{
  "id": "2011.10169",
  "version": "v1",
  "published": "2020-11-20T01:45:55.000Z",
  "updated": "2020-11-20T01:45:55.000Z",
  "title": "Alternative Theorem of Navier-Stokes Equations in $\\mathbb{R}^3$",
  "authors": [
    "Yongqian Han"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider Cauchy problem of the incompressible Navier-Stokes equations with initial data $u_0\\in L^1(\\mathbb{R}^3)\\cap L^{\\infty}(\\mathbb{R}^3)$. There exist a maximum time interval $[0,T_{max})$ and a unique solution $u\\in C\\big([0,T_{max}); L^2(\\mathbb{R}^3) \\cap L^p(\\mathbb{R}^3)\\big)$ ($\\forall p>3$). We find one of function class $S_{regular}$ defined by scaling invariant norm pair such that $T_{max}=\\infty$ provided $u_0\\in S_{regular}$. Especially, $\\|u_0\\|_{L^p}$ is arbitrarily large for any $u_0\\in S_{regular}$ and $p>3$. On the other hand, the alternative theorem is proved. It is that either $T_{max}= \\infty$ or $T_{max}\\in(T_l,T_r]$. Especially, $T_r<T_{max}<\\infty$ is disappearing. Here the explicit expressions of $T_l$ and $T_r$ are given. This alternative theorem is one kind of regular criterion which can be verified by computer. If $T_{max}=\\infty$, the solution $u$ is regular for any $(t,x)\\in(0,\\infty) \\times \\mathbb{R}^3$. As $t\\rightarrow\\infty$, the solution is decay. On the other hand, lower bound of blow up rate of $u$ is obtained again provided $T_{max}\\in (T_l,T_r]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-20T01:45:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "alternative theorem",
      "scaling invariant norm pair",
      "maximum time interval",
      "function class",
      "initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}