{
  "id": "1504.05067",
  "version": "v1",
  "published": "2015-04-20T14:15:21.000Z",
  "updated": "2015-04-20T14:15:21.000Z",
  "title": "The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion",
  "authors": [
    "Hakima Bessaih",
    "Benedetta Ferrario"
  ],
  "comment": "28 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized through a smoothing kernel of order $\\alpha$ in the nonlinear term and with a $\\beta$-fractional Laplacian; we are in the critical case $\\alpha+\\beta=\\frac{5}{4}$. The temperature equation is a pure transport equation. We prove regularity results when the initial velocity is in $H^r$ and the initial temperature is in $H^{r-\\beta}$ for $r>\\max \\left\\{ \\frac{5}{2}-2\\alpha, \\beta+1\\right\\}$ with $\\beta\\ge \\frac{1}{2}$ and $\\alpha\\ge 0$. This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions with respect to the initial conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-20T14:15:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "76D03",
      "35Q86"
    ],
    "keywords": [
      "regularized 3d boussinesq equations",
      "fractional laplacian",
      "3d regularized boussinesq equations",
      "pure transport equation",
      "initial conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150405067B"
    }
  }
}