Singularity Formation for the Two-dimensional Harmonic Map Flow into S

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere S2, ut = ∆u+ |∇u|u in Ω× (0, T ) u = φ on ∂Ω× (0, T ) u(·, 0) = u0 in Ω, where Ω is a bounded, smooth domain in R2, u : Ω×(0, T ) → S2, u0 : Ω̄ → S2 is smooth, and φ = u0 ∣∣ ∂Ω . Given any points q1, . . . , qk 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-corrotational harmonic map. We build a continuation after blowup as a H1-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.