Homogenization of Transport Equations: Weak Mean Field Approximation

We are interested, with respect to the small parameter $\epsilon$, in the behavior of solutions $\rho^\epsilon$ of the conservative advection-diffusion equation $\partial_t\rho^\epsilon + \nabla_x\cdot(\rho^\epsilon u^\epsilon)=\eta\Delta_x\rho^\epsilon$, driven by a large velocity field, $|u^\epsilon|=\mathcal O(1/\epsilon)$, which oscillates periodically with respect to time and space variables. The novelty of our approach compared to that of previous works is that we deal with the periodic case in its full generality. In particular, the cell equation which allows us to compute effective coefficients is parabolic and not elliptic. We also derive estimates on the homogenized solution via entropy methods.