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

TL;DR

This paper establishes long-time existence for solutions to a two-dimensional Whitham-Boussinesq system modeling surface waves, using dispersive and Strichartz estimates to handle low regularity initial data.

Contribution

It proves well-posedness with extended existence time for the 2D Whitham-Boussinesq system, introducing frequency localized estimates that depend on the shallowness parameter.

Findings

Existence time scales as 1/√ε for small ε

Well-posedness holds for low regularity initial data

Dispersive and Strichartz estimates are key analytical tools

Abstract

This paper is concerned with a two dimensional Whitham-Boussinesq system modelling 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 $\mathcal O(1/\sqrt{\epsilon})$, where $\epsilon>0$ is a shallowness parameter measuring the ratio of the amplitude of the wave to the mean depth of the fluid. The key ingredients in the proof are frequency loacalised dispersive and Strichartz estimates that depend on $\epsilon$ as well as bilinear estimates in some Strichartz norms.