{
  "id": "2210.16600",
  "version": "v1",
  "published": "2022-10-29T13:45:42.000Z",
  "updated": "2022-10-29T13:45:42.000Z",
  "title": "Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field",
  "authors": [
    "Hongxia Lin",
    "Jiahong Wu",
    "Yi Zhu"
  ],
  "comment": "46 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and damps electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in $\\mathbb R^3$. The velocity equation in this system is the 3D Navier-Stokes equation with dissipation only in the $x_1$-direction while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field $(0,1,0)$ is globally stable in the Sobolev setting $H^3(\\mathbb R^3)$. In addition, explicit decay rates in $H^2(\\mathbb R^3)$ are also obtained. When there is no presence of the magnetic field, the 3D anisotropic Navier-Stokes equation in $\\mathbb R^3$ is not well understood and the small data global well-posedness remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps stabilize the fluid.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-29T13:45:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35"
    ],
    "keywords": [
      "background magnetic field",
      "3d incompressible mhd equations",
      "partial dissipation",
      "large-time behavior",
      "field generates enhanced dissipation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}