Global smooth solutions to 3D MHD with mixed partial dissipation and magnetic diffusion

R E S E A R C H

Open Access

Global smooth solutions to 3D MHD with

mixed partial dissipation and magnetic

diffusion

Menglong Su

_{and Jian Wang}

*_{Correspondence:}

[email protected]

1_{School of Mathematical Sciences,}

Luoyang Normal University, Luoyang, 471022, P.R. China

2_{Key Laboratory of Symbolic}

Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun, 130012, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, we prove the existence of global smooth solutions to the Cauchy problem of 3D incompressible magnetohydrodynamics (MHD) flows with mixed partial dissipation and magnetic diffusion if the initial condition is suitably small.

Keywords: incompressible magnetohydrodynamics; global smooth solution; partial dissipation and magnetic diffusion

1 Introduction

In this paper, we consider the following D incompressible MHD equations with mixed partial dissipation and magnetic diffusion (see []),i.e.,

ut+u· ∇u= –∇p+μuxx+μuyy+b· ∇b, ()

bt+u· ∇b=ηbxx+ηbyy+b· ∇u, ()

divu= , divb= , ()

associated with the initial data

u(,x) =u, b(,x) =b. ()

Hereu= (u(t,x),u(t,x),u(t,x)) is the velocity field,b= (b(t,x),b(t,x),b(t,x)) is the magnetic field,p=p(t,x) is scalar pressure,μ>  is the kinematic viscosity,η>  is the magnetic diffusion. For more background, we refer the reader to [] for MHD and [, ] for MHD with mixed partial dissipation and magnetic diffusion. Without loss of generality, we assume thatμ=η=  in the remainder of the paper.

To state the main results, we first introduce the following conventions and notations which will be used throughout this paper. Set

fdx

R

fdxdydz,

and that · is theL_norm,i.e.,

f=

fdx

  ,

HmWm,R=

|α|≤m

Dαu dx

/

with the norm · Hm.

Our main result of this paper can be stated as follows.

Theorem . Assume that u∈H and b∈Hwithdivu=divb= anduH+

bH ≤ε, where ε is a sufficiently small positive number. Then ()-() admit global

smooth solutions.

Remark . Theorem . is Theorem . in [], which has not been proved in their paper.

We would also emphasize that our proof of Theorem . is clearer for deducing the desired a priori estimates in Lemma . (see the next section) than that of Proposition . in [].

The rest of the paper is organized as follows. In Section , we deduce the desireda prioriestimates to complete the proof of Theorem .. We finish the proof of Theorem . in Section  by the method of vanishing viscosities.

2 A priori estimates

In this section, we deduce the desireda prioriestimates in order to finish the main result. Before we begin to prove the main theorem of this paper, we first state the following useful lemma that was deduced in [].

Lemma . Assume that f,g,h,fx,gy,hzare all in L(R).Then we have

|fgh|dx≤Cf_g_h_fx_gy_hz_.

Clearly, the standard energy estimate shows that

 

d dt

u+b+ux+uy+bx+by= . ()

We denote thatω=∇ ×uandj=∇ ×b. Thus, applying the operator ‘∇×’ to () and (), together with (), we deduce that

ωt+u· ∇ω–ω· ∇u=ωxx+ωyy+b· ∇j–j· ∇b, ()

jt+u· ∇j=jxx+jyy+b· ∇ω+εijk(∂jbl∂luk–∂jul∂lbk), ()

whereεijkis defined as follows:

εijk=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

 if (i,j,k) is an even permutation, – if (i,j,k) is an odd permutation,

The first key lemma is the following.

Lemma . If(u,b)solves()-()and the initial data satisfies

u+b≤ 

C and ω

_+j ≤



, ()

where C is a suitable large number,then the vorticityωand the current density j satisfy

ω+j≤,

t



ωx+ωy+jx+jy

ds≤ for all t≥.

