# Well-posedness for a Whitham-Boussinesq system with surface tension

## TL;DR

This paper establishes the well-posedness of a Whitham-Boussinesq system modeling surface waves, using energy estimates and Hamiltonian conservation, with results on global solutions for small initial data in 1D.

## Contribution

It introduces a novel approach to proving well-posedness for a nonlocal dispersive surface wave model, adapting energy norms due to symmetry issues.

## Key findings

- Global well-posedness for small initial data in 1D
- Modified energy norm necessary for a priori estimates
- Hamiltonian conservation ensures long-term solution stability

## Abstract

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The proof of well-posedness relies on energy estimates. However, due to the symmetry lack of the nonlinear part, in order to close the a priori estimates one has to modify the traditional energy norm in use. Hamiltonian conservation provides with global well-posedness at least for small initial data in the one dimensional settings.