Acta mathematica scientia,Series A ›› 2019, Vol. 39 ›› Issue (5): 1136-1145.

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Regularity Criteria for the NS and MHD Equations in Terms of Horizontal Components

Hui Zhang*(),Juan Xu   

  1. School of Mathematical Sciences, Anqing Normal University, Anhui Anqing 246133
  • Received:2018-06-07 Online:2019-10-26 Published:2019-11-08
  • Contact: Hui Zhang
  • Supported by:
    the Anhui Education Bureau(AQKJ2014B009);the Doctor's Funding of Anqing Normal University(K050001309)


In this paper, we consider the regularity of weak solutions to the incompressible NS equations and MHD equations in the Triebel-Lizorkin space and multiplier space respectively. By using Littlewood-Paley decomposition and energy estimate methods, we proved that if horizontal velocity ũ=(u1, u2, 0) satisfies

then the weak solution is actually the unique strong solution on[0, T). For MHD equations, we prove that if horizontal velocity and magnetic field satisfies

or horizontal gradient satisfies

then the weak solution is actually unique strong solution on[0, T).

Key words: NS equations, MHD equations, Regularity criteria

CLC Number: 

  • O179.25