Sobolev estimates for the Keller-Segel system and applications to the JKO scheme

TL;DR

This paper establishes Sobolev estimates for the Keller-Segel system with linear diffusion, extends these estimates to the JKO scheme, and proves convergence results by leveraging functional inequalities and diffusion properties.

Contribution

It introduces new Sobolev estimates for the Keller-Segel system and applies them to analyze the convergence of the JKO scheme in this context.

Findings

Sobolev estimates valid in any dimension for Keller-Segel system

Extension of convergence results of JKO scheme to Keller-Segel system

Functional inequality inspired by Brezis-Gallouet-Wainger used in analysis

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-Gallou\"et-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.