy of weak oluion o he MHD equaion. They proved ha if he velociy u belong o he pace L p (,T; L q (R 3 )) wih 1 p + 3 2q 1 2, q>3 (excep for he criical cae when q 3,p ), hen he weak oluion o he MHD equaion i regular. Similar reul have been obained by Zhou [9]. In our paper, we fir conider an arbirary domain and ge he uniquene of weak oluion o he MHD equaion under he ame aumpion a in Mauda [5] for he Navier Soke equaion. We how ha he condiion of he righ coninuiy of velociy field and magneic field can be removed in ome pecial domain uch a he whole pace R n, he half pace R n +, a bounded domain or exerior domain. Indeed, by conrucing a rong oluion, we can prove ha every weak oluion (u, B) o he MHD equaion in he cla L (,T; L n (Ω)) i necearily righ coninuou wih value in L n (Ω) on [,T). I hould be poined ou, however, ha moivaed by he reul of Ecauriaza Seregin Sverák [7], when u L (,T; L n (Ω)), he oluion o he MHD equaion could be regular.

3 Energy Equaliy and Uniquene o MHD Equaion 85 The funcion pace ued in hi udy are inroduced a follow. C,σ denoe he e of all C vecor funcion φ (φ 1,...,φ n ) wih compac uppor in Ω uch ha φ. L r and for he uual (vecor-valued) L r pace over Ω, 1 <r<, wih he norm φ L r ( φ r dx) 1 r. L r σ i he cloure of C,σ wih repec o L r -norm L r. H 1,r,σ denoe he cloure of C,σ wih repec o he norm φ H 1,r φ L r + φ L 2,where φ ( φ i / x j,i,j 1,...,n). For an inerval I in R 1 and a Banach pace X, L p (I,X)andC m (I,X) denoe he uual Banach pace, wih he norm φ Lp (I,X) I φ Xd and φ Cm (I,X) up I φ X where 1 p, m, 1,... repecively, The paper i organized a follow. In Secion 2, we preen he main reul. In Secion 3, we udy he energy equaliy of he weak oluion o (MHD) in L (,T; L n (Ω)). In Secion 4, we inveigae he uniquene of he weak oluion o (MHD) in L (,T; L n (Ω)). 2 Main Reul In hi ecion, we preen he main reul. Throughou hi paper, we impoe he following aumpion. Aumpion 1 a, b L 2 σ(ω). Aumpion 2 Pf L 1 (,T; L 2 σ(ω)), where P i he projecion operaor from L 2 o L 2 σ. Aumpion 3 Ω belong o one of he following cae: (i) Ω i he whole pace R n (n 3); (ii) Ω i he half pace R n + (n 3); (iii) Ω i a bounded domain in R n (n 3) wih C 2+μ (μ>)-boundary Ω; (iv) Ω i an exerior domain in R n (n 3), i.e. a domain having a compac complemen in R n \ΩwihC 2+μ (μ>)-boundary Ω. Aumpion 4 a, b L 2 σ(ω); f L 1 (,T; L 2 (Ω)) L (,T; L q (Ω)), where 1 <<, n 3 <q n and 2 + n q < 3. Now we give he definiion of a weak oluion o (MHD) a follow (ee [5]): Definiion 2.1 Le Aumpion 1 and 2 hold. A pair of meaurable funcion (u, B) i called aweakoluiono(mhd) if (i) u, B L (,T; L 2 σ) L 2 (,T; H 1,2,σ ); (ii) { (u, Φ )+( u, Φ) + (u u, Φ) (B B,Φ)}d (f,φ)d +(a, Φ()), (2.1) { (B,Φ )+( B, Φ) + (u B,Φ) (B u, Φ)}d (b, Φ()), (2.2) for all Φ C 1 ([,T); Y ) wih Φ(T ),wherey H 1,2,σ Ln. Clearly, Y i a Banach pace wih he norm φ Y φ H 1,2 + φ L n. Moreover, i hould be noed ha under Aumpion 3, C,σ i dene in Y (ee Mauda [5], Propoiion 1; Giga [1], Appendix and Kozono Shor [11], Theorem 2). The main reul of hi paper are aed a follow: On he energy equaliy of weak oluion o (MHD), we have

4 86 Yong Y. and Jiu Q. S. Theorem 2.1 Le Ω be an arbirary domain in R n and Aumpion 1, 2 hold. Le (u, B) be aweakoluiono(mhd) on ome inerval [,T] wih <T. If u L (,T; L n (Ω)), hen he following energy equaliy hold : u() 2 2+ B() for any [,T). { u B 2 2}dτ a b (f,u)dτ, (2.3) Concerning he uniquene, we have he following wo heorem. When he domain i an arbirary domain, he reul read a: Theorem 2.2 Le Ω be an arbirary domain in R n and Aumpion 1, 2 hold. Le (u, B) and (v, N) be weak oluion o (MHD) on ome inerval [,T] wih <T. Suppoe ha (v, N) aifie he energy inequaliy v() N() a b { v N 2 2}dτ (f,v)dτ, <T, (2.4) and ha u, B L (,T; L n (Ω)). Ifu() and B() are righ coninuou for all in [,T) in he norm of L n,hen(u, B) (v, N). For ome pecial domain, he aumpion of he righ coninuiy of velociy field and magneic field in he above heorem can be removed. Theorem 2.3 Le Aumpion 3 and 4 hold. Le (u, B) and (v, N) be weak oluion o (MHD). Ifu, B L (,T; L n (Ω)), and(v, N) aifie he energy inequaliy, v() N() a b hen (u, B) (v, N). 3 Energy Equaliy { v N 2 2}dτ (f,v)dτ, <T, (2.5) In hi ecion, we prove ha he weak oluion o (MHD) aify he energy equaliy (2.3) when u L (,T; L n (Ω)), where Ω i an arbirary domain. To hi end, we fir prove ha if (u, B) i a weak oluion o (MHD), hen u() andb() are weak coninuou in he ene of L 2 σ. Lemma 3.1 Suppoe ha (u, B) i a weak oluion o (MHD). Thenu() and B() are weak coninuou in he ene of L 2 σ if we redefine he value of (u(),b()) in a e of zero-meaure. And he following equaliie hold : { (u, Φ τ )+( u, Φ) + (u u, Φ) (B B,Φ)}dτ (f,φ)dτ (u(), Φ()) + (a, Φ()), (3.1) { (B,Φ τ )+( B, Φ) + (u B,Φ) (B u, Φ)}dτ for all <T and for any Φ H 1 ([,); Y ). (B(), Φ()) + (b, Φ()), (3.2)

5 Energy Equaliy and Uniquene o MHD Equaion 87 Proof For any fixed (,T)andanyh >, le θ(τ) be a mooh funcion, aifying θ(τ) 1; θ(τ) 1,<τ<; θ(τ),τ > + h. Seing Ψ θ(τ)φ(x, τ) in (2.1) and (2.2), we ge { (u, Φ τ )+( u, Φ) + (u u, Φ) (B B,Φ) (f,φ)}θdτ (u, Φ)θ τ dτ +(a, Φ()), (3.3) { (B,Φ τ )+( B, Φ) + (u B,Φ) (B u, Φ)}θdτ (B,Φ)θ τ dτ +(b, Φ()). (3.4) To inveigae he righ-hand ide of (3.3), we oberve ha θ τ vanihe ouide he inerval (, + h), and aifie +h θ τ dτ 1. Since we may aume furher ha c/h θ τ wih c> a conan, i i hen a andard procedure o how ha (u, Φ)θ τdτ (u(), Φ()) for all belonging o he Lebegue e L of u() ah. In fac, we have +h +h (u, Φ)θ τ dτ ( (u(), Φ())) (u, Φ)θ τ dτ (u(), Φ())θ τ dτ +h c h, +h (u, Φ)() (u, Φ)(τ) θ τ dτ a h. Leing h in (3.3), we ge (3.1) for a.e. [,T]. (u, Φ)() (u, Φ)(τ) dτ Now we conider an arbirary inan of ime. Since u() L 2 i uniformly bounded, i follow from he weak compacne of L 2 ha here exi a vecor funcion U(x, ) anda equence { j } j1 in L, j (j ), uch ha (u(x, j ), Φ( j )) (U(x, ), Φ()) a j. Since (3.1) hold for all L, i i an eay conequence ha { (u, Φ τ )+( u, Φ) + (u u, Φ) (B B,Φ) (f,φ)}dτ (U(x, ), Φ()) + (a, Φ()), (3.5) for all T and for all Φ H 1 ([,); Y ). Now if i ielf in L, i i no difficul o ee ha U(x, ) mubeu(x, ) a a funcion in L 2. Hence he funcion U(x, ) i a redefiniion of u a mo a a e of value of of meaure zero. Replacing U by u in (3.5) yield (3.1). By a imilar argumen, we can prove ha (3.2) hold. Moreover, ince Y i dene in L 2 σ, i follow from (3.1) and (3.2) ha u(x, ) andb(x, ) are weak coninuou in L 2 σ a a funcion of ime. Thi complee he proof of Lemma 3.1. On he bai of Lemma 3.1, we now prove Theorem 2.1. Proof of Theorem 2.1 Sep 1 We divide he proof ino hree ep. We how ha he following ideniy hold: u() (B B,u)dτ +2 u 2 2dτ a (f,u)dτ. (3.6)

6 88 Yong Y. and Jiu Q. S. Le ρ C (R) wihupporin 1, uch ha ρ() ρ( ), ρ(), and + ρ()d 1. We e ρ h () h 1 ρ( h ) and define u h(τ) ρ h(τ σ)u(σ)dσ. Then u h H 1 (,; Y )and u h u L 2 (,; Y )ah. Take u h a he e funcion in (3.1) o obain (u, u h,τ )+( u, u h )+(u u, u h ) (B B,u h )dτ (f,u h )dτ (u(),u h ()) + (a, u h (). (3.7) Hereandinhefollowing,u h,τ d d u h i he derivaive of u h wih repec o τ; ρ h,τ, ρ h,σ and oher are imilarly defined in he following. For he fir erm on he lef-hand ide of (3.7), i follow from he Fubini heorem and he ymmery of ρ h ha (u, u h,τ )dτ ( u(τ), ( u(τ), ) ρ h,τ (τ σ)u(σ)dσ dτ ) ρ h,σ (τ σ)u(σ)dσ dτ ( ) ρ h,σ (τ σ)u(τ)dτ, u(σ) dσ (u, u h,τ )dτ, which implie (u, u h,τ)dτ. For he econd erm on he righ-hand ide of (3.7), we have ( ) (u(),u h ()) u(), ρ h ( σ)u(σ)dσ ( u(), h Since u() i weak coninuou in L 2 σ,wehave where ɛ(η) ah. (u(),u h ()) h Taking limi in he above equaliy yield Similarly we have ) ρ h (η)u( η)dη ρ h (η)(u(),u( η))dη ρ h (η)(u(),u( η))dη. ρ h (η){ u() ɛ(η)}dη, (3.8) (u(),u h ()) 1 2 u() 2 2, h. (3.9) (a, u h ()) 1 2 a 2 2, h. (3.1) Now we conider he fourh erm on he lef-hand ide of (3.7). Since u L (,T; L n (Ω)), here exi M > uch ha u h L n <Mand here exi a ubequence {u h } (wrien a

