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In the present article we are interested in further investigations for thebarotropic compressible Navier-Stokes system endowed with a non-localcapillarity we studied in [7]. Thanks to an accurate study of the associatedlinear system using a Lagrangian change of coordinates, we provide more preciseenergy estimates in terms of hybrid Besov spaces naturally depending on athreshold frequency (which is determined in function of the physical parameter)distinguishing the low and the high regimes. It allows us in particular toprove the convergence of the solutions from the non-local to the local Kortewegsystem. Another mathematical interest of this article is the study of theeffect of the Lagrangian change on the non-local capillary term.

On a Lagrangian method for the convergence from a non-local to a local Korteweg capillary fluid model