AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation38040210.1155/2011/380402380402Research ArticleRegularity Criteria for a Turbulent Magnetohydrodynamic ModelZhouYong1FanJishan2ReichSimeon1Department of MathematicsZhejiang Normal UniversityJinhua 321004Chinazjnu.edu.cn2Department of Applied MathematicsNanjing Forestry UniversityNanjing 210037Chinanjfu.edu.cn201121072011201123052011120720112011Copyright © 2011 Yong Zhou and Jishan Fan.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We establish some regularity criteria for a turbulent magnetohydrodynamic model. As a corollary, we prove that the smooth solution exists globally when the spatial dimension n satisfies 3n8.

1. Introduction

In this paper, we study the following simplified turbulent MHD model :tv-Δv+(u)u+π=(B)B,tH-ΔH+(u)B-(B)u=0,u=(1-α2Δ)u,H=(1-α2Δ)B,α>0,divv=divu=divH=divB=0,(v,H)(0)=(v0,H0)in  Rn  (n3). Here v is the fluid velocity field, u is the “filtered” fluid velocity, π is the pressure, H is the magnetic field, and B is the “filtered” magnetic field. α>0 is the length scale parameter that represents the width of the filter. For simplicity we will take α=1.

When n=3, the global well-posedness of the problem has been proved in . When H=B=0, (1.1) and (1.4) is the well-known Bardina model. Very recently, the authors  have proved that the Bardina model has a unique global-in-time weak solution when 3n8. Here we wold like to point out that by the same arguments, we can prove the following.

Theorem 1.1.

Let 3n8. Let (u0,B0)H1(n) with divu0=divB0=0 in n. Then for any T>0, the problem (1.1)–(1.5) has a unique weak solution satisfying 12u2+|u|2+B2+|B|2dx+0T|u|2+|Δu|2+|B|2+|ΔB|2dxdt12u02+|u0|2+B02+|B0|2dx.

The proof for Theorem 1.1 is similar to that for the Bardina model in , so we omit it here.

The aim of this paper is to study the regularity of the weak solutions. We will prove

Theorem 1.2.

Let n3. Let (v0,H0)Hs(n) with s>1 and divv0=divH0=0 in n. Let (v,H) be a local smooth solution to the problem (1.1)–(1.5) satisfying (v,H)L(0,T;Hs)L2(0,T;Hs+1), for any fixed T>0. Then (v,H) can be extended beyond T>0 provided that one of the following condition is satisfied: (1)  (u,B)Lp(0,T;Lq)with  2p+nq=3,n3<qn,(2)  (u,B)Lp(0,T;Lq(Rn))with  2p+nq=4,n4<qn2.

By (1.6) and (1.8), as a corollary, we have the following

Corollary 1.3.

Let 3n8. Let (v0,H0)Hs(n) with s>1 and divv0=divH0=0 in n. Then for any T>0, the problem (1.1)–(1.5) has a unique smooth solution (v,H) satisfying (1.7).

When n=9 or 10, we can get a better result as follows.

Theorem 1.4.

Let n=9 or 10 and let (v0,H0)L2(n) and divv0=divH0=0 in n. Let (v,H) be a local smooth solution to the problem (1.1)–(1.5) satisfying (v,H)L(0,T;L2)L2(0,T;H1), for any fixed T>0. Then (v,H) can be extended beyond T>0 if one of the following conditions is satisfied: (1)  uC([0,T];Ln/3),(2)  uLp(0,T;Lq)with  2p+nq=3,  n3<qn,(3)  uC([0,T];Ln/4),(4)  uLp(0,T;Lq)with  2p+nq=4  with  n4<qn2.

Remark 1.5.

If we delete the harmless lower order terms tu-Δu and tB-ΔB in (1.1) and (1.2), then we have -tΔu+Δ2u+(u)u+π=(B)B,-tΔB+Δ2B+(u)B-(B)u=0, then the system (1.15) has the following property: if (u,B,π) is a solution of (1.15), then for all λ>0, (uλ,Bλ,πλ)(x,t)=(λ3u,λ3B,λ6π)(λx,λ2t) is also a solution. In this sense, our conditions (1.8) and (1.11)–(1.14) are scaling invariant (optimal). Equations (1.8) and (1.12) do not hold true for q>n. But we also can establish regularity criteria for q>n in nonscaling invariant forms.

In Section 2, we will prove Theorem 1.2. In Section 3, we will prove Theorem 1.4.

