Abstract
We construct finite time blowup solutions to the 2dimensional harmonic map flow into the sphere S ^{2}, ut=Δu+∇u2uinΩ×(0,T)u=φon∂Ω×(0,T)u(·,0)=u0inΩ,where Ω is a bounded, smooth domain in R ^{2}, u: Ω × (0 , T) → S ^{2}, u: Ω ¯ → S ^{2} is smooth, and φ= u _{∂} _{Ω}. Given any k points q _{1}, … , q _{k} in the domain, we find initial and boundary data so that the solution blowsup precisely at those points. The profile around each point is close to an asymptotically singular scaling of a 1corotational harmonic map. We build a continuation after blowup 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 blowup. Furthermore, we prove the codimension one stability of the one point blowup phenomenon.
Original language  English 

Pages (fromto)  345466 
Number of pages  122 
Journal  Inventiones Mathematicae 
Volume  219 
Issue number  2 
Early online date  27 Jul 2019 
DOIs  
Publication status  Published  29 Feb 2020 
ASJC Scopus subject areas
 Mathematics(all)
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Juan Davila Bonczos
 Department of Mathematical Sciences  Professor  Royal Society Wolfson Fellow
Person: Research & Teaching