{
  "id": "2009.08648",
  "version": "v1",
  "published": "2020-09-18T06:27:06.000Z",
  "updated": "2020-09-18T06:27:06.000Z",
  "title": "On well-posedness and singularity formation for the Euler-Riesz system",
  "authors": [
    "Young-Pil Choi",
    "In-Jee Jeong"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we investigate the initial value problem for the Euler-Riesz system, where the interaction forcing is given by $\\nabla(-\\Delta)^{s}\\rho$ for some $-1<s<0$, with $s = -1$ corresponding to the classical Euler-Poisson system. We develop a functional framework to establish local-in-time existence and uniqueness of classical solutions for the Euler-Riesz system. In this framework, the fluid density could decay fast at infinity, and the Euler-Poisson system can be covered as a special case. Moreover, we prove local well-posedness for the pressureless Euler-Riesz system when the potential is repulsive, by observing hyperbolic nature of the system. Finally, we present sufficient conditions on the finite-time blowup of classical solutions for the isentropic/isothermal Euler-Riesz system with either attractive or repulsive interaction forces. The proof, which is based on estimates of several physical quantities, establishes finite-time blowup for a large class of initial data; in particular, it is not required that the density is of compact support.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-18T06:27:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singularity formation",
      "well-posedness",
      "isentropic/isothermal euler-riesz system",
      "classical solutions",
      "establishes finite-time blowup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}