Well-posedness of the Cahn–Hilliard equation with fractional free energy and its Fourier Galerkin approximation

Abstract

We study the well-posedness of the Cahn–Hilliard equation in which the gradient term in the free energy is replaced by a fractional derivative. We begin by establishing the existence and uniqueness of a Fourier-Galerkin approximation and then derive a number of priori estimates for the Fourier-Galerkin scheme. Compactness arguments are then used to deduce the existence and uniqueness of the solution to the fractional Cahn–Hilliard equation. An estimate for the rate of convergence of the Fourier-Galerkin approximation is then obtained. Finally, we present some numerical illustrations of typical solutions to the fractional Cahn–Hilliard equation and how they vary with the fractional order β and the parameter ε. In particular, we show how the width τ of the diffuse interface depends on ε and β, and derive the scaling law τ=O(ɛ1/β) which is verified numerically.