$H^2$-regularity for stationary and non-stationary Bingham problems with perfect slip boundary condition

$H^2$-spatial regularity of stationary and non-stationary problems for Bingham fluids formulated with the pseudo-stress tensor is discussed. The problem is mathematically described by an elliptic or parabolic variational inequality of the second kind, to which weak solvability in the Sobolev space $H^1$ is well known. However, higher regularity up to the boundary in a bounded smooth domain seems to remain open. This paper indeed shows such $H^2$-regularity if the problems are supplemented with the so-called perfect slip boundary condition and if the yield stress vanishes on the boundary. For the stationary Bingham--Stokes problem, the key of the proof lies in a priori estimates for a regularized problem avoiding investigation of higher pressure regularity, which seems difficult to get in the presence of a singular diffusion term. The $H^2$-regularity for the stationary case is then directly applied to establish strong solvability of the non-stationary Bingham--Navier--Stokes problem, based on discretization in time and on the truncation of the nonlinear convection term.