  In the present article we consider a capillary compressible system introduced
by C. Rohde after works of Bandon, Lin and Rogers, called the order-parameter
model, and whose aim is to reduce the numerical difficulties that one
encounters in the case of the classical local Korteweg system (involving
derivatives of order three) or the non-local system (also introduced by Rohde
after works of Van der Waals, and which involves a convolution operator). We
prove that this system has a unique global solution for initial data close to
an equilibrium and we precisely study the convergence of this solution towards
the local Korteweg model.
