The Cauchy problem on large time for a Boussinesq-Peregrine equation with large topography variations

TL;DR

This paper establishes long time existence results for a modified Boussinesq-Peregrine equation modeling shallow water waves over large topography variations, improving previous results by removing smallness constraints on bathymetry.

Contribution

It introduces a new model with the same accuracy as the Boussinesq-Peregrine equation and proves long time existence without small bathymetry assumptions.

Findings

Long time existence in 1D for the new model.

Local existence results for original and new models.

No smallness assumption on bathymetry for the long time result.

Abstract

We prove in this paper a long time existence result for a modified Boussinesq-Peregrine equation in one dimension, describing the motion of Water Waves in shallow water, in the case of a non flat bottom. We first give a local existence result for the original Boussinesq Peregrine equation as derived by Boussinesq and Peregrine in all dimensions. We then introduce a new model which has formally the same precision as the Boussinesq-Peregrine equation, and give a local existence result in all dimensions. We finally prove a long time existence result in dimension 1 for this new equation, without any assumption on the smallness of the bathymetry, which is an improvement of the long time existence result for the Boussinesq systems in the case of flat bottom.