{
  "id": "math/0112013",
  "version": "v1",
  "published": "2001-12-03T04:19:36.000Z",
  "updated": "2001-12-03T04:19:36.000Z",
  "title": "On a new scale of regularity spaces with applications to Euler's equations",
  "authors": [
    "Eitan Tadmor"
  ],
  "journal": "Nonlinearity 14 (2001) 513-532",
  "doi": "10.1088/0951-7715/14/3/305",
  "categories": [
    "math.AP"
  ],
  "abstract": "We introduce a new ladder of function spaces which is shown to fill in the gap between the weak $L^{p\\infty}$ spaces and the larger Morrey spaces, $M^p$. Our motivation for introducing these new spaces, denoted $\\V^{pq}$, is to gain a more accurate information on (compact) embeddings of Morrey spaces in appropriate Sobolev spaces. It is here that the secondary parameter q (-- and a further logarithmic refinement parameter $\\alpha$, denoted $\\V^{pq}(\\log \\V)^{\\alpha}$) gives a finer scaling, which allows us to make the subtle distinctions necessary for embedding in spaces with a fixed order of smoothness. We utilize an $H^{-1}$-stability criterion which we have recently introduced in {Lopes Filho M C, Nussenzveig Lopes H J and Tadmor E 2001 Approximate solution of the incompressible Euler equations with no concentrations Ann. Institut H Poincare C 17 371-412}, in order to study the strong convergence of approximate Euler solutions. We show how the new refined scale of spaces, $\\V^{pq}(\\log \\V)^{\\alpha}$, enables us approach the borderline cases which separate between $H^{-1}$-compactness and the phenomena of concentration-cancelation. Expressed in terms of their $\\V^{pq}(\\log \\V)^{\\alpha}$ bounds, these borderline cases are shown to be intimately related to uniform bounds of the total (Coulomb) energy and the related vorticity configuration.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-12-03T04:19:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "76B03",
      "65M12"
    ],
    "keywords": [
      "eulers equations",
      "regularity spaces",
      "applications",
      "borderline cases",
      "subtle distinctions necessary"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}