Long-time existence for a Whitham–Boussinesq system in two dimensions

Abstract

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 [Formula: see text], where [Formula: see text] and [Formula: see text] are small parameters related to the level of dispersion and nonlinearity, respectively. In particular, in the KdV regime [Formula: see text]}, the existence time is of order [Formula: see text]. The main ingredients in the proof are frequency loacalized dispersive estimates and bilinear Strichartz estimates that depend on the parameter [Formula: see text].