# Regularity Criteria for a Turbulent Magnetohydrodynamic Model

Authors: Yong Zhou; Jishan Fan
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We establish some regularity criteria for a turbulent magnetohydrodynamic model. As a corollary, we prove that the smooth solution exists globally when the spatial dimensionn satisfies 3≤n≤8.


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## Body

## 1. Introduction

In this paper, we study the following simplified turbulent MHD model [1]:(1.1)∂tv-Δv+(u⋅∇)u+∇π=(B⋅∇)B,(1.2)∂tH-ΔH+(u⋅∇)B-(B⋅∇)u=0,(1.3)u=(1-α2Δ)u,H=(1-α2Δ)B,α>0,(1.4)div⁡⁡v=div⁡⁡u=div⁡⁡H=div⁡⁡B=0,(1.5)(v,H)(0)=(v0,H0)inRn(n≥3).
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.Whenn=3, the global well-posedness of the problem has been proved in [1]. When H=B=0, (1.1) and (1.4) is the well-known Bardina model. Very recently, the authors [2] have proved that the Bardina model has a unique global-in-time weak solution when 3≤n≤8. Here we wold like to point out that by the same arguments, we can prove the following.Theorem 1.1.
Let3≤n≤8. Let (u0,B0)∈H1(ℝn) with divu0=divB0=0 in ℝn. Then for any T>0, the problem (1.1)–(1.5) has a unique weak solution satisfying
(1.6)12∫u2+|∇u|2+B2+|∇B|2dx+∫0T∫|∇u|2+|Δu|2+|∇B|2+|ΔB|2dxdt≤12∫u02+|∇u0|2+B02+|∇B0|2dx.The proof for Theorem1.1 is similar to that for the Bardina model in [2], so we omit it here.The aim of this paper is to study the regularity of the weak solutions. We will proveTheorem 1.2.
Letn≥3. Let (v0,H0)∈Hs(ℝn) with s>1 and divv0=divH0=0 in ℝn. Let (v,H) be a local smooth solution to the problem (1.1)–(1.5) satisfying
(1.7)(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:
(1.8)(1)(u,B)∈Lp(0,T;Lq)with2p+nq=3,n3<q≤n,(1.9)(2)(∇u,∇B)∈Lp(0,T;Lq(Rn))with2p+nq=4,n4<q≤n2.By (1.6) and (1.8), as a corollary, we have the followingCorollary 1.3.
Let3≤n≤8. Let (v0,H0)∈Hs(ℝn) with s>1 and divv0=divH0=0 in ℝn. Then for any T>0, the problem (1.1)–(1.5) has a unique smooth solution (v,H) satisfying (1.7).Whenn=9 or 10, we can get a better result as follows.Theorem 1.4.
Letn=9 or 10 and let (v0,H0)∈L2(ℝn) and divv0=divH0=0 in ℝn. Let (v,H) be a local smooth solution to the problem (1.1)–(1.5) satisfying
(1.10)(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:
(1.11)(1)u∈C([0,T];Ln/3),(1.12)(2)u∈Lp(0,T;Lq)with2p+nq=3,n3<q≤n,(1.13)(3)∇u∈C([0,T];Ln/4),(1.14)(4)∇u∈Lp(0,T;Lq)with2p+nq=4withn4<q≤n2.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
(1.15)-∂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,
(1.16)(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 Section2, we will prove Theorem 1.2. In Section 3, we will prove Theorem 1.4.


