{
  "id": "1806.09001",
  "version": "v1",
  "published": "2018-06-23T16:29:47.000Z",
  "updated": "2018-06-23T16:29:47.000Z",
  "title": "\"Life after death\" in ordinary differential equations with a non-Lipschitz singularity",
  "authors": [
    "Theodore D. Drivas",
    "Alexei A. Mailybaev"
  ],
  "comment": "27 pages, 8 figures",
  "categories": [
    "math.CA",
    "math-ph",
    "math.DS",
    "math.MP"
  ],
  "abstract": "We consider a class of ordinary differential equations in $d$-dimensions featuring a non-Lipschitz singularity at the origin. Solutions of such systems exist globally and are unique up until the first time they hit the origin, $t = t_b$, which we term `blowup'. However, infinitely many solutions may exist for longer times. To study continuation past blowup, we introduce physically motivated regularizations: they consist of smoothing the vector field in a $\\nu$--ball around the origin and then removing the regularization in the limit $\\nu\\to 0$. We show that this limit can be understood using a certain autonomous dynamical system obtained by a solution-dependent renormalization procedure. This procedure maps the pre-blowup dynamics, $t < t_b$, to the solution ending at infinitely large renormalized time. In particular, the asymptotic behavior as $t \\nearrow t_b$ is described by an attractor. The post-blowup dynamics, $t > t_b$, is mapped to a different renormalized solution starting infinitely far in the past. Consequently, it is associated with another attractor. The $\\nu$-regularization establishes a relation between these two different \"lives\" of the renormalized system. We prove that, in some generic situations, this procedure selects a unique global solution (or a family of solutions), which does not depend on the details of the regularization. We provide concrete examples and argue that these situations are qualitatively similar to post-blowup scenarios observed in infinite-dimensional models of turbulence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-23T16:29:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ordinary differential equations",
      "non-lipschitz singularity",
      "solution starting infinitely far",
      "regularization",
      "study continuation past blowup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}