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Journal of Applied Mathematics
Volume 2012, Article ID 541203, 13 pages
http://dx.doi.org/10.1155/2012/541203
Research Article

Logarithmically Improved Blow up Criterion for Smooths Solution to the 3D Micropolar Fluid Equations

1School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
2Department of Mathematical and Physical Sciences, Henan Institute of Engineering, Zhengzhou 451191, China

Received 8 January 2012; Accepted 24 April 2012

Academic Editor: Mina Abd-El-Malek

Copyright © 2012 Yin-Xia Wang and Hengjun Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 3𝜕𝑡𝜕𝑢(𝜇+𝜒)Δ𝑢+𝑢𝑢+𝑝𝜒×𝑤=0,𝑡𝑤𝛾Δ𝑤𝜅𝑤+2𝜒𝑤+𝑢𝑤𝜒×𝑢=0,𝑢=0(1.1) with the initial value𝑡=0𝑢=𝑢0(𝑥),𝑤=𝑤0(𝑥),(1.2) where 𝑢(𝑡,𝑥), 𝑤(𝑡,𝑥), and 𝑝(𝑡,𝑥) stand for the velocity field, microrotation field, and the scalar pressure, respectively. And 𝜈>0 is the Newtonian kinetic viscosity, 𝜅>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 [38]). 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 𝜅=0 and 𝑤=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 [1126]). 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 𝑢0,𝑤0𝐻𝑚(3)(𝑚3) with 𝑢0=0. Assume that (𝑢,𝑤) is a smooth solution to (1.1) and (1.2) on [0, 𝑇). If 𝑢 satisfies 𝑇0𝑢(𝑡)̇𝑀2/(1𝑟)2,3/𝑟1+ln𝑒+𝑢(𝑡)𝐿𝑑𝑡<,0<𝑟<1,(1.3) then the solution (𝑢,𝑤) can be extended beyond 𝑡=𝑇.

We have the following corollary immediately.

Corollary 1.2. Let 𝑢0,𝑤0𝐻𝑚(3)(𝑚3) with 𝑢0=0. Assume that (𝑢,𝑤) is a smooth solution to (1.1) and (1.2) on [0, 𝑇). Suppose that 𝑇 is the maximal existence time, then 𝑇0𝑢(𝑡)̇𝑀2/(1𝑟)2,3/𝑟1+ln𝑒+𝑢(𝑡)𝐿𝑑𝑡=,0<𝑟<1.(1.4)

Theorem 1.3. Let 𝑢0,𝑤0𝐻𝑚(3)(𝑚3) with 𝑢0=0. Assume that (𝑢,𝑤) is a smooth solution to (1.1) and (1.2) on [0, 𝑇). If 𝑢 satisfies 𝑇0𝑢(𝑡)̇𝑀2/(2𝑟)2,3/𝑟1+ln𝑒+𝑢(𝑡)𝐿𝑑𝑡<,0<𝑟<1,(1.5) then the solution (𝑢,𝑤) can be extended beyond 𝑡=𝑇.

One has the following corollary immediately.

