{"paper":{"title":"Random-data Cauchy Problem for the Periodic Navier-Stokes Equations with Initial Data in Negative-order Sobolev Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Chao Deng, Shangbin Cui","submitted_at":"2011-03-31T13:18:04Z","abstract_excerpt":"In this paper we study existence of solutions of the initial-boundary value problems of the Navier-Stokes equations with a periodic boundary value condition for initial data in the Sobolev spaces $\\mathcal{H}^{s}(\\mathbb{T}^N)$ with a negative order $-1<s<0$, where $N=2, 3$. By using the randomization approach of N. Burq and N. Tzvetkov, we prove that for almost all $\\omega\\in\\Omega$, where $\\Omega$ is the sample space of a probability space $(\\Omega,\\mathcal{A},p)$, for the randomized initial data $\\vec{f}^\\omega\\in\\mathcal{H}_{\\sigma}^{s}(\\mathbb{T}^N)$ with $-1<s<0$, such a problem has a un"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.6170","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"}