On Variational Models for Quasi-static Bingham Fluids

We study a quasi-static incompressible flow of Bingham type with constituent law \ \begin{array}{ll} T = p\left| {\cal E}u\right| ⁁{p-2}{\cal E}u+\beta \frac{{\cal E}u}{\left| {\cal E}u\right| } & \text{if }{\cal E}u\neq 0, \\ \left| T\right| \leq \beta & \text{if }{\cal E}u = 0, \end{array} \] T = p∣ℰu∣^p-2ℰu+β ℰu ∣ℰu∣ if ℰu≠0, ∣T∣⩽β if ℰu = 0, where p≥2 and β>0. Here ℰu denotes the strain velocity and T the corresponding stress. The problem admits a variational formulation in the sense that the velocity field u minimizes the energy I(u) = ∫_Ω∣ℰu∣^p+β∣ℰu∣dx in the space {v∈H^1,p(Ω,ℝⁿ): div v = 0} subject to appropriate boundary conditions. We then show smoothness of u on the set {x∈Ω: ℰu≠0}.