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Global conservative and dissipative solutions of the generalized Camassa-Holm equation. (English) Zbl 1263.65084

Summary: This paper is devoted to the continuation of solutions to the generalized Camassa-Holm equation beyond wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear system. This formulation allows one to continue the solution after collision time, giving either a global conservative solution where the energy is conserved for almost all times or a dissipative solution where energy may vanish from the system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global conservative or dissipative solutions, which depend continuously on the initial data.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35B10 Periodic solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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