Singularity formation for the harmonic map flow from a volume into S^2.



Consider a volume V ⊂ R3 generated by rotating around the Z axis a bounded smooth domain Ω ⊂ R2 that lives in the XZ plane. We construct a finite time blow-up solution to the harmonic map flow from volume V into the sphere S2, the problem is

vt = ∆v+|∇v|2vinV×(0,T) v = v∂V in∂V×(0,T)

v(·,0) = v0inV,

wherev:V ×[0,T)→S2,v0 :V →S2 issmoothandv∂V = v0|∂V :∂V →S2. Given a point q ∈ Ω we define the circumference c(q) generated by the rotation of q around the Z axis. We find initial and boundary data so that the solution v blows up at exactly the curve c(q) at a finite small time. The construction of the solution is done by reducing the problem to 2 dimensions and using the method of D ́avila, Del Pino and Wei [1] that transforms the problem into an inner-outer gluing system which separates the main effect of the equation near and far away from the singularity. We obtain a solution that at main order has the profile of a scaled 1-corrotational harmonic map near the singularity.

Date: Jul 12, 2018 at 16:00 h
Venue: Sala de Seminarios Felipe Álvarez Daziano, 5to. piso del DIM, U. de Chile, Beauchef 851.
Speaker: Catalina Pesce
Affiliation: Depto. de Ingeniería Matemática, Universidad de Chile
Coordinator: Hanne Van Den Bosch

Posted on Jul 5, 2018 in CAPDE, Seminars