A Regularity Criterion for the Magneto-Micropolar Fluid Equations in 𝐵̇⁻¹_∞,∞

Abstract

The paper is dedicated to study of the Cauchy problem for the magneto-micropolar fluid equations in three-dimensional spaces. A new logarithmically improved regularity criterion for the magneto-micropolar fluid equations is established in terms of the pressure in the homogeneous Besov space 𝐵̇⁻¹_∞,∞.

1. Introduction

This paper concerns with the regularity of weak solutions to the magneto-micropolar fluid equations in three dimensions as 𝜕_𝑡𝑣−𝜇+𝜒Δ𝑣+𝑣⋅∇𝑣−𝑏⋅∇𝑏+∇𝑝+𝑏²𝜕−𝜒∇×𝜔=0,_𝑡𝜔−𝛾Δ𝜔−𝜅∇div𝜕𝜔+2𝜒𝜔+𝑣⋅∇𝜔−𝜒∇×𝑣=0,_𝑡𝑏−𝜈Δ𝑏+𝑣⋅∇𝑏−𝑏⋅∇𝑣=0,div𝑣=div𝑣()𝑏=0,0,𝑥=𝑣₀(𝑥),𝜔(0,𝑥)=𝜔₀(𝑥),𝑏(0,𝑥)=𝑏₀(𝑥), (1) where 𝑣(𝑡,𝑥)=(𝑣₁(𝑡,𝑥),𝑣₂(𝑡,𝑥),𝑣₃(𝑡,𝑥))∈ℝ³ denotes the velocity of the fluid at a point 𝑥∈ℝ³,𝑡∈[0,𝑇), 𝜔(𝑡,𝑥)∈ℝ³,𝑏(𝑡,𝑥)∈ℝ³, and 𝑝(𝑡,𝑥)∈ℝ denote, respectively, the microrotational velocity, the magnetic field, and the hydrostatic pressure. 𝜇,𝜒,𝜅,𝛾,𝜈 are positive numbers associated to properties of the material: 𝜇 is the kinematic viscosity, 𝜒 is the vortex viscosity, 𝜅 and 𝛾 are spin viscosities, and 1/𝜈 is the magnetic Reynold. 𝑢₀,𝜔₀,𝑏₀ are initial data for the velocity, the angular velocity, and the magnetic field with properties div𝑢₀=0 and div𝑏₀=0. For more detailed background, we refer the readers to [1–3].

As we know, the problem of global regularity or finite time singularity for the weak solutions of the magneto-micropolar fluid equations model with large initial data still remains unsolved since (1) includes the 3D Navier-Stokes equations. It is of interest that the regularity of the weak solutions is under preassumption of certain growth conditions. There are a lot of lectures to study the regularity of weak solutions of the magneto-micropolar fluid equations (see, [4–6]). The purpose of this paper is to establish a new logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure in Besov space 𝐵̇⁻¹_∞,∞. Now we state the main results as follows.

Theorem 1. Let (𝑣₀(𝑥),𝜔₀(𝑥),𝑏₀(𝑥))∈𝐻¹(ℝ³). Let 𝑇>0 and (𝑣,𝜔,𝑏) be a weak solution to the system (1). If the pressure filed 𝑃 satisfies the following condition: ^𝑇₀‖𝑃(𝑡,⋅)‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑃(𝑡,⋅)‖_̇𝐵⁻¹_∞,∞𝑑𝑡<∞, (2) then the weak solution (𝑣,𝜔,𝑏) is regular on [0,𝑇].

Remark 2. Since the space 𝐵̇⁻¹_∞,∞ is wider than ℳ̇_2,3, so our result resolves the limit case 𝑟=1 in [7], which greatly improves the result in [7].

Remark 3. Since the space 𝐵̇⁻¹_∞,∞ is wider than 𝐿^3/𝑟,∞, hence our result extends and improves the recent results given by [4].

2. Preliminaries and Lemmas

Throughout this paper, we introduce some function spaces, notations, and important inequalities.

Let 𝑒^𝑡Δ denote the heat semigroup defined by 𝑒^𝑡Δ𝑓=𝐾_𝑡∗𝑓,𝐾_𝑡=(4𝜋𝑡)^−3/2−|𝑥|exp²4𝑡 (3) for 𝑡>0 and 𝑥∈ℝ³, where ∗ denotes the convolution of functions defined on ℝ³.