## 2. Proof of Theorem1.2

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 n≤4 is easier and similar and thus we omit the details here, we only deal with the case n≥5.Testing (1.1) by u, using (1.3) and (1.4), we find that(2.1)12ddt∫u2+|∇u|2dx+∫|∇u|2+|Δu|2dx=∫(B⋅∇)B⋅udx.
Testing (1.2) by B, using (1.3) and (1.4), we see that(2.2)12ddt∫B2+|∇B|2dx+∫|∇B|2+|ΔB|2dx=∫(B⋅∇)u⋅Bdx=-∫(B⋅∇)B⋅udx.Summing up (2.1) and (2.2), we easily get (1.6).(I)
Let (1.8) hold true.In the following calculations, we will use the product estimates due to Kato and Ponce [3]:(2.3)‖Λs(fg)‖Lp≤C(‖Λsf‖Lp1‖g‖Lq1+‖f‖Lp2‖Λsg‖Lq2),
with s>0, Λ≔(-Δ)1/2 and 1/p=1/p1+1/q1=1/p2+1/q2.The proof of the caseq=n is easier and similar, we omit the details here. Now we assume n/3<q<n.ApplyingΛs to (1.1), testing by Λsv, using (1.4), we deduce that(2.4)12ddt∫|Λsv|2dx+∫|Λs+1v|2dx=∫(Λs((B⋅∇)B)-Λs((u⋅∇)u))Λsvdx=∫Λsdiv⁡⁡(B⊗B-u⊗u)Λsvdx.
Similarly, applying Λs to (1.2), testing by ΛsH, using (1.4), we infer that(2.5)12ddt∫|ΛsH|2dx+∫|Λs+1H|2dx=∫Λscurl⁡⁡(u×B)⋅ΛsHdx.
Summing up (2.4) and (2.5), using (2.3), we get(2.6)12ddt∫|Λsv|2+|ΛsH|2dx+∫|Λs+1v|2+|Λs+1H|2dx=∫Λsdiv⁡⁡(B⊗B-u⊗u)Λsvdx+∫Λscurl⁡⁡(u×B)⋅ΛsHdx≤C(‖B‖Lq‖Λs+1B‖Lt1+‖u‖Lq‖Λs+1u‖Lt1)‖Λsv‖Lt2+C(‖u‖Lq‖Λs+1B‖Lt1+‖B‖Lq‖Λs+1u‖Lt1)‖ΛsH‖Lt2≤C(‖u‖Lq+‖B‖Lq)(‖Λs+1u‖Lt1+‖Λs+1B‖Lt1)(‖Λsv‖Lt2+‖ΛsH‖Lt2)=C‖(u,B)‖Lq‖Λs+1(u,B)‖Lt1‖Λs(v,H)‖Lt2(1q+1t1+1t2=1,2≤t2≤2nn-2<t1≤2nn-4)≤C‖(u,B)‖Lq‖Λs-1(v,H)‖Lt1‖Λs(v,H)‖Lt2≤C‖(u,B)‖Lq‖Λs(v,H)‖L21-θ1‖Λs+1(v,H)‖L2θ1‖Λs(v,H)‖L21-θ2‖Λs+1(v,H)‖L2θ2=C‖(u,B)‖Lq‖Λs(v,H)‖L22-θ1-θ2‖Λs+1(v,H)‖L2θ1+θ2≤12‖Λs+1(v,H)‖L22+C‖(u,B)‖Lq2/(2-θ1-θ2)‖Λs(v,H)‖L22,
which implies (2.7)‖(v,H)‖L∞(0,T;Hs)+‖(v,H)‖L2(0,T;Hs+1)≤C.
Here we have used the following Gagliardo-Nirenberg inequalities:(2.8)‖Λs-1(v,H)‖Lt1≤C‖Λs(v,H)‖L21-θ1‖Λs+1(v,H)‖L2θ1,(-nt1=(1-θ1)(1-n2)+θ1(2-n2))‖Λs(v,H)‖Lt2≤C‖Λs(v,H)‖L21-θ2‖Λs+1(v,H)‖L2θ2,(-nt2=(1-θ2)(-n2)+θ2(1-n2)).(II)
Let (1.9) hold true.In the following calculations, we will use the following commutator estimates due to Kato and Ponce [3]:(2.9)‖Λs(fg)-fΛsg‖Lp≤C(‖∇f‖Lp1‖Λs-1g‖Lq1+‖Λsf‖Lp2‖g‖Lq2),
with s>0 and 1/p=1/p1+1/q1=1/p2+1/q2.The proof of the caseq=n/2 is easier and similar, we omit the details here. Now we assume n/4<q<n/2.ApplyingΛs to (1.1), testing by Λsu, and using (1.3) and (1.4), we deduce that(2.10)12ddt∫|Λsu|2+|Λs+1u|2dx+∫|Λs+1u|2+|Λs+2u|2dx=-∫[Λs(u⋅∇u)-u⋅∇Λsu]Λsudx+∫[Λs(B⋅∇B)-B⋅∇ΛsB]Λsudx+∫(B⋅∇)ΛsB⋅Λsudx.ApplyingΛs to (1.2), testing by ΛsB, using (1.3) and (1.4), we infer that(2.11)12ddt∫|ΛsB|2+|Λs+1B|2dx+∫|Λs+1B|2+|Λs+2B|2dx=-∫[Λs(u⋅∇B)-u∇ΛsB]ΛsBdx+∫[Λs(B⋅∇u)-B⋅∇Λsu]ΛsBdx+∫(B⋅∇)Λsu⋅ΛsBdx.Summing up (2.10) and (2.11), noting that the last terms of (2.10) and (2.11) disappeared, and using (2.9), we obtain(2.12)12ddt∫|Λsu|2+|Λs+1u|2+|ΛsB|2+|Λs+1B|2dx+∫|Λs+1u|2+|Λs+2u|2+|Λs+1B|2+|Λs+2B|2dx=-∫[Λs(u⋅∇u)-u⋅∇Λsu]Λsudx+∫[Λs(B⋅∇B)-B⋅∇ΛsB]Λsudx-∫[Λs(u⋅∇B)-u⋅∇ΛsB]ΛsBdx+∫[Λs(B⋅∇u)-B⋅∇Λsu]ΛsBdx≤C‖∇u‖Lq‖Λsu‖L2q/(q-1)2+C‖∇B‖Lq‖ΛsB‖L2q/(q-1)‖Λsu‖L2q/(q-1)+C‖∇B‖Lq‖ΛsB‖L2q/(q-1)2+C‖∇u‖Lq‖ΛsB‖L2q/(q-1)2≤C‖(∇u,∇B)‖Lq‖Λs(u,B)‖L2q/(q-1)2≤C‖(∇u,∇B)‖Lq‖Λs+1(u,B)‖L22(1-θ)‖Λs+2(u,B)‖L22θ≤12‖Λs+2(u,B)‖L22+C‖(∇u,∇B)‖Lq1/(1-θ)‖Λs+1(u,B)‖L22,
which yields(2.13)‖(u,B)‖L∞(0,T;Hs+1)+‖(u,B)‖L2(0,T;Hs+2)≤C.
Here we have used the following Gagliardo-Nirenberg inequality: (2.14)‖Λs(u,B)‖L2q/(q-1)≤C‖Λs+1(u,B)‖L21-θ‖Λs+2(u,B)‖L2θ,
with(2.15)-q-12qn=(1-θ)(1-n2)+θ(2-n2),2nn-2≤2qq-1≤2nn-4.This completes the proof.


