Local in time results for local and non-local capillary Navier-Stokes systems with large data

TL;DR

This paper establishes local in time existence and convergence results for local and non-local capillary Navier-Stokes systems with large initial data, including models with non-stable reference states and density-dependent coefficients.

Contribution

It provides the first local existence results for large data in both local and non-local capillary Navier-Stokes models with non-stable reference states.

Findings

Unique local solutions for large initial data.

Convergence rates of non-local models to the local Korteweg model.

Extension framework for density-dependent coefficients.

Abstract

In this article we study three capillary compressible models (the classical local Navier-Stokes-Korteweg system and two non-local models) for large initial data, bounded away from zero, and with a reference pressure state $\bar{\rho}$ which is not necessarily stable ($P'(\bar{\rho})$ can be non-positive). We prove that these systems have a unique local in time solution and we study the convergence rate of the solutions of the non-local models towards the local Korteweg model. The results are given for constant viscous coefficients and we explain how to extend them for density dependant coefficients.