Convergence of capillary fluid models: from the non-local to the local Korteweg model

TL;DR

This paper proves the global well-posedness of a non-local capillarity fluid model and demonstrates its convergence to the local Korteweg system as the non-local parameter tends to zero, bridging non-local and local fluid models.

Contribution

It establishes the global well-posedness of the non-local capillarity model and rigorously shows its convergence to the local Korteweg system as the non-local parameter vanishes.

Findings

Proved global well-posedness of the non-local model.

Established convergence to the local Korteweg system as epsilon approaches zero.

Provided physical motivation related to non-classical shocks.

Abstract

In this paper we are interested in the barotropic compressible Navier-Stokes system endowed with a non-local capillarity tensor depending on a small parameter $\epsilon$ such that it heuristically tends to the local Korteweg system. After giving some physical motivations related to the theory of non-classical shocks (see [28]) we prove global well-posedness (in the whole space $R^d$ with $d\geq 2$) for the non-local model and we also prove the convergence, as $\epsilon$ goes to zero, to the solution of the local Korteweg system.