Abstract

Blow-up criteria of smooth solutions for the 3D micropolar fluid equations are investigated. Logarithmically improved blow-up criteria are established in the Morrey-Campanto space.

1. Introduction

This paper concerns the initial value problem for the micropolar fluid equations in ℝ3πœ•π‘‘πœ•π‘’βˆ’(πœ‡+πœ’)Δ𝑒+π‘’β‹…βˆ‡π‘’+βˆ‡π‘βˆ’πœ’βˆ‡Γ—π‘€=0,π‘‘π‘€βˆ’π›ΎΞ”π‘€βˆ’πœ…βˆ‡βˆ‡β‹…π‘€+2πœ’π‘€+π‘’β‹…βˆ‡π‘€βˆ’πœ’βˆ‡Γ—π‘’=0,βˆ‡β‹…π‘’=0(1.1) with the initial value𝑑=0βˆΆπ‘’=𝑒0(π‘₯),𝑀=𝑀0(π‘₯),(1.2) where 𝑒(𝑑,π‘₯), 𝑀(𝑑,π‘₯), and 𝑝(𝑑,π‘₯) stand for the velocity field, microrotation field, and the scalar pressure, respectively. And 𝜈>0 is the Newtonian kinetic viscosity, πœ…>0 is the dynamics micro-rotation viscosity, and 𝛼,𝛽,𝛾>0 are the angular viscosity (see, i.e., Lukaszewicz [1]).

The micropolar fluid equations were first proposed by Eringen [2]. It is a type of fluids which exhibits the micro-rotational effects and micro-rotational inertia and can be viewed as a non-Newtonian fluid. Physically, it may represent adequately the fluids consisting of bar-like elements. Certain anisotropic fluids, for example, liquid crystals that are made up of dumbbell molecules, are of the same type. For more background, we refer to [1] and references therein. Besides their physical applications, the micropolar fluid equations are also mathematically significant. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research, and many interesting results have been obtained (see [3–8]). Regularity criterion of weak solutions to (1.1) and (1.2) in terms of the pressure was obtained (see [4]). Gala [5] established a Serrin-type regularity criterion for the weak solutions to (1.1) and (1.2) in Morrey-Campanato space. Wang and Chen [7] established the regularity criteria of weak solutions to (1.1) and (1.2) via the derivative of the velocity in one direction. A new logarithmically improved blow-up criterion of smooth solutions to (1.1) and (1.2) in an appropriate homogeneous Besov space is established by Wang and Yuan [8].

If πœ…=0 and 𝑀=0, then (1.1) reduces to be the Navier-Stokes equations. Besides its physical applications, the Navier-Stokes equations are also mathematically significant. In the last century, Leray [9] and Hopf [10] constructed weak solutions to the Navier-Stokes equations. The solution is called the Leray-Hopf weak solution. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results are established (see [11–26]). Regularity criteria of weak solutions to the Navier-Stokes equations in Morrey space were obtained in [13, 21].

The main aim of this paper is to establish two logarithmically blow-up criteria of smooth solution to (1.1), (1.2). Our results state as follows.

Theorem 1.1. Let 𝑒0,𝑀0βˆˆπ»π‘š(ℝ3)(π‘šβ‰₯3) with βˆ‡β‹…π‘’0=0. Assume that (𝑒,𝑀) is a smooth solution to (1.1) and (1.2) on [0, 𝑇). If 𝑒 satisfies ξ€œπ‘‡0‖𝑒(𝑑)‖̇𝑀2/(1βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+‖𝑒(𝑑)β€–πΏβˆžξ€Έπ‘‘π‘‘<∞,0<π‘Ÿ<1,(1.3) then the solution (𝑒,𝑀) can be extended beyond 𝑑=𝑇.

We have the following corollary immediately.

Corollary 1.2. Let 𝑒0,𝑀0βˆˆπ»π‘š(ℝ3)(π‘šβ‰₯3) with βˆ‡β‹…π‘’0=0. Assume that (𝑒,𝑀) is a smooth solution to (1.1) and (1.2) on [0, 𝑇). Suppose that 𝑇 is the maximal existence time, then ξ€œπ‘‡0‖𝑒(𝑑)‖̇𝑀2/(1βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+‖𝑒(𝑑)β€–πΏβˆžξ€Έπ‘‘π‘‘=∞,0<π‘Ÿ<1.(1.4)

Theorem 1.3. Let 𝑒0,𝑀0βˆˆπ»π‘š(ℝ3)(π‘šβ‰₯3) with βˆ‡β‹…π‘’0=0. Assume that (𝑒,𝑀) is a smooth solution to (1.1) and (1.2) on [0, 𝑇). If 𝑒 satisfies ξ€œπ‘‡0β€–βˆ‡π‘’(𝑑)‖̇𝑀2/(2βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+β€–βˆ‡π‘’(𝑑)β€–πΏβˆžξ€Έπ‘‘π‘‘<∞,0<π‘Ÿ<1,(1.5) then the solution (𝑒,𝑀) can be extended beyond 𝑑=𝑇.

One has the following corollary immediately.

Corollary 1.4. Let 𝑒0,𝑀0βˆˆπ»π‘š(ℝ3)(π‘šβ‰₯3) with βˆ‡β‹…π‘’0=0. Assume that (𝑒,𝑀) is a smooth solution to (1.1) and (1.2) on [0, 𝑇). Suppose that 𝑇 is the maximal existence time, then ξ€œπ‘‡0β€–βˆ‡π‘’(𝑑)‖̇𝑀2/(2βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+β€–βˆ‡π‘’(𝑑)β€–πΏβˆžξ€Έπ‘‘π‘‘=∞,0<π‘Ÿ<1.(1.6)

The paper is organized as follows. We first state some important inequalities in Section 2, which play an important role in the proof of our main result. Then, we prove the main result in Section 3 and Section 4, respectively.

