A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent
The Li-Yau and Aronson-Bénilan estimates are fundamental differential inequalities for heat and porous-medium equations, yielding differential Harnack inequalities and sharp smoothing and decay rates. We develop analogues for the critical parabolic-elliptic Keller-Segel system on $\mathbb{R}^d$, where diffusion and aggregation act at the same scale. Our main tool is the effective pressure $v=p-u$, with $p=\logρ$ in $d=2$ and $p=\frac{m}{m-1}ρ^{m-1}$ for $d>2$, $m=2-\frac{2}{d}$. We prove quantitative lower bounds on $Δv$, showing that classical pressure estimates subsist in a suitable form despite the nonlocal interaction. These estimates unify previous results and yield new regularity. For nonnegative finite-measure initial data of sufficiently small mass, assuming only a logarithmic moment in $d=2$, we construct global solutions smooth for positive times and satisfying $\|ρ_t\|_{L^\infty}\le C/t$. For finite-free-energy data, we obtain global $L^\infty$ bounds of order $1+t^{-1}$ in the subcritical and critical mass regimes. We also introduce a new characterization of the critical mass as the minimal mass $M_c^{\mathrm{sub}}(d)$ of subsolutions of the elliptic Lane-Emden equation (Liouville in $d=2$). We prove that it equals $8π$ in dimension $2$ and, for $d\ge3$, coincides with the Keller-Segel critical mass given by the HLS inequality, by reducing the problem to a one-dimensional optimal-control problem. Along the way we prove tools of independent interest: $L^\infty$ estimates from lower bounds on $Δv$, critical log-HLS/HLS estimates away from Dirac masses, Hessian bounds for Newtonian potentials, and Lipschitz and Hessian estimates for bounded porous-medium solutions with drifts on the whole space.