Global Regularity Criterion for the Magneto-Micropolar Fluid Equations

Abstract

We are concerned with the magneto-micropolar fluid equations in ℝ³. Using Littlewood-Paley decomposition, we obtain an Osgood type global regularity criterion for the system.

1. Introduction

In this paper, we consider the following magneto-micropolar fluid equations in ℝ³: 𝜕_𝑡𝑢−𝜇+𝜒Δ𝑢+𝑢⋅∇𝑢−𝑏⋅∇𝑏+∇𝑝+𝑏²𝜕−𝜒∇×𝜔=0,_𝑡𝜕𝜔−𝛾Δ𝜔−𝜅∇div𝜔+2𝜒𝜔+𝑢⋅∇𝜔−𝜒∇×𝑢=0,_𝑡𝑢()𝑏−𝜈Δ𝑏+𝑢⋅∇𝑏−𝑏⋅∇𝑢=0,div𝑢=div𝑏=0,0,𝑥=𝑢₀(𝑥),𝜔(0,𝑥)=𝜔₀(𝑥),𝑏(0,𝑥)=𝑏₀(𝑥), (1) where 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),𝑢₂(𝑡,𝑥),𝑢₃(𝑡,𝑥))∈ℝ³ denotes the velocity of the fluid at a point 𝑥∈ℝ³,𝑡∈[0,𝑇), 𝜔(𝑡,𝑥)∈ℝ³and𝑏(𝑡,𝑥)∈ℝ³, and 𝑝(𝑡,𝑥)∈ℝ denote, respectively, the microrotational velocity, the magnetic field, and the hydrostatic pressure. 𝜇,𝜒,𝜅,𝛾,and𝜈 are positive numbers associated with 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,and𝑏₀ are initial data for the velocity, the angular velocity, and the magnetic field with properties div𝑢₀=0 and div𝑏₀=0.

It is well known that the question of global existence or finite time blowup of smooth solutions for the 3D incompressible Euler or Navier-Stokes equations has been one of the most outstanding open problems in applied analysis, as well as that for the 3D incompressible magneto-micropolar fluid equations. This challenging problem has attracted significant attention. Therefore, it is interesting to study the global regularity criterion of the solutions for system (1). But there are few theories about regularity and blow-up criteria of magneto-micropolar fluid equations. Some blow-up criterion are obtained by Yuan [1] in 2010. His paper implies that most classical blow-up criteria of smooth solutions to Navier-Stokes or magneto-hydrodynamic equations also hold for magneto-micropolar fluid equations. In particular, using Fourier frequency localization, Yuan proved the Beale-Kato-Majda criterion only relying on ∇𝑢; that is, if ^𝑇₀‖∇𝑢(𝑡)‖_̇𝐵⁰_∞,∞𝑑𝑡<∞, (2) then the solution (𝑢,𝜔,𝑏) can be extended past time 𝑡=𝑇. In 2008, Yuan [2] obtain the following blow-up criteria: if lim_𝜀→0sup_𝑗∈𝑍^𝑇_𝑇−𝜀Δ_𝑗(∇×𝑢)(𝑡)_∞𝑑𝑡<∞, (3) then the solution (𝑢,𝜔,𝑏) can be extended 𝑡=𝑇. Recently,  Xu [3] also studied the regularity of weak solutions to magneto-micropolar fluid equations in Besov spaces.

In this paper, we establish a refined global regularity criterion by means of Osgood norm which improves the results (2) and (3). As we know, Osgood condition plays an important role in solving uniqueness of solutions to the incompressible fluids equations. This induces us to apply it to global regularity criterion problems of smooth solution. To achieve this goal, taking full advantage of Fourier frequency localization method and using the low-high decomposition technique, we show the following main results.

