arXiv subjects

Kleber Carrapatoso

Publications and source records attributed to Kleber Carrapatoso.

24 records · Page 2Linked to original sources

arXiv2015

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 $D(f)$ by a weighted relative Fisher information of $f$ 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 ($H$- 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 $L^1$-moment. More precisely, if the initial data have some (large enough) polynomial $L^1$-moment, then we obtain an algebraic decay. If the initial data have a stretched exponential $L^1$-moment, then we recover a stretched exponential decay.

math.AP↗

arXiv2015

On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials

We investigate in this work the rate of convergence to equilibrium of solutions to the spatially homogeneous Landau equation with soft potentials. Firstly, we prove a polynomial in time convergence using an entropy method with some new a priori estimates. Finally we prove an exponential in time convergence towards the equilibrium with the optimal rate given by the spectral gap of the associated linearized operator, combining new decay estimates for the semigroup generated by the linearized Landau operator in weighted $L^p$-spaces togheter with the polynomial decay described above.

math.AP↗

arXiv2014

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) $L^p$-spaces, using a method develloped by Gualdani, Mischler and Mouhot \cite{GMM}.

math.AP↗

arXiv2014

Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules

We prove a quantitative propagation of chaos, uniformly in time, for the spatially homogeneous Landau equation in the case of Maxwellian molecules. We improve the results of Fontbona, Guérin and Méléard \cite{FonGueMe} and Fournier \cite{Fournier} where the propagation of chaos is proved for finite time. Moreover, we prove a quantitative estimate on the rate of convergence to equilibrium uniformly in the number of particles.

math.AP↗

arXiv2014

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.

math.PR↗

arXiv2013

Chaos and Entropic Chaos in Kac's Model Without High Moments

In this paper we present a new local Lévy Central Limit Theorem, showing convergence to stable states that are not necessarily the Gaussian, and use it to find new and intuitive entropically chaotic families with underlying one-particle function that has moments of order $2α$, with $1<α<2$. We also discuss a lower semi continuity result for the relative entropy with respect to our specific family of functions, and use it to show a form of stability property for entropic chaos in our settings.

math.PR↗