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arXiv · 2410.15095

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

Charles Elbar
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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.

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Charles Elbar. 2024-10-19. Sobolev estimates for the Keller-Segel system and applications to the JKO scheme. https://arxiv.org/abs/2410.15095

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