Longtime behavior for a modelof homogeneous incompressible two-phase flows

We consider a diffuse interface model for the evolution of aniso-thermal incompressible two-phase flow in a two-dimensional boundeddomain. The model consists of the Navier-Stokes equation for the fluidvelocity u coupled with a convective Allen-Cahn equation forthe order (phase) parameter φ, both endowed with suitable boundaryconditions. We analyze the asymptotic behavior of the solutions within thetheory of infinite-dimensional dissipative dynamical systems. We first provethat the initial and boundary value problem generates a strongly continuoussemigroup on a suitable phase space which possesses the global attractor A. Then we establish the existence of an exponential attractor E which entails that A has finite fractal dimension.This dimension is then estimated in terms of some model parameters.Moreover, assuming the potential to be real analytic, we demonstrate that,in absence of external forces, each trajectory converges to a singleequilibrium by means of a Łojasiewicz-Simon inequality. We also obtain aconvergence rate estimate. Finally, we discuss the case where φ isforced to take values in a bounded interval, e.g., by a so-called singularpotential.