Logarithmical Regularity Criteria of the Three-Dimensional Micropolar Fluid Equations in terms of the Pressure

Abstract

This paper is devoted to the regularity criterion of the three-dimensional micropolar fluid equations. Some new regularity criteria in terms of the partial derivative of the pressure in the Lebesgue spaces and the Besov spaces are obtained which improve the previous results on the micropolar fluid equations.

1. Introduction and Main Result

In this paper, we consider a three-dimensional micropolar fluid equations model described by the following equations: 𝜕∇⋅𝑢=0,_𝑡𝜕𝑢−Δ𝑢−∇×𝑤+∇𝜋+𝑢⋅∇𝑢=0,_𝑡𝑤−Δ𝑤−∇∇⋅𝑤+2𝑤−∇×𝑢+𝑢⋅∇𝑤=0, (1.1) associated with the initial condition 𝑢(𝑥,0)=𝑢₀,𝑤(𝑥,0)=𝑤₀, (1.2) where 𝑢(𝑥,𝑡)=(𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡),𝑢₃(𝑥,𝑡)) is the unknown velocity vector field, 𝜋(𝑥,𝑡) is the unknown scalar pressure field, and 𝑤(𝑥,𝑡)=(𝑤₁(𝑥,𝑡),𝑤₂(𝑥,𝑡),𝑤₃(𝑥,𝑡)) is the unknown microrotation vector field with ∇⋅𝑢₀=0 in the sense of distribution. 𝑢₀ and 𝑤₀ represent the prescribed initial data for the velocity and microrotation fields.

Here, ∇=𝜕_𝑥1,𝜕_𝑥2,𝜕_𝑥3,Δ=∇⋅∇,∇⋅𝑢=³_𝑖=1𝜕_𝑖𝑢_𝑖,𝑢⋅∇𝑤=³_𝑖=1𝑢_𝑖𝜕_𝑖𝑤_𝑗,𝑗=1,2,3. (1.3)

Micropolar fluid equations introduced by Eringen [1] in 1960’ are a special model of the non-Newtonian fluids which is coupled with the viscous incompressible Navier-Stokes model, microrotational effects and microrotational inertia. Physically, micropolar fluid may represent fluids that consist of rigid, randomly oriented (or spherical particles), suspended in a viscous medium, where the deformation of fluid particles is ignored. It can describe many phenomena appeared in a large number of complex fluids such as the suspensions, animal blood, and liquid crystals which cannot be characterized appropriately by the Navier-Stokes system [2]. When the microrotation effects are neglected or 𝑤=0, the micropolar fluid equations reduces to the classic Navier-Stokes flows (see [3]).

Due to the importance of both physics and mathematics, the questions on the well posedness and asymptotic behaviors of solutions for micropolar fluid equations have been attracted more and more attention. Galdi and Rionero [4], Łukaszewicz [5] considered the existence of weak solutions of the micropolar fluid flows (1.1)-(1.2). the existence of strong solutions with either local for large data or global for small data is considered by many authors [6–9]. One may also refer to the interesting results on the large time behaviors of micropolar fluid equations in [10–13].

However, similar to the classic Navier-Stokes equations, the question of global regularity of the weak solutions of the 3D micropolar fluid equations still remain a big open problem. It is of importance on the study of regularity of the weak solutions under additional critical growth conditions on the velocity or the pressure. Based on some analysis technique, some regularity criteria via the velocity of weak solutions in the Lebesgue spaces, multiplier spaces and Besov spaces have been obtained in [14–16]. In particular, as for the pressure criterion, Dong et al. [17] (see also Yuan [18]) showed that the weak solution becomes regular if the pressure satisfies 𝜋∈𝐿^𝑝0,𝑇;𝐿^𝑞,∞ℝ³2,for𝑝+3𝑞3≤2,2<𝑞≤∞, (1.4) or 𝜋∈𝐿¹0,𝑇;𝐵⁰_∞,∞ℝ³, (1.5) where 𝐿^𝑞,∞ and 𝐵⁰_∞,∞ denote the Lorentz space and Besov space. Later on, those results of pressure regularity criteria were further improved to the largest critical Besov spaces by Jia et al. [19] as 𝜋(𝑥,𝑡)∈𝐿^𝑝0,𝑇;𝐵^𝑟_𝑞,∞ℝ³, (1.6) where 2𝑝+3𝑞3=2+𝑟,2+𝑟<𝑞<∞,−1<𝑟≤1. (1.7)

One may also refer to some interesting results on the pressure criteria for Navier-Stokes equations [20–23] and MHD equations [24, 25].

