{
  "id": "2008.07756",
  "version": "v1",
  "published": "2020-08-18T06:16:28.000Z",
  "updated": "2020-08-18T06:16:28.000Z",
  "title": "Singularity formation for compressible Euler equations with time-dependent damping",
  "authors": [
    "Ying Sui",
    "Huimin Yu"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the compressible Euler equations with time-dependent damping \\frac{\\a}{(1+t)^\\lambda}u in one space dimension. By constructing 'decoupled' Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation of shock wave, if the derivatives of initial data are appropriately large at a point even when the damping coefficient goes to infinity with a algebraic growth rate. We study the case \\lambda\\neq1 and \\lambda=1 respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for 1<\\gamma<3 we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-18T06:16:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q05",
      "76L05",
      "G.0.0"
    ],
    "keywords": [
      "compressible euler equations",
      "time-dependent damping",
      "singularity formation",
      "initial data",
      "riccati type equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}