{"paper":{"title":"Explicit form of spatially linear Navier-Stokes velocity fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","nlin.CD","nlin.SI"],"primary_cat":"physics.flu-dyn","authors_text":"Gabriel Provencher Langlois, George Haller","submitted_at":"2015-08-27T20:46:17Z","abstract_excerpt":"We show that a smooth linear unsteady velocity field $u(x,t)=A(t)x+f(t)$ solves the incompressible Navier--Stokes equation if and only if the matrix $A(t)$ has zero trace, and $\\dot{{A}}(t)+A^{2}(t)$ is symmetric. In two dimensions, these constraints imply that $A(t)$ is the sum of an arbitrary time-dependent traceless symmetric matrix and an arbitrary constant skew-symmetric matrix. One can, therefore, verify by inspection if an unsteady spatially linear vector field is a Navier--Stokes solution. In three dimensions, we obtain a simple ordinary differential equation that $A(t)$ must solve. Ou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07024","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}