The aim of this paper is to study the logarithmical regularity criterion for the three-dimensional micropolar fluid equations in terms of the partial derivative of pressure in the Lebesgue space or pressure in Besov spaces. Before giving the main results, we first recall the definition of weak solutions for micropolar fluid equations (1.1)-(1.2).

Definition 1.1 (Lukaszewicz [5]). Let (𝑢₀,𝑤₀)∈𝐿²(ℝ³) and ∇⋅𝑢₀=0 in the sense of distribution. A pair vector field (𝑢(𝑥,𝑡),𝑤(𝑥,𝑡)) is termed as a weak solution of (1.1)-(1.2) on (0,𝑇), if (𝑢,𝑤) satisfies the following properties:(i)(𝑢,𝑤)∈𝐿^∞(0,𝑇;𝐿²(ℝ³))∩𝐿²(0,𝑇;𝐻¹(ℝ³));(ii)∇⋅𝑢=0 in the sense of distribution;(iii)(𝑢,𝑤) verifies (1.1) in the sense of distribution.

Now our result reads as follows.

Theorem 1.2. Suppose (𝑢₀,𝑤₀)∈𝐿⁴(ℝ³)∩𝐿²(ℝ³) and (𝑢,𝑤) is a weak solution of the 3𝐷 micropolar fluid equations (1.1)-(1.2). If the partial derivative of the pressure, for example, 𝜕₃𝜋 satisfies the logarithmical Serrin’s type condition ^𝑇₀𝜕₃𝜋^𝑝_𝐿𝑞1+ln𝑒+‖𝑤‖_𝐿42𝑑𝑠<∞,𝑝+3𝑞=74,127<𝑞≤∞, (1.8) then the weak solution (𝑢,𝑤) becomes a regular solution on (0,𝑇].

Theorem 1.3. On substitution of logarithmical Serrin’s type condition (1.8) by the following growth condition _𝑇0‖𝜋‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑤‖_𝐿4𝑑𝑠<∞, (1.9) the conclusion of Theorem 1.2 holds true.

Remark 1.4. It is easy to see that (1.8) on the partial derivative of pressure does not seem comparable with (1.4)–(1.6) on the total pressure at least there is no inclusion relation between them. Moreover, (1.9) can be viewed as an extension of (1.4)–(1.6). Since the growth condition (1.8) is about 𝜕₃𝜋 and (1.9) is on the Besov space 𝐵̇⁻¹_∞,∞, the methods in previous results in [17–19] are not available any more. In order to come over the additional difficulty, the rigorous analysis due to the new structure in nonlinear terms of the system (1.1)-(1.2) is made and some anisotropic function inequalities are also employed.

2. Preliminaries

Throughout this paper, we denote by 𝐶 the constant which may change from line to line. We also use the following usual function spaces 𝐿^𝑝(ℝ³), 1≤𝑝≤∞, denote the Lebesgue space of all 𝐿^𝑝 integral functions associated with the norm 𝑓_𝐿𝑝=∫_ℝ3|𝑓(𝑥)|^𝑝𝑑𝑥^1/𝑝,1≤𝑝<∞,esssup_𝑥∈ℝ3𝑓(𝑥),𝑝=∞. (2.1)𝐻^𝑘(ℝ³) denotes the Hilbert space {𝑢∈𝐿²(ℝ³);‖∇^𝑘𝑢‖_𝐿2<∞}.

