Parabolic Equations for Curves on Surfaces. 2. Intersections, Blow Up and Generalized Solutions
Abstract
This paper describes a theory for parabolic differential equations for immersed curves on surfaces, which generalizes the curve shortening or flow by mean curvature problem, as well as several models in the theory of phase transitions in two dimensions. The author describes a class of equations for which the initial value problem is well posed for rough initial data, for which one define generalized solutions, i.e. solutions which are smooth, except at a discrete set of times. The methods which are used in this paper are more geometrical than those of part I. By comparing arbitrary solutions with certain special solutions, and by considering the way they intersect, estimates for the curvature and the tangent are derived, which allow one to study the initial value problem, and the way solutions become singular. Keywords: Geometric heat equation.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1989
- Accession Number
- ADA212890
Entities
People
- Sigurd Angenent
Organizations
- University of Wisconsin–Madison