{
  "id": "2403.12383",
  "version": "v1",
  "published": "2024-03-19T02:47:58.000Z",
  "updated": "2024-03-19T02:47:58.000Z",
  "title": "New regularity criteria for NSE and SQG in critical spaces",
  "authors": [
    "Yiran Xu",
    "Ly Kim Ha",
    "Haina Li",
    "Zexi Wang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we investigate some priori estimates to provide the critical regularity criteria for incompressible Navier-Stokes equations on $\\mathbb{R}^3$ and super critical surface quasi-geostrophic equations on $\\mathbb{R}^2$. Concerning the Navier-Stokes equation, we demonstrate that a Leray-Hopf solution $u$ is regular if $u\\in L_T^{\\frac{2}{1-\\alpha}} \\dot{B}^{-\\alpha}_{\\infty,\\infty}(\\mathbb{R}^3)$, or $u$ in Lorentz space $ L_T^{p,r} \\dot{B}^{-1+\\frac{2}{p}}_{\\infty,\\infty}(\\mathbb{R}^3)$, with $4\\leq p\\leq r<\\infty$. Additionally, an alternative regularity condition is expressed as $u\\in L_{T}^{\\frac{2}{1-\\alpha}} \\dot{B}^{-\\alpha}_{\\infty,\\infty}(\\mathbb{R}^3)+{L_T^\\infty\\dot{B}^{-1}_{\\infty,\\infty}}(\\mathbb{R}^3)$($\\alpha\\in(0,1)$), contingent upon a smallness assumption on the norm $L_T^\\infty\\dot{B}^{-1}_{\\infty,\\infty}$. For the SQG equation, we derive that a Leray-Hopf weak solution $\\theta\\in L_T^{\\frac{\\alpha}{\\varepsilon}} \\dot{C}^{1-\\alpha+\\epsilon}(\\mathbb{R}^2)$ is smooth for any $\\varepsilon$ small enough. Similar to the case of Navier-Stokes equation, we derive regularity criterion in more refined spaces, i.e. Lorentz spaces $L_T^{\\frac{\\alpha}{\\epsilon},r}\\dot{C}^{1-\\alpha+\\epsilon}(\\mathbb{R}^2)$ and addition of two critical spaces $L_{T}^{\\frac{\\alpha}{\\epsilon}}\\dot{C}^{1-\\alpha+\\epsilon}(\\mathbb{R}^2)+{L_T^\\infty\\dot{C}^{1-\\alpha}(\\mathbb{R}^2)}$, with smallness assumption on $L_T^\\infty\\dot{C}^{1-\\alpha}(\\mathbb{R}^2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-19T02:47:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regularity criterion",
      "critical spaces",
      "navier-stokes equation",
      "super critical surface quasi-geostrophic equations",
      "lorentz space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}