Proof Multiplying () and () byωandj, respectively, then integrating the resulting equa-tions by parts, after adding the two equalities together, we finally deduce

 

d dt

ω+j+ωx+ωy+jx+jy

= ω· ∇ω·ω–j· ∇b·ω+εijk(∂jbl∂luk–∂jul∂lbk)ji

dx=I+J+K+L. ()

We have to estimate each term on the right-hand side of (). Some of the terms are the same as in [] and are proved here for completeness. First,Ican be written as

I=

ω· ∇u·ωdx=

ωi∂iujωjdx =

ω∂xujωjdx+

ω∂yujωjdx+

ω∂zujωjdx=I+I+I.

With the help of Lemma ., we deduce that

I =

ω∂xujωjdx≤Cω  _u

x 

_ω_ω x

 _ω

y  _ω

x  

≤ 

ωx ₊ 

ωy _+Cu

xω.

Similarly, we obtain that

I=

ω∂yujωjdx≤  ωx

₊  ωy

_+Cu

yω,

and

I=

ω∂zujωjdx=

(∂xu–∂yu)∂zujωjdx

≤ 

ωx ₊ 

ωy

_+Cu

x+uy

ω.

In order to boundJ, we rewrite the integrand explicitly as follows:

J= –

j· ∇b·ωdx= –

ji∂ibjωjdx

= –

j∂xbjωjdx–

j∂ybjωjdx–

Due to Lemma ., we see that

J= –

j∂xbjωjdx≤Cj 

_bx_ω_jx_∂xbz_ωy

≤ 

jx ₊ 

ωy _+Cb

x

ω+j.

Similarly,

J= –

j∂ybjωjdx≤  ωy

₊  jx

₊  jy

_+Cb y

ω+j,

J= –

j∂zbjωjdx= –

(∂xb–∂yb)∂zbjωjdx

≤ 

ωx ₊ 

jx ₊ 

jy

_+Cb

x+by

ω+j.

Now, let us turn to boundL,

L= –

εijk∂jul∂lukjidx

= –

εijk∂xul∂lukjidx–

εijk∂yul∂lukjidx–

εijk∂zul∂lukjidx =L+L+L.

By Lemma ., we have that

L= –

εijk∂xul∂lukjidx≤Cux 

_j_j_j x

 _j

y  _ω

x  

≤ 

ωx ₊ 

jx ₊ 

jy _+Cu

xj. Similarly,

L= –

εijk∂yul∂lukjidx≤  ωy

₊  jx

₊  jy

_+Cu yj. As forL, we should split it into three parts:

L= –

εijk∂zul∂lukjidx = –

εijk∂zu∂xukjidx–

εijk∂zu∂yukjidx–

εijk∂zu∂zukjidx =L+L+L,

L= –

εijk∂zu∂xukjidx

≤Cω_bx_j_ωx_∂xbz_jy

≤ 

ωx ₊ 

jx ₊ 

jy _+Cb

x

Similarly,

L= –

εijk∂zu∂yukjidx≤  ωx

₊  jy

_+Cb y

ω+j.

To boundL, using the incompressibility conditiondivu= , we deduce that

L= –

εijk∂zu∂zukjidx

=

εijk(∂xu+∂yu)∂zukjidx=L+L.

We get theL

andLas follows:

L_=

εijk∂xu∂zukjidx≤Cux 

_j_j_∂ xuz

 _j

x  _j

y  

≤ 

ωx ₊ 

jx ₊ 

jy _+Cu

xj, and

L_≤ 

ωy ₊ 

jx ₊ 

jy _+Cu

yj. To boundK, we should divide it into three parts:

K=

εijk∂jbl∂lukjidx

=

εik∂xbl∂lukjidx+

εik∂ybl∂lukjidx+

εik∂zbl∂lukjidx =K+K+K.

Similarly, we deduce that

K=

εik∂xbl∂lukjidx≤Cbx 

_ω_j_∂xbz_ωx_jy

≤ 

ωx ₊ 

jx ₊ 

jy _+Cb

x

ω+j,

and

K=

εik∂ybl∂lukjidx≤  ωx

₊  jy

_+Cb y

ω+j.

