Long-time behavior for the Hele–Shaw–Cahn–Hilliard system

Abstract

We study the Hele–Shaw–Cahn–Hilliard system that models two phase incompressible Darcian flow in porous media with matched density but arbitrary viscosity contrast. In the 3D case, we prove eventual regularity of weak solutions, as well as existence of global classical solutions if either the Peclet number is sufficiently small or the initial datum is close to one local energy minimizer of the free energy. In both 2D and 3D, we demonstrate that the ω-limit set of each trajectory consists of a single steady state. Finally, stability of local minimizers is established.