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Global dissipative solutions of the Camassa-Holm equation. (English) Zbl 1139.35378
Summary: This paper is devoted to the continuation of solutions to the Camassa-Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an ${𝐋}^{\infty }$ space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data $\overline{u}\in {H}^{1}\left(ℝ\right)$, and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.
##### MSC:
 35L65 Conservation laws 35L67 Shocks and singularities 35Q58 Other completely integrable PDE (MSC2000) 35B60 Continuation of solutions of PDE 35L45 First order hyperbolic systems, initial value problems 35L60 Nonlinear first-order hyperbolic equations
##### Keywords:
non-local source; continuation after wave breaking