## 3. Proof of Theorem1.4

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 have(3.1)12ddt∫v2+H2dx+∫|∇v|2+|∇H|2dx=∫(u⋅∇)u⋅Δudx-∫(B⋅∇)B⋅Δudx+∫(u⋅∇)B⋅ΔBdx-∫(B⋅∇)u⋅ΔBdx=I1+I2+I3+I4.Using (1.4), we see that(3.2)I1=∑i,j∫ui∂iu∂j2udx=-∑i,j∫∂jui∂iu∂judx,I2=-∑i,j∫Bi∂iB∂j2udx=∑i,j∫∂jBi∂iB∂judx-∑i,j∫Bi∂jB∂i∂judx,I3=∑i,j∫ui∂iB∂j2Bdx=-∑i,j∫∂jui∂iB∂jBdx,I4=-∑i,j∫Bi∂iu∂j2Bdx=∑i,j∫∂jBi∂iu∂jBdx+∑i,j∫Bi∂jB∂i∂judx.Inserting the above estimates into (3.1), noting that the last term of I2 and I4 disappeared, we have(3.3)12ddt∫v2+H2dx+∫|∇v|2+|∇H|2dx=-∑i,j∫∂iu∂ju∂juidx+∑i,j∫∂jui∂iB∂jBidx-∑i,j∫∂jui∂iB∂jBdx+∑i,j∫∂iu∂jB∂jBidx=J.The proofs of the cases (1.11) and (1.13) are similar, we omit the details here.(I)
Let (1.14) hold true,(3.4)J≤C‖∇u‖Lq‖∇(u,B)‖L2q/(q-1)2≤C‖∇u‖Lq‖Δ(u,B)‖L22(1-θ)‖Δ∇(u,B)‖L22θ≤C‖∇u‖Lq‖(v,H)‖L22(1-θ)‖∇(v,H)‖L22θ≤12‖∇(v,H)‖L22+C‖∇u‖Lq1/(1-θ)‖(v,H)‖L22.Inserting the above estimates into (3.3) and using the Gronwall inequality yields(3.5)‖(v,H)‖L∞(0,T;L2)+‖(v,H)‖L2(0,T;H1)≤C.Here we have used the Gagliardo-Nirenberg inequality:(3.6)‖w‖L2q/(q-1)≤C‖∇w‖L21-θ‖Δw‖L2θ
with (3.7)-q-12qn=(1-θ)(1-n2)+θ(2-n2),2nn-2≤2qq-1≤2nn-4.(II)
Let (1.12) hold true.Integrating by parts, using (1.4) we have(3.8)J=∑i,j∫u∂i∂ju∂juidx-∑i,j∫ui∂j(∂iB∂jBi)dx+∑i,j∫ui∂j(∂iB∂jB)dx-∑i,j∫u∂i∂jB∂jBidx≤C‖u‖Lq‖∇(u,B)‖Lt1‖Δ(u,B)‖Lt2(1q+1t1+1t2=1)≤C‖u‖Lq‖Δ(u,B)‖L21-θ1‖∇Δ(u,B)‖L2θ1‖Δ(u,B)‖L21-θ2‖∇Δ(u,B)‖L2θ2=C‖u‖Lq‖Δ(u,B)‖L22-θ1-θ2‖∇Δ(u,B)‖L2θ1+θ2≤C‖u‖Lq‖(v,H)‖L22-θ1-θ2‖∇(v,H)‖L2θ1+θ2≤12‖∇(v,H)‖L22+C‖u‖Lq2/(2-θ1-θ2)‖(v,H)‖L22,
which yields (3.5).Here we have used the Gagliardo-Nirenberg inequalities:(3.9)‖∇(u,B)‖Lt1≤C‖Δ(u,B)‖L21-θ1‖∇Δ(u,B)‖L2θ1,(-nt1=(1-θ1)(1-n2)+θ1(2-n2))‖Δ(u,B)‖Lt2≤C‖Δ(u,B)‖L21-θ2‖∇Δ(u,B)‖L2θ2,(-nt2=(1-θ2)(-n2)+θ2(1-n2)).This completes the proof.



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*Source: 380402-2011-09-08.xml*