{
  "id": "2301.07898",
  "version": "v1",
  "published": "2023-01-19T05:54:34.000Z",
  "updated": "2023-01-19T05:54:34.000Z",
  "title": "Spectral Submanifolds of the Navier-Stokes Equations",
  "authors": [
    "Gergely Buza"
  ],
  "comment": "54 pages, 16 figures",
  "categories": [
    "math.DS",
    "math.AP",
    "physics.flu-dyn"
  ],
  "abstract": "Spectral subspaces of a linear dynamical system identify a large class of invariant structures that highlight/isolate the dynamics associated to select subsets of the spectrum. The corresponding notion for nonlinear systems is that of spectral submanifolds -- manifolds invariant under the full nonlinear dynamics that are determined by their tangency to spectral subspaces of the linearized system. In light of the recently-emerged interest in their use as tools in model reduction, we propose an extension of the relevant theory to the realm of fluid dynamics. We show the existence of a large (and the most pertinent) subclass of spectral submanifolds and foliations - describing the behaviour of nearby trajectories - about fixed points and periodic orbits of the Navier-Stokes equations. Their uniqueness and smoothness properties are discussed in detail, due to their significance from the perspective of model reduction. The machinery is then put to work via a numerical algorithm developed along the lines of the parameterization method, that computes the desired manifolds as power series expansions. Results are shown within the context of 2D channel flows.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-19T05:54:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37L25",
      "37L65"
    ],
    "keywords": [
      "spectral submanifolds",
      "navier-stokes equations",
      "model reduction",
      "spectral subspaces",
      "power series expansions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}