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On a degenerate parabolic system describing the mean curvature flow of rotationally symmetric closed surfaces

Harald Garcke, Bogdan-Vasile Matioc

Abstract

We show that the mean curvature flow for a closed and rotationally symmetric surface can be formulated as an evolution problem consisting of an evolution equation for the square of the function whose graph is rotated and two ODEs describing the evolution of the points of the evolving surface that lie on the rotation axis. For the fully nonlinear and degenerate parabolic problem we establish the well-posedness property in the setting of classical solutions. Besides we prove that the problem features the effect of parabolic smoothing.

On a degenerate parabolic system describing the mean curvature flow of rotationally symmetric closed surfaces

Abstract

We show that the mean curvature flow for a closed and rotationally symmetric surface can be formulated as an evolution problem consisting of an evolution equation for the square of the function whose graph is rotated and two ODEs describing the evolution of the points of the evolving surface that lie on the rotation axis. For the fully nonlinear and degenerate parabolic problem we establish the well-posedness property in the setting of classical solutions. Besides we prove that the problem features the effect of parabolic smoothing.

Paper Structure

This paper contains 4 sections, 9 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. The transformed problem
  3. The generator property
  4. Parabolic smoothing

Key Result

Theorem 1.2

Let $\alpha\in(0,1)$ be a fixed Hölder exponent, $a_0< b_0\in\mathbb{R}$, and let $v_0\in {\rm C}^1([a_0,b_0])$ be positive in $(a_0,b_0)$ such that $v_0(a_0)=v_0(b_0)=0,$$v_0'(a)>0>v_0'(b),$ and Then the evolution problem Pv has a unique maximal solution $(v,a,b):=(v,a,b)(\,\cdot\,;(v_0,a_0,b_0))$ such that where and $t^+:=t^+(v_0,a_0,b_0)\in(0, \infty]$. Moreover, it holds that

Theorems & Definitions (20)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • ...and 10 more