Theorem 1. Suppose that for 𝑠>1/2, (𝑢,𝜔,𝑏)∈𝐶([0,𝑇);𝐻^𝑠)∩𝐶¹((0,𝑇);𝐻^𝑠)∩𝐶((0,𝑇);𝐻^𝑠+2) is the smooth solution to (1). If the following Osgood type condition: sup_2≤𝑞<∞^𝑇₀𝑆_𝑞∇𝑢(𝑡)_∞𝑞log𝑞𝑑𝑡<∞, (4) then the solution (𝑢,𝜔,𝑏) can be extended past time 𝑡=𝑇. Here one denotes that 𝑆_𝑞∶=∑^𝑞_𝑘=−𝑞Δ̇_𝑘.

Remark 2. The Osgood type condition (4) is weaker than (2) and (3). Note that, for 𝑞∈[2,∞), we have 𝑆_𝑞∇𝑢_∞≤∑𝑞log𝑞^{𝑞𝑘=−𝑞}Δ̇_𝑘(∇𝑢)_∞‖‖𝑞log𝑞≤𝐶∇𝑢_̇𝐵⁰_∞,∞. (5)

2. Preliminaries

The proof of the results presented in this paper is based on a dyadic partition of unity in Fourier variables, the so-called homogeneous Littlewood-Paley decomposition. So, we first introduce the Littlewood-Paley decomposition and review the so-called Beinstein estimate and commutator estimate, which are to be used in the proof of our theorem.

Let 𝒮(ℝ³) be the Schwartz class of rapidly decreasing functions. Given 𝑓∈𝒮(ℝ³), the Fourier transform of 𝑓 is defined by 𝑓(𝜉)=(2𝜋)^−(3/2)_ℝ3𝑒^{−𝑖𝑥⋅𝜉}𝑓(𝑥)𝑑𝑥. (6)

We consider that 𝜒,𝜑∈𝒮(ℝ³), respectively, support in 𝐵={𝜉∈ℝ³,|𝜉|≤4/3} and 𝒞={𝜉∈ℝ³,3/4≤|𝜉|≤8/3}, such that 𝜒(𝜉)+_𝑗≥0𝜑2^−𝑗𝜉=1,∀𝜉∈ℝ³,_𝑗∈ℤ𝜑2^−𝑗𝜉=1,∀𝜉∈ℝ³⧵{0}. (7) Setting 𝜑_𝑗=𝜑(2^−𝑗𝜉), then supp𝜑_𝑗∩supp𝜑^′_𝑗=∅ if |𝑗−𝑗^′|≥2 and supp𝜒∩supp𝜑^′_𝑗=∅ if |𝑗−𝑗^′|≥1. Let ℎ=𝐹⁻¹𝜑 and ℎ=𝐹⁻¹𝜒; the dyadic blocks are defined as follows: Δ_𝑗𝑓=𝜑2^−𝑗𝐷𝑓=2^3𝑗_ℝ3ℎ2^𝑗𝑦𝑓𝑆𝑥−𝑦𝑑𝑦,_𝑗𝑓=_{𝑘≤𝑗−1}Δ_𝑘𝑓=2^3𝑗_ℝ3ℎ2^𝑗𝑦𝑓𝑥−𝑦𝑑𝑦,𝑗∈ℤ. (8) Informally, Δ_𝑗=𝑆_𝑗+1−𝑆_𝑗 is a frequency projection to the annulus |𝜉|≈2^𝑗, while 𝑆_𝑗 is frequency projection to the ball |𝜉|≲2^𝑗. The details of Littlewood-Paley decomposition can be found in Triebel [4] and Chemin [5]. Now Besov spaces in ℝ³ can be defined as follows: 𝐵̇^𝑠_𝑝,𝑞=𝑓∈𝒵^′ℝ³∣𝑓_{̇𝐵𝑠𝑝,𝑞}=_𝑗∈ℤ2^𝑗𝑠𝑞Δ_𝑗𝑓^𝑞_𝑝^1/𝑞,𝐵̇<∞𝑞≠∞,_𝑠𝑝,∞=𝑓∈𝒵^′ℝ³∣𝑓_{̇𝐵𝑠𝑝,𝑞}=sup_𝑗∈ℤ2^𝑗𝑠Δ_𝑗𝑓_𝑝<∞, (9) where 𝒵^′ denotes the dual space of 𝒵={𝑓∈𝒮;𝐷^𝛼𝑓(0)=0;∀𝛼∈ℕ^𝑛multi-index}.

