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 Author Saez Trumper, Mariel Ines Title Relaxation of the curve shortening flow on the plane via the parabolic Ginzburg-Landau equation
 Descript 88 p Note Source: Dissertation Abstracts International, Volume: 66-08, Section: B, page: 4263 Adviser: Rafe Mazzeo Thesis (Ph.D.)--Stanford University, 2005 This thesis presents a method to represent curves evolving under curve shortening flow as nodal sets of the limit of solutions to the parabolic Ginzburg-Landau equation Consider family of compact curves Gamma(lambda,t) : [0, lambda) x [0, T) → R2 that depend on a time parameter t, have an extinction time T and satisfy the equation 6G6t l,t=kGn&d4; , 1 where kGamma is the spatial curvature of Gamma(lambda, t) and nˆ its unit normal Let u* be a solution to -u*xx +12W' u*=0 u*0= 0andlim x→+/-infinityu* x=+/-1 I construct a family of solutions to the parabolic Ginzburg-Landau equation: 6ue6t -Due+1 uWue 2e2 =0 2 such that lime→0 supx ,t∈R2x &sqbl0;0,infinity&parr0; uex,t- v&d5;*e x,t=0, where v&d5;*e is a function with the following features: Let d(x, t) be the signed distance to Gamma(lambda, t). Then for t < T there are neighborhoods U ' ⊂ U of Gamma(lambda, t) such that v&d5;*e x,t=u* dx,t e forx∈U', and v&d5;*e x,t≡ 1forx∈R 2\U For t ≥ T v&d5;*e x,t≡1 This result is proven by constructing approximate solutions v&d5;*e to the equation (2) and estimating epsilon( x, t) - v&d5;*e (x, t)| using fixed point methods School code: 0212 DDC Host Item Dissertation Abstracts International 66-08B Subject Mathematics 0405 Alt Author Stanford University
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