Regularity Criteria for a Turbulent Magnetohydrodynamic Model

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 dimension 𝑛 satisfies 3≤𝑛≤8.

1. Introduction

In this paper, we study the following simplified turbulent MHD model [1]:𝜕𝜕_𝑡𝑣−Δ𝑣+(𝑢⋅∇)𝑢+∇𝜋=(𝐵⋅∇)𝐵,(1.1)_𝑡𝑢=1𝐻−−𝛼₂Δ𝐻+(𝑢⋅∇)𝐵−(𝐵⋅∇)𝑢=0,(1.2)(𝑣d,i𝐻v𝑣)(=0Δ)d=i𝑢v,𝑢𝑣₀=𝐻d=iv𝐻1−=𝛼d²iΔv𝐵𝐵=,0,𝛼>0,((11..34)),𝐻₀inℝ^𝑛(𝑛≥3).(1.5) 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 [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. Let 3≤𝑛≤8. Let (𝑢₀,𝐵₀)∈𝐻¹(ℝ^𝑛) with div𝑢₀=div𝐵₀=0 in ℝ^𝑛. Then for any 𝑇>0, the problem (1.1)–(1.5) has a unique weak solution satisfying 12𝑢²+||∇𝑢||²+𝐵²+||∇𝐵||²𝑑𝑥+^𝑇0||∇𝑢||²≤1+||Δ𝑢||²+||∇𝐵||²+||Δ𝐵||²𝑑𝑥𝑑𝑡2𝑢²⁰+||∇𝑢₀||²+𝐵²⁰+||∇𝐵₀||²𝑑𝑥.(1.6)

The proof for Theorem 1.1 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 prove

Theorem 1.2. Let 𝑛≥3. Let (𝑣₀,𝐻₀)∈𝐻^𝑠(ℝ^𝑛) with 𝑠>1 and div𝑣₀=div𝐻₀=0 in ℝ^𝑛. Let (𝑣,𝐻) be a local smooth solution to the problem (1.1)–(1.5) satisfying (𝑣,𝐻)∈𝐿^∞(0,𝑇;𝐻^𝑠)∩𝐿²0,𝑇;𝐻^𝑠+1,(1.7) 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)∈𝐿^𝑝(0,𝑇;𝐿^𝑞)with2(2)(∇𝑢,∇𝐵)∈𝐿_𝑝𝑝+𝑞𝑛=3,3𝑛<𝑞≤𝑛,(1.8)(0,𝑇;𝐿^𝑞(ℝ^𝑛))with2𝑝+𝑞𝑛=4,4𝑛<𝑞≤2𝑛.(1.9)

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

Corollary 1.3. Let 3≤𝑛≤8. Let (𝑣₀,𝐻₀)∈𝐻^𝑠(ℝ^𝑛) with 𝑠>1 and div𝑣₀=div𝐻₀=0 in ℝ^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(ℝ^𝑛) and div𝑣₀=div𝐻₀=0 in ℝ^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,(1.10) for any fixed 𝑇>0. Then (𝑣,𝐻) can be extended beyond 𝑇>0 if one of the following conditions is satisfied: ((12))𝑢𝑢∈∈𝐶𝐿_𝑝[0,𝑇];𝐿^𝑛/3,(1.11)(0,𝑇;𝐿^𝑞)with2(3)∇𝑢∈𝐶[0,𝑇];𝐿_𝑛/4𝑝+𝑞𝑛=3,3𝑛<𝑞≤𝑛,(1.12)(4)∇𝑢∈𝐿_𝑝,(1.13)(0,𝑇;𝐿^𝑞)with2𝑝+𝑞𝑛=4with4𝑛<𝑞≤2𝑛.(1.14)

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 −−𝜕𝜕_𝑡𝑡Δ𝑢+Δ²𝑢+(𝑢⋅∇)𝑢+∇𝜋=(𝐵⋅∇)𝐵,Δ𝐵+Δ²𝐵+(𝑢⋅∇)𝐵−(𝐵⋅∇)𝑢=0,(1.15) then the system (1.15) has the following property: if (𝑢,𝐵,𝜋) is a solution of (1.15), then for all 𝜆>0, 𝑢_𝜆,𝐵_𝜆,𝜋_𝜆(𝑥,𝑡)=𝜆³𝑢,𝜆³𝐵,𝜆⁶𝜋𝜆𝑥,𝜆²𝑡(1.16) 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 1.2

