An improved result for the full justification of asymptotic models for the propagation of internal waves

TL;DR

This paper enhances the mathematical justification of asymptotic models for internal wave propagation by incorporating medium amplitude topography variations and strong nonlinearity, improving the model's applicability and robustness.

Contribution

It extends the full justification of internal wave models to include variable topography and strong nonlinearity, removing previous smallness constraints.

Findings

Model remains consistent and well-posed with new assumptions.

Solutions stay close to full Euler system solutions under extended conditions.

Improved stability results for more realistic wave scenarios.

Abstract

We consider here asymptotic models that describe the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with uneven bottoms. The aim of this paper is to show that the full justification result of the model obtained by Duch\^ene, Israwi and Talhouk [{\em SIAM J. Math. Anal.}, 47(1), 240--290], in the sense that it is consistent, well-posed, and that its solutions remain close to exact solutions of the full Euler system with corresponding initial data, can be improved in two directions. The first direction is taking into account medium amplitude topography variations and the second direction is allowing strong nonlinearity using a new pseudo-symmetrizer, thus canceling out the smallness assumptions of the Camassa-Holm regime for the well-posedness and stability results.