Now we introduce well-known Bernstetin's Lemma and commutator estimate, the proof is omitted here, and we can find the details in Chemin [5], Chemin and Lerner [6], and Kato and Ponce [7].

Lemma 3 (Bernstein's lemma). Let 1≤𝑝≤𝑞≤∞. Assume that 𝑓∈𝐿^𝑝, then there exist constants 𝐶,𝐶₁,and𝐶₂ independent of 𝑓,𝑗, such that sup_|𝛼|=𝑘𝜕^𝛼𝑓_𝑞≤𝐶2^{𝑗𝑘+3𝑗(1/𝑝−1/𝑞)}𝑓_𝑝supp𝑓⊂𝜉≲2^𝑗,𝐶₁2^𝑗𝑘𝑓_𝑝≤sup_|𝛼|=𝑘𝜕^𝛼𝑓_𝑝≤𝐶₂2^𝑗𝑘𝑓_𝑝supp𝑓⊂𝜉≈2^𝑗. (10)

Remark 4. From the above Beinstein estimate, we easily know that in ℝ³, for the Reisz transform 𝑅_𝑘(𝑘=1,2,3), it has for ∀1≤𝑝≤𝑞≤∞𝑅_𝑘Δ_𝑗𝑢_𝑞≤𝐶2^{3𝑗(1/𝑝−1/𝑞)}‖𝑢‖_𝑝. (11) If suppose vector valued funtion 𝑢 be divergence free, by Biot Savard law ∇𝑢=(−Δ)⁻¹∇∇×𝑣 with 𝑣=∇×𝑢 and the boundedness of Reisz transform on 𝐿^𝑝(1<𝑝<∞), we have, there exist constants 𝐶 independent 𝑢 such that ‖∇𝑢‖_𝑝≤𝐶‖𝑣‖_𝑝,∀1<𝑝<∞. (12) If the frequency of 𝑢 is restricted to annulus |𝜉|≈2^𝑗, then (11) implies that ‖∇𝑢‖_𝑝≤𝐶‖𝑣‖_𝑝,∀1≤𝑝≤∞. (13)

Now we denote that Λ=(𝐼−Δ)^1/2, which satisfies Λ𝑓(𝜉)=1+𝜉^21/2𝑓(𝜉). (14)Λ^𝑠(𝑠∈ℝ) can be defined in the same way as follows: Λ^𝑠𝑓(𝜉)=1+𝜉^2s/2𝑓(𝜉). (15) Using the perivious notation, we define the norm of Sobolev space 𝑊^𝑠,𝑝𝑓_{𝑊𝑠,𝑝}≜Λ^𝑠𝑓_𝐿𝑝, (16) especially by Fourier transform, and 𝐻^𝑠≜𝑊^𝑠,2 can be defined as 𝐻^𝑠≜𝑓∈𝒮^′ℝ³∣𝑓_𝐻𝑠<∞, (17) where 𝑓_𝐻𝑠≜Λ^𝑠𝑓_𝐿2ℝ31+𝜉^2𝑠𝑓(𝜉)²𝑑𝜉^1/2. (18)

Lemma 5 (Commutator estimate). Let 1<𝑝<∞,s>0; assume that 𝑓,𝑔∈𝑊^𝑠,𝑝, then there exists a constant 𝐶 independent of 𝑓,𝑔, such that Λ^𝑠,𝑓𝑔_𝐿𝑝≤𝐶∇𝑓_𝐿𝑝1𝑔_{𝑊2𝑠−1,𝑝}+𝑓_{𝑊3𝑠,𝑝}𝑔_𝐿𝑝4 (19) with 1<𝑝₂,𝑝₃<∞, such that 1𝑝=1𝑝₁+1𝑝₂=1𝑝₃+1𝑝₄. (20) Here [Λ^𝑠,𝑓]𝑔=Λ^𝑠(𝑓𝑔)−𝑓Λ^𝑠𝑔.

