On singularities of the heat flow for harmonic maps from surfaces into spheres
type: Item
instance of: scholarly article
title: On singularities of the heat flow for harmonic maps from surfaces into spheres
zbMATH Open document ID: 0868.58021
author: Jie Qing
published in: Communications in Analysis and Geometry
publication date: 1997-08-21T00:00:00Z
review text: This is an investigation of the local behavior of solutions of the heat equation for harmonic maps from surfaces into spheres near singular points. The author proves that there is no unaccounted energy loss for blow-ups of solutions. Any energy drop is shown to correspond precisely to the production of bubbles, i.e., nontrivial harmonic maps from 2-sphere into the target. It was shown earlier by \textit{M. Struwe} [Comment. Math. Helv. 60, 558-581 (1985; Zbl 0595.58013)] that global solutions exist and are regular except at possibly finitely many isolated points. More recently, \textit{K. C. Chang, W. Ding} and \textit{R. Ye} [J. Differ. Geom. 36, No. 2, 507-515 (1992; Zbl 0765.53026)] proved that these singular points indeed exist in some cases. The present paper thus offers an explanation what happens to the solutions near these singularities.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 856918
zbMATH Keywords: singularity
zbMATH Keywords: Palais-Smale sequence
zbMATH Keywords: heat equation
zbMATH Keywords: harmonic maps
zbMATH Keywords: energy loss
reviewed by: Dušan Repovš
MaRDI profile type: Publication
DOI: 10.4310/CAG.1995.V3.N2.A4
description: scientific article; zbMATH DE number 856918