Cahn-Hilliard Equations on Random Walk Spaces

TL;DR

This paper investigates a nonlocal Cahn-Hilliard model within random walk spaces, establishing existence, uniqueness, and asymptotic behavior of solutions, and linking it to gradient flow structures in a broad mathematical framework.

Contribution

It introduces a general nonlocal Cahn-Hilliard model on random walk spaces, proving fundamental properties and connecting it to gradient flow theory.

Findings

Existence and uniqueness of solutions in L^1

Cahn-Hilliard equation as a gradient flow of free energy

Analysis of asymptotic behavior of solutions

Abstract

In this paper we study a nonlocal Cahn-Hilliard model (CHE) in the framework of random walk spaces, which includes as particular cases, the CHE on locally finite weighted connected graphs, the CHE determined by finite Markov chains or the Cahn-Hilliard Equations driven by convolution integrable kernels. We consider different transitions for the phase and the chemical potential, and a large class of potentials including obstacle ones. We prove existence and uniqueness of solutions in $L^1$ of the Cahn-Hilliard Equation. We also show that the Cahn-Hilliard equation is the gradient flow of the Ginzburg-Landau free energy functional on an appropriate Hilbert space. We finally study the asymptotic behaviour of the solutions.