98 research outputs found
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} . In the physical regime of a small
inelasticity (that is for some constructive
) we prove uniqueness of the self-similar profile for given values
of the restitution coefficient , 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 , and
under some smallness condition on depending on the mass, energy
and 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 , 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}
for the inelasticity). In the physical regime of a small inelasticity (that is
for some constructive ) we prove
uniqueness of the stationary solution for given values of the restitution
coefficient , 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
-dimensional sphere as 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
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