The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities

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 existence of global attractor for a Cahn-Hilliard/Allen-Cahn‎ ‎equation

In this paper, we consider a Cahn-Hillard/Allen-Cahn equation. By using the semigroup and the classical existence theorem of global attractors, we give the existence of the global attractor in H^k(0

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...

Approximate inertial manifolds for the pattern formation Cahn-Hilliard equation

An approximate inertial manifold for an évolution équation is a finite dimensional smooth manifold such that the orbits enter, after a transient time, a very thin neighbourhood of the manifold In this paper, we consider the Cahn-Hilliard équation and we present a method which allows to construct several approximate inertial manifolds providing better and better order approximations to the orbit...