On the well-posedness of the Degasperis-Procesi equation

We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis-Procesi equation \begin{equation} \pt u-\ptxx u+4u\px u =3\px u\pxx u +u\pxxx u. \tag{DP} \end{equation} 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 $L^1 \cap BV$, while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class $L^2\cap L^4$. Finally, we extend our results to a class of generalized Degasperis-Procesi equations.