Hostname: page-component-5b777bbd6c-cp4x8 Total loading time: 0 Render date: 2025-06-20T11:13:01.258Z Has data issue: false hasContentIssue false
  • English
  • Français

LOG-TRANSFORM AND THE WEAK HARNACK INEQUALITY FOR KINETIC FOKKER-PLANCK EQUATIONS

Part of: Close-to-elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  16 May 2022

Jessica Guerand  and
Cyril Imbert Open the ORCID record for Cyril Imbert [Opens in a new window]
Jessica Guerand
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, CB3 0WB Cambridge, United Kingdom (jg900@cam.ac.uk)
Cyril Imbert*
Affiliation:
Département de Mathématiques et applications, École Normale Supérieure, CNRS, PSL Research University, 45 rue d’Ulm, 75005 Paris, France
*
cyril.imbert@ens.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

This article deals with kinetic Fokker–Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument due to S. N. Kružkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

Keywords

kinetic theoryFokker–Planck equationHölder continuityPoincaré inequalityHarnack inequality

MSC classification

Primary: 35H10: Hypoelliptic equations
Secondary: 35B65: Smoothness and regularity of solutions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 6 , November 2023 , pp. 2749 - 2774
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Anceschi, F., Eleuteri, M. and Polidoro, S., A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients, Commun. Contemp. Math. 21(7) (2019), 1850057.CrossRefGoogle Scholar
Anceschi, F., Polidoro, S. and Ragusa, M. A., Moser’s estimates for degenerate Kolmogorov equations with non-negative divergence lower order coefficients , Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 189 (19) (2019), 111568.10.1016/j.na.2019.07.001CrossRefGoogle Scholar
Anceschi, F. and Zhu, Y., On the nonlinear kinetic Fokker–Planck model, Cauchy Problem and Diffusion Asymptotics 2020, In preparation.Google Scholar
Armstrong, S. and Mourrat, J.-C., Variational methods for the kinetic Fokker–Planck equation, Preprint, 2019, arXiv:1902.04037.Google Scholar
Bramanti, M., Cerutti, M. C. and Manfredini, M., ${\mathbf{\mathcal{L}}}^p$ estimates for some ultraparabolic operators with discontinuous coefficients, J. Math. Anal. Appl. 200 (1996), 332–354.10.1006/jmaa.1996.0209CrossRefGoogle Scholar
Cinti, C., Pascucci, A. and Polidoro, S., Pointwise estimates for a class of non-homogeneous Kolmogorov equations, Math. Ann. 340 (2008), 237–264.CrossRefGoogle Scholar
De Giorgi, E., Sull’analiticità delle estremali degli integrali multipli, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 20 (1956), 438–441.Google Scholar
De Giorgi, E., Sulla differenziabilità e delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43.Google Scholar
DiBenedetto, E. and Trudinger, N. S., Harnack inequalities for quasi-minima of variational integrals, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984), 295–308.CrossRefGoogle Scholar
Evans, L. C., Partial Differential Equations, vol. 19, (American Mathematical Society, Providence, RI, 1998).Google Scholar
Ferretti, E. and Safonov, M. V., Growth theorems and Harnack inequality for second order parabolic equations, in Harmonic Analysis and Boundary Value Problems Providence, RI: American Mathematical Society (AMS), 2001, pp. 87–112.Google Scholar
Gianazza, U. and Vespri, V., Parabolic De Giorgi classes of order $p$ and the Harnack inequality, Calc. Var. Partial Differential Equations, 26 (2006), 379–399.10.1007/s00526-006-0022-4CrossRefGoogle Scholar
Golse, F., Imbert, C., Mouhot, C. and Vasseur, A. F., 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), 253–295.Google Scholar
Guerand, J. and Mouhot, C., Quantitative De Giorgi methods in kinetic theory (2021).CrossRefGoogle Scholar
Hörmander, L., Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.CrossRefGoogle Scholar
Imbert, C. and Mouhot, C., The Schauder estimate in kinetic theory with application to a toy nonlinear model (2020).10.5802/ahl.75CrossRefGoogle Scholar
Imbert, C. and Silvestre, L., An introduction to fully nonlinear parabolic equations, in An Introduction to the Kähler–Ricci Flow (Springer, Cham, 2013) pp. 7–88.Google Scholar
Imbert, C. and Silvestre, L., The weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc. (JEMS) 22 (2020), 507–592.10.4171/JEMS/928CrossRefGoogle Scholar
