{
  "id": "2203.13115",
  "version": "v1",
  "published": "2022-03-24T15:16:02.000Z",
  "updated": "2022-03-24T15:16:02.000Z",
  "title": "An Intermittent Onsager Theorem",
  "authors": [
    "Matthew Novack",
    "Vlad Vicol"
  ],
  "comment": "54 pages, no figures. arXiv admin note: text overlap with arXiv:2101.09278",
  "categories": [
    "math.AP"
  ],
  "abstract": "For any regularity exponent $\\beta<\\frac 12$, we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class $C^0_t (H^{\\beta} \\cap L^{\\frac{1}{(1-2\\beta)}})$. By interpolation, such solutions belong to $C^0_tB^{s}_{3,\\infty}$ for $s$ approaching $\\frac 13$ as $\\beta$ approaches $\\frac 12$. Hence this result provides a new proof of the flexible side of the Onsager conjecture, which is independent from that of Isett [A proof of Onsager's conjecture, Annals of Mathematics, 188(3):871, 2018]. Of equal importance is that the intermittent nature of our solutions matches that of turbulent flows, which are observed to possess an $L^2$-based regularity index exceeding $\\frac 13$. Our proof builds on the authors' previous joint work with Buckmaster and Masmoudi [Nonconservative $H^{\\frac 12-}$ weak solutions of the incompressible 3D Euler equations, arXiv 2101.09278, 2021], in which an intermittent convex integration scheme is developed for the 3D incompressible Euler equations. We employ a scheme with higher-order Reynolds stresses, which are corrected via a combinatorial placement of intermittent pipe flows of optimal relative intermittency.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-24T15:16:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "intermittent onsager theorem",
      "3d incompressible euler equations",
      "intermittent convex integration scheme",
      "intermittent pipe flows",
      "higher-order reynolds stresses"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}