Smooth finite time singularity formation without quantization

TL;DR

This paper constructs smooth solutions to the focusing energy critical wave equation that develop finite time singularities without quantization, by carefully assembling multiple solitons with controlled shrinking rates.

Contribution

It introduces a geometric singular-analytic approach to build smooth solutions with prescribed singularity formation rates, extending previous work on finite time blow-up.

Findings

Constructed $C^{ u/2-}$ regular approximate solutions

Achieved solutions with singularity rates $t^{ u}$ for any $ u>8$

Corrected approximate solutions to exact solutions using energy estimates

Abstract

We revisit the finite time singularity formation of Krieger-Schlag-Tataru [KST09] for the focusing energy critical wave equation in $\mathbb{R}^{3+1}$ from a geometric singular-analytic point of view, following Hintz [Hintz23]. We construct $C^{\nu/2-}$ regular approximate solutions that settle down to multiple solitons, shrinking at a rate $t^{\nu}$ with $\nu>1$, and approaching the origin on different geodesics $\{x=zt\}$. By fine tuning the velocities, sizes and signs of the solitons, we are able to construct smooth ans\"atze with any $\nu>8$. Using robust energy estimates, the ans\"atze are corrected to exact solutions.