{
  "id": "1801.04896",
  "version": "v1",
  "published": "2018-01-15T18:09:26.000Z",
  "updated": "2018-01-15T18:09:26.000Z",
  "title": "Magnetic helicity and subsolutions in ideal MHD",
  "authors": [
    "Daniel Faraco",
    "Sauli Lindberg"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We show that ideal 2D MHD does not possess weak solutions (or even subsolutions) with compact support in time and non-trivial magnetic field. We also show that the $\\Lambda$-convex hull of ideal MHD has empty interior in both 2D and 3D; this is seen by finding suitable $\\Lambda$-convex functions. As a consequence we show that mean-square magnetic potential is conserved in 2D by subsolutions and weak limits of solutions in the physically natural energy space $L^\\infty_t L^2_x$, and in 3D we show the conservation of magnetic helicity by $L^3$-integrable subsolutions and weak limits of solutions. However, in 3D the $\\Lambda$-convex hull is shown to be large enough that nontrivial smooth, compactly supported strict subsolutions exist.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-15T18:09:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76W05",
      "35Q35",
      "76B03"
    ],
    "keywords": [
      "ideal mhd",
      "magnetic helicity",
      "weak limits",
      "convex hull",
      "physically natural energy space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}