On stability of blow up solutions for the critical co-rotational Wave Maps problem

Abstract

We show that the finite time blow up solutions for the co-rotational Wave Maps problem constructed in [7,15] are stable under suitably small perturbations within the co-rotational class, provided the scaling parameter $\lambda (t)=t^{-1-\nu }$ is sufficiently close to $t^{-1}$, i. e. the constant $\nu$ is sufficiently small and positive. The method of proof is inspired by [3,12], but takes advantage of geometric structures of the Wave Maps problem already used in [1,21] to simplify the analysis. In particular, we heavily exploit that the resonance at zero satisfies a natural first order differential equation.

arXiv comment66 pages

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• Quantized slow blow-up dynamics for the energy-critical corotational wave maps problem
• Global, Non-Scattering Solutions to the Energy Critical Yang-Mills Problem