𝐵̇^𝑠_𝑝,𝑞(ℝ³) denotes the homogeneous Besov space which is defined by the full-dyadic decomposition such as 𝐵̇^𝑠_𝑝,𝑞ℝ³=𝒮𝑓∈^′ℝ³𝒫ℝ³∶𝑓_{̇𝐵𝑠𝑝,𝑞}<∞, (2.2) where 𝑓_{̇𝐵𝑠𝑝,𝑞}=^{∞𝑗=−∞}2^𝑗𝑠𝑞Δ_𝑗𝑓^𝑞_𝐿𝑝^1/𝑞,1≤𝑞<∞,sup_𝑗∈𝑍2^𝑗𝑠Δ_𝑗𝑓_𝐿𝑝,𝑞=∞, (2.3) and 𝒮^′(ℝ³), 𝒫(ℝ³) are the spaces of all tempered distributions on ℝ³ and the set of all scalar polynomials defined on ℝ³, respectively. It is worthy to note that the homogeneous Besov space 𝐵̇^𝑠_2,2(ℝ³) is equivalent to the Hilbert space 𝐻^𝑠(ℝ³).

In order to prove the main results, we need to recall some anisotropic inequalities. The following Sobolev inequality is due to Cao and Wu [26].

Lemma 2.1 (Cao and Wu [26]). Let the three constants 1≤𝜇,𝜆,𝛾<∞ satisfy 31+𝛾=1𝜇+2𝜆. (2.4) Assume 𝜙∈𝐻¹(ℝ³), 𝜕₁𝜙,𝜕₂𝜙∈𝐿^𝜆(ℝ³) and 𝜕₃𝜙∈𝐿^𝜇(ℝ³), then there exists a constant 𝐶=𝐶(𝜇,𝜆) such that 𝜙_𝐿𝛾≤𝐶𝜕₁𝜙_𝐿^1/3_𝜆𝜕₂𝜙_𝐿^1/3_𝜆𝜕₃𝜙_𝐿^1/3_𝜇. (2.5)

We also recall an interpolation inequality in Besov space due to Meyer [27].

Lemma 2.2 (Meyer [27]). Let 𝛼>0, 𝛽>0 and 1≤𝑝<𝑞<∞, we have for all 𝑓∈𝒮(ℝ³), 𝑓_𝐿𝑞≤𝐶𝑓_̇𝐵^1−𝜃_{−𝛼∞,∞}𝑓^𝜃_{̇𝐵𝛽𝑝,𝑝}𝑞for𝛽=𝛼𝑝𝑞−1,𝜃=𝑝. (2.6)

The following local existence and uniqueness of the strong solution for 3D micropolar fluid equations obtained by Dong et al. [17].

Lemma 2.3 (Dong et al. [17]). Assume 3<𝑝<∞ and (𝑢₀,𝑤₀)∈𝐿^𝑝(ℝ³) with ∇⋅𝑢₀=0 in the sense of distributions. Then there exist a constant 𝑇>0 and a unique strong solution (𝑢,𝑤) of the 3D micropolar fluid equations (1.1)-(1.2) such that 𝑢∈𝐵𝐶[0,𝑇);𝐿^𝑝ℝ³,𝑡^1/2∇𝑢∈𝐵𝐶[0,𝑇);𝐿^𝑝ℝ³. (2.7)

3. Proof of Main Results

According to the local existence result in Lemma 2.3, it allows us to construct a unique 𝐿⁴-strong solution (𝑢,𝑤) of micropolar fluid equations (1.1)-(1.2) under the initial data (𝑢₀,𝑤₀)∈𝐿²(ℝ³)∩𝐿⁴(ℝ³). Moreover, the strong solution can be proved to exist on a maximal time interval using the standard local solution extension technique. For the notation simplicity, we may suppose that the maximal time interval is [0,𝑇). Thus, to prove the main results, it suffices to show that lim_𝑡→𝑇‖𝑢(𝑡)‖_𝐿4+‖𝑤(𝑡)‖_𝐿4<∞. (3.1)

This boundedness will be obtained by the following steps.

3.1. Uniform Estimates of Solutions

In order to prove the main results, we first need some basic estimates of the solutions for the 3D micropolar fluid equations (1.1)-(1.2).

