The Schauder estimate in kinetic theory with application to a toy nonlinear model

Imbert, Cyril; Mouhot, Clément

Annales Henri Lebesgue, Tome 4 (2021), pp. 369-405.

Cet article traite de l’estimation de Schauder pour l’équation de Fokker–Planck cinétique linéaire avec coefficients Hölder-continus. Cette équation a une structure hypoelliptique. Comme exemple d’application de cette estimation de Schauder, nous démontrons le caractère bien posé globalement en temps d’un modéle jouet nonlinéaire en théorie cinétique. Ce modèle nonlinéaire est une équation de Fokker–Planck cinétique dont les équilibres sont maxwelliens mais dont la diffusion en la variable de vitesse est proportionnelle à la masse locale de la solution.

This article is concerned with the Schauder estimate for linear kinetic Fokker–Planck equations with Höder continuous coefficients. This equation has an hypoelliptic structure. As an application of this Schauder estimate, we prove the global well-posedness of a toy nonlinear model in kinetic theory. This nonlinear model consists in a non-linear kinetic Fokker–Planck equation whose steady states are Maxwellian and whose diffusion in the velocity variable is proportional to the mass of the solution.

Classification : 35B65, 35Q84, 82C40
Mots clés : Fokker–Planck equation, hypoelliptic, Schauder estimate, nonlinear kinetic equation

[Bau17] Baudoin, Fabrice Bakry–Émery meet Villani, J. Funct. Anal., Volume 273 (2017) no. 7, pp. 2275-2291 | DOI | MR | Zbl

[BB07] Bramanti, Marco; Brandolini, Luca Schauder estimates for parabolic nondivergence operators of Hörmander type, J. Differ. Equations, Volume 234 (2007) no. 1, pp. 177-245 | DOI | MR | Zbl

[BKM84] Beale, J. Thomas; Kato, Tosio; Majda, Andrew J. Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Commun. Math. Phys., Volume 94 (1984) no. 1, pp. 61-66 | DOI | MR | Zbl

[Bol72] Boltzmann, Ludwig Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Sitzungs. Further Studies on the Thermal Equilibrium of Gas Molecules, Kinetic Theory, Volume 2 (1872), pp. 88-174 | Zbl

[BÉ85] Bakry, Dominique; Émery, Michel Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84 (Lecture Notes in Mathematics), Volume 1123, Springer, 1985, pp. 177-206 | DOI | Numdam | MR | Zbl

[Cha08] Chavanis, Pierre-Henri Nonlinear mean field Fokker–Planck equations. Application to the chemotaxis of biological populations, Eur. Phys. J. B, Condens. Matter Complex Syst., Volume 62 (2008) no. 2, pp. 179-208 | DOI | Zbl

[DFP06] Di Francesco, Marco; Polidoro, Sergio Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form, Adv. Differ. Equ., Volume 11 (2006) no. 11, pp. 1261-1320 | MR | Zbl

[GIMV19] Golse, François; Imbert, Cyril; Mouhot, Clément; Vasseur, Alexis F. Harnack inequality for kinetic Fokker–Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 19 (2019) no. 1, pp. 253-295 | MR | Zbl

[GW12] Guillin, Arnaud; Wang, Feng-Yu Degenerate Fokker–Planck equations: Bismut formula, gradient estimate and Harnack inequality, J. Differ. Equations, Volume 253 (2012) no. 1, pp. 20-40 | DOI | MR | Zbl

[HS20] Henderson, Christopher; Snelson, Stanley $C^{\infty }$ smoothing for weak solutions of the inhomogeneous Landau equation, Arch. Ration. Mech. Anal., Volume 236 (2020) no. 1, pp. 113-143 | DOI | MR | Zbl

[IS19] Imbert, Cyril; Silvestre, Luis Global regularity estimates for the Boltzmann equation without cut-off (2019) (https://arxiv.org/abs/1909.12729)

[IS20] Imbert, Cyril; Silvestre, Luis The weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc., Volume 22 (2020) no. 2, pp. 507-592 | DOI | MR | Zbl

[IS21] Imbert, Cyril; Silvestre, Luis The Schauder estimate for kinetic integral equations, Anal. PDE, Volume 14 (2021) no. 1, pp. 171-204 | DOI | MR

[Kac56] Kac, Mark Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, University of California Press, Berkeley and Los Angeles (1956), pp. 171-197 | MR | Zbl

[KL06] Kiessling, Michael; Lancellotti, Carlo The linear Fokker–Planck equation for the Ornstein–Uhlenbeck process as an (almost) nonlinear kinetic equation for an isolated $N$-particle system, J. Stat. Phys., Volume 123 (2006) no. 3, pp. 525-546 | DOI | MR | Zbl

[Kol34] Kolmogoroff, Andreĭ N. Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. Math., Volume 35 (1934) no. 1, pp. 116-117 | DOI | MR | Zbl

[Kry96] Krylov, Nicolaĭ V. Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, 12, American Mathematical Society, 1996 | DOI | MR | Zbl

[Lan36] Landau, Lev D. The transport equation in the case of Coulomb interactions – Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung, Phys. Z. Sowjetunion, Volume 10 (1936), pp. 154-164 | Zbl

[Lun97] Lunardi, Alessandra Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in ${\mathbf{R}}^n$, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 24 (1997) no. 1, pp. 133-164 | MR | Zbl

[Man97] Manfredini, Maria The Dirichlet problem for a class of ultraparabolic equations, Adv. Differ. Equ., Volume 2 (1997) no. 5, pp. 831-866 | MR | Zbl

[Max67] Maxwell, James Clerk On the dynamical theory of gases, Philosophical Transactions of the Royal Society of London Series I, Volume 157 (1867), pp. 49-88

[Pol94] Polidoro, Sergio On a class of ultraparabolic operators of Kolmogorov–Fokker–Planck type, Matematiche, Volume 49 (1994) no. 1, pp. 53-105 | MR | Zbl

[Rad08] Radkevich, Evgeniĭ V. Equations with nonnegative characteristic form. II, Sovrem. Mat. Prilozh. (2008) no. 56, pp. 3-147 (Differentsialʼ nye Uravneniya s Chastnymi Proizvodnymi) | DOI | MR | Zbl

[Saf84] Safonov, Mikhail V. The classical solution of the elliptic Bellman equation, Dokl. Akad. Nauk SSSR, Volume 278 (1984) no. 4, pp. 810-813 | MR

[Sil16] Silvestre, Luis A new regularization mechanism for the Boltzmann equation without cut-off, Commun. Math. Phys., Volume 348 (2016) no. 1, pp. 69-100 | DOI | MR | Zbl

[Vil09] Villani, Cédric Hypocoercivity, Memoirs of the American Mathematical Society, 950, American Mathematical Society, 2009 | DOI | MR | Zbl

[WZ09] Wang, WenDong; Zhang, LiQun The $C^{\alpha }$ regularity of a class of non-homogeneous ultraparabolic equations, Sci. China, Ser. A, Volume 52 (2009) no. 8, pp. 1589-1606 | DOI | MR | Zbl