Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres

We consider the spatially homogeneous Boltzmann equation for {\em inelastic hard spheres}, in the framework of so-called {\em constant normal restitution coefficients} $\alpha \in [0,1]$. In the physical regime of a small inelasticity (that is $\alpha \in [\alpha_*,1)$ for some constructive $\alpha_*>0$) we prove uniqueness of the self-similar profile for given values of the restitution coefficient $\alpha \in [\alpha_*,1)$, the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation). Moreover, if the initial datum lies in $L^1_3$, and under some smallness condition on $(1-\alpha_*)$ depending on the mass, energy and $L^1_3$ norm of this initial datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the self-similar solution (the so-called {\em homogeneous cooling state}). These uniqueness, stability and convergence results are expressed in the self-similar variables and then translate into corresponding results for the original Boltzmann equation. The proofs are based on the identification of a suitable elastic limit rescaling, and the construction of a smooth path of self-similar profiles connecting to a particular Maxwellian equilibrium in the elastic limit, together with tools from perturbative theory of linear operators. Some universal quantities, such as the "quasi-elastic self-similar temperature" and the rate of convergence towards self-similarity at first order in terms of $(1-\alpha)$, are obtained from our study. These results provide a positive answer and a mathematical proof of the Ernst-Brito conjecture [16] in the case of inelastic hard spheres with small inelasticity.Comment: 73 page

Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media

We consider a space-homogeneous gas of {\it inelastic hard spheres}, with a {\it diffusive term} representing a random background forcing (in the framework of so-called {\em constant normal restitution coefficients} $\alpha \in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $\alpha \in [\alpha_*,1)$ for some constructive $\alpha_* \in [0,1)$) we prove uniqueness of the stationary solution for given values of the restitution coefficient $\alpha \in [\alpha_*,1)$, the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution

Quantitative uniform in time chaos propagation for Boltzmann collision processes

This paper is devoted to the study of mean-field limit for systems of indistinguables particles undergoing collision processes. As formulated by Kac \cite{Kac1956} this limit is based on the {\em chaos propagation}, and we (1) prove and quantify this property for Boltzmann collision processes with unbounded collision rates (hard spheres or long-range interactions), (2) prove and quantify this property \emph{uniformly in time}. This yields the first chaos propagation result for the spatially homogeneous Boltzmann equation for true (without cut-off) Maxwell molecules whose "Master equation" shares similarities with the one of a L\'evy process and the first {\em quantitative} chaos propagation result for the spatially homogeneous Boltzmann equation for hard spheres (improvement of the %non-contructive convergence result of Sznitman \cite{S1}). Moreover our chaos propagation results are the first uniform in time ones for Boltzmann collision processes (to our knowledge), which partly answers the important question raised by Kac of relating the long-time behavior of a particle system with the one of its mean-field limit, and we provide as a surprising application a new proof of the well-known result of gaussian limit of rescaled marginals of uniform measure on the $N$-dimensional sphere as $N$ goes to infinity (more applications will be provided in a forthcoming work). Our results are based on a new method which reduces the question of chaos propagation to the one of proving a purely functional estimate on some generator operators ({\em consistency estimate}) together with fine stability estimates on the flow of the limiting non-linear equation ({\em stability estimates})

Fast algorithms for computing the Boltzmann collision operator

The development of accurate and fast numerical schemes for the five fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the so-called hard spheres model in dimension three, we are able to derive spectral methods that can be evaluated through fast algorithms. These algorithms are based on a suitable representation and approximation of the collision operator. Explicit expressions for the errors in the schemes are given and spectral accuracy is proved. Parallelization properties and adaptivity of the algorithms are also discussed.Comment: 22 page

The Schauder estimate in kinetic theory with application to a toy nonlinear model

This article is concerned with the Schauder estimate for linear kinetic Fokker-Planck equations with H\"older continuous coefficients. This equation has an hypoelliptic structure. As an application of this Schauder estimate, we prove the global well-posedness of a toy nonlinear model in kinetic theory. This nonlinear model consists in a non-linear kinetic Fokker-Planck equation whose steady states are Maxwellian and whose diffusion in the velocity variable is proportional to the mass of the solution