On the regularity criterion for the 3D generalized MHD equations in Besov spaces
© Zhang and Qiu; licensee Springer. 2014
Received: 12 May 2014
Accepted: 2 July 2014
Published: 24 September 2014
In this paper, we consider the three-dimensional generalized MHD equations, a system of equations resulting from replacing the Laplacian in the usual MHD equations by a fractional Laplacian . We obtain a regularity criterion of the solution for the generalized MHD equations in terms of the summation of the velocity field and the magnetic field by means of the Littlewood-Paley theory and the Bony paradifferential calculus, which extends the previous result.
MSC: 76B03, 76D03.
where denotes the fluid velocity vector field, is the magnetic field, is the scalar pressure; while and are the given initial velocity and initial magnetic fields, respectively, in the sense of distributions, with ; are the parameters. The generalized MHD equations generalize the usual MHD equations by replacing the Laplacian by a general fractional Laplacian , . As , the generalized MHD equations reduce to the usual MHD equations; when and , the generalized MHD equations reduce to the Navier-Stokes equations. Moreover, it has similar scaling properties and energy estimate to the Navier-Stokes equations and the MHD equations. The study of system (1) will improve our understanding of the Navier-Stokes equations and the MHD equations.
with , , , provided that .
In this paper, we are interested in what kind of confluence the integrability of or brings to the weak solution of the system (1). Furthermore, we shall make efforts to establish some new regularity criterion of weak solutions of the system (3) in terms of or . When , He and Wang , Gala , Dong et al. establish some regularity criteria in Lorentz spaces, the multiplier space, and the nonhomogeneous Besov space, respectively.
The purpose of this paper is to deal with the case of the system (3), to establish certain kind of regularity criteria. The tools we use here are the Littlewood-Paley theory and the Bony paraproduct decomposition. Before stating our main result, we firstly recall the definition of weak solutions to the 3D generalized MHD equations (1) as follows.
and in the sense of a distribution;
- (3)for any with and , one has
The weak solution to the system (3) can be defined in a similar way as follows.
and in the sense of a distribution;
- (3)for any with and , one has
From the above two definitions as regards weak solutions, the systems (1) and (3) are equivalent. It is easy to see that also verifies the system (3) in the sense of distribution, provided that is a weak solution to the system (1) as .
Now our main result can be stated.
with, , , . Then the weak solutionremains smooth on.
Our result here improves the recent result obtained by Dong et al. as except that . However, the method of  cannot also be applied to the case in this paper. We shall consider this problem in the future.
Throughout the paper, stands for generic constant. We use the notation to denote the relation , and the notation to denote the relations and . For convenience, given a Banach space , we denote its norm by . If there is no ambiguity, we omit the domain of function spaces.
This paper is structured as follows. In Section 2, we introduce the Littlewood-Paley decomposition and the Bony paradifferential calculus. In Section 3, we give the proof of Theorem 1.3 by means of the Littlewood-Paley theory and the Bony paradifferential calculus.
2 The Littlewood-Paley theory
In this section, we will provides the Littlewood-Paley theory and the related facts.
and the set of temperate distributions is the dual set of for the usual pairing.
We now introduce the inhomogeneous Besov spaces.
The following lemmas will be useful in the proof of our main result.
(Bernstein Inequality, )
(Embedding Results, )
Let , , and . Assume that either or and . Then .
If and , then .
3 Proof of Theorem 1.3
This section is devoted to the proof of Theorem 1.3. Since by Lemma 2.3, we only need to prove that Theorem 1.3 holds in the case , that is, to prove that if satisfies , then the weak solution to the system (1) is regular on .
The proof of Theorem 1.3 in the rest of this section is divided into two cases.
The authors are highly grateful for the referees’ careful reading and comments on this paper. The work of Zhang is partially supported by the NNSF of China under the grant 11101337, Doctoral Foundation of Ministry of Education of China grant-20110182120013, Fundamental Research Funds for the Central Universities grant-XDJK2011C046, and Doctoral Fund of Southwest University (SWU110035). The work of Qiu is partially supported by the NNSF of China grant-11126266, the NSF of Guangdong grant-S2013010013608, Foundation for Distinguished Young Talents in Higher Education of Guangdong, China grant 2012LYM_0030 and Pearl River New Star Program grant-2012J2200016.
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