{
  "id": "2208.06808",
  "version": "v1",
  "published": "2022-08-14T09:02:02.000Z",
  "updated": "2022-08-14T09:02:02.000Z",
  "title": "Asymptotically autonomous robustness in probability of random attractors for stochastic Navier-Stokes equations on unbounded Poincaré domains",
  "authors": [
    "Renhai Wang",
    "Kush Kinra",
    "Manil T. Mohan"
  ],
  "categories": [
    "math.PR",
    "math.DS"
  ],
  "abstract": "The asymptotically autonomous robustness of random attractors of stochastic fluid equations defined on \\emph{bounded} domains has been considered in the literature. In this article, we initially consider this topic (almost surely and in probability) for a non-autonomous stochastic 2D Navier-Stokes equation driven by additive and multiplicative noise defined on some \\emph{unbounded Poincar\\'e domains}. There are two significant keys to study this topic: what is the asymptotically autonomous limiting set of the time-section of random attractors as time goes to negative infinity, and how to show the precompactness of a time-union of random attractors over an \\emph{infinite} time-interval $(-\\infty,\\tau]$. We guess and prove that such a limiting set is just determined by the random attractor of a stochastic Navier-Stokes equation driven by an autonomous forcing satisfying a convergent condition. The uniform \"tail-smallness\" and \"flattening effecting\" of the solutions are derived in order to justify that the usual asymptotically compactness of the solution operators is \\emph{uniform} over $(-\\infty,\\tau]$. This in fact leads to the precompactness of the time-union of random attractors over $(-\\infty,\\tau]$. The idea of uniform tail-estimates due to Wang \\cite{UTE-Wang} is employed to overcome the noncompactness of Sobolev embeddings on unbounded domains. Several rigorous calculations are given to deal with the pressure terms when we derive these uniform tail-estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-14T09:02:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random attractor",
      "stochastic navier-stokes equation",
      "asymptotically autonomous robustness",
      "probability",
      "stochastic 2d navier-stokes equation driven"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}