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On Solutions of the Modified Boltzmann Equation

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Abstract

We study the self-similar solutions and related questions for the modified Boltzmann equation. This equation formally coincides with the classical spatially homogeneous Boltzmann equation in the presence of an artificial force term proportional to a matrix A. The modification is connected with applications to homoenergetic solutions of the spatially inhomogeneous Boltzmann equation. Our study is restricted to the case of Maxwell-type interactions. We investigate existence and uniqueness of self-similar solutions and their role as attractors for large values of time. Similar questions were studied recently under assumption of sufficient smallness of norm \( \Vert A \Vert \) without explicit estimates of that smallness. In this paper we fill this gap and prove, in particular, that all important facts related to self-similar solutions remain valid for moderately small values \( \Vert A \Vert = O (10^{-1})\) in appropriate dimensionless units.

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Acknowledgements

The author thanks Alessia Nota and Juan Velazquez for valuable discussions.

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  1. Keldysh Institute of Applied Mathematics RAS, Moscow, Russia

    A. V. Bobylev

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  1. A. V. Bobylev
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Correspondence to A. V. Bobylev.

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Communicated by Isabelle Gallagher.

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Bobylev, A.V. On Solutions of the Modified Boltzmann Equation. J Stat Phys 190, 24 (2023). https://doi.org/10.1007/s10955-022-03010-5

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