2. Preliminaries

Firstly, we recall the definition and some properties of the space that we are going to use. The space plays an important role in studying the regularity of solutions to nonlinear differential equations.

Definition 2.1. For 1<π‘β‰€π‘žβ‰€+∞, the Morrey-Campanato space ̇𝑀𝑝,π‘ž is defined by ̇𝑀𝑝,π‘ž=ξƒ―π‘“βˆˆπΏπ‘locℝ3ξ€Έβˆ£β€–π‘“β€–Μ‡π‘€π‘,π‘ž=supπ‘₯βˆˆβ„3sup𝑅>0𝑅3/π‘žβˆ’3/𝑝‖𝑓‖𝐿𝑝(𝐡(π‘₯,𝑅))ξƒ°<∞,(2.1) where 𝐡(π‘₯,𝑅) denotes the ball of center π‘₯ with radius 𝑅.
It is easy to verify that ̇𝑀𝑝,π‘ž is a Banach space under the norm ‖⋅‖̇𝑀𝑝,π‘ž. Furthermore, it is easy to check the following: ‖‖𝑓(πœ†β‹…)̇𝑀𝑝,π‘ž=πœ†βˆ’3/π‘žβ€–π‘“β€–Μ‡π‘€π‘,π‘ž,πœ†>0.(2.2) Morrey-Campanato spaces can be seen as a complement to 𝐿𝑝 spaces. In fact, for π‘β‰€π‘ž, one has πΏπ‘ž=Μ‡π‘€π‘ž,π‘žβŠ‚Μ‡π‘€π‘,π‘ž.(2.3) one has the following comparison between Lorentz spaces and Morrey-Campanato spaces: for 𝑝β‰₯2, 𝐿3/π‘Ÿξ€·β„3ξ€ΈβŠ‚πΏ3/π‘Ÿ,βˆžξ€·β„3ξ€ΈβŠ‚Μ‡π‘€π‘,3/π‘Ÿξ€·β„3ξ€Έ,(2.4) where 𝐿𝑝,∞ denotes the usual Lorentz (weak 𝐿𝑝) space.
In the proof of our main result, we need the following lemma which was given in [27].

Lemma 2.2. For 0β‰€π‘Ÿ<3/2, the space Μ‡π‘π‘Ÿ is defined as the space of 𝑓(π‘₯)∈𝐿2loc(ℝ3) such that β€–π‘“β€–Μ‡π‘π‘Ÿ=supβ€–π‘”β€–Μ‡π΅π‘Ÿ2,1≀1‖𝑓𝑔‖𝐿2<∞.(2.5) Then Μ‡π‘€π‘“βˆˆ2,3/π‘Ÿ if and only if Μ‡π‘π‘“βˆˆπ‘Ÿ with equivalence of norms. And the fact that 𝐿2̇𝐻1βŠ‚Μ‡π΅π‘Ÿ2,1βŠ‚Μ‡π»π‘Ÿ,0<π‘Ÿ<1,(2.6) one has Μ‡π‘‹π‘ŸβŠ‚Μ‡π‘€2,3/π‘Ÿ,(2.7) where Μ‡π‘‹π‘Ÿ denotes the pointwise multiplier space from Μ‡π»π‘Ÿ to 𝐿2.

We need the following lemma that is basically established in [28]. For completeness, the proof will be also sketched here.

Lemma 2.3. For 0<π‘Ÿ<1, the inequality β€–π‘“β€–Μ‡π΅π‘Ÿ2,1≀𝐢‖𝑓‖𝐿1βˆ’π‘Ÿ2β€–βˆ‡π‘“β€–π‘ŸπΏ2(2.8) holds, where 𝐢 is a positive constant that depends on π‘Ÿ.

Proof. It follows from the definition of Besov spaces that β€–π‘“β€–Μ‡π΅π‘Ÿ2,1=ξ“π‘–βˆˆβ„€2π‘–π‘Ÿβ€–β€–Ξ”π‘–π‘“β€–β€–πΏ2≀𝑖≀𝑗2π‘–π‘Ÿβ€–β€–Ξ”π‘–π‘“β€–β€–πΏ2+𝑖>𝑗2𝑖(π‘Ÿβˆ’1)2𝑖‖‖Δ𝑖𝑓‖‖𝐿2≀𝑖≀𝑗22π‘–π‘Ÿξƒͺ1/2𝑖≀𝑗‖‖Δ𝑖𝑓‖‖2𝐿2ξƒͺ1/2+𝑖≀𝑗22𝑖(π‘Ÿβˆ’1)ξƒͺ1/2𝑖>𝑗22𝑖‖‖Δ𝑖𝑓‖‖2𝐿2ξƒͺ1/2ξ€·2β‰€πΆπ‘—π‘Ÿβ€–π‘“β€–πΏ2+2𝑗(π‘Ÿβˆ’1)‖𝑓‖̇𝐻1ξ€Έξ€·2=πΆπ‘—π‘Ÿπ΄βˆ’π‘Ÿ+2𝑗(π‘Ÿβˆ’1)𝐴1βˆ’π‘Ÿξ€Έβ€–π‘“β€–πΏ1βˆ’π‘Ÿ2β€–π‘“β€–π‘ŸΜ‡π»1,(2.9) where 𝐴=(‖𝑓‖̇𝐻1)/(‖𝑓‖𝐿2). Choosing 𝑗 such that 1/2≀2π‘—π‘Ÿπ΄βˆ’π‘Ÿβ‰€1, from (2.9) we get β€–π‘“β€–Μ‡π΅π‘Ÿ2,1≀1+2𝑗(π‘Ÿβˆ’1)𝐴1βˆ’π‘Ÿξ€Έβ€–π‘“β€–πΏ1βˆ’π‘Ÿ2β€–π‘“β€–π‘ŸΜ‡π»1ξ‚΅ξ‚€1≀𝐢1+2ξ‚βˆ’1/π‘Ÿξ‚Άβ€–π‘“β€–πΏ1βˆ’π‘Ÿ2β€–βˆ‡π‘“β€–π‘ŸπΏ2.(2.10) Therefore, we have completed the proof of Lemma 2.3.

