Advertisement

Stability of Global Solution for the Relativistic Enskog Equation near Vacuum

  • Zhigang Wu
Article

Abstract

The Cauchy problem of the relativistic Enskog equation with near-vacuum data is considered in this paper. Under the same assumption as that in Jiang (J. Stat. Phys. 127:805–812, 2007) for the relativistic Enskog equation, we obtain the uniform L -stability of the solution. What’s more important, is that for two new types of the scattering cross section σ, we give the global existence and L 1(x,v)-stability for mild solution when the initial data lies in the space L 1(x,v). As a corollary, we have a BV-type estimate. It is worth mentioning that the stability results in this paper can be applied to the case in Jiang (J. Stat. Phys. 127:805–812, 2007).

Keywords

Relativistic Enskog equation Cauchy problem (Weighted) stability Vacuum 

References

  1. 1.
    Arkeryd, L.: On the Enskog equation in two space variables. Trans. Theor. Stat. Phys. 15, 673–691 (1986) MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Arkeryd, L.: On the Enskog equation with large initial data. SIAM J. Math. Anal. 21, 631–646 (1990) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bellomo, N., Lachowicz, M., Polewczak, J., Toscani, G.: Mathematical Topics in Nonlinear Kinetic Theory II. Series on Advances in Mathematics for Applied Sciences, vol. 1. World Scientific, Singapore (1991). ISBN 981-02-0447-7. MR1119780 (92m:82110) MATHGoogle Scholar
  4. 4.
    Boltzmann, L.: Weitere Studienüber das Wärmegleichgewicht unter Gasmolekülen. Sitzungsber. Akad. Wiss. Wien 66, 275–370 (1872) Google Scholar
  5. 5.
    Cercignani, C.: The Boltzmann Equation and its Applications. Springer, New York (1988) MATHGoogle Scholar
  6. 6.
    Cercignani, C., Kremer, G.: The Relativistic Boltzmann Equation, Theory and Applications. Birkhauser, Boston (2002) MATHGoogle Scholar
  7. 7.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer, New York (1994) MATHGoogle Scholar
  8. 8.
    Cercignani, C.: Small data existence for the Enskog equation in L 1. J. Stat. Phys. 51, 291–297 (1988) MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    de Groot, S.R., van Leeuwen, W.A., van Weert, C.G.: Relativistic Kinetic Theory. North-Holland, Amsterdam (1980) Google Scholar
  10. 10.
    DiPerna, R., Lions, P.L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130, 321–366 (1989) CrossRefMathSciNetGoogle Scholar
  11. 11.
    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, 1–28 (2006) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dudyński, M., Ekiel-Jezewska, M.: Global existence proof for relativistic Boltzmann equation. J. Stat. Phys. 66, 991–1001 (1992) MATHCrossRefADSGoogle Scholar
  13. 13.
    Enskog, D.: Kinetische Theorie. Kgl. Svenska Akad. Handl. 63(4) (1921) [English transl. in S. Brush, Kinetic Theory, Vol. 3 (Pergamon Press, New York, 1972)] Google Scholar
  14. 14.
    Esteban, M.J., Perthame, B.: On the modified Enskog equation for elastic and inelastic collisions, models with spin. Ann. Inst. H. Poincare Anal. Non Linaire 8, 289–308 (1991) MATHMathSciNetGoogle Scholar
  15. 15.
    Glassey, R.: The Cauchy Problem in Kinetic Theory. SIAM, Philadelphia (1996) MATHGoogle Scholar
  16. 16.
    Glassey, R.: Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data. Commun. Math. Phys. 26, 705–724 (2006) CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Glassey, R., Strauss, W.A.: Asymptotic stability of the relativistic Maxwellian. Publ. RIMS, Kyoto Univ. 29, 301–347 (1993) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Glassey, R., Strauss, W.A.: Asymptotic stability of the relativistic Maxwellian via fourteen moments. Trans. Theor. Stat. Phys. 24, 657–678 (1995) MATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Ha, S.Y.: L 1 stability of the Boltzmann equation for the hard sphere model. Arch. Ration. Mech. Anal. 173, 25–42 (2004) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Ha, S.Y.: Nonlinear functionals for the Boltzmann equation and uniform stability estimates. J. Differ. Equ. 215, 178–205 (2005) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ha, S.Y.: Lyapunov functionals for the Enskog-Boltzmann equation. Indiana Univ. Math. J. 54, 997–1014 (2005) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hamdache, K.: Quelques résultats pour l’équation de Boltzmann. C. R. Acad. Sci. Paris I 299, 431–434 (1984) MATHMathSciNetGoogle Scholar
  23. 23.
    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
  24. 24.
    Jiang, Z.L.: Global solution to the relativistic Enskog equation with near-vacuum data. J. Stat. Phys. 127, 805–812 (2007) MATHCrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Kaniel, S., Shinbrot, M.: The Boltzmann equation, uniqueness and local existence. Commun. Math. Phys. 58, 65–84 (1978) MATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Lachowicz, M.: On the local existence and uniqueness of solution of initial-value problem for the Enskog equation. Bull. Pol. Acad. Sci. 31, 89–96 (1983) MATHMathSciNetGoogle Scholar
  27. 27.
    Polewczak, J.: Classical solution of the nonlinear Boltzmann equation in all R 3: asymptotic behavior of solutions. J. Stat. Phys. 50, 611–632 (1988) MATHCrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Polewczak, J.: Global existence and asymptotic behavior for the nonlinear Enskog equation. SIAM J. Appl. Math. 49, 952–959 (1989) MATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Polewczak, J.: Global existence in L 1 for the modified nonlinear Enskog equation in R 3. J. Stat. Phys. 56, 159–173 (1989) MATHCrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Polewczak, J.: Global existence in L 1 for the generalized Enskog equation. J. Stat. Phys. 59, 461–500 (1989) CrossRefMathSciNetADSGoogle Scholar
  31. 31.
    Stewart, J.: Non-Equilibrium Relativistic Kinetic Theory. Lecture Notes in Physics, vol. 10. Springer, New York (1971) CrossRefGoogle Scholar
  32. 32.
    Toscani, G., Bellomo, N.: The Enskog-Boltzmann equation in the whole space R 3: Some global existence, uniqueness and stability results. Comput. Math. Appl. 13, 851–859 (1987) MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Ukai, S.: Solutions of the Boltzmann equation. Stud. Math. Appl. 18, 37–96 (1986) CrossRefMathSciNetGoogle Scholar
  34. 34.
    Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Fridlander, S., Serre, D. (eds.) Handbook of Fluid Mechanics (2002) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina

Personalised recommendations