{
  "id": "1911.05038",
  "version": "v1",
  "published": "2019-11-12T17:57:29.000Z",
  "updated": "2019-11-12T17:57:29.000Z",
  "title": "Rough solutions of the 3-D compressible Euler equations",
  "authors": [
    "Qian Wang"
  ],
  "categories": [
    "math.AP",
    "gr-qc",
    "math.DG"
  ],
  "abstract": "We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\\varrho, \\fw) \\in H^s\\times H^s\\times H^{s'}$, $2<s'<s$. The classical local well-posedness result for the compressible Euler equations in $3$-D holds for the initial data $v, \\varrho \\in H^{s+\\f12},\\, s>2$. Due to the works of Smith-Tataru and Wang for the irrotational isentropic case, the local well-posedness can be achieved if the data satisfy $v, \\varrho \\in H^{s}$, with $s>2$. In the incompressible case the solution is proven to be ill-posed for the datum $\\fw\\in H^\\frac{3}{2}$ by Bourgain-Li. The solution of the compressible Euler equations is not expected to be well-posed if the data merely satisfy $v, \\varrho\\in H^{s}, s>2$ with a general rough vorticity. By decomposing the velocity into the term $(I-\\Delta_e)^{-1}\\curl \\fw$ and a wave function verifying an improved wave equation, with a series of cancellations for treating the latter, we achieve the $H^s$-energy bound and complete the linearization for the wave functions by using the $H^{s-\\f12}, \\, s>2$ norm for the vorticity. The propagation of energy for the vorticity typically requires $\\curl \\fw\\in C^{0, 0+}$ initially, stronger than our assumption by 1/2-derivative. We perform trilinear estimates to gain regularity by observing a div-curl structure when propagating the energy of the normalized double-curl of the vorticity, and also by spacetime integration by parts. To prove the Strichartz estimate for the linearized wave in the rough spacetime, we encounter a strong Ricci defect requiring the bound of $\\|\\curl \\fw\\|_{L_x^\\infty L_t^1}$ on null cones. This difficulty is solved by uncovering the cancellation structures due to the acoustic metric on the angular derivatives of Ricci and the second fundamental form.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-12T17:57:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compressible euler equations",
      "rough solutions",
      "wave function",
      "perform trilinear estimates",
      "general rough vorticity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}