Calculus of Variations and Geometric Measure Theory

L. De Luca - M. Goldman - M. Strani

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

created by goldman on 21 Feb 2018
modified by deluca on 23 Oct 2019


Published Paper

Inserted: 21 feb 2018
Last Updated: 23 oct 2019

Journal: Math. Annalen
Volume: 374
Number: 3-4
Pages: 2041-2081
Year: 2019
Doi: 10.1007/s00208-018-1765-x


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 $\dot H^{-1}$ distance to a kink. This leads to a scale separation of the dynamics into two different stages: a first {\it fast} phase of the order $t^{-\frac 1 2}$ where one sees convergence to some kink, followed by a {\it slow} relaxation phase with rate $t^{-\frac 1 4}$ where convergence to the centered kink is observed.