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We consider weak solutions to very singular parabolic equations involving aone-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 themotion of a Bingham flow. It has been a long-standing open problem of whetherthe spatial gradients of weak solutions are continuous in space and time. Thispaper aims to give an affirmative answer for a wide class of such equations.This equation becomes no longer uniformly parabolic near the facet, the placewhere the spatial gradient vanishes. To achieve our goal, we show local apriori H\"{o}lder continuity of gradients suitably truncated near facets. Forthis purpose, we consider a parabolic approximate problem and appeal tostandard methods, including De Giorgi's truncation and comparisons withDirichlet heat flows. Our method is a parabolic adjustment of our methoddeveloped to prove the corresponding statements for stationary problems.

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