On Solutions of the Modified Boltzmann Equation

• Published in: Journal of statistical physics Vol. 190; no. 1
• Main Author: Bobylev, A. V.
• Format: Journal Article
• Language: English
• Published: New York Springer US 2023; Springer; Springer Nature B.V
• Subjects: Boltzmann transport equation; Mathematical analysis; Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Questions; Self-similarity; Statistical Physics and Dynamical Systems; Theoretical; Boltzmann equation; Shear flow; Homoenergetic solutions
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-022-03010-5

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
‖
A
‖
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
‖
A
‖
=
O
(
10
-
1
)
in appropriate dimensionless units.