Well-posedness of the Cahn-Hilliard equation with fractional free energy and its Fourier Galerkin approximation
Well-posedness of the Cahn-Hilliard equation with fractional free energy and its Fourier Galerkin approximation
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DOI：
10.1016/j.chaos.2017.05.022
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发表时间：
2017-09-01
影响因子：
7.8
通讯作者：
Mao, Zhiping
中科院分区：
文献类型：
--
作者：
Ainsworth, Mark;Mao, Zhiping
研究了自由能中的梯度项被分数阶导数取代的Cahn-Hilliard方程的适定性。我们开始建立的存在性和唯一性的Fourier-Galerkin逼近，然后推导出一些先验估计的Fourier-Galerkin计划。紧性参数，然后推导出的分数Cahn-Hilliard方程的解的存在性和唯一性。然后，得到的Fourier-Galerkin逼近的收敛速度的估计。最后，我们给出了分数阶Cahn-Hilliard方程的典型解的一些数值例子，以及它们是如何随分数阶β和参数β而变化的。特别是，我们展示了扩散界面的宽度τ如何依赖于β和β，并推导出标度律τ = O（β（1/β）），这是数值验证。(C)2017爱思唯尔有限公司版权所有
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. (C) 2017 Elsevier Ltd. All rights reserved.