A Galerkin type method for kinetic Fokker Planck equations based on Hermite expansions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929391408250880 |
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| author | Avelin, Benny Hou, Mingyi Nyström, Kaj |
| author_facet | Avelin, Benny Hou, Mingyi Nyström, Kaj |
| contents | In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain $(0, T) \times D \times \mathbb{R}^d$, where $D$ is either $\mathbb{T}^d$ or $\mathbb{R}^d$. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from $\href{arXiv:1902.04037v2}{AAMN21}$ and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_14234 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Galerkin type method for kinetic Fokker Planck equations based on Hermite expansions Avelin, Benny Hou, Mingyi Nyström, Kaj Analysis of PDEs In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain $(0, T) \times D \times \mathbb{R}^d$, where $D$ is either $\mathbb{T}^d$ or $\mathbb{R}^d$. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from $\href{arXiv:1902.04037v2}{AAMN21}$ and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term. |
| title | A Galerkin type method for kinetic Fokker Planck equations based on Hermite expansions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2305.14234 |