Energy-dissipation for time-fractional phase-field equations

TL;DR

This paper establishes energy dissipation laws for time-fractional phase-field models, including Allen-Cahn and Cahn-Hilliard equations, by proving positivity and positive-definiteness of related kernels, advancing understanding of non-local energy dynamics.

Contribution

It introduces new criteria for kernel positive-definiteness and proves fractional energy dissipation laws for non-local phase-field models.

Findings

Proved weighted positivity results for Caputo derivatives

Established fractional energy dissipation laws

Identified criteria for kernel positive-definiteness

Abstract

We consider a class of time-fractional phase field models including the Allen-Cahn and Cahn-Hilliard equations. We establish several weighted positivity results for functionals driven by the Caputo time-fractional derivative. Several novel criterions are examined for showing the positive-definiteness of the associated kernel functions. We deduce strict energy-dissipation for a number of non-local energy functionals, thereby proving fractional energy dissipation laws.