{
  "id": "2404.11878",
  "version": "v1",
  "published": "2024-04-18T03:31:40.000Z",
  "updated": "2024-04-18T03:31:40.000Z",
  "title": "Transition threshold for the 2-D Couette flow in whole space via Green's function",
  "authors": [
    "Gaofeng Wang",
    "Weike Wang"
  ],
  "comment": "20pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we investigate the transition threshold problem concerning the 2-D Navier-Stokes equations in the context of Couette flow $(y,0)$ at high Reynolds number $Re$ in whole space. By utilizing Green's function estimates for the linearized equations around Couette flow, we initially establish refined dissipation estimates for the linearized Navier-Stokes equations with a precise decay rate $(1+t)^{-1}.$ As an application, we prove that if the initial perturbation of vorticity satisfies$$\\|\\omega_{0}\\|_{H^{1}\\cap L^1}\\leq c_0\\nu^{\\frac{3}{4}}$$ for some small constant $c_0$ independent of the viscosity $\\nu$, then we can reach the conclusion that the solution remains within $O\\left( \\nu ^{\\frac{3}{4}}\\right) $ of the Couette flow.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-18T03:31:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30"
    ],
    "keywords": [
      "couette flow",
      "establish refined dissipation estimates",
      "high reynolds number",
      "utilizing greens function estimates",
      "transition threshold problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}