{
  "id": "2108.11940",
  "version": "v1",
  "published": "2021-08-26T17:53:01.000Z",
  "updated": "2021-08-26T17:53:01.000Z",
  "title": "Large time behavior of solutions to the $3$D anisotropic Navier-Stokes equation",
  "authors": [
    "MIkihiro Fujii"
  ],
  "comment": "28 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the large time behavior of the solution to the $3$D Navier-Stokes equation with horizontal viscosity $\\Delta{\\rm h} u=\\partial_1^2 u+\\partial_2^2 u$ and show that the $L^p$ decay rate of the horizontal components of the velocity fields coinside to that of the $2$D heat kernel, while the vertical component behaves like the $3$D heat kernel. Moreover, we find that the asymptotic profile of the horizontal components of the velocity field is different from that of the vertical component.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-26T17:53:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "35B40",
      "35Q35"
    ],
    "keywords": [
      "large time behavior",
      "anisotropic navier-stokes equation",
      "horizontal components",
      "heat kernel",
      "velocity fields coinside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}