{"paper":{"title":"Dynamical Behavior for the Solutions of the Navier-Stokes Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Baoxiang Wang, Kuijie Li, Tohru Ozawa","submitted_at":"2016-08-24T01:09:53Z","abstract_excerpt":"We study the Cauchy problem for the incompressible Navier-Stokes equations (NS) in three and higher spatial dimensions: \\begin{align} u_t -\\Delta u+u\\cdot \\nabla u +\\nabla p=0, \\ \\ {\\rm div} u=0, \\ \\ u(0,x)= u_0(x). \\label{NSa} \\end{align} Leray and Giga obtained that for the weak and mild solutions $u$ of NS in $L^p(\\mathbb{R}^d)$ which blow up at finite time $T>0$, respectively, one has that for $d <p \\leq \\infty$, $$ \\|u(t)\\|_p \\gtrsim ( T-t )^{-(1-d/p)/2}, \\ \\ 0< t<T. $$ We will obtain the blowup profile and the concentration phenomena in $L^p(\\mathbb{R}^d)$ with $d\\leq p\\leq \\infty$ for t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.06680","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"}