The following Lemma comes from [29].

Lemma 2.4. Assume that 1<𝑝<∞. For 𝑓,π‘”βˆˆπ‘Šπ‘š,𝑝, and 1<π‘žβ‰€βˆž, 1<π‘Ÿ<∞, one has β€–βˆ‡π›Ό(𝑓𝑔)βˆ’π‘“βˆ‡π›Όπ‘”β€–πΏπ‘ξ€·β‰€πΆβ€–βˆ‡π‘“β€–πΏπ‘ž1β€–β€–βˆ‡π›Όβˆ’1π‘”β€–β€–πΏπ‘Ÿ1+β€–π‘”β€–πΏπ‘ž2β€–βˆ‡π›Όπ‘“β€–πΏπ‘Ÿ2ξ€Έ,(2.11) where 1β‰€π›Όβ‰€π‘š and 1/𝑝=1/π‘ž1+1/π‘Ÿ1=1/π‘ž2+1/π‘Ÿ2.

In order to prove Theorem 1.1, we need the following interpolation inequalities in three space dimensions.

Lemma 2.5. In three space dimensions, the following inequalities β€–βˆ‡π‘“β€–πΏ4≀𝐢‖𝑓‖𝐿1/82β€–β€–βˆ‡2𝑓‖‖𝐿7/82‖𝑓‖𝐿4≀𝐢‖𝑓‖𝐿3/42β€–β€–βˆ‡3𝑓‖‖𝐿1/42β€–β€–βˆ‡2𝑓‖‖𝐿4≀𝐢‖𝑓‖𝐿1/122β€–β€–βˆ‡3𝑓‖‖𝐿11/122β€–β€–βˆ‡2𝑓‖‖𝐿2≀𝐢‖𝑓‖𝐿1/32β€–β€–βˆ‡3𝑓‖‖𝐿2/32(2.12) hold.

