{
  "id": "1910.08911",
  "version": "v1",
  "published": "2019-10-20T06:44:14.000Z",
  "updated": "2019-10-20T06:44:14.000Z",
  "title": "New Regularity Criteria for the Navier-Stokes Equations in Terms of Pressure",
  "authors": [
    "Benjamin Pineau",
    "Xinwei Yu"
  ],
  "comment": "9 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we generalize the main results of [1] and [31] to Lorentz spaces, using a simple procedure. The main results are the following. Let $n\\geq 3$ and let $u$ be a Leray-Hopf solution to the $n$-dimensional Navier-Stokes equations with viscosity $\\nu$ and divergence free initial condition $u_0\\in L^2(\\mathbb{R}^n)\\cap L^{k}(\\mathbb{R}^n)$ (where $k=k(s)$ is sufficiently large). Then there exists a constant $c>0$ such that if \\begin{equation} \\|p\\|_{L^{r,\\infty}(0,\\infty;L^{s,\\infty}(\\mathbb{R}^n))}<c\\hspace{10mm}\\frac{n}{s}+\\frac{2}{r}\\leq 2,\\hspace{5mm}s>\\frac{n}{2} \\end{equation} or \\begin{equation} \\|\\nabla p\\|_{L^{r,\\infty}(0,\\infty;L^{s,\\infty}(\\mathbb{R}^n))}<c\\hspace{10mm}\\frac{n}{s}+\\frac{2}{r}\\leq 3,\\hspace{5mm}s>\\frac{n}{3} \\end{equation} then $u$ is smooth on $(0, \\infty) \\times \\mathbb{R}^n$. Partial results in the case $n=3$ were obtained in [32], [33] and then recently extended to all appropriate pairs of $r,s$ in [14]. Our results present a unified proof which works for all dimensions $n\\geq 3$ and the full range or admissible pairs, $(s,r)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-20T06:44:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regularity criteria",
      "divergence free initial condition",
      "main results",
      "dimensional navier-stokes equations",
      "appropriate pairs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}