# Trend to Equilibrium for a Delay Vlasov-Fokker-Planck Equation and Explicit Decay Estimates

@article{Klar2017TrendTE, title={Trend to Equilibrium for a Delay Vlasov-Fokker-Planck Equation and Explicit Decay Estimates}, author={A. Klar and L. Kreusser and O. Tse}, journal={SIAM J. Math. Anal.}, year={2017}, volume={49}, pages={3277-3298} }

In this paper, a delay Vlasov-Fokker-Planck equation associated to a stochastic interacting particle system with delay is investigated analytically. Under certain restrictions on the parameters well-posedness and ergodicity of the mean-field equation are shown and an exponential rate of convergence towards the unique stationary solution is proven as long as the delay is finite. For infinte delay i.e., when all the history of the solution paths are taken into consideration polynomial decay of… Expand

#### One Citation

Convergence to Equilibrium in Wasserstein Distance for Damped Euler Equations with Interaction Forces

- Mathematics
- 2017

We develop tools to construct Lyapunov functionals on the space of probability measures in order to investigate the convergence to global equilibrium of a damped Euler system under the influence of… Expand

#### References

SHOWING 1-10 OF 40 REFERENCES

Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation

- Mathematics
- 2009

We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation… Expand

Stationary solutions of the vlasov‐fokker‐planck equation

- Mathematics
- 1987

We study stationary solutions of the Vlasov-Fokker-Planck Equation, a modification of the Vlasov-Poisson Equation, obtained by adding a diffusion term with respect to velocity. From a physical point… Expand

On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems : The linear Fokker-Planck equation

- 2004

In this interesting article the authors study the long-time behaviour of kinetic equations with a degenerate collision operator action only on the velocity variable v. The main result proves a decay… Expand

The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics

- Physics, Mathematics
- 2012

The present work establishes the mean-field limit of a N-particle system towards a regularized variant of the relativistic Vlasov-Maxwell system, following the work of Braun-Hepp [Commun Math Phys… Expand

HYPOCOERCIVITY FOR LINEAR KINETIC EQUATIONS CONSERVING MASS

- Physics, Mathematics
- 2010

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… Expand

A Stochastic Model and Associated Fokker-Planck Equation for the Fiber Lay-Down Process in Nonwoven Production Processes

- Mathematics, Computer Science
- SIAM J. Appl. Math.
- 2007

A stochastic model and its associated Fokker–Planck equation for the lay-down of fibers on a conveyor belt in the production process of nonwoven materials is presented and used to compute the distribution of functionals of the process that are important for the quality assessment of industrial fabrics. Expand

Exponential Rate of Convergence to Equilibrium for a Model Describing Fiber Lay-Down Processes

- Mathematics
- 2012

This paper is devoted to the adaptation of the method developed in [4,3] to a Fokker-Planck equation for fiber lay-down which has been studied in [1,5]. Exponential convergence towards a unique… Expand

On the Dynamics of Large Particle Systems in the Mean Field Limit

- Mathematics, Physics
- 2016

This course explains how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics—such as the Vlasov-Poisson system, the vorticity formulation of the… Expand

Long-time effect of relaxation for hyperbolic conservation laws

- Physics
- 1995

The hyperbolic conservation laws with relaxation appear in many physical systems such as nonequilibrium gas dynamics, flood flow with friction, viscoelasticity, magnetohydrodynamics, etc. This… Expand

Hypocoercivity for kinetic equations with linear relaxation terms

- Mathematics, Physics
- 2008

Abstract This Note is devoted to a simple method for proving the hypocoercivity associated to a kinetic equation involving a linear time relaxation operator. It is based on the construction of an… Expand