Renormalization and blow up for wave maps from $S^2\times \RR$ to $S^2$

TL;DR

This paper constructs a family of finite time blow-up solutions for the co-rotational wave maps from $S^2\times \mathbb{R}$ to $S^2$, demonstrating energy concentration and specific blow-up rates near singularity.

Contribution

It introduces a novel one-parameter family of blow-up solutions for wave maps from $S^2\times \mathbb{R}$ to $S^2$, with detailed analysis of their structure and energy behavior.

Findings

Existence of finite time blow-up solutions with controlled energy concentration.

Precise blow-up rate characterized by $\lambda(t)=t^{-1-\nu}$.

Solutions exhibit regularity $H^{1+\nu-}$ up to blow-up time.

Abstract

We construct a one parameter family of finite time blow ups to the co-rotational wave maps problem from $S^2\times \RR$ to $S^2,$ parameterized by $\nu\in(1/2,1].$ The longitudinal function $u(t,\alpha)$ which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from $\RR^2$ to $S^2.$ The domain of this harmonic map is identified with a neighborhood of the north pole in the domain $S^2$ via the exponential coordinates $(\alpha,\theta).$ In these coordinates $u(t,\alpha)=Q(\lambda(t)\alpha)+\mathcal{R}(t,\alpha),$ where $Q(r)=2\arctan{r},$ is the standard co-rotational harmonic map to the sphere, $\lambda(t)=t^{-1-\nu},$ and $\mathcal{R}(t,\alpha)$ is the error with local energy going to zero as $t\rightarrow 0.$ Blow up will occur at $(t,\alpha)=(0,0)$ due to energy concentration, and up to this point the solution will have regularity $H^{1+\nu-}.$