数学物理学报 ›› 2019, Vol. 39 ›› Issue (5): 1136-1145.

• 论文 • 上一篇    

涉及水平分量的NS方程与MHD方程的正则性准则

张辉, 许娟   

  1. 安庆师范大学数学与计算科学学院 安徽安庆 246133
  • 收稿日期:2018-06-07 修回日期:2018-11-15 发布日期:2019-11-08
  • 通讯作者: 张辉 E-mail:zhangaqtc@126.com
  • 基金资助:
    安徽省教育厅项目(AQKJ2014B009)和安庆师范大学博士科研启动基金(K050001309)

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)

摘要: 该文在Triebel-Lizorkin空间和乘子空间中分别考虑了三维Navier-Stokes(NS)方程与Magneto-hydrodynamics(MHD)方程的正则性准则.利用Littlewood-Paley分解与能量不等式的方法获得了一些结果.关于NS方程,证明了如果水平速度场ũ=(u1u2,0)的水平梯度
hũLp(0,Tq,2q/30(R3)),2/p+3/q=2,3/2< q ≤ ∞,
则弱解是存在区间[0,T)上的唯一强解,其中▽h=(∂1,∂2,0).关于MHD方程,证明了如果水平速度场与水平磁场
ũ,b)∈ L2/1-r(0,T;Ẋr(R3)),r ∈[0,1),
或者相应的水平梯度
(▽hũ,▽hb)∈ L2/2-r(0,T;Ẋr(R3)),r ∈[0,1],
则弱解是存在区间[0,T)上的唯一强解.

关键词: Navier-Stokes方程, MHD方程, 正则性准则

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

中图分类号: 

  • O179.25