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

In this paper we study the asymptotic behavior (∈→0) of the Ginzburg-Landau equation: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaDa% aaleaacaWGSbaabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYt% UvgaiuaacqWF1oG8aaGccqGHsislcqqHuoarcaWG1bWaaWbaaSqabe% aacqWF1oG8aaGccqGHRaWkdaWcaaqaaiaaigdaaeaacqWF1oG8daah% aaWcbeqaaGqaaiaa+jdaaaaaaOGaamOzaiaacIcacaWG1bWaaWbaaS% qabeaacqWF1oG8aaGccaGGPaGaeyypa0JaaGimaiaac6caaaa!565F! $$u_l^\varepsilon - \Delta u^\varepsilon + \frac{1}{{\varepsilon ^2 }}f(u^\varepsilon ) = 0.$$ . 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 % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaCa% aaleqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaa% cqWF1oG8daWgaaadbaGaamOBaaqabaaaaOGaeyOKH4kcbeGaa4xmam% aaBaaaleaacqWFpepuaeqaaOGaa4xlaiaa+fdadaWgaaWcbaGae8xd% X7eabeaakiaacYcaruqqYLwySbacgaGaa0hiaiaa9bcacaqFGaGaa0% hiaGqaaiaa8fgacaaFZbGaa0hiaGqaciaa75gacqGHsgIRcqGHEisP% caWEUaGaa0hiaaaa!595E! $$u^{\varepsilon _n } \to 1_\mathcal{P} - 1_\mathcal{N} , as n \to \infty . $$ 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 % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaCa% aaleqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaa% cqWF1oG8daWgaaadbaGaamOBaaqabaaaaOGaaiikaiaadshacaGGSa% GaamiEaiaacMcacqGH9aqpcaWGXbWaaeWaaeaadaWcaaqaaiaadsga% caGGOaGaamiDaiaacYcacaWG4bGaaiykaiabgUcaRiaad+eacaGGOa% Gae8xTdi-aaSbaaSqaaGqaciaa+5gaaeqaaOGaaiykaaqaaiab-v7a% YpaaBaaaleaacaGFUbaabeaaaaaakiaawIcacaGLPaaacaGGUaaaaa!5AE5! $$u^{\varepsilon _n } (t,x) = q\left( {\frac{{d(t,x) + O(\varepsilon _n )}}{{\varepsilon _n }}} \right).$$ 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, e∈(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.