3. Proof of the Theorem 1

First we go on with the 𝐻¹ estimates of the solution (𝑢,𝜔,𝑏) under the condition (3). Denote that 𝐻=∇×𝑢,𝐼=∇×𝜔,  and 𝐽=∇×𝑏 we take curl on both sides of (1); we get the following equation: 𝜕_𝑡𝜕𝐻−𝜇+𝜒Δ𝐻+𝑢⋅∇𝐻−𝐻⋅∇𝑢−𝑏⋅∇𝐽+𝐽⋅∇𝑏−𝜒∇×𝐼=0,_𝑡𝜕𝐼−𝛾Δ𝐼+2𝜒𝐼+𝑢⋅∇𝐼−𝐻⋅∇𝜔−𝜒∇×𝐻=0,_𝑡𝐽−𝜈Δ𝐽+𝑢⋅∇𝐽−𝐻⋅∇𝑏−𝑏⋅∇𝐻+𝐽⋅∇𝑢=2𝑇(𝑏,𝑢) (21) with 𝑇()=𝜕𝑏,𝑢²𝑏⋅𝜕₃𝑢−𝜕₃𝑏⋅𝜕₂𝑢𝜕₃𝑏⋅𝜕₁𝑢−𝜕₁𝑏⋅𝜕₃𝑢𝜕₁𝑏⋅𝜕₂𝑢−𝜕₂𝑏⋅𝜕₁𝑢 (22) which uses the fact ∇×∇div𝜔=0.

Multiplying the three equations with (𝐻,𝐼,𝐽), respectively, integrating by parts over ℝ³ about the variable 𝑥, then adding the resulting equations yields that 12𝑑‖𝐻‖𝑑𝑡²²+‖𝐼‖²₂+‖𝐽‖²₂+𝜇+𝜒‖∇𝐻‖²₂+𝛾‖∇𝐼‖²₂+𝜈‖∇𝐽‖²₂+2𝜒‖𝐼‖²₂=_ℝ3(𝐻⋅∇)𝑢⋅𝐻𝑑𝑥+_ℝ3()+𝐻⋅∇𝜔⋅𝐼𝑑𝑥_ℝ3()−𝐽⋅∇𝑢⋅𝐽𝑑𝑥_ℝ3(𝐽⋅∇)𝑏⋅𝐻𝑑𝑥+_ℝ3(𝐻⋅∇)𝑏⋅𝐽𝑑𝑥+2𝜒_ℝ3(∇×𝐻)⋅𝐼𝑑𝑥+2_ℝ3𝑇(𝑏,𝑢)⋅𝐽𝑑𝑥=𝐼𝐼₁+𝐼𝐼₂+𝐼𝐼₃+𝐼𝐼₄+𝐼𝐼₅+𝐼𝐼₆+𝐼𝐼₇, (23) where we use the following fact due to the divergence free condition of 𝑢,𝑏: _ℝ3(𝑢⋅∇)𝐻⋅𝐻𝑑𝑥=_ℝ3()=𝑢⋅∇𝐼⋅𝐼𝑑𝑥_ℝ3(𝑢⋅∇)𝐽⋅𝐽𝑑𝑥=0,_ℝ3(𝑏⋅∇)𝐽⋅𝐻𝑑𝑥+_ℝ3(𝑏⋅∇)𝐻⋅𝐽𝑑𝑥=0,_ℝ3(∇×𝐻)⋅𝐼𝑑𝑥=_ℝ3(∇×𝐼)⋅𝐻𝑑𝑥. (24)