3. Proof of Theorem 1.1

Proof. Multiplying the first equation of (1.1) by 𝑒 and integrating with respect to π‘₯ over ℝ3, using integration by parts, we obtain 12𝑑(𝑑𝑑‖𝑒𝑑)β€–2𝐿2+(πœ‡+πœ’)β€–βˆ‡π‘’(𝑑)β€–2𝐿2ξ€œ=πœ’β„3(βˆ‡Γ—π‘€)⋅𝑒𝑑π‘₯(3.1) Similarly, we get 12𝑑(𝑑𝑑‖𝑀𝑑)β€–2𝐿2+π›Ύβ€–βˆ‡π‘€(𝑑)β€–2𝐿2+πœ…β€–βˆ‡β‹…π‘€β€–2𝐿2+2πœ’β€–π‘€β€–2𝐿2ξ€œ=πœ’β„3(βˆ‡Γ—π‘’)⋅𝑀𝑑π‘₯.(3.2) Summing up (3.1) and (3.2), we deduce thats 12𝑑‖𝑑𝑑‖𝑒(𝑑)2𝐿2+‖‖𝑀(𝑑)2𝐿2‖+(πœ‡+πœ’)β€–βˆ‡π‘’(𝑑)2𝐿2+π›Ύβ€–βˆ‡π‘€(𝑑)β€–2𝐿2+πœ…β€–βˆ‡β‹…π‘€β€–2𝐿2+2πœ’β€–π‘€β€–2𝐿2ξ€œ=πœ’β„3ξ€œ(βˆ‡Γ—π‘€)⋅𝑒𝑑π‘₯+πœ’β„3(βˆ‡Γ—π‘’)⋅𝑀𝑑π‘₯.(3.3) Using integration by parts and Cauchy’s inequality, we obtain πœ’ξ€œβ„3(ξ€œβˆ‡Γ—π‘€)⋅𝑒𝑑π‘₯+πœ’β„3(βˆ‡Γ—π‘’)⋅𝑀𝑑π‘₯β‰€πœ’β€–βˆ‡π‘’β€–2𝐿2+πœ’β€–π‘€β€–2𝐿2.(3.4) Combining (3.3) and (3.4) yields 12𝑑‖𝑑𝑑‖𝑒(𝑑)2𝐿2+‖‖𝑀(𝑑)2𝐿2‖+πœ‡β€–βˆ‡π‘’(𝑑)2𝐿2β€–+π›Ύβ€–βˆ‡π‘€(𝑑)2𝐿2+πœ…β€–βˆ‡β‹…π‘€β€–2𝐿2+πœ’β€–π‘€β€–2𝐿2≀0.(3.5) Integrating with respect to 𝑑, we have (‖𝑒𝑑)β€–2𝐿2+‖𝑀(𝑑)β€–2𝐿2ξ€œ+2𝑑0ξ‚€πœ‡β€–βˆ‡π‘’(𝜏)β€–2𝐿2+π›Ύβ€–βˆ‡π‘€(𝜏)β€–2𝐿2ξ‚ξ€œπ‘‘πœ+2πœ…π‘‘0β€–βˆ‡β‹…π‘€(𝜏)β€–2𝐿2ξ€œπ‘‘πœ+2πœ’π‘‘0‖𝑀(𝜏)β€–2𝐿2β€–β€–π‘’π‘‘πœβ‰€0β€–β€–2𝐿2+‖‖𝑀0β€–β€–2𝐿2.(3.6)
Taking βˆ‡ to the first equation of (1.1), then multiplying the resulting equation by βˆ‡π‘’ and using integration by parts, we obtain 12𝑑(π‘‘π‘‘β€–βˆ‡π‘’π‘‘)β€–2𝐿2β€–β€–βˆ‡+(πœ‡+πœ’)2‖‖𝑒(𝑑)2𝐿2ξ€œ=βˆ’β„3ξ€œβˆ‡(π‘’β‹…βˆ‡π‘’)βˆ‡π‘’π‘‘π‘₯+πœ’β„3βˆ‡(βˆ‡Γ—π‘€)βˆ‡π‘’π‘‘π‘₯.(3.7) Similarly, we get 12π‘‘β€–π‘‘π‘‘β€–βˆ‡π‘€(𝑑)2𝐿2β€–β€–βˆ‡+𝛾2‖‖𝑀(𝑑)2𝐿2+πœ…β€–βˆ‡β‹…βˆ‡π‘€β€–2𝐿2+2πœ’β€–βˆ‡π‘€β€–2𝐿2ξ€œ=βˆ’β„3ξ€œβˆ‡(π‘’β‹…βˆ‡π‘€)β‹…βˆ‡π‘€π‘‘π‘₯+πœ’β„3βˆ‡(βˆ‡Γ—π‘’)β‹…βˆ‡π‘€π‘‘π‘₯.(3.8) Combining (3.7) and (3.8) yields 12π‘‘ξ‚€β€–π‘‘π‘‘β€–βˆ‡π‘’(𝑑)2𝐿2+β€–β€–βˆ‡π‘€(𝑑)2𝐿2ξ‚β€–β€–βˆ‡+(πœ‡+πœ’)2‖‖𝑒(𝑑)2𝐿2β€–β€–βˆ‡+𝛾2‖‖𝑀(𝑑)2𝐿2+πœ…β€–βˆ‡βˆ‡β‹…π‘€β€–2𝐿2+2πœ’β€–βˆ‡π‘€β€–2𝐿2ξ€œ=βˆ’β„3ξ€œβˆ‡(π‘’β‹…βˆ‡π‘’)βˆ‡π‘’π‘‘π‘₯+πœ’β„3βˆ’ξ€œβˆ‡(βˆ‡Γ—π‘€)βˆ‡π‘’π‘‘π‘₯ℝ3ξ€œβˆ‡(π‘’β‹…βˆ‡π‘€)βˆ‡π‘€π‘‘π‘₯+πœ’β„3βˆ‡(βˆ‡Γ—π‘’)βˆ‡π‘€π‘‘π‘₯.