Phase transition of anisotropic Ginzburg--Landau 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.