7 Energy Equaliy and Uniquene o MHD Equaion 89 ielf) uch ha u h uweak* in L (,T; L n (Ω)) a h. On he oher hand, by virue of Hölder inequaliy and Sobolev inequaliy, we have B B L1 (L n ) B L 2 (L ) B L2 (L 2 ) C B 2 L 2 (H 1 ) <, where 1 n + 1 n 1, n. Conequenly, Similarly, we can prove lim h lim h Now leing h in (3.7) yield (B B,u h )dτ (u u, u h )dτ { u 2 2 +(u u, u) (B B,u)}dτ (B B,u)dτ. (u u, u)dτ. (f,u)dτ 1 2 u() a 2 2. Noing ha (u u, u) ( 1 2 u 2,u) 1 2 ( u 2, u) in he above equaliy, we ge (3.6). Sep 2 We will how ha B() (B B,u)dτ +2 B 2 2dτ b 2 2. (3.11) Since C,σ i dene in H 1,σ and C,σ Y H 1,σ, wehavehay i dene in H 1,σ. Thereexi a equence {B k } k1 L2 (,T; Y ), uch ha B k B in L 2 (,T; H 1,σ) ak. For any given <<T, we define B h (τ) ρ h (τ σ)b(σ)dσ, B k h(τ) ρ h (τ σ)b k (σ)dσ. (3.12) Then i follow from Mauda ([5], Lemma 2.1) ha B h H 1 (,; H 1,σ ),Bk h H1 (,; Y ), and ha B h B in L 2 (,; H 1,σ )ah, Bk h B h in H 1 (,; H 1,σ )ak. Taking Bk h a he e funcion in (3.2), we obain { (B,B k h,τ )+( B, B k h)+(u B,B k h) (B u, B k h)}dτ (B(),B k h()) + (b, B k h()). (3.13) Since (B u, Bh k) (B Bk h,u), i follow from he Hölder inequaliy and Gargliado Nirenberg inequaliy ha (B Bh,u)dτ k (B B h,u)dτ (B (Bh k B h ),u) dτ a k,where 1 r + 1 n 1 2.Conequenly lim k C B L r u L n B k h B h L 2dτ B H 1 u L n B k h B h L 2dτ C u L (L n ) B L2 (H 1 ) B k h B h L2 (L 2 ), (B u, B k h)dτ (B B h,u)dτ.

8 81 Yong Y. and Jiu Q. S. By a imilar argumen o ha in Sep 1, leing k in (3.13) yield { (B,B h,τ )+( B, B h )+(u B,B h )+(B B h,u)}dτ (B(),B h ()) + (b, B h ()). Then aking limi h in he above equaliy, we obain (3.11). Sep 3 Adding (3.6) over (3.11) yield he energy equaliy (2.3). Thi complee he proof of Theorem 1.1. Remark From he energy equaliy (2.3) in Theorem 2.1, i i eay o ge: u() B() for all <T. 4 Uniquene { u B 2 2}dτ u() B() (f,u)dτ, In hi ecion, we udy he uniquene of he weak oluion o (MHD) in L (,T; L n (Ω)). 4.1 General Domain Fir, we conider he cae when Ω i an arbirary domain in R n. To hi end, we need he following lemma of which he proof i refered o [5]. Lemma 4.1 ([5]) Le w L 2 (, T ; H,σ 1 ) and u L (, T ; L n ).Suppoe w 2 2d > for any (, T ). Suppoe alo ha u i righ coninuou for a in he norm of L n.then for any ɛ>, we have (w w, u) d ɛ where M i a conan independen of. Remark one ha (ee [5]) for all T. w 2 2d + M w 2 2d, T, Under aumpion of Lemma 4.1, in which w i replaced by w 1 and w 2 repecively, (w 1 w 2,u) d ɛ Now we give he proof of Theorem 2.2. Proof of Theorem 2.2 ( w w 2 2 2)d + M We divide our proof ino five ep. ( w w 2 2 2)d, Sep 1 Define vh k in a imilar way o ha in (3.12) in Sep 2 of he proof of Theorem 1.1. Take vh k a he e funcion in (3.1) o ge { (u, v k h,τ )+( u, v k h)+(u u, v k h) (B B,v k h)}dτ (f,v k h)d (u(),v k h()) + (a, v k h()). (4.1) Taking he limi k,noinghav k h v h in H 1 (,; H 1 )ak,wehave { (u, v h,τ )+( u, v h )+(u u, v h ) (B B,v h )}dτ (f,v h )dτ (u(),v h ()) + (a, v h ()). (4.2)

