Gradient continuity for the parabolic $(1,\,p)$-Laplace system

This paper deals with the parabolic $(1,\,p)$-Laplace system, a parabolic system that involves the one-Laplace and $p$-Laplace operators with $p\in(1,\,\infty)$. We aim to prove that a spatial gradient is continuous in space and time. An external force term is treated under the optimal regularity assumption in the parabolic Lebesgue spaces. We also discuss a generalized parabolic system with the Uhlenbeck structure. A main difficulty is that the uniform ellipticity of the $(1,\,p)$-Laplace operator is violated on a facet, or the degenerate region of a spatial gradient. The gradient continuity is proved by showing local H\"{o}lder continuity of a truncated gradient, whose support is far from the facet. This is rigorously demonstrated by considering approximate parabolic systems and deducing various regularity estimates for approximate solutions by classical methods such as De Giorgi's truncation, Moser's iteration, and freezing coefficient arguments. A weak maximum principle is also utilized when $p$ is not in the supercritical range.