H\"older-Zygmund Estimates for Degenerate Parabolic Systems

We consider energy solutions of the inhomogeneous parabolic $p$-Laplacien system ${\mathrm{\partial }}_{t}u-\text{div}(\mathrm{\mid }Du{\mathrm{\mid }}^{p-2}Du)=-\text{div}g$. We show in the case $p\ge 2$ that if the right hand side $g$ is locally in $L^{\mathrm{\infty }}(\text{BMO})$, then $u$ is locally in $L^{\mathrm{\infty }}({\mathcal{C}}^1)$, where ${\mathcal{C}}^1$ is the 1-H"older–Zygmund space. This is the borderline case of the Calder'on-Zygmund theorey. We provide local quantitative estimates. We also show that finer properties of $g$ are conserved by $Du$, e.g.\ H"older continuity. Moreover, we prove a new decay for gradients of $p$-caloric solutions for all $\frac{2n}{n+2}<p<\mathrm{\infty }$.