Regularity Criteria for a Turbulent Magnetohydrodynamic Model
Yong Zhou1and Jishan Fan2
Academic Editor: Simeon Reich
Received23 May 2011
Accepted12 Jul 2011
Published08 Sept 2011
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 βfilteredβ fluid velocity, is the pressure, is the magnetic field, and is the βfilteredβ 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)β(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)β(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)β(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)β(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)β(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.
Since it is easy to prove that the problem (1.1)β(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
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, βLarge eddy simulation for turbulent magnetohydrodynamic flows,β Journal of Mathematical Analysis and Applications, vol. 377, no. 2, pp. 516β533, 2011.
T. Kato and G. Ponce, βCommutator estimates and the Euler and Navier-Stokes equations,β Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891β907, 1988.