Asymptotic Analysis on the Sharp Interface Limit of the Time-Fractional Cahn--Hilliard Equation

TL;DR

This paper derives and analyzes sharp interface limit models for the time-fractional Cahn--Hilliard equation, revealing different interface motions and coarsening rates influenced by fractional derivatives, supported by asymptotic analysis and numerical consistency.

Contribution

It introduces asymptotic models for the TFCHE's interface dynamics and coarsening rates, extending classical results to fractional time derivatives with theoretical and numerical validation.

Findings

Sharp interface limit models are fractional Stefan and Mullins--Sekerka problems.

Coarsening rate scales as α/3, with crossover to α/4 for degenerated mobility.

Results align well with numerical experiments.

Abstract

In this paper, we aim to study the motions of interfaces and coarsening rates governed by the time-fractional Cahn--Hilliard equation (TFCHE). It is observed by many numerical experiments that the microstructure evolution described by the TFCHE displays quite different dynamical processes comparing with the classical Cahn--Hilliard equation, in particular, regarding motions of interfaces and coarsening rates. By using the method of matched asymptotic expansions, we first derive the sharp interface limit models. Then we can theoretically analyze the motions of interfaces with respect to different timescales. For instance, for the TFCHE with the constant diffusion mobility, the sharp interface limit model is a fractional Stefan problem at the time scale $t = O(1)$. However, on the time scale $t = O({\varepsilon}^{1/{\alpha}} )$ the sharp interface limit model is a fractional Mullins--Sekerka model. Similar asymptotic regime results are also obtained for the case with one-sided degenerated mobility. Moreover, scaling invariant property of the sharp interface models suggests that the TFCHE with constant mobility preserves an ${\alpha}/3$ coarsening rate and a crossover of the coarsening rates from ${\alpha}/3$ to ${\alpha}/4$ is obtained for the case with one-sided degenerated mobility, which are in good agreement with the numerical experiments.