Ill-posedness of the Boltzmann-BGK model in the exponential class

Donghyun Lee; Sungbin Park; Seok-Bae Yun

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arXiv · 2408.08163

Ill-posedness of the Boltzmann-BGK model in the exponential class

Donghyun Lee·Sungbin Park·Seok-Bae Yun

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Abstract

BGK (Bhatnagar-Gross-Krook) model is a relaxation-type model of the Boltzmann equation, which is popularly used in place of the Boltzmann equation in physics and engineering. In this paper, we address the ill-posedness problem for the BGK model, in which the solution instantly escapes the initial solution space. For this, we propose two ill-posedness scenarios, namely, the homogeneous and the inhomogeneous ill-posedness mechanisms. In the former case, we find a class of spatially homogeneous solutions to the BGK model, where removing the small velocity part of the initial data triggers ill-posedness by increasing temperature. For the latter, we construct a spatially inhomogeneous solution to the BGK model such that the local temperature constructed from the solution has a polynomial growth in spatial variable. These ill-posedness properties for the BGK model pose a stark contrast with the Boltzmann equation for which the solution map is, at least for a finite time, stable in the corresponding solution spaces.

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Donghyun Lee, Sungbin Park, Seok-Bae Yun. 2024-08-15. Ill-posedness of the Boltzmann-BGK model in the exponential class. https://arxiv.org/abs/2408.08163

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