  Continuum modelling of granular flow has been plagued with the issue of
ill-posed equations for a long time. Equations for incompressible,
two-dimensional flow based on the Coulomb friction law are ill-posed regardless
of the deformation, whereas the rate-dependent $\mu(I)$-rheology is ill-posed
when the non-dimensional strain-rate $I$ is too high or too low. Here,
incorporating ideas from Critical-State Soil Mechanics, we derive conditions
for well-posedness of PDEs that combine compressibility with $I$-dependent
rheology. When the $I$-dependence comes from a specific friction coefficient
$\mu(I)$, our results show that, with compressibility, the equations are
well-posed for all deformation rates provided that $\mu(I)$ satisfies certain
minimal, physically natural, inequalities.
