Critical Keller-Segel meets Burgers on ${\mathbb{S}}^1$: large-time smooth solutions

We show that solutions to the parabolic-elliptic Keller-Segel system on ${\mathbb{S}}^1$ with critical fractional diffusion $(-\mathrm{\Delta }{)}^{\frac{1}{2}}$ remain smooth for any initial data and any positive time. This disproves, at least in the periodic setting, the large-data-blowup conjecture by Bournaveas and Calvez. As a tool, we show smoothness of solutions to a modified critical Burgers equation via a generalization of the method of moduli of continuity by Kiselev, Nazarov and Shterenberg. over a setting where the considered equation has no scaling. This auxiliary result may be interesting by itself. Finally, we study the asymptotic behavior of global solutions, improving the existing results.