(3.9) Using integration by parts and Cauchy’s inequality, we obtain πœ’ξ€œβ„3ξ€œβˆ‡(βˆ‡Γ—π‘€)β‹…βˆ‡π‘’π‘‘π‘₯+πœ’β„3β€–β€–βˆ‡βˆ‡(βˆ‡Γ—π‘’)β‹…βˆ‡π‘€π‘‘π‘₯β‰€πœ’2𝑒‖‖2𝐿2+πœ’β€–βˆ‡π‘€β€–2𝐿2.(3.10) Using HΓΆlder’s inequality, (2.8), and Young’s inequality, we obtain βˆ’ξ€œβ„3βˆ‡(π‘’β‹…βˆ‡π‘’)βˆ‡π‘’π‘‘π‘₯β‰€β€–βˆ‡π‘’β€–πΏ2β€–βˆ‡π‘’βˆ‡π‘’β€–πΏ2β‰€πΆβ€–βˆ‡π‘’β€–Μ‡π‘€2,3/π‘Ÿβ€–βˆ‡π‘’β€–Μ‡π΅π‘Ÿ2,1β€–βˆ‡π‘’β€–πΏ2β‰€πΆβ€–βˆ‡π‘’β€–Μ‡π‘€2,3/π‘Ÿβ€–βˆ‡π‘’β€–Μ‡π΅2βˆ’π‘Ÿπ‘Ÿ2,1β€–β€–βˆ‡2π‘’β€–β€–π‘ŸπΏ2β‰€πœ‡2β€–β€–βˆ‡2‖‖𝑒(𝑑)2𝐿2+πΆβ€–βˆ‡π‘’β€–Μ‡π‘€2/(2βˆ’π‘Ÿ)2,3/π‘Ÿβ€–βˆ‡π‘’β€–2𝐿2.(3.11) Similarly, we have the following estimate: βˆ’ξ€œβ„3βˆ‡(π‘’β‹…βˆ‡π‘€)βˆ‡π‘€π‘‘π‘₯β‰€β€–βˆ‡π‘€β€–πΏ2β€–βˆ‡π‘’βˆ‡π‘€β€–πΏ2β‰€πΆβ€–βˆ‡π‘’β€–Μ‡π‘€2,3/π‘Ÿβ€–βˆ‡π‘€β€–Μ‡π΅π‘Ÿ2,1β€–βˆ‡π‘€β€–πΏ2β‰€πΆβ€–βˆ‡π‘’β€–Μ‡π‘€2,3/π‘Ÿβ€–βˆ‡π‘€β€–Μ‡π΅2βˆ’π‘Ÿπ‘Ÿ2,1β€–β€–βˆ‡2π‘€β€–β€–π‘ŸπΏ2≀𝛾2β€–β€–βˆ‡2𝑀‖‖(𝑑)2𝐿2+πΆβ€–βˆ‡π‘’β€–Μ‡π‘€2/(2βˆ’π‘Ÿ)2,3/π‘Ÿβ€–βˆ‡π‘€β€–2𝐿2.(3.12) Combining (3.9)-(3.12) yields π‘‘ξ‚€β€–π‘‘π‘‘β€–βˆ‡π‘’(𝑑)2𝐿2+β€–β€–βˆ‡π‘€(𝑑)2𝐿2ξ‚β€–β€–βˆ‡+πœ‡2‖‖𝑒(𝑑)2𝐿2β€–β€–βˆ‡+𝛾2𝑀‖‖2𝐿2+πœ…β€–βˆ‡βˆ‡β‹…π‘€β€–2𝐿2+πœ’β€–βˆ‡π‘€β€–2𝐿2β‰€πΆβ€–βˆ‡π‘’β€–Μ‡π‘€2/(2βˆ’π‘Ÿ)2,3/π‘Ÿξ‚€β€–βˆ‡π‘’β€–2𝐿2+β€–βˆ‡π‘€β€–2𝐿2ξ‚β‰€πΆβ€–βˆ‡π‘’β€–Μ‡π‘€2/(2βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+β€–βˆ‡π‘’β€–πΏβˆžξ€Έξ‚€β€–βˆ‡π‘’β€–2𝐿2+β€–βˆ‡π‘€β€–2𝐿21+ln𝑒+β€–βˆ‡π‘’β€–πΏβˆžξ€Έξ€Έβ‰€πΆβ€–βˆ‡π‘’β€–Μ‡π‘€2/(2βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+β€–βˆ‡π‘’β€–πΏβˆžξ€Έξ‚€β€–βˆ‡π‘’β€–2𝐿2+β€–βˆ‡π‘€β€–2𝐿2ξ‚ξ€·ξ€·β€–β€–βˆ‡1+ln𝑒+3𝑒‖‖𝐿2+β€–β€–βˆ‡3𝑀‖‖𝐿2,ξ€Έξ€Έ(3.13) where we have used 𝐻2ℝ3ξ€Έβ†ͺπΏβˆžξ€·β„3ξ€Έ.(3.14) For any 𝑇0≀𝑑≀𝑇, we set πœ—(𝑑)=sup𝑇0β‰€πœβ‰€π‘‘ξ‚€β€–β€–βˆ‡3‖‖𝑒(𝜏)2𝐿2+β€–β€–βˆ‡3‖‖𝑀(𝜏)2𝐿2.(3.15) Thus, from (3.13), we have π‘‘ξ‚€π‘‘π‘‘β€–βˆ‡π‘’(𝑑)β€–2𝐿2+β€–βˆ‡π‘€(𝑑)β€–2𝐿2ξ‚β€–β€–βˆ‡+πœ‡2‖‖𝑒(𝑑)2𝐿2β€–β€–βˆ‡+𝛾2𝑀‖‖2𝐿2+πœ…β€–βˆ‡βˆ‡β‹…π‘€β€–2𝐿2+πœ’β€–βˆ‡π‘€β€–2𝐿2β€–β‰€πΆβˆ‡π‘’β€–Μ‡π‘€2/(2βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+β€–βˆ‡π‘’β€–πΏβˆžξ€Έξ‚€β€–βˆ‡π‘’β€–2𝐿2+β€–βˆ‡π‘€β€–2𝐿2(1+ln(𝑒+πœ—(𝑑))),βˆ€π‘‡0≀𝑑<𝑇.