We now recall the definition of the homogeneous Besov space with negative indices 𝐵̇^−𝛼_∞,∞ on ℝ^𝑛 and the homogeneous Sobolev space 𝐻̇^𝛼_𝑞 of exponent 𝛼>0. It is known (p. 192 of [8]) that 𝑓∈𝒮^′(ℝ³) belongs to 𝐵̇^−𝛼_∞,∞ if and only if 𝑒^𝑡Δ∈𝐿^∞ for all 𝑡>0 and 𝑡^𝛼/2‖𝑒^𝑡Δ‖_∞∈𝐿^∞(0,∞;𝐿^∞). The norm of 𝐵̇^−𝛼_∞,∞ is defined, up to equivalence, by 𝑓_{̇𝐵−𝛼∞,∞}=sup_𝑡>0𝑡^𝛼/2𝑒^𝑡Δ_∞. (4)

We introduce now the homogeneous Sobolev space 𝐻̇^𝛼_𝑞(ℝ³), which is defined by the set of functions 𝑓∈𝐿^𝑟(ℝ³),1/𝑟=(1/𝑞)−(𝑠/3) such that (−Δ)^𝑠/2𝑓∈𝐿^𝑞(ℝ³). This space is endowed with the norm 𝑓_{̇𝐻𝛼𝑞}=(−Δ)^𝑠/2𝑓_𝐿𝑞, (5) and when 𝑞=2, we just let 𝐻̇^𝛼₂(ℝ³)=𝐻̇^𝛼(ℝ³). Additionally, we have the following inclusion relations (see, e.g., [9]): 𝐻̇^1/2ℝ³⊂𝐿³ℝ³⊂𝐿^3,∞ℝ³⊂𝐵̇⁻¹_∞,∞ℝ³,𝐻̇^1/2ℝ³⊂𝐿³ℝ³⊂ℳ̇_2,3ℝ³⊂𝐵̇⁻¹_∞,∞ℝ³ (6) with continuous injection.

Lemma 4 (see [10]). Let 1<𝑝<𝑞<∞ and 𝑠=𝛼((𝑞/𝑝)−1)>0. Then there exists a constant 𝐶 depending only on 𝛼, 𝑝, and 𝑞 such that for all 𝑓∈𝐻̇^𝛼_𝑝(ℝ³)∩𝐵̇^−𝛼_∞,∞(ℝ³), 𝑓_𝐿𝑞≤𝐶(−Δ)^𝑠/2𝑓_𝐿^𝑝/𝑞_𝑝𝑓_̇𝐵^{1−(𝑝/𝑞)}_{−𝛼∞,∞}. (7)

In particular, for 𝑠=1,𝑝=2, and 𝑞=4, we get 𝛼=1 and 𝑓_𝐿4≤𝐶𝑓_̇𝐻^1/2₁𝑓_̇𝐵^1/2_−1∞,∞. (8)

Lemma 5 (see [11]). Let 𝑓∈𝑊^1,𝑠(ℝ³)(𝑠≥1), and 𝑟≥1, then there exists a positive constant 𝐶 independent of 𝑓 such that 𝑓_𝐿𝛾≤𝐶𝑓_𝐿^1−𝛼₂∇𝑓^𝛼_𝐿2, (9) where ()−𝛼=1/𝑟1/𝛾()−()−().1/31/𝑠1/𝑟 (10)

3. Proof of Theorem 1

For given initial data (𝑣₀,𝜔₀,𝑏₀)∈𝐻¹(ℝ³), the weak solution is the same as the local strong solution (𝑣,𝜔,𝑏) in a local interval (0,𝑇) as in the discussion of Navier-Stokes equations. For the uniqueness and existence of local strong solution, we refer to [1]. Thus, it proves that Theorem 1 is reduced to establish a priori estimates uniformly in (0,𝑇) for strong solutions. With the use of the a priori estimates, the local strong solution (𝑣,𝜔,𝑏) can be continuously extended to 𝑡=𝑇 by a standard process to obtain global regularity of the weak solution. Therefore, we assume that the solution (𝑣,𝜔,𝑏) is sufficiently smooth on (0,𝑇).

