Uniqueness of conservative solutions to the camassa-holm equation via characteristics

Alberto Bressan; Geng Chen; Qingtian Zhang

doi:10.3934/dcds.2015.35.25

Uniqueness of conservative solutions to the camassa-holm equation via characteristics

Alberto Bressan, Geng Chen, Qingtian Zhang

Research output: Contribution to journal › Article › peer-review

74 Scopus citations

Abstract

The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution u = u(t, x), an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities u and v = 2 arctan u_x along each characteristic, it is proved that the Cauchy problem with general initial data u₀∈ H¹(R) has a unique solution, globally in time.

Original language	English (US)
Pages (from-to)	25-42
Number of pages	18
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	35
Issue number	1
DOIs	https://doi.org/10.3934/dcds.2015.35.25
State	Published - Jan 1 2015

All Science Journal Classification (ASJC) codes

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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abstract = "The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution u = u(t, x), an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities u and v = 2 arctan ux along each characteristic, it is proved that the Cauchy problem with general initial data u0∈ H1(R) has a unique solution, globally in time.",

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AB - The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution u = u(t, x), an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities u and v = 2 arctan ux along each characteristic, it is proved that the Cauchy problem with general initial data u0∈ H1(R) has a unique solution, globally in time.

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Uniqueness of conservative solutions to the camassa-holm equation via characteristics

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