# American Institute of Mathematical Sciences

January  2015, 35(1): 25-42. doi: 10.3934/dcds.2015.35.25

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

 1 Department of Mathematics, Penn State University, University Park, Pa.16802, United States 2 School of Mathematics, Georgia Institute of Technology, Atlanta, Ga. 30332, United States

Received  January 2014 Revised  January 2014 Published  August 2014

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.
Citation: Alberto Bressan, Geng Chen, Qingtian Zhang. Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 25-42. doi: 10.3934/dcds.2015.35.25
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