Taking the inner product of the second equation of (1.1) with 𝑢|𝑢|² and integrating by parts, it follows that 14𝑑‖𝑢‖𝑑𝑡^4𝐿4+‖|𝑢|∇𝑢‖²_𝐿2+12∇|𝑢|²²_𝐿2=_ℝ3(∇×𝑤)⋅𝑢|𝑢|²𝑑𝑥−_ℝ3𝑢⋅∇𝜋|𝑢|²𝑑𝑥, (3.2) where we have used the following facts: _ℝ3(𝑢⋅∇𝑢)⋅𝑢|𝑢|²()⋅∇𝑢|𝑢|𝑑𝑥=0,∇𝑢²=|∇𝑢|²|𝑢|²+12∇|𝑢|²². (3.3)

Similarly, taking the inner product of the second equation of (1.1) with |𝑤|²𝑤 and integrating by parts yield 14𝑑‖𝑤‖𝑑𝑡^4𝐿4+‖|𝑤|∇𝑤‖²_𝐿2+12‖|𝑤|∇⋅𝑤‖²_𝐿2+∇|𝑤|²²_𝐿2=_ℝ3(∇×𝑢)⋅𝑤|𝑤|²𝑑𝑥−2‖𝑤‖⁴_𝐿4, (3.4) where we have used _ℝ3(𝑢⋅∇𝑤)⋅𝑤|𝑤|²()⋅∇𝑤|𝑤|𝑑𝑥=0,∇𝑤²=|∇𝑤|²|𝑤|²+12∇|𝑤|²²,_ℝ3(∇⋅𝑤)∇⋅𝑤|𝑤|²𝑑𝑥=_ℝ3|∇⋅𝑤|²|𝑤|²𝑑𝑥+_ℝ3(∇⋅𝑤)𝑤⋅∇|𝑤|²≥1𝑑𝑥2_ℝ³|∇⋅𝑤|²|𝑤|²1𝑑𝑥−2_ℝ³∇|𝑤|²²𝑑𝑥. (3.5)

Combination of (3.2) and (3.4) derives 14𝑑‖𝑢‖𝑑𝑡^4𝐿4+‖𝑤‖⁴_𝐿4+‖|𝑢|∇𝑢‖²_𝐿2+12∇|𝑢|²²_𝐿2+‖|𝑤|∇𝑤‖²_𝐿2+2‖𝑤‖⁴_𝐿4≤_ℝ3(∇×𝑤)⋅𝑢|𝑢|²𝑑𝑥+_ℝ3(∇×𝑢)⋅𝑤|𝑤|²𝑑𝑥−_ℝ3𝑢⋅∇𝜋|𝑢|²𝑑𝑥. (3.6)

With the use of Hölder’s inequality, Young’s inequality, and integration by parts, the first two terms on the right-hand side of (3.6) is bounded by _ℝ3(∇×𝑤)⋅𝑢|𝑢|²𝑑𝑥+_ℝ3(∇×𝑢)⋅𝑤|𝑤|²≤‖𝑤‖𝑑𝑥_𝐿4‖|𝑢|∇𝑢‖_𝐿2‖𝑢‖_𝐿4+‖𝑢‖_𝐿4‖|𝑤|∇𝑤‖_𝐿2‖𝑤‖_𝐿4≤𝐶‖𝑢‖⁴_𝐿4+2‖𝑤‖⁴_𝐿4+12‖|𝑤|∇𝑤‖²_𝐿2+‖|𝑢|∇𝑢‖²_𝐿2. (3.7)

Inserting (3.7) into (3.6), we obtain 𝑑‖𝑢‖𝑑𝑡^4𝐿4+‖𝑤‖⁴_𝐿4+2‖|𝑢|∇𝑢‖²_𝐿2+2∇|𝑢|²²_𝐿2≤𝐶‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4+𝐶_ℝ3𝑢⋅∇𝜋|𝑢|²‖𝑢‖𝑑𝑥∶=𝐶⁴_𝐿4+‖𝑤‖⁴_𝐿4+𝐼. (3.8)

In order to estimate 𝐼, we first establish some estimates between the pressure and the velocity. Taking the operator div on both sides of the second equation of (1.1) gives 𝜋=(−Δ)³⁻¹_𝑖,𝑗=1𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗𝑢_𝑖𝑢_𝑗,∇𝜋=(−Δ)³⁻¹_𝑖,𝑗=1𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗∇𝑢_𝑖𝑢_𝑗. (3.9)

