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

Mark Ainsworth and Zhiping Mao

Chaos, Solitons & Fractals, 2017, vol. 102, issue C, 264-273

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.

Keywords: Fractional Cahn–Hilliard equation; Non-local energy; Well-posedness; Fourier spectral method; Error estimates (search for similar items in EconPapers)
Date: 2017
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0960077917302114
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:102:y:2017:i:c:p:264-273

DOI: 10.1016/j.chaos.2017.05.022

Access Statistics for this article

Chaos, Solitons & Fractals is currently edited by Stefano Boccaletti and Stelios Bekiros

More articles in Chaos, Solitons & Fractals from Elsevier
Bibliographic data for series maintained by Thayer, Thomas R. ().