{
  "id": "2309.15744",
  "version": "v1",
  "published": "2023-09-27T16:00:20.000Z",
  "updated": "2023-09-27T16:00:20.000Z",
  "title": "Statistically self-similar mixing by Gaussian random fields",
  "authors": [
    "Michele Coti Zelati",
    "Theodore D. Drivas",
    "Rishabh S. Gvalani"
  ],
  "comment": "9 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.AP",
    "math.DS",
    "math.MP"
  ],
  "abstract": "We study the passive transport of a scalar field by a spatially smooth but white-in-time incompressible Gaussian random velocity field on $\\mathbb{R}^d$. If the velocity field $u$ is homogeneous, isotropic, and statistically self-similar, we derive an exact formula which captures non-diffusive mixing. For zero diffusivity, the formula takes the shape of $\\mathbb{E}\\ \\| \\theta_t \\|_{\\dot{H}^{-s}}^2 = \\mathrm{e}^{-\\lambda_{d,s} t} \\| \\theta_0 \\|_{\\dot{H}^{-s}}^2$ with any $s\\in (0,d/2)$ and $\\frac{\\lambda_{d,s}}{D_1}:= s(\\frac{\\lambda_{1}}{D_1}-2s)$ where $\\lambda_1/D_1 = d$ is the top Lyapunov exponent associated to the random Lagrangian flow generated by $u$ and $ D_1$ is small-scale shear rate of the velocity. Moreover, the mixing is shown to hold $\\textit{uniformly}$ in diffusivity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-27T16:00:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian random fields",
      "statistically self-similar mixing",
      "incompressible gaussian random velocity field",
      "white-in-time incompressible gaussian random velocity",
      "small-scale shear rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}