From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

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

Upper Bounds for Coarsening for the Degenerate Cahn-hilliard Equation

The long time behavior for the degenerate Cahn-Hilliard equation [4, 5, 10], ut = ∇ · (1− u)∇ [Θ 2 {ln(1 + u)− ln(1− u)} − αu− 4u ] , is characterized by the growth of domains in which u(x, t) ≈ u±, where u± denote the ”equilibrium phases;” this process is known as coarsening. The degree of coarsening can be quantified in terms of a characteristic length scale, l(t), where l(t) is prescribed vi...

Interface Solutions of a Degenerate Cahn - Hilliard Equation

We present an analysis of the equilibrium diiusive interfaces in a model for the interaction of layers of pure polymers. The discussion focuses on the important qualitative features of the solutions of the nonlinear singular Cahn-Hilliard equation with degenerate mobility for the Flory-Huggins-de Gennes free energy model. The spatial structure of possible equilibrium phase separated solutions a...

On the Cahn{hilliard Equation with Degenerate Mobility

An existence result for the Cahn{Hilliard equation with a concentration dependent diiusional mobility is presented. In particular the mobility is allowed to vanish when the scaled concentration takes the values 1 and it is shown that the solution is bounded by 1 in magnitude. Finally applications of our method to other degenerate fourth order parabolic equations are discussed.