Corollary 1.4. Let 𝑢0,𝑤0𝐻𝑚(3)(𝑚3) with 𝑢0=0. Assume that (𝑢,𝑤) is a smooth solution to (1.1) and (1.2) on [0, 𝑇). Suppose that 𝑇 is the maximal existence time, then 𝑇0𝑢(𝑡)̇𝑀2/(2𝑟)2,3/𝑟1+ln𝑒+𝑢(𝑡)𝐿𝑑𝑡=,0<𝑟<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<𝑝𝑞+, the Morrey-Campanato space ̇𝑀𝑝,𝑞 is defined by ̇𝑀𝑝,𝑞=𝑓𝐿𝑝loc3𝑓̇𝑀𝑝,𝑞=sup𝑥3sup𝑅>0𝑅3/𝑞3/𝑝𝑓𝐿𝑝(𝐵(𝑥,𝑅))<,(2.1) where 𝐵(𝑥,𝑅) denotes the ball of center 𝑥 with radius 𝑅.
It is easy to verify that ̇𝑀𝑝,𝑞 is a Banach space under the norm ̇𝑀𝑝,𝑞. Furthermore, it is easy to check the following: 𝑓(𝜆)̇𝑀𝑝,𝑞=𝜆3/𝑞𝑓̇𝑀𝑝,𝑞,𝜆>0.(2.2) Morrey-Campanato spaces can be seen as a complement to 𝐿𝑝 spaces. In fact, for 𝑝𝑞, one has 𝐿𝑞=̇𝑀𝑞,𝑞̇𝑀𝑝,𝑞.(2.3) one has the following comparison between Lorentz spaces and Morrey-Campanato spaces: for 𝑝2, 𝐿3/𝑟3𝐿3/𝑟,3̇𝑀𝑝,3/𝑟3,(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 0𝑟<3/2, the space ̇𝑍𝑟 is defined as the space of 𝑓(𝑥)𝐿2loc(3) such that 𝑓̇𝑍𝑟=sup𝑔̇𝐵𝑟2,11𝑓𝑔𝐿2<.(2.5) Then ̇𝑀𝑓2,3/𝑟 if and only if ̇𝑍𝑓𝑟 with equivalence of norms. And the fact that 𝐿2̇𝐻1̇𝐵𝑟2,1̇𝐻𝑟,0<𝑟<1,(2.6) one has ̇𝑋𝑟̇𝑀2,3/𝑟,(2.7) where ̇𝑋𝑟 denotes the pointwise multiplier space from ̇𝐻𝑟 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<𝑟<1, the inequality 𝑓̇𝐵𝑟2,1𝐶𝑓𝐿1𝑟2𝑓𝑟𝐿2(2.8) holds, where 𝐶 is a positive constant that depends on 𝑟.

Proof. It follows from the definition of Besov spaces that 𝑓̇𝐵𝑟2,1=𝑖2𝑖𝑟Δ𝑖𝑓𝐿2𝑖𝑗2𝑖𝑟Δ𝑖𝑓𝐿2+𝑖>𝑗2𝑖(𝑟1)2𝑖Δ𝑖𝑓𝐿2𝑖𝑗22𝑖𝑟1/2𝑖𝑗Δ𝑖𝑓2𝐿21/2+𝑖𝑗22𝑖(𝑟1)1/2𝑖>𝑗22𝑖Δ𝑖𝑓2𝐿21/22𝐶𝑗𝑟𝑓𝐿2+2𝑗(𝑟1)𝑓̇𝐻12=𝐶𝑗𝑟𝐴𝑟+2𝑗(𝑟1)𝐴1𝑟𝑓𝐿1𝑟2𝑓𝑟̇𝐻1,(2.9) where 𝐴=(𝑓̇𝐻1)/(𝑓𝐿2). Choosing 𝑗 such that 1/22𝑗𝑟𝐴𝑟1, from (2.9) we get 𝑓̇𝐵𝑟2,11+2𝑗(𝑟1)𝐴1𝑟𝑓𝐿1𝑟2𝑓𝑟̇𝐻11𝐶1+21/𝑟𝑓𝐿1𝑟2𝑓𝑟𝐿2.(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,𝑔𝑊𝑚,𝑝, and 1<𝑞, 1<𝑟<, one has 𝛼(𝑓𝑔)𝑓𝛼𝑔𝐿𝑝𝐶𝑓𝐿𝑞1𝛼1𝑔𝐿𝑟1+𝑔𝐿𝑞2𝛼𝑓𝐿𝑟2,(2.11) where 1𝛼𝑚 and 1/𝑝=1/𝑞1+1/𝑟1=1/𝑞2+1/𝑟2.

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 𝑓𝐿4𝐶𝑓𝐿1/822𝑓𝐿7/82𝑓𝐿4𝐶𝑓𝐿3/423𝑓𝐿1/422𝑓𝐿4𝐶𝑓𝐿1/1223𝑓𝐿11/1222𝑓𝐿2𝐶𝑓𝐿1/323𝑓𝐿2/32(2.12) hold.