Let us begin with estimating 𝐼𝐼₁ and 𝐼𝐼₃. Using Littlewood-Paley decomposition to ∇𝑢, we have 𝐼𝐼₁=_{𝑗<−𝑁ℝ3}(𝐻⋅∇)Δ_𝑗𝑢⋅𝐻𝑑𝑥+_{−𝑁≤𝑗≤𝑁ℝ3}(𝐻⋅∇)Δ_𝑗𝑢⋅𝐻𝑑𝑥+_{𝑗>𝑁ℝ3}(𝐻⋅∇)Δ_𝑗𝑢⋅𝐻𝑑𝑥,=𝐼𝐼¹₁+𝐼𝐼²₁+𝐼𝐼³₁. (25) For the terms 𝐼𝐼¹₁ and 𝐼𝐼²₁, using Hölder's inequality, Beinstein's inequality, and (12), (13), we obtain 𝐼𝐼¹₁≤‖𝐻‖²_{2𝑗<−𝑁}∇Δ_𝑗𝑢_∞≤𝐶‖𝐻‖²_{2𝑗<−𝑁}2^(3/2)𝑗Δ_𝑗𝐻₂≤𝐶2^−(3/2)𝑁‖𝐻‖³₂,𝐼𝐼²₁≤‖𝐻‖²_{2−𝑁≤𝑗≤𝑁}∇Δ_𝑗𝑢_∞≤𝐶‖𝐻‖²₂𝑆_𝑁∇𝑢_∞, (26) for 𝐼𝐼³₁, we get 𝐼𝐼³₁≤‖𝐻‖²_3𝑗>𝑁∇Δ_𝑗𝑢₃≤𝐶‖𝐻‖²_3𝑗>𝑁2^𝑗/2Δ_𝑗𝐻₂≤𝐶‖𝐻‖²_3𝑗>𝑁2^{−(𝑗/2)⋅21/2}_𝑗>𝑁2^𝑗⋅2‖𝐻‖²₂^1/2≤𝐶2^−(𝑁/2)‖𝐻‖₂‖∇𝐻‖²₂, (27) where we use the interpolation inequality ‖𝐻‖₃≤𝐶‖𝐻‖₂^1/2‖∇𝐻‖₂^1/2. (28) Summing up (26)-(27), we have 𝐼𝐼₁≤𝐶2^−(3/2)𝑁‖𝐻‖³₂+‖𝐻‖²₂𝑆_𝑁∇𝑢_∞+2^−(N/2)‖𝐻‖₂‖∇𝐻‖²₂. (29)𝐼𝐼₃ can be treated in the same way, and we decompose it as 𝐼𝐼₃=_{𝑗<−𝑁ℝ3}(𝐽⋅∇)Δ_𝑗𝑢⋅𝐽𝑑𝑥+_{−𝑁≤𝑗≤𝑁ℝ3}(𝐽⋅∇)Δ_𝑗𝑢⋅𝐽𝑑𝑥+_{𝑗>𝑁ℝ3}(𝐽⋅∇)Δ_𝑗𝑢⋅𝐽𝑑𝑥, (30) then we obtain 𝐼𝐼₃≤𝐶2^−(3/2)𝑁‖𝐽‖²₂‖𝐻‖₂+‖𝐽‖²₂𝑆_𝑁∇𝑢_∞+2^−(N/2)‖𝐽‖₂‖∇𝐽‖₂‖∇𝐻‖₂. (31) Now we study 𝐼𝐼₂, we decompose 𝐻 by using Littlewood-Paley theory; that is, 𝐼𝐼₂=_{𝑗<−𝑁ℝ3}Δ_𝑗𝐻⋅∇𝜔⋅𝐼𝑑𝑥+_{−𝑁≤𝑗≤𝑁ℝ3}Δ_𝑗𝐻⋅∇𝜔⋅𝐼𝑑𝑥+_{𝑗>𝑁ℝ3}Δ_𝑗𝐻⋅∇𝜔⋅𝐼𝑑𝑥, (32) then 𝐼𝐼₂≤𝐶2^−(3/2)𝑁‖𝐼‖²₂‖𝐻‖₂+‖𝐼‖²₂𝑆_𝑁∇𝑢_∞+2^−(N/2)‖𝐼‖₂‖∇𝐼‖₂‖∇𝐻‖₂. (33) Similarly for 𝐼𝐼₄,𝐼𝐼₅, and 𝐼𝐼₇, we have 𝐼𝐼₄+𝐼𝐼₅+𝐼𝐼₇≤𝐶2^−(3/2)𝑁‖𝐽‖²₂‖𝐻‖₂+‖𝐽‖²₂𝑆_𝑁∇𝑢_∞+2^−(N/2)‖𝐽‖₂‖∇𝐽‖₂‖∇𝐻‖₂. (34) Using Young's inequality, the term 𝐼𝐼₆ can be written as 𝐼𝐼₆≤2𝜒‖∇×𝐻‖₂‖𝐼‖₂≤𝜒2‖∇𝐻‖²₂+2𝜒‖𝐼‖²₂. (35) Summing up (29)–(35) and taking the sum into (23), by Young's inequality, we get 𝑑‖𝐻‖𝑑𝑡²²+‖𝐼‖²₂+‖𝐽‖²₂+2𝜇+𝜒‖∇𝐻‖²₂+2𝛾∇𝐼²₂+2𝜈‖∇𝐽‖²₂≤𝐶2^−(3/2)𝑁‖𝐻‖³₂+‖𝐼‖³₂+‖𝐽‖³₂+𝑆_𝑁∇𝑢_∞‖𝐻‖²₂+‖𝐼‖²₂+‖𝐽‖²₂+2^−(N/2)‖𝐻‖₂+‖𝐼‖₂+‖𝐽‖₂×‖∇𝐻‖²₂+‖∇𝐼‖²₂+‖∇𝐽‖²₂. (36) If we let 2^−(N/2)(‖𝐻‖₂+‖𝐼‖₂+‖𝐽‖₂)≤min(𝜇,𝛾,𝜈), that is, if we choose 2𝑁≥log2log⁺𝐶‖𝐻‖min𝜇,𝛾,𝜈²+‖𝐼‖₂+‖𝐽‖₂+1, (37) where [𝑎] stands for the integral parts of 𝑎∈ℝ, log⁺(𝑥)=log(𝑥+𝑒), then we have 𝑑‖𝐻‖𝑑𝑡²²+‖𝐼‖²₂+‖𝐽‖²₂+𝜇+𝜒‖∇𝐻‖²₂+𝛾‖∇𝐼‖²₂+𝜈‖∇𝐽‖²₂≤𝐶𝑆_𝑁∇𝑢_∞‖𝐻‖²₂+‖𝐼‖²₂+‖𝐽‖²₂+𝐶≤𝐶𝑓_𝑁𝑁log𝑁‖𝐻‖²₂+‖𝐼‖²₂+‖𝐽‖²₂+𝐶≤𝐶𝑓_𝑁log⁺‖𝐻‖₂+‖𝐼‖₂+‖𝐽‖₂×log⁺log⁺‖𝐻‖₂+‖𝐼‖₂+‖𝐽‖₂×‖𝐻‖²₂+‖𝐼‖²₂+‖𝐽‖²₂+𝐶, (38) where 𝑓_𝑁=‖𝑆_𝑁∇𝑢‖_∞/𝑁log𝑁.

