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Non-local hydrodynamics as a slow manifold for the one-dimensional kinetic equation

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Abstract

We prove an explicit, non-local hydrodynamic closure for the linear one-dimensional kinetic equation independent on the size of the relaxation time. We compare this dynamical equation to the local approximations obtained from the Chapman–Enskog expansion for small relaxation times. Our results rely on the spectral theory of Jacobi operators with rank-one perturbations.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55. US Government printing office, Washington (1948)

    MATH  Google Scholar 

  2. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94(3), 511 (1954)

  3. Bobylev, A.: The Chapman-Enskog and Grad methods for solving the Boltzmann equation. Akademiia Nauk SSSR Doklady 262, 71–75 (1982)

    ADS  MathSciNet  Google Scholar 

  4. Burnett, D.: The Distribution of Molecular Velocities and the Mean Motion in a Non-Uniform Gas. Proc. Lond. Math. Soc. 40(1), 382–435 (1936)

    Article  MathSciNet  Google Scholar 

  5. Colosqui, C.E.: High-order hydrodynamics via lattice Boltzmann methods. Phys. Rev. E 81(2), 026702 (2010)

    Article  ADS  Google Scholar 

  6. Ellis, R.S., Pinsky, M.A.: The first and second fluid approximations to the linearized Boltzmann equation. J. Math. Pures Appl. 54(9), 125–156 (1975)

    MathSciNet  MATH  Google Scholar 

  7. Gorban, A., Karlin, I.: Hilbert’s 6th problem: exact and approximate hydrodynamic manifolds for kinetic equations. Bull. Am. Math. Soc. 51(2), 187–246 (2014)

    Article  MathSciNet  Google Scholar 

  8. Gorban, A.N., Karlin, I.V.: Short-wave limit of hydrodynamics: a soluble example. Phys. Rev. Lett. 77(2), 282 (1996)

    Article  ADS  Google Scholar 

  9. Gorban, A.N., Karlin, I.V.: Invariant Manifolds for Physical and Chemical Kinetics, vol. 660. Springer, Berlin (2005)

    MATH  Google Scholar 

  10. Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331–407 (1949)

    Article  MathSciNet  Google Scholar 

  11. Hislop, P., Sigal, I.: Introduction to Spectral Theory: With Applications to Schrödinger Operators. Applied Mathematical Sciences. Springer, New York (2012)

    MATH  Google Scholar 

  12. Karlin, I., Chikatamarla, S., Kooshkbaghi, M.: Non-perturbative hydrodynamic limits: a case study. Phys. A Stat. Mech. Appl. 403, 189–194 (2014)

    Article  MathSciNet  Google Scholar 

  13. Kogelbauer, F.: Hydrodynamic manifolds for the one-dimensional nonlinear BGK system. (to appear)

  14. Kogelbauer, F.: Slow hydrodynamic manifolds for the three-component linearized Grad system. Continuum Mech. Thermodyn. 32, 1141–1146 (2020). https://doi.org/10.1007/s00161-019-00819-6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Kuramoto, Y.: Diffusion-induced chaos in reaction systems. Prog. Theor. Phys. Suppl. 64, 346–367 (1978)

    Article  ADS  Google Scholar 

  16. Kurasov, P., Kuroda, S.-T.: Krein’s resolvent formula and perturbation theory. J. Oper. Theory 1, 321–334 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Perthame, B.: Global existence to the BGK model of Boltzmann equation. J. Differ. Eq. 82(1), 191–205 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  18. Perthame, B., Pulvirenti, M.: Weighted \({L}^{\infty }\) bounds and uniqueness for the Boltzmann BGK model. Archive Ration. Mech. Anal. 125(3), 289–295 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  19. Saint-Raymond, L.: Some recent results about the sixth problem of Hilbert. In: Calgaro, C., Coulombel, J.-F., Goudon, T. (eds.) Analysis and Simulation of Fluid Dynamics, pp. 183–199. Birkhäuser Basel, Basel (2007)

    MATH  Google Scholar 

  20. Saint-Raymond, L.: A mathematical PDE perspective on the Chapman-Enskog expansion. Bull. Am. Math. Soc. 51(2), 247–275 (2014)

    Article  MathSciNet  Google Scholar 

  21. Sivashinsky, G.: Nonlinear analysis of hydrodynamic instability in laminar flames-I. derivation of basic equations. AcAau 4(11), 1177–1206 (1977)

  22. Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices, vol. 72. American Mathematical Soc, Providence (2000)

    MATH  Google Scholar 

  23. Yun, S.-B.: Cauchy problem for the Boltzmann-BGK model near a global Maxwellian. J. Math. Phys. 51(12), 123514 (2010)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank Gerald Teschl for very useful discussions on Jacobi operators and the spectral theory of rank-one perturbations. The author would like to thank Ilya Karlin for pointing out this direction of research and several useful discussions and comments in the context of statistical physics and the CE series. Furthermore, the author would like to thank the anonymous reviewers for several useful comments and suggestions.

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  1. Institute for Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Florian Kogelbauer

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  1. Florian Kogelbauer
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Correspondence to Florian Kogelbauer.

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Communicated by Andreas Öchsne.

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Kogelbauer, F. Non-local hydrodynamics as a slow manifold for the one-dimensional kinetic equation. Continuum Mech. Thermodyn. 33, 431–444 (2021). https://doi.org/10.1007/s00161-020-00913-0

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