Nonlinear Analysis: Theory, Methods & Applications
A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field
Introduction
In this paper, we consider the following 3D viscous incompressible MHD equations where is the velocity field, is the magnetic field, is a scalar pressure, and with in the sense of distribution are the initial velocity and magnetic fields.
It is well known [1] that the problem (1.1) is locally well posed for any given initial datum , . But whether this unique local solution can exist globally is a problem that presents a serious challenge. Some fundamental Serrin’s-type regularity criteria in terms of only the velocity was studied in [2], [3] independently. Recently, some improvement and extension was made based on these two basic papers. Part of them are listed here: Chen, Miao and Zhang [4] proved regularity by adding condition on ; Zhou and Gala [5] proved regularity for and in the multiplier spaces; Wu [6] considered the velocity field being in the homogeneous Besov space; regularity was obtained by imposing condition on the pressure in [7] (a complete result was established in [8]); in [9] direction of vorticity field was discussed (see also [2]).
The purpose of this work is to improve the criterion on regularity of weak solutions under vorticity with . More precisely, we will prove
Theorem 1.1 Let , . Suppose that is a weak solution of MHD equations(1.1)on with . If the vorticity field satisfiesthen the corresponding weak solution to(1.1)is smooth.
Section snippets
Preliminaries
Before stating our main result, we recall the definition and some properties of the space that we are going to use. These spaces play an important role in studying the regularity of solutions to partial differential equations (see e.g., [10], [11]).
Definition 2.1 For , the Morrey–Campanato space is defined by : where denotes the ball of center with radius .
It is easy to verify that is a Banach space under the
Proof of Theorem 1.1
In order to prove Theorem 1.1, first we should establish if (1.2) holds.
First we consider the case of . We multiply the both sides of the second equation of (1.1) by , the integration over to get, with the help of integration by parts, that
Acknowledgements
The authors would like to thank the referee for his/her careful reading and constructive suggestions, which greatly improved the paper. This work is partially supported by the Program for New Century Excellent Talents in Universities (Grant No. NCET 07-0299), ZJNSF (Grant No. R6090109) and NSFC (Grant No. 10971197).
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