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

Alberto Bressan, Geng Chen, Qingtian Zhang

Research output: Contribution to journalArticle

26 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)25-42
Number of pages18
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume35
Issue number1
DOIs
StatePublished - Jan 1 2015

Fingerprint

Camassa-Holm Equation
Uniqueness
Characteristic Curve
Uniqueness of Solutions
Unique Solution
Cauchy Problem

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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Uniqueness of conservative solutions to the camassa-holm equation via characteristics. / Bressan, Alberto; Chen, Geng; Zhang, Qingtian.

In: Discrete and Continuous Dynamical Systems- Series A, Vol. 35, No. 1, 01.01.2015, p. 25-42.

Research output: Contribution to journalArticle

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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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