Ill Posedness Results for Generalized Water Wave Models

Abstract

In the first part of the study, the weak asymptotic method is used to find singular solutions of the shallow water system in both one and two space dimensions. The singular solutions so constructed are allowed to contain Dirac-delta; distributions (Espinosa & Omel'yanov, 2005). The idea is to con- struct complex-valued approximate solutions which become real-valued in the distributional limit. The approach, which extends the range f possible singular solutions, is used to construct solutions which contain combinations of hyperbolic shock waves and Dirac-delta; distributions. It is shown in the second part that the Cauchy problem for Korteweg-de Vries (KdV) type equations is locally ill-posed in a negative Sobolev space. The method is used to construct a solution which does not depend continuously on its initial data in H^{s_epsilon}, s_epsilon = -1/2 - epsilon