2. Proof of Theorem <xref ref-type="statement" rid="thm1.2">1.2</xref>

Since it is easy to prove that the problem (1.1)–(1.5) has a unique local smooth solution, we only need to establish the a priori estimates. The proof of the case n4 is easier and similar and thus we omit the details here, we only deal with the case n5.

Testing (1.1) by u, using (1.3) and (1.4), we find that12ddtu2+|u|2dx+|u|2+|Δu|2dx=(B)Budx. Testing (1.2) by B, using (1.3) and (1.4), we see that12ddtB2+|B|2dx+|B|2+|ΔB|2dx=(B)uBdx=-(B)Budx.

Summing up (2.1) and (2.2), we easily get (1.6).

Let (1.8) hold true.

In the following calculations, we will use the product estimates due to Kato and Ponce :Λs(fg)LpC(ΛsfLp1gLq1+fLp2ΛsgLq2), with s>0, Λ(-Δ)1/2 and 1/p=1/p1+1/q1=1/p2+1/q2.

The proof of the case q=n is easier and similar, we omit the details here. Now we assume n/3<q<n.

Applying Λs to (1.1), testing by Λsv, using (1.4), we deduce that12ddt|Λsv|2dx+|Λs+1v|2dx=(Λs((B)B)-Λs((u)u))Λsvdx=Λsdiv(BB-uu)Λsvdx. Similarly, applying Λs to (1.2), testing by ΛsH, using (1.4), we infer that12ddt|ΛsH|2dx+|Λs+1H|2dx=Λscurl(u×B)ΛsHdx. Summing up (2.4) and (2.5), using (2.3), we get12ddt|Λsv|2+|ΛsH|2dx+|Λs+1v|2+|Λs+1H|2dx=Λsdiv(BB-uu)Λsvdx+Λscurl(u×B)ΛsHdxC(BLqΛs+1BLt1+uLqΛs+1uLt1)ΛsvLt2+C(uLqΛs+1BLt1+BLqΛs+1uLt1)ΛsHLt2C(uLq+BLq)(Λs+1uLt1+Λs+1BLt1)(ΛsvLt2+ΛsHLt2)=C(u,B)LqΛs+1(u,B)Lt1Λs(v,H)Lt2(1q+1t1+1t2=1,  2t22nn-2<t12nn-4)C(u,B)LqΛs-1(v,H)Lt1Λs(v,H)Lt2C(u,B)LqΛs(v,H)L21-θ1Λs+1(v,H)L2θ1Λs(v,H)L21-θ2Λs+1(v,H)L2θ2=C(u,B)LqΛs(v,H)L22-θ1-θ2Λs+1(v,H)L2θ1+θ212Λs+1(v,H)L22+C(u,B)Lq2/(2-θ1-θ2)Λs(v,H)L22, which implies (v,H)L(0,T;Hs)+(v,H)L2(0,T;Hs+1)C. Here we have used the following Gagliardo-Nirenberg inequalities:Λs-1(v,H)Lt1CΛs(v,H)L21-θ1Λs+1(v,H)L2θ1,(-nt1=(1-θ1)(1-n2)+θ1(2-n2))Λs(v,H)Lt2CΛs(v,H)L21-θ2Λs+1(v,H)L2θ2,(-nt2=(1-θ2)(-n2)+θ2(1-n2)).

Let (1.9) hold true.

In the following calculations, we will use the following commutator estimates due to Kato and Ponce :Λs(fg)-fΛsgLpC(fLp1Λs-1gLq1+ΛsfLp2gLq2), with s>0 and 1/p=1/p1+1/q1=1/p2+1/q2.

The proof of the case q=n/2 is easier and similar, we omit the details here. Now we assume n/4<q<n/2.

Applying Λs to (1.1), testing by Λsu, and using (1.3) and (1.4), we deduce that12ddt|Λsu|2+|Λs+1u|2dx+|Λs+1u|2+|Λs+2u|2dx=-[Λs(uu)-uΛsu]Λsudx+[Λs(BB)-BΛsB]Λsudx+(B)ΛsBΛsudx.

Applying Λs to (1.2), testing by ΛsB, using (1.3) and (1.4), we infer that12ddt|ΛsB|2+|Λs+1B|2dx+|Λs+1B|2+|Λs+2B|2dx=-[Λs(uB)-uΛsB]ΛsBdx+[Λs(Bu)-BΛsu]ΛsBdx+(B)ΛsuΛsBdx.

