On correctors for linear elliptic homogenization in the presence of local defects: the case of advection-diffusion

TL;DR

This paper extends homogenization techniques to advection-diffusion equations with locally perturbed coefficients, establishing the existence of invariant measures and transforming the problem into divergence form for analysis.

Contribution

It introduces a method to handle advection-diffusion equations with perturbed coefficients by constructing invariant measures, enabling the use of divergence form homogenization techniques.

Findings

Existence of a unique invariant measure for the perturbed advection-diffusion equations.

Transformation of the original problem into divergence form using the invariant measure.

Applicability of previous divergence form homogenization methods to more general advection-diffusion equations.

Abstract

We follow-up on our works devoted to homogenization theory for linear second-order elliptic equations with coefficients that are perturbations of periodic coefficients. We have first considered equations in divergence form in [6, 7, 8]. We have next shown, in our recent work [9], using a slightly different strategy of proof than in our earlier works, that we may also address the equation --aij$\partial$iju = f. The present work is devoted to advection-diffusion equations: --aij$\partial$iju + bj$\partial$ju = f. We prove, under suitable assumptions on the coefficients aij, bj, 1 $\le$ i, j $\le$ d (typically that they are the sum of a periodic function and some perturbation in L p , for suitable p < +$\infty$), that the equation admits a (unique) invariant measure and that this measure may be used to transform the problem into a problem in divergence form, amenable to the techniques we have previously developed for the latter case.