Renormalization and blow up for charge one equivariant critical wave maps

Abstract

Abstract. We prove the existence of equivariant finite time blow-up solutions for the wave map problem from R2+1 → S2 of the form u(t, r) = Q(λ(t)r)+R(t, r) where u is the polar angle on the sphere, Q(r) = 2arctan r is the ground state harmonic map, λ(t) = t−1−ν, and R(t, r) is a radiative error with local energy going to zero as t → 0. The number ν> 1 can be described arbitrarily. This is accomplished by first ”renormalizing” 2 the blow-up profile, followed by a perturbative analysis. 1.