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Journal of Statistical Physics

, Volume 165, Issue 5, pp 866–906 | Cite as

Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff

  • Yoshinori Morimoto
  • Shuaikun Wang
  • Tong Yang
Article

Abstract

A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.

Keywords

Boltzmann equation Homogenuous Measure valued solutions Characteristic functions 

Mathematics Subject Classification

Primary 35Q20 76P05 Secondary 35H20 82B40 82C40 

Notes

Acknowledgements

Authors would like to express their hearty gratitude to anonymous referees for many suggestions and advises which improved the submitted manuscript. The research of the first author was supported in part by Grant-in-Aid for Scientific Research No.25400160, Japan Society for the Promotion of Science. The research of the third author was supported in part by the General Research Fund of Hong Kong, CityU No. 11303614.

References

  1. 1.
    Alexandre, R., Desvillettes, L., Villani, C., Wennberg, B.: Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152, 327–355 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Boltzmann equation without angular cutoff in the whole space: qualitative properties of solutions. Arch. Ration. Mech. Anal. 202, 599–661 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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. J. Funct. Anal. 262, 915–1010 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff. Kyoto J. Math. 52, 433–463 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bobylev, A.V.: The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules. Dokl. Akad. Nauk SSSR 225(6), 1041–1044 (1975)ADSMathSciNetGoogle Scholar
  6. 6.
    Bobylev, A.V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Math. Phys. Rev. 7, 111–233 (1988)MathSciNetMATHGoogle Scholar
  7. 7.
    Cannone, M., Karch, G.: Infinite energy solutions to the homogeneous Boltzmann equation. Commun. Pure Appl. Math. 63, 747–778 (2010)MathSciNetMATHGoogle Scholar
  8. 8.
    Carlen, E.A., Gabetta, E., Toscani, G.: Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Commun. Math. Phys. 199, 521–546 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fournier, N.: Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition. Ann. Appl. Probab. 25, 860–897 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fournier, N., Guérin, H.: On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity. J. Stat. Phys. 131, 749–781 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gabetta, E., Toscani, G., Wennberg, B.: Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys. 81, 901–934 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Huo, Z.H., Morimoto, Y., Ukai, S., Yang, T.: Regularity of solutions for spatially homogeneous Boltzmann equation without Angular cutoff. Kinet. Relat. Models 1, 453–489 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jacob, N.: Pseudo-Differential Operators and Markov Processes. Fourier Analysis and Semigroups, vol. 1. Imperial College Press, London (2001)Google Scholar
  15. 15.
    Lu, X., Mouhot, C.: On measure solutions of the Boltzmann equation, part I: moment production and stability estimates. J. Differ. Equ. 252, 3305–3363 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lu, X., Wennberg, B.: Solutions with increasing energy for the spatially homogeneous Boltzmann equation. Nonlinear Anal. Real World Appl. 3, 243–258 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 467–501 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Morimoto, Y.: A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules. Kinet. Relat. Models 5, 551–561 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Discret. Contin. Dyn. Syst. Ser. A 24, 187–212 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Morimoto, Y., Wang, S., Yang, T.: A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation. J. Math. Pures Appl. 103, 809–829 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Morimoto, Y., Wang, S., Yang, T.: Moment classification of infinite energy solutions to the homogeneous Boltzmann equation. Anal. Appl. (2015). doi: 10.1142/S0219530515500232, arXiv:1506.06493
  22. 22.
    Morimoto, Y., Yang, T.: Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 429–442 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Pulvirenti, A., Toscani, G.: The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation. Ann. Math. Pura Appl. 171, 181–204 (1996)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Pulvirenti, A., Wennberg, B.: Lower bounds for the solutions to the Kac and the Boltzmann equation. In: Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), pp. 437-446 (1996)Google Scholar
  25. 25.
    Pulvirenti, A., Wennberg, B.: A Maxwellian lower bound for solutions to the Boltzmann equation. Commun. Math. Phys. 183, 145–160 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Toscani, G., Villani, C.: Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas. J. Stat. Phys. 94, 619–637 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    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)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of Fluid Mathematical Fluid Dynamics. Elsevier Science, New York (2002)Google Scholar
  29. 29.
    C. Villani, Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, (2003)Google Scholar
  30. 30.
    Zhang, X., Zhang, X.: Probability approaches to spatially homogeneous Boltzmann equations. Stoch. Anal. Appl. 25(6), 1129–1150 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Graduate School of Human and Environmental StudiesKyoto UniversityKyotoJapan
  2. 2.Department of MathematicsCity University of Hong KongHong KongPeople’s Republic of China
  3. 3.Department of MathematicsJinan UniversityGuangzhouPeople’s Republic of China

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