Weak Well-posedness for the Generalized Camassa-Holm Equation

Chen Rong-ning; Wei Xue-mei

doi:10.12052/gdutxb.190053

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Weak Well-posedness for the Generalized Camassa-Holm Equation

更新时间：2026-05-27

- Weak Well-posedness for the Generalized Camassa-Holm Equation
- Vol. 37, Issue 2, Pages: 80-86(2020)
- 作者机构：
  
  广东工业大学应用数学学院,广东,广州,510520
- 作者简介：
- 基金信息：
- DOI：10.12052/gdutxb.190053
  CLC： O175
- Online First：10 March 2020，
  
  Published：2020
- 稿件说明：
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陈荣宁, 卫雪梅. 广义的Camassa-Holm方程的弱适定性[J]. 广东工业大学学报, 2020, 37(2): 80-86.

Chen Rong-ning, Wei Xue-mei. Weak Well-posedness for the Generalized Camassa-Holm Equation[J]. 2020, 37(2): 80-86.
陈荣宁, 卫雪梅. 广义的Camassa-Holm方程的弱适定性[J]. 广东工业大学学报, 2020, 37(2): 80-86. DOI： 10.12052/gdutxb.190053.

Chen Rong-ning, Wei Xue-mei. Weak Well-posedness for the Generalized Camassa-Holm Equation[J]. 2020, 37(2): 80-86. DOI： 10.12052/gdutxb.190053.

摘要

本文主要研究广义的Camassa-Holm方程Cauchy问题当初值

${u_0}$

img text_id='M1' class='formula-img' style='display:none;' src='gdgydxxb-37-2-80_M1.jpg'>

在空间

${H^1}({\mathbb{R}}) \cap {W^{1

\infty }}({\mathbb{R}})$

img text_id='M2' class='formula-img' style='display:none;' src='gdgydxxb-37-2-80_M2.jpg'>

时解的弱适定性。首先运用特征线把广义的Camassa-Holm方程转化成类似常微分方程(Ordinary Differential Equation，ODE)的方程。其次运用ODE理论证明新方程解的局部存在唯一性。最后利用新方程与原方程的关系

证明原方程解的局部存在唯一性并且给出解对初值的弱连续依赖性。

Abstract

The weak well-posedness for the generalized Camassa-Holm equation with initial data

${u_0} \in $

img text_id='M32047' class='formula-img' style='display:none;' src='gdgydxxb-37-2-80_M32047.jpg'>

${H^1}({\mathbb{R}}) \cap {W^{1

\infty }}({\mathbb{R}})$

img text_id='M3' class='formula-img' style='display:none;' src='gdgydxxb-37-2-80_M3.jpg'>

is mainly considered. First

by introducing the characteristics

the generalized Camassa-Holm equation is transformed into an ODE(Ordinary Differential Equation)-similar equation. Then applying the ODE theory

the local existence and uniqueness of the solution to the new equation are proved. Finally

by using the relationship between the new equation and the original equation

the local existence and uniqueness of the solution are investigated

deriving the conclusion of the weak continuous dependence on initial data for the original equation.

关键词

Keywords

references

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