{
  "id": "1809.06712",
  "version": "v1",
  "published": "2018-09-15T19:19:55.000Z",
  "updated": "2018-09-15T19:19:55.000Z",
  "title": "Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces",
  "authors": [
    "Hongjie Dong",
    "Kunrui Wang"
  ],
  "comment": "32 pages, submitted. arXiv admin note: text overlap with arXiv:0903.1461",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove if $u\\in L_{\\infty}^tL_d^x((0,T)\\times \\mathbb{R}^d_+)$ is a Leray-Hopf weak solution vanishing on the boundary and the pressure $p$ satisfies a local condition $\\|p\\|_{L_{2-1/d}(Q(z_0,1)\\cap (0,T)\\times \\mathbb{R}^d_+)}\\leq K$ for some constant $K>0$ uniformly in $z_0$, then $u$ is regular up to the boundary in $(0,T)\\times \\mathbb{R}^d_+$. Furthermore, when $T=\\infty$, $u$ tends to zero as $t\\rightarrow \\infty$. We also study the local problem in half unit cylinder $Q^+$ and prove that if $u\\in L^t_{\\infty}L^x_d(Q^+)$ and $ p\\in L_{2-1/d}(Q^+)$, then $u$ is H\\\"{o}lder continuous in the closure of the set $Q^+(1/4)$. This generalizes a result by Escauriaza, Seregin, and \\v{S}ver\\'{a}k to higher dimensions and domains with boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-15T19:19:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical lebesgue spaces",
      "boundary regularity",
      "dimensional incompressible navier-stokes equations",
      "study regularity criteria",
      "half unit cylinder"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}