Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off

This paper deals with the long time behavior of solutions to the spatially homogeneous Boltzmann equation. The interactions considered are the so-called (non cut-off and non mollified) hard potentials. We prove an exponential in time convergence towards the equilibrium, improving results of Villani from \cite{Vill1} where a polynomial decay to equilibrium is proven. The basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a $L^1$ space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by Gualdani and al in \cite{GMM}. We then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish.Comment: 22 page

Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting

In this paper, we consider the spatially inhomogeneous diffusively driven inelastic Boltzmann equation in different cases: the restitution coefficient can be constant or can depend on the impact velocity (which is a more physically relevant case), including in particular the case of viscoelastic hard spheres. In the weak thermalization regime, i.e. when the diffusion parameter is sufficiently small, we prove existence of global solutions considering the close-to-equilibrium regime as well as the weakly inhomogeneous regime in the case of a constant restitution coefficient. It is the very first existence theorem of global solution in an inelastic "collision regime" (that is excluding \cite{AR} where an existence theorem is proven in a near to the vacuum regime). We also study the long-time behavior of these solutions and prove a convergence to equilibrium with an exponential rate. The basis of the proof is the study of the linearized equation. We obtain a new result on it, we prove existence of a spectral gap in weighted (stretched exponential and polynomial) Sobolev spaces and a result of exponential stability for the semigroup generated by the linearized operator. To do that, we develop a perturbative argument around the spatially inhomogeneous equation for elastic hard spheres and we take advantage of the recent paper \cite{GMM}where this equation has been considered. We then link the linearized theory with the nonlinear one in order to handle the full non-linear problem thanks to new bilinear estimates on the collision operator that we establish. As far as the case of a constant coefficient is concerned, the present paper largely improves similar results obtained in \cite{MM2} in a spatially homogeneous framework. Concerning the case of a non-constant coefficient, this kind of results is new and we use results on steady states of the linearized equation from \cite{AL3}.Comment: arXiv admin note: text overlap with arXiv:1112.0096 by other author

Landau damping for the linearized Vlasov Poisson equation in a weakly collisional regime

In this paper, we consider the linearized Vlasov-Poisson equation around an homogeneous Maxwellian equilibrium in a weakly collisional regime: there is a parameter \eps in front of the collision operator which will tend to $0$. Moreover, we study two cases of collision operators, linear Boltzmann and Fokker-Planck. We prove a result of Landau damping for those equations in Sobolev spaces uniformly with respect to the collision parameter \eps as it goes to $0$

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

Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off

In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a moderate angular singularity). We prove a result of existence and uniqueness of solutions in a close-to-equilibrium regime for this equation in weighted Sobolev spaces with a polynomial weight, contrary to previous works on the subject, all developed with a weight prescribed by the equilibrium. It is the first result in this more physically relevant frameworkfor this equation. Moreover, 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 of the linearized equation. Let us highlight the fact that a key point of the development of our Cauchy theory is the proof of new regularization estimates in short time for the linearized operator thanks to pseudo-differential tools.Comment: arXiv admin note: text overlap with arXiv:1709.0994

On the convergence of smooth solutions from Boltzmann to Navier-Stokes

International audienceIn this work, we are interested in the link between strong solutions of the Boltzmann and the Navier-Stokes equations. To justify this connection, our main idea is to use information on the limit system (for instance the fact that the Navier-Stokes equations are globally wellposed in two space dimensions or when the initial data is small). In particular we prove that the life span of the solutions to the rescaled Boltzmann equation is bounded from below by that of the Navier-Stokes system. We deal with general initial data in the whole space in dimensions 2 and 3, and also with well-prepared data in the case of periodic boundary conditions

Existence et stabilité de solutions fortes en théorie cinétique des gaz

The topic of this thesis is the study of models coming from kinetic theory. In all the problems that are addressed, the associated linear or linearized problem is analyzed from a spectral point of view and from the point of view of semigroups. Tothat, we add the study of the nonlinear stability when the equation is nonlinear. More precisely, to begin with, we treat the problem of trend to equilibrium for the fractional Fokker-Planck and Boltzmann without cut-off equations, proving an exponential decay to equilibrium in spaces of type L1 with polynomial weights. Concerning the inhomogeneous Landau equation, we develop a Cauchy theory of perturbative solutions in spaces of type L2 with various weights such as polynomial and exponential weights and we also prove the exponential stability of these solutions. Then, we prove similar results for the inhomogeneous inelastic diffusively driven Boltzmann equation in a small inelasticity regime in L1 spaces with polynomial weights. Finally, we study in the same and uniform framework from the spectral analysis point of view with a semigroup approach several Fokker-Planck equations which converge towards the classical one.Cette thèse est centrée sur l’étude d’équations issues de la théorie cinétique des gaz. Dans tous les problèmes qui y sont explorés, une analyse des problèmes linéaires ou linéarisés associés est réalisée d’un point de vue spectral et du point de vue des semi-groupes. A cela s’ajoute une analyse de la stabilité non linéaire lorsque le modèle est non linéaire. Plus précisément, dans une première partie, nous nous intéressons aux équations de Fokker-Planck fractionnaire et Boltzmann sans cut-off homogène en espace et nous prouvons un retour vers l’équilibre des solutions de ces équations avec un taux exponentiel dans des espaces de type L1 à poids polynomial. Concernant l’équation de Landau inhomogène en espace, nous développons une théorie de Cauchy de solutions perturbatives dans des espaces de type L2 avec différents poids (polynomiaux ou exponentiels) et nous prouvons également la stabilité exponentielle de ces solutions.Nous démontrons ensuite pour l’équation de Boltzmann inélastique inhomogène avec terme diffusif le même type de résultat dans des espaces L1 à poids polynomial dans un régime de faible inélasticité. Pour finir, nous étudions dans un cadre général et uniforme des modèles qui convergent vers l’équation de Fokker-Planck du point de vue de l’analyse spectrale et des semi-groupes