Applying 𝐿^𝑝(1<𝑝<∞) boundedness of the singular integral operators yields ‖𝜋‖_𝐿𝑝≤𝐶‖𝑢‖²_𝐿𝑝,‖∇𝜋‖_𝐿𝑝≤𝐶‖𝑢⋅∇𝑢‖_𝐿𝑝. (3.10)

Then employing the integration by parts and Hölder’s inequality, we estimate 𝐼 as 𝐼=𝐶_ℝ3𝑢⋅∇𝜋|𝑢|²𝑑𝑥=𝐶_ℝ3𝜋𝑢⋅∇|𝑢|²𝑑𝑥≤𝐶_ℝ3|𝜋𝑢|∇|𝑢|²‖‖𝑑𝑥≤𝐶𝜋𝑢_𝐿2‖∇|𝑢|²‖_𝐿2≤𝐶‖𝜋‖_𝐿4‖𝑢‖_𝐿4∇|𝑢|²_𝐿2. (3.11)

Now we begin to prove Theorems 1.2 and 1.3, respectively.

3.2. Proof of Theorem 1.2

Applying Lemma 2.1 to ‖𝜋‖_𝐿4 by choosing 𝛾=4, 𝜇=𝑞such that 1𝑞+2𝜆=74. (3.12) Thus, we have ‖𝜋‖_𝐿4≤𝐶𝜕₁𝜋_𝐿^1/3_𝜆𝜕₂𝜋_𝐿^1/3_𝜆𝜕₃𝜋_𝐿^1/3_𝑞≤𝐶‖∇𝜋‖_𝐿^2/3_𝜆𝜕₃𝜋_𝐿^1/3_𝑞. (3.13) Thanks to 12/7<𝑞≤∞, then 877≤𝜆<6. (3.14) Using (3.10) together with Hölder’s inequality and Gagliardo-Nirenberg inequality derives that ‖∇𝜋‖_𝐿𝜆≤𝐶‖𝑢⋅∇𝑢‖_𝐿𝜆≤𝐶‖𝑢‖_{𝐿2𝜆/(2−𝜆)}‖∇𝑢‖_𝐿2=𝐶|𝑢|²_𝐿^1/2_{𝜆/(2−𝜆)}‖∇𝑢‖_𝐿2≤𝐶|𝑢|²_𝐿^{(3/𝜆)−(7/4)}₂∇|𝑢|²_𝐿^{(9/4)−(3/𝜆)}₂‖∇𝑢‖_𝐿2≤𝐶‖𝑢‖_𝐿^{(6/𝜆)−(7/2)}₄∇|𝑢|²_𝐿^{(9/4)−(3/𝜆)}₂‖∇𝑢‖_𝐿2. (3.15)

Plugging (3.13) and (3.15) into (3.11) and employing Young’s inequality yield that 𝐼≤𝐶𝜕₃𝜋_𝐿^1/3_𝑞‖∇𝑢‖_𝐿^2/3₂‖𝑢‖_𝐿^{(4/𝜆)−(4/3)}₄∇|𝑢|²_𝐿^{(5/2)−(2/𝜆)}₂≤12∇|𝑢|²²_𝐿2+𝐶𝜕₃𝜋_𝐿^{4𝜆/3(4−𝜆)}_𝑞‖∇𝑢‖_𝐿^{8𝜆/(12−7𝜆)}₂‖𝑢‖_𝐿^{16(3−𝜆)/3(4−𝜆)}₄≤12∇|𝑢|²²_𝐿2+𝐶𝜕₃𝜋_𝐿^{4𝜆/(12−7𝜆)}_𝑞+‖∇𝑢‖²_𝐿2‖𝑢‖_𝐿^{16(3−𝜆)/3(4−𝜆)}₄≤12∇|𝑢|²²_𝐿2+𝐶𝜕₃𝜋^𝑝_𝐿𝑞+‖∇𝑢‖²_𝐿2‖𝑢‖⁴_𝐿4+1, (3.16) where we have used the following facts in the last line: 16(3−𝜆)3()24−𝜆<4,𝑝+3𝑞=74,thatis𝑝=4𝜆.12−7𝜆 (3.17)

