{
  "id": "2305.04052",
  "version": "v1",
  "published": "2023-05-06T13:47:15.000Z",
  "updated": "2023-05-06T13:47:15.000Z",
  "title": "Asymptotic stability of Couette flow in a strong uniform magnetic field for the Euler-MHD system",
  "authors": [
    "Weiren Zhao",
    "Ruizhao Zi"
  ],
  "comment": "74 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we prove the asymptotic stability of Couette flow in a strong uniform magnetic field for the Euler-MHD system, when the perturbations are in Gevrey-$\\frac{1}{s}$, $(\\frac12<s\\leq 1)$ and of size smaller than the resistivity coefficient $\\mu$. More precisely, we prove (1) the $\\mu^{-\\frac13}$-amplification of the perturbed vorticity, namely, the size of the vorticity grows from $\\|\\omega_{\\mathrm{in}}\\|_{\\mathcal{G}^{\\lambda_{0}}}\\lesssim \\mu$ to $\\|\\omega_{\\infty}\\|_{\\mathcal{G}^{\\lambda'}}\\lesssim \\mu^{\\frac23}$; (2) the polynomial decay of the perturbed current density, namely, $\\left\\|j_{\\neq}\\right\\|_{L^2}\\lesssim \\frac{c_0 }{\\langle t\\rangle^2 }\\min\\left\\{\\mu^{-\\frac13},\\langle t \\rangle\\right\\}$; (3) and the damping for the perturbed velocity and magnetic field, namely, \\[ \\left\\|(u^1_{\\neq},b^1_{\\neq})\\right\\|_{L^2}\\lesssim \\frac{c_0\\mu }{\\langle t\\rangle }\\min\\left\\{\\mu^{-\\frac13},\\langle t \\rangle\\right\\}, \\quad \\left\\|(u^2,b^2)\\right\\|_{L^2}\\lesssim \\frac{c_0\\mu }{\\langle t\\rangle^2 }\\min\\left\\{\\mu^{-\\frac13},\\langle t \\rangle\\right\\}. \\] We also confirm that the strong uniform magnetic field stabilizes the Euler-MHD system near Couette flow.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-06T13:47:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "couette flow",
      "euler-mhd system",
      "asymptotic stability",
      "strong uniform magnetic field stabilizes",
      "resistivity coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 74,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}