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| We will present a work concerning the quasi-elastic hydrodynamic limit of the diffusively excited granular gases equation. The granular gases equation is a Boltzmann-like kinetic equation arising when one wants to give a statistical description of 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.
As a first step to prove mathematically the validity of the quasi-elastic hydrodynamic limit of this equation, we will present results inspired from the seminal paper of Ellis and Pinsky about the spectrum of the linearized collision operator. The main differences in our case concern the lack of energy conservation and the use of an exponentially weighted Banach $L^1$ setting instead of the classical Hilbertian $L^2$ one with Gaussian weights. 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. |
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