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arXiv · 2202.09074

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

Bernhard Kepka
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Abstract

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 $ \gamma>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.

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Bernhard Kepka. 2022-02-18. Longtime behavior for homoenergetic solutions in the collision dominated regime for hard potentials. https://doi.org/10.2140/paa.2024.6.415

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