Relaxation of the curve shortening flow on the plane via the parabolic Ginzburg-Landau equation.

摘要

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 |uepsilon( x, t) - v&d5;*e (x, t)| using fixed point methods.