On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems

Well-posedness of the Hele-Shaw-Cahn-Hilliard system

We study the well-posedness of the Hele-Shaw-Cahn-Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. Well-posedness that is global in time in the two dimensional case and local in time in the three dimensional case are established. Several blow-up criterions in the three dimensional case are provided as well.

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

We study the long time behavior of the Hele-Shaw-Cahn-Hilliard system (HSCH) which models two phase incompressible Darcian flow in porous media with matched density but arbitrary viscosity contrast. We demonstrate that the ω-limit set of each trajectory is a single stationary solution of the system via Lojasiewicz-Simon type technique. Moreover, a rate of convergence has been established. Event...

On Nonlocal Cahn-Hilliard-Navier-Stokes Systems in Two Dimensions

We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. He...

The Cahn-hilliard Equation

1. Steady states. There are many two component systems in which phase separation can be induced by rapidly cooling the system. Thus, if a two component system, which is spatially uniform at temperature T1, is rapidly cooled to a second sufficiently lower temperature T2, then the cooled system will separate into regions of higher and lower concentration. A phenomenological description of the beh...

The Cahn–Hilliard Equation