A gradient flow approach to relaxation rates for the multi-dimensional Cahn-Hilliard equation

TL;DR

This paper investigates the relaxation dynamics of the multi-dimensional Cahn-Hilliard equation using a gradient flow framework, revealing a two-stage convergence process with distinct rates.

Contribution

It extends the gradient flow approach to higher dimensions, establishing new relationships between energy, dissipation, and distance to a kink, and identifies the two-phase relaxation rates.

Findings

Fast convergence to a kink at rate t^{-1/2}

Slow relaxation to the centered kink at rate t^{-1/4}

Differential relationships between energy, dissipation, and distance

Abstract

The aim of this paper is to study relaxation rates for the Cahn-Hilliard equation in dimension larger than one. We follow the approach of Otto and Westdickenberg based on the gradient flow structure of the equation and establish differential and algebraic relationships between the energy, the dissipation, and the squared $H^{--1}$ distance to a kink. This leads to a scale separation of the dynamics into two different stages: a first fast phase of the order $t^{ -- 1/2}$ where one sees convergence to some kink, followed by a slow relaxation phase with rate $t^{-- 1/ 4}$ where convergence to the centered kink is observed.