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 equationequation(DP)∂tu-∂txx3u+4u∂xu=3∂xu∂xx2u+u∂xxx3u.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 L1L1 stability (uniqueness) results for entropy weak solutions belonging to the class L1∩BVL1∩BV, while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class L2∩L4L2∩L4. Finally, we extend our results to a class of generalized Degasperis–Procesi equations.