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Communications in Mathematical Physics

, Volume 199, Issue 3, pp 521–546 | Cite as

Propagation of Smoothness and the Rate of Exponential Convergence to Equilibrium for a Spatially Homogeneous Maxwellian Gas

  • E. A. Carlen
  • E. Gabetta
  • G. Toscani

Abstract:

We prove an inequality for the gain term in the Boltzmann equation for Maxwellian molecules that implies a uniform bound on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. We then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm. These results are then combined, using interpolation inequalities, to obtain the optimal rate of exponential convergence in the strong L 1 norm, as well as various Sobolev norms. These results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non-linear Boltzmann equation, even for initial data that is far from equilibrium.

Keywords

Initial Data Sobolev Space Boltzmann Equation Optimal Rate Collision Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • E. A. Carlen
    • 1
  • E. Gabetta
    • 2
  • G. Toscani
    • 2
  1. 1.School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USAUS
  2. 2.Dipartimento di Matematica – Università di Pavia, via Abbiategrasso 215, 27100 Pavia, ItalyIT

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