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In this paper, we study some regularity issues concerning the gradient ofweak solutions of $u_t - {\rm div} \mathcal{A}(x,t,\nabla u) = g$, where$\mathcal{A}(x,t,\nabla u)$ is modeled after the $p$-Laplace operator. The mainresults we are interested in is to obtain optimal conditions on the datum $g$(independent of time) such that borderline higher integrability of the gradientand Lipschitz estimates for the weak solution holds. Moreover, we develop atheory where we can obtain elliptic type estimates using parabolic theory,which gives improved potential estimates for the elliptic systems.

Borderline gradient regularity estimates for quasilinear parabolic systems with data independent of time