Parabolic equations for curves on surfaces Part I. Curves with $p$-integrable curvature

@article{Angenent1990ParabolicEF,
  title={Parabolic equations for curves on surfaces Part I. Curves with \$p\$-integrable curvature},
  author={Sigurd B. Angenent},
  journal={Annals of Mathematics},
  year={1990},
  volume={132},
  pages={451-483}
}
  • S. Angenent
  • Published 1 November 1990
  • Mathematics
  • Annals of Mathematics
This is the first of a two-part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so-called curve shortening of flow-by-mean-curvature problem, as well as to a number of models for phase transitions in two dimensions. We introduce a class of equations for which the initial value problem is solvable for initial data with p-integrable curvature, and we also give estimates for the rate at which the p-norms of the curvature must blow up, if… 
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References

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The normalized curve shortening flow and homothetic solutions
The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradient
Nonlinear analytic semiflows
In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems ( x ′( t ) = f ( x ( t ))), and quasilinear initial value problems ( x ′( t )= A ( x ( t
The heat equation shrinks embedded plane curves to round points
Soit C(•,0):S 1 →R 2 une courbe lisse plongee dans le plan. Alors C:S 1 ×[0,T)→R 2 existe en satisfaisant δC/δt=K•N, ou K est la courbure de C, et N est son vecteur unite normal entrant. C(•,t) est
A stable manifold theorem for the curve shortening equation
On presente une famille de solutions homothetiques de l'equation pour une courbe planaire ∂X/∂τ=KN et on demontre l'existence de varietes non lineaires stables et instables autour de telles solutions