Phase Transition of Parabolic Ginzburg--Landau Equation with Potentials of High-Dimensional Wells

TL;DR

This paper investigates the dynamic behavior and interface evolution of parabolic Ginzburg--Landau systems with complex potentials, extending previous static results to include time-dependent phenomena and boundary conditions.

Contribution

It generalizes prior static interface results to a dynamical setting, deriving harmonic heat flows and boundary conditions using energy and convergence methods.

Findings

Derived limiting harmonic heat flows in bulk regions

Established a non-standard boundary condition for the flows

Extended static interface results to dynamical cases

Abstract

In this work, we study the co-dimensional one interface limit and geometric motions of parabolic Ginzburg--Landau systems with potentials of high-dimensional wells. The main result generalizes the one by Lin et al. (Comm. Pure Appl. Math., 65(6):833-888, 2012) to a dynamical case. In particular combining modulated energy methods and weak convergence methods, we derive the limiting harmonic heat flows in the inner and outer bulk regions segregated by the sharp interface, and a non-standard boundary condition for them. These results are valid provided that the initial datum of the system is well-prepared under natural energy assumptions.