Critical Keller-Segel meets Burgers on S1: Large-time smooth solutions

Abstract

We show that solutions to the parabolic–elliptic Keller–Segel system on S1 with critical fractional diffusion (Delta)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 [15]. As a tool, we show smoothness of solutions to a modified critical Burgers equation via a generalization of the ingenious method of moduli of continuity by Kiselev, Nazarov and Shterenberg [35] 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 corresponding to small initial data, improving the existing results.JB thanks Tomasz Cieślak and Grzegorz Karch for fruitful discussions. JB is supported by the ‘Mobilność Plus’ grant no 1289/MOB/IV/2015/0. RGB is funded by the Labex MILYON and the Grant MTM2014-59488-P from the Ministerio de Economía y Competitividad (MINECO, Spain)