{"paper":{"title":"Remarks on the Liouville type problem in the stationary 3D Navier-Stokes equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dongho Chae","submitted_at":"2015-02-17T04:17:16Z","abstract_excerpt":"We study the Liouville type problem for the stationary 3D Navier-Stokes equations on $\\Bbb R^3$. Specifically, we prove that if $v$ is a smooth solution to (NS) satisfying $\\omega={\\rm curl}\\,v \\in L^q (\\Bbb R^3) $ for some $\\frac32 \\leq q< 3$, and $|v(x)|\\to 0$ as $|x|\\to +\\infty$, then either $v=0$ on $\\Bbb R^3$, or $\\int_{\\Bbb R^6} \\Phi_+ dxdy=\\int_{\\Bbb R^6} \\Phi_- dxdy=+\\infty$, where $\\Phi(x,y) :=\\frac{1}{4\\pi}\\frac{\\omega (x)\\cdot(x-y)\\times (v(y)\\times \\omega(y) )}{|x-y|^3} $, and $\\Phi_\\pm:=\\max\\{ 0, \\pm \\Phi\\}$. The proof uses crucially the structure of nonlinear term of the equation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.04793","kind":"arxiv","version":1},"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"}