Then inserting (3.16) into (3.8) obtains 𝑑‖𝑢‖𝑑𝑡^4𝐿4+‖𝑤‖⁴_𝐿4+∇|𝑢|²²_𝐿2≤𝐶1+𝜕₃𝜋^𝑝_𝐿𝑞+‖∇𝑢‖²_𝐿2‖𝑢‖⁴_𝐿4+1+𝐶‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4, (3.18) or 𝑑‖𝑢‖𝑑𝑡𝑒+⁴_𝐿4+‖𝑤‖⁴_𝐿4+∇|𝑢|²²_𝐿2≤𝐶1+𝜕₃𝜋^𝑝_𝐿𝑞+‖∇𝑢‖²_𝐿2𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4≤𝐶𝜕₃𝜋^𝑝_𝐿𝑞‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4+𝐶1+‖∇𝑢‖²_𝐿2𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4. (3.19)

Denote 𝐽=𝜕₃𝜋^𝑝_𝐿𝑞𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4, (3.20) and thanks to 1+ln𝑒+‖𝑤‖_𝐿41≤1+ln𝑒+4‖𝑤‖⁴_𝐿4+431≤1+ln2𝑒+‖𝑤‖⁴_𝐿41≤1+ln2+ln𝑒+‖𝑤‖⁴_𝐿4≤1+ln𝑒+‖𝑤‖⁴_𝐿4, (3.21) we have for 𝐽𝜕𝐽=³𝜋^𝑝_𝐿𝑞1+ln𝑒+‖𝑤‖_𝐿4𝑒+‖𝑢‖⁴_𝐿4+𝑋‖𝑢‖⁴_𝐿41+ln𝑒+‖𝑤‖_𝐿4≤𝜕₃𝜋^𝑝_𝐿𝑞1+ln𝑒+‖𝑤‖_𝐿4𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿41+ln𝑒+‖𝑤‖⁴_𝐿4≤𝜕₃𝜋^𝑝_𝐿𝑞1+ln𝑒+‖𝑤‖_𝐿4𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿41+ln𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4. (3.22)

Thus, plugging (3.22) into (3.19) and denoting 𝒦=𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4, (3.23) we have 𝑑𝜕𝑑𝑡𝒦≤𝐶³𝜋^𝑝_𝐿𝑞1+ln𝑒+‖𝑤‖_𝐿4{1+ln𝒦}𝒦+𝐶1+‖∇𝑢‖²_𝐿2𝒦. (3.24)

Taking Gronwall’s inequality into consideration derives 𝒦≤ln𝑒+𝑢₀⁴_𝐿4+𝑤₀⁴_𝐿4×exp𝐶^𝑡₀𝜕₃𝜋^𝑝_𝐿𝑞1+ln𝑒+‖𝑤‖_𝐿4{1+ln𝒦}𝑑𝑠×exp^𝑡₀𝐶1+‖∇𝑢‖²_𝐿2𝑑𝑠. (3.25)

Thanks to (𝑢,𝑤) is a weak solution of the 3D micropolar equations (1.1)-(1.2), then ^𝑇₀𝐶1+‖∇𝑢‖²_𝐿2𝑑𝑠<𝐶𝑇, (3.26)

we rewrite (3.25) as 1+ln𝒦≤𝐶𝑇+ln𝑒+𝑢₀⁴_𝐿4+𝑤₀⁴_𝐿4+𝐶^𝑡₀𝜕₃𝜋^𝑝_𝐿𝑞1+ln𝑒+‖𝑤‖_𝐿4{1+ln𝒦}𝑑𝑠. (3.27)

Applying Gronwall’s inequality again yields ln𝒦≤𝐶𝑢₀,𝑤₀,𝑇exp𝐶^𝑇₀𝜕₃𝜋^𝑝_𝐿𝑞1+ln𝑒+‖𝑤‖_𝐿4𝑑𝑠, (3.28) which implies that lim_𝑡→𝑇‖𝑢(𝑡)‖_𝐿4+‖𝑤(𝑡)‖_𝐿4<∞. (3.29)

Hence, we complete the proof of Theorem 1.2.

