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Accepted · first revision · record

Title: A well-posed continuum theory of dense granular flow
Plain: 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: 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.
Settled by: 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.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: partially-resolved
Status 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).
Posed since: 2015