Quantitative De Giorgi methods in kinetic theory | Vidéo | Carmin.tv

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Quantitative De Giorgi methods in kinetic theory

By Clément Mouhot

Appears in collection : Jean Morlet Chair 2021- Conference: Kinetic Equations: From Modeling Computation to Analysis / Chaire Jean-Morlet 2021 - Conférence : Equations cinétiques : Modélisation, Simulation et Analyse

We consider hypoelliptic equations of kinetic Fokker-Planck type, also sometimes called of Kolmogorov or Langevin type, with rough coefficients in the drift-diffusion operator in velocity. We present novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities (which imply Hölder continuity with quantitative estimates). This is a joint work with Jessica Guerand.

Information about the video

Date of recording 23/03/2021
Date of publication 09/04/2021
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19735303
Cite this video Mouhot, Clément (23/03/2021). Quantitative De Giorgi methods in kinetic theory. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19735303
URL https://dx.doi.org/10.24350/CIRM.V.19735303

Domain(s)

Analysis of PDEs

Bibliography

GUERAND, Jessica et MOUHOT, Clément. Quantitative De Giorgi methods in kinetic theory. arXiv preprint arXiv:2103.09646, 2021. - https://arxiv.org/abs/2103.09646
BOUIN, Emeric, DOLBEAULT, Jean, MISCHLER, Stéphane, et al. Hypocoercivity without confinement. Pure and Applied Analysis, 2020, vol. 2, no 2, p. 203-232. - https://doi.org/10.2140/paa.2020.2.203
DE GIORGI, Ennio. Sull'analiticità delle estremali degli integrali multipli. Consiglio Nazionale delle Ricerche, 1956.
DE GIORGI, Memoria di Ennio. Sulla differenziabilitae l’analiticita delle estremali degli integrali multipli regolari. Ennio De Giorgi, 1957, p. 167.
EVANS, Lawrence Craig et GARIEPY, Ronald F. Measure theory and fine properties of functions. CRC press, 2015.
GUERAND, Jessica et IMBERT, Cyril. Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations. arXiv preprint arXiv:2102.04105, 2021. - https://arxiv.org/abs/2102.04105
GOLSE, François, IMBERT, Cyril, MOUHOT, Clément, et al. Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. 19 (2019), no. 1, 253–295 - http://dx.doi.org/10.2422/2036-2145.201702_001
GUERAND, Jessica. Quantitative regularity for parabolic De Giorgi classes. arXiv preprint arXiv:1903.07421, 2019. - https://arxiv.org/abs/1903.07421
HÖRMANDER, Lars. Hypoelliptic second order differential equations. Acta Mathematica, 1967, vol. 119, no 1, p. 147-171. - https://doi.org/10.1007/BF02392081
IMBERT, Cyril et SILVESTRE, Luis. The weak Harnack inequality for the Boltzmann equation without cut-off. Journal of the European Mathematical Society, 2019, vol. 22, no 2, p. 507-592. - https://doi.org/10.4171/jems/928
KOLMOGOROFF, Andrey. Zufallige bewegungen (zur theorie der Brownschen bewegung). Annals of Mathematics, 1934, p. 116-117.
KRUZHKOV, Stanislav Nikolaevich. A priori bounds for generalized solutions of second-order elliptic and parabolic equations. In : Doklady Akademii Nauk. Russian Academy of Sciences, 1963. p. 748-751.
KRUZHKOV, Stanislav Nikolaevich. A priori bounds and some properties of solutions of elliptic and parabolic equations. Matematicheskii Sbornik, 1964, vol. 107, no 4, p. 522-570.
LI, Dongsheng et ZHANG, Kai. A note on the Harnack inequality for elliptic equations in divergence form. Proceedings of the American Mathematical Society, 2017, vol. 145, no 1, p. 135-137. - http://dx.doi.org/10.1090/proc/13174
MOSER, Jürgen. A Harnack inequality for parabolic differential equations. Communications on pure and applied mathematics, 1964, vol. 17, no 1, p. 101-134.
PASCUCCI, Andrea et POLIDORO, Sergio. The Moser's iterative method for a class of ultraparabolic equations. Communications in Contemporary Mathematics, 2004, vol. 6, no 03, p. 395-417. - https://doi.org/10.1142/S0219199704001355
VASSEUR, Alexis F. The De Giorgi method for elliptic and parabolic equations and some applications. Lectures on the analysis of nonlinear partial differential equations, 2016, vol. 4.
WANG, WenDong et ZHANG, LiQun. The C α regularity of a class of non-homogeneous ultraparabolic equations. Science in China Series A: Mathematics, 2009, vol. 52, no 8, p. 1589-1606. - http://dx.doi.org/10.1007/s11425-009-0158-8
WANG, Wendong et ZHANG, Liqun. The $ C^{\alpha} $ regularity of weak solutions of ultraparabolic equations. Discrete & Continuous Dynamical Systems-A, 2011, vol. 29, no 3, p. 1261. - http://dx.doi.org/10.3934/dcds.2011.29.1261

MSC codes

Document(s)

https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Clement_MOUHOT.pdf

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