On a kinetic Poincaré inequality and beyond
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866915564521259008 |
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| author | Niebel, Lukas Zacher, Rico |
| author_facet | Niebel, Lukas Zacher, Rico |
| contents | In this article, we give a trajectorial proof of a kinetic Poincaré inequality which plays an important role in the De Giorgi-Nash-Moser theory for kinetic equations. The present work improves a result due to J. Guerand and C. Mouhot [10] in several directions. We use kinetic trajectories along the vector fields $\partial_t + v \cdot \nabla_x$ and $\partial_{v_i}$, $i = 1,\dots, d$ and do not rely on higher-order commutators such as $[\partial_{v_i},\partial_t + v \cdot \nabla_x] = \partial_{x_i}$ or on the fundamental solution. The presented method also applies to more general hypoelliptic equations. We illustrate this by studying a Kolmogorov equation with $k$ steps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_03199 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On a kinetic Poincaré inequality and beyond Niebel, Lukas Zacher, Rico Analysis of PDEs 35K70, 26D10, 35B45, 35B65, 35H10 In this article, we give a trajectorial proof of a kinetic Poincaré inequality which plays an important role in the De Giorgi-Nash-Moser theory for kinetic equations. The present work improves a result due to J. Guerand and C. Mouhot [10] in several directions. We use kinetic trajectories along the vector fields $\partial_t + v \cdot \nabla_x$ and $\partial_{v_i}$, $i = 1,\dots, d$ and do not rely on higher-order commutators such as $[\partial_{v_i},\partial_t + v \cdot \nabla_x] = \partial_{x_i}$ or on the fundamental solution. The presented method also applies to more general hypoelliptic equations. We illustrate this by studying a Kolmogorov equation with $k$ steps. |
| title | On a kinetic Poincaré inequality and beyond |
| topic | Analysis of PDEs 35K70, 26D10, 35B45, 35B65, 35H10 |
| url | https://arxiv.org/abs/2212.03199 |