Using Gronwall's inequality, we have ‖𝐻‖²₂+‖𝐼‖²₂+‖𝐽‖²₂+^𝑡₀𝜇+𝜒‖∇𝐻‖²₂+𝛾‖∇𝐼‖²₂+𝜈‖∇𝐽‖²₂𝐶𝑑𝑡≤expexpexp^𝑡₀𝑓_𝑁𝑡^′𝑑𝑡^′. (39) On the other hand, by multiplying (𝑢,𝜔,𝑏), it can be easily derived from magneto-micropolar fluid equation (1) that ‖𝑢‖²₂+‖𝜔‖²₂+‖𝑏‖²₂+2𝜇^𝑡₀‖∇𝑢‖²₂𝑡^′+2𝛾^𝑡₀‖∇𝜔‖²₂𝑑𝑡^′+2𝜈^𝑡₀‖∇𝑏‖²₂𝑑𝑡^′+2𝜅^𝑡₀‖div𝜔‖²₂𝑑𝑡^′+2𝜒^𝑡₀‖𝜔‖²₂𝑑𝑡^′≤𝑢₀²₂+𝜔₀²₂+𝑏₀²₂. (40) Equation (39) along with (40) implies that the 𝐻¹ estimate of solution (𝑢,𝜔,𝑏).