Kolmogoroff, A. N., Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35 (1934), 116–117.CrossRefGoogle Scholar
Kružkov, S., A priori estimates and certain properties of the solutions of elliptic and parabolic equations, AMS Transl. Ser. 2(68) (1968), 169–220.Google Scholar
Kružkov, S. N., A priori bounds for generalized solutions of second-order elliptic and parabolic equations, Dokl. Akad. Nauk SSSR 150 (1963), 748–751.Google Scholar
Kružkov, S. N., A priori bounds and some properties of solutions of elliptic and parabolic equations, Mat. Sb. (N.S.) 65 (107) (1964), 522–570.Google Scholar
Krylov, N. V. and Safonov, M. V., A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 161–175, 239.Google Scholar
Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural’tseva, N. N., Linear and Quasi-Linear Equations of Parabolic Type, Vol. 23 (American Mathematical Society (AMS), Providence, RI, 1968). Translated from the Russian by Smith, S..10.1090/mmono/023CrossRefGoogle Scholar
Landau, L. D. and Lifshitz, E. M., Statistical Physics, Course of Theoretical Physics. Vol. 5 (Pergamon Press Ltd., London-Paris; Addison-Wesley Publishing Company, Inc., Reading, MA, 1958). Translated from the Russian by Peierls, E. and Peierls, R. F..Google Scholar
Li, D. and Zhang, K., A note on the Harnack inequality for elliptic equations in divergence form, Proceedings of the American Mathematical Society 145 (2017), 135–137.10.1090/proc/13174CrossRefGoogle Scholar
Lieberman, G. M., Second Order Parabolic Differential Equations (World Scientific Publishing Co., Inc., River Edge, NJ, 1996).CrossRefGoogle Scholar
Litsgård, M. and Nyström, K., The Dirichlet problem for Kolmogorov–Fokker–Planck type equations with rough coefficients (2020).10.1016/j.jfa.2021.109226CrossRefGoogle Scholar
Manfredini, M. and Polidoro, S., Interior regularity for weak solutions of ultraparabolic equations in divergence form with discontinuous coefficients, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. 1(8) (1998), 651–675.Google Scholar
Moser, J., On Harnack’s theorem for elliptic differential equations, Commun. Pure Appl. Math. 14 (1961), 577–591.CrossRefGoogle Scholar
Moser, J., A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134.10.1002/cpa.3160170106CrossRefGoogle Scholar
Nash, J., Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954.10.2307/2372841CrossRefGoogle Scholar
Pascucci, A. and Polidoro, S., The Moser’s iterative method for a class of ultraparabolic equations, Commun. Contemp. Math. 6 (2004), 395–417.CrossRefGoogle Scholar
Polidoro, S. and Ragusa, M. A., Sobolev–Morrey spaces related to an ultraparabolic equation, Manuscr. Math. 96 (1998), 371–392.CrossRefGoogle Scholar
Polidoro, S. and Ragusa, M. A., Hölder regularity for solutions of ultraparabolic equations in divergence form, Potential Anal. 14 (2001), 341–350.CrossRefGoogle Scholar
Schwab, R. W. and Silvestre, L., Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE 9 (2016), 727–772.10.2140/apde.2016.9.727CrossRefGoogle Scholar
Trudinger, N. S., Pointwise estimates and quasilinear parabolic equations, Commun. Pure Appl. Math. 21 (1968), 205–226.CrossRefGoogle Scholar
Wang, G. L., Harnack Inequalities for Functions in De Giorgi Parabolic Class, in Partial Differential Equations (Tianjin, 1986), Vol. 1306 of Lecture Notes in Math. (Springer, Berlin, 1988), 182–201.Google Scholar
Wang, G. L. and Sun, A. X., Weak Harnack inequality for functions from class ${\mathbf{\mathcal{B}}}_2$ and application to parabolic $Q$ -minima, Chinese J. Contemp. Math. 14 (1993), 75–84.Google Scholar
Wang, W. and Zhang, L., The ${C}^{\alpha }$ regularity of a class of non-homogeneous ultraparabolic equations, Sci. China Ser. A 52 (2009), 1589–1606.CrossRefGoogle Scholar
Wang, W. and Zhang, L., The ${C}^{\alpha }$ regularity of weak solutions of ultraparabolic equations, Discrete Contin. Dyn. Syst. 29 (2011), 1261–1275.CrossRefGoogle Scholar
Wang, W. and Zhang, L., ${C}^{\alpha }$ regularity of weak solutions of non-homogeneous ultraparabolic equations with drift terms, Preprint, 2017. arXiv:1704.05323.Google Scholar
Zhu, Y., Velocity averaging and Hölder regularity for kinetic Fokker–Planck equations with general transport operators and rough coefficients (2020).CrossRefGoogle Scholar