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In the present article we consider a capillary compressible system introducedby C. Rohde after works of Bandon, Lin and Rogers, called the order-parametermodel, and whose aim is to reduce the numerical difficulties that oneencounters in the case of the classical local Korteweg system (involvingderivatives of order three) or the non-local system (also introduced by Rohdeafter works of Van der Waals, and which involves a convolution operator). Weprove that this system has a unique global solution for initial data close toan equilibrium and we precisely study the convergence of this solution towardsthe local Korteweg model.

Convergence of a low order non-local Navier-Stokes-Korteweg system: the order-parameter model