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# 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

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n6.a6

#### Authors

#### Abstract

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.

#### Keywords

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

#### 2010 Mathematics Subject Classification

Primary 35L65. Secondary 35L45.