%0 Journal Article
%T A Remark on the Regularity Criterion for the MHD Equations via Two Components in Morrey-Campanato Spaces
%A Zujin Zhang
%A Tong Tang
%A Fumin Zhang
%J Journal of Difference Equations
%D 2014
%R 10.1155/2014/364269
%X We consider the regularity criterion for the 3D MHD equations. It is proved that if the horizontal components of the velocity and magnetic fields satisfy with , then the solution smooth. This improves the result given by Gala (2012). 1. Introduction In this paper, we consider the following three-dimensional ( D) magnetohydrodynamic (MHD) equations: where is the fluid velocity field, is the magnetic field, is a scalar pressure, and are the prescribed initial data satisfying in the distributional sense. Physically, (1) governs the dynamics of the velocity and magnetic fields in electrically conducting fluids, such as plasmas, liquid metals, and salt water. Moreover, (1)1 reflects the conservation of momentum, (1)2 is the induction equation, and (1)3 specifies the conservation of mass. Besides its physical applications, the MHD system (1) is also mathematically significant. Duvaut and Lions [1] constructed a global weak solution to (1) for initial data with finite energy. However, the issue of regularity and uniqueness of such a given weak solution remains a challenging open problem. Many sufficient conditions (see, e.g., [2¨C14] and the references therein) were derived to guarantee the regularity of the weak solution. Among these results, we are interested in regularity criteria involving only partial components of the velocity , the magnetic field , the pressure gradient , and so forth. Cao and Wu [2] proved the following regularity criterion: Jia and Zhou showed that if then the solution is regular. These results were improved by Zhang [15] to be Once only partial components of the velocity field are concerned, we have combinatoric regularity criterion involving partial components of the magnetic field also. This is due partially to the strong coupling of the velocity and magnetic fields. Let us list some recent progress. Gala and Lemari¨¦-Rieusset [6] established the following two regularity conditions: Recently, Ni et al. in [10] showed that each of the following three conditions or or ensures the smoothness of the solution. Here, is the horizontal gradient operator. The motivation of this paper is to give another contribution in this direction. Motivated by [8], we would like to show the following regularity condition for (1): Here, and in what follows, we denote by the horizontal components of the velocity and magnetic fields, respectively, and by the horizontal gradient operator. Before stating the precise result, let us recall the weak formulation of the MHD equations (1). Definition 1. Let with and . A measurable -valued pair is called a weak
%U http://www.hindawi.com/journals/jde/2014/364269/