Mathematics

We review the state of the art in Grad's validity problem for a mathematical justification of fluid equations based on fundamental laws of classical mechanics. With the techniques currently available, such problem can be faced in some simple case for perfect gases, using the kinetic theory of Boltzmann as an intermediate step. We will discuss a recent result establishing the connection between microscopic and hydrodynamic scales, for perturbations of a global equilibrium. More precisely, we consider dynamical fluctuations of a system of hard spheres at low density (Boltzmann-Grad limit), with random initial data distributed according to the corresponding Gibbs measure. The asymptotics of the fluctuations were conjectured by Spohn in the Eighties. Lanford and collaborators proved partial results for short times. However, the small time restriction intrinsic to the perturbation theory can be lifted in this case. The main feature of the proof is an $L^2$ (entropy-type) bound together with a suitable coupling of trajectories. The method allows to reach diffusive time scales and obtain the fluctuating hydrodynamics.