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

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