Parabolic Equations for Curves on Surfaces. 1. Curves with p-Integrable Curvature

Abstract

This is the first of two papers in which the author develops a theory of parabolic equations for curves on surfaces which can be applied to the so- called curve shortening or flow by mean curvature problem, as well as to a number of models for phase transitions in two dimensions. This document introduces 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 the curve becomes singular in finite time. A detailed discussion of the way in which solutions can become singular and a method for continuing the solution through a singularity will be the subject of the second part.

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Document Details

Document Type
Technical Report
Publication Date
Nov 01, 1988
Accession Number
ADA204093

Entities

People

  • Sigurd Angenent

Organizations

  • University of Wisconsin–Madison

Tags

Communities of Interest

  • Air Platforms
  • C4I

DTIC Thesaurus Topics

  • Air Force
  • Banach Space
  • Differential Equations
  • Equations
  • Equations Of State
  • Geometric Forms
  • Heat Energy
  • Inequalities
  • Intervals
  • Lines (Geometry)
  • Partial Differential Equations
  • Periodic Functions
  • Personal Information Managers
  • Phase
  • Phase Transformations
  • Time Intervals
  • Two Dimensional

Fields of Study

  • Mathematics

Readers

  • Calculus or Mathematical Analysis
  • Fluid Dynamics.