{
  "id": "2304.01129",
  "version": "v1",
  "published": "2023-04-03T16:46:30.000Z",
  "updated": "2023-04-03T16:46:30.000Z",
  "title": "Diffusive Limit of the Boltzmann Equation in Bounded Domains",
  "authors": [
    "Zhimeng Ouyang",
    "Lei Wu"
  ],
  "comment": "60 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "The rigorous justification of the hydrodynamic limits of kinetic equations in bounded domains has been actively investigated in recent years. In spite of the progress for the diffuse-reflection boundary case, the more challenging in-flow boundary case, in which the leading-order boundary layer effect is non-negligible, still remains open. In this work, we consider the stationary and evolutionary Boltzmann equation with the in-flow boundary in general (convex or non-convex) bounded domains, and demonstrate their incompressible Navier-Stokes-Fourier (INSF) limits in $L^2$. Our method relies on a novel and surprising gain of $\\varepsilon^{\\frac{1}{2}}$ in the kernel estimate, which is rooted from a key cancellation of delicately chosen test functions and conservation laws. We also introduce the boundary layer with grazing-set cutoff and investigate its BV regularity estimates to control the source terms of the remainder equation with the help of Hardy's inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-03T16:46:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bounded domains",
      "diffusive limit",
      "leading-order boundary layer effect",
      "diffuse-reflection boundary case",
      "challenging in-flow boundary case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}