{
  "id": "2311.06064",
  "version": "v1",
  "published": "2023-11-10T13:58:08.000Z",
  "updated": "2023-11-10T13:58:08.000Z",
  "title": "Non-uniqueness of forced active scalar equations with even drift operators",
  "authors": [
    "Mimi Dai",
    "Susan Friedlander"
  ],
  "comment": "30 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider forced active scalar equations with even and homogeneous degree 0 drift operator on $\\mathbb T^d$. Inspired by the non-uniqueness construction for dyadic fluid models, by implementing a sum-difference convex integration scheme we obtain non-unique weak solutions for the active scalar equation in space $C_t^0C_x^\\alpha$ with $\\alpha<\\frac{1}{2d+1}$. We note that in 1D, the regularity $\\alpha<\\frac13$ is sharp as the energy identity is satisfied for solutions in $C^\\alpha$ with $\\alpha>\\frac13$. Without external forcing, Isett and Vicol constructed non-unique weak solutions for such active scalar equations with spatial regularity $C_x^\\alpha$ for $\\alpha<\\frac{1}{4d+1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-10T13:58:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "35Q86",
      "76D03"
    ],
    "keywords": [
      "forced active scalar equations",
      "drift operator",
      "non-uniqueness",
      "sum-difference convex integration scheme",
      "vicol constructed non-unique weak solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}