From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit

Thierry Bodineau (1) , Isabelle Gallagher (2) , Laure Saint-Raymond (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 IMJ - Institut de Mathématiques de Jussieu
3 DMA - Département de Mathématiques et Applications - ENS Paris

We derive the linear acoustic and Stokes-Fourier equations as the limiting dynamics of a system of N hard spheres of diameter $\epsilon$ in two space dimensions, when N $\rightarrow$ $\infty$, $\epsilon$ $\rightarrow$ 0, N $\epsilon$ = $\alpha$ $\rightarrow$ $\infty$, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford's strategy [18], and on the pruning procedure developed in [5] to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform L 2 a pri-ori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.

Domaines: Equations aux dérivées partielles [math.AP]

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond. From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit. Annals of PDE, 2017, 3 (1), ⟨10.1007/s40818-016-0018-0⟩. ⟨hal-01226094v3⟩