Next, we will show how to deduce 𝐻^𝑠 estimates based on the 𝐻¹ estimates. We apply operator Λ^𝑠 on the two sides of (1), multiply (Λ^𝑠𝑢,Λ^𝑠𝜔,Λ^𝑠𝑏) by the resulting equations and integrate the final form over ℝ³, and ge 12𝑑Λ𝑑𝑡^𝑠𝑢²₂+Λ^𝑠𝜔²₂+Λ^𝑠𝑏²₂+𝜇+𝜒∇Λ^𝑠𝑢²₂+𝛾∇Λ^𝑠𝜔²₂+𝜈∇Λ^𝑠𝑏²₂+𝜅divΛ^𝑠𝜔²₂+2𝜒Λ^𝑠𝑏²₂=−_ℝ3Λ^𝑠(𝑢⋅∇𝑢)Λ^𝑠𝑢𝑑𝑥−_ℝ3Λ^𝑠(𝑢⋅∇𝜔)Λ^𝑠−𝜔𝑑𝑥_ℝ3Λ^𝑠(𝑢⋅∇𝑏)Λ^𝑠𝑏𝑑𝑥+_ℝ3Λ^𝑠(𝑏⋅∇𝑏)Λ^𝑠+𝑢𝑑𝑥_ℝ3Λ^𝑠(𝑏⋅∇𝑢)Λ^𝑠𝑏𝑑𝑥−2𝜒_ℝ3Λ^𝑠(∇×𝑢)Λ^𝑠𝜔𝑑𝑥, (41) where we use the fact _ℝ3Λ^𝑠(∇×𝜔)Λ^𝑠𝑢𝑑𝑥=_ℝ3Λ^𝑠(∇×𝑢)Λ^𝑠𝜔𝑑𝑥. (42) Furthermore, the divergence free conditions of (𝑢,𝑏) imply that _ℝ3𝑢⋅∇Λ^𝑠𝑢Λ^𝑠=𝑢𝑑𝑥_ℝ3𝑢⋅∇Λ^𝑠𝜔Λ^𝑠𝜔𝑑𝑥=_ℝ3𝑢⋅∇Λ^𝑠𝑏Λ^𝑠𝑏𝑑𝑥=0,_ℝ3𝑏⋅∇Λ^𝑠𝑏Λ^𝑠𝑢𝑑𝑥+_ℝ3𝑏⋅∇Λ^𝑠𝑢Λ^𝑠𝑏𝑑𝑥=0, (43) then 12𝑑Λ𝑑𝑡^𝑠𝑢²₂+Λ^𝑠𝜔²₂+Λ^𝑠𝑏²₂+𝜇+𝜒∇Λ^𝑠𝑢²₂+𝛾∇Λ^𝑠𝜔²₂+𝜈∇Λ^𝑠𝑏²₂+𝜅divΛ^𝑠𝜔²₂+2𝜒Λ^𝑠𝜔²₂=−_ℝ3Λ^𝑠,𝑢∇𝑢Λ^𝑠𝑢𝑑𝑥−_ℝ3Λ^𝑠,𝑢∇𝜔Λ^𝑠−𝜔𝑑𝑥_ℝ3Λ^𝑠,𝑢∇𝑏Λ^𝑠+𝑏𝑑𝑥_ℝ3Λ^𝑠,𝑏∇𝑏Λ^𝑠𝑢+Λ^𝑠,𝑏∇𝑢Λ^𝑠𝑏𝑑𝑥+−2𝜒_ℝ3Λ^𝑠(∇×𝑢)Λ^𝑠𝜔𝑑𝑥.