Proof of Theorem 1. We show that Theorem 1 holds under condition (1). To prove the theorem, we need the 𝐿⁴-estimate. For this purpose, taking the inner product of the first equation of (1) with |𝑢|²𝑢 and integrating by parts, it can be deduced that 14𝑑‖𝑢‖𝑑𝑡^4𝐿4+𝜇+𝜒‖|∇𝑢||𝑢|‖²_𝐿2+12𝜇+𝜒∇|𝑢|²²_𝐿2≤2_ℝ3|𝑃||𝑢|²|∇𝑢|𝑑𝑥+3𝜒_ℝ3|𝑤||𝑢|²||−∇𝑢𝑑𝑥_ℝ3|𝑏|∇|𝑢|²𝑢|𝑏|𝑑𝑥, (11) where we used the following relations by the divergence-free condition div𝑢=0: 𝑢⋅∇𝑢⋅|_ℝ3𝑢|²1𝑢𝑑𝑥=2_ℝ³𝑢⋅∇|𝑢|⁴𝑑𝑥=0,_ℝ3Δ𝑢⋅|𝑢|²𝑢𝑑𝑥=−_ℝ3|∇𝑢|²|𝑢|²1𝑑𝑥−2_ℝ³∇|𝑢|²²𝑑𝑥,_ℝ3∇×𝜔⋅|𝑢|²𝑢𝑑𝑥=−_ℝ3|𝑢|²𝜔⋅∇×𝑢𝑑𝑥−_ℝ3𝜔⋅∇|𝑢|²||≤||,|∇|𝑢≤||.×𝑢𝑑𝑥,∇×𝑢∇𝑢||∇𝑢 (12) Similarly, taking the inner product of the second equation of (1) with |𝜔|²𝜔 and integrating by parts, it can be inferred that 14𝑑‖𝜔‖𝑑𝑡^4𝐿4+𝛾‖|∇𝜔||𝜔|‖²_𝐿2+𝛾2∇|𝜔|²²_𝐿2+𝑘‖div𝜔‖²_𝐿2+2𝜒‖𝜔‖⁴_𝐿4≤3𝜒_ℝ3|𝑢||𝜔|²|∇𝜔|𝑑𝑥. (13) Using an argument similar to that used in deriving the estimate (11)–(13), it can be obtained for the third equation of (1) that 14𝑑‖𝑏‖𝑑𝑡^4𝐿4+‖|∇𝑏||𝑏|‖²_𝐿2+2‖∇|𝑏||𝑏|‖²_𝐿2≤_ℝ3|𝑏|∇|𝑏|²𝑏|𝑢|𝑑𝑥. (14) Adding up (11), (13), and (14), then we obtain 14𝑑‖𝑢‖𝑑𝑡^4𝐿4+‖𝜔‖⁴_𝐿4+‖𝑏‖⁴_𝐿4+𝜇+𝜒‖|∇𝑢||𝑢|‖²_𝐿2+12𝜇+𝜒∇|𝑢|²²_𝐿2+𝛾‖|∇𝜔||𝜔|‖²_𝐿2+𝛾2∇|𝜔|²²_𝐿2+𝑘‖div𝜔‖²_𝐿2+2𝜒‖𝜔‖⁴_𝐿4+‖|∇𝑏||𝑏|‖²_𝐿2+2‖∇|𝑏||𝑏|‖²_𝐿2≤2_ℝ3|𝑃||𝑢|²|∇𝑢|𝑑𝑥+3𝜒_ℝ3|𝑤||𝑢|²|∇𝑢|𝑑𝑥+3𝜒_ℝ3|𝑢||𝜔|²|∇𝜔|𝑑𝑥−_ℝ3|𝑏|∇|𝑢|²𝑢𝑏|+|𝑑𝑥_ℝ3|𝑏|∇|𝑏|²𝑏|𝑢|𝑑𝑥≜𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅. (15) Applying the Hölder inequality