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

In this paper we study the asymptotic behavior (ε → 0) of the Ginzburg-Landau equation: u₁^ε - Δu^ε + 1/ε² f (u^ε) = 0, where the unknown u^ε is a real-valued function of [0, ∞) x R^d, and the given nonlinear function f (u) = 2u (u² - 1) is the derivative of a potential W (u) = (u² - 1)²/2 with two minima of equal depth. We prove that there are a subsequence ε_n and two disjoint, open subsets P, N of (0, ∞) x R^d satisfying u^εn → 1_P - 1_N, as n → ∞, uniformly in P and N (here 1_A is the indicator of the set A ). Furthermore, the Hausdorff dimension of the interface Γ = complement of (P ∪ N) ⊂ (0, ∞) x R^d is equal to d 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 that u^εn has an expansion of the form u^εn(t, x) = q (d (t, x) + O (ε_n)/ε_n), where q (r) = tanh (r) is the traveling wave associated to the cubic nonlinearity f, O (ε) → 0 as ε → 0, and d (t, x) is the signed distance of x to the t-section of Γ. We prove these results under fairly general assumptions on the initial data, u₀. In particular we do not assume that u^ε (,0 x) = q (d(0, x)/ε), nor that we assume that the initial energy, ε^ε (u^ε (0, •)), is uniformly bounded in ε. 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.