JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel: $L^p$ and Sobolev bounds

Di Marino, Simone; Santambrogio, Filippo

doi:10.4171/aihpc/36

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JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel:

L^{p}

and Sobolev bounds

Di Marino, Simone ; Santambrogio, Filippo

Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 6, pp. 1485-1517

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Résumé

We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the $L^{p}$ and $L^{\infty}$ norms of the iterated solutions in terms of the previous norms, essentially recovering well-known results obtained on the continuous-in-time equations. Then we pass to higher-order results, and in particular to some specific BV and Sobolev estimates, where the JKO scheme together with the so-called “five gradients inequality” allows us to recover some estimates that can be deduced from the Bakry–Émery theory for diffusion operators, but also to obtain some novel ones, in particular for the Keller–Segel chemotaxis model.

Accepté le : 2021-09-09
Publié le : 2022-06-24

DOI : 10.4171/aihpc/36

Classification : 35-XX, 58J35, 49Q22
Keywords: Wasserstein space, Sobolev bounds, parabolic PDEs

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     author = {Di Marino, Simone and Santambrogio, Filippo},
     title = {JKO estimates in linear and non-linear {Fokker{\textendash}Planck} equations, and {Keller{\textendash}Segel:} $L^p$ and {Sobolev} bounds},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1485--1517},
     volume = {39},
     number = {6},
     year = {2022},
     doi = {10.4171/aihpc/36},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/36/}
}

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Di Marino, Simone; Santambrogio, Filippo. JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel: $L^p$ and Sobolev bounds. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 6, pp. 1485-1517. doi: 10.4171/aihpc/36

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