{
  "id": "2309.04096",
  "version": "v1",
  "published": "2023-09-08T03:17:28.000Z",
  "updated": "2023-09-08T03:17:28.000Z",
  "title": "Kinetic description of scalar conservation laws with Markovian data",
  "authors": [
    "Fraydoun Rezakhanlou"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We derive a kinetic equation to describe the statistical structure of solutions $\\rho$ to scalar conservation laws $\\rho_t=H(x,t,\\rho )_x$, with certain Markov initial conditions. When the Hamiltonian function is convex and increasing in $\\rho$, we show that the solution $\\rho(x,t)$ is a Markov process in $x$ (respectively $t$) with $t$ (respectively $x$) fixed. Two classes of Markov conditions are considered in this article. In the first class, the initial data is characterize by a drift $b$ which satisfies a linear PDE, and a jump density $f$ which satisfies a kinetic equation as time varies. In the second class, the initial data is a concatenation of fundamental solutions that are characterized by a parameter $y$, which is a Markov jump process with a jump density $g$ satisfying a kinetic equation. When $H$ is not increasing in $\\rho$, the restriction of $\\rho$ to a line in $(x,t)$ plane is a Markov process of the same type, provided that the slope of the line satisfies an inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-08T03:17:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scalar conservation laws",
      "markovian data",
      "kinetic description",
      "kinetic equation",
      "jump density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}