Longtime behavior for homoenergetic solutions in the collision dominated regime for hard potentials
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866914430160207872 |
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| author | Kepka, Bernhard |
| author_facet | Kepka, Bernhard |
| contents | In this paper, 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. 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. This local stability result is a consequence of a dominant shear and the homogeneity $ γ>0 $ of the collision operator with respect to relative velocities. The proof relies on an ansatz which is motivated by a Hilbert-type expansion. We consider both non-cutoff and cutoff kernels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_09074 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Longtime behavior for homoenergetic solutions in the collision dominated regime for hard potentials Kepka, Bernhard Analysis of PDEs Mathematical Physics 35Q20 (Primary) 82C40, 35C20 (Secondary) In this paper, 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. 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. This local stability result is a consequence of a dominant shear and the homogeneity $ γ>0 $ of the collision operator with respect to relative velocities. The proof relies on an ansatz which is motivated by a Hilbert-type expansion. We consider both non-cutoff and cutoff kernels. |
| title | Longtime behavior for homoenergetic solutions in the collision dominated regime for hard potentials |
| topic | Analysis of PDEs Mathematical Physics 35Q20 (Primary) 82C40, 35C20 (Secondary) |
| url | https://arxiv.org/abs/2202.09074 |