ForK, we have

K=

εik∂zbl∂lukjidx

=

εik∂zb∂xukjidx+

εik∂zb∂yukjidx+

Thus,

K=

εik∂zb∂xukjidx≤Cj 

_ux_j_ωx_jx_jy

≤ 

ωx ₊ 

jx ₊ 

jy _+Cu

xj, and

K=

εik∂zb∂yukjidx≤  ωy

₊  jx

₊  jy

_+Cu yj. As forK, usingdivb= , we obtain

K=

εik∂zb∂zukjidx= –

εik(∂xb+∂yb)∂zukjidx

≤ 

ωx ₊ 

jy

_+Cb

x+by

ω+j.

Substituting all the above estimates into (), we conclude that

 

d dt

ω+j+ 

ωx+ωy+jx+jy

≤Cux+uy+bx+by

ω+j.

Letω_+j_{≤, we deduce, with the assumption () on the initial data, that}

ω+j+

t



ωx+ωy+jx+jy

ds≤.

Thus, the proof of Lemma . is completed.

Now, we turn to deduce the higher order estimates about the solution.

Lemma . If(u,b)is the solution of ()-(),then

∇ω+∇j+

t



∇ωx+∇ωy+∇jx+∇jy

ds≤C. ()

Proof Multiplying () and () byωandj, respectively, then integrating the resultant equations by parts, after adding the two equalities together, we finally obtain

 

d dt

∇ω+∇j+∇ωx+∇ωy+∇jx+∇jy

= u· ∇ω·ω+u· ∇j·j–ω· ∇ω·ω+j· ∇b·ω–b· ∇j·ω

–b· ∇ω·j–εijk(∂jbl∂luk–∂jul∂lbk)ji

dx

Now, we turn to bound each term on the right-hand side of (). Similar as the proof of Lemma ., keeping in mind Lemma . and the divergence-free property ofuandb, we deduce

M=

u· ∇ω·ωdx=

ui∂iωj∂kkωjdx= –

∂kui∂iωj∂kωjdx

= –

∂xui∂iωj∂xωjdx–

∂yui∂iωj∂yωjdx–

∂zui∂iωj∂zωjdx =M+M+M.

We estimate each term as follows:

M= –

∂xui∂iωj∂xωjdx≤Cux 

∇_ω∇_ω∇_ω x



∇_ω

y  _∂

xuz  

≤ 

∇ωx ₊ 

∇ωy _+Cu

xωx∇ω

≤ 

∇ωx ₊ 

∇ωy

_+Cu

x+ωx

∇ω.

Similarly,

M= –

∂yui∂iωj∂yωjdx

≤Cuy 

∇_ω∇_ω∇_ωx∇_ωy_∂yuz

≤ 

∇ωx ₊ 

∇ωy

_+Cu

y+ωy

∇ω.

Now, we turn to boundM,

M= –

∂zui∂iωj∂zωjdx

= –

∂zu∂xωj∂zωjdx–

∂zu∂yωj∂zωjdx–

∂zu∂zωj∂zωjdx =M+M+M.

Similarly, we can deduce that

M= –

∂zu∂xωj∂zωjdx

≤Cω_ωx∇_ω_ωx∇_ωx∇_ωy

≤ 

∇ωx ₊ 

∇ωy

_+Cω_+ω x

∇ω,

M= –

∂zu∂yωj∂zωjdx

≤Cω_ωy∇_ω_ωx∇_ωy∇_ωy

≤ 

∇ωy

_+Cω_+ω x

M= –

∂zu∂zωj∂zωjdx=

(∂xu+∂yu)∂zωj∂zωjdx

≤Cux 

∇_ω∇_ω_∂xuz∇_ωx∇_ωy +Cuy



∇_ω∇_ω_∂ yuz



∇_ω

x 

∇_ω

y  

≤ 

∇ωx ₊ 

∇ωy

_+Cu

x+uy+ωx+ωy

∇ω.

