{
  "id": "1808.08736",
  "version": "v1",
  "published": "2018-08-27T08:48:40.000Z",
  "updated": "2018-08-27T08:48:40.000Z",
  "title": "Transition threshold for the 2-D Couette flow in a finite channel",
  "authors": [
    "Qi Chen",
    "Te Li",
    "Dongyi Wei",
    "Zhifei Zhang"
  ],
  "comment": "46 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Couette flow $(y,0)$ at large Reynolds number $Re$ in a finite channel. We develop a systematic method to establish the resolvent estimates of the linearized operator and space-time estimates of the linearized Navier-Stokes equations. In particular, three kinds of important effects: enhanced dissipation, inviscid damping and boundary layer, are integrated into the space-time estimates in a sharp form. As an application, we prove that if the initial velocity $v_0$ satisfies $\\|v_0-(y, 0)\\|_{H^2}\\le cRe^{-\\frac 12}$ for some small $c$ independent of $Re$, then the solution of the 2-D Navier-Stokes equations remains within $O(Re^{-\\frac 12})$ of the Couette flow for any time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-27T08:48:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "couette flow",
      "finite channel",
      "space-time estimates",
      "large reynolds number",
      "transition threshold problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}