Sobolev estimates for the Keller-Segel system and applications to the JKO scheme
Charles Elbar
TL;DR
This paper develops a Brezis-Gallouët-Wainger–type inequality to control the Hessian of the KS potential by the Laplacian and Hölder norms, enabling $L^{\infty}_{t}W^{1,p}_{x}$ regularity for the KS system with linear diffusion in any dimension. By embedding this into a Fisher-information–type functional $\mathcal{F}_{p}[\rho]$, the authors derive Sobolev bounds that persist in the discrete JKO scheme and prove strong convergence in $L^{2}(0,T;H^{2}(\Omega))$ to a KS weak solution under small $L^{d/2}$ initial data. The analysis extends previous results for the Fokker-Planck equation to the aggregation-diffusion KS system and provides a rigorous link between continuous dynamics and its time-discrete variational approximation. The results enhance the theoretical understanding of global well-posedness and numerical approximation for KS-type models in subcritical regimes, with potential implications for long-time behavior and numerical schemes for related nonlinear diffusion-aggregation systems.
Abstract
We prove $L^{\infty}_{t}W^{1,p}_{x}$ Sobolev estimates in the Keller-Segel system with linear diffusion in any dimensionby proving a functional inequality, inspired by the Brezis-Gallouët-Wainger inequality. These estimates are also valid at the discrete level in the Jordan-Kinderlehrer-Otto (JKO) scheme. By coupling this result with the diffusion properties of a functional according to Bakry-Emery theory, we deduce the $L^2_t H^{2}_{x}$ convergence of the scheme, thereby extending the recent result of Santambrogio and Toshpulatov in the context of the Fokker-Planck equation to the Keller-Segel system.