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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 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 u ¯H 1 (), 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:
35L65Conservation laws
35L67Shocks and singularities
35Q58Other completely integrable PDE (MSC2000)
35B60Continuation of solutions of PDE
35L45First order hyperbolic systems, initial value problems
35L60Nonlinear first-order hyperbolic equations