The nonlocal Cahn–Hilliard–Hele–Shaw system with logarithmic potential

We study a diffuse interface model of the two-component Hele–Shaw flow. This is an advective Cahn–Hilliard equation for the relative concentration , where the incompressible velocity field is determined by the Darcy's law depending on the Korteweg force . Here μ is the derivative of a nonlocal non-convex free energy characterized by a logarithmic potential. The system is subject to no-flux boundary conditions for and μ along with an initial condition for . First of all, we prove that the corresponding problem is globally well posed with respect to a natural notion of weak solution. Also, we establish the existence of global strong solutions. In dimension two, we show the validity of the so-called instantaneous separation property. This means that any solution, which is not a pure phase initially, stays away from the pure phases, uniformly with respect to the initial energy and total mass. Finally we prove the existence of the global attractor for the corresponding dynamical system as well as the convergence to a single equilibrium of any weak solution in the two-dimensional case.