Convergence rates in homogenization of parabolic systems with locally periodic coefficients

In this paper we study the quantitative homogenization of second-order parabolic systems with locally periodic (in both space and time) coefficients. The O(?) scale-invariant error estimate in L2(0,T;L2dd?1(?)) is established C1,1 cylinders under minimum smoothness conditions on This process relies critical estimates smoothing operators. We also develop a new construction flux correctors manner sharp for temporal boundary layers.