{
  "id": "1608.06680",
  "version": "v1",
  "published": "2016-08-24T01:09:53.000Z",
  "updated": "2016-08-24T01:09:53.000Z",
  "title": "Dynamical Behavior for the Solutions of the Navier-Stokes Equation",
  "authors": [
    "Kuijie Li",
    "Tohru Ozawa",
    "Baoxiang Wang"
  ],
  "comment": "45 Pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "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 the blowup mild solution. On the other hand, if the Fourier support has the form ${\\rm supp} \\ \\widehat{u_0} \\subset \\{\\xi\\in \\mathbb{R}^n: \\xi_1\\geq L \\}$ and $\\|u_0\\|_{\\infty} \\ll L$ for some $L >0$, then \\eqref{NSa} has a unique global solution $u\\in C(\\mathbb{R}_+, L^\\infty)$. Finally, if the blowup rate is of type I: $$ \\|u(t)\\|_p \\sim ( T-t )^{-(1-d/p)/2}, \\ for \\ 0< t<T<\\infty, \\ d<p<\\infty $$ in 3 dimensional case, then we can obtain a minimal blowup solution $\\Phi$ for which $$ \\inf \\{\\limsup_{t \\to T}(T-t)^{(1-3/p)/2}\\|u(t)\\|_{L^p_x}: \\ u\\in C([0,T); L^p) \\mbox{\\ solves \\eqref{NSa}}\\} $$ is attainable at some $\\Phi \\in L^\\infty (0,T; \\ \\dot B^{-1+6/p}_{p/2,\\infty})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-24T01:09:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "navier-stokes equation",
      "dynamical behavior",
      "unique global solution",
      "minimal blowup solution",
      "blowup mild solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}