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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 whether aspatial gradient of a weak solution is continuous in space and time. This paperaims to give an affirmative answer for a wide class of such equations. Thisequation becomes no longer uniformly parabolic near a facet, the place where aspatial gradient vanishes. To achieve our goal, we show local a prioriH\"{o}lder continuity of gradients suitably truncated near facets. For thispurpose, we consider a parabolic approximate problem and appeal to standardmethods, including De Giorgi's truncation and comparisons with Dirichlet heatflows. Our method is a parabolic adjustment of our method developed to provethe corresponding statements for stationary problems.

Continuity of a spatial gradient of a weak solution to a very singular parabolic equation involving the one-Laplacian