Abstract: We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis-Procesi equation
ut- utxx+4uux =3uxuxx +uuxxx. (1)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 $L1$ stability (uniqueness) results for entropy weak solutions belonging to the class $L1 \cap BV$, while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class $L2\cap L4$. Finally, we extend our results to a class of generalized Degasperis-Procesi equations.
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