Abstract
We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis–Procesi equation (DP) ∂ t u - ∂ txx 3 u + 4 u ∂ x u = 3 ∂ x u ∂ xx 2 u + u ∂ xxx 3 u . This equation can be regarded as a model for shallow water dynamics and its asymptotic accuracy is the same as for the Camassa–Holm equation (one order more accurate than the KdV equation). We prove existence and L 1 stability (uniqueness) results for entropy weak solutions belonging to the class L 1 ∩ BV , while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class L 2 ∩ L 4 . Finally, we extend our results to a class of generalized Degasperis–Procesi equations.
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