Related papers: Hypocoercivity and sub-exponential local equilibri…
We introduce in this paper a new approach to the problem of the convergence to equilibrium for kinetic equations. The idea of the approach is to prove a 'weak' coercive estimate, which implies exponential or polynomial convergence rate. Our…
This paper is concerned with the hypercoercivity property of solutions to the Cauchy problem on the linear Boltzmann equation with a confining potential force. We obtain the exponential time rate of solutions converging to the steady state…
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 establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary…
A compressible Oldroyd--B type model with stress diffusion is derived from a compressible Navier--Stokes--Fokker--Planck system arising in the kinetic theory of dilute polymeric fluids, where polymer chains immersed in a barotropic,…
We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier-Stokes equations in a bounded Lipschitz domain,…
This paper is devoted to the linearized Vlasov-Poisson-Fokker-Planck system in presence of an external potential of confinement. We investigate the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar…
In this article we develop a new abstract strategy for proving ergodicity with explicit computable rate of convergence for diffusions associated with a degenerate Kolmogorov operator L. A crucial point is that the evolution operator L may…
We develop and analyze a class of structure-preserving discontinuous Galerkin schemes for the nonlinear Vlasov-Poisson-Fokker-Planck model, reformulated as a hyperbolic system through a Hermite expansion in the velocity variable. We…
We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ or on the whole space $(x,v) \in…
A kinetic model for semiconductor devices is considered on a flat torus. We prove exponential decay to equilibrium for this non-linear kinetic model by hypocoercivity estimates. This seems to be the first hypocoercivity result for this…
This paper is devoted to Fokker-Planck and linear kinetic equations with very weak confinement corresponding to a potential with an at most logarithmic growth and no integrable stationary state. Our goal is to understand how to measure the…
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…
We consider Kramers-Fokker-Planck operators with general degenerate coefficients. We prove semiclassical hypocoercivity estimates for a large class of such operators. Then, we manage to prove Eyring-Kramers formulas for the bottom of the…
In this paper, we provide a general framework to study general class of linear and nonlinear kinetic equations with random uncertainties from the initial data or collision kernels, and their stochastic Galerkin approximations, in both…
This paper concerns the existence of global weak solutions \`a la Leray for compressible Navier-Stokes equations with a pressure law that depends on the density and on time and space variables $t$ and $x$. The assumptions on the pressure…
We revisit Villani's approach to the study of hypocoercive diffusion operators by applying a variant of the Bakry-\'Emery machinery. The method relies on a generalized Bakry-\'Emery type criterion that applies to Kolmogorov type operators.…
We study a nonlocal approximation of the Fokker-Planck equation in which we can estimate the speed of convergence to equilibrium in a way which does not degenerate as we approach the local limit of the equation. This uniform estimate cannot…
We study the relaxation to equilibrium for a class linear one-dimensional Fokker-Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker-Planck equations is that, for…
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…