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

• 论文 • 上一篇    下一篇

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 出版日期:2019-10-26 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);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

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 E-mail:zhangaqtc@126.com
  • Supported by:
    the Anhui Education Bureau(AQKJ2014B009);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

则弱解是存在区间[0,T)上的唯一强解,其中▽h=(12,0).关于MHD方程,证明了如果水平速度场与水平磁场

或者相应的水平梯度

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

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

中图分类号: 

  • O179.25