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Well-Posedness of Spatially Homogeneous Boltzmann Equation with Full-Range Interaction

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Abstract

In the present work, we give the coercivity estimates for the Boltzmann collision operator Q(·, ·) without angular cut-off to clarify in which case the functional \({\langle -Q(g, f), f\rangle}\) will become the truly sub-elliptic. Based on this observation and commutator estimates in Alexandre et al. (Arch Rat Mech Anal 198:39–123, 2010), the upper bound estimates for the collision operator in Chen and He (Arch Rat Mech Anal 201(2):501–548, 2011) and the stability results in Desvillettes and Mouhot (Arch Rat Mech Anal 193(2):227–253, 2009), in the function space \({L^1_q\cap H^N}\) , we establish global well-posedness or local well-posedness for the spatially homogeneous Boltzmann equation with full-range interaction(covering most of physical collision kernels).

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References

  1. Alexandre R., Desvillettes L., Villani C., Wennberg B.: Entropy dissipation and long- range interactions. Arch. Rat. Mech. Anal. 152(4), 327–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexandre R., Morimoto Y., Ukai S., Xu C.-J., Yang T.: Regularizing effect and local existence for non-cutoff Boltzmann equation. Arch. Rat. Mech. Anal. 198, 39–123 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alexandre R., Morimoto Y., Ukai S., Xu C.-J., Yang T.: Global existence and full regularity of the Boltzmann equation without angular cutoff. Commun. Math. Phys. 304(2), 513–581 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: The Boltzmann equation without angular cutoff in the whole space I: Global existence for soft potential, to appear in J. Funct. Anal.

  5. Alexandre R., Morimoto Y., Ukai S., Xu C.-J., Yang T.: The Boltzmann equation without angular cutoff in the whole space: II, Global existence for hard potential. Anal. Appl. (Singap.) 9(2), 113–134 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

  7. 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 

  8. Arkeryd L.: On the Boltzmann equation. Arch. Rat. Mech. Anal. 45, 1–34 (1972)

    MathSciNet  MATH  Google Scholar 

  9. Chen Y., He L.: Smoothing estimates for Boltzmann equation with full-range interactions I: spatially homogeneous case. Arch. Rat. Mech. Anal. 201(2), 501–548 (2011)

    Article  MathSciNet  Google Scholar 

  10. Cianchi, A.: Some results in the theory of Orlicz spaces and application to variational problems. In: Nonlinear Analysis, Function Spaces and Applications, Proceedings of the Spring School held in Prague, May 31-June 6, 1998. Vol. 6, Praha: Czech Academy of Sciences, Mathematical Institute, 1999, pp. 50–92

  11. Desvillettes L., Wennberg B.: Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Comm. Part. Diff. Eqs. 29(1-2), 133–155 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Desvillettes L., Mouhot C.: Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions. Arch. Rat. Mech. Anal. 193(2), 227–253 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fournier N.: Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff. J. Stat. Phys. 125, 927–946 (2006)

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Fournier N., Guerin H.: On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity. J. Stat. Phys 131, 781–949 (2008)

    Article  MathSciNet  Google Scholar 

  16. Gressman P.T., Strain R.M.: Global Classical Solutions of the Boltzmann Equation without Angular Cut-off. J. Amer. Math. Soc. 24(3), 771–847 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gressman P.T., Strain R.M.: Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production. Adv. Math. 227(6), 2349–2384 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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 

  19. Lu X., Mouhot C.: On Measure Solutions of the Boltzmann Equation, part I: Moment Production and Stability Estimates. J. Diff. Eqs. 252(4), 3305–3363 (2012)

    Article  MATH  Google Scholar 

  20. Mischler S., Wennberg B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. PoincaréAnal. Non Linéaire 16, 467–501 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Mouhot C., Strain R.M.: Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J. Math. Pures Appl. (9) 87(5), 515–535 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Toscani G., Villani C.: Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys. 94, 619–637 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of mathematical fluid dynamics, vol. I, Amsterdam: North Holland, 2002, pp. 71–305

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  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P. R. China

    Lingbing He

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  1. Lingbing He
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Correspondence to Lingbing He.

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Communicated by P. Constantin

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He, L. Well-Posedness of Spatially Homogeneous Boltzmann Equation with Full-Range Interaction. Commun. Math. Phys. 312, 447–476 (2012). https://doi.org/10.1007/s00220-012-1481-4

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  • DOI: https://doi.org/10.1007/s00220-012-1481-4

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