涉及水平分量的NS方程与MHD方程的正则性准则

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

涉及水平分量的NS方程与MHD方程的正则性准则

张辉*(),许娟

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

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/Abstract

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=（u₁，u₂，0）的水平梯度

$ \nabla_{h}\tilde{u}\in L^{p}(0,T; \dot{F}^{0}_{q,\frac{2q}{3}}(\mathbb{R} ^{3})), ~~~~~\frac{2}{p}+\frac{3}{q}=2,~~~~\frac{3}{2}<q\leq\infty, $

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

$ (\tilde{u},\tilde{b})\in L^{\frac{2}{1-r}}(0, T;\dot{X}_{r}(\mathbb{R} ^{3})),\hspace{0.2cm}r\in[0, 1), $

或者相应的水平梯度

$(\nabla_{h}\tilde{u},\nabla_{h}\tilde{b})\in L^{\frac{2}{2-r}}(0, T;\dot{X}_{r}(\mathbb{R} ^{3})),\hspace{0.2cm}r\in[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

$ \nabla_{h}\tilde{u}\in L^{p}(0,T; \dot{F}^{0}_{q,\frac{2q}{3}}(\mathbb{R} ^{3})), ~~~~~\frac{2}{p}+\frac{3}{q}=2,~~~~\frac{3}{2}<q\leq\infty, $

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

$ (\tilde{u},\tilde{b})\in L^{\frac{2}{1-r}}(0, T;\dot{X}_{r}(\mathbb{R} ^{3})),\hspace{0.2cm}r\in[0, 1), $

or horizontal gradient satisfies

$(\nabla_{h}\tilde{u},\nabla_{h}\tilde{b})\in L^{\frac{2}{2-r}}(0, T;\dot{X}_{r}(\mathbb{R} ^{3})),\hspace{0.2cm}r\in[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

张辉,许娟. 涉及水平分量的NS方程与MHD方程的正则性准则[J]. 数学物理学报, 2019, 39(5): 1136-1145.

Hui Zhang,Juan Xu. Regularity Criteria for the NS and MHD Equations in Terms of Horizontal Components[J]. Acta mathematica scientia,Series A, 2019, 39(5): 1136-1145.

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