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Longtime behavior of homoenergetic solutions in the collision dominated regime for hard potentials

Bernhard Kepka

Vol. 6 (2024), No. 2, 415–454
Abstract

We consider a particular class of solutions to the Boltzmann equation which are referred to as homoenergetic solutions. They describe the dynamics of a dilute gas due to collisions and the action of either a shear, a dilation or a combination of both. More precisely, we study the case in which the shear is dominant compared with the dilation and the collision operator has homogeneity γ > 0. We prove that solutions with initially high temperature remain close and converge to a Maxwellian distribution with temperature going to infinity. Furthermore, we give precise asymptotic formulas for the temperature. The proof relies on an ansatz which is motivated by a Hilbert-type expansion. We consider both noncutoff and cutoff kernels.

Keywords
Boltzmann equation, Homoenergetic solutions, long-range interactions, nonequilibrium, hard potentials
Mathematical Subject Classification
Primary: 35C20, 35Q20, 82C40
Milestones
Received: 18 March 2023
Revised: 19 September 2023
Accepted: 14 November 2023
Published: 16 May 2024
Authors
Bernhard Kepka
Institute for Applied Mathematics
University of Bonn
Bonn
Germany