Singularity formation for the two-dimensional harmonic map flow into S2

• 成果类型: Article
• 署名作者: Davila, Juan; del Pino, Manuel; Wei, Juncheng
• 署名单位: Universidad de Antioquia; Universidad de Chile; University of Bath; University of British Columbia
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00908-y
• 发表日期: 2020
• 页码: 345-466
• 关键词: blow-up dynamics; heat-flow; asymptotics; STABILITY; EXISTENCE; mappings

摘要

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere S-2, u(t) = Delta u + vertical bar del u vertical bar(2)u in Omega x (0, T) u = phi on partial derivative Omega x (0, T) u(., 0) = u(0) in Omega, where Omega is a bounded, smooth domain in R-2, u : Omega x (0, T) -> S-2, u(0) : (Omega) over bar -> S-2 is smooth, and phi = u(0)vertical bar(partial derivative Omega). Given any k points q(1), ..., q(k) in the domain, we find initial and boundary data so that the solution blows-up precisely at those points. The profile around each point is close to an asymptotically singular scaling of a 1-corotational harmonic map. We build a continuation after blow-up as a H-1-weak solution with a finite number of discontinuities in space-time by reverse bubbling, which preserves the homotopy class of the solution after blow-up. Furthermore, we prove the codimension one stability of the one point blow-up phenomenon.