{
  "id": "1703.10874",
  "version": "v1",
  "published": "2017-03-31T12:10:30.000Z",
  "updated": "2017-03-31T12:10:30.000Z",
  "title": "A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff",
  "authors": [
    "Nicolas Fournier"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution $(f_t)_{t\\geq 0}$, once the initial condition $f_0$ with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a $(0,\\infty)\\times\\mathbb{R}^3$ random variable $(M_t,V_t)$ such that $E[M_t {\\bf 1}_{\\{V_t \\in \\cdot\\}}]=f_t$. We also write down a series expansion of $f_t$. Although both the algorithm and the series expansion might be theoretically interesting in that they explicitly express $f_t$ in terms of $f_0$, we believe that the algorithm is not very efficient in practice and that the series expansion is rather intractable. This is a tedious extension to non-Maxwellian molecules of Wild's sum and of its interpretation by McKean.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-31T12:10:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82C40",
      "60K35"
    ],
    "keywords": [
      "series expansion",
      "hard potentials",
      "angular cutoff",
      "recursive algorithm",
      "unique conservative weak solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}