Workshop - Problèmes mathématiques et modélisation en théorie cinétique

Thomas Rey

Hydrodynamic limits of granular gases equation

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.




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