Stochastic Homogenization of a Convection-Diffusion Equation
Hakima Bessaih, Yalchin R. Efendiev, Florian Maris · SIAM Journal on Mathematical Analysis · 2021
In this paper, we study the homogenization of a coupled diffusion-convection system. We consider a highly heterogeneous diffusion-convection equation in space, coupled with a stochastic differential equation with highly heterogeneous temporal scales that accounts for media change. Our main result consists of deriving the macroscale equation. We show that the resulting macroscale equation is deterministic with a nonlinear convective term. The convergence analysis involves the cell problem and the invariant measure of the stochastic differential equation. Although homogenization with changing media properties has been previously studied in other settings, this is the first result of this type, that includes a convective term. The latter makes the coupling between the homogenization and the time averaging much stronger. We show that the convergence to the limiting equation is in probability and we also find a corrector for it that gives stronger convergence.