Abstract and Applied AnalysisVolume 2011 (2011), Article ID 380402, 9 pagesdoi:10.1155/2011/380402

Research Article

Regularity Criteria for a Turbulent Magnetohydrodynamic Model

Yong Zhou¹ and Jishan Fan²

¹Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China²Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China

Received 23 May 2011; Accepted 12 July 2011

Academic Editor: Simeon Reich

Copyright © 2011 Yong Zhou and Jishan Fan. 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 establish some regularity criteria for a turbulent magnetohydrodynamic model. As a corollary, we prove that the smooth solution exists globally when the spatial dimension satisfies .

1. Introduction

In this paper, we study the following simplified turbulent MHD model [1]: Here is the fluid velocity field, is the “filtered” fluid velocity, is the pressure, is the magnetic field, and is the “filtered” magnetic field. is the length scale parameter that represents the width of the filter. For simplicity we will take .

When , the global well-posedness of the problem has been proved in [1]. When , (1.1) and (1.4) is the well-known Bardina model. Very recently, the authors [2] have proved that the Bardina model has a unique global-in-time weak solution when . Here we wold like to point out that by the same arguments, we can prove the following.

Theorem 1.1. Let . Let with in . Then for any , the problem (1.1)–(1.5) has a unique weak solution satisfying

The proof for Theorem 1.1 is similar to that for the Bardina model in [2], so we omit it here.

The aim of this paper is to study the regularity of the weak solutions. We will prove

Theorem 1.2. Let . Let with and in . Let be a local smooth solution to the problem (1.1)–(1.5) satisfying for any fixed . Then can be extended beyond provided that one of the following condition is satisfied:

By (1.6) and (1.8), as a corollary, we have the following

Corollary 1.3. Let . Let with and in . Then for any , the problem (1.1)–(1.5) has a unique smooth solution satisfying (1.7).

When or 10, we can get a better result as follows.

Theorem 1.4. Let or 10 and let and in . Let be a local smooth solution to the problem (1.1)–(1.5) satisfying for any fixed . Then can be extended beyond if one of the following conditions is satisfied:

Remark 1.5. If we delete the harmless lower order terms and in (1.1) and (1.2), then we have then the system (1.15) has the following property: if is a solution of (1.15), then for all , is also a solution. In this sense, our conditions (1.8) and (1.11)–(1.14) are scaling invariant (optimal). Equations (1.8) and (1.12) do not hold true for . But we also can establish regularity criteria for in nonscaling invariant forms.

In Section 2, we will prove Theorem 1.2. In Section 3, we will prove Theorem 1.4.

2. Proof of Theorem 1.2

Since it is easy to prove that the problem (1.1)–(1.5) has a unique local smooth solution, we only need to establish the a priori estimates. The proof of the case is easier and similar and thus we omit the details here, we only deal with the case .

Testing (1.1) by , using (1.3) and (1.4), we find that Testing (1.2) by , using (1.3) and (1.4), we see that

Summing up (2.1) and (2.2), we easily get (1.6).(I)Let (1.8) hold true.

In the following calculations, we will use the product estimates due to Kato and Ponce [3]: with , and .

The proof of the case is easier and similar, we omit the details here. Now we assume .

Applying to (1.1), testing by , using (1.4), we deduce that Similarly, applying to (1.2), testing by , using (1.4), we infer that Summing up (2.4) and (2.5), using (2.3), we get which implies Here we have used the following Gagliardo-Nirenberg inequalities:(II)Let (1.9) hold true.

In the following calculations, we will use the following commutator estimates due to Kato and Ponce [3]: with and .

The proof of the case is easier and similar, we omit the details here. Now we assume .

Applying to (1.1), testing by , and using (1.3) and (1.4), we deduce that

Applying to (1.2), testing by , using (1.3) and (1.4), we infer that

Summing up (2.10) and (2.11), noting that the last terms of (2.10) and (2.11) disappeared, and using (2.9), we obtain which yields Here we have used the following Gagliardo-Nirenberg inequality: with

This completes the proof.

3. Proof of Theorem 1.4

We only need to prove the a priori estimates.

Testing (1.1) and (1.2) by , using (1.3) and (1.4), and summing up the results, we have

Using (1.4), we see that

Inserting the above estimates into (3.1), noting that the last term of and disappeared, we have

The proofs of the cases (1.11) and (1.13) are similar, we omit the details here.(I)Let (1.14) hold true,

Inserting the above estimates into (3.3) and using the Gronwall inequality yields

Here we have used the Gagliardo-Nirenberg inequality: with (II)Let (1.12) hold true.

Integrating by parts, using (1.4) we have which yields (3.5).

Here we have used the Gagliardo-Nirenberg inequalities:

This completes the proof.

Acknowledgments

This paper is partially supported by Zhejiang Innovation Project (Grant no. T200905), ZJNSF (Grant no. R6090109), and NSFC (Grant no. 10971197).

References

A. Labovsky and C. Trenchea, “Large eddy simulation for turbulent magnetohydrodynamic flows,” Journal of Mathematical Analysis and Applications, vol. 377, no. 2, pp. 516–533, 2011.

Y. Zhou and J. Fan, “Global well-posedness of a Bardina model,” Applied Mathematics Letters, vol. 24, no. 5, pp. 605–607, 2011.

T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988.