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

TL;DR

This paper proves the stability of finite time blow-up solutions for the critical co-rotational Wave Maps problem under small perturbations, focusing on the behavior of the scaling parameter and leveraging geometric structures for analysis.

Contribution

It demonstrates stability of blow-up solutions in the co-rotational class for specific scaling parameters, utilizing geometric insights to simplify the proof.

Findings

Finite time blow-up solutions are stable under small perturbations.

Stability holds when the scaling parameter is close to $t^{-1}$ with small positive $ u$.

The analysis exploits geometric structures and resonance properties of the 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.