Continuity of the spatial gradient of weak solutions to very singular parabolic equations involving the one-Laplacian

We consider weak solutions to very singular parabolic equations involving a one-Laplace-type operator, which is singular and degenerate, and a $p$-Laplace-type operator with $\frac{2n}{n+2}<p<\infty$, where $n\ge 2$ denotes the space dimension. This type of equation is used to describe the motion of a Bingham flow. It has been a long-standing open problem of whether the spatial gradients of weak solutions are continuous in space and time. This paper aims to give an affirmative answer for a wide class of such equations. This equation becomes no longer uniformly parabolic near the facet, the place where the spatial gradient vanishes. To achieve our goal, we show local a priori H\"{o}lder continuity of gradients suitably truncated near facets. For this purpose, we consider a parabolic approximate problem and appeal to standard methods, including De Giorgi's truncation and comparisons with Dirichlet heat flows. Our method is a parabolic adjustment of our method developed to prove the corresponding statements for stationary problems.