THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM*

doi:10.1007/s10473-023-0225-1

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 942-958.doi: 10.1007/s10473-023-0225-1

THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM^*

Boling Guo¹, Yamin Xiao^2,†

1. Institute of Applied Physics and Computational Mathematics, Beijing 100088, China;
2. The Graduate School of China Academy of Engineering Physics, Beijing 100088, China

收稿日期:2022-06-08 修回日期:2022-08-10 出版日期:2023-03-25 发布日期:2023-04-12
通讯作者: †Yamin Xiao, E-mail: xiaoyamin20@gscaep.ac.cn.
作者简介:Boling Guo, E-mail: gbl@iapcm.ac.cn

THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM^*

Boling Guo¹, Yamin Xiao^2,†

1. Institute of Applied Physics and Computational Mathematics, Beijing 100088, China;
2. The Graduate School of China Academy of Engineering Physics, Beijing 100088, China

Received:2022-06-08 Revised:2022-08-10 Online:2023-03-25 Published:2023-04-12
Contact: †Yamin Xiao, E-mail: xiaoyamin20@gscaep.ac.cn.
About author:Boling Guo, E-mail: gbl@iapcm.ac.cn

摘要/Abstract

摘要： This paper examines the existence of weak solutions to a class of the high-order Korteweg-de Vries (KdV) system in $\mathbb{R}^n$. We first prove, by the Leray-Schauder principle and the vanishing viscosity method, that any initial data $N$-dimensional vector value function $u_0(x)$ in Sobolev space $H^{s}(\mathbb{R}^n)$ $(s\geq1)$ leads to a global weak solution. Second, we investigate some special regularity properties of solutions to the initial value problem associated with the KdV type system in $\mathbb{R}^2$ and $\mathbb{R}^3$.

关键词: global existence, propagation regularity, KdV system

Abstract: This paper examines the existence of weak solutions to a class of the high-order Korteweg-de Vries (KdV) system in $\mathbb{R}^n$. We first prove, by the Leray-Schauder principle and the vanishing viscosity method, that any initial data $N$-dimensional vector value function $u_0(x)$ in Sobolev space $H^{s}(\mathbb{R}^n)$ $(s\geq1)$ leads to a global weak solution. Second, we investigate some special regularity properties of solutions to the initial value problem associated with the KdV type system in $\mathbb{R}^2$ and $\mathbb{R}^3$.

Key words: global existence, propagation regularity, KdV system

Boling Guo, Yamin Xiao. THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM^*[J]. 数学物理学报(英文版), 2023, 43(2): 942-958.

Boling Guo, Yamin Xiao. THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM^*[J]. Acta mathematica scientia,Series B, 2023, 43(2): 942-958.

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THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM^*

THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM^*

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