Continuity of the Spatial Gradient of Weak Solutions to Very Singular Parabolic Equations Involving the One-Laplacian

Abstract. We consider weak solutions to very singular parabolic equations involving a one-Laplace-type operator, which is singular and degenerate, and a [Formula: see text]-Laplace-type operator with [Formula: see text], where [Formula: see text] 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 Hölder 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.