and the Young inequality for 𝐼₂, it follows that 𝐼₂≤𝜒+𝜇2‖|∇𝑢||𝑢|‖²_𝐿2+𝐶‖𝑢‖⁴_𝐿4+‖𝜔‖⁴_𝐿4. (16) Arguing similarly to above, it can be derived for 𝐼₃ that 𝐼₃≤𝛾2‖|∇𝜔||𝜔|‖²_𝐿2+𝐶‖𝑢‖⁴_𝐿4+‖𝜔‖⁴_𝐿4. (17) Considering the term 𝐼₁, by virtue of the Cauchy inequality, we have 𝐼₁≤12_ℝ³∇|𝑣|²²1𝑑𝑥+2_ℝ³|𝑃|²|𝑣|²𝑑𝑥. (18) Let us bound the integral (1/2)∫_ℝ3|𝑃|²|𝑣|²𝑑𝑥. Applying the divergence operator div to the first equation of (1), one formally has 𝑃=∑³_𝑖,𝑗=1𝑅_𝑖𝑅_𝑗(𝑢_𝑖𝑢_𝑗−𝑏_𝑖𝑏_𝑗), where 𝑅_𝑗 denotes the 𝑗th Riesz operator. By the Calderon-Zygmund inequality, we have ‖∇𝑃‖_𝐿²≤𝐶‖|𝑣||∇𝑣|‖_𝐿²+‖|𝑏||∇𝑏|‖_𝐿². (19) With the help of (8) and (19), by the Hölder inequality and the Young inequality, we deduce that 12_ℝ³|𝑃|²|𝑣|²≤1𝑑𝑥2‖𝑃‖²_𝐿4‖𝑣‖²_𝐿4≤𝐶‖∇𝑃‖_𝐿²‖𝑃‖_̇𝐵⁻¹_∞,∞‖𝑣‖²_𝐿4≤𝐶‖|𝑣||∇𝑣|‖_𝐿²+‖|𝑏||∇𝑏|‖_𝐿²‖𝑃‖_̇𝐵⁻¹_∞,∞‖𝑣‖²_𝐿4=‖|𝑣||∇𝑣|‖_𝐿²+‖|𝑏||∇𝑏|‖_𝐿²𝐶‖𝑃‖²_{̇𝐵−1∞,∞}‖𝑣‖⁴_𝐿4^1/2≤14‖|𝑣||∇𝑣|‖²_𝐿2+‖|𝑏||∇𝑏|‖²_𝐿2+𝐶‖𝑃‖²_{̇𝐵−1∞,∞}‖𝑣‖⁴_𝐿4. (20) So the term 𝐼₁ can be estimated as 𝐼₁≤12_ℝ³∇|𝑣|²²1𝑑𝑥+4‖|𝑣||∇𝑣|‖²_𝐿2+‖|𝑏||∇𝑏|‖²_𝐿2+𝐶‖𝑃‖²_{̇𝐵−1∞,∞}‖𝑣‖⁴_𝐿4. (21) Next we have the following estimate for the term 𝐼₄: 𝐼₄≤_ℝ3|𝑏|²|𝑢|∇|𝑢|²𝑑𝑥. (22) Since 𝑢∈𝐿²(ℝ³)∩𝐿⁶(ℝ³) and using Cauchy inequality, generalized Hölder inequality, Gagliardo-Nirenberg inequality, and Sobolev imbedding theorem, we obtain 𝐼₄≤𝐶|𝑏|²|𝑢|_𝐿2∇|𝑢|²_𝐿2≤𝐶|𝑏|²|𝑢|²_𝐿2+𝜒+𝜇4∇|𝑢|²²_𝐿2≤𝐶|𝑏|²²_𝐿6‖𝑢‖²_𝐿3+𝜒+𝜇4∇|𝑢|²²_𝐿2≤𝐶∇|𝑏|²²_𝐿2‖𝑢‖_𝐿²‖∇𝑢‖_𝐿²+𝜒+𝜇4∇|𝑢|²²_𝐿2≤𝐶‖|𝑏|∇|𝑏|‖²_𝐿2+𝜒+𝜇4∇|𝑢|²²_𝐿2. (23) The