(3.16) It follows from (3.8) and Gronwall’s inequality that β€–β€–βˆ‡π‘’(𝑑)2𝐿2β€–+β€–βˆ‡π‘€(𝑑)2𝐿2β‰€ξ‚€β€–β€–ξ€·π‘‡βˆ‡π‘’0ξ€Έβ€–β€–2𝐿2+β€–β€–ξ€·π‘‡βˆ‡π‘€0ξ€Έβ€–β€–2𝐿2ξ‚βŽ§βŽͺ⎨βŽͺβŽ©ξ€œexp𝐢(1+ln(𝑒+πœ—(𝑑)))𝑑𝑇0β€–βˆ‡π‘’(𝜏)‖̇𝑀2/(2βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+β€–βˆ‡π‘’β€–πΏβˆžξ€ΈβŽ«βŽͺ⎬βŽͺβŽ­π‘‘πœβ‰€πΆ0[]}exp{πΆπœ€1+ln(𝑒+πœ—(𝑑))≀𝐢0[]}exp{2πΆπœ€ln(𝑒+πœ—(𝑑))≀𝐢0(𝑒+πœ—(𝑑))2πΆπœ€,(3.17) provided that ξ€œπ‘‘π‘‡0β€–βˆ‡π‘’(𝜏)‖̇𝑀2/(2βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+β€–βˆ‡π‘’β€–πΏβˆžξ€Έπ‘‘πœ<πœ€β‰ͺ1,(3.18) where 𝐢0=β€–βˆ‡π‘’(𝑇0)β€–2𝐿2+β€–βˆ‡π‘€(𝑇0)β€–2𝐿2.
Applying βˆ‡π‘š to the first equation of (1.1), then multiplying the resulting equation by βˆ‡π‘šπ‘’ and using integration by parts, we have 12π‘‘π‘‘π‘‘β€–βˆ‡π‘šπ‘’(𝑑)β€–2𝐿2β€–β€–βˆ‡+(πœ‡+πœ’)π‘š+1‖‖𝑒(𝑑)2𝐿2ξ€œ=βˆ’β„3βˆ‡π‘š(π‘’β‹…βˆ‡π‘’)βˆ‡π‘šξ€œπ‘’π‘‘π‘₯+πœ’β„3βˆ‡π‘š(βˆ‡Γ—π‘€)βˆ‡π‘šπ‘’π‘‘π‘₯.(3.19) Likewise, from the second equation of (1.1), we obtain 12π‘‘π‘‘π‘‘β€–βˆ‡π‘šβ€–π‘€(𝑑)2𝐿2β€–β€–βˆ‡+π›Ύπ‘š+1‖‖𝑀(𝑑)2𝐿2+πœ…β€–βˆ‡π‘šβ€–βˆ‡β‹…π‘€2𝐿2+2πœ’β€–βˆ‡π‘šβ€–π‘€(𝑑)2𝐿2ξ€œ=βˆ’β„3βˆ‡π‘š(π‘’β‹…βˆ‡π‘€)βˆ‡π‘šξ€œπ‘€π‘‘π‘₯+πœ’β„3βˆ‡π‘š(βˆ‡Γ—π‘’)βˆ‡π‘šπ‘€π‘‘π‘₯.(3.20) Using βˆ‡β‹…π‘’=0 and (3.19) and (3.20), we have 12π‘‘ξ€·π‘‘π‘‘β€–βˆ‡π‘šβ€–π‘’(𝑑)2𝐿2+β€–βˆ‡π‘šβ€–π‘€(𝑑)2𝐿2ξ€Έβ€–β€–βˆ‡+(πœ‡+πœ’)π‘š+1‖‖𝑒(𝑑)2𝐿2β€–β€–βˆ‡+π›Ύπ‘š+1‖‖𝑀(𝑑)2𝐿2+πœ…β€–βˆ‡π‘šβ€–βˆ‡β‹…π‘€2𝐿2+2πœ’β€–βˆ‡π‘šβ€–π‘€(𝑑)2𝐿2ξ€œ=βˆ’β„3ξ€Ίβˆ‡π‘š(π‘’β‹…βˆ‡π‘’)βˆ’π‘’β‹…βˆ‡βˆ‡π‘šπ‘’ξ€»βˆ‡π‘šξ€œπ‘’π‘‘π‘₯+πœ’β„3βˆ‡π‘š(βˆ‡Γ—π‘€)βˆ‡π‘šβˆ’ξ€œπ‘’π‘‘π‘₯ℝ3ξ€Ίβˆ‡π‘š(π‘’β‹…βˆ‡π‘€)βˆ’π‘’β‹…βˆ‡βˆ‡π‘šπ‘€ξ€»βˆ‡π‘šξ€œπ‘€π‘‘π‘₯+πœ’β„3βˆ‡π‘š(βˆ‡Γ—π‘’)βˆ‡π‘šπ‘€π‘‘π‘₯.(3.21) In what follows, for simplicity, we will set π‘š=3.
By HΓΆlder’s inequality, (2.11), (2.12), and Young’s inequality, we obtain