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Entropy production estimates for Boltzmann equations with physically realistic collision kernels

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Abstract

We establish strict entropy production bounds for the Boltzmann equation with the hard-sphere collision kernel. Using these entropy production bounds, we prove results asserting that the rate at which strongL 1 convergence to equilibrium occurs is uniform in wide classes of initial data. This extends our previous results in this direction, which applied only to a very special collision kernel. Moreover, the present results provide computable lower bounds; compactness arguments are entirely avoided. The uniformity is an important ingredient in our study of scaling limits of solutions of the non-spatially homogeneous Boltzmann equation, and is the main focus of this paper. However, the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous Boltzmann equation with initial data far from equilibrium to reach any given small strongL 1 neighborhood of equilibrium.

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Author notes

  1. M. C. Carvalho

    Present address: C.F.M.C. and Departamento de Matemática da Faculdade de Ciencias de Lisboa, 1700, Lisboa codex, Portugal

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, 30332, Atlanta, Georgia

    E. A. Carlen & M. C. Carvalho

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Carlen, E.A., Carvalho, M.C. Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J Stat Phys 74, 743–782 (1994). https://doi.org/10.1007/BF02188578

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