{
  "id": "2209.07142",
  "version": "v1",
  "published": "2022-09-15T08:41:53.000Z",
  "updated": "2022-09-15T08:41:53.000Z",
  "title": "Structure of vanishing viscosity limits initiated by $δ$-measures from two point sources",
  "authors": [
    "Abhishek Das"
  ],
  "comment": "34 pages, 14 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article, we consider the one-dimensional zero-pressure gas dynamics system \\[ u_t + \\left( {u^2}/{2} \\right)_x = 0,\\ \\rho_t + (\\rho u)_x = 0 \\] in the upper-half plane with a linear combination of two $\\delta$-measures \\[ u|_{t=0} = u_a\\ \\delta_{x=a} + u_b\\ \\delta_{x=b},\\ \\rho|_{t=0} = \\rho_c\\ \\delta_{x=c} + \\rho_d\\ \\delta_{x=d} \\] as initial data. Here $a$, $b$, $c$, $d$ are distinct points on the real line ordered as $a < c < b < d$. Our objective is to provide a detailed analysis of the structure of the vanishing viscosity limits of this system utilizing the corresponding modified adhesion model \\[ u^\\epsilon_t + \\left({(u^\\epsilon)^2}/{2} \\right)_x =\\frac{\\epsilon}{2} u^\\epsilon_{xx},\\ \\rho^\\epsilon_t + (\\rho^\\epsilon u^\\epsilon)_x = \\frac{\\epsilon}{2} \\rho^\\epsilon_{xx}. \\] For this purpose, we extensively use the various asymptotic properties of the function erfc $: z \\longmapsto \\int_{z}^{\\infty} e^{-s^2}\\ ds$ along with suitable Hopf-Cole transformations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-15T08:41:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35F25",
      "35B25",
      "35L67",
      "35R05"
    ],
    "keywords": [
      "vanishing viscosity limits",
      "point sources",
      "one-dimensional zero-pressure gas dynamics system",
      "suitable hopf-cole transformations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}