From hard sphere dynamics to the Stokes-Fourier equations: An $L^2$ analysis of the Boltzmann-Grad limit

Abstract

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.