The Inviscid Limit for the Navier–Stokes Equations with Slip Condition on Permeable Walls

Abstract

We consider the Navier–Stokes equations in a 2D-bounded domain with general non-homogeneous Navier slip boundary conditions prescribed on permeable boundaries, and study the vanishing viscosity limit. We prove that solutions of the Navier–Stokes equations converge to solutions of the Euler equations satisfying the same Navier slip boundary condition on the inflow region of the boundary. The convergence is strong in Sobolev’s spaces \(W^{1}_{p}, p>2\), which correspond to the spaces of the data.

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Acknowledgements

We would like to thank the referees for several corrections and comments which improved the presentation of the article.

N.V. Chemetov thanks for support from FCT and FEDER through the project POCTI/ISFL/209 of Centro de Matemática e Aplicações Fundamentais da Universidade de Lisboa (CMAF/UL). The research work of F. Cipriano was supported by the projects FCT-PTDC/MAT/104173/2008 and EURATOM/IST.

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Correspondence to N. V. Chemetov.

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Communicated by Andrew Szeri.

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Chemetov, N.V., Cipriano, F. The Inviscid Limit for the Navier–Stokes Equations with Slip Condition on Permeable Walls. J Nonlinear Sci 23, 731–750 (2013). https://doi.org/10.1007/s00332-013-9166-5

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Keywords

  • Navier–Stokes equations
  • Euler equations
  • Navier slip boundary conditions
  • Vanishing viscosity
  • Boundary layer
  • Turbulence

Mathematics Subject Classification (2000)

  • 35D05
  • 76B03
  • 76B47
  • 76D09