Singularity formation for the two-dimensional harmonic map flow into S-2

Abstract

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.