Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (4): 1033-1047.doi: 10.1016/S0252-9602(17)30056-5

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LOCAL REGULARITY CRITERIA OF A SUITABLE WEAK SOLUTION TO MHD EQUATIONS

Jae-Myoung KIM   

  1. Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
  • Received:2016-03-04 Revised:2016-05-23 Online:2017-08-25 Published:2017-08-25
  • About author:Jae-Myoung KIM,E-mail:cauchy02@naver.com
  • Supported by:

    Jae-Myoung Kim is also partly supported by BK21 PLUS SNU Mathematical Sciences Division and Basic Science Research Program through the National Research Foundation of Korea (NRF) (NRF-2016R1D1A1B03930422).

Abstract:

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are Hölder continuous near boundary provided that the scaled mixed Lx,tp,q -norm of the velocity vector field with 3/p + 2/q ≤ 2, 2 < q < ∞ is sufficiently small near the boundary. Also, we will investigate that for this solution uLx,tp,q with 1 ≤ 3/p + 2/q ≤ 3/2, 3 < p < ∞, the Hausdorff dimension of its singular set is no greater than max{p,q}(3/p + 2/q -1).

Key words: local regularity criteria, suitable weak solution, MHD equations

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