Global solutions for the one-dimensional Boussinesq-Peregrine system under small bottom variation

TL;DR

This paper proves the global well-posedness of the one-dimensional Boussinesq-Peregrine system with small bottom variation, extending previous flat bottom results to more general topographies in Sobolev spaces.

Contribution

It establishes the first global existence results for the Boussinesq-Peregrine system with small bottom variation in Sobolev spaces, using a new functional setting.

Findings

Global well-posedness in Sobolev spaces for small bottom variation

Extension of flat bottom results to general topography

Existence of weak low-regularity solutions

Abstract

The Boussinesq-Peregrine system is derived from the water waves system in presence of topographic variation under the hypothesis of shallowness and small amplitude regime. The system becomes significantly simpler (at least in the mathematical sens) under the hypothesis of small topographic variation. In this work we study the long time and global well-posedness of the Boussinesq-Peregrine system. We start by showing the intermediate time well-posedness in the case of general topography (i.e. the amplitude of the bottom graph $\beta=O(1)$). The novelty resides in the functional setting, $H^s({\mathbb R}), \, s> \frac {1} {2}$. Then we show our main result establishing that the global existence result obtained in Molinet-Talhouk-Zaiter in the flat bottom case is still valid for the Boussinesq-Peregrine system under the hypothesis of small amplitude bottom variation (i.e. $\beta =O(\mu)$). More precisely we prove that this system is unconditionally globally well-posed in the Sobolev spaces of type $ H^s ({\mathbb R}), \, s> \frac {1} {2}$. Finally, we show the existence of a weak global solution in the Schonbek sense, i.e. existence of low regularity entropic solutions of the small bottom amplitude Boussinesq-Pelegrine equations emanating from $ u_0 \in H^1 $ and $ \zeta_0 $ in an Orlicz class as weak limits of regular solutions.