Sobolev estimates for the Keller-Segel system and applications to the JKO scheme
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912839023722496 |
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| author | Elbar, Charles |
| author_facet | Elbar, Charles |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15095 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sobolev estimates for the Keller-Segel system and applications to the JKO scheme Elbar, Charles Analysis of PDEs 35A35, 35D30, 35K55, 35Q92, 49Q22 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. |
| title | Sobolev estimates for the Keller-Segel system and applications to the JKO scheme |
| topic | Analysis of PDEs 35A35, 35D30, 35K55, 35Q92, 49Q22 |
| url | https://arxiv.org/abs/2410.15095 |