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Regularization properties of the 2D homogeneous Boltzmann equation without cutoff
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  • Published: 13 July 2010

Regularization properties of the 2D homogeneous Boltzmann equation without cutoff

  • Vlad Bally1 &
  • Nicolas Fournier2 

Probability Theory and Related Fields volume 151, pages 659–704 (2011)Cite this article

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Abstract

We consider the 2-dimensional spatially homogeneous Boltzmann equation for hard potentials. We assume that the initial condition is a probability measure that has some exponential moments and is not a Dirac mass. We prove some regularization properties: for a class of very hard potentials, the solution instantaneously belongs to H r, for some \({r\in (-1,2)}\) depending on the parameters of the equation. Our proof relies on the use of a well-suited Malliavin calculus for jump processes.

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References

  1. Alexandre R.: A review of Botzmann equation with singular kernels. Kinet. Relat. Models 2(4), 551–646 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexandre R., Desvillettes L., Villani C., Wennberg B.: Entropy dissipation and long-range interactions. Arch. Rat. Mech. Anal. 152, 327–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alexandre R., El Safadi M.: Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff case and Maxwellian molecules. Math. Models Methods Appl. Sci. 15(6), 907–920 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alexandre R., El Safadi M.: Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. II. Non cutoff case and non Maxwellian molecules. Discrete Contin. Dyn. Syst. 24(1), 1–11 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bally, V., Clément, E.: Integration by parts formula and applications to equations with jumps. Preprint, arXiv:0911.3017v1

  6. Bhatt A., Karandikar R.: Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21(4), 2246–2268 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bichteler K., Gravereaux J.B., Jacod J.: Malliavin calculus for processes with jumps. Stochastics Monographs, vol 2. Gordon and Breach Science Publishers, New York (1987)

    Google Scholar 

  8. Bobylev A.V.: Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Stat. Phys. 88(5–6), 1183–1214 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Cercignani, C.: The Boltzmann equation and its applications. In: Applied Mathematical Sciences, vol. 67, xii+455 pp. Springer, New York (1988)

  10. Desvillettes L.: About the regularizing properties of the non cut-off Kac equation. Comm. Math. Phys. 168(2), 417–440 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Desvillettes L.: Regularization properties of the 2-dimensional non-radially symmetric non- cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules. Transp. Theory Stat. Phys. 26(3), 341–357 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Desvillettes L., Villani C.: On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness. Comm. Partial Differ. Equ. 25(1–2), 179–259 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Desvillettes L., Wennberg B.: Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Comm. Partial Differ. Equ. 29(1), 133–155 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fournier N.: Existence and regularity study for 2D Boltzmann equation without cutoff by a probabilistic approach. Ann. Appl. Probab. 10, 434–462 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fournier N.: A new regularization possibility for the Boltzmann equation with soft potentials. Kinet. Relat. Models 1, 405–414 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fournier N., Guérin H.: On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity. J. Stat. Phys. 131(4), 749–781 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fournier N., Mouhot C.: On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity. Comm. Math. Phys. 289, 803–824 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Graham C., Méléard S.: Existence and regularity of a solution of a Kac equation without cutoff using the stochastic calculus of variations. Comm. Math. Phys. 205(3), 551–569 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huo Z., Morimoto Y., Ukai S., Yang T.: Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff. Kinet. Relat. Models 1(3), 453–489 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mouhot C., Villani C.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rational Mech. Anal. 173, 169–212 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. und Verw. Gebiete 46, 67–105 (1978/79)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Villani C.: Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off. Rev. Matem. Iberoam. 15, 335–352 (1999)

    MathSciNet  MATH  Google Scholar 

  24. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of Mathematical Fluid Dynamics, vol. I, pp. 71–305. North-Holland, Amsterdam (2002)

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Authors and Affiliations

  1. LAMA UMR 8050, Université Paris Est, Cité Descartes, 5 boulevard Descartes, Champs sur Marne, 77454, Marne la Vallée Cedex, France

    Vlad Bally

  2. LAMA UMR 8050, Faculté de Sciences et Technologies, Université Paris Est, 61, avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Nicolas Fournier

Authors
  1. Vlad Bally
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  2. Nicolas Fournier
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Corresponding author

Correspondence to Vlad Bally.

Additional information

The second author of this work was supported by a grant of the Agence Nationale de la Recherche numbered ANR-08-BLAN-0220-01.

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Cite this article

Bally, V., Fournier, N. Regularization properties of the 2D homogeneous Boltzmann equation without cutoff. Probab. Theory Relat. Fields 151, 659–704 (2011). https://doi.org/10.1007/s00440-010-0311-x

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  • Received: 13 November 2009

  • Revised: 26 April 2010

  • Published: 13 July 2010

  • Issue Date: December 2011

  • DOI: https://doi.org/10.1007/s00440-010-0311-x

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Keywords

  • Kinetic equations
  • Hard potentials without cutoff
  • Malliavin calculus
  • Jump processes

Mathematics Subject Classification (2000)

  • 60H07
  • 82C40
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