| Abstract: |
| In the talk, an initial-boundary value problem for the incompressible Navier-Stokes-Cahn-Hilliard sys-
tem with non-constant density, proposed by the speaker together with Abels and Gr\un in 2012, will be analyzed. This model arises in the diffuse
interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-
known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak
solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes toward an
equilibrium state for large times.
The analysis hinges upon the following key points: a global regularity result (with
explicit bounds) for the Cahn-Hilliard equation with divergence-free velocity,
the energy dissipation of the system, the separation property at large times, a weak-strong uniqueness type result,
and the Lojasiewicz-Simon inequality.
We then discuss mixed regularity
conditions on the velocity field and its gradient, under which the global weak solution conserves its
energy for all times. Finally, we show Lyapunov
stability for each steady state consisting of a zero velocity together with a local energy minimizer of the Ginzburg-Landau functional. |
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