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Approximation of a Solution to the Euler Equation by Solutions of the Navier–Stokes Equation

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Abstract

We show that a smooth solution u 0 of the Euler boundary value problem on a time interval (0, T 0) can be approximated by a family of solutions of the Navier–Stokes problem in a topology of weak or strong solutions on the same time interval (0, T 0). The solutions of the Navier–Stokes problem satisfy Navier’s boundary condition, which must be “naturally inhomogeneous” if we deal with the strong solutions. We provide information on the rate of convergence of the solutions of the Navier–Stokes problem to the solution of the Euler problem for ν → 0. We also discuss possibilities when Navier’s boundary condition becomes homogeneous.

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Authors and Affiliations

  1. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Prague, Czech Republic

    Jiří Neustupa

  2. Mathématique, Université du Sud Toulon–Var, BP 20132, 83957, La Garde Cedex, France

    Patrick Penel

Authors
  1. Jiří Neustupa
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  2. Patrick Penel
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Correspondence to Jiří Neustupa.

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Communicated by G. P. Galdi

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Neustupa, J., Penel, P. Approximation of a Solution to the Euler Equation by Solutions of the Navier–Stokes Equation. J. Math. Fluid Mech. 15, 179–196 (2013). https://doi.org/10.1007/s00021-012-0125-y

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  • DOI: https://doi.org/10.1007/s00021-012-0125-y

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