Ginzburg-Landau equation and motion by mean curvature, I: Convergence

In this paper we study the asymptotic behavior (∈→0) of the Ginzburg-Landau equation:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u_l^\varepsilon - \Delta u^\varepsilon + \frac{1}{{\varepsilon ^2 }}f(u^\varepsilon ) = 0.$$ \end{document}. where the unknownu∈ is a real-valued function of [0. ∞)×Rd, and the given nonlinear functionf(u) = 2u(u2−1) is the derivative of a potential W(u) = (u2−l)2/2 with two minima of equal depth. We prove that there are a subsequence ∈n and two disjoint, open subsetsP, N of (0, ∞) ×Rd satisfying\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u^{\varepsilon _n } \to 1_\mathcal{P} - 1_\mathcal{N} , as n \to \infty . $$ \end{document} uniformly inP andN (here 1A is the indicator of the setA). Furthermore, the Hausdorff dimension of the interface Γ = complement of (P∪N) ⊂ (0, ∞)×Rd is equal tod and it is a weak solution of the mean curvature flow as defined in [13,92]. If this weak solution is unique, or equivalently if the level-set solution of the mean curvature flow is “thin,” then the convergence is on the whole sequence. We also show thatu∈n has an expansion of the form\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u^{\varepsilon _n } (t,x) = q\left( {\frac{{d(t,x) + O(\varepsilon _n )}}{{\varepsilon _n }}} \right).$$ \end{document} whereq(r) = tanh(r) is the traveling wave associated to the cubic nonlinearityf, O(∈) → 0 as ∈ → 0, andd(t, x) is the signed distance ofx to thet-section of Γ. We prove these results under fairly general assumptions on the initial data,u0. In particular we donot assume thatu∈(0.x) = q(d(0,x)/∈), nor that we assume that the initial energy, ε∈(u∈(0, .)), is uniformly bounded in ∈. Main tools of our analysis are viscosity solutions of parabolic equations, weak viscosity limit of Barles and Perthame, weak solutions of mean curvature flow and their properties obtained in [13] and Ilmanen’s generalization of Huisken’s monotonicity formula.