三维MHD方程轴对称弱解的正则准则

陈芳如

三维MHD方程轴对称弱解的正则准则

陈芳如

(温州大学数理学院，浙江温州 325035)

MHD方程组；正则性准则；轴对称；Besov空间

本文考虑的三维粘性不可压缩磁流体动力学方程（Magnetohydrodynamics，MHD）形式如下：

He等人在文献[1]中仅根据速度场建立了基本的Serrin型规律性准则，准确地说，表明了速度场若满足

或

得到方程组（1）的弱解是光滑的．

得到方程组（1）的弱解是光滑的．同时，也有轴对称Navier-Stokes方程的分量正则性准则，如文献[10]，正则性准则只作用于涡场的两个分量．受Navier-Stokes方程结果的启发，我们将注意力转向轴对称MHD方程．Liu等人在文献[11]中构造了具有特定形式磁场的三维不可压缩MHD方程的一类轴对称解的正则性判据．

1 预备知识

2 定理及其证明

定理1的证明．

首先对（1）式进行先验估计．更准确地说，我们将展示下面的一个先验估计．

在（1）式上求其旋度，得到：

（17）式右侧第一项可估计为：

将（20）式―（22）式加起来，可得：

同样地，有以下估计

由此可得：

然后将（23）式、（24）式和（25）式代入（18）式，就得到了

现在估计（17）式右边的第二项，

得到：

它遵循类似的方法，

组合（17）式、（26）式、（27）式、（28）式得到：

同样地，得到

证明完成．

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[12] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Interals [M]. Princeton: Princeton University Press, 1993:285-288.

Regularity Criteria of Axisymmetric Weak Solutions to the 3D MHD Equations

CHEN Fangru

(College of Mathematics and Physics, Wenzhou University, Wenzhou, China 325035)

MHD Equations; Regularity Criteria; Axisymmetric; Besov Space

O175

A

1674-3563(2022)01-0025-09

10.3875/j.issn.1674-3563.2022.01.004

本文的PDF文件可以从www.wzu.edu.cn/wzdxxb.htm获得

2020-10-21

陈芳如(1995― )，甘肃定西人，硕士研究生，研究方向：微分方程与动力系统

(编辑：王一芳)

(英文审校：黄璐)