{
  "id": "2304.14785",
  "version": "v1",
  "published": "2023-04-28T11:52:10.000Z",
  "updated": "2023-04-28T11:52:10.000Z",
  "title": "Improved estimates for the sharp interface limit of the stochastic Cahn-Hilliard equation with space-time white noise",
  "authors": [
    "Banas Lubomir",
    "Mukam Jean Daniel"
  ],
  "categories": [
    "math.PR",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "We study the sharp interface limit of the stochastic Cahn-Hilliard equation with cubic double-well potential and additive space-time white noise $\\epsilon^{\\sigma}\\dot{W}$ where $\\epsilon>0$ is an interfacial width parameter. We prove that, for sufficiently large scaling constant $\\sigma >0$, the stochastic Cahn-Hilliard equation converges to the deterministic Mullins-Sekerka/Hele-Shaw problem for $\\epsilon\\rightarrow 0$. The convergence is shown in suitable fractional Sobolev norms as well as in the $L^p$-norm for $p\\in (2, 4]$ in spatial dimension $d=2,3$. This generalizes the existing result for the space-time white noise to dimension $d=3$ and improves the existing results for smooth noise, which were so far limited to $p\\in \\left(2, frac{2d+8}{d+2}\\right]$ in spatial dimension $d=2,3$. As a byproduct of the analysis of the stochastic problem with space-time white noise, we identify minimal regularity requirements on the noise which allow convergence to the sharp interface limit in the $\\mathbb{H}^1$-norm and also provide improved convergence estimates for the sharp interface limit of the deterministic problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-28T11:52:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "space-time white noise",
      "sharp interface limit",
      "spatial dimension",
      "stochastic cahn-hilliard equation converges",
      "existing result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}