article

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

Authors: Frederic Charve and Boris Haspot

Issue: Volume 60 (2011), Issue 6, 2021-2060

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 formally tends to the local Korteweg system. After giving some explanations about the capillarity (physical justification and purpose, motivations related to the theory of non-classical shocks (see [P.G. LeFloch, \textit{Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves}, Lectures in Mathematics ETH Z\"urich, Birkh\"auser Verlag, Basel, 2002])), we prove global well-posedness (in the whole space $\mathbb{R}^d$ with $d \geq 2$) for the non-local model, as well as the convergence, as $\epsilon$ goes to zero, to the solution of the local Korteweg system.$