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In this paper, we provide the first rigorous derivation of hydrodynamicequations from the Boltzmann equation for inelastic hard spheres with smallinelasticity. The hydrodynamic system that we obtain is an incompressibleNavier-Stokes-Fourier system with self-consistent forcing terms and is thus thefirst hydrodynamic system that properly describes rapid granular flows. To dothat, we write our Boltzmann equation in nondimensional form introducing thedimensionless Knudsen number which is intended to tend to 0. The difficultiesare then manyfold, the first one coming from the fact that the originalBoltzmann equation is free-cooling and thus requires a self-similar change ofvariables to work with an equation that has an homogeneous steady state. Thelatter is not explicit and is heavy-tailed, which is a major obstacle to adaptenergy estimates and spectral analysis. One of the main challenges here is tounderstand the relation between the restitution coefficient (which quantifiesthe loss of energy at the microscopic level) and the Knudsen number. This isdone identifying the correct nearly elastic regime to capture nontrivialhydrodynamic behavior. We are then able to prove exponential stabilityuniformly with respect to the Knudsen number of the solution of our rescaledBoltzmann equation in a close to equilibrium regime. Finally, we prove that oursolution to the Boltzmann equation converges in some very specific weak sensetowards some hydrodynamic solution which depends on time and space variablesonly through macroscopic quantities. Such macroscopic quantities are solutionsto a suitable modification of the incompressible Navier-Stokes-Fourier systemwhich appears to be new in this context.

Fluid dynamic limit of Boltzmann equation for granular hard--spheres in a nearly elastic regime