Asymptotic limit of a vector-valued Allen-Cahn equation for phase transition dynamics

TL;DR

This paper analyzes the asymptotic behavior of solutions to a vector-valued Allen-Cahn equation, showing that as the parameter approaches zero, the solutions converge to a two-phase flow system with interfaces evolving by mean curvature.

Contribution

It rigorously establishes the sharp interface limit of the vector-valued Allen-Cahn equation, combining matched asymptotic expansions with spectral analysis techniques.

Findings

Interface evolves by mean curvature flow

Solutions follow harmonic map heat flow in bulk

Mixed boundary conditions on the interface

Abstract

In this paper, we study the asymptotic limit, as $\varepsilon\to 0$, of solutions to a vector-valued Allen-Cahn equation $$ \partial_t u = \Delta u - \frac{1}{\varepsilon^2} \partial_u F(u), $$ where $u: \Omega \subset \mathbb{R}^m \to \mathbb{R}^n$ and $F(u): \mathbb{R}^n \to \mathbb{R}$ is a nonnegative radial function which vanishes precisely on two concentric spheres. This equation, proposed and studied by Bronsard and Stoth [Trans. Amer. Math. Soc. 1998] for the case $n=2$, serves as a typical example for a general reaction-diffusion equation introduced by Rubinstein, Sternberg, and Keller to model chemical reactions and diffusions as well as phase transitions. We establish that the sharp interface limit is a two-phase flow system: (i) The interface evolves by mean curvature flow; (ii) Within the bulk phase regions, the solution follows the harmonic map heat flow into $\mathbb{S}^{n-1}$; (iii) Across the interface, the $\mathbb{S}^{n-1}$-valued vectors on the two sides satisfy a mixed boundary condition. Furthermore, we rigorously justify this limit using the matched asymptotic expansion method. First, we employ the idea of ``quasi-minimal connecting orbits'' developed in Fei, Lin, Wang, and Zhang [Invent. Math. 2023] to construct approximated solutions up to arbitrary order. Second, we derive a uniform spectral lower bound for the linearized operator around the approximate solution, which relies on a novel application of the boundary condition. To achieve this, we introduce a suitable decomposition which can reduce the problem to spectral analysis of two scalar one-dimensional linear operators and some singular product estimates.