\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A Galerkin type method for kinetic Fokker-Planck equations based on Hermite expansions

  • *Corresponding author: Mingyi Hou

    *Corresponding author: Mingyi Hou 

B. Avelin and M. Hou were supported by [Swedish Research Council dnr: 2019-04098]. K. Nyström was supported by [Swedish Research Council dnr: 2022-03106]

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35Q84, 65N30, 65M15; Secondary: 35Kxx, 35B65.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. DolbeaultC. 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. FokB. 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. HérauM. 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. HérauJ. 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. SarnaJ. 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. WienerThe Fourier Integral and Certain of its Applications, Cambridge University Press, Cambridge, 1988.  doi: 10.1017/CBO9780511662492.
  • 加载中
SHARE

Article Metrics

HTML views(1629) PDF downloads(329) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return