{
  "id": "2309.05844",
  "version": "v1",
  "published": "2023-09-11T22:07:15.000Z",
  "updated": "2023-09-11T22:07:15.000Z",
  "title": "On well-posedness of a mildly dissipative family of active scalar equations in borderline Sobolev spaces",
  "authors": [
    "Anuj Kumar",
    "Vincent R. Martinez"
  ],
  "comment": "48 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper considers a family of active scalar equations which modify the generalized surface quasi-geostrophic (gSQG) equations through its constitutive law or dissipative perturbations. These modifications are characteristically mild in the sense that they are logarithmic. The problem of well-posedness, in the sense of Hadamard, is then studied in a borderline setting of regularity in analog to the scaling-critical spaces of the gSQG equations. A novelty of the system considered is the nuanced form of smoothing provided by the proposed mild form of dissipation, which is able to support global well-posedness at the Euler endpoint, but in a setting where the inviscid counterpart is known to be ill-posed. A novelty of the analysis lies in the simultaneous treatment of modifications in the constitutive law, dissipative mechanism, and functional setting, which the existing literature has typically treated separately. A putatively sharp relation is identified between each of the distinct system-modifiers that is consistent with previous studies that considered these modifications in isolation. This unified perspective is afforded by the introduction of a linear model equation, referred to as the protean system, that successfully incorporates the more delicate commutator structure collectively possessed by the gSQG family and upon which each facet of well-posedness can be reduced to its study.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-11T22:07:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76B03",
      "35Q35",
      "35Q86",
      "35B45"
    ],
    "keywords": [
      "active scalar equations",
      "borderline sobolev spaces",
      "mildly dissipative family",
      "delicate commutator structure",
      "linear model equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}