2017/02/19 by Davila, Juan, del Pino, Manuel, Wei, Juncheng · 2 citations
#35K55 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1702.05801
We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere S2, ut amp; = Δu + |∇ u|2 u in Ω×(0,T)
u amp;= φ on ∂ Ω×(0,T)
u(⋅,0) amp;= u0 in Ω, where Ω is a bounded, smooth domain in ℝ2, u: Ω×(0,T)→ S2, u0:Ω→ 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 blow-up 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.