Smooth finite time singularity formation without quantization

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.