Dynamics and hydrodynamic limits of the inelastic Boltzmann equation

We investigate the macroscopic description of a dilute, gas-like system of particles, which interact through binary collisions that conserve momentum and mass, but which dissipate energy, as in the case of granular media with inelastic collisions. Our starting point is on the mesoscopic level, through the Boltzmann equation. We deduce hydrodynamic equations for the macroscopic description that would reduce to the compressible Navier-Stokes equations if there were no energy dissipation. We do this in a regime where both the Knudsen number (the ratio of mesoscopic to macroscopic length scales) and the restitution deficit (which measures the inelasticity) are small but non-zero. In this regime, we show that for small values of the Knudsen number and small inelasticity it is possible to relate the actual dynamics to a reduced dynamics on a 'slow manifold', which in the limit of zero inelasticity and zero Knudsen number is simply the 'manifold' of local Maxwellians. Instead of expanding the Boltzmann equation itself, we expand this manifold in terms of these two small parameters. In this way, a number of ideas from the theory of dynamical systems, and especially geometric singular perturbation theory, enter our analysis. We discuss the resulting hydrodynamic equations, and compare them with those obtained by other researchers using other methods (suited to other regimes). As we explain here, the particular regime we investigate is especially interesting in the context of pattern formation in driven granular media.