Hydrodynamic limit of granular gases to pressureless Euler in dimension 1

Together with my collaborator Pierre-Emmanuel Jabin, we’ve just submitted a paper entitled Hydrodynamic limit of granular gases to pressureless Euler in dimension 1. This works aims to provide the first rigorous proof of the hydrodynamic limit of the kinetic, granular gases equation with strong inelasticity towards the fluid, pressureless Euler system, in dimension 1 of space and velocity. More precisely,

We investigate the behavior of granular gases in the limit of small Knudsen number, that is very frequent collisions. We deal with the strongly inelastic case, in one dimension of space and velocity. We are able to prove the convergence toward the pressureless Euler system. The proof relies on dispersive relations at the kinetic level, which leads to the so-called Oleinik property at the limit.

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Spectral study of the granular gases operator

I have just submitted a new paper, entitled A Spectral Study of the Linearized Boltzmann Equation for Diffusively Excited Granular Media.

In this work, we are interested in the spectrum of the diffusively excited granular gases equation, in a space inhomogeneous setting, linearized around an homogeneous equilibrium.
We perform a study which generalizes to a non-hilbertian setting and to the inelastic case the seminal work of Ellis and Pinsky about the spectrum of the linearized Boltzmann operator. We first give a precise localization of the spectrum, which consists in an essential part lying on the left of the imaginary axis and a discrete spectrum, which is also of nonnegative real part for small values of the inelasticity parameter. We then give the so-called inelastic “dispersion relations”, and compute an expansion of the branches of eigenvalues of the linear operator, for small Fourier (in space) frequencies and small inelasticity.
One of the main novelty in this work, apart from the study of the inelastic case, is that we consider an exponentially weighted L1(m−1) Banach setting instead of the classical L2(M1,0,1-1) Hilbertian case, endorsed with Gaussian weights. We prove in particular that the results of Ellis and Pinsky hold also in this space.

A Rescaling Velocity Method for Dissipative Kinetic Equations

I finished a new paper with my collaborator (and advisor) Francis Filbet: A Rescaling Velocity Method for Dissipative Kinetic Equations – Applications to Granular Media.
We present in this paper a new numerical algorithm based on a relative energy scaling for collisional kinetic equations allowing to study numerically their long time behavior, without the usual problems related to the change of scales in velocity variables. Several applications are presented for Boltzmann like equations. This method is particularly efficient for numerical simulations of the granular gases equation with dissipative energy: it allows to study accurately the long time behavior of this equation and is very well suited for the study of clustering phenomena.