Advertisement

Journal of Statistical Physics

, 137:1147 | Cite as

Distributional and Classical Solutions to the Cauchy Boltzmann Problem for Soft Potentials with Integrable Angular Cross Section

  • Ricardo J. Alonso
  • Irene M. Gamba
Article

Abstract

This paper focuses on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming S n−1 integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel–Shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes the large data near local Maxwellian regime with possibly infinite initial mass. We study the propagation of regularity using a recent estimate for the positive collision operator given in (Alonso et al. in Convolution inequalities for the Boltzmann collision operator. arXiv:0902.0507 [math.AP]) , by E. Carneiro and the authors, that allows us to show such propagation without additional conditions on the collision kernel. Finally, an L p -stability result (with 1≤p≤∞) is presented assuming the aforementioned condition.

Boltzmann equation for soft potentials Generalized and classical solutions Stability in Lp spaces 

References

  1. 1.
    Alexandre, R., Villani, C.: On the Boltzmann equation for long-range interactions. Commun. Pure Appl. Math. 55, 30–70 (2002) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alonso, R.: 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) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alonso, R., Carneiro, E.: Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform. Adv. Math. (to appear) Google Scholar
  4. 4.
    Alonso, R., Carneiro, E., Gamba, I.: Convolution inequalities for the Boltzmann collision operator. arXiv:0902.0507 [math.AP] (submitted for publication)
  5. 5.
    Bellomo, N., Toscani, G.: On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic behavior. J. Math. Phys. 26, 334–338 (1985) MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Boudin, L., Desvillettes, L.: On the singularities of the global small solutions of the full Boltzmann equation. Monatsh. Math. 131, 91–108 (2000) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Caflisch, R.: The Boltzmann equation with a soft potential (II). Commun. Math. Phys. 74, 97–109 (1980) MATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Appl. Math. Sci. Springer, Berlin (1994) MATHGoogle Scholar
  9. 9.
    Diperna, R., Lions, P.-L.: On the Cauchy problem for the Boltzmann equation. Ann. Math. 130, 321–366 (1989) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Glassey, R.: Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data. Commun. Math. Phys. 264, 705–724 (2006) MATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Goudon, T.: Generalized invariant sets for the Boltzmann equation. Math. Models Methods Appl. Sci. 7, 457–476 (1997) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169, 305–353 (2003) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Guo, Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153, 593–630 (2003) MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Ha, S.-Y.: Nonlinear functionals of the Boltzmann equation and uniform stability estimates. J. Differ. Equ. 215, 178–205 (2005) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ha, S.-Y., Yun, S.-B.: Uniform L 1-stability estimate of the Boltzmann equation near a local Maxwellian. Phys. Nonlinear Phenom. 220, 79–97 (2006) MATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Hamdache, K.: Existence in the large and asymptotic behavior for the Boltzmann equation. Jpn. J. Appl. Math. 2, 1–15 (1985) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hamdache, K.: Initial boundary value problems for Boltzmann equation. Global existence of week solutions. Arch. Ration. Mech. Anal. 119, 309–353 (1992) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Illner, R., Shinbrot, M.: The Boltzmann equation, global existence for a rare gas in an infinite vacuum. Commun. Math. Phys. 95, 217–226 (1984) MATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Kaniel, S., Shinbrot, M.: The Boltzmann equation I. Uniqueness and local existence. Commun. Math. Phys. 58, 65–84 (1978) MATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Mischler, S.: On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 210, 447–466 (2000) MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Mischler, S., Perthame, B.: Boltzmann equation with infinite energy: renormalized solutions and distributional solutions for small initial data and initial data close to Maxwellian. SIAM J. Math. Anal. 28, 1015–1027 (1997) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Palczewski, A., Toscani, G.: Global solution of the Boltzmann equation for rigid spheres and initial data close to a local Maxwellian. J. Math. Phys. 30, 2445–2450 (1989) MATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Toscani, G.: On the nonlinear Boltzmann equation in unbounded domains. Arch. Ration. Mech. Anal. 95, 37–49 (1986) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Toscani, G.: Global solution of the initial value problem for the Boltzmann equation near a local Maxwellian. Arch. Ration. Mech. Anal. 102, 231–241 (1988) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Ukai, S., Asano, K.: On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. Res. Inst. Math. Sci. 18, 477–519 (1982) (57–99) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    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) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Computational & Applied MathematicsRice UniversityHoustonUSA
  2. 2.Department of Mathematics & ICESUniversity of Texas at AustinAustinUSA

Personalised recommendations