Full blow-up range for co-rotaional wave maps to surfaces of revolution

TL;DR

This paper constructs polynomial blow-up solutions for energy critical wave maps into surfaces of revolution, extending the blow-up rate range previously established and generalizing earlier results on spherical targets.

Contribution

It extends the blow-up range for wave maps into surfaces of revolution from b1
2 to b1a0, broadening the understanding of singularity formation in these equations.

Findings

Constructed blow-up solutions with polynomial rate for b1a0b1a0

Extended blow-up rate range to b1a0b1a0

Generalized previous results from spherical targets to surfaces of revolution

Abstract

We construct blow-up solutions of the energy critical wave map equation on $\mathbb{R}^{2+1}\to \mathcal N$ with polynomial blow-up rate ($t^{-1-\nu}$ for blow-up at $t=0$) in the case when $\mathcal{N}$ is a surface of revolution. Here we extend the blow-up range found by C\^arstea ($\nu>\frac 12$) based on the work by Krieger, Schlag and Tataru to $\nu>0$. This work relies on and generalizes the recent result of Krieger and the author where the target manifold is chosen as the standard sphere.