In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain $ (0, T) \times D \times \mathbb{R}^d $, where $ D $ is either $ \mathbb{T}^d $ or $ \mathbb{R}^d $. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from [2] and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term.
Citation: |
[1] |
R. A. Adams, Sobolev Spaces, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
![]() ![]() |
[2] |
D. Albritton, S. Armstrong, J.-C. Mourrat and M. Novack, Variational methods for the kinetic Fokker-Planck equation, preprint, 2021. arXiv: 1902.04037v2.
![]() |
[3] |
D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Springer, Cham, 2014.
doi: 10.1007/978-3-319-00227-9.![]() ![]() ![]() |
[4] |
A. Blaustein and F. Filbet, On a discrete framework of hypocoercivity for kinetic equations, to appear, Math. Comp..
![]() |
[5] |
J.-F. Bony, D. L. Peutrec and L. Michel, Eyring-Kramers law for Fokker-Planck type differential operators, preprint, 2022. arXiv: 2201.01660.
![]() |
[6] |
H. C. Brinkman, Brownian motion in a field of force and the diffusion theory of chemical reactions. Ⅱ, Physica, 22 (1956), 149-155.
doi: 10.1016/S0031-8914(56)80019-0.![]() ![]() |
[7] |
Y. Cao, J. Lu and L. Wang, On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamics, Arch. Rational Mech. Anal., 247 (2023), Paper No. 90, 34 pp.
doi: 10.1007/s00205-023-01922-4.![]() ![]() ![]() |
[8] |
G. Chai and T. Wang, Mixed generalized Hermite-Fourier spectral method for Fokker-Planck equation of periodic field, Appl. Numer. Math., 133 (2018), 25-40.
doi: 10.1016/j.apnum.2017.10.006.![]() ![]() ![]() |
[9] |
J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.
doi: 10.1090/S0002-9947-2015-06012-7.![]() ![]() ![]() |
[10] |
L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010.
![]() |
[11] |
J. C. M. Fok, B. Guo and T. Tang, Combined Hermite spectral-finite difference method for the Fokker-Planck equation, Math. Comp., 71 (2002), 1497-1528.
doi: 10.1090/S0025-5718-01-01365-5.![]() ![]() ![]() |
[12] |
I. M. Gamba and S. Rjasanow, Galerkin-Petrov approach for the Boltzmann equation, J. Comput. Phys., 366 (2018), 341-365.
doi: 10.1016/j.jcp.2018.04.017.![]() ![]() ![]() |
[13] |
H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403.![]() ![]() ![]() |
[14] |
B.-Y. Guo and T.-J. Wang, Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel, Math. Comp., 78 (2009), 129-151.
doi: 10.1090/S0025-5718-08-02152-2.![]() ![]() ![]() |
[15] |
F. Hérau, M. Hitrik and J. Sjöstrand, Tunnel effect for Kramers-Fokker-Planck type operators: Return to equilibrium and applications, Int. Math. Res. Not. IMRN, 2008 (2008), Art. ID rnn057, 48 pp.
doi: 10.1093/imrn/rnn057.![]() ![]() ![]() |
[16] |
F. Hérau, M. Hitrik and J. Sjöstrand, Tunnel effect and symmetries for Kramers-Fokker-Planck type operators, J. Inst. Math. Jussieu, 10 (2011), 567-634.
doi: 10.1017/S1474748011000028.![]() ![]() ![]() |
[17] |
F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.
doi: 10.1007/s00205-003-0276-3.![]() ![]() ![]() |
[18] |
F. Hérau, J. Sjöstrand and C. C. Stolk, Semiclassical analysis for the Kramers-Fokker-Planck equation, Comm. Partial Differential Equations, 30 (2005), 689-760.
doi: 10.1081/PDE-200059278.![]() ![]() ![]() |
[19] |
M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilités, XXXIII, Lecture Notes in Math., 1709, Springer, Berlin, 1999,120-216.
doi: 10.1007/BFb0096511.![]() ![]() ![]() |
[20] |
R. Li, Y. Ren and Y. Wang, Hermite spectral method for Fokker-Planck-Landau equation modeling collisional plasma, J. Comput. Phys., 434 (2021), Paper No. 110235, 22 pp.
doi: 10.1016/j.jcp.2021.110235.![]() ![]() ![]() |
[21] |
J. Meyer and J. Schröter, Comments on the Grad Procedure for the Fokker-Planck Equation, J. Statist. Phys., 32 (1983), 53-69.
doi: 10.1007/BF01009419.![]() ![]() ![]() |
[22] |
G. A. Pavliotis, Stochastic Processes and Applications, Springer, New York, 2014.
doi: 10.1007/978-1-4939-1323-7.![]() ![]() ![]() |
[23] |
H. Risken, The Fokker-Planck Equation, 2$^nd$ edition, Springer Berlin, Heidelberg, 1996
![]() |
[24] |
N. Sarna, J. Giesselmann and M. Torrilhon, Convergence analysis of Grad's Hermite expansion for linear kinetic equations, SIAM J. Numer. Anal., 58 (2020), 1164-1194.
doi: 10.1137/19M1270884.![]() ![]() ![]() |
[25] |
C. Schmeiser and A. Zwirchmayr, Convergence of moment methods for linear kinetic equations, SIAM J. Numer. Anal., 36 (1999), 74-88.
doi: 10.1137/S0036142996304516.![]() ![]() ![]() |
[26] |
J. M. Tölle, Uniqueness of weighted Sobolev spaces with weakly differentiable weights, J. Funct. Anal., 263 (2012), 3195-3223.
doi: 10.1016/j.jfa.2012.08.002.![]() ![]() ![]() |
[27] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), no.950, ⅳ+141 pp.
doi: 10.1090/S0065-9266-09-00567-5.![]() ![]() ![]() |
[28] |
N. Wiener, The Fourier Integral and Certain of its Applications, Cambridge University Press, Cambridge, 1988.
doi: 10.1017/CBO9780511662492.![]() ![]() ![]() |