βˆ’ξ€œβ„3ξ€Ίβˆ‡3(π‘’β‹…βˆ‡π‘’)βˆ’π‘’β‹…βˆ‡βˆ‡3π‘’ξ€»βˆ‡3β‰€β€–β€–βˆ‡π‘’π‘‘π‘₯3(π‘’β‹…βˆ‡π‘’)βˆ’π‘’β‹…βˆ‡βˆ‡3𝑒‖‖𝐿2β€–β€–βˆ‡3𝑒‖‖𝐿2β‰€πΆβ€–βˆ‡π‘’β€–πΏ4β€–β€–βˆ‡3𝑒‖‖𝐿4β€–β€–βˆ‡3𝑒‖‖𝐿2β‰€πΆβ€–βˆ‡π‘’β€–πΏ3/42β€–β€–βˆ‡4𝑒‖‖𝐿1/42β€–βˆ‡π‘’β€–πΏ1/122β€–β€–βˆ‡4𝑒‖‖𝐿11/122β€–βˆ‡π‘’β€–πΏ1/32β€–β€–βˆ‡4𝑒‖‖𝐿2/32β‰€πΆβ€–βˆ‡π‘’β€–πΏ7/62β€–β€–βˆ‡4𝑒‖‖𝐿11/62β‰€πœ‡4β€–β€–βˆ‡4𝑒‖‖2𝐿2+πΆβ€–βˆ‡π‘’β€–πΏ142β‰€πœ‡4β€–β€–βˆ‡4𝑒‖‖2𝐿2+𝐢(𝑒+πœ—(𝑑))14πΆπœ€,βˆ’ξ€œ(3.22)ℝ3ξ€Ίβˆ‡3(π‘’β‹…βˆ‡π‘€)βˆ’π‘’β‹…βˆ‡βˆ‡3π‘€ξ€»βˆ‡3β‰€β€–β€–βˆ‡π‘€π‘‘π‘₯3(π‘’β‹…βˆ‡π‘€)βˆ’π‘’β‹…βˆ‡βˆ‡3𝑀‖‖𝐿2β€–β€–βˆ‡3𝑀‖‖𝐿2β‰€πΆβ€–βˆ‡π‘’β€–πΏ4β€–β€–βˆ‡3𝑀‖‖𝐿4β€–β€–βˆ‡3𝑀‖‖𝐿2+β€–βˆ‡π‘€β€–πΏ4β€–β€–βˆ‡3𝑒‖‖𝐿4β€–β€–βˆ‡3𝑀‖‖𝐿2β‰€πΆβ€–βˆ‡π‘’β€–πΏ3/42β€–β€–βˆ‡4𝑒‖‖𝐿1/42β€–βˆ‡π‘€β€–πΏ1/122β€–β€–βˆ‡4𝑀‖‖𝐿11/122β€–βˆ‡π‘€β€–πΏ1/32β€–β€–βˆ‡4𝑀‖‖𝐿2/32+πΆβ€–βˆ‡π‘€β€–πΏ3/42β€–β€–βˆ‡4𝑀‖‖𝐿1/42β€–βˆ‡π‘’β€–πΏ1/122β€–β€–βˆ‡4𝑒‖‖𝐿11/122β€–βˆ‡π‘€β€–πΏ1/32β€–β€–βˆ‡4𝑀‖‖𝐿2/32β‰€πœ‡4β€–β€–βˆ‡4𝑒‖‖2𝐿2+𝛾2β€–β€–βˆ‡4𝑀‖‖2𝐿2+πΆβ€–βˆ‡π‘’β€–9𝐿2β€–βˆ‡π‘€β€–5𝐿2+πΆβ€–βˆ‡π‘’β€–πΏ2β€–βˆ‡π‘€β€–πΏ132β‰€πœ‡4β€–β€–βˆ‡4𝑒‖‖2𝐿2+𝛾2β€–β€–βˆ‡4𝑀‖‖2𝐿2+𝐢(𝑒+πœ—(𝑑))14πΆπœ€.(3.23) It follows from integration by parts and Cauchy’s inequality that πœ’ξ€œβ„3βˆ‡3(βˆ‡Γ—π‘€)βˆ‡3ξ€œπ‘’π‘‘π‘₯+πœ’β„3βˆ‡3(βˆ‡Γ—π‘’)βˆ‡3β€–β€–βˆ‡π‘€π‘‘π‘₯β‰€πœ’4‖‖𝑒(𝑑)2𝐿2β€–β€–βˆ‡+πœ’3‖‖𝑀(𝑑)2𝐿2.(3.24) Combining (3.21)-(3.24) yields 12π‘‘ξ€·π‘‘π‘‘β€–βˆ‡π‘šβ€–π‘’(𝑑)2𝐿2+β€–βˆ‡π‘šβ€–π‘€(𝑑)2𝐿2ξ€Έβ€–β€–βˆ‡+(πœ‡+πœ’)π‘š+1‖‖𝑒(𝑑)2𝐿2β€–β€–βˆ‡+π›Ύπ‘š+1‖‖𝑀(𝑑)2𝐿2+πœ…β€–βˆ‡π‘šβ€–βˆ‡β‹…π‘€2𝐿2+2πœ’β€–βˆ‡π‘šβ€–π‘€(𝑑)2𝐿2≀𝐢(𝑒+πœ—(𝑑))14πΆπœ€,βˆ€π‘‡0≀𝑑<𝑇.(3.25) Taking πœ€ small enough yields π‘‘ξ‚€β€–β€–βˆ‡π‘‘π‘‘3𝑒‖‖2𝐿2+β€–β€–βˆ‡3𝑀‖‖2𝐿2≀𝐢(𝑒+πœ—(𝑑)),𝑇0≀𝑑<𝑇,(3.26) for all 𝑇0≀𝑑<𝑇.
Integrating (3.26) with respect to time from 𝑇0 to 𝜏, we have β€–β€–βˆ‡π‘’+3‖‖𝑒(𝜏)2𝐿2+β€–β€–βˆ‡3‖‖𝑀(𝜏)2𝐿2β€–β€–βˆ‡β‰€π‘’+3𝑒𝑇0ξ€Έβ€–β€–2𝐿2+β€–β€–βˆ‡3𝑀𝑇0ξ€Έβ€–β€–2𝐿2+𝐢2ξ€œπœπ‘‡0(𝑒+πœ—(𝑠))𝑑𝑠.(3.27) Owing to (3.27), we get β€–β€–βˆ‡π‘’+πœ—(𝑑)≀𝑒+3𝑒𝑇0ξ€Έβ€–β€–2𝐿2+β€–β€–βˆ‡3𝑀𝑇0ξ€Έβ€–β€–2𝐿2+𝐢2ξ€œπ‘‘π‘‡0(𝑒+πœ—(𝜏))π‘‘πœ(3.28) For all 𝑇0≀𝑑<𝑇, with help of Gronwall inequality and (3.28), we have β€–β€–βˆ‡π‘’+3‖‖𝑒(𝑑)2𝐿2+β€–β€–βˆ‡3‖‖𝑀(𝑑)2𝐿2≀𝐢,(3.29) where 𝐢 depends on β€–βˆ‡π‘’(𝑇0)β€–2𝐿2+β€–βˆ‡π‘€(𝑇0)β€–2𝐿2. From (3.29) and (3.5), we know that (𝑒,𝑀)∈𝐿∞(0,𝑇;𝐻3(ℝ3)). Thus, (𝑒,𝑀) can be extended smoothly beyond 𝑑=𝑇. We have completed the proof of Theorem 1.1.