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 that12𝑑𝑑𝑡𝑢²+||∇𝑢||²𝑑𝑥+||∇𝑢||²+||Δ𝑢||²𝑑𝑥=(𝐵⋅∇)𝐵⋅𝑢𝑑𝑥.(2.1) Testing (1.2) by 𝐵, using (1.3) and (1.4), we see that12𝑑𝑑𝑡𝐵²+||∇𝐵||²𝑑𝑥+||∇𝐵||²+||Δ𝐵||²𝑑𝑥=(𝐵⋅∇)𝑢⋅𝐵𝑑𝑥=−(𝐵⋅∇)𝐵⋅𝑢𝑑𝑥.(2.2)

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]:‖Λ^𝑠(𝑓𝑔)‖_𝐿^𝑝≤𝐶‖Λ^𝑠𝑓‖_𝐿^𝑝1‖𝑔‖_𝐿^𝑞1+‖𝑓‖_𝐿^𝑝2‖Λ^𝑠𝑔‖_𝐿^𝑞2,(2.3) 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 Λ^𝑠 to (1.1), testing by Λ^𝑠𝑣, using (1.4), we deduce that12𝑑𝑑𝑡||Λ^𝑠𝑣||²𝑑𝑥+||Λ^𝑠+1𝑣||²𝑑𝑥==Λ(Λ^𝑠((𝐵⋅∇)𝐵)−Λ^𝑠((𝑢⋅∇)𝑢))Λ^𝑠𝑣𝑑𝑥_𝑠div(𝐵⊗𝐵−𝑢⊗𝑢)Λ^𝑠𝑣𝑑𝑥.(2.4) Similarly, applying Λ^𝑠 to (1.2), testing by Λ^𝑠𝐻, using (1.4), we infer that12𝑑𝑑𝑡||Λ^𝑠𝐻||²𝑑𝑥+||Λ^𝑠+1𝐻||²𝑑𝑥=Λ^𝑠curl(𝑢×𝐵)⋅Λ^𝑠𝐻𝑑𝑥.(2.5) Summing up (2.4) and (2.5), using (2.3), we get12𝑑𝑑𝑡||Λ^𝑠=Λ𝑣||²+||Λ^𝑠𝐻||²𝑑𝑥+||Λ^𝑠+1𝑣||²+||Λ^𝑠+1𝐻||²𝑑𝑥_𝑠≤𝐶‖𝐵di‖v(_𝐿𝑞𝐵⊗𝐵−𝑢⊗𝑢)Λ^𝑠𝑣𝑑𝑥+Λ^𝑠curl(𝑢×𝐵)⋅Λ^𝑠𝐻𝑑𝑥‖‖Λ^𝑠+1𝐵‖‖_𝐿𝑡1+‖𝑢‖_𝐿^𝑞‖‖Λ^𝑠+1𝑢‖‖_𝐿𝑡1‖Λ^𝑠𝑣‖_𝐿^𝑡2+𝐶‖𝑢‖_𝐿^𝑞‖‖Λ^𝑠+1𝐵‖‖_𝐿𝑡1+‖𝐵‖_𝐿^𝑞‖‖Λ^𝑠+1𝑢‖‖_𝐿𝑡1‖Λ^𝑠𝐻‖_𝐿^𝑡2=≤𝐶𝐶‖(‖𝑢𝑢,‖𝐵)‖_𝐿^𝑞+_𝐿𝑞‖𝐵‖_𝐿^𝑞‖‖Λ^𝑠+1𝑢‖‖_𝐿𝑡1+‖‖Λ^𝑠+1𝐵‖‖_𝐿𝑡1‖Λ^𝑠𝑣‖_𝐿^𝑡2+‖Λ^𝑠𝐻‖_𝐿^𝑡2‖‖Λ^𝑠+1(𝑢,𝐵)‖‖_𝐿𝑡1‖Λ^𝑠(𝑣,𝐻)‖_𝐿^𝑡21𝑞+𝑡1₁+1𝑡₂=1,2≤𝑡₂≤2𝑛𝑛−2<𝑡¹≤2𝑛≤𝐶‖(𝑢,𝐵)‖_𝐿𝑞𝑛−4‖‖Λ^𝑠−1(𝑣,𝐻)‖‖_𝐿𝑡1‖Λ^𝑠(𝑣,𝐻)‖_𝐿^𝑡2≤𝐶‖(𝑢,𝐵)‖_𝐿^𝑞‖Λ^𝑠(𝑣,𝐻)‖^1−𝜃1_𝐿2‖‖Λ^𝑠+1(𝑣,𝐻)‖‖^𝜃1_𝐿2‖Λ^𝑠(𝑣,𝐻)‖^1−𝜃2_𝐿2‖‖Λ^𝑠+1(𝑣,𝐻)‖‖^𝜃2_𝐿2=𝐶‖(𝑢,𝐵)‖_𝐿^𝑞‖Λ^𝑠(𝑣,𝐻)‖^{2−𝜃1−𝜃2}_𝐿2‖‖Λ^𝑠+1(𝑣,𝐻)‖‖^𝜃1+𝜃2_𝐿2≤12‖‖Λ^𝑠+1(𝑣,𝐻)‖‖^2𝐿₂+𝐶‖(𝑢,𝐵)‖^{2/(2−𝜃1−𝜃2)𝐿}_𝑞‖Λ^𝑠(𝑣,𝐻)‖^2𝐿₂,(2.6) which implies ‖(𝑣,𝐻)‖_𝐿^∞_(0,𝑇;𝐻^𝑠₎+‖(𝑣,𝐻)‖_𝐿²_(0,𝑇;𝐻^𝑠+1₎≤𝐶.(2.7) Here we have used the following Gagliardo-Nirenberg inequalities:‖‖Λ^𝑠−1(𝑣−,𝐻𝑛)‖‖_𝐿𝑡1≤𝐶‖Λ^𝑠(𝑣,𝐻)‖^1−𝜃1_𝐿2‖‖Λ^𝑠+1(𝑣,𝐻)‖‖^𝜃1_𝐿2,𝑡₁=1−𝜃₁1−𝑛‖Λ_𝑠2+𝜃¹2−2𝑛(𝑣,𝐻−)𝑛‖_𝐿^𝑡2≤𝐶‖Λ^𝑠(𝑣,𝐻)‖^1−𝜃2_𝐿2‖‖Λ^𝑠+1(𝑣,𝐻)‖‖^𝜃2_𝐿2,𝑡₂=1−𝜃₂−𝑛2+𝜃²1−2𝑛.(2.8)(II)Let (1.9) hold true.

