Communications in Mathematical Sciences

Volume 14 (2016)

Number 6

Uniqueness of conservative solutions to the two-component Camassa–Holm system via characteristics

Pages: 1581 – 1597



Mingjie Li (College of Science, Minzu University of China, Beijing, China)

Qingtian Zhang (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)


The paper is concerned with a direct proof the uniqueness of global conservative solutions to the two-component Camassa–Holm system, based on characteristics. Given a conservative solution $u=u(t,x)$ and $\rho = \rho(t,x)$, an equation is introduced to single out a unique characteristic curve through each initial point. It is proved that the Cauchy problem with general initial data $u_0 \in H^1 (R), \rho_0 \in L^2 (R)$ has a unique global conservative solution.


Camassa–Holm equation, uniqueness, singularity, large data, characteristic

2010 Mathematics Subject Classification

Primary 35L65. Secondary 35L45.

Published 12 August 2016