Uniform regularity estimates in parabolic homogenization

We consider a family of second-order parabolic systems in divergence formwith rapidly oscillating and time-dependent periodic coefficients, arising in the theory of homogenization. We obtain uniform interior W^1,p, Holder, and Lipschitz estimates as well as boundary W^1,p and Holder estimates, using compactness methods. As a consequence, we establish uniform W^1,p estimates for the initial-Dirichlet problems in C¹ cylinders.