Renormalization and blow up for charge one equivariant critical wave maps

摘要

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