The Schauder estimate in kinetic theory with application to a toy nonlinear model
Annales Henri Lebesgue, Volume 4 (2021) , pp. 369-405.

KeywordsFokker–Planck equation, hypoelliptic, Schauder estimate, nonlinear kinetic equation

### Abstract

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.

### References

[Bau17] Baudoin, Fabrice Bakry–Émery meet Villani, J. Funct. Anal., Volume 273 (2017) no. 7, pp. 2275-2291 | Article | MR 3677826 | Zbl 1373.35061

[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 | Article | MR 2298970 | Zbl 1113.35033

[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 | Article | MR 763762 | Zbl 0573.76029

[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 04.0566.01

[BÉ85] Bakry, Dominique; Émery, Michel Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84 (Lecture Notes in Mathematics), Volume 1123, Springer, 1985, pp. 177-206 | Article | Numdam | MR 889476 | Zbl 0561.60080

[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 | Article | Zbl 1189.35335

[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 2277064 | Zbl 1153.35312

[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 3923847 | Zbl 1431.35016

[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 | Article | MR 2917400 | Zbl 1306.60121

[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 | Article | MR 4072211 | Zbl 1435.35101

[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 | Article | MR 4049224 | Zbl 07174139

[IS21] Imbert, Cyril; Silvestre, Luis The Schauder estimate for kinetic integral equations, Anal. PDE, Volume 14 (2021) no. 1, pp. 171-204 | Article | MR 4229202

[Kac56] Kac, Mark Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III (1956), pp. 171-197 | MR 0084985 | Zbl 0072.42802

[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 | Article | MR 2252155 | Zbl 1101.82025

[Kol34] Kolmogoroff, Andreĭ N. Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. Math., Volume 35 (1934) no. 1, p. 116-117 | Article | MR 1503147 | Zbl 0008.39906

[Kry96] Krylov, Nicolaĭ V. Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, 12, American Mathematical Society, 1996 | Article | MR 1406091 | Zbl 0865.35001

[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 0015.38202

[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 1475774 | Zbl 0887.35062

[Man97] Manfredini, Maria The Dirichlet problem for a class of ultraparabolic equations, Adv. Differ. Equ., Volume 2 (1997) no. 5, pp. 831-866 | MR 1751429 | Zbl 1023.35518

[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 1386366 | Zbl 0845.35059

[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) | Article | MR 2675371 | Zbl 1200.35157

[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 765302

[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 | Article | MR 3551261 | Zbl 1352.35091

[Vil09] Villani, Cédric Hypocoercivity, Memoirs of the American Mathematical Society, 950, American Mathematical Society, 2009 | Article | MR 2562709 | Zbl 1197.35004

[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 | Article | MR 2530175 | Zbl 1181.35139