Skip to main content
Springer Nature Link
Log in

Sharpening of Decay Rates in Fourier Based Hypocoercivity Methods

  • Conference paper
  • First Online:
Recent Advances in Kinetic Equations and Applications

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

  • 596 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Achleitner, F., Arnold, A., Carlen, E.A.: The hypocoercivity index for the short and large time behavior of ODEs. Preprint arXiv (2021). https://arxiv.org/abs/2109.10784

  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). https://doi.org/10.1007/978-3-319-32144-8_1

  3. Achleitner, F., Arnold, A., Carlen, E.A.: On multi-dimensional hypocoercive BGK models. Kinet. Relat. Models 11(4), 953–1009 (2018). https://doi.org/10.3934/krm.2018038

    Article  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). https://doi.org/10.1007/978-3-030-15096-9_6

  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). https://doi.org/10.1007/s10955-021-02784-4

  6. Armstrong, S., Mourrat, J.C.: Variational methods for the kinetic Fokker-Planck equation. Preprint arXiv (2019). https://arxiv.org/abs/1409.5425

  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). https://doi.org/10.1007/s10955-021-02702-8

    Article  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). https://doi.org/10.1016/j.jde.2018.01.052

    Article  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). https://arxiv.org/abs/1409.5425

  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). https://doi.org/10.1016/j.jde.2019.08.047

    Article  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). https://arxiv.org/abs/2003.01405

  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). https://doi.org/10.1007/s10955-013-0825-6

    Article  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). https://doi.org/10.1007/s10955-020-02631-y

    Article  MathSciNet  Google Scholar 

  14. Bouin, E., Dolbeault, J., Lafleche, L., Schmeiser, C.: Hypocoercivity and sub-exponential local equilibria. Monatshefte für Mathematik (2020). https://doi.org/10.1007/s00605-020-01483-8

  15. Bouin, E., Dolbeault, J., Mischler, S., Mouhot, C., Schmeiser, C.: Hypocoercivity without confinement. Pure Appl. Anal. 2(2), 203–232 (2020). https://doi.org/10.2140/paa.2020.2.203

    Article  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). https://doi.org/10.3934/krm.2020012

    Article  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). https://doi.org/10.4171/rlm/886

    MathSciNet  MATH  Google Scholar 

  18. Calvez, V., Raoul, G.: Confinement by biased velocity jumps: aggregation of escherichia coli. Kinet. Relat. Models 8, 651 (2015). https://doi.org/10.3934/krm.2015.8.651

  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). https://doi.org/10.1093/amrx/abs015

  20. Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for kinetic equations with linear relaxation terms. C. R. Math. 347(9–10), 511–516 (2009). https://doi.org/10.1016/j.crma.2009.02.025

    Article  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). https://doi.org/10.1090/s0002-9947-2015-06012-7

    Article  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). https://doi.org/10.1007/s10955-020-02503-5

    Article  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). https://doi.org/10.3934/krm.2017042

    Article  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). https://doi.org/10.1016/j.anihpc.2016.12.005

  25. Hérau, F.: Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal. 46(3–4), 349–359 (2006). https://content.iospress.com/articles/asymptotic-analysis/asy741

    MathSciNet  MATH  Google Scholar 

  26. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013). https://doi.org/10.1017/CBO9780511810817

  27. Kawashima, S.: The Boltzmann equation and thirteen moments. Jpn. J. Appl. Math. 7(2), 301–320 (1990). https://doi.org/10.1007/BF03167846

    Article  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). https://doi.org/10.1088/0951-7715/19/4/011

    Article  MathSciNet  Google Scholar 

  29. Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958). https://doi.org/10.2307/2372841

    Article  MathSciNet  Google Scholar 

  30. Neumann, L., Schmeiser, C.: A kinetic reaction model: decay to equilibrium and macroscopic limit. Kinet. Relat. Models 9, 571 (2016). https://doi.org/10.3934/krm.2016007

    Article  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). https://doi.org/10.1007/978-1-4612-5561-1

  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). https://doi.org/10.14492/hokmj/1381757663

    Article  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). https://doi.org/10.1007/s00205-012-0508-5

    Article  MathSciNet  Google Scholar 

  34. Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202(950), iv+ 141 (2009). https://doi.org/10.1090/S0065-9266-09-00567-5

Download references

Acknowledgements

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.

Ⓒ 2020 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Author information

Authors and Affiliations

  1. Technische Universität Wien, Institut für Analysis und Scientific Computing, Wien, Austria

    Anton Arnold & Tobias Wöhrer

  2. CEREMADE (CNRS UMR n∘7534), PSL University, Université Paris-Dauphine, Place de Lattre de Tassigny, Paris, France

    Jean Dolbeault

  3. Fakultät für Mathematik, Universität Wien, Wien, Austria

    Christian Schmeiser

Authors
  1. Anton Arnold
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean Dolbeault
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian Schmeiser
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tobias Wöhrer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean Dolbeault .

Editor information

Editors and Affiliations

  1. De Vinci Pôle Universitaire Research Center, Courbevoie, France

    Francesco Salvarani

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

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. https://doi.org/10.1007/978-3-030-82946-9_1

Download citation

Publish with us

Policies and ethics