On the stability of blowup solutions for the critical corotational wave-map problem

Abstract

We show that the finite time blowup solutions for the corotational wave-map problem constructed by the first author along with Gao, Schlag, and Tataru are stable under suitably small perturbations within the corotational class, provided that the scaling parameter $\lambda (t)=t^{-1-\nu }$ is sufficiently close to $t^{-1}$; that is, the constant $\nu$ is sufficiently small and positive. The method of proof is inspired by recent work by the first author and Burzio, but takes advantage of geometric structures of the wave-map problem, already used in previous work by the first author, Bejenaru, Tataru, Raphaël, and Rodnianski, to simplify the analysis. In particular, we heavily exploit the fact that the resonance at zero satisfies a natural first-order differential equation.