9 Energy Equaliy and Uniquene o MHD Equaion 811 On he oher hand, aking u h a he e funcion in (3.1) wih u replaced by v, B replaced by N, wehave { (v, u h,τ )+( v, u h )+(v v, u h ) (N N,u h )}dτ (f,u h )dτ (v(),u h ()) + (a, u h ()). (4.3) By virue of Fubini heorem and ymmery of he kernel ρ h,iieayoge Adding (4.2) over (4.3) yield (u, v h,τ )dτ (v, u h,τ )dτ. {( u, v h )+( v, u h )+(u u, v h )+(v v, u h ) (B B,v) (N N,u)}dτ (f,v h )dτ + Leing h in he above ideniy, we obain (u(),v()) + 2 (f,u h )dτ (u(),v h ()) (v(),u h ()) + (a, v h ()) + (a, u h ()). ( u, v)dτ + (B B,v) (N N,u)}dτ {(u u, v)+(v v, u) Sep 2 By a imilar argumen o ha in Sep 1, we ge (B(),N()) + 2 ( B, N)dτ + (f,v)dτ + (f,u)dτ + a 2 2. (4.4) {(u B,N)+(v N,B) (B u, N) (N v, B)}dτ b 2 2. (4.5) Sep 3 Since u L (,T; L n (Ω)), i follow from Theorem 2.2 ha u B u 2 2dτ +2 B 2 2dτ a b (f,u)dτ. (4.6) Muliplying 2 on boh ide of (4.4) and (4.5), adding he reuling equaliie wih (2.4) and (4.6), we obain ω() W () (u u, v)dτ 2 (v N,B)dτ +2 (B u, N)dτ +2 ω() 2 2dτ +2 (v v, u)dτ 2 W () 2 2dτ (B B,v)dτ +2 (u B,N)dτ (N N,u)dτ (N v, B)dτ, (4.7) where w u v, W B N. I i eay o ee ha (u u, v)+(v v, u) (w w, u) due o (w u, u). Similarly, we have (u B,N)+(v N,B) (w W, B), (B B,v)+(N N,u)+(B u, N)+(N v, B) (W w, B)+(W W, u).

10 812 Yong Y. and Jiu Q. S. Conequenly, (4.7) can be rewrien a w() W () (w w, u)dτ 2 (W w, B)dτ +2 w() 2 2dτ +2 (w W, B)dτ W () 2 2dτ (W W, u)dτ. (4.8) Sep 4 Claim: here exi an abolue conan δ>, uch ha (u, B) (v, N), (,δ); if no, we have w 2 2dτ >, By virue of Lemma 4.1 and i remark, we have (w W, B)dτ ɛ (W ω, B)dτ ɛ (w ω, u)dτ ɛ (W W, u)dτ ɛ Hence i follow from (4.8) ha Now we ake ɛ 1 3 W 2 2dτ >, (,δ). { w W 2 2 }dτ + M { w W 2 2 }dτ, { w W 2 2}dτ + M w 2 2dτ + M W 2 2dτ + M w 2 2dτ, W 2 2dτ. { w W 2 2}dτ, w() W () w() 2 2dτ +2 W () 2 2dτ { } 6ɛ { w W 2 2}dτ +6M w W 2 2 dτ. and ge w() W () 2 2 2M where M i a conan independen of w. Then i follow from Gronwall inequaliy ha ω() W () 2 2. { w W 2 2}dτ, Thi lead o a conradicion. The claim i proved. Sep 5 By exenion, i i eay o ee ha (u, B) (v, N) for all [,T). Thi complee he proof of Theorem Special Domain In hi ecion, we hall how ha he aumpion on he righ coninuiy of w() in he norm of L n can be removed, and he cla L (,T; L n (Ω)) i ufficien o enure he uniquene in ome pecial domain uch a he whole pace R n, he half pace R n +, a bounded domain or exerior domain. Indeed, by conrucing a rong oluion o (MHD), we prove ha every weak oluion (u, B) o (MHD) in he cla L (,T; L n (Ω)) i necearily righ coninuou on [,T) wih value in L n (Ω).