As forN, integrating by parts, we deduce that

N=

u· ∇j·jdx=

ui∂ijj∂kkjjdx= –

∂kui∂ijj∂kjjdx = –

∂xui∂ijj∂xjjdx–

∂yui∂ijj∂yjjdx–

∂zui∂ijj∂zjjdx =N+N+N.

Similarly, we have

N= –

∂xui∂ijj∂xjjdx

≤Cux 

∇_j_j x

 _ω

x  ∇_j

x  ∇_j

y  

≤ 

∇jx ₊ 

∇jy

_+Cu

x+ωx

∇j,

N= –

∂yui∂ijj∂yjjdx

≤Cuy 

∇_j_jy_ωy∇_jx∇_jy

≤ 

∇jx ₊ 

∇jy

_+Cu

y+ωy

∇j.

As forN, we obtain

N= –

∂zui∂ijj∂zjjdx

= –

∂zu∂xjj∂zjjdx–

∂zu∂yjj∂zjjdx–

∂zu∂zjj∂zjjdx =N+N+N.

Thus, we have

N= –

∂zu∂xjj∂zjjdx

≤Cω_jx∇_j_ωx∇_jx∇_jy

≤ 

∇jx ₊ 

∇jy

_+Cω_+ω x

N= –

∂zu∂yjj∂zjjdx

≤Cω_jy∇_j_ωx∇_jy∇_jy

≤ 

∇jy

_+Cω_+ω x

∇j,

N= –

∂zu∂zjj∂zjjdx

≤Cux 

∇_j∇_j∇_jx∇_jy_ωx +Cuy



∇_j∇_j∇_jx∇_jy_ωy

≤ 

∇jx ₊ 

∇jy

_+Cu

x+uy+ωx+ωy

∇j.

We now turn to boundP,

P= –

ω· ∇ω·ωdx= –

ωi∂iuj∂kkωjdx =

∂kωi∂iuj∂kωjdx+

ωi∂k∂iuj∂kωjdx=P+P.

ForP, we have

P=

∂kωi∂iuj∂kωjdx =

∂xωi∂iuj∂xωjdx+

∂yωi∂iuj∂yωjdx+

∂zωi∂iuj∂zωjdx =P+P+P,

P=

∂xωi∂iuj∂xωjdx

≤Cωx 

_ω_ωx_ωx∇_ωx∇_ωx

≤ 

∇ωx

_+Cω_+ω x

∇ω,

P=

∂yωi∂iuj∂yωjdx

≤Cωy 

_ω_ω y

 _ω

x 

∇_ω

y 

∇_ω

y  

≤ 

∇ωy

_+Cω_+ω x

∇ω.

ForP, we see that

P=

∂zωi∂iuj∂zωjdx =

∂zω∂xuj∂zωjdx+

∂zω∂yuj∂zωjdx+

Thus, we can boundPas follows:

P_=

∂zω∂xuj∂zωjdx≤C∇ω 

_ux∇_ω∇_ωx∇_ωy_ωx

≤ 

∇ωx ₊ 

∇ωy

_+Cu

x+ωx

∇ω,

P_=

∂zω∂yuj∂zωjdx≤C∇ω  _u

y 

∇_ω∇_ω x



∇_ω

y  _ω

y  

≤ 

∇ωx ₊ 

∇ωy

_+Cu

y+ωy

∇ω,

P_=

∂zω∂zuj∂zωjdx

=

∂z(∂xu–∂yu)∂zuj∂zωjdx

≤Cωx 

_ω∇_ω∇_ωx_ωx∇_ωy

+Cωy 

_ω∇_ω∇_ω y

 _ω

y 

∇_ω

x  

≤ 

∇ωx ₊ 

∇ωy

_+Cω_+ω

x+ωy

∇ω.

Now, we turn toP,

P=

ωi∂k∂iuj∂kωjdx

=

ω∂k∂xuj∂kωjdx+

ω∂k∂yuj∂kωjdx+

ω∂k∂zuj∂kωjdx =P+P+P.

