Compensated integrability on tori; a priori estimate for space-periodic gas flows
We extend our theory of Compensated Integrability of positive symmetric tensors, to the case where the domain is the product of a linear space $\mathbb{R}^k$ and of a torus $\mathbb{R}^m/\Lambda $, $\Lambda $ being a lattice of $\mathbb{R}^m$. We apply our abstract results in two contexts, for which...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.654/ |
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| author | Serre, Denis |
| author_facet | Serre, Denis |
| author_sort | Serre, Denis |
| collection | DOAJ |
| description | We extend our theory of Compensated Integrability of positive symmetric tensors, to the case where the domain is the product of a linear space $\mathbb{R}^k$ and of a torus $\mathbb{R}^m/\Lambda $, $\Lambda $ being a lattice of $\mathbb{R}^m$. We apply our abstract results in two contexts, for which $k=1$ is associated with a time variable, while $m=d$ is a space dimension. On the one hand to $d$-dimensional inviscid gas dynamics, governed by the Euler equations, when the initial data is space-periodic; we obtain an a priori space-time estimate of our beloved quantity $\rho ^{\frac{1}{d}}p$. On the other hand to hard spheres dynamics in a periodic box $L\mathbb{T}_d$. We obtain a weighted estimate of the average number of collisions per unit time, provided that the “linear density” $Na/L$ ($N$ particles of radius $a$) is smaller than some threshold. |
| format | Article |
| id | doaj-art-16aa702ba17440d1b09453c61dc99c13 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-16aa702ba17440d1b09453c61dc99c132025-02-07T11:23:50ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G111425144410.5802/crmath.65410.5802/crmath.654Compensated integrability on tori; a priori estimate for space-periodic gas flowsSerre, Denis0École Normale Supérieure de Lyon, U.M.P.A., UMR CNRS–ENSL # 5669. 46 allée d’Italie, 69364 Lyon cedex 07. FranceWe extend our theory of Compensated Integrability of positive symmetric tensors, to the case where the domain is the product of a linear space $\mathbb{R}^k$ and of a torus $\mathbb{R}^m/\Lambda $, $\Lambda $ being a lattice of $\mathbb{R}^m$. We apply our abstract results in two contexts, for which $k=1$ is associated with a time variable, while $m=d$ is a space dimension. On the one hand to $d$-dimensional inviscid gas dynamics, governed by the Euler equations, when the initial data is space-periodic; we obtain an a priori space-time estimate of our beloved quantity $\rho ^{\frac{1}{d}}p$. On the other hand to hard spheres dynamics in a periodic box $L\mathbb{T}_d$. We obtain a weighted estimate of the average number of collisions per unit time, provided that the “linear density” $Na/L$ ($N$ particles of radius $a$) is smaller than some threshold.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.654/Compensated integrabilityperfect gasbilliardperiodic data |
| spellingShingle | Serre, Denis Compensated integrability on tori; a priori estimate for space-periodic gas flows Comptes Rendus. Mathématique Compensated integrability perfect gas billiard periodic data |
| title | Compensated integrability on tori; a priori estimate for space-periodic gas flows |
| title_full | Compensated integrability on tori; a priori estimate for space-periodic gas flows |
| title_fullStr | Compensated integrability on tori; a priori estimate for space-periodic gas flows |
| title_full_unstemmed | Compensated integrability on tori; a priori estimate for space-periodic gas flows |
| title_short | Compensated integrability on tori; a priori estimate for space-periodic gas flows |
| title_sort | compensated integrability on tori a priori estimate for space periodic gas flows |
| topic | Compensated integrability perfect gas billiard periodic data |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.654/ |
| work_keys_str_mv | AT serredenis compensatedintegrabilityontoriaprioriestimateforspaceperiodicgasflows |