Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics

Pages: 25 - 42, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.25

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Alberto Bressan - Department of Mathematics, Penn State University, University Park, Pa.16802, United States (email)
Geng Chen - School of Mathematics, Georgia Institute of Technology, Atlanta, Ga. 30332, United States (email)
Qingtian Zhang - Department of Mathematics, Penn State University, University Park, Pa. 16802, United States (email)

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 $u_0\in H^1(\mathbb{R})$ has a unique solution, globally in time.

Keywords:  Camassa-Holm equation, uniqueness, singularity, large data, characteristic.
Mathematics Subject Classification:  Primary: 35L65; Secondary: 35L45.

Received: January 2014;      Revised: January 2014;      Available Online: August 2014.