Similarly,

P=

ω∂k∂xuj∂kωjdx

≤Cω_ωx∇_ω_ωx∇_ωy∇_ωx

≤ 

∇ωx ₊ 

∇ωy

_+Cω_+ω x

∇ω,

P=

ω∂k∂yuj∂kωjdx

≤Cω_ωy∇_ω_ωx∇_ωy∇_ωy

≤ 

∇ωy

_+Cω_+ω x

∇ω,

P=

ω∂k∂zuj∂kωjdx

≤Cux 

∇_ω∇_ω∇_ω x



∇_ω

y  _ω

x  

+Cuy 

∇_ω∇_ω∇_ωx∇_ωy_ωy

≤ 

∇ωx ₊ 

∇ωy

_+Cu

x+uy+ωx+ωy

To boundQ, we see that

Q=

j· ∇b·ωdx=

ji∂ibj∂kkωjdx

= –

∂kji∂ibj∂kωjdx–

ji∂k∂ibj∂kωjdx=Q+Q,

Q= –

∂xji∂ibj∂xωjdx–

∂yji∂ibj∂yωjdx–

∂zji∂ibj∂zωjdx =Q+Q+Q,

Q= –

∂xji∂ibj∂xωjdx≤Cjx 

_j_ωx∇_jx∇_j∇_ωx

≤ 

∇ωx ₊ 

∇jx

_+Cj_+j x

∇ω+∇j,

Q= –

∂yji∂ibj∂yωjdx≤Cjy 

_j_ωx∇_jy_jx∇_ωx

≤ 

∇ωx ₊ 

∇jy

_+Cj_+j y

∇ω+∇j.

As forQ, we see that

Q= –

∂zji∂ibj∂zωjdx = –

∂zj∂xbj∂zωjdx–

∂zj∂ybj∂zωjdx–

∂zj∂zbj∂zωjdx =Q_+Q_+Q_.

Thus, we can deduce that

Q_= –

∂zj∂xbj∂zωjdx

≤C∇j_bx∇_ω∇_jx_jx∇_ωy

≤ 

∇ωy ₊ 

∇jx

_+Cb

x+jx

∇ω+∇j,

Q_= –

∂zj∂ybj∂zωjdx

≤C∇j_bx∇_ω∇_jx_jy∇_ωy

≤ 

∇ωy ₊ 

∇jx

_+Cb

y+jy

∇ω+∇j,

Q_= –

∂zj∂zbj∂zωjdx= –

∂z(∂xb–∂yb)∂zbj∂zωjdx

≤Cjx 

_j∇_ω∇_j x

 _j

y 

∇_ω

x  

+Cjy 

_j∇_ω∇_jy_jy∇_ωx

≤ 

∇ωx ₊ 

∇jx ₊ 

∇jy

_+Cj_+j

x+jy

Now, we turn toQ,

Q= –

b· ∇j·ωdx= –

ji∂k∂ibj∂kωjdx

= –

j∂k∂xbj∂kωjdx–

j∂k∂ybj∂kωjdx–

j∂k∂zbj∂kωjdx =Q+Q+Q.

We deduce that

Q= –

j∂k∂xbj∂kωjdx

≤Cj_jx∇_ω∇_ωx_jy∇_jx

≤ 

∇ωx ₊ 

∇jx

_+Cj_+j x

∇ω+∇j,

Q= –

j∂k∂ybj∂kωjdx

≤Cj_j y



∇_ω∇_ω x

 _j

y  ∇_j

y  

≤ 

∇ωx ₊ 

∇jy

_+Cj_+j y

∇ω+∇j,

Q= –

j∂k∂zbj∂kωjdx

≤Cbx 

∇_j∇_ω∇_ωx∇_jy_jx +Cby



∇_j∇_ω∇_ω x

 ∇_j

y  _j

y  

≤ 

∇ωx ₊ 

∇jy

_+Cb

x+jx+by+jy

∇ω+∇j.

Here we start to estimateRas follows:

R= –

b· ∇j·ωdx–

b· ∇ω·jdx

= –

bi∂ijj∂kkωjdx–

bi∂iωj∂kkjjdx

=

∂kbi∂ijj∂kωjdx+

∂kbi∂iωj∂kjjdx=R+R.

