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

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 compared 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 timescale $t=O(1)$. However, on the timescale $t=O(\varepsilon^{-\frac1\alpha})$, 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, the 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 $\frac{\alpha}{3}$ to $\frac\alpha4$ is obtained for the case with one-sided degenerated mobility, in good agreement with the numerical experiments.