35 research outputs found
Exponential convergence to equilibrium for the homogeneous Landau equation with hard potentials
This paper deals with the long time behaviour of solutions to the spatially
homogeneous Landau equation with hard potentials . We prove an exponential in
time convergence towards the equilibrium with the optimal rate given by the
spectral gap of the associated linearized operator. This result improves the
polynomial in time convergence obtained by Desvillettes and Villani
\cite{DesVi2}. Our approach is based on new decay estimates for the semigroup
generated by the linearized Landau operator in weighted (polynomial or
stretched exponential) -spaces, using a method develloped by Gualdani,
Mischler and Mouhot \cite{GMM}.Comment: 20 pages. Minor corrections, improvement on the presentatio
Quantitative and qualitative Kac's chaos on the Boltzmann's sphere
We investigate the construction of chaotic probability measures on the
Boltzmann's sphere, which is the state space of the stochastic process of a
many-particle system undergoing a dynamics preserving energy and momentum.
Firstly, based on a version of the local Central Limit Theorem (or Berry-Esseen
theorem), we construct a sequence of probabilities that is Kac chaotic and we
prove a quantitative rate of convergence. Then, we investigate a stronger
notion of chaos, namely entropic chaos introduced in \cite{CCLLV}, and we
prove, with quantitative rate, that this same sequence is also entropically
chaotic. Furthermore, we investigate more general class of probability measures
on the Boltzmann's sphere. Using the HWI inequality we prove that a Kac chaotic
probability with bounded Fisher's information is entropically chaotic and we
give a quantitative rate. We also link different notions of chaos, proving that
Fisher's information chaos, introduced in \cite{HaurayMischler}, is stronger
than entropic chaos, which is stronger than Kac's chaos. We give a possible
answer to \cite[Open Problem 11]{CCLLV} in the Boltzmann's sphere's framework.
Finally, applying our previous results to the recent results on propagation of
chaos for the Boltzmann equation \cite{MMchaos}, we prove a quantitative rate
for the propagation of entropic chaos for the Boltzmann equation with
Maxwellian molecules.Comment: 51 pages, to appear in Ann. Inst. H. Poincar\'e Probab. Sta
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
Estimates for the large time behavior of the Landau equation in the Coulomb case
This work deals with the large time behaviour of the spatially homogeneous
Landau equation with Coulomb potential. Firstly, we obtain a bound from below
of the entropy dissipation by a weighted relative Fisher information of
with respect to the associated Maxwellian distribution, which leads to a
variant of Cercignani's conjecture thanks to a logarithmic Sobolev inequality.
Secondly, we prove the propagation of polynomial and stretched exponential
moments with an at most linearly growing in time rate. As an application of
these estimates, we show the convergence of any (- or weak) solution to the
Landau equation with Coulomb potential to the associated Maxwellian equilibrium
with an explicitly computable rate, assuming initial data with finite mass,
energy, entropy and some higher -moment. More precisely, if the initial
data have some (large enough) polynomial -moment, then we obtain an
algebraic decay. If the initial data have a stretched exponential -moment,
then we recover a stretched exponential decay
Hydrodynamic limit for the non-cutoff Boltzmann equation
This work deals with the non-cutoff Boltzmann equation with hard potentials,
in both the torus and in the whole space , under
the incompressible Navier-Stokes scaling. We first establish the well-posedness
and decay of global mild solutions to this rescaled Boltzmann equation in a
perturbative framework, that is for solutions close to the Maxwellian,
obtaining in particular integrated-in-time regularization estimates. We then
combine these estimates with spectral-type estimates in order to obtain the
strong convergence of solutions to the non-cutoff Boltzmannn equation towards
the incompressible Navier-Stokes-Fourier system
On the derivation of a Stokes-Brinkman problem from Stokes equations around a random array of moving spheres
We consider the Stokes system in deprived of spheres of radius completed by constant boundary conditions on the spheres. This problem models the instantaneous response of a viscous fluid to an immersed cloud of moving solid spheres. We assume that the centers of the spheres and the boundary conditions are given randomly and we compute the asymptotic behavior of solutions when the parameter diverges. Under the assumption that the distribution of spheres/centers is chaotic, we prove convergence in mean to the solution of a Stokes-Brinkman problem