ForR, we have

R=

∂kbi∂ijj∂kωjdx =

∂xbi∂ijj∂xωjdx+

∂ybi∂ijj∂yωjdx+

We deduce that

R=

∂xbi∂ijj∂xωjdx

≤Cbx 

∇_j_ωx∇_jx∇_ωx_jx

≤ 

∇ωx ₊ 

∇jx

_+Cb

x+ωx

∇j,

R=

∂ybi∂ijj∂yωjdx

≤Cby 

∇_j_ω y

 ∇_j

x 

∇_ω

y  _j

y  

≤ 

∇jx ₊ 

∇ωy

_+Cb

y+ωy

∇j.

ForR, we have

R=

∂zbi∂ijj∂zωjdx

=

∂zb∂xjj∂zωjdx+

∂zb∂yjj∂zωjdx+

∂zb∂zjj∂zωjdx=R+R+R.

Similarly,

R_=

∂zb∂xjj∂zωjdx

≤Cj_jx∇_ω∇_ωx∇_jx_jy

≤ 

∇ωx ₊ 

∇jx

_+Cj_+j x

∇ω+∇j,

R_=

∂zb∂yjj∂zωjdx

≤Cj_j y



∇_ω∇_ω x

 _j

y  ∇_j

y  

≤ 

∇ωx ₊ 

∇jy

_+Cj_+j y

∇ω+∇j,

R_=

∂zb∂zjj∂zωjdx

= –

(∂xb+∂yb)∂zjj∂zωjdx

≤Cbx 

∇_j∇_ω∇_ω x

 ∇_j

y  _j

x  

+Cby 

∇_j∇_ω∇_ωx∇_jy_jy

≤ 

∇ωx ₊ 

∇jy

_+Cj_+j y

∇ω+∇j.

Now, we turn toR,

R=

∂kbi∂iωj∂kjjdx

=

∂xbi∂iωj∂xjjdx+

∂ybi∂iωj∂yjjdx+

Thus, similarly, we deduce that

R=

∂xbi∂iωj∂xjjdx≤Cbx 

∇_ω_j x



∇_ω

x  ∇_j

x  _j

x  

≤ 

∇ωx ₊ 

∇jx

_+Cb

x+jx

∇ω+∇j,

R=

∂ybi∂iωj∂yjjdx≤Cby 

∇_ω_jy∇_ωx∇_jy_jy

≤ 

∇ωx ₊ 

∇jy

_+Cb

y+jy

∇ω+∇j.

ForR, we have

R=

∂zbi∂iωj∂zjjdx

=

∂zb∂xωj∂zjjdx+

∂zb∂yωj∂zjjdx+

∂zb∂zωj∂zjjdx =R_+R_+R_.

Similarly,

R_=

∂zb∂xωj∂zjjdx≤Cj  _ω

x 

∇_j∇_ω x

 ∇_j

x  _j

y  

≤ 

∇ωx ₊ 

∇jx

_+Cj_+ω x

∇j,

R_=

∂zb∂yωj∂zjjdx≤Cj 

_ωx∇_j∇_jx_jy∇_ωy

≤ 

∇jx ₊ 

∇ωy

_+Cj_+j y

∇ω+∇j,

R_=

∂zb∂zωj∂zjjdx

= –

(∂xb+∂yb)∂zωj∂zjjdx

≤Cbx 

∇_ω∇_j∇_ωx∇_jy_jx +Cby



∇_ω∇_j∇_ω x

 ∇_j

y  _j

y  

≤ 

∇ωx ₊ 

∇jy

_+Cb

x+by+jx+jy

∇ω+∇j.

Finally, for the last termS, we have

S= –

εijk(∂jbl∂luk–∂jul∂lbk)jidx =

We first considerS,

S=

εijk∂m(∂jbl∂luk)∂mjidx

=

εijk∂x(∂jbl∂luk)∂xjidx+

εijk∂y(∂jbl∂luk)∂yjidx+

εijk∂z(∂jbl∂luk)∂zjidx =S+S+S.

