陳芳如

三維MHD方程軸對稱弱解的正則準(zhǔn)則

陳芳如

(溫州大學(xué)數(shù)理學(xué)院，浙江溫州 325035)

MHD方程組；正則性準(zhǔn)則；軸對稱；Besov空間

本文考慮的三維粘性不可壓縮磁流體動力學(xué)方程（Magnetohydrodynamics，MHD）形式如下：

He等人在文獻[1]中僅根據(jù)速度場建立了基本的Serrin型規(guī)律性準(zhǔn)則，準(zhǔn)確地說，表明了速度場若滿足

或

得到方程組（1）的弱解是光滑的．

得到方程組（1）的弱解是光滑的．同時，也有軸對稱Navier-Stokes方程的分量正則性準(zhǔn)則，如文獻[10]，正則性準(zhǔn)則只作用于渦場的兩個分量．受Navier-Stokes方程結(jié)果的啟發(fā)，我們將注意力轉(zhuǎn)向軸對稱MHD方程．Liu等人在文獻[11]中構(gòu)造了具有特定形式磁場的三維不可壓縮MHD方程的一類軸對稱解的正則性判據(jù)．

1 預(yù)備知識

2 定理及其證明

定理1的證明．

首先對（1）式進行先驗估計．更準(zhǔn)確地說，我們將展示下面的一個先驗估計．

在（1）式上求其旋度，得到：

（17）式右側(cè)第一項可估計為：

將（20）式―（22）式加起來，可得：

同樣地，有以下估計

由此可得：

然后將（23）式、（24）式和（25）式代入（18）式，就得到了

現(xiàn)在估計（17）式右邊的第二項，

得到：

它遵循類似的方法，

組合（17）式、（26）式、（27）式、（28）式得到：

同樣地，得到

證明完成．

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[8] Chae D, Choe J. Regularity of solutions to the Navier-Stokes Equations [J]. Electronic Journal of Differential Equations, 1999, 56: 1-7.

[9] Kozono H, Ogawa T, Taniuchi Y. The Critical Sobolev Inequalities in Besov Spaces and Regularity Criterion to Some Semi-linear Evolutions [J]. Mathematische Zeitschrift, 2002, 242: 251-278.

[10] Chae D, Lee J. On the Regularity of the Axisymmetric Solutions of the Navier-Stokes Equations [J]. Mathematische Zeitschrift, 2002, 239: 645-671.

[11] Liu J. On Regularity Criterion to the 3D Axisymmetric Incompressible MHD Equations [J]. Mathematical Methods in the Applied Sciences, 2016, 39: 4535-4544.

[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文件可以從www.wzu.edu.cn/wzdxxb.htm獲得

2020-10-21

陳芳如(1995― )，甘肅定西人，碩士研究生，研究方向：微分方程與動力系統(tǒng)

(編輯：王一芳)

(英文審校：黃璐)