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 $C^{\hat{I}½/2-}$ regular approximate solutions that settle down to multiple solitons, shrinking at a rate $t^{\hat{I}}½$ with $\hat{I}½>1$, 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 ansätze with any $\hat{I}½>8$. Using robust energy estimates, the ansätze are corrected to exact solutions.

49 pages. All comments are welcome!