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The Boltzmann equation with a soft potential

I. Linear, spatially-homogeneous

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Abstract

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 −λ t β 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.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, 94305, Stanford, CA, USA

    Russel E. Caflisch

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  1. Russel E. Caflisch
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Communicated by J.L. Lebowitz

Supported by the National Science Foundation under Grant MCS78-09525 and MCS76-07039 and by the United States Army under Contract DAAG29-75-C-0024

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Caflisch, R.E. The Boltzmann equation with a soft potential. Commun.Math. Phys. 74, 71–95 (1980). https://doi.org/10.1007/BF01197579

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  • DOI: https://doi.org/10.1007/BF01197579

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