Phase transition of anisotropic Ginzburg--Landau equation

TL;DR

This paper investigates the sharp interface limit of an anisotropic Ginzburg--Landau equation, showing it converges to mean curvature flow and relates to liquid crystal models, using energy and weak convergence methods.

Contribution

It establishes the geometric evolution of solutions as the interface sharpens, connecting anisotropic Ginzburg--Landau equations to mean curvature flow and liquid crystal models.

Findings

Solutions develop a sharp interface following mean curvature flow.

The bulk solutions are of unit length on one side and zero on the other.

The vector field is tangent to the interface and satisfies a liquid crystal evolution equation.

Abstract

We study the effective geometric motions of an anisotropic Ginzburg--Landau equation with a small parameter $\varepsilon>0$ which characterizes the width of the transition layer. For well-prepared initial datum, we show that as $\varepsilon$ tends to zero the solutions will develop a sharp interface limit which evolves under mean curvature flow. The bulk limits of the solutions correspond to a vector field $\mathbf{u}(x,t)$ which is of unit length on one side of the interface, and is zero on the other side. The proof combines the modulated energy method and weak convergence methods. In particular, by a (boundary) blow-up argument we show that $\mathbf{u}$ must be tangent to the sharp interface. Moreover, it solves a geometric evolution equation for the Oseen--Frank model in liquid crystals.