Lei Zhang

Abstract: This talk addresses the issue of homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available [2].

We develop the method in [1] and consider divergence form linear parabolic operators in $ \Omega\in\mathbb{R}^{n}$ with $ L^{\infty}(\Omega\times(0,T))$ coefficients. Actually, we can show that under a parabolic Cordes condition the first order time derivative and second order space derivatives are $ L^{2}$ w.r.t harmonic coordinates (instead of $ H^{-1}$ in Euclidean coordinates). Therefore we can approximate the solution space in $ H^{1}$ norm with a functional space of dimension $ n$. It follows that once we solved the equation at least $ n$-times, we can homogenize them both in time and space.

Similar idea can also be applied to acoustic wave equation [3].