{
  "id": "2510.01166",
  "version": "v1",
  "published": "2025-10-01T17:52:51.000Z",
  "updated": "2025-10-01T17:52:51.000Z",
  "title": "A viscosity solution approach to the large deviation principle for stochastic convective Brinkman-Forchheimer equations",
  "authors": [
    "Sagar Gautam",
    "Manil T. Mohan"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "This article develops the viscosity solution approach to the large deviation principle for the following two- and three-dimensional stochastic convective Brinkman-Forchheimer equations on the torus $\\mathbb{T}^d,\\ d\\in\\{2,3\\}$ with small noise intensity: \\begin{align*} \\mathrm{d}\\boldsymbol{u}_n+[-\\mu\\Delta\\boldsymbol{u}_n+ (\\boldsymbol{u}_n\\cdot\\nabla)\\boldsymbol{u}_n +\\alpha\\boldsymbol{u}_n+\\beta|\\boldsymbol{u}_n|^{r-1}\\boldsymbol{u}_n+\\nabla p_n]\\mathrm{d} t=\\boldsymbol{f}\\mathrm{d} t+\\frac{1}{\\sqrt{n}}\\mathrm{Q}^{\\frac12}\\mathrm{d}\\mathrm{W}, \\ \\nabla\\cdot\\boldsymbol{u}_n=0, \\end{align*} where $\\mu,\\alpha,\\beta>0$, $r\\in[1,\\infty)$, $\\mathrm{Q}$ is a trace class operator and $\\mathrm{W}$ is Hilbert-valued calendrical Wiener process. We build our analysis on the framework of Varadhan and Bryc, together with the techniques of [J. Feng et.al., Large Deviations for Stochastic Processes, American Mathematical Society (2006) vol. \\textbf{131}]. By employing the techniques from the comparison principle, we identify the Laplace limit as the convergence of the viscosity solution of the associated second-order singularly perturbed Hamilton-Jacobi-Bellman equation. A key advantage of this method is that it establishes a Laplace principle without relying on additional sufficient conditions such as Bryc's theorem, which the literature commonly requires. For $r>3$ and $r=3$ with $2\\beta\\mu\\geq1$, we also derive the exponential moment bounds without imposing the classical orthogonality condition $((\\boldsymbol{u}_n\\cdot\\nabla)\\boldsymbol{u}_n,\\mathrm{A}\\boldsymbol{u}_n)=0$, where $\\mathrm{A}=-\\Delta$, in both two-and three-dimensions. We first establish the large deviation principle in the Skorohod space. Then, by using the $\\mathrm{C}-$exponential tightness, we finally establish the large deviation principle in the continuous space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-10-01T17:52:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic convective brinkman-forchheimer equations",
      "large deviation principle",
      "viscosity solution approach",
      "second-order singularly perturbed hamilton-jacobi-bellman",
      "singularly perturbed hamilton-jacobi-bellman equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}