Ill-posedness of the Boltzmann-BGK model in the exponential class
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917310791417856 |
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| author | Lee, Donghyun Park, Sungbin Yun, Seok-Bae |
| author_facet | Lee, Donghyun Park, Sungbin Yun, Seok-Bae |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_08163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ill-posedness of the Boltzmann-BGK model in the exponential class Lee, Donghyun Park, Sungbin Yun, Seok-Bae Analysis of PDEs 82C40, 35Q20, 82C40, 35A01, 35A02 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. |
| title | Ill-posedness of the Boltzmann-BGK model in the exponential class |
| topic | Analysis of PDEs 82C40, 35Q20, 82C40, 35A01, 35A02 |
| url | https://arxiv.org/abs/2408.08163 |