{
  "id": "2212.05809",
  "version": "v1",
  "published": "2022-12-12T10:08:48.000Z",
  "updated": "2022-12-12T10:08:48.000Z",
  "title": "Liouville--type Theorems for Steady MHD and Hall--MHD Equations in $\\R^2 \\times \\T$",
  "authors": [
    "Wentao Hu",
    "Zhengce Zhang"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the Liouville--type theorems for three--dimensional stationary incompressible MHD and Hall--MHD systems in a slab with periodic boundary condition. We show that, under the assumptions that $(u^\\theta,b^\\theta)$ or $(u^r,b^r)$ is axisymmetric, or $(ru^r,rb^r)$ is bounded, any smooth bounded solution to the MHD or Hall--MHD system with local Dirichlet integral growing as an arbitrary power function must be constant. This hugely improves the result of \\cite[Theorem 1.2]{pan2021Liouville}, where the Dirichlet integral of $\\mathbf{u}$ is assumed to be finite. Motivated by \\cite[Bang--Gui--Wang--Xie, 2022, {\\it arXiv:2205.13259}]{bang2022Liouvilletype}, our proof relies on establishing Saint--Venant's estimates associated with our problem, and the result in the current paper extends that for stationary Navier--Stokes equations shown by \\cite{bang2022Liouvilletype} to MHD and Hall--MHD equations. To achieve this, more intricate estimates are needed to handle the terms involving $\\mathbf{b}$ properly.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-12T10:08:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B53",
      "35Q30",
      "35B10",
      "76D05",
      "76W05"
    ],
    "keywords": [
      "hall-mhd equations",
      "liouville-type theorems",
      "steady mhd",
      "hall-mhd system",
      "periodic boundary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}