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

Xavier Blanc; C. Le Bris; P. -L Lions

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arXiv · 1801.10330

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

Xavier Blanc·C. Le Bris·P. -L Lions

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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.

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Xavier Blanc, C. Le Bris, P. -L Lions. 2018-01-31. On correctors for linear elliptic homogenization in the presence of local defects: the case of advection-diffusion. https://arxiv.org/abs/1801.10330

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