{
  "id": "2011.07419",
  "version": "v1",
  "published": "2020-11-15T00:13:56.000Z",
  "updated": "2020-11-15T00:13:56.000Z",
  "title": "Incompressible Navier Stokes Equations",
  "authors": [
    "Terry E. Moschandreou",
    "Keith C. Afas"
  ],
  "comment": "8 pages, 1 figure",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "A closely related problem to The Clay Math Institute \"Navier-Stokes, breakdown of smooth solutions here on an arbitrary cube subset of three dimensional space with periodic boundary conditions is examined. The incompressible Navier-Stokes Equations are presented in a new and conventionally different way here, by naturally reducing them to an operator form which is then further analyzed. It is shown that a reduction to a general 2D N-S system decoupled from a 1D non-linear partial differential equation is possible to obtain. This is executed using integration over n-dimensional compact intervals which allows decoupling. Here we extract the measure-zero points in the domain where singularities may occur and are left with a pde that exhibits finite time singularity. The operator form is considered in a physical geometric vorticity case, and a more general case. In the general case, the solution is revealed to have smooth solutions which exhibit finite-time blowup on a fine measure zero set and using the Gagliardo-Nirenberg inequalities it is shown that for any non zero measure set in the form of cube subset of 3D there is finite time blowup for the non-starred parametrized velocity in the z-direction of flow. It is proposed that for the non-dimensional quantity $\\delta$ used in the reparametrization of the Navier-Stokes equations, the starred velocity in the $z^*$ direction does not blowup and this corresponds to $\\delta$ approaching infinity, that is, regular defined flow at infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-15T00:13:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30"
    ],
    "keywords": [
      "incompressible navier stokes equations",
      "1d non-linear partial differential equation",
      "operator form",
      "navier-stokes equations",
      "general 2d n-s system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}