Hydrodynamic Limit for a Fokker-Planck Equation with Coefficients in Sobolev Spaces

TL;DR

This paper investigates the hydrodynamic limit of a Fokker-Planck equation with coefficients in Sobolev spaces, analyzing two methods of convergence for particle systems in viscous flows.

Contribution

It introduces two approaches for proving hydrodynamic convergence, one based on weak convergence in weighted L2 spaces and another on entropic methods leading to L1 convergence.

Findings

Demonstrates convergence using weak L2 methods

Establishes L1 convergence via entropy techniques

Provides insights into particle system behavior in viscous flows

Abstract

In this paper we study the hydrodynamic (small mass approximation) limit of a Fokker-Planck equation. This equation arises in the kinetic description of the evolution of a particle system immersed in a viscous Stokes flow. We discuss two different methods of hydrodynamic convergence. The first method works with initial data in a weighted L2 space and uses weak convergence and the extraction of convergent subsequences. The second uses entropic initial data and gives an L1 convergence to the solution of the limit problem via the study of the relative entropy.