Rem arkson Liouville Type Resu lt for the 3D Hall-M HD System

ZHANG Zu jin,YANG Xianand QIU Shu linSchoolofMathematicsand ComputerSciences,Gannan NormalUniversity,

Ganzhou 341000,China.

2Foreign LanguageDepartment,Ganzhou TeachersCollege,Ganzhou 341000,China. Received 17M ay 2015;Accep ted 31 Ju ly 2015

Rem arkson Liouville Type Resu lt for the 3D Hall-M HD System

ZHANG Zu jin1,?,YANG Xian2and QIU Shu lin11SchoolofMathematicsand ComputerSciences,Gannan NormalUniversity,

Ganzhou 341000,China.

2Foreign LanguageDepartment,Ganzhou TeachersCollege,Ganzhou 341000,China. Received 17M ay 2015;Accep ted 31 Ju ly 2015

.In this paper,w e consider the 3D Hall-M HD system,and p rovide an imp roved Liouv ille type resu lt for its stationary version.

Hall-MHD system;Liouville theorem.

1 In troduction

This paper concerns itselfw ith the th ree-d im ensional(3D)Hall-m agnetohyd rodynam ics system(Hall-MHD):

w here u is fl uid velocity field,B is them agnetic field,and π is a scalar p ressu re.We p rescribe the initialdata to satisfy the cond ition

The fi rst system atic study of the Hall-MHD system is p ioneered by Ligh thill[1]follow ed by Cam pos[2].Com paring w ith the usualMHD equations,the Hall-MHD systemhas the Hall term ?×[(? × B)× B]in(1.1)3,w hichm ay becom e signifi cant for such p roblem s asm agnetic reconnection in geo-dynam o[3],star form ation[4,5],neu tron stars[6] and space p lasm as[7,8].

Mathem atically,the Hall-MHD system can be derived from either tw o-fl uids or kineticm odels(see[3]),and the global existence of w eak solu tions,local existence and uniquenessof sm ooth solu tions,blow-up criteria and sm alldata globalexistenceof classical so lu tions w ere established in[9,10].For the fractional Hall-M HD,the reader is referred to[11].

The stationary version of(1.1)is

And in[9],theauthorsestablished the follow ing Liouville type theorem.

Theorem 1.1.([9])Let u,B be C2(R3)solutions to(1.3)satisfying

Then wehave u=B=0.

It isnotnatu ral to assum e that theboundednessof the solu tion u,B(see[12,13]),and the aim of this paper is to im proving Theorem 1.1 as

Theorem 1.2.Let u,B be C2(R3)solutions to(1.3)satisfying

Then wehave u=B=0.

2 Proof of Theorem 1.2

In th is section,w e shall p rove Theorem 1.2.

We fi rst derive an estim ate o f the p ressure.Taking the d ivergence o f(1.3)1,and using the vector iden tity

w e obtain

Classical ellip tic regu larity resu lts then yields

and 0 ≤ σ(|x|)≤ 1 for 1 ＜ |x|＜ 2.For each R ＞ 0,define σR(x)= σ(|x|/R),x ∈ R3.

Taking the inner p roduct of(1.3)1w ith uσR,(1.3)3w ith BσRin L2(R3),add ing the resu lting equations together,and in tegrating by parts,w e obtain

Now,consider a standard rad ial cu t-off function σ ∈ Cc∞(R3)such that

We successively estim ate Iifor i=1,2,···,7.For I1,H¨older inequality im p lies

For I2,w e invoke(2.1)to deduce

Then for I3,by Sobolev im bedd ing theorem,

The term I4can be sim ilarly estim ated as I1,

Exactly asestim ate for I3,w em ay dom inate I5,

Finally,w em ay bound I6and I7sim ultaneously as

Therefore,passing R → ∞ in(2.2),w e get

Levi’sm onotone convergence theorem then yields

Consequen tly,u,B are constantvector,and both are zero due to the fact that u,B∈ L92(R3).

The p roo f of Theorem 1.2 is com p lete.

A cknow ledgem en ts

Thisw orkw as partially supported by the Natu ral Science Foundation of JiangxiProvince (20151BAB201010).

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?Correspond ing au thor.Emailaddresses:zhangzuj in361@163.com(Z.J.Zhang),yangxianxisu@163.com(X. Yang),qiushul in2003@163.com(S.L.Q iu)

AM SSub ject Classifi cations:35Q35,35B65

Chinese Library Classifi cations:O 175.29,O175.28