Phase transition of anisotropic Ginzburg--Landau equation

Abstract

In this work, we study the co-dimensional 1 interface limit of an anisotropic Ginzburg-Landau equation under parabolic scalings. This is a semi-linear parabolic system of a planar vector field with a small parameter ε > 0 corresponding to the transition layer width. For well-prepared initial datum, as ε tends to 0, we show the solution gradient will concentrate on a closed simple curve evolving by curveshortening flow. Moreover, the limiting solution satisfies an anchoring boundary condition when restricted on the curve. These results hold as long as the limiting curvature flow remains smooth. The main ingredient of the proof is an estimate of the level set of the quasi-distance function corresponding to the solution.

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

“…A generalization to matrix-valued case has been done by Laux-Liu [21] to study the isotropic-nematic transition in Landau-De Gennes model of liquid crystals, which essentially corresponds to the case when m + = RP 2 and m − = 0. More recently, in [26] the author used these methods, together with those developed recently by Lin-Wang [23], to attack the convergence problem of an anisotropic 2D Ginzburg-Landau model. In particular, he derived some delicate convergence results of the level sets of the solutions, which are crucial to obtain anchoring boundary conditions of the limiting solutions on the moving interface.…”

“…Such an estimate, derived by applying the maximum principle to (1.7a), enables us to avoid several technical complications in the passage of the limit ε ↓ 0. Indeed, even in the case when d = 2, severe difficulties arise in the anisotropic model considered in [26] where an estimate like (1.22) is not available. Secondly, we also get a bound like sup t∈[0,T ] B[u ε |Σ](t) ≤ Cε.…”