Communications in Mathematical Physics

, Volume 74, Issue 1, pp 71–95 | Cite as

The Boltzmann equation with a soft potential

I. Linear, spatially-homogeneous
  • Russel E. Caflisch


The initial value problem for the linearized spatially-homogeneous Boltzmann equation has the form ∂f/∂t+Lf=0 withf(ξ,t=0) given. The linear operatorL operates only on the ξ variable and is non-negative, but, for the soft potentials considered here, its continuous spectrum extends to the origin. Thus one cannot expect exponential decay forf, but in this paper it is shown thatf decays likee−λtβ with β<1. This result will be used in Part II to show existence of solutions of the initial value problem for the full nonlinear, spatially dependent problem for initial data that is close to equilibrium.


Neural Network Statistical Physic Complex System Initial Data Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Russel E. Caflisch
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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