{
  "id": "1510.05733",
  "version": "v1",
  "published": "2015-10-20T01:38:24.000Z",
  "updated": "2015-10-20T01:38:24.000Z",
  "title": "Ill-posedness of the Navier-Stokes and magneto-hydrodynamics systems",
  "authors": [
    "Alexey Cheskidov",
    "Mimi Dai"
  ],
  "categories": [
    "math.AP",
    "physics.flu-dyn"
  ],
  "abstract": "We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed due to the discontinuity of weak solutions in a wide range of spaces. Specifically, we construct initial data which has finite energy and is small in certain spaces, such that any Leray-Hopf type of weak solution to the MHD system starting from this initial data is discontinuous at time $t=0$ in such spaces. An analogous result is also obtained for the Navier-Stokes equation which extends the previous result of ill-posedness in $\\dot B^{-1}_{\\infty,\\infty}$ by Cheskidov and Shvydkoy to spaces that are not necessarily critical. The region of the spaces where the norm inflation occurs almost touches $L^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-20T01:38:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76D03",
      "35Q35"
    ],
    "keywords": [
      "magneto-hydrodynamics systems",
      "ill-posedness",
      "weak solution",
      "construct initial data",
      "norm inflation occurs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}