Archive for Rational Mechanics and Analysis

, Volume 225, Issue 1, pp 375–424 | Cite as

Global Well-Posedness of the Boltzmann Equation with Large Amplitude Initial Data

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Abstract

The global well-posedness of the Boltzmann equation with initial data of large amplitude has remained a long-standing open problem. In this paper, by developing a new \({L^\infty_xL^1_{v}\cap L^\infty_{x,v}}\) approach, we prove the global existence and uniqueness of mild solutions to the Boltzmann equation in the whole space or torus for a class of initial data with bounded velocity-weighted \({L^\infty}\) norm under some smallness condition on the \({L^1_xL^\infty_v}\) norm as well as defect mass, energy and entropy so that the initial data allow large amplitude oscillations. Both the hard and soft potentials with angular cut-off are considered, and the large time behavior of solutions in the \({L^\infty_{x,v}}\) norm with explicit rates of convergence are also studied.

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References

  1. 1.
    Baranger C., Mouhot C.: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Rev. Mat. Iberoamericana 21(3), 819–841 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bellomo N., Palczewski A., Toscani G.: Mathematical Topics in Nonlinear Kinetic Theory. World Scientific Publishing, Singapore (1988)MATHGoogle Scholar
  3. 3.
    Briant M., Guo Y.: Asymptotic stability of the Boltzmann equation with Maxwell boundary conditions. J. Differ. Equ. 261(12), 7000–7079 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Carleman T.: Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60(1), 91–146 (1933)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cercignani C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute Gases. Springer, New York (1994)CrossRefMATHGoogle Scholar
  6. 6.
    Duan R.J., Yang T., Zhao H.J.: The Vlasov–Poisson–Boltzmann system for soft potentials. Math. Models Methods Appl. Sci. 23(6), 979–1028 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    DiPerna R.J., Lions P.-L.: On the Cauchy problem for Boltzmann equation: global existence and weak stability. Ann. Math. 130, 321–366 (1989)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Desvillettes L., Villani C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems:The Boltzmann equation. Invent. Math. 159, 243–316 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ellis R., Pinsky M.A.: The first and second fluid approximations to the linearized Boltzmann equation. J. Math. Pures Appl. 54(9), 125–156 (1975)MathSciNetMATHGoogle Scholar
  10. 10.
    Glassey R.T.: The Cauchy Problem in Kinetic Theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1996)CrossRefMATHGoogle Scholar
  11. 11.
    Grad, H.: Asymptotic theory of the Boltzmann equation. In: Laurmann, J.A. (ed.) Rarefied Gas Dynamics, vol. 1, pp. 26–59. Academic Press, New York, 1963Google Scholar
  12. 12.
    Gualdani, M.P., Mischler, S., Mouhot, C.: Factorization for Non-symmetric Operators and Exponential H-Theorem. arXiv:1006.5523
  13. 13.
    Guo Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169(4), 305–353 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Guo Y.: Decay and continuity of the Boltzmann equation in Bounded domains. Arch. Rational. Mech. Anal. 197, 713–809 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Guo Y.: Bounded solutions for the Boltzmann equation. Q. Appl. Math. 68(1), 143–148 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Huang, F.M., Wang, Y.: Macroscopic Regularity for the Boltzmann Equation. arXiv:1512.08608
  17. 17.
    Illner R., Shinbrot M.: Global existence for a rare gas in an infinite vacuum. Comm. Math. Phys. 95, 217–226 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kaniel S., Shinbrot M.: The Boltzmann equation I: uniqueness and local existence. Comm. Math. Phys. 58, 65–84 (1978)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kim C.: Boltzmann equation with a large potential in a periodic box. Comm. Partial Differ. Equ. 39, 1393–1423 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Liu T., Yang T., Yu S.H.: Energy method for the Boltzmann equation. Phys. D 188, 178–192 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lu X.-G., Mouhot C.: On measure solutions of the Boltzmann equation, part II: rate of convergence to equilibrium. J. Differ. Equ. 258(11), 3742–3810 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Strain R.M.: Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinet. Rel. Models 5, 583–613 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Strain R.M., Guo Y.: Exponential decay for soft potentials near Maxwellian. Arch. Rational. Mech. Anal. 187, 287–339 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ukai S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Jpn. Acad. 50, 179–184 (1974)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ukai S., Yang T.: The Boltzmann equation in the space \({L^2\cap L^\infty_\beta}\): global and time-periodic solutions. Anal. Appl. 4, 263–310 (2006)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Vidav I.: Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl. 30, 264–279 (1970)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Villani, C.: A Review of Mathematical Topics in Collisional Kinetic Theory. Handbook of Mathematical Fluid Dynamics, vol. I, pp. 71–305. North-Holland, Amsterdam, 2002Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Renjun Duan
    • 1
  • Feimin Huang
    • 2
    • 3
  • Yong Wang
    • 3
  • Tong Yang
    • 4
  1. 1.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.Institute of Applied MathematicsAMSS, CASBeijingPeople’s Republic of China
  4. 4.Department of MathematicsCity University of Hong KongKowloonHong Kong

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