Gradient Estimates for Parabolic Systems from Composite Material

In this paper we derive $W^{1,\mathrm{\infty }}$ and piecewise $C^{1,\alpha }$ estimates for solutions, and their $t-$derivatives, of divergence form parabolic systems with coefficients piecewise H"older continuous in space variables $x$ and smooth in $t$. This is an extension to parabolic systems of results of Li and Nirenberg on elliptic systems. These estimates depend on the shape and the size of the surfaces of discontinuity of the coefficients, but are independent of the distance between these surfaces.