We will present a work concerning hydrodynamic limits of the granular
gases equation, in various physical regimes. The granular gases equation
is a Boltzmann-like kinetic equation describing a rarefied gas composed
of macroscopic particles, interacting via energy-dissipative binary
collisions (pollen flow in a fluid, or planetary rings for example). The
purpose of an hydrodynamic limit is to give a reduced description of
this equation, using a fluid approximation.
We shall first present results inspired from the seminal paper of Ellis
and Pinsky about the spectrum of the linearized collision operator, for
the quasi-elastic regime. We will give a precise localization of the
spectrum, and an expansion of the branches of eigenvalues of this
operator, for small Fourier (in space) frequencies and small
inelasticity, allowing to explain some of the classical features of this
equation and its hydrodynamic limit, such as the clustering instability.
If time permits, we shall then deal with the strongly inelastic case, in
one dimension of space and velocity. Using a nonlinear functional, we
will establish the hydrodynamic limit of our equation toward the
presureless Euler system. This is a joint work with P-E. Jabin.