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.