A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent

TL;DR

This work establishes Li–Yau and Aronson–Bénilan type lower bounds for the parabolic–elliptic Keller–Segel system at the critical exponent, yielding pointwise control of the pressure and, consequently, uniform $L^{\infty}$ bounds on the density under subcritical and critical mass regimes. The authors develop a framework built around the evolution of the Laplacian of a pressure-like quantity $v$ and a key quantity $Q(u)$ measuring Hessian differences of the Newtonian potential, carefully treating $d=2$ (linear diffusion) and $d>2$ (porous-medium-type diffusion). A central pillar is the translation of subsolution properties of Liouville and Lane–Emden equations into minimal-mass thresholds, which underpin the global existence results for small, subcritical, and critical masses; in particular, they prove $M^{sub}_c(2)=M_c(2)=8\pi$ and $M^{sub}_c(3)=M_c(3)$, with conjectured equality in higher dimensions. The work also introduces an optimal-control perspective to analyze subsolutions and provides a robust justification ensuring the full rigor of the maximum-principle-based arguments, including smoothing effects and entropy/Ln-HLS-type controls at critical mass. Overall, the results strengthen the understanding of KS dynamics by connecting classical diffusion inequalities to nonlocal chemotactic drift, yielding explicit global existence criteria and decay bounds that enhance prior literature on critical-mass KS dynamics.

Abstract

We prove Li-Yau and Aronson-Bénilan 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.

Figures (1)

• Figure 1: Left: Plot of $M(f(\gamma,\cdot))(R_{\gamma})$ with respect to $\gamma\in[10^{-6}, 1]$ in dimension $d=4$. The graph appears to be increasing and a closer look shows that for small $\gamma$, let say $\gamma\le 10^{-1}$ the values are the same up to $10^{-10}$. Right: Same plot, zooming on the zone $\gamma\in [10^{-6}, 10^{-3}]$. Similar results were obtained in higher dimensions.

Theorems & Definitions (80)

• Theorem 1.2: Li--Yau estimate for Keller--Segel in dimension 2
• Theorem 1.4: Aronson--Bénilan estimate for Keller--Segel in dimension $d>2$
• Lemma 1.5: See Remark \ref{['rem:corollary']}
• Theorem 1.6: See Theorem \ref{['thm:new_proof_hls']}
• Theorem 1.7: See Theorem \ref{['thm:new_proof_hls_multid']}
• Proposition 2.1
• Proposition 2.3
• proof
• Proposition 2.4
• proof