奇异对流方程组非常弱解的梯度正则性

数学物理学报 ›› 2023, Vol. 43 ›› Issue (3): 713-732.

奇异对流方程组非常弱解的梯度正则性

陈淑红^1,^*(),谭忠²

¹武夷学院数学与计算机学院福建武夷山 354300
²厦门大学数学科学学院福建厦门 361005

收稿日期:2022-04-11 修回日期:2022-10-19 出版日期:2023-06-26 发布日期:2023-06-01
通讯作者: 陈淑红 E-mail:shiny0320@163.com
基金资助:
国家自然科学基金(11571159);国家自然科学基金(12231016);武夷学院引进人才科研启动项目(YJ202118)

Gradient Regularity of Very Weak Solution to Elliptic Equations with Singular Convection

Chen Shuhong^1,^*(),Tan Zhong²

¹School of Mathematics and Computer, Wuyi University,Fujian Wuyishan 354300
²School of Mathematical Science, Xiamen University, Fujian Xiamen 361005

Received:2022-04-11 Revised:2022-10-19 Online:2023-06-26 Published:2023-06-01
Contact: Shuhong Chen E-mail:shiny0320@163.com
Supported by:
NSFC(11571159);NSFC(12231016);Foundation of Wuyi University(YJ202118)

摘要/Abstract

摘要：

该文主要考虑奇异对流方程组非常弱解的梯度部分正则性. 首先, 结合Lorentz空间及其与Lebesgue 空间之间的关系, 推出奇异对流方程组在 $L^p$ 空间存在非常弱解. 接着, 通过Hodge分解证明 Dirichlet 问题的非常弱解实际上就是古典弱解. 最后, 利用 A - 调和逼近技巧, 建立了奇异对流方程组非常弱解的梯度部分正则性结果, 最重要的是, 由此所得到的正则性结果是最优的.

关键词: 非常弱解, Hodge 分解, 奇异对流, A -调和逼近引理

Abstract:

This paper deals with the partial regularity of very weak solutions to elliptic equations with singular convective. By the properties of Lorentz space and its relation to Lebesgue space, we conclude that the elliptic systems with singular convection have very weak solutions in $L^p$ space. Then, it can be found from Hodge decomposition that the very weak solutions of Dirichlet problem are actually the classical weak solutions. Finally, combining with A-harmonic approximation technique, we further find that the obtained weak solution has partial regularity; especially, the regularity is optimal.

Key words: Very weak solution, Hodge composition, Singular convection, A-harmonic approximation technique

中图分类号:

O175.2

陈淑红,谭忠. 奇异对流方程组非常弱解的梯度正则性[J]. 数学物理学报, 2023, 43(3): 713-732.

Chen Shuhong,Tan Zhong. Gradient Regularity of Very Weak Solution to Elliptic Equations with Singular Convection[J]. Acta mathematica scientia,Series A, 2023, 43(3): 713-732.

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