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• W2144911403 abstract "In this paper we study the asymptotic behavior (c —• 0) of the Ginzburg Landau equation: ul Au l p 1#, as ii • oo, uniformly in 7> and AT (here 1^ is the indicator of the set A ). Furthermore the Hausdorff dimension of the interface T = complement of (V U A) C (0, oo) x7i is equal to d and it is a weak solution of the mean curvature flow as defined in [13, 93]. 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 ti» has an expansion of the form where q(r) = tanh(r) is the travelling wave associated to the cubic nonlinearity / , O(e) —• 0 as e —• 0, and d(t,x) is the signed distance of x to the {-section of r. We prove these results under fairly general assumptions on the initial data, «oIn particular we do not assume that ti(0,x) = q(d(0,z)/€), nor that we assume that the initial energy, £'(«'((),•)), is uniformly bounded in c. Main tools of our analysis are viscosity solutions of parabolic equations, weak viscosity limit of Barles-Perthame, weak solutions of mean curvature flow and their properties obtained in [13] and Umanen's generalization of Huisken's monotonidty formula." @default.
• W2144911403 created "2016-06-24" @default.
• W2144911403 creator A5050033426 @default.
• W2144911403 date "1993-01-01" @default.
• W2144911403 modified "2023-09-27" @default.
• W2144911403 title "Ginzburg-Landau equation and motion by mean curvature." @default.
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