last term of (15) can be treated in the same way as 𝐼₅≤𝐶_ℝ3|𝑏|²|𝑢|∇|𝑏|²𝑑𝑥≤𝐶|𝑏|²|𝑢|²_𝐿2+18∇|𝑏|²_𝐿2≤𝐶‖|𝑏|∇|𝑏|‖²_𝐿2. (24) Inserting the estimates (15) and (21) into (14), it follows that 𝑑‖𝑣‖𝑑𝑡⁴⁴+‖𝜔‖⁴₄+‖𝑏‖⁴₄≤𝐶‖𝑃‖²_{̇𝐵−1∞,∞}‖𝑣‖⁴_𝐿4+𝐶‖𝑣‖⁴₄+‖𝜔‖⁴₄+‖𝑏‖⁴₄≤𝐶‖𝑃‖²_{̇𝐵−1∞,∞}‖𝑣‖⁴₄+‖𝜔‖⁴₄+𝐶‖𝑣‖⁴₄+‖𝜔‖⁴₄+‖𝑏‖⁴₄‖𝑃(≤𝐶1+𝑡,⋅)‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑃(𝑡,⋅)‖_̇𝐵⁻¹_∞,∞×1+ln𝑒+‖𝑃(𝑡,⋅)‖_̇𝐵⁻¹_∞,∞‖𝑣‖⁴₄+‖𝜔‖⁴₄+‖𝑏‖⁴₄‖𝑃(≤𝐶1+𝑡,⋅)‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑃(𝑡,⋅)‖_̇𝐵⁻¹_∞,∞×1+ln𝑒+‖𝑃(𝑡,⋅)‖_𝐿³‖𝑣‖⁴₄+‖𝜔‖⁴₄+‖𝑏‖⁴₄‖𝑃(≤𝐶1+𝑡,⋅)‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑃(𝑡,⋅)‖_̇𝐵⁻¹_∞,∞×1+ln𝑒+‖𝑣(𝑡,⋅)‖²_𝐿6‖𝑣‖⁴₄+‖𝜔‖⁴₄+‖𝑏‖⁴₄‖𝑃(≤𝐶1+𝑡,⋅)‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑃(𝑡,⋅)‖_̇𝐵⁻¹_∞,∞×1+ln𝑒+𝑦(𝑡)‖𝑣‖⁴₄+‖𝜔‖⁴₄+‖𝑏‖⁴₄, (25) where 𝑦(𝑡) is defined by 𝑦(𝑡)=∶sup_{𝑇0≤𝑠≤𝑡}Λ³𝑣²_𝐿2+Λ³𝜔²_𝐿2+Λ³𝑏²_𝐿2. (26) Applying Gronwall's inequality on (25) for the interval [𝑇₀,𝑡], one has sup_{𝑇0≤𝑠≤𝑡}‖𝑣‖⁴₄+‖𝜔‖⁴₄+‖𝑏‖⁴₄≤𝐶₀exp𝐶𝜀1+ln𝑒+𝑦(𝑡)≤𝐶₀exp2𝐶𝜀ln𝑒+𝑦(𝑡)≤𝐶₀𝑒+𝑦(𝑡)^2𝐶𝜀 (27) provided that ^𝑡_𝑇0‖𝑃(𝑡,⋅)‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑃(𝑡,⋅)‖_̇𝐵⁻¹_∞,∞𝑑𝑠<𝜀≪1, (28) where 𝐶₀ is a positive constant depending on 𝑇₀.
