Compensated integrability on tori; a priori estimate for space-periodic gas flows - Serre, Denis (Comptes Rendus. Mathématique)
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.