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

Xavier Blanc (1) , Claude Le Bris (2) , Pierre Louis Lions (3)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 MATHERIALS - MATHematics for MatERIALS
3 Collège de France - Chaire Équations aux dérivées partielles et applications

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 L p , 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.

Domaines: Equations aux dérivées partielles [math.AP]

Xavier Blanc, Claude Le Bris, Pierre Louis Lions. On correctors for linear elliptic homogenization in the presence of local defects: the case of advection-diffusion. Journal de Mathématiques Pures et Appliquées, 2019, ⟨10.1016/j.matpur.2018.04.010⟩. ⟨hal-01697105v2⟩