Long time well-posedness of Whitham-Boussinesq systems

TL;DR

This paper proves long-time well-posedness for three bi-directional shallow water wave models with surface tension, extending the understanding of their validity over longer times and under specific initial conditions.

Contribution

It establishes the first long-time well-posedness results for these systems, including one for which it was previously unknown, and refines conditions for models with surface tension.

Findings

Well-posedness on time scale of order 1/ε for three systems

New result for one system even for short time

Extended validity of water wave models with surface tension

Abstract

Consideration is given to three different full dispersion Boussinesq systems arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water. We prove that, under a non-cavitation condition on the initial data, these three systems are well-posed on a time scale of order $\mathcal{O}(\frac1\varepsilon)$, where $\varepsilon$ is a small parameter measuring the weak non-linearity of the waves. This result seems new for one of these systems, even for short time. The two other systems involve surface tension, and for one of them, the non-cavitation condition has to be sharpened when the surface tension is small. The proof relies on suitable symmetrizers and the classical theory of hyperbolic systems. However, we have to track the small parameters carefully in the commutator estimates to get the long time well-posedness. Finally, combining our results with the recent ones of Emerald provide a full justification of these systems as water wave models in a larger range of regimes than the classical $(a,b,c,d)$-Boussinesq systems.