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

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 ε 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 ≥ 2) for the non-local model and we also prove the convergence, as ε goes to zero, to the solution of the local Korteweg system.