A Li-Yau and Aronson-B\'enilan approach for the Keller-Segel system with critical exponent

TL;DR

This paper develops Li-Yau and Aronson-Bénilan type estimates for the Keller-Segel system at critical exponent, leading to new bounds and global existence results for solutions depending on initial conditions and mass.

Contribution

It introduces novel estimates for the Keller-Segel system at critical exponent, enabling proof of global existence under specific initial data conditions.

Findings

Establishes lower bounds on the Laplacian of pressure for the system.

Derives $L^{ abla ext{infty}}$ bounds on density based on initial mass.

Proves global existence of smooth solutions for small or subcritical initial mass.

Abstract

We prove Li-Yau and Aronson-B\'enilan type estimates for the parabolic-elliptic Keller-Segel system with critical exponent $m=2-\frac 2d$, i.e. lower bounds on the Laplacian of a suitable notion of pressure in any dimension. We show that these estimates entail $L^{\infty}$ bounds on the density, depending on its initial mass, up to the critical mass case for $d \in \{ 2, 3 \}$. We deduce from these results the global existence of smooth solutions in two cases: first, when the initial data is merely a measure but has sufficiently small mass; and second, when the initial free energy is bounded, and the mass is subcritical or critical. Our argument requires a careful study of the subsolutions of the Liouville and Lane-Emden equations arising in the model.