11 Energy Equaliy and Uniquene o MHD Equaion 813 Lemma 4.2 Le Aumpion 3 and 4 hold. Aume moreover a, b L n σ. Then here exi <T T and a weak oluion (u, B) of (MHD) wih he propery u, B BC([,T ); L n σ(ω)). Remark The rericion of he domain i due o he following fac. Le A be he Soke operaor which i defined by A P, wherep i he projecion operaor from L 2 o L 2 σ. Then o prove Lemma 4.2, he following eimae will be ued. For any 1 <p r<, here hold e A a r C n 2 ( 1 p 1 r ) a p, (4.9) e A a r C( ) n 2 ( 1 p 1 r ) a p, (4.1) for all a L p σ, >, C C(n, p, r). Here n i he dimenion of he pace. The econd eimae (4.1) demand 1 <p r<nwhen he domain i an exerior domain. The deail proof of Lemma 4.2 can be een in [6] and [12], and we omi i here. Now we give he proof of Theorem 2.3 a follow. Proof of Theorem 2.3 In view of Theorem 2.2, we only need o prove ha u() andb() are righ coninuou on [,T) wih value in L n σ. To hi end, we fir prove ha u(),b() L n σ. Indeed, ince u L (,T; L n ), we have ha for any fixed, here i a equence { k } k1 uch ha k (k ), and uch ha u( k ) L n. Then i follow ha u( k ) L n C(C independen of ), here exi a ubequence of { k } k1 (denoed by ielf for impliciy) and a funcion ω uch ha (u( k ),φ) (ω, φ)(k ), Recalling ha u() i weak coninuou in L 2,wehave (u( k ),φ) (u(),φ)(k ), φ C,σ. φ C,σ. Hence, we ge u() ω and u() L n σ. Similarly we prove ha B() L n σ. Then i follow from Lemma 4.2 ha here exi T > andaweakoluion(ũ, B) o(mhd)onheinerval (, + T ) aifying he iniial value (ũ, B) (u(),b()) and ũ, B BC([, + T ); L n σ(ω)). Then ũ, B are righ coninuou on [,T)wihvalueinL n. On he oher hand, by Theorem 1.1 and i remark, (u, B) aifie he energy equaliy of he rong form u B η a b u 2 2dτ +2 η η B 2 2dτ (f,u)dτ, η<t. Now, aking (u(),b()) a he iniial value, i follow from Theorem 2.2 ha (u, B) (ũ, B) on [, + T ). Since [,T) can be aken arbirarily, we conclude ha u, B are righ coninuou on he whole inerval [,T)wihvalueinL n σ. Thi complee he proof of Theorem 2.3. Acknowledgmen The auhor expre heir graiude o Profeor Zhouping Xin for hi conan encouragemen and o Profeor Cheng He for hi valuable dicuion. Reference [1] Sohr, H.: The Navier Soke Equaion: an elemenary funcaional analyic approach, Birkhäuer, Boon, 21 [2] Serrin, J.: The iniial problem for he Navier Soke equaion. Nonlinear Problem, R. E. Langer ed., Madion:Univeriy of Wiconin Pre, 69 98, 1963

12 814 Yong Y. and Jiu Q. S. [3] Foia, C.: Une remarque ur l unicié de oluion de equaion de Navier Soke en dimenion. Bull. Soc. Mah. France, 89, 1 8 (1961) [4] Sohr, H., Von. Wahl, W.: On he ingular e and he uniquene of weak oluion of he Navier Soke equaion. Manucripa Mah., 49, (1984) [5] Mauda, K.: Weak oluion of Navier Soke equaion. Tohoku Mah. J., 36, (1984) [6] Kozono, H., Sohr, H.: Remark on uniquene of weak oluion o he Navier Soke equaion. Analyi, 16, (1996) [7] Ecauriaza, L., Seregin, G., Sverák, V.: On L 3 oluion o he Navier Soke equaion and backward uniquene. Ru. Mah. Surv., 58(2), (23) [8] He, C., Xin, Z. P.: On he regulariy of weak oluion o he Magneohydrodynamic equaion. J. Diff. Eqn., 213, (25) [9] Zhou, Y.: Remark on regulariy for he 3D MHD equaion. Dicree And Coninuou Dynamical Syem, 12(5), (25) [1] Giga, Y., Shor, H.: On he Soke operaor in exerior domain. J. Fac. Sci. Univ. Tokyo, Sec IA, 36, (1989) [11] Kozono, H., Sohr, H.: Deniy properie for olenoidal vecor field, wih applicaion o he Navier Soke equaion in exerior domain. J.Mah.Soc.Japan, 44, (1992) [12] Yong, Y.: Energy equaliy and uniquene of MHD Equaion in L ([,T]; L n (Ω)), Maer Thei, Capial Normal Univeriy, 26