3.3. Proof of Theorem 1.3

Applying Lemma 2.2 to ‖𝜋‖_𝐿4 by choosing 1𝑞=2,𝑝=4,𝛼=1,𝛽=1,𝜃=2, (3.30) we have ‖𝜋‖_𝐿4≤𝐶‖𝜋‖_̇𝐵^1/2_−1∞,∞‖𝜋‖_̇𝐵^1/2_12,2≤𝐶‖𝜋‖_̇𝐵^1/2_−1∞,∞‖𝜋‖_̇𝐻^1/2₁=𝐶‖𝜋‖_̇𝐵^1/2_−1∞,∞‖∇𝜋‖_𝐿^1/2₂. (3.31)

Plugging the above inequality into (3.11) together with Young’s inequality implies 𝐼≤𝐶‖𝜋‖_𝐿4‖𝑢‖_𝐿4∇|𝑢|²_𝐿2≤𝐶‖𝜋‖²_𝐿4‖𝑢‖²_𝐿4+12∇|𝑢|²²_𝐿2≤𝐶‖𝜋‖_{̇𝐵−1∞,∞}‖∇𝜋‖_𝐿2‖𝑢‖²_𝐿4+12∇|𝑢|²²_𝐿2≤𝐶‖𝜋‖_{̇𝐵−1∞,∞}‖|𝑢|∇𝑢‖_𝐿2‖𝑢‖²_𝐿4+12∇|𝑢|²²_𝐿2≤𝐶‖𝜋‖²_{̇𝐵−1∞,∞}‖𝑢‖⁴_𝐿4+12‖|𝑢|∇𝑢‖²_𝐿2+12∇|𝑢|²²_𝐿2. (3.32)

Then inserting (3.32) into (3.8) derives 𝑑‖𝑢‖𝑑𝑡^4𝐿4+‖𝑤‖⁴_𝐿4≤𝐶‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4+𝐶‖𝜋‖²_{̇𝐵−1∞,∞}‖𝑢‖⁴_𝐿4. (3.33)

Thanks to ‖𝜋‖²_{̇𝐵−1∞,∞}‖𝑢‖⁴_𝐿4=𝐶‖𝜋‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑤‖_𝐿4‖𝑢‖⁴_𝐿41+ln𝑒+‖𝑤‖_𝐿4≤𝐶‖𝜋‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑤‖_𝐿2‖𝑢‖⁴_𝐿41+ln𝑒+‖𝑤‖⁴_𝐿4≤𝐶‖𝜋‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑤‖_𝐿2𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿41+ln𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4, (3.34) inserting the above inequality into (3.33) and, similarly, denoting 𝒦=𝑒+‖𝑢‖⁴_𝐿4+‖𝑤‖⁴_𝐿4, (3.35) then we have 𝑑‖𝜋‖𝑑𝑡𝒦≤𝐶^{2̇𝐵−1∞,∞}1+ln𝑒+‖𝑤‖_𝐿4{1+ln𝒦}𝒦. (3.36)

Taking Gronwall, inequality into consideration derives 𝒦≤ln𝑒+𝑢₀⁴_𝐿4+𝑤₀⁴_𝐿4exp𝐶^𝑡₀‖𝜋‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑤‖_𝐿4{1+ln𝒦}𝑑𝑠, (3.37) or 1+ln𝒦≤ln𝑒+𝑢₀⁴_𝐿4+𝑤₀⁴_𝐿4+𝐶^𝑡₀‖𝜋‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑤‖_𝐿4{1+ln𝒦}𝑑𝑠. (3.38)

Likewise, applying Gronwall’s inequality again implies ln𝒦≤𝐶𝑢₀,𝑤₀exp𝐶^𝑇₀‖𝜋‖²_{̇𝐵−1∞,∞}1+ln𝑒+‖𝑤‖_𝐿4𝑑𝑠. (3.39)

Hence, we derive that lim_𝑡→𝑇‖𝑢(𝑡)‖_𝐿4+‖𝑤(𝑡)‖_𝐿4<∞, (3.40) which completes the proof of Theorem 1.3.

Acknowledgments

This work is partially supported by the NSF of China (10801001), NSF of Anhui Province (11040606M02), and is also financed by the 211 Project of Anhui University (KJTD002B, KJJQ005).