3. Proof of Theorem 1.1

Proof. Multiplying the first equation of (1.1) by 𝑢 and integrating with respect to 𝑥 over 3, using integration by parts, we obtain 12𝑑(𝑑𝑡𝑢𝑡)2𝐿2+(𝜇+𝜒)𝑢(𝑡)2𝐿2=𝜒3(×𝑤)𝑢𝑑𝑥(3.1) Similarly, we get 12𝑑(𝑑𝑡𝑤𝑡)2𝐿2+𝛾𝑤(𝑡)2𝐿2+𝜅𝑤2𝐿2+2𝜒𝑤2𝐿2=𝜒3(×𝑢)𝑤𝑑𝑥.(3.2) Summing up (3.1) and (3.2), we deduce thats 12𝑑𝑑𝑡𝑢(𝑡)2𝐿2+𝑤(𝑡)2𝐿2+(𝜇+𝜒)𝑢(𝑡)2𝐿2+𝛾𝑤(𝑡)2𝐿2+𝜅𝑤2𝐿2+2𝜒𝑤2𝐿2=𝜒3(×𝑤)𝑢𝑑𝑥+𝜒3(×𝑢)𝑤𝑑𝑥.(3.3) Using integration by parts and Cauchy’s inequality, we obtain 𝜒3(×𝑤)𝑢𝑑𝑥+𝜒3(×𝑢)𝑤𝑑𝑥𝜒𝑢2𝐿2+𝜒𝑤2𝐿2.(3.4) Combining (3.3) and (3.4) yields 12𝑑𝑑𝑡𝑢(𝑡)2𝐿2+𝑤(𝑡)2𝐿2+𝜇𝑢(𝑡)2𝐿2+𝛾𝑤(𝑡)2𝐿2+𝜅𝑤2𝐿2+𝜒𝑤2𝐿20.(3.5) Integrating with respect to 𝑡, we have (𝑢𝑡)2𝐿2+𝑤(𝑡)2𝐿2+2𝑡0𝜇𝑢(𝜏)2𝐿2+𝛾𝑤(𝜏)2𝐿2𝑑𝜏+2𝜅𝑡0𝑤(𝜏)2𝐿2𝑑𝜏+2𝜒𝑡0𝑤(𝜏)2𝐿2𝑢𝑑𝜏02𝐿2+𝑤02𝐿2.(3.6)
Taking to the first equation of (1.1), then multiplying the resulting equation by 𝑢 and using integration by parts, we obtain 12𝑑(𝑑𝑡𝑢𝑡)2𝐿2+(𝜇+𝜒)2𝑢(𝑡)2𝐿2=3(𝑢𝑢)𝑢𝑑𝑥+𝜒3(×𝑤)𝑢𝑑𝑥.(3.7) Similarly, we get 12𝑑𝑑𝑡𝑤(𝑡)2𝐿2+𝛾2𝑤(𝑡)2𝐿2+𝜅𝑤2𝐿2+2𝜒𝑤2𝐿2=3(𝑢𝑤)𝑤𝑑𝑥+𝜒3(×𝑢)𝑤𝑑𝑥.(3.8) Combining (3.7) and (3.8) yields 12𝑑𝑑𝑡𝑢(𝑡)2𝐿2+𝑤(𝑡)2𝐿2+(𝜇+𝜒)2𝑢(𝑡)2𝐿2+𝛾2𝑤(𝑡)2𝐿2+𝜅𝑤2𝐿2+2𝜒𝑤2𝐿2=3(𝑢𝑢)𝑢𝑑𝑥+𝜒3(×𝑤)𝑢𝑑𝑥3(𝑢𝑤)𝑤𝑑𝑥+𝜒3(×𝑢)𝑤𝑑𝑥.(3.9) Using integration by parts and Cauchy’s inequality, we obtain 𝜒3(×𝑤)𝑢𝑑𝑥+𝜒3(×𝑢)𝑤𝑑𝑥𝜒2𝑢2𝐿2+𝜒𝑤2𝐿2.