BLOW-UP FOR THE 3-DIMENSIONAL AXIALLY SYMMETRIC HARMONIC MAP FLOW INTO S2

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere S-2, u(t) =Delta u + vertical bar del vertical bar(2)u in Omega x (0, T) u = u(b) on partial derivative Omega x (0, T) u(., 0) =u(0) in Omega, with u(x, t) : (Omega) over bar x [0, T) -> S-2 . Here Omega is a bounded, smooth axially symmetric domain in R-3. We prove that for any circle Gamma subset of Omega with the same axial symmetry, and any sufficiently small T > 0 there exist initial and boundary conditions such that u(x, t) blows-up exactly at time T and precisely on the curve Gamma, in fact vertical bar del u(., t)vertical bar(2 )-> vertical bar del u(*)vertical bar(2) + 8 pi delta(Gamma) as t -> T. for a regular function u(*)(x), where delta(Gamma) denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [5, 6].