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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 \(\mathbf L^{\infty}\) 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:

35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
35Q58 Other completely integrable PDE (MSC2000)
35B60 Continuation and prolongation of solutions to PDEs
35L45 Initial value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
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