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Sharpening of Decay Rates in Fourier Based Hypocoercivity Methods

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Part of the Springer INdAM Series book series (SINDAMS,volume 48)


This paper is dealing with two L2 hypocoercivity methods based on Fourier decomposition and mode-by-mode estimates, with applications to rates of convergence or decay in kinetic equations on the torus and on the whole Euclidean space. The main idea is to perturb the standard L2 norm by a twist obtained either by a nonlocal perturbation build upon diffusive macroscopic dynamics, or by a change of the scalar product based on Lyapunov matrix inequalities. We explore various estimates for equations involving a Fokker–Planck and a linear relaxation operator. We review existing results in simple cases and focus on the accuracy of the estimates of the rates. The two methods are compared in the case of the Goldstein–Taylor model in one-dimension.


  • Hypocoercivity
  • Linear kinetic equations
  • Entropy–entropy production inequalities
  • Goldstein–Taylor model
  • Fokker–Planck operator
  • Linear relaxation operator
  • Linear BGK operator
  • Transport operator
  • Fourier modes decomposition
  • Pseudo-differential operators
  • Nash’s inequality

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  1. Achleitner, F., Arnold, A., Carlen, E.A.: The hypocoercivity index for the short and large time behavior of ODEs. Preprint arXiv (2021).

  2. Achleitner, F., Arnold, A., Carlen, E.A.: On linear hypocoercive BGK models. In: From Particle Systems to Partial Differential Equations III, pp. 1–37. Springer, Berlin (2016).

  3. Achleitner, F., Arnold, A., Carlen, E.A.: On multi-dimensional hypocoercive BGK models. Kinet. Relat. Models 11(4), 953–1009 (2018).

    CrossRef  MathSciNet  Google Scholar 

  4. Achleitner, F., Arnold, A., Signorello, B.: On optimal decay estimates for ODEs and PDEs with modal decomposition. In: Stochastic Dynamics Out of Equilibrium. Springer Proc. Math. Stat., vol. 282, pp. 241–264. Springer, Cham (2019).

  5. Addala, L., Dolbeault, J., Li, X., Tayeb, M.L.: L 2-hypocoercivity and large time asymptotics of the linearized Vlasov–Poisson–Fokker-Planck system. J. Stat. Phys. 184, 34 (2021).

  6. Armstrong, S., Mourrat, J.C.: Variational methods for the kinetic Fokker-Planck equation. Preprint arXiv (2019).

  7. Arnold, A., Einav, A., Signorello, B., Wöhrer, T.: Large time convergence of the non-homogeneous Goldstein-Taylor equation. J. Stat. Phys. 182, 35 (2021).

    CrossRef  MathSciNet  Google Scholar 

  8. Arnold, A., Einav, A., Wöhrer, T.: On the rates of decay to equilibrium in degenerate and defective Fokker-Planck equations. J. Differ. Equ. 264(11), 6843–6872 (2018).

    CrossRef  MathSciNet  Google Scholar 

  9. Arnold, A., Erb, J.: Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift. Preprint arXiv (2014).

  10. Arnold, A., Jin, S., Wöhrer, T.: Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: from ODEs to linear kinetic equations. J. Differ. Equ. 268(3), 1156–1204 (2020).

    CrossRef  MathSciNet  Google Scholar 

  11. Arnold, A., Schmeiser, C., Signorello, B.: Propagator norm and sharp decay estimates for Fokker-Planck equations with linear drift. Preprint arXiv (2020).

  12. Bernard, É., Salvarani, F.: Optimal estimate of the spectral gap for the degenerate Goldstein-Taylor model. J. Stat. Phys. 153(2), 363–375 (2013).

    CrossRef  MathSciNet  Google Scholar 

  13. Bernard, É., Salvarani, F.: Correction to: Optimal estimate of the spectral gap for the degenerate Goldstein-Taylor model. J. Stat. Phys. 181(4), 1–2 (2020).

    CrossRef  MathSciNet  Google Scholar 

  14. Bouin, E., Dolbeault, J., Lafleche, L., Schmeiser, C.: Hypocoercivity and sub-exponential local equilibria. Monatshefte für Mathematik (2020).

