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Large time asymptotics for Fermi–Dirac statistics coupled to a Poisson equation

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Abstract

This paper deals with the large time behaviour of the solutions of nonlinear Vlasov–Poisson–Fokker–Planck system in presence of an external potential of confinement. The statistics of collisions we are considering here, is the Fermi–Dirac operator with the Pauli exclusion term and without the detailed balance principle. A hypocoercivity method and a notion of scalar product adapted to the presence of a Poisson coupling are used to prove the exponential convergence of the solution of our system.

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References

  1. Addala, L., Dolbeault, J., Li, X., Lazhar Tayeb, M.: \(L^2\)-hypocoercivity and large time asymptotics of the linearized Vlasov–Poisson–Fokker–Planck system. J. Stat. Phys. 184(1), 4–34 (2021)

    Article  MATH  Google Scholar 

  2. Carrillo, J.A., Laurencot, P., Rosado, J.: Fermi–Dirac–Fokker-Planck equation: well-posedness and long-time asymptotics. J. Differ. Equ. 247(8), 2209–2234 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carrillo, J.A., Soler, J., Vázquez, J.L.: Asymptotic behaviour and self-similarity for the three-dimensional Vlasov–Poisson–Fokker–Planck system. J. Funct. Anal. 141, 99–132 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carrillo, J.A., Soler, J., Vasquez, J.L.: Asymptotic behavior and selfsimilarity for the three dimensional Vlasov–Poisson–Fokker–Planck system. J. Funct. Anal. 141, 99–132 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carrillo, J.A., Rosado, J., Salvarani, F.: 1D Nonlinear Fokker–Planck equations for Fermions and Bosons. Appl. Math. Lett. (to appear)

  6. Dolbeault, J.: Free energy and solutions of the Vlasov–Poisson–Fokker–Planck system: external potential and confinement (large time behavior and steady states). J. Math. Pure Appl. 9(78), 121–157 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for kinetic equations with linear relaxation terms. Comptes Rendus Mathématique 347, 511–516 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Eckmann, J.-P., Hairer, M.: Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235, 233–253 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to equilibrium for the Fokker–Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171, 151–218 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hérau, F., Thomann, L.: On global existence and trend to the equilibrium for the Vlasov–Poisson–Fokker–Planck system with exterior confining potential. J. Funct. Anal. 271, 1301–1340 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Herda, M., Rodrigues, L.M.: Large-time behavior of solutions to Vlasov–Poisson–Fokker–Planck equations: from evanescent collisions to diffusive limit. J. Stat. Phys. 170, 895–931 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  14. Il’in, A.M., Has’minskiĭ, R.Z.: On the equations of Brownian motion. Teor. Verojatnost. i Primenen 9, 466–491 (1964)

    MathSciNet  Google Scholar 

  15. Kolmogoroff, A.: Zufällige Bewegungen (zur Theorie der Brownschen Bewegung). Ann. Math. 2(35), 116–117 (1934)

    Article  MATH  Google Scholar 

  16. Li, H.-L., Matsumura, A.: Behaviour of the Fokker–Planck–Boltzmann equation near a Maxwellian. Arch. Ration. Mech. Anal. 189(1), 1–44 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, T.-P., Yang, T., Yu, S.-H.: Energy method for the Boltzmann equation. Physica D 188(3/4), 178–192 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Luo, L., Zhang, X.: Global classical solutions for quantum kinetic Fokker–Planck equations. Acta Math. Sci. 35B(1), 140–156 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Neumann, L., Sparber, C.: Stability of steady states in kinetic Fokker–Planck equations for Bosons and Fermions. Commun. Math. Sci. 5(4), 765–777 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Villani, C.: Hypocoercive diffusion operators. In: International Congress of Mathematicians, vol. III, pp. 473–498. Eur. Math. Soc., Zürich (2006)

  22. Villani, C.: Hypocoercivity, vol. 202, p. iv+141. Memoirs of the American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  23. Yang, T., Yu, H.-J.: Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space. J. Differ. Equ. 248, 1518–1560 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, T., Yu, H.-J.: Global classical solutions for the Vlasov–Maxwell–Fokker–Planck system. SIAM J. Math. Anal. 42(1), 459–488 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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  1. ISSAT Sousse: Institut Superieur des Sciences Appliquees et de Technologie de Sousse, Sousse, Tunisia

    Lanoir Addala

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  1. Lanoir Addala
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Addala, L. Large time asymptotics for Fermi–Dirac statistics coupled to a Poisson equation. SeMA 80, 381–391 (2023). https://doi.org/10.1007/s40324-022-00303-3

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  • DOI: https://doi.org/10.1007/s40324-022-00303-3

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