We deduce that

S=

εijk∂x(∂jbl∂luk)∂xjidx=

εijk∂x∂jbl∂luk∂xjidx+

εijk∂jbl∂x∂luk∂xjidx

≤Cjx  _ω_j

x  _ω

x  ∇_j

x  ∇_j

y  

+Cj_ω x

 _j

x  _j

x 

∇_ω

x  ∇_j

x  

≤ 

∇ωx ₊ 

∇jx ₊ 

∇jy _+Cj

x+ω+j

∇ω+∇j,

S=

εijk∂y(∂jbl∂luk)∂yjidx=

εijk∂y∂jbl∂luk∂yjidx+

εijk∂jbl∂y∂luk∂yjidx

≤Cjy  _ω_j

y  ∇_j

x  ∇_j

y  ∇_ω

+Cj_ω y

 _j

y  _j

x 

∇_ω

y  ∇_j

x  

≤ 

∇ωy ₊ 

∇jx ₊ 

∇jy 

+Cω+jy+j+jx

∇ω+∇j.

As forS, we see that

S=

εijk∂z(∂jbl∂luk)∂zjidx =

εijk∂z∂jbl∂luk∂zjidx+

εijk∂jbl∂z∂luk∂zjidx =

εijk∂z∂xbl∂luk∂zjidx+

εijk∂z∂ybl∂luk∂zjidx+

εijk∂z∂zbl∂luk∂zjidx +

εijk∂xbl∂z∂luk∂zjidx+

εijk∂ybl∂z∂luk∂zjidx+

εijk∂zbl∂z∂luk∂zjidx =S_+S_+S_+S_+S_+S_.

We deduce each term step by step as follows:

S_=

εijk∂z∂xbl∂luk∂zjidx

≤Cjx 

_ω∇_j∇_jy_ωx∇_jx

≤ 

∇jx ₊ 

∇jy

_+Cω_+j x

S_=

εijk∂z∂ybl∂luk∂zjidx

≤Cjy 

_ω∇_j∇_jx_ωy∇_jy

≤ 

∇jx ₊ 

∇jy

_+Cω_+j y

∇ω+∇j.

ForS_, we have

S_=

εijk∂z∂zbl∂luk∂zjidx =

εijk∂z∂zb∂xuk∂zjidx+

εijk∂z∂zb∂yuk∂zjidx+

εijk∂z∂zb∂zuk∂zjidx =S_+S_+S_.

Thus, we deduce that

S_=

εijk∂z∂zb∂xuk∂zjidx

≤C∇j_ux∇_j∇_jx∇_jy_ωx

≤ 

∇jx ₊ 

∇jy

_+Cu

x+ωx

∇j,

S_=

εijk∂z∂zb∂yuk∂zjidx

≤C∇j_u y



∇_j∇_j x

 ∇_j

y  _ω

x  

≤ 

∇jx ₊ 

∇jy

_+Cu

y+ωy

∇j,

S_=

εijk∂z∂zb∂zuk∂zjidx

= –

εijk∂z(∂xb+∂yb)∂zuk∂zjidx

≤Cjx 

_ω∇_j∇_jx∇_jy∇_ω +Cjy



_ω∇_j∇_j x

 ∇_j

y  ∇_ω

≤ 

∇jx ₊ 

∇jy

_+Cω_+j

x+jy

∇ω+∇j,

S_=

εijk∂xbl∂z∂luk∂zjidx

≤Cbx 

∇_ω∇_j∇_ωx∇_jy_jx

≤ 

∇ωx ₊ 

∇jy

_+Cb

x+jx

∇ω+∇j,

S_=

εijk∂ybl∂z∂luk∂zjidx

≤Cby 

∇_ω∇_j∇_ωx∇_jy_jy

≤ 

∇ωx ₊ 

∇jy

_+Cb

y+jy

For the last termS_, we see that

S_=

εijk∂zbl∂z∂luk∂zjidx

=

εijk∂zb∂z∂xuk∂zjidx+

εijk∂zb∂z∂yuk∂zjidx+

εijk∂zb∂z∂zuk∂zjidx =S_+S_+S_.

