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arXiv · 1504.00955

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

Jan Burczak·Rafael Granero-Belinchón
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Abstract

We show that solutions to the parabolic-elliptic Keller-Segel system on ${\mathbb S}^1$ with critical fractional diffusion $(-Δ)^\frac{1}{2}$ 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.

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Jan Burczak, Rafael Granero-Belinchón. 2016-12-02. Critical Keller-Segel meets Burgers on ${\mathbb S}^1$: large-time smooth solutions. https://doi.org/10.1088/0951-7715%2F29%2F12%2F3810

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