{
  "id": "2301.13645",
  "version": "v1",
  "published": "2023-01-31T13:58:00.000Z",
  "updated": "2023-01-31T13:58:00.000Z",
  "title": "Existence, uniqueness and $L^2 _t (H_x ^2) \\cap L^\\infty _t (H^1 _x) \\cap H^1 _t (L^2 _x) $ regularity of the gradient flow of the Ambrosio-Tortorelli functional",
  "authors": [
    "Tommaso Cortopassi"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the gradient flow of the Ambrosio-Tortorelli functional at fixed $\\epsilon>0$, proving existence, uniqueness and $L^2 _t (H_x ^2) \\cap L^\\infty _t (H^1 _x) \\cap H^1 _t (L^2 _x) $ regularity in dimension 2. In particular we improve a previous result where such regularity was known only up to a finite number of space time points, which diverged as $\\epsilon \\to 0$. By employing a different technique for the crucial $L^2 _t (H^2 _x)$ estimates we can see how in fact the desired regularity holds everywhere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-31T13:58:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gradient flow",
      "ambrosio-tortorelli functional",
      "uniqueness",
      "space time points",
      "finite number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}