{
  "id": "2107.04157",
  "version": "v1",
  "published": "2021-07-09T00:26:54.000Z",
  "updated": "2021-07-09T00:26:54.000Z",
  "title": "The energy conservation and regularity for the Navier-Stokes equations",
  "authors": [
    "W. Tan",
    "Z. Yin"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies $\\lim_{t\\to T}\\sqrt{T-t}||u(t)||_{BMO}<\\infty$ and $\\lim_{t\\to T}\\sqrt{T-t}||u(t)||_{L^\\infty}=\\infty$ to demonstrate that the Type II singularity is admissible in the endpoint case $u\\in L^{2,\\infty}(BMO)$. Secondly, we prove that if a suitable weak solution $u(t,x)$ satisfying $||u||_{L^{2,\\infty}([0,T];BMO(\\Omega))}<\\infty$ for arbitrary $\\Omega\\subseteq\\mathbb{R}^3$ then the local energy equality is valid on $[0,T]\\times\\Omega$. As a corollary, we also prove $||u||_{L^{2,\\infty}([0,T];BMO(\\mathbb{R}^3))}<\\infty$ implies the global energy equality on $[0,T]$. Thirdly, we show that as the solution $u$ approaches a finite blowup time $T$, the norm $||u(t)||_{BMO}$ must blow up at a rate faster than $\\frac{c}{\\sqrt{T-t}}$ with some absolute constant $c>0$. Furthermore, we prove that if $||u_3||_{L^{2,\\infty}([0,T];BMO(\\mathbb{R}^3))}=M<\\infty$ then there exists a small constant $c_M$ depended on $M$ such that if $||u_h||_{L^{2,\\infty}([0,T];BMO(\\mathbb{R}^3))}\\leq c_M$ then $u$ is regular on $(0,T]\\times\\mathbb{R}^3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-09T00:26:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "energy conservation",
      "navier-stokes equations",
      "regularity",
      "weak solution",
      "endpoint case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}