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Zbl 1139.35378
Bressan, Alberto; Constantin, Adrian
Global dissipative solutions of the Camassa-Holm equation. (English)
[J] Anal. Appl., Singap. 5, No. 1, 1-27 (2007). ISSN 0219-5305

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 $\bold L^{\infty}$ 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 $\overline u \in H^1(\mathbb R)$, 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 2000:: *35L65 Conservation laws
35L67 Shocks, etc.
35Q58 Other completely integrable PDE
35B60 Continuation of solutions of PDE
35L45 First order hyperbolic systems, initial value problems
35L60 First-order nonlinear hyperbolic equations

Keywords: non-local source; continuation after wave breaking

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