Schauder estimates for parabolic $p$-Laplace systems

We establish the local Hölder regularity of the spatial gradient of bounded weak solutions $u\text{\hspace{0.17em}⁣}:E_T\to {\mathbb{R}}^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big( a(x,t)\big(μ^2+|Du|^2\big)^\frac{p-2}2Du\Big)=0 \qquad\mbox{in $E_T$}, \end{equation*} where $p>1$, $\mu \in [0,1]$, and the coefficient $a\in L^{\mathrm{\infty }}(E_T)$ is bounded below by a positive constant and is Hölder continuous in the space variable $x$. As an application, we prove Hölder estimates for the gradient of weak solutions to a doubly non-linear parabolic equation in the super-critical fast diffusion regime.