Refined description and stability for singular solutions of the 2D Keller-Segel system

We construct solutions to the two dimensional parabolic-elliptic Keller-Segel model for chemotaxis that blow up in finite time $T$. The solution is decomposed as the sum of a stationary state concentrated at scale $\lambda$ and of a perturbation. We rely on a detailed spectral analysis for the linearized dynamics in the parabolic neighbourhood of the singularity performed by the authors, providing a refined expansion of the perturbation. Our main result is the construction of a stable dynamics in the full nonradial setting for which the stationary state collapses with the universal law $\lambda \sim 2e^{-\frac{2+\gamma}{2}}\sqrt{T-t}e^{-\sqrt{\frac{|\ln (T-t)|}{2}}}$ where $\gamma$ is the Euler constant. This improves on the earlier result by Raphael and Schweyer 2014 and gives a new robust approach to so-called type II singularities for critical parabolic problems. A by-product of the spectral analysis we developed is the existence of unstable blowup dynamics with speed $\lambda_\ell \sim C_0(T-t)^{\frac{\ell}{2}} |\ln(T-t)|^{-\frac{\ell}{2(\ell - 1)}}$ for $\ell \geq 2$ integer.