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Semigroup Theory for the Stokes Operator with Navier Boundary Condition on Lp Spaces

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Waves in Flows

Part of the book series: Advances in Mathematical Fluid Mechanics ((LNMFM))

Abstract

We consider the incompressible Navier–Stokes equations in a bounded domain with \(\mathcal {C}^{1,1}\) boundary, completed with slip boundary condition. We study the general semigroup theory in L p-spaces related to the Stokes operator with Navier boundary condition where the slip coefficient α is a non-smooth scalar function. It is shown that the strong and weak Stokes operators with Navier conditions admit analytic semigroup with bounded pure imaginary powers. We also show that for α large, the weak and strong solutions of both the linear and nonlinear systems are bounded uniformly with respect to α. This justifies mathematically that the solution of the Navier–Stokes problem with slip condition converges in the energy space to the solution of the Navier–Stokes with no-slip boundary condition as α →∞.

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Author information

Authors and Affiliations

  1. Université de Pau et des Pays de l’Adour, Pau Cedex, France

    Chérif Amrouche & Amrita Ghosh

  2. Departamento de Matemáticas, Universidad del País Vasco, Lejona, Spain

    Miguel Escobedo

Authors
  1. Chérif Amrouche
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  2. Miguel Escobedo
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  3. Amrita Ghosh
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Corresponding author

Correspondence to Chérif Amrouche .

Editor information

Editors and Affiliations

  1. Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University, Prague, Czech Republic

    Tomáš Bodnár

  2. Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Giovanni P. Galdi

  3. Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

    Šárka Nečasová

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Amrouche, C., Escobedo, M., Ghosh, A. (2021). Semigroup Theory for the Stokes Operator with Navier Boundary Condition on Lp Spaces. In: Bodnár, T., Galdi, G.P., Nečasová, Š. (eds) Waves in Flows. Advances in Mathematical Fluid Mechanics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-68144-9_1

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