Then,

S_=

εijk∂zb∂z∂xuk∂zjidx≤Cj 

_ωx∇_j∇_jx_jy∇_ωx

≤ 

∇ωx ₊ 

∇jx

_+Cj_+ω x

∇j,

S_=

εijk∂zb∂z∂yuk∂zjidx≤Cj  _ω

y 

∇_j∇_j x

 _j

y 

∇_ω

y  

≤ 

∇ωy ₊ 

∇jx

_+Cj_+ω y

∇j,

S_=

εijk∂zb∂z∂zuk∂zjidx = –

εijk(∂xb+∂yb)∂z∂zuk∂zjidx

≤Cbx 

∇_ω∇_j∇_ωx∇_jy_jy +Cby



∇_ω∇_j∇_ω x

 ∇_j

y  _j

y  

≤ 

∇ωx ₊ 

∇jy

_+Cb

x+by+jy

∇j.

As forS, deduced by similar methods, we can obtain the following inequality (for sim-plicity we omit the details here):

S≤  ∇ωx

₊  ∇ωy

₊  ∇jx



+ 

∇jy _+ω

x+ω+j+ωy

+bx+jx+by+jy+ux+uy+ωy

∇ω+∇j.

Then, substituting all the above estimates into (), we finally deduce that

 

d dt

∇ω+∇j+ 

∇ωx+∇ωy+∇jx+∇jy

≤Cωx+ω+j+ωy+bx+jx+by+jy +ux+uy+ωy

∇ω+∇j.

3 Proof of Theorem 1.1

In this section, we prove Theorem . by using the method of vanishing viscosity. To this end, we consider the following regularized problem:

uε_t +uε· ∇uε= –∇pε+uε_xx+uε_yy+εuε_zz+bε· ∇bε, ()

bε_t+uε· ∇bε=bε_xx+bε_yy+εbε_zz+bε· ∇uε, ()

divuε= , divbε= , ()

with smooth initial data

uε(,x) =ψε∗u, bε(,x) =ψε∗b, () whereψε(x,y) =ε–ψ(x/ε,y/ε) is the standard mollifier satisfying

ψ≥, ψ∈C_∞R and

ψdx= .

Now, an application of the classical result shows that for anyT> , there exists a unique global smooth solution (uε_,bε_{) of ()-() onR}_{×(,T) satisfying the global bounds} stated in Lemma . and ., which are uniform inε. So, by standard compactness argu-ments, we can extract a subsequence (uεj_,bεj_{) and pass to the limit asj}→ ∞_{to get that the} limit function (u,b) is indeed a global smooth solution of the problem ()-(). The proof of Theorem . is therefore completed.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MS carried out the main work and drafted the manuscript. JW participated in completing the proof of Lemma 2.2. All authors read and approved the final manuscript.

Author details

1_{School of Mathematical Sciences, Luoyang Normal University, Luoyang, 471022, P.R. China.}2_{Key Laboratory of Symbolic}

Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun, 130012, P.R. China.

3_{School of Mathematical Sciences, Ocean University of China, Qingdao, 266000, P.R. China.}

Acknowledgements

This work was supported by the National Nature Science Foundation of China (No.11026079), the Youth Backbone Teacher Foundation of Henan Province (No. 2010GGJS-167), the Natural Science Foundation of Henan Province (No.122300410261), and the Foundation of Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education (No. 93K172012K07).

Received: 7 January 2013 Accepted: 10 July 2013 Published: 26 July 2013

References

1. Wang, F, Wang, K: Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion. Nonlinear Anal., Real World Appl.14, 525-536 (2013)

2. Cabannes, H, Theoretical Magnetofluiddynamics. Academic Press, New York (1970)

3. Cao, C, Wu, J: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math.226, 1803-1822 (2011)

doi:10.1186/1029-242X-2013-345