=𝐼𝐼𝐼₁+𝐼𝐼𝐼₂+𝐼𝐼𝐼₃+𝐼𝐼𝐼₄+𝐼𝐼𝐼₅. (44) By Lemma 5, Hölder's inequality, Gagliardo-Nirenberg's inequality 𝑓_𝑊𝑠,4≤𝑓_𝑊^1/4_𝑠,2∇𝑓_𝑊^3/4_𝑠,2, (45) and Young's inequality 1𝑎𝑏≤𝑝𝑎^𝑝+1𝑞𝑏^𝑞,1𝑝+1𝑞=1, (46) we deduce that 𝐼𝐼𝐼₁≤Λ^𝑠,𝑢∇𝑢_4/3Λ^𝑠𝑢₄≤𝐶‖∇𝑢‖₂‖∇𝑢‖_𝑊^𝑠−1,4+‖𝑢‖_𝑊^𝑠,4‖∇𝑢‖₂‖𝑢‖_𝑊^𝑠,4≤𝐶‖∇𝑢‖₂‖𝑢‖_𝐻^1/2_𝑠‖∇𝑢‖_𝐻^3/2_𝑠≤𝐶‖∇𝑢‖⁴₂‖𝑢‖²_𝐻^𝑠+𝜇4‖∇𝑢‖²_𝐻^𝑠. (47) Similarly, we estimate 𝐼𝐼𝐼₂,𝐼𝐼𝐼₃,and𝐼𝐼𝐼₄ as follows: 𝐼𝐼𝐼₂+𝐼𝐼𝐼₃+𝐼𝐼𝐼₄≤𝐶‖∇𝑢‖⁴₂+‖∇𝜔‖⁴₂+‖∇𝑏‖⁴₂×‖𝑢‖²_𝐻^𝑠+‖𝜔‖²_𝐻^𝑠+‖𝑏‖²_𝐻^𝑠+𝜇4‖∇𝑢‖²_𝐻^𝑠+𝛾2‖∇𝜔‖²_𝐻^𝑠+𝜈2‖∇𝑏‖²_𝐻^𝑠. (48) Finally, we estimate the last term 𝐼𝐼𝐼₅≤2𝜒Λ^𝑠(∇×𝑢)₂Λ^𝑠𝜔₂≤𝜒2‖∇𝑢‖²_𝐻^𝑠+2𝜒‖𝜔‖²_𝐻^𝑠. (49) Summing up (47)–(49) with (44), we obtain 𝑑‖𝑢‖𝑑𝑡^2𝐻𝑠+‖𝜔‖²_𝐻^𝑠+‖𝑏‖²_𝐻^𝑠+𝜇+𝜒‖∇𝑢‖²_𝐻^𝑠+𝛾‖∇𝜔‖²_𝐻^𝑠+𝜈‖∇𝑏‖²_𝐻^𝑠+𝜅‖div𝜔‖²_𝐻^𝑠≤𝐶‖𝑢‖⁴_𝐻1+‖𝜔‖⁴_𝐻1+‖𝑏‖⁴_𝐻1×‖𝑢‖²_𝐻^𝑠+‖𝜔‖²_𝐻^𝑠+‖𝑏‖²_𝐻^𝑠. (50) Using Gronwall's inequality we obtain ‖𝑢‖²_𝐻^𝑠+‖𝜔‖²_𝐻^𝑠+‖𝑏‖²_𝐻^𝑠+^𝑡₀𝜇+𝜒‖∇𝑢‖²_𝐻^𝑠+𝛾‖∇𝜔‖²_𝐻^𝑠+𝜈‖∇𝑏‖²_𝐻^𝑠+𝜅‖div𝜔‖²_𝐻^𝑠𝑡^′𝑑𝑡^′≤𝐶𝑢₀²_𝐻𝑠+𝜔₀²_𝐻𝑠+𝑏₀²_𝐻𝑠×exp𝑡sup_{𝑡′∈[)0,𝑡}‖𝑢‖⁴_𝐻1+‖𝜔‖⁴_𝐻1+‖𝑏‖⁴_𝐻1. (51) Hence by (39) and (51), we can get the 𝐻^𝑠 regularity at time 𝑡=𝑇; that is, the smooth solution (𝑢,𝜔,𝑏) can be extended past time 𝑇.