(3.10) Using Hölder’s inequality, (2.8), and Young’s inequality, we obtain 3(𝑢𝑢)𝑢𝑑𝑥𝑢𝐿2𝑢𝑢𝐿2𝐶𝑢̇𝑀2,3/𝑟𝑢̇𝐵𝑟2,1𝑢𝐿2𝐶𝑢̇𝑀2,3/𝑟𝑢̇𝐵2𝑟𝑟2,12𝑢𝑟𝐿2𝜇22𝑢(𝑡)2𝐿2+𝐶𝑢̇𝑀2/(2𝑟)2,3/𝑟𝑢2𝐿2.(3.11) Similarly, we have the following estimate: 3(𝑢𝑤)𝑤𝑑𝑥𝑤𝐿2𝑢𝑤𝐿2𝐶𝑢̇𝑀2,3/𝑟𝑤̇𝐵𝑟2,1𝑤𝐿2𝐶𝑢̇𝑀2,3/𝑟𝑤̇𝐵2𝑟𝑟2,12𝑤𝑟𝐿2𝛾22𝑤(𝑡)2𝐿2+𝐶𝑢̇𝑀2/(2𝑟)2,3/𝑟𝑤2𝐿2.(3.12) Combining (3.9)-(3.12) yields 𝑑𝑑𝑡𝑢(𝑡)2𝐿2+𝑤(𝑡)2𝐿2+𝜇2𝑢(𝑡)2𝐿2+𝛾2𝑤2𝐿2+𝜅𝑤2𝐿2+𝜒𝑤2𝐿2𝐶𝑢̇𝑀2/(2𝑟)2,3/𝑟𝑢2𝐿2+𝑤2𝐿2𝐶𝑢̇𝑀2/(2𝑟)2,3/𝑟1+ln𝑒+𝑢𝐿𝑢2𝐿2+𝑤2𝐿21+ln𝑒+𝑢𝐿𝐶𝑢̇𝑀2/(2𝑟)2,3/𝑟1+ln𝑒+𝑢𝐿𝑢2𝐿2+𝑤2𝐿21+ln𝑒+3𝑢𝐿2+3𝑤𝐿2,(3.13) where we have used 𝐻23𝐿3.(3.14) For any 𝑇0𝑡𝑇, we set 𝜗(𝑡)=sup𝑇0𝜏𝑡3𝑢(𝜏)2𝐿2+3𝑤(𝜏)2𝐿2.(3.15) Thus, from (3.13), we have 𝑑𝑑𝑡𝑢(𝑡)2𝐿2+𝑤(𝑡)2𝐿2+𝜇2𝑢(𝑡)2𝐿2+𝛾2𝑤2𝐿2+𝜅𝑤2𝐿2+𝜒𝑤2𝐿2𝐶𝑢̇𝑀2/(2𝑟)2,3/𝑟1+ln𝑒+𝑢𝐿𝑢2𝐿2+𝑤2𝐿2(1+ln(𝑒+𝜗(𝑡))),𝑇0𝑡<𝑇.(3.16) It follows from (3.8) and Gronwall’s inequality that 𝑢(𝑡)2𝐿2+𝑤(𝑡)2𝐿2𝑇𝑢02𝐿2+𝑇𝑤02𝐿2exp𝐶(1+ln(𝑒+𝜗(𝑡)))𝑡𝑇0𝑢(𝜏)̇𝑀2/(2𝑟)2,3/𝑟1+ln𝑒+𝑢𝐿𝑑𝜏𝐶0[]}exp{𝐶𝜀1+ln(𝑒+𝜗(𝑡))𝐶0[]}exp{2𝐶𝜀ln(𝑒+𝜗(𝑡))𝐶0(𝑒+𝜗(𝑡))2𝐶𝜀,(3.17) provided that 𝑡𝑇0𝑢(𝜏)̇𝑀2/(2𝑟)2,3/𝑟1+ln𝑒+𝑢𝐿𝑑𝜏<𝜀1,(3.18) where 𝐶0=𝑢(𝑇0)2𝐿2+𝑤(𝑇0)2𝐿2.
Applying 𝑚 to the first equation of (1.1), then multiplying the resulting equation by 𝑚𝑢 and using integration by parts, we have 12𝑑𝑑𝑡𝑚𝑢(𝑡)2𝐿2+(𝜇+𝜒)𝑚+1𝑢(𝑡)2𝐿2=3𝑚(𝑢𝑢)𝑚𝑢𝑑𝑥+𝜒3𝑚(×𝑤)𝑚𝑢𝑑𝑥.