In the following calculations, we will use the following commutator estimates due to Kato and Ponce [3]:‖Λ^𝑠(𝑓𝑔)−𝑓Λ^𝑠𝑔‖_𝐿^𝑝≤𝐶‖∇𝑓‖_𝐿^𝑝1‖‖Λ^𝑠−1𝑔‖‖_𝐿𝑞1+‖Λ^𝑠𝑓‖_𝐿^𝑝2‖𝑔‖_𝐿^𝑞2,(2.9) 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 Λ^𝑠 to (1.1), testing by Λ^𝑠𝑢, and using (1.3) and (1.4), we deduce that12𝑑𝑑𝑡||Λ^𝑠=−𝑢||²Λ+_𝑠||Λ^𝑠+1𝑢||²𝑑𝑥+||Λ^𝑠+1𝑢||²+||Λ^𝑠+2𝑢||²𝑑𝑥+(𝐵⋅∇(𝑢)⋅Λ∇_𝑠𝑢)−𝑢⋅∇Λ^𝑠𝑢Λ^𝑠𝑢𝑑𝑥+Λ^𝑠(𝐵⋅∇𝐵)−𝐵⋅∇Λ^𝑠𝐵Λ^𝑠𝑢𝑑𝑥𝐵⋅Λ^𝑠𝑢𝑑𝑥.(2.10)

