{
  "id": "1904.05142",
  "version": "v1",
  "published": "2019-04-10T12:50:41.000Z",
  "updated": "2019-04-10T12:50:41.000Z",
  "title": "Uniqueness of the Non-Equilibrium Steady State for a $1$d BGK model in kinetic theory",
  "authors": [
    "Eric Carlen",
    "Raffaele Esposito",
    "Joel Leibowitz",
    "Rossana Marra",
    "Clement Mouhot"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We continue our investigation of kinetic models of a one-dimensional gas in contact with homogeneous thermal reservoirs at different temperatures. Nonlinear collisional interactions between particles are modeled by a so-called BGK dynamics which conserves local energy and particle density. Weighting the nonlinear BGK term with a parameter $\\alpha\\in [0,1]$, and the linearinteraction with the reservoirs by $(1-\\alpha)$, we prove that for all $\\alpha$ close enough to zero, the explicit spatially uniform non-equilibrium stable state (NESS) is \\emph{unique}, and there are no spatially non-uniform NESS with a spatial density $\\rho$ belonging to $L^p$ for any $p>1$. We also show that for all $\\alpha\\in [0,1]$, the spatially uniform NESS is dynamically stable, with small perturbation converging to zero exponentially fast.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-10T12:50:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q20"
    ],
    "keywords": [
      "non-equilibrium steady state",
      "bgk model",
      "kinetic theory",
      "uniqueness",
      "nonlinear bgk term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}