Subdifferential Operator Approach to the Cahn-Hilliard Equation with Constraint

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

The critical mass constraint in the Cahn-Hilliard equation

When the mass constraint of the Cahn-Hilliard equation in two dimensions is lowered to the order of ǫ, where ǫ is the interface thickness parameter, the existence of droplet solutions becomes conditional. For interior single droplet solutions, there is a critical value for the mass constraint such that above this value two interior single droplet solutions exist, and below this value interior s...

A Gamma-convergence approach to the Cahn–Hilliard equation

We study the asymptotic dynamics of the Cahn–Hilliard equation via the “Gamma-convergence” of gradient flows scheme initiated by Sandier and Serfaty. This gives rise to an H1-version of a conjecture by De Giorgi, namely, the slope of the Allen–Cahn functional with respect to the H−1-structure Gamma-converges to a homogeneous Sobolev norm of the scalar mean curvature of the limiting interface. W...

The convective Cahn-Hilliard equation

We consider the convective Cahn–Hilliard equation with periodic boundary conditions as an infinite dimensional dynamical system and establish the existence of a compact attractor and a finite dimensional inertial manifold that contains it. Moreover, Gevrey regularity of solutions on the attractor is established and used to prove that four nodes are determining for each solution on the attractor...