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
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References

SHOWING 1-10 OF 40 REFERENCES
Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation
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 interpretationExpand
Stationary solutions of the vlasov‐fokker‐planck equation
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 pointExpand
On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems : The linear Fokker-Planck equation
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 decayExpand
The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics
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 PhysExpand
HYPOCOERCIVITY FOR LINEAR KINETIC EQUATIONS CONSERVING MASS
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 collisionExpand
A Stochastic Model and Associated Fokker-Planck Equation for the Fiber Lay-Down Process in Nonwoven Production Processes
TLDR
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
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 uniqueExpand
On the Dynamics of Large Particle Systems in the Mean Field Limit
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 theExpand
Long-time effect of relaxation for hyperbolic conservation laws
The hyperbolic conservation laws with relaxation appear in many physical systems such as nonequilibrium gas dynamics, flood flow with friction, viscoelasticity, magnetohydrodynamics, etc. ThisExpand
Hypocoercivity for kinetic equations with linear relaxation terms
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 anExpand
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