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

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere S ², ut=Δu+|∇u|2uinΩ×(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 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.