Applying Λ^𝑠 to (1.2), testing by Λ^𝑠𝐵, using (1.3) and (1.4), we infer that12𝑑𝑑𝑡||Λ^𝑠=−𝐵||²Λ+_𝑠||Λ^𝑠+1𝐵||²𝑑𝑥+||Λ^𝑠+1𝐵||²+||Λ^𝑠+2𝐵||²𝑑𝑥+(𝐵⋅∇(𝑢)⋅Λ∇_𝑠𝐵)−𝑢∇Λ^𝑠𝐵Λ^𝑠𝐵𝑑𝑥+Λ^𝑠(𝐵⋅∇𝑢)−𝐵⋅∇Λ^𝑠𝑢Λ^𝑠𝐵𝑑𝑥𝑢⋅Λ^𝑠𝐵𝑑𝑥.(2.11)

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||Λ^𝑠+||Λ𝑢||²+||Λ^𝑠+1𝑢||²+||Λ^𝑠𝐵||²+||Λ^𝑠+1𝐵||²𝑑𝑥_𝑠+1=−Λ_𝑠𝑢||²+||Λ^𝑠+2𝑢||²+||Λ^𝑠+1𝐵||²+||Λ^𝑠+2𝐵||²𝑑𝑥−Λ(𝑢⋅∇𝑢)−𝑢⋅∇Λ^𝑠𝑢Λ^𝑠𝑢𝑑𝑥+Λ^𝑠(𝐵⋅∇𝐵)−𝐵⋅∇Λ^𝑠𝐵Λ^𝑠𝑢𝑑𝑥_𝑠≤𝐶‖∇𝑢‖(𝑢_𝐿𝑞⋅∇𝐵)−𝑢⋅∇Λ^𝑠𝐵Λ^𝑠𝐵𝑑𝑥+Λ^𝑠(𝐵⋅∇𝑢)−𝐵⋅∇Λ^𝑠𝑢Λ^𝑠𝐵𝑑𝑥‖Λ^𝑠𝑢‖^2𝐿_{2𝑞/(𝑞−1)}+𝐶‖∇𝐵‖_𝐿^𝑞‖Λ^𝑠𝐵‖_𝐿^{2𝑞/(𝑞−1)}‖Λ^𝑠𝑢‖_𝐿^{2𝑞/(𝑞−1)}+𝐶‖∇𝐵‖_𝐿^𝑞‖Λ^𝑠𝐵‖^2𝐿_{2𝑞/(𝑞−1)}+𝐶‖∇𝑢‖_𝐿^𝑞‖Λ^𝑠𝐵‖^2𝐿_{2𝑞/(𝑞−1)}≤𝐶‖(∇𝑢,∇𝐵)‖_𝐿^𝑞‖Λ^𝑠(𝑢,𝐵)‖^2𝐿_{2𝑞/(𝑞−1)}≤𝐶‖(∇𝑢,∇𝐵)‖_𝐿^𝑞‖‖Λ^𝑠+1(𝑢,𝐵)‖‖^{𝐿2(1−𝜃)}₂‖‖Λ^𝑠+2(𝑢,𝐵)‖‖^𝐿2𝜃₂≤12‖‖Λ^𝑠+2(𝑢,𝐵)‖‖^2𝐿₂+𝐶‖(∇𝑢,∇𝐵)‖^{𝐿1/(1−𝜃)}_𝑞‖‖Λ^𝑠+1(𝑢,𝐵)‖‖^2𝐿₂,(2.12) which yields‖(𝑢,𝐵)‖_𝐿^∞_(0,𝑇;𝐻^𝑠+1₎+‖(𝑢,𝐵)‖_𝐿²_(0,𝑇;𝐻^𝑠+2₎≤𝐶.(2.13) Here we have used the following Gagliardo-Nirenberg inequality: ‖Λ^𝑠(𝑢,𝐵)‖_𝐿^{2𝑞/(𝑞−1)}≤𝐶‖‖Λ^𝑠+1(𝑢,𝐵)‖‖^𝐿1−𝜃₂‖‖Λ^𝑠+2(𝑢,𝐵)‖‖^𝜃𝐿₂,(2.14) with−𝑞−12𝑞𝑛=2(𝑛1−𝜃)1−2𝑛+𝜃2−2𝑛,𝑛−2≤𝑞2−𝑞1≤𝑛2−𝑛4.(2.15)

This completes the proof.

