Spinodal decomposition and coarsening fronts in the Cahn-Hilliard equation

TL;DR

This paper investigates the dynamics of spinodal decomposition and coarsening in the Cahn-Hilliard equation, analyzing invasion fronts, their speeds, and patterns through theoretical and numerical methods.

Contribution

It provides a rigorous existence proof of nonlinear invasion fronts and compares theoretical predictions with numerical simulations in the context of phase separation.

Findings

Existence of nonlinear invasion fronts confirmed.

Linear predictions for front speeds and wavenumbers validated.

Bifurcation phenomena observed in simulations.

Abstract

We study spinodal decomposition and coarsening when initiated by localized disturbances in the Cahn-Hilliard equation. Spatio-temporal dynamics are governed by multi-stage invasion fronts. The first front invades a spinodal unstable equilibrium and creates a spatially periodic unstable pattern. Secondary fronts invade this unstable pattern and create a coarser pattern in the wake. We give linear predictions for speeds and wavenumbers in this process and show existence of corresponding nonlinear fronts. The existence proof is based on Conley index theory, a priori estimates, and Galerkin approximations. We also compare our results and predictions with direct numerical simulations and report on some interesting bifurcations.