{
  "id": "1412.6587",
  "version": "v1",
  "published": "2014-12-20T03:12:19.000Z",
  "updated": "2014-12-20T03:12:19.000Z",
  "title": "Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions",
  "authors": [
    "Milton C. Lopes Filho",
    "Helena J. Nussenzveig Lopes",
    "Edriss S. Titi",
    "Aibin Zang"
  ],
  "comment": "20pages,1figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: $\\alpha > 0$, corresponding to the elastic response, and $\\nu > 0$, corresponding to viscosity. Formally setting these parameters to $0$ reduces the equations to the incompressible Euler equations of ideal fluid flow. In this article we study the limits $\\alpha, \\nu \\to 0$ of solutions of the second-grade fluid system, in a smooth, bounded, two-dimensional domain with no-slip boundary conditions. This class of problems interpolates between the Euler-$\\alpha$ model ($\\nu = 0$), for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the Navier-Stokes case ($\\alpha = 0$), for which the vanishing viscosity limit is an important open problem. We prove three results. First, we establish convergence of the solutions of the second-grade model to those of the Euler equations provided $\\nu = \\mathcal{O}(\\alpha^2)$, as $\\alpha \\to 0$, extending the main result in [19]. Second, we prove equivalence between convergence (of the second-grade fluid equations to the Euler equations) and vanishing of the energy dissipation in a suitably thin region near the boundary, in the asymptotic regime $\\nu = \\mathcal{O}(\\alpha^{6/5})$, $\\nu/\\alpha^2 \\to \\infty$ as $\\alpha \\to 0$. This amounts to a convergence criterion similar to the well-known Kato criterion for the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations. Finally, we obtain an extension of Kato's classical criterion to the second-grade fluid model, valid if $\\alpha = \\mathcal{O}(\\nu^{3/2})$, as $\\nu \\to 0$. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-20T03:12:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "76D05",
      "76D10"
    ],
    "keywords": [
      "second-grade fluid equations",
      "dirichlet boundary conditions",
      "2d euler equations",
      "vanishing viscosity limit",
      "incompressible euler equations"
    ],
    "publication": {
      "doi": "10.1007/s00021-015-0207-8",
      "journal": "Journal of Mathematical Fluid Mechanics",
      "year": 2015,
      "month": "Jun",
      "volume": 17,
      "number": 2,
      "pages": 327
    },
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JMFM...17..327L"
    }
  }
}