Download this article
 Download this article For screen
For printing
Recent Issues
Volume 6, Issue 2
Volume 6, Issue 1
Volume 5, Issue 4
Volume 5, Issue 3
Volume 5, Issue 2
Volume 5, Issue 1
Volume 4, Issue 4
Volume 4, Issue 3
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 4
Volume 3, Issue 3
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 4
Volume 1, Issue 3
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN (electronic): 2578-5885
ISSN (print): 2578-5893
Author Index
To Appear
Other MSP Journals
Longtime behavior of homoenergetic solutions in the collision dominated regime for hard potentials

Bernhard Kepka

Vol. 6 (2024), No. 2, 415–454

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.

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