Motion of a body immersed in a vlasov system

In this chapter we study the motion of a body immersed in a Vlasov system. Such a choice for the medium allows to overcome problems connected to the irregular motion of the body occurring when it interacts with a gas of point particles. On the contrary, in case of a Vlasov system, the motion is expected to be regular. The interaction body/medium is assumed to be hard core, which implies the existence of a stationary motion for any initial data and any intensity of external constant force acting on the body.Moreover,we investigate the asymptotic approach of the body velocity to the limiting one, showing that in case of not self-interacting medium the approach is proportional to an inverse power of time. Such a behavior, surprising for not being exponential as in many viscous friction problems, is due to the recollisions that a single particle of the medium can deliver with the body.