Summing up (2.10) and (2.11), noting that the last terms of (2.10) and (2.11) disappeared, and using (2.9), we obtain12ddt|Λsu|2+|Λs+1u|2+|ΛsB|2+|Λs+1B|2dx+|Λs+1u|2+|Λs+2u|2+|Λs+1B|2+|Λs+2B|2dx=-[Λs(uu)-uΛsu]Λsudx+[Λs(BB)-BΛsB]Λsudx-[Λs(uB)-uΛsB]ΛsBdx+[Λs(Bu)-BΛsu]ΛsBdxCuLqΛsuL2q/(q-1)2+CBLqΛsBL2q/(q-1)ΛsuL2q/(q-1)+CBLqΛsBL2q/(q-1)2+CuLqΛsBL2q/(q-1)2C(u,B)LqΛs(u,B)L2q/(q-1)2C(u,B)LqΛs+1(u,B)L22(1-θ)Λs+2(u,B)L22θ12Λs+2(u,B)L22+C(u,B)Lq1/(1-θ)Λs+1(u,B)L22, which yields(u,B)L(0,T;Hs+1)+(u,B)L2(0,T;Hs+2)C. Here we have used the following Gagliardo-Nirenberg inequality: Λs(u,B)L2q/(q-1)CΛs+1(u,B)L21-θΛs+2(u,B)L2θ, with-q-12qn=(1-θ)(1-n2)+θ(2-n2),2nn-22qq-12nn-4.

This completes the proof.

3. Proof of Theorem <xref ref-type="statement" rid="thm1.3">1.4</xref>

We only need to prove the a priori estimates.

Testing (1.1) and (1.2) by (v,H), using (1.3) and (1.4), and summing up the results, we have12ddtv2+H2dx+|v|2+|H|2dx=(u)uΔudx-(B)BΔudx+(u)BΔBdx-(B)uΔBdx=I1+I2+I3+I4.

Using (1.4), we see thatI1=i,juiiuj2udx=-i,jjuiiujudx,I2=-i,jBiiBj2udx=i,jjBiiBjudx-i,jBijBijudx,I3=i,juiiBj2Bdx=-i,jjuiiBjBdx,I4=-i,jBiiuj2Bdx=i,jjBiiujBdx+i,jBijBijudx.

Inserting the above estimates into (3.1), noting that the last term of I2 and I4 disappeared, we have12ddtv2+H2dx+|v|2+|H|2dx=-i,jiujujuidx+i,jjuiiBjBidx-i,jjuiiBjBdx+i,jiujBjBidx=J.

The proofs of the cases (1.11) and (1.13) are similar, we omit the details here.

Let (1.14) hold true,

JCuLq(u,B)L2q/(q-1)2CuLqΔ(u,B)L22(1-θ)Δ(u,B)L22θCuLq(v,H)L22(1-θ)(v,H)L22θ12(v,H)L22+CuLq1/(1-θ)(v,H)L22.

Inserting the above estimates into (3.3) and using the Gronwall inequality yields(v,H)L(0,T;L2)+(v,H)L2(0,T;H1)C.

Here we have used the Gagliardo-Nirenberg inequality:wL2q/(q-1)CwL21-θΔwL2θ with -q-12qn=(1-θ)(1-n2)+θ(2-n2),2nn-22qq-12nn-4.

Let (1.12) hold true.

Integrating by parts, using (1.4) we haveJ=i,juijujuidx-i,juij(iBjBi)dx+i,juij(iBjB)dx-i,juijBjBidxCuLq(u,B)Lt1Δ(u,B)Lt2(1q+1t1+1t2=1)CuLqΔ(u,B)L21-θ1Δ(u,B)L2θ1Δ(u,B)L21-θ2Δ(u,B)L2θ2=CuLqΔ(u,B)L22-θ1-θ2Δ(u,B)L2θ1+θ2CuLq(v,H)L22-θ1-θ2(v,H)L2θ1+θ212(v,H)L22+CuLq2/(2-θ1-θ2)(v,H)L22, which yields (3.5).

Here we have used the Gagliardo-Nirenberg inequalities:(u,B)Lt1CΔ(u,B)L21-θ1Δ(u,B)L2θ1,(-nt1=(1-θ1)(1-n2)+θ1(2-n2))Δ(u,B)Lt2CΔ(u,B)L21-θ2Δ(u,B)L2θ2,(-nt2=(1-θ2)(-n2)+θ2(1-n2)).

This completes the proof.

Acknowledgments

This paper is partially supported by Zhejiang Innovation Project (Grant no. T200905), ZJNSF (Grant no. R6090109), and NSFC (Grant no. 10971197).

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