Calculus of Variations and Geometric Measure Theory

C. Elbar

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

created by elbar on 06 Nov 2024

[BibTeX]

preprint

Inserted: 6 nov 2024
Last Updated: 6 nov 2024

Year: 2024

ArXiv: 2410.15095 PDF

Abstract:

We prove $L^{\infty}_{t}W^{1,p}_{x}$ Sobolev estimates in the Keller-Segel system by 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.

Tags: EYAWKAJKOS