{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"04f225a9ae87640c81ee3888d2a8f0ac1280d52bf21694f5b93f33da2f1ffc6d","created":"2026-10-03T07:17:57Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"945b8e31f5f2dac9379aaea7b5b332b4f269530ca4df15f6e37ad90ab89fee04","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"bio.granular-rheology.mu-i-wellposed","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"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.","posed_since":"2015","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.","problem_ref":null,"references":"","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.","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).","title":"A well-posed continuum theory of dense granular flow","topic_ref":"ed3f93b43bbd83eba41e4cf73d3470ba3da603db0979939bf3fb8e305f26445f"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"d78e25de6f49ee6a8172c8680bfdffb47c1fe4a3f18c9ff84f37e94699fdbc59","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"5af1b61679f15fe165882e4deeeacbf3d33c00a8bac531c7663a55f85d93f779","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"QGohvB2rxl2g-CwGakdOdNlLDvB7mzPPRT3JzdZ02cn_MbLdMbJ_G0G__8lCl_TDcio9yg7_qb9o7uSDM0bmCQ"},"schema":"pubphys.envelope/1"},"record_hash":"d78e25de6f49ee6a8172c8680bfdffb47c1fe4a3f18c9ff84f37e94699fdbc59","leaf_index":783}