(3.19) Likewise, from the second equation of (1.1), we obtain 12𝑑𝑑𝑡𝑚𝑤(𝑡)2𝐿2+𝛾𝑚+1𝑤(𝑡)2𝐿2+𝜅𝑚𝑤2𝐿2+2𝜒𝑚𝑤(𝑡)2𝐿2=3𝑚(𝑢𝑤)𝑚𝑤𝑑𝑥+𝜒3𝑚(×𝑢)𝑚𝑤𝑑𝑥.(3.20) Using 𝑢=0 and (3.19) and (3.20), we have 12𝑑𝑑𝑡𝑚𝑢(𝑡)2𝐿2+𝑚𝑤(𝑡)2𝐿2+(𝜇+𝜒)𝑚+1𝑢(𝑡)2𝐿2+𝛾𝑚+1𝑤(𝑡)2𝐿2+𝜅𝑚𝑤2𝐿2+2𝜒𝑚𝑤(𝑡)2𝐿2=3𝑚(𝑢𝑢)𝑢𝑚𝑢𝑚𝑢𝑑𝑥+𝜒3𝑚(×𝑤)𝑚𝑢𝑑𝑥3𝑚(𝑢𝑤)𝑢𝑚𝑤𝑚𝑤𝑑𝑥+𝜒3𝑚(×𝑢)𝑚𝑤𝑑𝑥.(3.21) In what follows, for simplicity, we will set 𝑚=3.
By Hölder’s inequality, (2.11), (2.12), and Young’s inequality, we obtain 33(𝑢𝑢)𝑢3𝑢3𝑢𝑑𝑥3(𝑢𝑢)𝑢3𝑢𝐿23𝑢𝐿2𝐶𝑢𝐿43𝑢𝐿43𝑢𝐿2𝐶𝑢𝐿3/424𝑢𝐿1/42𝑢𝐿1/1224𝑢𝐿11/122𝑢𝐿1/324𝑢𝐿2/32𝐶𝑢𝐿7/624𝑢𝐿11/62𝜇44𝑢2𝐿2+𝐶𝑢𝐿142𝜇44𝑢2𝐿2+𝐶(𝑒+𝜗(𝑡))14𝐶𝜀,(3.22)33(𝑢𝑤)𝑢3𝑤3𝑤𝑑𝑥3(𝑢𝑤)𝑢3𝑤𝐿23𝑤𝐿2𝐶𝑢𝐿43𝑤𝐿43𝑤𝐿2+𝑤𝐿43𝑢𝐿43𝑤𝐿2𝐶𝑢𝐿3/424𝑢𝐿1/42𝑤𝐿1/1224𝑤𝐿11/122𝑤𝐿1/324𝑤𝐿2/32+𝐶𝑤𝐿3/424𝑤𝐿1/42𝑢𝐿1/1224𝑢𝐿11/122𝑤𝐿1/324𝑤𝐿2/32𝜇44𝑢2𝐿2+𝛾24𝑤2𝐿2+𝐶𝑢9𝐿2𝑤5𝐿2+𝐶𝑢𝐿2𝑤𝐿132𝜇44𝑢2𝐿2+𝛾24𝑤2𝐿2+𝐶(𝑒+𝜗(𝑡))14𝐶𝜀.(3.23) It follows from integration by parts and Cauchy’s inequality that 𝜒33(×𝑤)3𝑢𝑑𝑥+𝜒33(×𝑢)3𝑤𝑑𝑥𝜒4𝑢(𝑡)2𝐿2+𝜒3𝑤(𝑡)2𝐿2.(3.24) Combining (3.21)-(3.24) yields 12𝑑𝑑𝑡𝑚𝑢(𝑡)2𝐿2+𝑚𝑤(𝑡)2𝐿2+(𝜇+𝜒)𝑚+1𝑢(𝑡)2𝐿2+𝛾𝑚+1𝑤(𝑡)2𝐿2+𝜅𝑚𝑤2𝐿2+2𝜒𝑚𝑤(𝑡)2𝐿2𝐶(𝑒+𝜗(𝑡))14𝐶𝜀,𝑇0𝑡<𝑇.(3.25) Taking 𝜀 small enough yields 𝑑𝑑𝑡3𝑢2𝐿2+3𝑤2𝐿2𝐶(𝑒+𝜗(𝑡)),𝑇0𝑡<𝑇,(3.26) for all 𝑇0𝑡<𝑇.
