Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions

TL;DR

This paper introduces a time-fractional Cahn-Hilliard model derived from continuum theory, analyzes its mathematical properties, and develops numerical schemes to simulate phase separation with memory effects.

Contribution

It provides the first rigorous analysis of well-posedness for the time-fractional Cahn-Hilliard equation with degenerating mobility and free energies, along with numerical methods and simulations.

Findings

Existence and uniqueness of weak solutions established.

Numerical simulations demonstrate the impact of fractional order on phase separation.

A fractional chain inequality for semiconvex functions is proved.

Abstract

In this paper, we derive the time-fractional Cahn-Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn-Hilliard equation. In this regard, we consider degenerating mobility functions and free energies of Landau, Flory--Huggins and double-obstacle type. We apply the Faedo-Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form. In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions. The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power. Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization.