Convergence along mean flows

TL;DR

This paper introduces a new asymptotic expansion technique along mean flows for homogenizing convection-dominated parabolic equations with rapidly oscillating coefficients, utilizing ergodic algebra theory.

Contribution

It develops a novel multiple scale asymptotic expansion method along mean flows and defines weak multiple scale convergence for homogenization problems.

Findings

Derived effective diffusion coefficients via averaging along mean flow orbits.

Applied ergodic algebra theory to rigorously justify the homogenization process.

Extended homogenization techniques to convection-dominated equations with fast oscillations.

Abstract

We develop a technique of multiple scale asymptotic expansions along mean flows and a corresponding notion of weak multiple scale convergence. These are applied to homogenize convection dominated parabolic equations with rapidly oscillating, locally periodic coefficients and $\mathcal{O}(\eps^{-1})$ mean convection term. Crucial to our analysis is the introduction of a fast time variable, $\tau=\frac{t}{\eps}$, not apparent in the heterogeneous problem. The effective diffusion coefficient is expressed in terms of the average of Eulerian cell solutions along the orbits of the mean flow in the fast time variable. To make this notion rigorous, we use the theory of ergodic algebras with mean value.