4. Proof of Theorem 1.3

We start to estimate every term on the right of (3.9). Using integration by parts, HΓΆlder inequality, (2.8) and Young inequality, we obtainβˆ’ξ€œβ„3β‰€β€–β€–βˆ‡βˆ‡(π‘’β‹…βˆ‡π‘’)βˆ‡π‘’π‘‘π‘₯2𝑒‖‖𝐿2β€–π‘’βˆ‡π‘’β€–πΏ2≀𝐢‖𝑒‖̇𝑀2,3/π‘Ÿβ€–βˆ‡π‘’β€–Μ‡π΅π‘Ÿ2,1β€–β€–βˆ‡2𝑒‖‖𝐿2≀𝐢‖𝑒‖̇𝑀2,3/π‘Ÿβ€–βˆ‡π‘’β€–Μ‡π΅1βˆ’π‘Ÿπ‘Ÿ2,1β€–β€–βˆ‡2𝑒‖‖𝐿1+π‘Ÿ2β‰€πœ‡2β€–β€–βˆ‡2‖‖𝑒(𝑑)2𝐿2+𝐢‖𝑒‖̇𝑀2/(1βˆ’π‘Ÿ)2,3/π‘Ÿβ€–βˆ‡π‘’β€–2𝐿2.(4.1) Similarly, we have the following estimateβˆ’ξ€œβ„3β‰€β€–β€–βˆ‡βˆ‡(π‘’β‹…βˆ‡π‘€)βˆ‡π‘€π‘‘π‘₯2𝑀‖‖𝐿2β€–π‘’βˆ‡π‘€β€–πΏ2≀𝐢‖𝑒‖̇𝑀2,3/π‘Ÿβ€–βˆ‡π‘€β€–Μ‡π΅π‘Ÿ2,1β€–β€–βˆ‡2𝑀‖‖𝐿2≀𝐢‖𝑒‖̇𝑀2,3/π‘Ÿβ€–βˆ‡π‘€β€–Μ‡π΅1βˆ’π‘Ÿπ‘Ÿ2,1β€–β€–βˆ‡2𝑀‖‖𝐿1+π‘Ÿ2≀𝛾2β€–β€–βˆ‡2‖‖𝑀(𝑑)2𝐿2+𝐢‖𝑒‖̇𝑀2/(1βˆ’π‘Ÿ)2,3/π‘Ÿβ€–βˆ‡π‘€β€–2𝐿2.(4.2) Thus from (3.9), (3.10), (4.1), and (4.2), we obtainπ‘‘ξ‚€β€–π‘‘π‘‘β€–βˆ‡π‘’(𝑑)2𝐿2+β€–β€–βˆ‡π‘€(𝑑)2𝐿2ξ‚β€–β€–βˆ‡+πœ‡2‖‖𝑒(𝑑)2𝐿2β€–β€–βˆ‡+𝛾2𝑀‖‖2𝐿2+πœ…β€–βˆ‡βˆ‡β‹…π‘€β€–2𝐿2+πœ’β€–βˆ‡π‘€β€–2𝐿2‖≀𝐢𝑒‖̇𝑀2/(1βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+β€–π‘’β€–πΏβˆžξ€Έξ‚€β€–βˆ‡π‘’β€–2𝐿2+β€–βˆ‡π‘€β€–2𝐿2(1+ln(𝑒+πœ—(𝑑))),βˆ€π‘‡0≀𝑑<𝑇.(4.3) It follows from (4.3) and Gronwall’s inequality thatβ€–β€–βˆ‡π‘’(𝑑)2𝐿2β€–+β€–βˆ‡π‘€(𝑑)2𝐿2β‰€ξ‚€β€–β€–ξ€·π‘‡βˆ‡π‘’0ξ€Έβ€–β€–2𝐿2+β€–β€–ξ€·π‘‡βˆ‡π‘€0ξ€Έβ€–β€–2𝐿2ξ‚βŽ§βŽͺ⎨βŽͺβŽ©ξ€œexp𝐢(1+ln(𝑒+πœ—(𝑑)))𝑑𝑇0‖𝑒(𝜏)‖̇𝑀2/(1βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+β€–π‘’β€–πΏβˆžξ€ΈβŽ«βŽͺ⎬βŽͺβŽ­π‘‘πœβ‰€πΆ0[]}exp{πΆπœ€1+ln(𝑒+πœ—(𝑑))≀𝐢0[]}exp{2πΆπœ€ln(𝑒+πœ—(𝑑))≀𝐢0(𝑒+πœ—(𝑑))2πΆπœ€,(4.4) provided thatξ€œπ‘‘π‘‡0‖𝑒(𝜏)‖̇𝑀2/(2βˆ’π‘Ÿ)2,3/π‘Ÿξ€·1+ln𝑒+β€–π‘’β€–πΏβˆžξ€Έπ‘‘πœ<πœ€β‰ͺ1,(4.5) where 𝐢0=β€–βˆ‡π‘’(𝑇0)β€–2𝐿2+β€–βˆ‡π‘€(𝑇0)β€–2𝐿2.

From (4.4), π»π‘š estimate for Theorem 1.3 is same as that for Theorem 1.1. Thus, Theorem 1.3 is proved.

Acknowledgments

This work was supported in part by the NNSF of China (Grant no. 11101144) and Research Initiation Project for High-level Talents (201031) of North China University of Water Resources and Electric Power.