86 research outputs found
Kac's chaos and Kac's program
In this note I present the main results about the quantitative and
qualitative propagation of chaos for the Boltzmann-Kac system obtained in
collaboration with C. Mouhot in \cite{MMinvent} which gives a possible answer
to some questions formulated by Kac in \cite{Kac1956}. We also present some
related recent results about Kac's chaos and Kac's program obtained in
\cite{MMWchaos,HaurayMischler,KleberSphere} by K. Carrapatoso, M. Hauray, C.
Mouhot, B. Wennberg and myself
Kinetic equations with Maxwell boundary conditions
We prove global stability results of {\sl DiPerna-Lions} renormalized
solutions for the initial boundary value problem associated to some kinetic
equations, from which existence results classically follow. The (possibly
nonlinear) boundary conditions are completely or partially diffuse, which
includes the so-called Maxwell boundary conditions, and we prove that it is
realized (it is not only a boundary inequality condition as it has been
established in previous works). We are able to deal with Boltzmann,
Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace
theorems of the kind previously introduced by the author for the Vlasov
equations, new results concerning weak-weak convergence (the renormalized
convergence and the biting -weak convergence), as well as the
Darroz\`es-Guiraud information in a crucial way
Spectral analysis of semigroups and growth-fragmentation equations
The aim of this paper is twofold: (1) On the one hand, the paper revisits the
spectral analysis of semigroups in a general Banach space setting. It presents
some new and more general versions, and provides comprehensible proofs, of
classical results such as the spectral mapping theorem, some (quantified)
Weyl's Theorems and the Krein-Rutman Theorem. Motivated by evolution PDE
applications, the results apply to a wide and natural class of generators which
split as a dissipative part plus a more regular part, without assuming any
symmetric structure on the operators nor Hilbert structure on the space, and
give some growth estimates and spectral gap estimates for the associated
semigroup. The approach relies on some factorization and summation arguments
reminiscent of the Dyson-Phillips series in the spirit of those used in
[87,82,48,81]. (2) On the other hand, we present the semigroup spectral
analysis for three important classes of "growth-fragmentation" equations,
namely the cell division equation, the self-similar fragmentation equation and
the McKendrick-Von Foerster age structured population equation. By showing that
these models lie in the class of equations for which our general semigroup
analysis theory applies, we prove the exponential rate of convergence of the
solutions to the associated remarkable profile for a very large and natural
class of fragmentation rates. Our results generalize similar estimates obtained
in \cite{MR2114128,MR2536450} for the cell division model with (almost)
constant total fragmentation rate and in \cite{MR2832638,MR2821681} for the
self-similar fragmentation equation and the cell division equation restricted
to smooth and positive fragmentation rate and total fragmentation rate which
does not increase more rapidly than quadratically. It also improves the
convergence results without rate obtained in \cite{MR2162224,MR2114413} which
have been established under similar assumptions to those made in the present
work
Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation
The present paper deals with the parabolic-parabolic Keller-Segel equation in
the plane inthe general framework of weak (or "free energy") solutions
associated to an initial datum with finite mass M\textless{} 8\pi, finite
second log-moment and finite entropy. The aim of the paper is twofold:(1) We
prove the uniqueness of the "free energy" solution. The proof uses a
DiPerna-Lions renormalizing argument which makes possible to get the "optimal
regularity" as well as an estimate of the difference of two possible solutions
in the critical Lebesgue norm similarly as for the vorticity
Navier-Stokes equation. (2) We prove a radially symmetric and polynomial
weighted exponential stability of the self-similar profile in the quasi
parabolic-elliptic regime. The proof is based on a perturbation argument which
takes advantage of the exponential stability of the self-similar profile for
the parabolic-elliptic Keller-Segel equation established by Campos-Dolbeault
and Egana-Mischler
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
Uniform semigroup spectral analysis of the discrete, fractional \& classical Fokker-Planck equations
In this paper, we investigate the spectral analysis (from the point of view
of semi-groups) of discrete, fractional and classical Fokker-Planck equations.
Discrete and fractional Fokker-Planck equations converge in some sense to the
classical one. As a consequence, we first deal with discrete and classical
Fokker-Planck equations in a same framework, proving uniform spectral estimates
using a perturbation argument and an enlargement argument. Then, we do a
similar analysis for fractional and classical Fokker-Planck equations using an
argument of enlargement of the space in which the semigroup decays. We also
handle another class of discrete Fokker-Planck equations which converge to the
fractional Fokker-Planck one, we are also able to treat these equations in a
same framework from the spectral analysis viewpoint, still with a semigroup
approach and thanks to a perturbative argument combined with an enlargement
one. Let us emphasize here that we improve the perturbative argument introduced
in [7] and developed in [11], relaxing the hypothesis of the theorem, enlarging
thus the class of operators which fulfills the assumptions required to apply
it
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})
On Kac's Chaos And Related Problems
This paper is devoted to establish quantitative and qualitative estimates
related to the notion of chaos as firstly formulated by M. Kac in his study of
mean-field limit for systems of undistinguishable particles. First, we
quantitatively liken three usual measures of Kac's chaos, some involving the
all variables, other involving a finite fixed number of variables. Next, we
define the notion of entropy chaos and Fisher information chaos in a similar
way as defined by Carlen et al (KRM 2010). We show that Fisher information
chaos is stronger than entropy chaos, which in turn is stronger than Kac's
chaos. More importantly, with the help of the HWI inequality of Otto-Villani,
we establish a quantitative estimate between these quantities, which in
particular asserts that Kac's chaos plus Fisher information bound implies
entropy chaos. We then extend the above quantitative and qualitative results
about chaos in the framework of probability measures with support on the Kac's
spheres. Additionally to the above mentioned tool, we use and prove an optimal
rate local CLT in norm for distributions with finite 6-th moment and
finite norm, for some . Last, we investigate how our techniques can
be used without assuming chaos, in the context of probability measures mixtures
introduced by De Finetti, Hewitt and Savage. In particular, we define the
(level 3) Fisher information for mixtures and prove that it is l.s.c. and
affine, as that was done previously for the level 3 Boltzmann's entropy.Comment: 80 pages. Last version before publicatio
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
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