Fractional time differential equations as a singular limit of the Kobayashi-Warren-Carter system

TL;DR

This paper rigorously derives a fractional time differential system as the singular limit of the Kobayashi-Warren-Carter phase field model, revealing fractional derivatives emerge in the zero interface thickness limit.

Contribution

It provides the first rigorous derivation of a fractional time derivative system from a classical phase field model in a one-dimensional setting.

Findings

Limit system involves fractional time derivatives

Derivation is rigorous in one-dimensional gradient flow setting

Connects phase field models with fractional calculus

Abstract

This paper is concerned with a singular limit of the Kobayashi-Warren-Carter system, a phase field system modelling the evolutions of structures of grains. Under a suitable scaling, the limit system is formally derived when the interface thickness parameter tends to zero. Different from many other problems, it turns out that the limit system is a system involving fractional time derivatives, although the original system is a simple gradient flow. A rigorous derivation is given when the problem is reduced to a gradient flow of a single-well Modica-Mortola functional in a one-dimensional setting.