A well-posed continuum theory of dense granular flow
In plain words
The standard equations for flowing grains predict, for some flow conditions, that tiny ripples grow infinitely fast, which means the equations have no sensible solutions there (they are ill-posed). A version that is mathematically sound in every regime and still matches experiments is not known.
Precise statement
The incompressible $\mu(I)$ rheology is linearly ill-posed (unbounded growth rate as wavenumber $k \to \infty$, a Hadamard instability) at small and at large $I$. Find a minimal extension (dilatancy through a volume-fraction law $\Phi(I)$, nonlocal or higher-gradient terms) that is well-posed for all $I$ and matches measured steady and transient flows, and prove well-posedness of the resulting equations.
What would settle it
A proof of linear (and preferably nonlinear) well-posedness for all I of a model that reproduces inclined-plane, shear-cell and silo flow data.
Status in the literature
Unverified note
Compressible models with $\mu(I)$ and $\Phi(I)$ were shown well-posed over a restricted parameter range (late 2010s); no model is known to be well-posed for all $I$ while fitting all data (2026).