  15. Bouin, E., Dolbeault, J., Mischler, S., Mouhot, C., Schmeiser, C.: Hypocoercivity without confinement. Pure Appl. Anal. 2(2), 203–232 (2020).

    CrossRef  MathSciNet  Google Scholar 

  16. Bouin, E., Dolbeault, J., Schmeiser, C.: Diffusion and kinetic transport with very weak confinement. Kinet. Relat. Models 13(2), 345–371 (2020).

    CrossRef  MathSciNet  Google Scholar 

  17. Bouin, E., Dolbeault, J., Schmeiser, C.: A variational proof of Nash’s inequality. Rend. Lincei Mate. Appl. 31(1), 211–223 (2020).

    MathSciNet  MATH  Google Scholar 

  18. Calvez, V., Raoul, G.: Confinement by biased velocity jumps: aggregation of escherichia coli. Kinet. Relat. Models 8, 651 (2015).

  19. Dolbeault, J., Klar, A., Mouhot, C., Schmeiser, C.: Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes. Appl. Math. Res. eXpress (2012).

  20. Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for kinetic equations with linear relaxation terms. C. R. Math. 347(9–10), 511–516 (2009).

    CrossRef  MathSciNet  Google Scholar 

  21. Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for linear kinetic equations conserving mass. Trans. Am. Math. Soc. 367(6), 3807–3828 (2015).

    CrossRef  MathSciNet  Google Scholar 

  22. Favre, G., Schmeiser, C.: Hypocoercivity and fast reaction limit for linear reaction networks with kinetic transport. J. Stat. Phys. 178(6), 1319–1335 (2020).

    CrossRef  MathSciNet  Google Scholar 

  23. Fellner, K., Prager, W., Tang, B.Q.: The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinet. Relat. Models 10(4), 1055–1087 (2017).

    CrossRef  MathSciNet  Google Scholar 

  24. Goudon, T., Alonso, R.J., Vavasseur, A.: Damping of particles interacting with a vibrating medium. Ann. Inst. Henri Poincaré (C) Non Linear Anal. (2016).

  25. Hérau, F.: Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal. 46(3–4), 349–359 (2006).

    MathSciNet  MATH  Google Scholar 

  26. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013).

  27. Kawashima, S.: The Boltzmann equation and thirteen moments. Jpn. J. Appl. Math. 7(2), 301–320 (1990).

    CrossRef  MathSciNet  Google Scholar 

  28. Mouhot, C., Neumann, L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity 19(4), 969–998 (2006).

    CrossRef  MathSciNet  Google Scholar 

  29. Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958).

    CrossRef  MathSciNet  Google Scholar 

  30. Neumann, L., Schmeiser, C.: A kinetic reaction model: decay to equilibrium and macroscopic limit. Kinet. Relat. Models 9, 571 (2016).

    CrossRef  MathSciNet  Google Scholar 

  31. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983).

  32. Shizuta, Y., Kawashima, S.: Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14(2), 249–275 (1985).

    CrossRef  MathSciNet  Google Scholar 

  33. Ueda, Y., Duan, R., Kawashima, S.: Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application. Arch. Ration. Mech. Anal. 205(1), 239–266 (2012).

    CrossRef  MathSciNet  Google Scholar 

  34. Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202(950), iv+ 141 (2009).

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This work has been partially supported by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR) and the Amadeus project Hypocoercivity no. 39453PH. J.D. and C.S. thank E. Bouin, S. Mischler and C. Mouhot for stimulating discussions that took place during the preparation of [15]: some questions raised at this occasion are the origin for this contribution. A.A., C.S., and T.W. were partially supported by the FWF (Austrian Science Fund) funded SFB F65.

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Correspondence to Jean Dolbeault .

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Arnold, A., Dolbeault, J., Schmeiser, C., Wöhrer, T. (2021). Sharpening of Decay Rates in Fourier Based Hypocoercivity Methods. In: Salvarani, F. (eds) Recent Advances in Kinetic Equations and Applications. Springer INdAM Series, vol 48. Springer, Cham.

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