2D Phase Diagram for Minimizers of a Cahn-Hilliard Functional with Long-Range Interactions

TL;DR

This paper combines asymptotic analysis and numerical methods to explore the phase diagram of minimizers in a 2D Cahn-Hilliard model with long-range interactions, revealing good agreement between theory and simulations.

Contribution

It introduces a hybrid numerical approach and asymptotic estimates to analyze the phase diagram of a complex nonlocal Cahn-Hilliard functional.

Findings

Asymptotic estimates accurately predict stability regions.

Numerical simulations reveal detailed phase structures.

Good agreement between asymptotic predictions and numerical results.

Abstract

This paper presents a detailed asymptotic and numerical investigation of the phase diagram for global minimizers to a Cahn-Hilliard functional with long-range interactions in two space dimensions. We introduce a small parameter measuring perturbation from the minimal orderdisorder transition, and derive asymptotic estimates for stability regions as the parameter tends to zero. Based upon the H^-1 gradient ow, we introduce a hybrid numerical method to navigate through the complex energy landscape and access the ground state of the functional. We use this method to numerically compute the phase diagram. Our asymptotic predictions show surprisingly good agreement with our numerical results.