Local Well-Posedness of the Two-Layer Shallow Water Model with Free Surface

Abstract

In this paper, we address the question of the hyperbolicity and the local well-posedness of the two-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 $\mathcal{H}^s(\mathbb{R}^2)$, with $s>2$. Then, we analyze rigorously the eigenstructure associated with this model and prove a more general criterion for hyperbolicity and local well-posedness, under a weak density-stratification assumption. Finally, we consider a new conservative two-layer shallow water model, prove the hyperbolicity and the local well-posedness, and relate it to the basic two-layer shallow water model.