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On Solution to the Navier-Stokes Equations with Navier Slip Boundary Condition for Three Dimensional Incompressible Fluid

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Abstract

In this article, we prove the existence and uniqueness of solutions of the Navier-Stokes equations with Navier slip boundary condition for incompressible fluid in a bounded domain of ℝ3. The results are established by the Galerkin approximation method and improved the existing results.

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References

  1. Matthews M T, Hill J M. Flow around nanospheres and nanocylinders. Quart J Mech Appl Math, 2006, 59: 191–210

    Article  MathSciNet  Google Scholar 

  2. Matthews M T, Hill J M. Newtonian flow with nonlinear Navier boundary condition. Acta Mechanica, 2007, 191: 195–217

    Article  Google Scholar 

  3. Dussan V E B. The moving contact line: the slip boundary conditions. J Fluid Mech, 1976, 77: 665–684

    Article  Google Scholar 

  4. Richardson S. On the no-slip boundary condition. J Fluid Mech, 1973, 59: 707–719

    Article  Google Scholar 

  5. Navier C L M H. Sur les lois du mouvement des fluides. Mem Acad R Sci Inst Fr, 1827, 6: 389–440

    Google Scholar 

  6. Maxwell J C. On stresses in rarefied gases arising from inequalities of temperature. Phil Trans R Soc London, 1879, 170: 231–256

    Article  Google Scholar 

  7. Amrouche C, Rejaiba A. Navier-Stokes equations with Navier boundary condition. Math Methods Appl Sci, 2016, 39(17): 5091–5112

    Article  MathSciNet  Google Scholar 

  8. Beirão da Veiga, Crispo F. The 3-D inviscid limit result under slip boundary conditions, A negative answer. J Math Fluid Mech, 2012, 14(1): 55–59

    Article  MathSciNet  Google Scholar 

  9. Iftimie D, Sueur F. Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions. Arch Ration Mech Anal, 2011, 199(1): 145–175

    Article  MathSciNet  Google Scholar 

  10. Jager W, Mikelić A. On the roughness-induced effective boundary conditions for an incompressible viscous flow. J Differential Equations, 2001, 170(1): 96–122

    Article  MathSciNet  Google Scholar 

  11. Kashiwabara T. On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type. J Differential Equations, 2013, 254(2): 756–778

    Article  MathSciNet  Google Scholar 

  12. Xiao Y L, Xin Z P. On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Comm Pure Appl Math, 2007, 60(7): 1027–1055

    Article  MathSciNet  Google Scholar 

  13. Xiao Y L, Xin Z P. On 3D Lagrangian Navier-Stokes α model with a class of vorticity-slip boundary conditions. J Math Fluid Mech, 2013, 15(2): 215–247

    MathSciNet  MATH  Google Scholar 

  14. Zajaczkowski W M. Global special regular solutions to the Navier-Stokes equations in a cylindrical domain without the axis of symmetry. Topol Methods Nonlinear Anal, 2004, 24(1): 69–105

    Article  MathSciNet  Google Scholar 

  15. Zhong X. Vanishing viscosity limits for the 3D Navier-Stokes equations with a slip boundary condition. Proc Amer Math Soc, 2017, 145(4): 1615–1628

    MathSciNet  MATH  Google Scholar 

  16. Masmoudi N, Rousset F. Uniform regularity for the Navier-Stokes equation with Navier boundary condition. Arch Ration Mech Anal, 2012, 203(2): 529–575

    Article  MathSciNet  Google Scholar 

  17. Clopeau T, Mikelić A, Robert R. On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions. Nonlinearity, 1998, 11(6): 1625–1636

    Article  MathSciNet  Google Scholar 

  18. Lopes Filho M C, Nussenzveig Lopes H J, Planas G. On the inviscid limit for two-dimensional incompressible flow with Navier friction condition. SIAM J Math Anal, 2005, 36(4): 1130–1141

    Article  MathSciNet  Google Scholar 

  19. Kelliher J P. Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J Math Anal, 2006, 38(1): 210–232

    Article  MathSciNet  Google Scholar 

  20. Lions J L. Quelques Methodes de Resolution des Problemes aux Limites non Lineaires. Dunod; Gauthier-Villars, Paris, 1969

    MATH  Google Scholar 

  21. Beirão da Veiga H, Crispo F. Sharp inviscid limit results under Navier type boundary conditions: An L p theory. J Math Fluid Mech, 2010, 12(3): 397–411

    Article  MathSciNet  Google Scholar 

  22. Temam R. Navier-Stokes Equations. Amsterdam: North-Holland, 1979

    MATH  Google Scholar 

  23. Sohr H. The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Modern Birkhauser Classics. Basel: Birkhauser/Springer Basel AG, 2001

    MATH  Google Scholar 

  24. Necas J. Direct Methods in the Theory of Elliptic Equations. Springer, 2012

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Acknowledgements

The author would like to thank Prof Jiří Neustupa for his valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tezpur University, Sonitpur Assam, 784028, India

    Subha Pal & Rajib Haloi

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  1. Subha Pal
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  2. Rajib Haloi
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Correspondence to Rajib Haloi.

Additional information

The first author was supported by National Board for Higher Mathematics (02011/9/2019NBHM (R.P.)/R and D II/1324).

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Pal, S., Haloi, R. On Solution to the Navier-Stokes Equations with Navier Slip Boundary Condition for Three Dimensional Incompressible Fluid. Acta Math Sci 39, 1628–1638 (2019). https://doi.org/10.1007/s10473-019-0613-8

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  • DOI: https://doi.org/10.1007/s10473-019-0613-8

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