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On the regularity criterion for the 3D generalized MHD equations in Besov spaces
Boundary Value Problems volume 2014, Article number: 178 (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.
In this paper, we are concerned with the following three-dimensional generalized MHD equations:
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.
For the 3D generalized MHD equations (1), Wu  showed that the system (1) possesses a global weak solutions corresponding to any initial data. Yet, just like the 3D Navier-Stokes equations and the 3D MHD equations, whether there exists a global smooth solution for the 3D generalized MHD equations (1) or not is an open problem. Recently, many authors studied the regularity problem for the 3D generalized MHD equations (1) intensively. Wu ,  obtained some regularity criteria only relying on the velocity . Zhou  considered the following two cases: and and established the Serrin-type criteria involving velocity . Wu  obtained the classical Beal-Kato-Majda criterion for the system (1). By means of the Fourier localization technique and the Bony paraproduct decomposition, Yuan  extended the Serrin-type criterion to
with , , , provided that .
On the other hand, suggested by the results in , , one may presume upon that there should have some cancelation properties between the velocity field and the magnetic field . When , plus and minus the first equation of (1) and the second one, respectively, the system (1) can be rewritten as
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.
Suppose that . The vector-valued function is called a weak solution to the system (1) on , if it satisfies the following properties:
and in the sense of a distribution;
for any with and , one has
The weak solution to the system (3) can be defined in a similar way as follows.
Suppose that . The vector-valued function is called a weak solution to the system (3) on , if it satisfies the following properties:
and in the sense of a distribution;
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.
Suppose that, the initial velocity and magnetic fieldandin the sense of distribution. Assume thatis a weak solution to the system (1) on some intervalwith. Ifsatisfies the following condition:
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.
Let be the Schwarz functions of rapidly decreasing functions. Given , its Fourier transformation is defined by
Take two nonnegative radial functions supported respectively in and such that
Let and . The frequency localization operator is defined by
Formally, is a frequency projection to the annulus , and is a frequency projection to the ball . The above dyadic decomposition has a nice quasi-orthogonality, with the choice of and ; namely, for any , we have the following properties:
Let , the homogeneous Sobolev space is defined by
and the set of temperate distributions is the dual set of for the usual pairing.
When dealing with our problems, we will use some paradifferential calculus , . It is a nice way to define a generalized product between temperate distributions, which is continuous in fractional Sobolev spaces, and which yet does not make any sense for the usual product. Let , be two temperate distributions. We denote
At least, we have the following the Bony decomposition:
where the paraproduct is a bilinear continuous operator. For simplicity, we denote
We now introduce the inhomogeneous Besov spaces.
Let , . The inhomogeneous Besov space is defined by
The following lemmas will be useful in the proof of our main result.
(Bernstein Inequality, )
Let. Assume that, then there exists a constantindependent of, such that
(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 .
Now, we denote , , , and take . Then we have . Applying the operator to both sides of (3), we get
Then multiplying the first and the second equation of (9) by , , respectively, and by Lemma 2.2, we obtain for
Denote . Adding the two equations in (10), we have
Then (11) can be rewritten as
where the commutator operator . By the Bony decomposition (8), the term can be rewritten as
From the definition of , we have
Then by the Minkowski inequality, we get
For the term ,
where is replaced by in the last equality. Noticing that for , then by Lemma 2.2 and the Minkowski inequality, we have
Using the support of the Fourier transformation of the term , we get
By the Minkowski inequality and Lemma 2.2, we have
With the incompressibility condition , we have
then by Lemma 2.2, we get
Combining the above estimates for -, we have
The proof of Theorem 1.3 in the rest of this section is divided into two cases.
Case I: . Take , then . Let
Multiplying (15) with and integrating with respect to , we have
By Lemma 2.2 and the Hölder inequality, we have
Combining the above estimates for - with (16) and the Young inequality, we have
which implies that
Then the Gronwall inequality gives
Case II: . Let
Multiplying (15) by , and integrating with respect to , it follows that
By Lemma 2.2 and the Hölder inequality, we have
Inserting the above estimates for - into (18) and by the Young inequality, we have
Again, by the Gronwall inequality, we have
By Lemma 2.3, as , for some , the following embedding holds:
According to the local existence with the condition , thus can be taken as . When , for , one has , which infers that
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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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Zhang, S., Qiu, H. On the regularity criterion for the 3D generalized MHD equations in Besov spaces. Bound Value Probl 2014, 178 (2014). https://doi.org/10.1186/s13661-014-0178-3
- generalized MHD equations
- regularity criterion
- Littlewood-Paley decomposition