PARABOLIC EQUATIONS FOR CURVES ON SURFACES .2. INTERSECTIONS, BLOW-UP AND GENERALIZED SOLUTIONS

• 成果类型: Article
• 署名作者: ANGENENT, S
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2944327
• 发表日期: 1991
• 页码: 171-215
• 关键词: plane-curves

摘要

We describe a theory for parabolic 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. A class of equations is described for which the initial value problem is well-posed for rough initial data, for which one can give a description of the way a smooth solution becomes singular, and for which one can 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, we derive estimates for the curvature and the tangent, which allow one to study the initial value problem, and the way solutions become singular.