On the local well-posedness for a full dispersion Boussinesq system with surface tension

TL;DR

This paper establishes local well-posedness for a fully dispersive Boussinesq system modeling water waves, using energy estimates and a modified energy approach to handle non-symmetry in nonlinear terms.

Contribution

It proves local-in-time well-posedness for a complex dispersive water wave model, extending analysis techniques to non-symmetric nonlinear systems.

Findings

Proved local well-posedness in 2D and 3D.

Developed a modified energy method for non-symmetric nonlinearities.

Extended dispersive water wave analysis to more general systems.

Abstract

In this note, we prove local-in-time well-posedness for a fully dispersive Boussinesq system arising in the context of free surface water waves in two and three spatial dimensions. Those systems can be seen as a weak nonlocal dispersive perturbation of the shallow-water system. Our method of proof relies on energy estimates and a compactness argument. However, due to the lack of symmetry of the nonlinear part, those traditional methods have to be supplemented with the use of a modified energy in order to close the a priori estimates.