A family of systems including the Herschel-Bulkley fluid equations

We analyze the Navier-Stokes equations for incompressible fluids with the {\lq\lq}viscous stress tensor{\rq\rq} $\mathbb{S}$ in a family which includes the Bingham model for viscoplastic fluids (more generally, the Herschel-Bulkley model). $\mathbb{S}$ is the subgradient of a convex potential $V=V(x,t,X)$, allowing that $V$ can depend on the space-time variables $(x,t)$. The potential has its one-sided directional derivatives $V'(X,X)$ uniformly bounded from below and above by a $p$-power function of the matrices $X$. For $p\geqslant 2.2$ we solve an initial boundary value problem for those fluid systems, in a bounded region in $\mathbb{R}^3$. We take a nonlinear boundary condition, which encompasses the Navier friction/slip boundary condition.