Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition

Hongjie Dong; Yan Guo; Timur Yastrzhembskiy

doi:10.3934/krm.2022003

Article Contents

2022, Volume 15, Issue 3: 467-516. Doi: 10.3934/krm.2022003

This issue Previous Article From kinetic to fluid models of liquid crystals by the moment method Next Article The Boltzmann-Grad limit for the Lorentz gas with a Poisson distribution of obstacles

Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition

Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA

Received: October 31, 2021

Published: June 2022

H. Dong’s research is supported in part by the Simons Foundation, grant no. 709545, a Simons fellowship, grant no. 007638, and the NSF under agreement DMS-2055244.
Y. Guo's research is supported in part by NSF DMS-grant 2106650

Abstract Full Text(HTML) Related Papers Cited by

Abstract

We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the $ S_p $ estimate of [7], we prove regularity in the kinetic Sobolev spaces $ S_p $ and anisotropic Hölder spaces for such weak solutions. Such $ S_p $ regularity leads to the uniqueness of weak solutions.

Keywords:

Kinetic Kolmogorov-Fokker-Planck equations,
specular boundary condition,
Landau equation,
plasma physics,
$ S_p $ estimates.

Mathematics Subject Classification: Primary: 35Q84, 35K70, 35H10, 35B45, 34A12; Secondary: 35Q70.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	D. Albritton, S. Armstrong, J.-C. Mourrat and M. Novack, Variational methods for the kinetic Fokker-Planck equation, arXiv: 1902.04037.
[2]	R. Beals and V. Protopopescu, Abstract time-dependent transport equations, J. Math. Anal. Appl., 121 (1987), 370-405. doi: 10.1016/0022-247X(87)90252-6.
[3]	M. Bramanti, M.C. Cerutti, M. Manfredini, $L^p$ estimates for some ultraparabolic operators with discontinuous coefficients, J. Math. Anal. Appl., 200 (1996), no. 2,332–354.
[4]	M. Bramanti, G. Cupini, E. Lanconelli, E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients., Math. Nachr., 286 (2013), no. 11-12, 1087–1101.
[5]	H. Dong, Timur Yastrzhembskiy, Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in divergence form, in preparation.
[6]	H. Dong, Y. Guo and Z. Ouyang, The Vlasov-Poisson-Landau system with the specular reflection boundary condition, arXiv: 2010.05314.
[7]	H. Dong and T. Yastrzhembskiy, Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in nondivergence form, arXiv: 2107.08568, to appear in Arch. Ration. Mech. Anal..
[8]	R. Duan, S. Liu, S. Sakamoto and R. M. Strain, Global mild solutions of the Landau and non-cutoff Boltzmann equations, Comm. Pure Appl. Math., 74 (2021), 932-1020. doi: 10.1002/cpa.21920.
[9]	Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 245-263. doi: 10.1007/BF02096997.
[10]	Y. Guo, Singular solutions of the Vlasov-Maxwell system on a half Line, Arch. Rational Mech. Anal., 131 (1995), 241-304. doi: 10.1007/BF00382888.
[11]	Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.
[12]	Y. Guo, H. J. Hwang, J. W. Jang and Z. Ouyang, The Landau equation with the specular reflection boundary condition, Arch. Ration. Mech. Anal., 236 (2020), 1389-1454. doi: 10.1007/s00205-020-01496-5.
[13]	H. J. Hwang, J. Jang and J. Jung, The Fokker-Planck equation with absorbing boundary conditions in bounded domains, SIAM J. Math. Anal., 50 (2018), 2194-2232. doi: 10.1137/16M1109928.
[14]	H. J. Hwang, J. Jang and J. J. L. Velázquez, The Fokker-Planck equation with absorbing boundary conditions, Arch. Ration. Mech. Anal., 214 (2014), 183-233. doi: 10.1007/s00205-014-0758-5.
[15]	H. J. Hwang, J. Jang and J. J. L. Velázquez, Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions, Arch. Ration. Mech. Anal., 231 (2019), 1309-1400. doi: 10.1007/s00205-018-1299-0.
[16]	J. Kim, Y. Guo and H. J. Hwang, An $L^2$ to $L^{\infty}$ framework for the Landau equation, Peking Math. J., 3 (2020), 131-202. doi: 10.1007/s42543-019-00018-x.
[17]	N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, 96. American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096.
[18]	N. V. Krylov, On a representation of fully nonlinear elliptic operators in terms of pure second order derivatives and its applications, J. Math. Sci. (N.Y.), 177 (2011), 1-26. doi: 10.1007/s10958-011-0445-0.
[19]	M. Litsgård and K. Nyström, The Dirichlet problem for Kolmogorov-Fokker-Planck type equations with rough coefficients, J. Funct. Anal., 281 (2021), Paper No. 109226, 39 pp. doi: 10.1016/j.jfa.2021.109226.
[20]	T. S. Motzkin and W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Physics, 31 (1953), 253-259.
[21]	L. Niebel and R. Zacher, Kinetic maximal $L^p$-regularity with temporal weights and application to quasilinear kinetic diffusion equations., J. Differential Equations, 307 (2022), 29-82.
[22]	S. Polidoro and M. A. Ragusa, Sobolev-Morrey spaces related to an ultraparabolic equation, Manuscripta Math., 96 (1998), 371-392. doi: 10.1007/s002290050072.