Convergence of the One-Dimensional Cahn--Hilliard Equation

Convergence of the One-Dimensional Cahn-Hilliard Equation

We consider the Cahn-Hilliard equation in one space dimension with scaling parameter ε, i.e. ut = (W ′(u) − εuxx)xx, where W is a nonconvex potential. In the limit ε ↓ 0, under the assumption that the initial data are energetically well-prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard e...

Relaxation to Equilibrium in the One-Dimensional Cahn-Hilliard Equation

We study the stability of a so-called kink profile for the one-dimensional Cahn– Hilliard problem on the real line. We derive optimal bounds on the decay to equilibrium under the assumption that the initial energy is less than three times the energy of a kink and that the initial Ḣ−1 distance to a kink is bounded. Working with the Ḣ−1 distance is natural, since the equation is a gradient flow w...

A semidiscrete scheme for a one-dimensional Cahn-Hilliard equation

We analyze a semidiscrete scheme for the Cahn-Hilliard equation in one space dimension, when the interface length parameter is equal to zero. We prove convergence of the scheme for a suitable class of initial data, and we identify the limit equation. We also characterize the long-time behavior of the limit solutions. keywords: Nonconvex functionals, forward-backward parabolic equations, finite ...

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