3. Proof of Theorem 1.4

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 have12𝑑𝑑𝑡𝑣²==𝐼₁(+𝑢𝐻⋅∇²)𝑑𝑢𝑥⋅+Δ𝑢||𝑑∇𝑥𝑣−||²+||(∇𝐵𝐻⋅∇||²𝑑)𝑥𝐵⋅Δ𝑢𝑑𝑥+(𝑢⋅∇)𝐵⋅Δ𝐵𝑑𝑥−(𝐵⋅∇)𝑢⋅Δ𝐵𝑑𝑥+𝐼₂+𝐼₃+𝐼₄.(3.1)

Using (1.4), we see that𝐼₁=_𝑖,𝑗𝑢^𝑖𝜕_𝑖𝑢𝜕^2𝑗𝑢𝑑𝑥=−_𝑖,𝑗𝜕^𝑗𝐼𝑢_𝑖𝜕_𝑖𝑢𝜕_𝑗𝑢𝑑𝑥,₂=−_𝑖,𝑗𝐵^𝑖𝜕_𝑖𝐵𝜕^2𝑗𝑢𝑑𝑥=_𝑖,𝑗𝜕^𝑗𝐵_𝑖𝜕_𝑖𝐵𝜕_𝑗𝑢𝑑𝑥−_𝑖,𝑗𝐵^𝑖𝐼𝜕_𝑗𝐵𝜕_𝑖𝜕_𝑗𝑢𝑑𝑥,₃=_𝑖,𝑗𝑢^𝑖𝜕_𝑖𝐵𝜕^2𝑗𝐵𝑑𝑥=−_𝑖,𝑗𝜕^𝑗𝐼𝑢_𝑖𝜕_𝑖𝐵𝜕_𝑗𝐵𝑑𝑥,₄=−_𝑖,𝑗𝐵^𝑖𝜕_𝑖𝑢𝜕^2𝑗𝐵𝑑𝑥=_𝑖,𝑗𝜕^𝑗𝐵_𝑖𝜕_𝑖𝑢𝜕_𝑗𝐵𝑑𝑥+_𝑖,𝑗𝐵^𝑖𝜕_𝑗𝐵𝜕_𝑖𝜕_𝑗𝑢𝑑𝑥.(3.2)

Inserting the above estimates into (3.1), noting that the last term of 𝐼₂ and 𝐼₄ disappeared, we have12𝑑𝑑𝑡𝑣²+𝐻²𝑑𝑥+||∇𝑣||²+||∇𝐻||²𝑑𝑥=−_𝑖,𝑗𝜕^𝑖𝑢𝜕_𝑗𝑢𝜕_𝑗𝑢_𝑖𝑑𝑥+_𝑖,𝑗𝜕^𝑗−𝑢_𝑖𝜕_𝑖𝐵𝜕_𝑗𝐵_𝑖𝑑𝑥_𝑖,𝑗𝜕^𝑗𝑢_𝑖𝜕_𝑖𝐵𝜕_𝑗𝐵𝑑𝑥+_𝑖,𝑗𝜕^𝑖=𝐽.𝑢𝜕_𝑗𝐵𝜕_𝑗𝐵_𝑖𝑑𝑥(3.3)

