Homogenization and hypocoercivity for Fokker-Planck equations driven by weakly compressible shear flows

TL;DR

This paper analyzes the long-time behavior of 2D Fokker-Planck equations with shear flows, providing explicit homogenization and decay rates, and connecting these results to enhanced diffusion phenomena in fluid mechanics.

Contribution

It introduces explicit homogenization rates and hypocoercivity-based decay estimates for Fokker-Planck equations with weakly compressible shear flows, linking stochastic homogenization and fluid dynamics.

Findings

Explicit homogenization rates depending on flow parameters.

Quantitative decay rates to invariant measure.

Connection to enhanced diffusion in fluid flows.

Abstract

We study the long-time dynamics of two-dimensional linear Fokker-Planck equations driven by a drift that can be decomposed in the sum of a large shear component and the gradient of a regular potential depending on one spatial variable. The problem can be interpreted as that of a passive scalar advected by a slightly compressible shear flow, and undergoing small diffusion. For the corresponding stochastic differential equation, we give explicit homogenization rates in terms of a family of time-scales depending on the parameter measuring the strength of the incompressible perturbation. This is achieved by exploiting an auxiliary Poisson problem, and by computing the related effective diffusion coefficients. Regarding the long-time behaviour of the solution of the Fokker-Planck equation, we provide explicit decay rates to the unique invariant measure by employing a quantitative version of the classical hypocoercivity scheme. From a fluid mechanics perspective, this turns out to be equivalent to quantifying the phenomenon of enhanced diffusion for slightly compressible shear flows.