{
  "id": "1706.01549",
  "version": "v1",
  "published": "2017-06-05T21:57:56.000Z",
  "updated": "2017-06-05T21:57:56.000Z",
  "title": "On the Endpoint Regularity in Onsager's Conjecture",
  "authors": [
    "Philip Isett"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Onsager's conjecture states that the conservation of energy may fail for $3D$ incompressible Euler flows with H\\\"older regularity below $1/3$. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question. In this work, we construct energy non-conserving solutions to the $3D$ incompressible Euler equations with space-time H\\\"older regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents $[0,1/3)$. This result improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a general convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of [Isett-Oh, 2014] to modify the convex integration scheme.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-05T21:57:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "endpoint regularity",
      "general convex integration scheme",
      "small spatial scales",
      "endpoint case remains",
      "onsagers conjecture states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}