Next we will estimate the 𝐿²-norm of ∇𝑣, ∇𝜔, and ∇𝑏. We multiply both sides of the first equation of (1) by (−Δ𝑣), the second equation of (1) by (−Δ𝜔), and the third equation of (1) by (−Δ𝑏), by integration by parts over ℝ³, we get 12𝑑‖‖𝑑𝑡∇𝑣²_𝐿2+‖Δ𝑣‖²_𝐿2=_ℝ3()+𝑣⋅∇𝑣⋅Δ𝑣𝑑𝑥_ℝ3(𝑏⋅∇)𝑏⋅Δ𝑣𝑑𝑥−_ℝ3≤‖𝑣‖curl𝜔Δ𝑣𝑑𝑥_𝐿4‖∇𝑣‖_𝐿⁴‖Δ𝑣‖_𝐿²+‖𝑏‖_𝐿⁴‖∇𝑏‖_𝐿⁴‖Δ𝑣‖_𝐿²+‖∇𝜔‖_𝐿²‖Δ𝑣‖_𝐿²≤‖𝑣‖_𝐿⁴‖𝑣‖_𝐿^1/8₂‖Δ𝑣‖_𝐿^7/8₂‖Δ𝑣‖_𝐿²+‖𝑏‖_𝐿⁴‖𝑏‖_𝐿^1/8₂‖Δ𝑏‖_𝐿^7/8₂‖Δ𝑣‖_𝐿²+1‖‖16Δ𝑣²_𝐿2+𝐶‖∇𝜔‖²_𝐿2≤18‖Δ𝑣‖²_𝐿2+18‖Δ𝜔‖²_𝐿2+18‖Δ𝑏‖²_𝐿2+𝐶‖𝑏‖_𝐿¹⁶₄‖𝑏‖²_𝐿2+𝐶‖𝑣‖_𝐿¹⁶₄‖𝑣‖²_𝐿2+𝐶‖𝜔‖²_𝐿2,12𝑑‖‖𝑑𝑡∇𝜔²_𝐿2+‖Δ𝜔‖²_𝐿2+‖∇div𝜔‖²_𝐿2+2‖∇𝜔‖²_𝐿2=()≤‖𝑣‖𝑣⋅∇𝜔⋅Δ𝜔𝑑𝑥−curl𝑣Δ𝜔𝑑𝑥_𝐿4‖∇𝜔‖_𝐿⁴‖Δ𝜔‖_𝐿²+‖∇𝑣‖_𝐿²‖Δ𝜔‖_𝐿²≤‖𝑣‖_𝐿⁴‖𝜔‖_𝐿^1/8₂‖Δ𝜔‖_𝐿^7/8₂‖Δ𝜔‖_𝐿²+𝐶‖𝑣‖_𝐿^1/2₂‖Δ𝑣‖_𝐿^1/2₂‖Δ𝜔‖_𝐿²≤18‖Δ𝑣‖²_𝐿2+18‖Δ𝜔‖²_𝐿2+𝐶‖𝑣‖_𝐿¹⁶₄‖𝜔‖²_𝐿2+𝐶‖𝑣‖²_𝐿2,12𝑑‖‖𝑑𝑡∇𝑏²_𝐿2+‖Δ𝑏‖²_𝐿2=_ℝ3(𝑣⋅∇𝑏)Δ𝑏𝑑𝑥−_ℝ3()≤‖𝑣‖𝑏⋅∇𝑣Δ𝑏𝑑𝑥_𝐿4‖∇𝑏‖_𝐿⁴‖Δ𝑏‖_𝐿²+‖𝑏‖_𝐿⁴‖∇𝑣‖_𝐿⁴‖Δ𝑏‖_𝐿²≤‖𝑣‖_𝐿⁴‖𝑏‖_𝐿^1/8₂‖Δ𝑏‖_𝐿^7/8₂‖Δ𝑣‖_𝐿²+‖𝑏‖_𝐿⁴‖𝑣‖_𝐿^1/8₂‖Δ𝑣‖_𝐿^7/8₂‖Δ𝑏‖_𝐿²≤18‖Δ𝑣‖²_𝐿2+18‖Δ𝑏‖²_𝐿2+𝐶‖𝑣‖_𝐿¹⁶₄‖𝑏‖²_𝐿2+𝐶‖𝑏‖_𝐿¹⁶₄‖𝑣‖²_𝐿2, (29)(30)(31) where we have used the Gagliardo-Nirenberg inequality: ∇𝑓_𝐿4≤𝐶𝑓_𝐿^1/8₂Δ𝑓_𝐿^7/8₂. (32) Combining (29), (30), and (31) and using the definition of the weak solution, we deduce that ‖∇𝑣‖²_𝐿2+‖∇𝜔‖²_𝐿2+‖∇𝑏‖²_𝐿2≤𝐶𝐶₀𝑒+𝑦(𝑡)^6𝐶𝜀𝑡−𝑇₀+∇𝑣⋅,𝑇₀²_𝐿2+∇𝜔⋅,𝑇₀²_𝐿2. (33) Finally we go to the estimate for 𝐻³-norm of 𝑣, 𝜔, and 𝑏. In the following calculations, we will use the following commutator estimate due to Kato and Ponce [12]: Λ^𝑠𝑓𝑔−𝑓Λ^𝑠𝑔_𝐿𝑝≤∇𝑓_𝐿𝑝1Λ^𝑠−1𝑔_𝐿𝑞1+Λ^𝑠𝑓_𝐿𝑝2𝑔_𝐿𝑞2, (34) with 𝑠>0, Λ^𝑠=(−Δ)^𝑠/2 and (1/𝑝)=(1/𝑝₁)+(1/𝑞₁)=(1/𝑝₂)+(1/𝑞₂). Taking the operation Λ³ on both sides