Continuity equation and characteristic flow for scalar Hencky plasticity

We investigate uniqueness issues for a continuity equation arising out of the simplest model for plasticity, Hencky plasticity. The associated system is of the form $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}(\mu \sigma )=0$ where $\mu$ is a nonnegative measure and $\sigma$ a two-dimensional divergence free unit vector field. After establishing the Sobolev regularity of that field, we provide a precise description of all possible geometries of the characteristic flow, as well as of the associated solutions.