Abstract and Applied Analysis

Volume 2012 (2012), Article ID 420721, 8 pages

doi:10.1155/2012/420721

Research Article

Logarithmically Improved Regularity Criteria for a Fluid System with the Linear Soret Effect

Minglei Zang¹ and Xiaoyan Guan²

¹School of Mathematics and Information Science, Yantai University, Yantai 264005, China

²State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, China

Received 27 July 2012; Accepted 21 August 2012

Academic Editor: Xinguang Zhang

Copyright © 2012 Minglei Zang and Xiaoyan Guan. 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.

We consider the 3D fluid system with the linear Soret effect. We obtain a logarithmically improved regularity criterion in the BMO space.

1. Introduction and Main Results

In this paper, we consider the regularity of the following fluid system with the linear Soret effect:

where denotes the fluid velocity vector field, is the scalar pressure, is the scalar temperature, and is the concentration field, , while , , and are the given initial velocity, initial temperature, and initial concentration, respectively, with . The term in (1.1) is the linear Soret effect (see page 102 in [1], [2–4]).

The question of global existence or blow-up in finite time 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 magnetohydrodynamics (MHD) equations. This challenging problem has attracted significant attention. In the absence of the global well-posedness, the development of blow-up or non-blow-up theory is of major importance for both theoretical and practical purposes.

When , (1.1) is the well-known Euler system, Beale-Kato-Majda [5] and Kozono-Taniuchi [6] showed the following regularity criteria, respectively:

When , (1.1) is the well-known Boussinesq system with zero viscosity, Fan and Zhou [7] proved the following blow-up criterion:

Here denotes the homogeneous Besov space.

Recently, Chan and Vasseur [8] and Zhou and Lei [9] proved some logarithmically improved regularity criterion for the 3D Navier-Stokes equations. Qiu et al. [10] obtained Serrin-type blow-up criteria of smooth solution for the 3D viscous Boussinesq equations. They showed that smooth solution for remains smooth at time , provided that the following condition holds:

Motivated by the previous results on the regularity criteria of the fluid dynamics equations, the purpose of this paper is to establish a logarithmically improved regularity criterion in terms of the vorticity field for (1.1) in the BMO space, which is defined in Section 2.

Now we state our main results as follows.

Theorem 1.1. Let , and with in . Assume that is a local smooth solution of (1.1). Then the solution can be smoothly extended after time provided that (1.4) is satisfied.

Remark 1.2. In the process of proof in [7, 10], they used the important fact that

Due to the linear Soret effect, we cannot prove

Thus the method in [7, 10] cannot be used here. We need new interpolation inequalities.

Remark 1.3. In [10], authors researched the regularity for the 3D viscous Boussinesq equations under condition (1.4). However, when , (1.1) is the well-known Boussinesq system with zero viscosity, so our results improve result of [10].

2. Preliminaries and Lemmas

First, we recall some definitions and lemmas, which play an important role in studying the regularity of solution to partial differential equations.

Definition 2.1. denotes the space of functions of bounded mean oscillation of John and Nirenberg associated with the norm

In order to prove Theorem 1.1, we need the following Gagliardo-Nirenberg inequality.

Lemma 2.2. There exists a uniform positive constant such that

holds for all .

Lemma 2.3 (see [9]). There exists a uniform positive constant such that

Lemma 2.4 (see [11]). The following calculus inequality holds:

with , and .

Lemma 2.5 (see [12]). In three-dimensional space, the following inequalities

hold.

3. Proofs of the Main Results

In this section, we prove Theorem 1.1.

Proof of Theorem 1.1. Multiplying both the sides of the second equation of (1.1) by and the third equation of (1.1) by , respectively, and integrating by parts over , we obtain

Combining (3.1) with (3.2), and using Gronwall's inequality, we infer that

Multiplying both the sides of the first equation of (1.1) by , and integrating by parts over , by (3.3), we get

which implies

Next we go to estimate -norm of , and . Multiplying the first equation, the second equation and the third equation of (1.1) by , , and , respectively, then integrating by parts over , we deduce that

Now combining (3.6), (3.7) with (3.8) and using Gronwall's inequality, we obtain

Noting (1.4), one concludes that for any small constant , there exists such that

Set

where .

It follows from (3.9), (3.10), and Lemma 2.3 that

where depends on , and is an absolute positive constant.

Finally we go to estimate for -norm of , , and . Applying the operation on both the sides of the first equation of (1.1), then multiplying , and integrating by parts over , by (2.4), (3.12), Hölder's inequality, and Young's inequality, we have

Similarly, by (2.4), Hölder's inequality, and Young's inequality, we obtain

Combining Lemma 2.5, Young's inequality with (3.12), we deduce

Using Lemmas 2.2 and 2.5, Young's inequality, and (3.12), we obtain

Inserting the above estimates and into (3.14), we get

provided that , which can be achieved by the absolute continuous property of integral (1.4).

Similarly, for the -equation, we have

Combining (3.13), (3.17) with (3.18), we easily conclude that

By Lemma 2.3 and (3.19), we infer that

Using Gronwall's inequality, we have

Therefore, can be bounded provided that

This completes the proof of Theorem 1.1.

Appendix

In this section, we give the following local well-posedness of the system (1.1).

Theorem A.1 (local well-posedness). Let , and . Then there exist and a unique solution of the system (1.1) such that

Proof. The proof is similar to Theorem 3.1 in [4, 13], so we omit it here.

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