On the well-posedness of the Degasperis-Procesi equation

Abstract

We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis-Procesi equation ∂tu - ∂txx3u + 4u∂xu = 3∂x u∂xx2u + u∂xxx3u. This equation can b e 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 stability L1 (uniqueness) results for entropy weak solutions belonging to the class L1 ∩ BV, while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class L2 ∩ L4. Finally, we extend our results to a class of generalized Degasperis-Procesi equations. © 2005 Elsevier Inc. All rights reserved.