High friction limits of Euler–Navier–Stokes–Korteweg equations for multicomponent models

In this paper we analyze the high friction regime for the Navier-Stokes-Korteweg equations for multicomponent systems. According to the shape of the mixing and friction terms, we shall perform two limits: the high friction limit toward an equilibrium system for the limit densities and the barycentric velocity, and, after an appropriate time scaling, the diffusive relaxation toward parabolic, gradient flow equations for the limit densities. The rigorous justification of these limits is done by means of relative entropy techniques in the framework of weak, finite energy solutions of the relaxation models, rewritten in the enlarged formulation in terms of the drift velocity, toward smooth solutions of the corresponding equilibrium dynamics. Finally, since our estimates are uniform for small viscosity, the results are also valid for the Euler-Korteweg multicomponent models, and the corresponding estimates can be obtained by sending the viscosity to zero.