Homoenergetic solutions of the Boltzmann equation: the case of simple-shear deformations

Alessia Nota; Juan J. L. Velázquez; Alessia Nota; Juan J. L. Velázquez

doi:10.3934/mine.2023019

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-25. doi: 10.3934/mine.2023019

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Homoenergetic solutions of the Boltzmann equation: the case of simple-shear deformations

Alessia Nota ^{1
,
,},
Juan J. L. Velázquez ²

1.
Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, 67100 L'Aquila, Italy
2.
Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Received: 20 December 2021 Revised: 05 February 2022 Accepted: 05 February 2022 Published: 22 March 2022

In these notes we review some recent results on the homoenergetic solutions for the Boltzmann equation obtained in ^[4,20,21,22]. These solutions are a particular class of non-equilibrium solutions of the Boltzmann equation which are useful to describe the dynamics of Boltzmann gases under shear, expansion or compression. Therefore, they do not behave asymptotically for long times as Maxwellian distributions, at least for all the choices of the collision kernels, and their behavior strongly depends on the homogeneity of the collision kernel and on the particular form of the hyperbolic terms which describe the deformation taking plance in the gas. We consider here the case of simple shear deformation and present different possible long-time asymptotics of these solutions. We discuss the current knowledge about the long-time behaviour of the homoenergetic solutions as well as some conjectures and open problems.
- Boltzmann equation,
- homoenergetic solutions,
- simple shear deformations,
- non-equilibrium,
- self-similar profiles,
- long-time asymptotics
Citation: Alessia Nota, Juan J. L. Velázquez. Homoenergetic solutions of the Boltzmann equation: the case of simple-shear deformations[J]. Mathematics in Engineering, 2023, 5(1): 1-25. doi: 10.3934/mine.2023019

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Abstract

In these notes we review some recent results on the homoenergetic solutions for the Boltzmann equation obtained in ^[4,20,21,22]. These solutions are a particular class of non-equilibrium solutions of the Boltzmann equation which are useful to describe the dynamics of Boltzmann gases under shear, expansion or compression. Therefore, they do not behave asymptotically for long times as Maxwellian distributions, at least for all the choices of the collision kernels, and their behavior strongly depends on the homogeneity of the collision kernel and on the particular form of the hyperbolic terms which describe the deformation taking plance in the gas. We consider here the case of simple shear deformation and present different possible long-time asymptotics of these solutions. We discuss the current knowledge about the long-time behaviour of the homoenergetic solutions as well as some conjectures and open problems.

References

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