Critical homogenization of the Navier--Stokes--Cahn--Hilliard system in perforated domains

We consider the three-dimensional incompressible Navier–Stokes–Cahn–Hilliard (NSCH) system in domains perforated by no-slip obstacles of diameter of order ${\epsilon }^3$ separated by distances of order $\epsilon$. Both viscosity and mobility may depend on the phase variable. At this critical Stokes-capacity scale, the limiting velocity $\mathit{u}$ and phase field $\phi$ satisfy an NSCH system with the additional Brinkman resistance $\nu (\phi )\mathbf{B}\mathit{u}$, where $\mathbf{B}$ is determined by the exterior Stokes capacity of the reference obstacle. The principal analytical difficulty is the interaction between the phase-dependent viscosity and the order-one energy concentration of the critical Stokes correctors. The variational chemical-potential identity yields, after scalar extension, strong convergence of the phase in $L^2(0,T;H^1(\mathrm{\Omega }))$, and hence strong convergence of the phase-dependent viscosity in the same topology. Combined with a cellwise Hardy multiplier estimate, this compactness can be transferred through the concentrated corrector layer. The resulting weighted-capacity statement applies to uniformly positive and bounded coefficient sequences converging strongly in $L^2(0,T;H^1(\mathrm{\Omega }))$ and simultaneously identifies the effective Brinkman force and the corresponding viscous dissipation lower bound. We also investigate a vanishing capillary coefficient ${\lambda }_{\epsilon }\to 0$. After the natural velocity scaling, the limit is an unsteady Stokes–Brinkman equation coupled to an unadvected Cahn–Hilliard equation.