On a class of nonlinear BGK-type kinetic equations with density dependent collision rates
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917227928748032 |
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| author | Evans, Josephine Morris, Daniel Yoldaş, Havva |
| author_facet | Evans, Josephine Morris, Daniel Yoldaş, Havva |
| contents | We consider a class of nonlinear, spatially inhomogeneous kinetic equations of BGK-type with density dependent collision rates. These equations share the same superlinearity as the Boltzmann equation, and fall into the class of run and tumble equations appearing in mathematical biology. We prove that the Cauchy problem is well-posed, and the solutions propagate Maxwellian bounds over time. Moreover, we show that the solutions approach to equilibrium with an exponential rate, known as a hypocoercivity result. Lastly, we derive a class of nonlinear diffusion equations as the hydrodynamic limit of the kinetic equations in the diffusive scaling, employing both hypocoercivity and relative entropy methods. The limit equations cover a wide range of nonlinear diffusion equations including both the porous medium and the fast diffusion equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14225 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a class of nonlinear BGK-type kinetic equations with density dependent collision rates Evans, Josephine Morris, Daniel Yoldaş, Havva Analysis of PDEs 35A01, 82C40, 35B40, 35Q35 We consider a class of nonlinear, spatially inhomogeneous kinetic equations of BGK-type with density dependent collision rates. These equations share the same superlinearity as the Boltzmann equation, and fall into the class of run and tumble equations appearing in mathematical biology. We prove that the Cauchy problem is well-posed, and the solutions propagate Maxwellian bounds over time. Moreover, we show that the solutions approach to equilibrium with an exponential rate, known as a hypocoercivity result. Lastly, we derive a class of nonlinear diffusion equations as the hydrodynamic limit of the kinetic equations in the diffusive scaling, employing both hypocoercivity and relative entropy methods. The limit equations cover a wide range of nonlinear diffusion equations including both the porous medium and the fast diffusion equations. |
| title | On a class of nonlinear BGK-type kinetic equations with density dependent collision rates |
| topic | Analysis of PDEs 35A01, 82C40, 35B40, 35Q35 |
| url | https://arxiv.org/abs/2503.14225 |