Skip to main content
SpringerLink
Log in

On the Cauchy Problem for the Inelastic Boltzmann Equation with External Force

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In this paper, the Cauchy problem for the inelastic Boltzmann equation with external force is considered for near vacuum data. Under the assumptions on the bicharacteristic generated by external force which can be arbitrarily large, we prove the global existence of mild solution for initial data small enough with respect to the sup norm with exponential weight by using the contraction mapping theorem. Furthermore, we prove the uniform L 1 stability of the mild solution following from the exponential decay estimate and the Gronwall’s inequality for the case of soft potentials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alonso, R.J.: Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data. Indiana Univ. Math. J. 58, 999–1022 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alonso, R.J., Gamba, I.M.: Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section. J. Stat. Phys. 137(5–6), 1147–1165 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Arkeryd, L.: Stability in L 1 for the spatially homogeneous Boltzmann equation. Arch. Ration. Mech. Anal. 103, 151–168 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Asano, K.: Local solutions to the initial and initial boundary value problem for the Boltzmann equation with an external force. J. Math. Kyoto Univ. 24(2), 225–238 (1984)

    MathSciNet  MATH  Google Scholar 

  5. Bobylev, A.V., Carrillo, J.A., Gamba, I.M.: On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys. 98, 743–773 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bobylev, A.V., Cercignani, C.: Self-similar solutions of the Boltzmann equation and their applications. J. Stat. Phys. 106, 1039–1071 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bobylev, A.V., Cercignani, C.: Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions. J. Stat. Phys. 110, 333–375 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bobylev, A.V., Cercignani, C., Toscani, G.: Proof of an asymptotic property of selfsimilar solutions of the Boltzmann equation for granular materials. J. Stat. Phys. 111, 403–417 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bobylev, A.V., Cercignani, C., Gamba, I.M.: Generalized kinetic Maxwell type models of granular gases. http://arxiv.org/abs/math-ph/0901.3864v1 (2009)

  10. Bobylev, A.V., Gamba, I.M., Panferov, V.: Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions. J. Stat. Phys. 116, 1651–1682 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Bellomo, N., Toscani, G.: On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic behaviour. J. Math. Phys. 26, 334–338 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Bellomo, N., Lachowicz, M., Palczewski, A., Toscani, G.: On the initial value problem for the Boltzmann equation with a force term. Transp. Theory Stat. Phys. 18(1), 87–102 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Bellomo, N., Palczewski, A., Toscani, G.: Mathematical Topics in Nonlinear Kinetic Theory. World Scientific, Singapore (1988)

    MATH  Google Scholar 

  14. Bisi, M., Carrillo, J.A., Toscani, G.: Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model. J. Stat. Phys. 118, 301–331 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)

    Book  MATH  Google Scholar 

  16. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer Series in Applied Mathematical Sciences, vol. 106. Springer, Berlin (1994)

    MATH  Google Scholar 

  17. Cheng, C.H.: Uniform stability of solutions of Boltzmann equation for soft potential with external force. J. Math. Anal. Appl. 352, 724–732 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. DiPerna, R.J., Lions, P.L.: On the Cauchy problem for Boltzmann equations. Ann. Math. 130, 321–366 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Duan, R.J., Yang, T., Zhu, C.J.: Global existence to Boltzmann equation with external force in infinite vacuum. J. Math. Phys. 46, 053307 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  20. Duan, R.J., Yang, T., Zhu, C.J.: Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete Contin. Dyn. Syst. 16(1), 253–277 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Duan, R.J., Yang, T., Zhu, C.J.: L 1 and BV-type stability of the Boltzmann equation with external forces. J. Differ. Equ. 227(1), 1–28 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ernst, M.H., Trizac, E., Barrat, A.: The Boltzmann equation for driven systems of inelastic soft spheres. J. Stat. Phys. 124, 549–586 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Gamba, I.M., Panferov, V., Villani, C.: On the Boltzmann equation for diffusively excited granular media. Commun. Math. Phys. 246, 503–541 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Guo, Y.: The Vlasov-Poisson-Boltzmann system near vacuum. Commun. Math. Phys. 218(2), 293–313 (2001)

    Article  ADS  MATH  Google Scholar 

  25. Ha, S.Y.: L 1 stability of the Boltzmann equation for the hard sphere model. Arch. Ration. Mech. Anal. 171, 279–296 (2004)

    Article  Google Scholar 

  26. Ha, S.Y.: Nonlinear functionals of the Boltzmann equation and uniform stability estimates. J. Differ. Equ. 215, 178–205 (2005)

    Article  MATH  Google Scholar 

  27. Ha, S.Y.: L 1-stability of the Boltzmann equation for Maxwellian molecules. Nonlinearity 18, 981–1001 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Hamdache, K.: Existence in the large and asymptotic behaviour for the Boltzmann equation. Jpn. J. Appl. Math. 2, 1–15 (1984)

    Article  MathSciNet  Google Scholar 

  29. Illner, R., Shinbrot, M.: Global existence for a rare gas in an infinite vacuum. Commun. Math. Phys. 95, 217–226 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Kaniel, S., Shinbrot, M.: The Boltzmann equation: uniqueness and local existence. Commun. Math. Phys. 58, 65–84 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Mischler, S., Mouhot, C., Ricard, M.R.: Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem. J. Stat. Phys. 124, 655–702 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Polewczak, J.: Classical solution of the nonlinear Boltzmann equation in all R3: asymptotic behavior of solutions. J. Stat. Phys. 50, 611–632 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Toscani, G.: On the Boltzmann equation in unbounded domains. Arch. Ration. Mech. Anal. 95, 37–49 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ukai, S.: Solutions of Boltzmann equation. Patterns and waves-qualitative analysis of nonlinear differential equations. Stud. Appl. Math. 18, 37–96 (1986)

    Article  MathSciNet  Google Scholar 

  35. Villani, C.: Mathematics of granular materials. J. Stat. Phys. 124, 781–822 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Wennberg, B.: Stability and exponential convergence in L p for the spatially homogeneous Boltzmann equation. Nonlinear Anal., Theory Methods Appl. 20, 935–964 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wu, Z.G.: L 1 and BV-type stability of the inelastic Boltzmann equation near vacuum. Contin. Mech. Thermodyn. 22, 239–249 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. Yun, S.B.: L p stability estimate of the Boltzmann equation around a traveling local Maxwellian. J. Differ. Equ. 251, 45–57 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, P.R. China

    Jinbo Wei & Xianwen Zhang

Authors
  1. Jinbo Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xianwen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinbo Wei.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wei, J., Zhang, X. On the Cauchy Problem for the Inelastic Boltzmann Equation with External Force. J Stat Phys 146, 592–609 (2012). https://doi.org/10.1007/s10955-011-0410-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-011-0410-9

Keywords

Search

Navigation