Journal of Statistical Physics

, Volume 123, Issue 4, pp 753–762 | Cite as

Weak Solutions of the Boltzmann Equation Without Angle Cutoff

  • Carlo Cercignani


The definition of the concept of weak solution of the nonlinear Boltzmann equation, recently introduced by the author, is used to prove that, without any cutoff in the collision kernel, the Boltzmann equation for Maxwell molecules in the one-dimensional case has a global weak solution in this sense. Global conservation of energy follows.


Boltzmann equation Energy conservation Global solution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Bony, Existence globale et diffusion en théorie cinétique discrète. In R. Gatignol and Soubbarameyer, editors, Advances in Kinetic Theory and Continuum Mechanics, pp. 81–90 (Springer-Verlag, Berlin, 1991).Google Scholar
  2. 2.
    C. Cercignani, Weak solutions of the Boltzmann equation and energy conservation. Appl. Math. Lett. 8: 53 (1995).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    C. Cercignani, Theory and Application of the Boltzmann equation (Springer Verlag, New York, 1988).Google Scholar
  4. 4.
    C. Cercignani, Global weak solutions of the Boltzmann Equation. J. Stat. Phys. 118: 333 (2005).MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    C. Cercignani, Estimating the solutions of the Boltzmann equation. Submitted to J. Stat. Phys. (2005).Google Scholar
  6. 6.
    C. Cercignani and R. Illner, Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions. Arch. Rational Mech. Anal. 134: 1 (1996).MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases (Springer Verlag, New York, 1994).MATHGoogle Scholar
  8. 8.
    R. DiPerna and P.L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. 130: 321 (1989).CrossRefMathSciNetGoogle Scholar
  9. 9.
    E. Ikenberry and C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell’s kinetic theory, I. J. Rat. Mech. Anal. 5: 1–54 (1956).MathSciNetGoogle Scholar
  10. 10.
    J. C. Maxwell, On the dynamical theory of gases. Phil. Trans. Roy Soc. (London) 157: 49–88 (1866).Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Carlo Cercignani
    • 1
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

Personalised recommendations