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On a Diffuse Interface Model for Two-Phase Flows of Viscous, Incompressible Fluids with Matched Densities
We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids of the same density in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. This leads to a coupled Navier-Stokes/Cahn-Hilliard system, which is capable to describe the evolution of droplet formation and collision during the flow. We prove existence of weak solutions of the non-stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities. Moreover, we present some results on regularity and uniqueness of weak solutions. In particular, we obtain that unique "strong" solutions exist in two dimensions globally in time and in three dimensions locally in time. Finally, we prove that any weak solution converges as $t\to \infty$ to a solution of the stationary system.