Integrating (3.26) with respect to time from 𝑇0 to 𝜏, we have 𝑒+3𝑢(𝜏)2𝐿2+3𝑤(𝜏)2𝐿2𝑒+3𝑢𝑇02𝐿2+3𝑤𝑇02𝐿2+𝐶2𝜏𝑇0(𝑒+𝜗(𝑠))𝑑𝑠.(3.27) Owing to (3.27), we get 𝑒+𝜗(𝑡)𝑒+3𝑢𝑇02𝐿2+3𝑤𝑇02𝐿2+𝐶2𝑡𝑇0(𝑒+𝜗(𝜏))𝑑𝜏(3.28) For all 𝑇0𝑡<𝑇, with help of Gronwall inequality and (3.28), we have 𝑒+3𝑢(𝑡)2𝐿2+3𝑤(𝑡)2𝐿2𝐶,(3.29) where 𝐶 depends on 𝑢(𝑇0)2𝐿2+𝑤(𝑇0)2𝐿2. From (3.29) and (3.5), we know that (𝑢,𝑤)𝐿(0,𝑇;𝐻3(3)). Thus, (𝑢,𝑤) can be extended smoothly beyond 𝑡=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, Hölder inequality, (2.8) and Young inequality, we obtain3(𝑢𝑢)𝑢𝑑𝑥2𝑢𝐿2𝑢𝑢𝐿2𝐶𝑢̇𝑀2,3/𝑟𝑢̇𝐵𝑟2,12𝑢𝐿2𝐶𝑢̇𝑀2,3/𝑟𝑢̇𝐵1𝑟𝑟2,12𝑢𝐿1+𝑟2𝜇22𝑢(𝑡)2𝐿2+𝐶𝑢̇𝑀2/(1𝑟)2,3/𝑟𝑢2𝐿2.(4.1) Similarly, we have the following estimate3(𝑢𝑤)𝑤𝑑𝑥2𝑤𝐿2𝑢𝑤𝐿2𝐶𝑢̇𝑀2,3/𝑟𝑤̇𝐵𝑟2,12𝑤𝐿2𝐶𝑢̇𝑀2,3/𝑟𝑤̇𝐵1𝑟𝑟2,12𝑤𝐿1+𝑟2𝛾22𝑤(𝑡)2𝐿2+𝐶𝑢̇𝑀2/(1𝑟)2,3/𝑟𝑤2𝐿2.(4.2) Thus from (3.9), (3.10), (4.1), and (4.2), we obtain𝑑𝑑𝑡𝑢(𝑡)2𝐿2+𝑤(𝑡)2𝐿2+𝜇2𝑢(𝑡)2𝐿2+𝛾2𝑤2𝐿2+𝜅𝑤2𝐿2+𝜒𝑤2𝐿2𝐶𝑢̇𝑀2/(1𝑟)2,3/𝑟1+ln𝑒+𝑢𝐿𝑢2𝐿2+𝑤2𝐿2(1+ln(𝑒+𝜗(𝑡))),𝑇0𝑡<𝑇.(4.3) It follows from (4.3) and Gronwall’s inequality that𝑢(𝑡)2𝐿2+𝑤(𝑡)2𝐿2𝑇𝑢02𝐿2+𝑇𝑤02𝐿2exp𝐶(1+ln(𝑒+𝜗(𝑡)))𝑡𝑇0𝑢(𝜏)̇𝑀2/(1𝑟)2,3/𝑟1+ln𝑒+𝑢𝐿𝑑𝜏𝐶0[]}exp{𝐶𝜀1+ln(𝑒+𝜗(𝑡))𝐶0[]}exp{2𝐶𝜀ln(𝑒+𝜗(𝑡))𝐶0(𝑒+𝜗(𝑡))2𝐶𝜀,(4.4) provided that𝑡𝑇0𝑢(𝜏)̇𝑀2/(2𝑟)2,3/𝑟1+ln𝑒+𝑢𝐿𝑑𝜏<𝜀1,(4.5) where 𝐶0=𝑢(𝑇0)2𝐿2+𝑤(𝑇0)2𝐿2.

From (4.4), 𝐻𝑚 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.

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