Local well-posedness of the multi-layer shallow-water model with free surface

TL;DR

This paper establishes conditions for the hyperbolicity and local well-posedness of multi-layer shallow water models with free surface, including new criteria and analysis under specific regimes and assumptions.

Contribution

It provides a general symmetrizability criterion for the model, analyzes eigenstructure, and introduces a new conservative model with proven well-posedness.

Findings

Symmetrizability implies hyperbolicity and well-posedness in H^s(R^2).

New criteria under asymptotic regimes and weak stratification.

A conservative model also shown to be well-posed.

Abstract

In this paper, we address the question of the hyperbolicity and the local well- posedness of the multi-layer shallow water model, with free surface, in two dimensions. We first provide a general criterion that proves the symmetrizability of this model, which implies hyperbolicity and local well-posedness in H^s(R^2), with s > 2. Then, we analyze rigorously the eigenstructure associated to this model and prove a more general criterion of hyperbolicity and local well-posedness, under a particular asymptotic regime and a weak stratification assumptions of the densities and the velocities. Finally, we consider a new conservative multi-layer shallow water model, we prove the symmetrizability, the hyperbolicity and the local well-posedness and rely it to the basic multi-layer shallow water model.