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

Zhang Hui, Xu Juan   

  1. School of Mathematical Sciences, Anqing Normal University, Anhui Anqing 246133
  • Received:2018-06-07 Revised:2018-11-15 Published:2019-11-08
  • Supported by:
    Supported by the Anhui Education Bureau (AQKJ2014B009) and the Doctor's Funding of Anqing Normal University (K050001309)

Abstract: 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
hũLp(0,T;q,2q/30(R3)),2/p+3/q=2,3/2< q ≤ ∞.
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
(ũ,b)∈ L2/1-r(0,T;Ẋr(R3)),r ∈[0,1),
or horizontal gradient satisfies
(▽hũ,▽hb)∈ L2/2-r(0,T;Ẋr),r ∈[0,1],
then the weak solution is actually unique strong solution on[0, T).

Key words: NS equations, MHD equations, Regularity criteria

CLC Number: 

  • O179.25