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Cauchy Problem and Exponential Stability for the Inhomogeneous Landau Equation

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An Erratum to this article was published on 25 November 2016

Abstract

This work deals with the inhomogeneous Landau equation on the torus in the cases of hard, Maxwellian and moderately soft potentials. We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct solutions in a close-to-equilibrium setting. Finally, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay.

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References

  1. Alexandre R., Villani C.: On the Landau approximation in plasma physics. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(1), 61–95 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  2. Baranger C., Mouhot C.: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Rev. Matem. Iberoam. 21, 819–841 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carrapatoso, K.: Exponential convergence to equilibrium for the homogeneous Landau equation with hard potentials. Bull. Sci. Math. (2014). doi:10.1016/j.bulsci.2014.12.002

  4. Carrapatoso, K.: On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials. J. Math. Pures Appl. (2015). doi:10.1016/j.matpur.2015.02.008

  5. Degond P., Lemou M.: Dispersion relations for the linearized Fokker-Planck equation. Arch. Ration. Mech. Anal. 138, 137–167 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Desvillettes L., Villani C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. DiPerna R., Lions P.-L.: On the Cauchy problem for the Boltzmann equation: global existence and weak stability. Ann. Math. 130, 312–366 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gualdani, M., Mischler, S., Mouhot, C.: Factorization for non-symmetric operators and exponential H-Theorem. arXiv:1006.5523

  9. Guo Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231, 391–434 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Hérau F.: Short and long time behavior of the Fokker-Planck equation in a confining potential and applications. J. Funct. Anal. 244(1), 95–118 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mischler, S., Mouhot, C.: Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation. Preprint. arXiv:1412.7487

  12. Mischler, S., Scher, J.: Spectral analysis of semigroups and growth-fragmentation equations. Preprint. arXiv:1310.7773

  13. Mouhot C.: Explicit coercivity estimates for the linearized boltzmann and landau operators. Commun. Partial Differ. Equ. 261, 1321–1348 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mouhot C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Commun. Math. Phys. 261, 629–672 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Mouhot C., Strain R.: Spectral gap and coercivity estimates for the linearized boltzmann collision operator without angular cutoff. J. Math. Pures Appl. 87, 515–535 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Strain R.M., Guo Y.: Almost exponential decay near Maxwellian. Commun. Partial Differ. Equ. 31(1–3), 417–429 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Strain R.M., Guo Y.: Exponential decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal. 187(2), 287–339 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tristani, I.: Fractional fokker-planck equation. Commun. Math. Sci. (2013, To appear in).

  20. Tristani I.: Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off. J. Stat. Phys. 157(3), 474–496 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Villani C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143(3), 273–307 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

  23. Wu, K.-C.: Exponential time decay estimates for the Landau equation on torus. arXiv:1301.0734

  24. Wu, K.-C.: Pointwise Description for the Linearized Fokker-Planck-Boltzmann Model. J. Stat. Phys. (2015). doi:10.1007/s10955-015-1206-0

  25. Yu H.: The exponential decay of global solutions to the generalized Landau equation near Maxwellians. Quart. Appl. Math. 64(1), 29–39 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. École Normale Supérieure de Cachan, CMLA (UMR 8536), 61 av. du président Wilson, 94235, Cachan, France

    Kleber Carrapatoso

  2. Université Paris Dauphine, Ceremade (UMR 7534), Place du Maréchal de Lattre de Tassigny, 75775, Paris, France

    Isabelle Tristani

  3. Department of Mathematics, National Cheng Kung University, Tainan, 70101, Taiwan

    Kung-Chien Wu

  4. National Center for Theoretical Sciences, National Taiwan University, 106, Taipei, Taiwan

    Kung-Chien Wu

Authors
  1. Kleber Carrapatoso
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  2. Isabelle Tristani
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  3. Kung-Chien Wu
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Corresponding author

Correspondence to Kleber Carrapatoso.

Additional information

Communicated by P.-L. Lions

An erratum to this article is available at http://dx.doi.org/10.1007/s00205-016-1064-1.

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Carrapatoso, K., Tristani, I. & Wu, KC. Cauchy Problem and Exponential Stability for the Inhomogeneous Landau Equation. Arch Rational Mech Anal 221, 363–418 (2016). https://doi.org/10.1007/s00205-015-0963-x

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  • DOI: https://doi.org/10.1007/s00205-015-0963-x

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