Sharp universal rate for stable blow-up of corotational wave maps

TL;DR

This paper establishes a universal blow-up rate for stable solutions of energy-critical corotational wave maps into the two-sphere, refining previous results by identifying a precise constant in the blow-up rate.

Contribution

It proves that the blow-up rate converges to a universal constant, improving understanding of the asymptotic behavior of solutions near singularity in wave maps.

Findings

The blow-up rate converges to 2e^{-1} times (T-t) times an exponential factor.

The solutions exhibit a universal contraction rate near blow-up.

The method involves explicit invariant subspace decomposition for the linearized operator.

Abstract

We consider the energy-critical (corotational) 1-equivariant wave maps into the two-sphere. By the seminal work [53] of Rapha\"el and Rodnianski, there is an open set of initial data whose forward-in-time development blows up in finite time with the blow-up rate $\lambda(t)=(T-t)e^{-\sqrt{|\log(T-t)|}+O(1)}$. In this paper, we show that this $e^{O(1)}$-factor in fact converges to the universal constant $2e^{-1}$, and hence these solutions contract at the universal rate $\lambda(t)=2e^{-1}(T-t)e^{-\sqrt{|\log(T-t)|}}(1+o_{t\to T}(1))$. Our proof is inspired by recent works on type-II blow-up dynamics for parabolic equations. The key improvement is in the construction of an explicit invariant subspace decomposition for the linearized operator perturbed by the scaling generator in the dispersive case, from which we obtain a more precise ODE system determining $\lambda(t)$.