Regularity theory for sub-critical $p$-parabolic systems with measurable coefficients

A quantitative regularity theory is developed for weak solutions to the parabolic system $${\mathrm{\partial }}_{t}u-\mathrm{d}\mathrm{i}\mathrm{v}\mathsf{A}(x,t,Du)=0\text{in }{E}_T\subset {\mathbb{R}}^N\times \mathbb{R},$$ which features the $p$-Laplacian with measurable coefficients. We focus on the sub-critical range $1<p\le \frac{2N}{N+2}$ and obtain two main results. \emph{Local boundedness:} starting from an $L^{\mathsf{r}}$-control of $u$ with $\mathsf{r}>\frac{N(2-p)}{p}$, we derive sharp, scale-invariant $L^{\mathrm{\infty }}$-estimates. \emph{Higher integrability of the gradient:} $\mathrm{\mid }Du\mathrm{\mid }$ self-improves from $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p$ to $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{p(1+\epsilon )}$ for some $\epsilon >0$ depending only on the data. The same results still hold given proper source terms.