On well-posedness of a dispersive system of the Whitham--Boussinesq type

TL;DR

This paper proves local well-posedness for a bidirectional Whitham system modeling surface water waves, using energy methods and compactness arguments, confirming its mathematical stability and relevance to Euler equations.

Contribution

It provides the first rigorous proof of local well-posedness for this specific dispersive water wave system, advancing the mathematical understanding of its stability.

Findings

The system is mathematically well-posed locally in time.

The proof employs energy estimates and compactness techniques.

Supports the system's validity as an approximation to Euler equations.

Abstract

The initial-value problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system was recently introduced in [4]. It is numerically shown to be stable and a good approximation to the incompressible Euler equations. Here we prove local in time well-posedness. Our proof relies on an energy method and a compactness argument.