3 years ago

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

X. Blanc, C. Le Bris, P.-l. Lions

Publication date: Available online 21 April 2018

Source: Journal de Mathématiques Pures et Appliquées

Author(s): X. Blanc, C. Le Bris, P.-L. Lions

##### 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 $−aij∂iju=f$. The present work is devoted to advection–diffusion equations: $−aij∂iju+bj∂ju=f$. We prove, under suitable assumptions on the coefficients $aij$, $bj$, $1≤i,j≤d$ (typically that they are the sum of a periodic function and some perturbation in $Lp$, for suitable $p<+∞$), 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.

##### Résumé

Nous continuous ici notre série de travaux sur l'homogénéisation d'équations linéaires elliptiques du second ordre dont les coefficients sont des perturbations de coefficients périodiques. Nous avons considéré les équations sous forme divergence dans [6], [7], [8], et, grâce à une stratégie de preuve légèrement différente, l'équation $−aij∂iju=f$ dans notre travail plus récent [9]. Ici, nous étudions les équations d'advection–diffusion : $−aij∂iju+bj∂ju=f$. Nous montrons, sous des hypothèses adéquates sur les coefficients $aij$, $bj$, $1≤i,j≤d$ (typiquement qu'ils sont somme d'une fonction périodique et d'une fonction dans $Lp$, pour $p<+∞$ convenable), que l'équation admet une (unique) mesure invariante qui peut être utilisée pour transformer l'équation en une équation sous forme divergence, équation qu'on

-Abstract Truncated-

DOI: S0021782418300540

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