Nonhomogeneous Cahn–Hilliard fluids

Abstract In this paper we are interested in the study of a model of nonhomogeneous diphasic incompressible flow. More precisely we consider a coupling of a Cahn–Hilliard and an incompressible Navier–Stokes equations where the densities of the phases are different. For this general model we can only show the local existence of a unique very regular solution and the existence of weaker solutions is still an open problem. But, if we look at the behavior of the system when the densities tends to be equal (slightly nonhomogeneous case), we show the existence of a global weak solution and of a unique local strong solution (which is in fact global in 2D). Finally, an asymptotic stability result for the metastable states is shown in this slightly nonhomogeneous case.

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