Uniform Regularity Estimates in Parabolic Homogenization

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

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