Global weak solutions to the Camassa-Holm equation

Zhenhua Guo; Mina Jiang; Zhian Wang; Gao Feng Zheng

Global weak solutions to the Camassa-Holm equation

Zhenhua Guo, Mina Jiang, Zhian Wang, Gao Feng Zheng

Research output: Journal article publication › Journal article › Academic research › peer-review

Abstract

The existence of a global weak solution to the Cauchy problem for a one-dimensional Camassa-Holm equation is established. In this paper, we assume that the initial condition u0 (x) has end states u±, which has much weaker constraints than that u0 (x) ε H1 (ℝ) discussed in [30]. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution as a limit of viscous approximation under the assumption u- < u+.

Original language	English
Pages (from-to)	883-906
Number of pages	24
Journal	Discrete and Continuous Dynamical Systems
Volume	21
Issue number	3
Publication status	Published - 1 Jul 2008
Externally published	Yes

Keywords

Camassa-Holm equation
Global weak solution
Vanishing viscosity method
Young measure

ASJC Scopus subject areas

General Mathematics
Discrete Mathematics and Combinatorics
Applied Mathematics
Analysis

Cite this

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title = "Global weak solutions to the Camassa-Holm equation",

abstract = "The existence of a global weak solution to the Cauchy problem for a one-dimensional Camassa-Holm equation is established. In this paper, we assume that the initial condition u0 (x) has end states u±, which has much weaker constraints than that u0 (x) ε H1 (ℝ) discussed in [30]. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution as a limit of viscous approximation under the assumption u- < u+.",

keywords = "Camassa-Holm equation, Global weak solution, Vanishing viscosity method, Young measure",

author = "Zhenhua Guo and Mina Jiang and Zhian Wang and Zheng, {Gao Feng}",

year = "2008",

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journal = "Discrete and Continuous Dynamical Systems",

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T1 - Global weak solutions to the Camassa-Holm equation

AU - Guo, Zhenhua

AU - Jiang, Mina

AU - Wang, Zhian

AU - Zheng, Gao Feng

PY - 2008/7/1

Y1 - 2008/7/1

N2 - The existence of a global weak solution to the Cauchy problem for a one-dimensional Camassa-Holm equation is established. In this paper, we assume that the initial condition u0 (x) has end states u±, which has much weaker constraints than that u0 (x) ε H1 (ℝ) discussed in [30]. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution as a limit of viscous approximation under the assumption u- < u+.

AB - The existence of a global weak solution to the Cauchy problem for a one-dimensional Camassa-Holm equation is established. In this paper, we assume that the initial condition u0 (x) has end states u±, which has much weaker constraints than that u0 (x) ε H1 (ℝ) discussed in [30]. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution as a limit of viscous approximation under the assumption u- < u+.

KW - Camassa-Holm equation

KW - Global weak solution

KW - Vanishing viscosity method

KW - Young measure

UR - http://www.scopus.com/inward/record.url?scp=45849100919&partnerID=8YFLogxK

M3 - Journal article

SN - 1078-0947

VL - 21

SP - 883

EP - 906

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

IS - 3

ER -

Global weak solutions to the Camassa-Holm equation

Abstract

Keywords

ASJC Scopus subject areas

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