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

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 (μ3/2 - 2+), where μ and are small parameters related to the level of dispersion and nonlinearity, respectively. In particular, in the KdV regime {μ}, the existence time is of order -1/2. The main ingredients in the proof are frequency loacalized dispersive estimates and bilinear Strichartz estimates that depend on the parameter μ.