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 partial derivative(t)u-partial derivative(txx)(3)u+4u partial derivative(x)u=3 partial derivative(x)u partial derivative(xx)(2)u+u partial derivative(xxx)(3)u. (DP) 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 boolean AND BV, while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class L-2 boolean AND L-4. Finally, we extend our results to a class of generalized Degasperis-Procesi equations. (C) 2005 Elsevier Inc. All rights reserved.