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

作者：Ainsworth Mark^*; Mao Zhiping

来源：Chaos Solitons & Fractals, 2017, 102: 264-273.

DOI：10.1016/j.chaos.2017.05.022

• Fractional Cahn-Hilliard equation
• Non-local energy
• Well-posedness
• Fourier spectral method
• Error estimates

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 beta and the parameter epsilon. In particular, we show how the width tau of the diffuse interface depends on epsilon and beta, and derive the scaling law tau = O (epsilon(1/beta)) which is verified numerically.

• 出版日期2017-9