of (1), then multiplying them by Λ³𝑣, Λ³𝜔, and Λ³𝑏, and integrating by parts over ℝ³, we have 12𝑑𝑑𝑡_ℝ³Λ³𝑣²+Λ³𝜔²+Λ³𝑏²+𝑑𝑥_ℝ3Λ⁴𝑣²𝑑𝑥+_ℝ3Λ⁴𝜔²𝑑𝑥+_ℝ3Λ⁴𝑏²+𝑑𝑥_ℝ3Λ³div𝑣²𝑑𝑥+2_ℝ3Λ³𝜔²𝑑𝑥=−_ℝ3Λ³(𝑣⋅∇𝑣)−𝑣⋅∇Λ³𝑣Λ³+𝑣𝑑𝑥_ℝ3Λ³curl𝜔⋅Λ³−𝑣𝑑𝑥_ℝ3Λ³(𝑣⋅∇𝜔)−𝑣⋅∇Λ³𝜔Λ³+𝜔𝑑𝑥_ℝ3Λ³curl𝑣⋅Λ³+𝜔𝑑𝑥_ℝ3Λ³(𝑏⋅∇𝑏)−𝑏⋅∇Λ³𝑏Λ³𝑣𝑑𝑥−_ℝ3Λ³(𝑣⋅∇𝑏)−𝑣⋅∇Λ³𝑏Λ³𝑏𝑑𝑥+_ℝ3Λ³(𝑏⋅∇𝑣)−𝑏⋅∇Λ³𝑣Λ³𝑏𝑑𝑥≡𝐴₁+𝐴₂+𝐴₃+𝐴₄+𝐴₅+𝐴₆+𝐴₇. (35) Hence 𝐴₁ can be estimated as 𝐴₁≤𝐶‖∇𝑣‖_𝐿³Λ³𝑣²_𝐿3≤𝐶‖∇𝑣‖_𝐿^13/2₂Λ³𝑣_𝐿^1/4₂Λ⁴𝑣_𝐿^5/3₂≤16Λ⁴𝑣²_𝐿2+𝐶‖∇𝑣‖_𝐿^13/2₂Λ³𝑣_𝐿^3/2₂, (36) where we used (33) with 𝑠=3,𝑝=3/2,𝑝₁=𝑞₁=𝑝₂=𝑞₂=3 and the following inequalities: ‖∇𝑣‖_𝐿³≤𝐶‖∇𝑣‖_𝐿^3/4₂Λ³𝑣_𝐿^1/4₂,Λ³𝑣_𝐿3≤𝐶‖∇𝑣‖_𝐿^1/6₂Λ⁴𝑣_𝐿^5/6₂. (37) If we use the existing estimate (31) for 𝑇₀<𝑡<𝑇, (36) reduces to 𝐴₁≤16Λ⁴𝑣²_𝐿2+𝐶𝐶₀𝑒+𝑦(𝑡)^{(3/4)+(39/2)𝐶𝜀}. (38) Using (37) again, we have 𝐴₃+𝐴₅+𝐴₆+𝐴₇≤16Λ⁴𝑣²_𝐿2+Λ⁴𝜔²_𝐿2+Λ⁴𝑏²_𝐿2+𝐶𝐶₀𝑒+𝑦(𝑡)^{(3/4)+(39/2)𝐶𝜀}. (39) For 𝐴₂ and 𝐴₄, we have 𝐴₂+𝐴₄≤16Λ⁴𝑣²_𝐿2+Λ⁴𝜔²_𝐿2+𝐶Λ³𝑣²_𝐿2+Λ³𝜔²_𝐿2≤16Λ⁴𝑣²_𝐿2+Λ⁴𝜔²_𝐿2+𝐶𝐶₀𝑒+𝑦(𝑡). (40) Inserting the above estimates (38)–(40) into (35), we obtain 𝑑𝑑𝑡_ℝ³Λ³𝑣²+Λ³𝜔²𝑑𝑥≤𝐶𝐶₀𝑒+𝑦(𝑡)^{(3/4)+(39/2)𝐶𝜀}+𝐶𝐶₀𝑒+𝑦(𝑡). (41) Gronwall's inequality implies the boundness of 𝐻³-norm of 𝑣, 𝜔, and 𝑏 provided that 39𝐶𝜀<(1/2), which can be achieved by the absolute continuous property of integral (2). This completes the proof of Theorem 1.

Acknowledgment

The authors thank Professor Xiaohong Fan for his profitable discussion and suggestions.