Kleber Carrapatoso

Quantitative and qualitative Kac’s chaos on the Boltzmann’s sphere

We investigate the construction of chaotic probability measures on the Boltzmanns 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 Boltzmanns sphere. Using the HWI inequality we prove that a Kac chaotic probability with bounded Fishers information is entropically chaotic and we give a quantitative rate. We also link different notions of chaos, proving that Fishers information chaos, introduced in \cite{HaurayMischler}, is stronger than entropic chaos, which is stronger than Kacs chaos. We give a possible answer to \cite[Open Problem 11]{CCLLV} in the Boltzmanns spheres 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.

Cauchy Problem and Exponential Stability for the Inhomogeneous Landau Equation

Abstract This work deals with the inhomogeneous Landau equation on the torus in the cases of hard, Maxwellian and moderately soft potentials. We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct solutions in a close-to-equilibrium setting. Finally, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay.

Landau equation for very soft and Coulomb potentials near Maxwellians

This work deals with the Landau equation for very soft and Coulomb potentials near the associated Maxwellian equilibrium. We first investigate the corresponding linearized operator and develop a method to prove strong asymptotical (but not uniformly exponential) stability estimates of its associated semigroup in large functional spaces. We then deduce existence, uniqueness and fast decay of the solutions to the nonlinear equation in a close-to-equilibrium framework. Our result drastically improves the set of initial data compared to the one considered by Guo and Strain who established similar results in Guo (Commun Math Phys 231:391–434, 2002) and Strain and Guo (Commun Partial Differ Equ 31(1–3):417–429, 2006; Arch Ration Mech Anal 187(2):287–339, 2008). Our functional framework is compatible with the non perturbative frameworks developed by Villani (Arch Ration Mech Anal 143(3):273–307 1998), Desvillettes and Villani (Invent Math 159(2):245–316, 2005), Desvillettes (J Funct Anal 269(5):1359–1403, 2015) and Carrapatoso et al. (arXiv:1510.08704, 2016), and our main result then makes possible to improve the speed of convergence to the equilibrium established therein.

Uniqueness and long time asymptotics for the parabolic–parabolic Keller–Segel equation

ABSTRACT The present paper deals with the parabolic–parabolic Keller–Segel equation in the plane in the general framework of weak (or “free energy”) solutions associated to initial data with finite mass M<8π, 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 L4∕3 Lebesgue norm similarly as for the 2d vorticity Navier–Stokes equation. (2) We prove a radially symmetric and polynomial weighted exponential stability of the self-similar profile in the quasiparabolic–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.

The parabolic-parabolic Keller-Segel equation

I present in this note recent results on the uniqueness and stability for the parabolic-parabolic Keller-Segel equation on the plane, obtained in collaboration with S. Mischler in [11].