The Allen-Cahn equation with generic initial datum

TL;DR

This paper studies the Allen-Cahn equation with random initial data, showing that under certain conditions, the evolving fronts resemble nodal sets of a Gaussian field and follow mean curvature flow.

Contribution

It demonstrates that for the Allen-Cahn equation with Gaussian initial conditions, the resulting interfaces are described by Gaussian field nodal sets and evolve via mean curvature flow, linking stochastic initial data to geometric evolution.

Findings

Fronts are described by nodal sets of the Bargmann-Fock Gaussian field.

The interfaces evolve according to mean curvature flow.

The results hold when the initial amplitude is not too large.

Abstract

We consider the Allen-Cahn equation $\partial_t u- \Delta u=u-u^3$ with a rapidly mixing Gaussian field as initial condition. We show that provided that the amplitude of the initial condition is not too large, the equation generates fronts described by nodal sets of the Bargmann-Fock Gaussian field, which then evolve according to mean curvature flow.