[Méthodes à la De Giorgi quantitatives en théorie cinétique]
Nous considérons des équations hypoelliptiques de type Fokker-Planck cinétique, également appelées équations de Kolmogorov ou ultraparaboliques, avec des coefficients sans régularité dans l’opérateur de dérive-diffusion. Nous donnons de nouvelles preuves quantitatives du lemme des valeurs intermédiaires de De Giorgi ainsi que des inégalités de Harnack faibles et fortes. Cela implique la continuité höldérienne avec bornes explicites. L’article ne fait pas appel à des résultats précédents.
We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies Hölder continuity with quantitative estimates. The paper is self-contained.
Accepté le :
Publié le :
Keywords: Hypoelliptic equations, kinetic theory, Fokker-Planck equation, ultraparabolic equations, Kolmogorov equation, Hölder continuity, De Giorgi method, Moser iteration, averaging lemma, weak Harnack inequality, trajectories
Mot clés : Équations hypoelliptiques, théorie cinétique, équation de Fokker-Planck, équations ultraparaboliques, équation de Kolmogorov, continuité höldérienne, méthode de De Giorgi, itération de Moser, lemme de moyenne, inégalité de Harnack faible, trajectoires
@article{JEP_2022__9__1159_0,
author = {Guerand, Jessica and Mouhot, Cl\'ement},
title = {Quantitative {De~Giorgi} methods in kinetic theory},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1159--1181},
publisher = {Ecole polytechnique},
volume = {9},
year = {2022},
doi = {10.5802/jep.203},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jep.203/}
}
TY - JOUR AU - Guerand, Jessica AU - Mouhot, Clément TI - Quantitative De Giorgi methods in kinetic theory JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 1159 EP - 1181 VL - 9 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.203/ DO - 10.5802/jep.203 LA - en ID - JEP_2022__9__1159_0 ER -
%0 Journal Article %A Guerand, Jessica %A Mouhot, Clément %T Quantitative De Giorgi methods in kinetic theory %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 1159-1181 %V 9 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.203/ %R 10.5802/jep.203 %G en %F JEP_2022__9__1159_0
Guerand, Jessica; Mouhot, Clément. Quantitative De Giorgi methods in kinetic theory. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1159-1181. doi : 10.5802/jep.203. http://www.numdam.org/articles/10.5802/jep.203/
[AP20] A survey on the classical theory for Kolmogorov equation, Matematiche (Catania), Volume 75 (2020) no. 1, pp. 221-258 | DOI | MR | Zbl
[BDM + 20] Hypocoercivity without confinement, Pure Appl. Anal., Volume 2 (2020) no. 2, pp. 203-232 | DOI | MR | Zbl
[DG56] Sull’analiticità delle estremali degli integrali multipli, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), Volume 20 (1956), pp. 438-441 | MR | Zbl
[DG57] Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., Volume 3 (1957), pp. 25-43 | Zbl
[EG15] Measure theory and fine properties of functions, Textbooks in Math., CRC Press, Boca Raton, FL, 2015 | DOI
[GI21] Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations, 2021 | arXiv
[GIMV19] Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to Landau equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 19 (2019) no. 5, pp. 253-295 | MR | Zbl
[Gue20] Quantitative regularity for parabolic De Giorgi classes, 2020 | HAL
[GV15] Hölder regularity for hypoelliptic kinetic equations with rough diffusion coefficients, 2015 | arXiv
[Hör67] Hypoelliptic second order differential equations, Acta Math., Volume 119 (1967), pp. 147-171 | DOI | MR | Zbl
[IS20] The weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc. (JEMS), Volume 22 (2020) no. 2, pp. 507-592 | DOI | MR | Zbl
[Kol34] Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2), Volume 35 (1934) no. 1, pp. 116-117 | DOI | Zbl
[Kru63] A priori bounds for generalized solutions of second-order elliptic and parabolic equations, Dokl. Akad. Nauk SSSR, Volume 150 (1963), pp. 748-751 | MR
[Kru64] A priori bounds and some properties of solutions of elliptic and parabolic equations, Mat. Sb. (N.S.), Volume 65 (1964), pp. 522-570 | MR
[LZ17] A note on the Harnack inequality for elliptic equations in divergence form, Proc. Amer. Math. Soc., Volume 145 (2017) no. 1, pp. 135-137 | DOI | MR | Zbl
[Mos64] A Harnack inequality for parabolic differential equations, Comm. Math. Phys., Volume 17 (1964), pp. 101-134 | MR | Zbl
[PP04] The Moser’s iterative method for a class of ultraparabolic equations, Commun. Contemp. Math., Volume 6 (2004) no. 3, pp. 395-417 | DOI | MR | Zbl
[Vas16] The De Giorgi method for elliptic and parabolic equations and some applications, Lectures on the analysis of nonlinear partial differential equations. Part 4 (Morningside Lect. Math.), Volume 4, Int. Press, Somerville, MA, 2016, pp. 195-222 | MR | Zbl
[WZ09] The regularity of a class of non-homogeneous ultraparabolic equations, Sci. China Ser. A, Volume 52 (2009) no. 8, pp. 1589-1606 | DOI | MR | Zbl
[WZ11] The regularity of weak solutions of ultraparabolic equations, Discrete Contin. Dynam. Systems, Volume 29 (2011) no. 3, pp. 1261-1275 | DOI | MR | Zbl
Cité par Sources :
