Compensated integrability on tori; a priori estimate for space-periodic gas flows

Serre, Denis

Comptes Rendus. Mathématique, Tome 362 (2024) no. G11, pp. 1425-1444

We extend our theory of Compensated Integrability of positive symmetric tensors, to the case where the domain is the product of a linear space ${ℝ}^k$ and of a torus ${ℝ}^m/\Lambda$, $\Lambda$ being a lattice of ${ℝ}^m$. 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 $L{𝕋}_d$. 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.

Nous étendons notre théorie d’Intégrabilité par Compensation au cas des domaines ${ℝ}^k\times ({ℝ}^m/\Lambda )$, produits d’un facteur linéaire et d’un tore plat. Nous appliquons les résultats abstraits à deux contextes, pour lesquels $k=1$ est associé à une variable de temps, tandis que $m=d$ est la dimension de l’espace physique ambiant. Le premier est la dynamique des gaz non visqueux, gouvernée par les équations d’Euler, lorsque les données initiales sont périodiques en espace. Nous obtenons une estimation a priori de notre quantité favorite ${\rho }^{\frac{1}{d}}p$. Le second est la dynamique des sphères dures, dans une boîte périodique $L{𝕋}_d$. Nous obtenons une estimation pondérée du nombre moyen de collisions par unité de temps, pourvu que la «  densité linéique  » $Na/L$ ($N$ particules de rayon $a$) soit inférieure à un certain seuil.

Classification : 35B09, 35B45, 76N15, 37C83
Keywords: Compensated integrability, perfect gas, billiard, periodic data
Mots-clés : Intégrabilité par compensation, gaz parfait, billard, données périodiques

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