{
  "id": "2012.14658",
  "version": "v1",
  "published": "2020-12-29T08:39:30.000Z",
  "updated": "2020-12-29T08:39:30.000Z",
  "title": "Existence and non-existence for the collision-induced breakage equation",
  "authors": [
    "Ankik Kumar Giri",
    "Philippe Laurençot"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "A mathematical model for collision-induced breakage is considered. Existence of weak solutions to the continuous nonlinear collision-induced breakage equation is shown for a large class of unbounded collision kernels and daughter distribution functions, assuming the collision kernel $K$ to be given by $K(x,y)= x^{\\alpha} y^{\\beta} + x^{\\beta} y^{\\alpha}$ with $\\alpha \\le \\beta \\le 1$. When $\\alpha + \\beta \\in [1,2]$, it is shown that there exists at least one weak mass-conserving solution for all times. In contrast, when $\\alpha + \\beta \\in [0,1)$ and $\\alpha \\ge 0$, global mass-conserving weak solutions do not exist, though such solutions are constructed on a finite time interval depending on the initial condition. The question of uniqueness is also considered. Finally, for $\\alpha <0$ and a specific daughter distribution function, the non-existence of mass-conserving solutions is also established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-29T08:39:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-existence",
      "collision kernel",
      "specific daughter distribution function",
      "global mass-conserving weak solutions",
      "mass-conserving solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}