A Fokker-Planck Study Motivated by a Problem in Fluid-Particle Interactions

Abstract

This dissertation is a study of problems that relate to a Fokker-Planck (Klein-Kramers)

equation with hypoelliptic structure. The equation describes the statistics of motion

of an ensemble of particles in a viscous fluid that follows the Stokes ’ equations of

fluid motion. The significance in this problem is that it relates to a variety of phenomena

besides its obvious connection to the study of macromolecular chains that are composed by

particle “ units ” in creeping flows. Such phenomena range from Kramers escape probability

(for a particle trapped in a potential well), to stellar dynamics. The problem can also be

seen as a simplified version of the Vlasov-Poisson-Fokker-Planck system that mainly describes

electrostatic models in plasma physics and gravitational forces between galaxies.

Well-posedeness of the equation has been studied by many authors, including the

case of irregular coefficients (Lions-Le Bris). The study of Sobolev regularity

is interesting in its own right and can be performed with fairly elementary tools (He\'rau,Villani,…).

We are interested here with short time estimates and with how smoothing proceeds in time.

Different types of Lyapunov functionals can be constructed depending on the

type of initial data to show regularization. Of particular interest is a recent

technique developed by C.Villani that builds upon a system of differential inequalities

and is being implemented here for the slightly more involved case of non constant friction.

The question of asymptotic convergence to a stationary state is also discussed, with

techniques that are similar to certain extend to the ones used in regularization but which in general

involve more computations.

Finally, we examine the hydrodynamic (zero mass) limit of the parametrized version of the

Fokker-Planck equation. We discuss two different approaches of hydrodynamic convergence.

The first uses weak compactness principles

of extracting subsequences that are shown to converge to a solution of the limit problem, and

works with initial data in weighted L^{2} setting. The second is based

on the study of relative entropy, gives L^{1} convergence to a solution of the limit problem, and

uses entropic initial data.