Long-time existence for a whitham boussinesq system in two dimensions

Abstract

A bstract . This paper is concerned with a two dimensional Whitham–Boussinesq system modeling surface waves of an inviscid incompressible fluid layer. We prove that the associated Cauchy problem is well-posed for initial data of low regularity, with existence time of scale O (cid:16) µ 3 / 2 − ǫ − 2 + (cid:17) , where µ and ǫ are small parameters related to the level of dispersion and nonlinearity, respectively. In particular, in the KdV regime { µ ∼ ǫ }, the existence time is of order ǫ − 1 / 2 . The main ingredients in the proof are frequency loacalised dispersive estimates and bilinear Strichartz estimates that depend on the parameter µ .