{"paper":{"title":"Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Guohuan Zhao, Xicheng Zhang","submitted_at":"2019-04-08T22:39:11Z","abstract_excerpt":"In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely, for any Leray's solution ${\\mathbf u}$ of 3D-NSE and each $(s,x)\\in\\mathbb{R}_+\\times\\mathbb{R}^3$, we show the existence of weak solutions to the following SDE, which has a density $\\rho_{s,x}(t,y)$ belonging to $\\mathbb{H}^{1,p}_q$ provided $p,q\\in[1,2)$ with $\\frac{3}{p}+\\frac{2}{q}>4$: $$ \\mathrm{d} X_{s,t}={\\mathbf u} (s,X_{s,t})\\mathrm{d} t+\\sqrt{2\\nu}\\mathrm{d} W_t,\\ \\ X_{s,s}=x,\\ \\ t\\geq s, $$ where $W$ is a three dimensional standard Brow"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.04387","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1904.04387/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}