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

Abstract

In this paper we study the asymptotic behavior (∈→0) of the Ginzburg-Landau equation:

$$u_l^\varepsilon - \Delta u^\varepsilon + \frac{1}{{\varepsilon ^2 }}f(u^\varepsilon ) = 0.$$

. where the unknownu ^∈ is a real-valued function of [0. ∞)×^Rd, and the given nonlinear functionf(u) = 2u(u ²−1) is the derivative of a potential W(u) = (u ²−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, ∞) ×R ^d satisfying

$$u^{\varepsilon _n } \to 1_\mathcal{P} - 1_\mathcal{N} , as n \to \infty . $$

uniformly inP andN (here 1 _A is the indicator of the setA). Furthermore, the Hausdorff dimension of the interface Γ = complement of (P∪N) ⊂ (0, ∞)×R ^d 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

$$u^{\varepsilon _n } (t,x) = q\left( {\frac{{d(t,x) + O(\varepsilon _n )}}{{\varepsilon _n }}} \right).$$

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,u ⁰. 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.