The proofs of the cases (1.11) and (1.13) are similar, we omit the details here.(I)Let (1.14) hold true,𝐽≤𝐶‖∇𝑢‖_𝐿^𝑞‖∇(𝑢,𝐵)‖^2𝐿_{2𝑞/(𝑞−1)}≤𝐶‖∇𝑢‖_𝐿^𝑞‖Δ(𝑢,𝐵)‖^{𝐿2(1−𝜃)}₂‖Δ∇(𝑢,𝐵)‖^𝐿2𝜃₂≤𝐶‖∇𝑢‖_𝐿^𝑞‖(𝑣,𝐻)‖^{𝐿2(1−𝜃)}₂‖∇(𝑣,𝐻)‖^𝐿2𝜃₂≤12‖∇(𝑣,𝐻)‖^2𝐿2+𝐶‖∇𝑢‖^{𝐿1/(1−𝜃)}_𝑞‖(𝑣,𝐻)‖^2𝐿₂.(3.4)

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.(3.5)

Here we have used the Gagliardo-Nirenberg inequality:‖𝑤‖_𝐿^{2𝑞/(𝑞−1)}≤𝐶‖∇𝑤‖^𝐿1−𝜃₂‖Δ𝑤‖^𝜃𝐿₂(3.6) with −𝑞−12𝑞𝑛=2(𝑛1−𝜃)1−2𝑛+𝜃2−2𝑛,𝑛−2≤𝑞2−𝑞1≤𝑛2−𝑛4.(3.7)(II)Let (1.12) hold true.

Integrating by parts, using (1.4) we have𝐽=_𝑖,𝑗𝑢𝜕^𝑖𝜕_𝑗𝑢𝜕_𝑗𝑢_𝑖𝑑𝑥−_𝑖,𝑗𝑢^𝑖𝜕_𝑗𝜕_𝑖𝐵𝜕_𝑗𝐵_𝑖𝑑𝑥+_𝑖,𝑗𝑢^𝑖𝜕_𝑗𝜕_𝑖𝐵𝜕_𝑗𝐵𝑑𝑥−_𝑖,𝑗𝑢𝜕^𝑖≤𝐶‖𝑢‖_𝐿𝑞𝜕_𝑗𝐵𝜕_𝑗𝐵_𝑖𝑑𝑥‖∇(𝑢,𝐵)‖_𝐿^𝑡1‖Δ(𝑢,𝐵)‖_𝐿^𝑡21𝑞+𝑡1₁+1𝑡₂≤𝐶‖𝑢‖_𝐿𝑞=1‖Δ(𝑢,𝐵)‖^1−𝜃1_𝐿2‖∇Δ(𝑢,𝐵)‖^𝜃1_𝐿2‖Δ(𝑢,𝐵)‖^1−𝜃2_𝐿2‖∇Δ(𝑢,𝐵)‖^𝜃2_𝐿2=𝐶‖𝑢‖_𝐿^𝑞‖Δ(𝑢,𝐵)‖^{2−𝜃1−𝜃2}_𝐿2‖∇Δ(𝑢,𝐵)‖^𝜃1+𝜃2_𝐿2≤𝐶‖𝑢‖_𝐿^𝑞‖(𝑣,𝐻)‖^{2−𝜃1−𝜃2}_𝐿2‖∇(𝑣,𝐻)‖^𝜃1+𝜃2_𝐿2≤12‖∇(𝑣,𝐻)‖^2𝐿2+𝐶‖𝑢‖^{2/(2−𝜃1−𝜃2)𝐿}_𝑞‖(𝑣,𝐻)‖^2𝐿₂,(3.8) which yields (3.5).

Here we have used the Gagliardo-Nirenberg inequalities:‖∇(𝑢−,𝐵𝑛)‖_𝐿^𝑡1≤𝐶‖Δ(𝑢,𝐵)‖^1−𝜃1_𝐿2‖∇Δ(𝑢,𝐵)‖^𝜃1_𝐿2,𝑡₁=1−𝜃₁1−𝑛‖Δ(𝑢,𝐵)‖_𝐿𝑡22+𝜃¹2−2𝑛−𝑛≤𝐶‖Δ(𝑢,𝐵)‖^1−𝜃2_𝐿2‖∇Δ(𝑢,𝐵)‖^𝜃2_𝐿2,𝑡₂=1−𝜃₂−𝑛2+𝜃²1−2𝑛.(3.9)

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).