  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.
