Related papers: $\mathrm L^2$ Hypocoercivity methods for kinetic F…
Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the…
We consider the relativistic, spatially inhomogeneous Fokker-Planck equation with an external confining potential. We prove the exponential time decay of solutions towards the global equilibrium in weighted $L^2$ and Sobolov spaces. Our…
This paper is dealing with two $L^2$ hypocoercivity methods based on Fourier decomposition and mode-by-mode estimates, with applications to rates of convergence or decay in kinetic equations on the torus and on the whole Euclidean space.…
We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining…
The purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in $H^1$ space in Villani's memoir \cite{Villani} to higher order Sobolev spaces. As in the $L^2$ and $H^1$ setting, there is lack of…
We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or…
We study time averages for the norm of solutions to kinetic Fokker--Planck equations associated with general Hamiltonians. We provide fully explicit and constructive decay estimates for systems subject to a confining potential, allowing…
We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modelling the fibre lay-down process in the production of non-woven textiles. Following a micro-macro decomposition, we use hypocoercivity…
This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker-Planck equations with non-quadratic confinement potentials in whole space. We extend…
We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the torus with Maxwellian or fat tail local equilibria. Results based on weak norms have recently been achieved by S. Armstrong and J.-C. Mourrat in case of Maxwellian…
In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic L\'evy-Fokker-Planck equation, for which we adapt hypocoercivity…
In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these discretizations of velocity and space, we prove the exponential convergence…
The recent work [11] developed a general framework to show hypocoercivity for a stationary Gibbs state and allowed spatial degeneracy, confining potentials and boundary conditions. In this work, we show that the explicit energy approach in…
In recent work, Chow, Huang, Li and Zhou introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When $N\ge 2$ is the number of vertices of the graph, they show that the corresponding…
In this paper, we study the hypocoercivity for a class of linear kinetic equations with both transport and degenerately dissipative terms. As concrete examples, the relaxation operator, Fokker-Planck operator and linearized Boltzmann…
This memoir attempts at a systematic study of convergence to stationary state for certain classes of degenerate diffusive equations, by means of well-chosen Lyapunov functionals. Typical examples are the kinetic Fokker--Planck and Boltzmann…
A class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered. After formulating the problem in a geometric setting, the question of the…
This paper is devoted to $\phi$-entropies applied to Fokker-Planck and kinetic Fokker-Planck equations in the whole space, with confinement. The so-called $\phi$-entropies are Lyapunov functionals which typically interpolate between Gibbs…
In this note, we consider a kinetic Fokker-Planck-Alignment equation with Rayleigh-type friction and self-propulsion force which is derived from general environmental averaging models. We show the exponential relaxation in time toward…
This paper is devoted to kinetic equations without confinement. We investigate the large time behaviour induced by collision operators with fat tailed local equilibria. Such operators have an anomalous diffusion limit. In the appropriate…