Cauchy problem for viscous shallow water equations with a term of capillarity

TL;DR

This paper investigates the well-posedness and existence of solutions for a compressible Navier-Stokes model with capillarity and variable viscosity, relevant to shallow water and related fluid systems.

Contribution

It establishes global and local existence results for solutions in critical regularity spaces, extending the understanding of such models with capillarity effects.

Findings

Proves global existence of solutions near stable equilibrium.

Establishes local existence for general initial data.

Demonstrates uniqueness of solutions.

Abstract

In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients and a term of capillarity introduced by Coquel et al in \cite{5CR}. This model includes at the same time the barotropic Navier-Stokes equations with variable viscosity coefficients, shallow-water system and the model of Rohde. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possibleto the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence for solutions with general initial data. Uniqueness is also obtained.