  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.
