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On 3D Lagrangian Navier–Stokes α Model with a Class of Vorticity-Slip Boundary Conditions

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Abstract

This paper concerns the 3-dimensional Lagrangian Navier–Stokes α model and the limiting Navier–Stokes system on smooth bounded domains with a class of vorticity-slip boundary conditions and the Navier-slip boundary conditions. It establishes the spectrum properties and regularity estimates of the associated Stokes operators, the local well-posedness of the strong solution and global existence of weak solutions for initial boundary value problems for such systems. Furthermore, the vanishing α limit to a weak solution of the corresponding initial-boundary value problem of the Navier–Stokes system is proved and a rate of convergence is shown for the strong solution.

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References

  1. Achdou Y., Pironneau O., Valentin F.: Effective boundary conditions for laminar flow over periodic rough boundaries. J. Comput. Phys. 147, 187–218 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Beirão da Veiga H.: Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions. Adv. Differ. Equ. 9, 1079–1114 (2004)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Beirão da Veiga H., Crispo F.: Concerning the W k,p-inviscid limit for 3D flows under a slip boundary condition. J. Math. Fluid Mech. 13, 117–135 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  5. Beirão da Veiga H., Berselli L.C.: Navier-Stokes equations: Green’s matrices, vorticity direction, and regularity up to the boundary. J. Differ. Equ. 246, 597–628 (2009)

    Article  MATH  Google Scholar 

  6. Beavers G.S., Joseph D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207 (1967)

    Article  ADS  Google Scholar 

  7. Bellout H., Neustupa J., Penel P.: On a ν continous family of strong solution to the Euler or Navier-Stokes equations with the Navier type boundary condition. Discrete Contin. Dyn. Syst. 27(4), 1353–1373 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bellout H., Neustupa J., Penel P.: On the Navier-Stokes equations with boundary conditions based on vorticity. Math. Nachr. 269/270, 59C72 (2004)

    Article  MathSciNet  Google Scholar 

  9. Bourguignon J.P., Brezis H.: Remarks on the Euler equation. J. Funct. Anal. 15, 341–363 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Busuioc A.V., Ratiu T.S.: Second grade fluid and averaged Euler equations with Navier slip boundary condition. Nonlinearity 16, 1119–1149 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Cantarella J., De Turk D., Gluck H.: Vector calculus and the topology of domains in 3-space. Am. Math. Mon. 109, 257–286 (2002)

    Article  Google Scholar 

  12. Cao Y., Titi E.S.: On the rate of convergence of the two-dimensional α− models of the turbulence to the Navier-Stokes equations. Num. Funct. Anal. Optim. 30(11–12), 1231–1271 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen G., Osborneb D., Qian Z.: The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries. Acta Math. Sci. 29(4), 919–948 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Chen S.Y., Foias C., Holm D.D., Olson E.J., Titi E.S., Wynne S.: The Camassa-Holm equations as a closure model for turbulent channel and pipe flows. Phys. Rev. Lett. 81, 5338–5341 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Chen S.Y., Foias C., Holm D.D, Olson E.J., Titi E.S., Wynne S.: The Camassa-Holm equations and turbulence in pipes and channels. Physica D 133, 49–65 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Chen S.Y., Foias C., Holm D.D, Olson E.J., Titi E.S., Wynne S.: The Camassa-Holm equations and turbulence in pipes and channels. Phys. Fluids 11, 2343–2353 (1999)

    Article  MathSciNet  ADS  MATH  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 11, 1625–1636 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Constantin P., Iyer G.: A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations. Commun. Pure Appl. Math. 61(3), 330–345 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Constantin P.: An Eulerian-Lagrangian approach to the Navier-Stokes equations. Commun. Math. Phys. 216(3), 663–686 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Constantin P., Foias C.: Navier Stokes Equation. University of Chicago Press, Chicago (1988)

    Google Scholar 

  21. Foias C., Holm D.D., Titi E.S.: The three-dimensional viscous Camassa-Holm equations and their relation to the Navier-Stokes equations and turbulence theory. J. Dyn. Differ. Equ. 14(1), 1–35 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Foias C., Temam R.: Remarques sur les equations de Navier–Stokes stationaires et les phenomenes successifs de bifurcation. Ann. Sc. Norm. Syper. Pisa 5, 29–63 (1978)

    MathSciNet  MATH  Google Scholar 

  23. Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Linearized steady problems, vol. I. Springer Tracts in Natural Philosophy, vol. 38. Springer, Berlin (1998)

  24. Gibbon J.D., Holm D.D.: Length-scale estimate for the LANS-α equations in terms of the Reynolds number. Physica D 222(2), 69–78 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  25. Gibbon J.D., Holm D.D.: Estimates for the LANS-α, Leray-α and Bardina models in terms of a Navier-Stokes Reynolds number. Indiana Univ. Math. J. 57(6), 2761–2773 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Girault V., Raviart P.-A.: Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  27. Holm D.D., Marsden J.E., Ratiu T.S.: Euler Poincare models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80, 4173–4176 (1998)

    Article  ADS  Google Scholar 

  28. Holm D.D.: Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion. Physica D 133, 215–269 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Holm D.D., Putkavadge V., Weiman P.D., Wingate B.A.: Boundary effects on exact solutions of the Lagrangian-averaged Navier-Stokes α equations. J. Stat. Phys. 113(516), 841–854 (2003)

    Article  MATH  Google Scholar 

  30. Iftimie D.S., Sueur F.: Viscosity boundary layers for the Navier-Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal. 199(1), 145C175 (2011)

    Article  MathSciNet  Google Scholar 

  31. Itoh S., Tanaka N., Tani A.: The initial value problem for the Navier-Stokes equations with general slip boundary condition in Hölder spaces. J. Math. Fluid Mech. 5, 275–301 (2003)

    MathSciNet  ADS  MATH  Google Scholar 

  32. Jäger W., Mikelić A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. John V.: Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equation-numerical test and aspect of the implementation. J. Comput. Appl. Math. 147, 287–300 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Kato, T.: Remarks on zero viscosity limit for non-stationary Navier-Stokes flows with boundary. In: Chen, S.S. (ed.) Seminar on PDE, pp. 85–98. Springer, New York (1984)

  35. Lions J.-L.: Quelques Méthodes de Résolution des Problemès aux Limites non Linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  36. Majda A.J., Bertozzi A.L.: Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  37. Marsden J.E., Shkoller S.: Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains. Phil. Trans. R. Soc. Lond. A 359, 1449–1468 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966)

  39. Navier C.L.M.H.: Sur les lois de l’équilibre et du mouvement des corps élastiques. Mem. Acad. R. Sci. Inst. France 6, 369 (1827)

    Google Scholar 

  40. Prizjev N.V., Troian S.M.: Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces. J. Fluid Mech. 554, 25–46 (2006)

    Article  ADS  Google Scholar 

  41. Schwarz G.: Hodge decomposition: a method for solving boundary value problems. Lecture Notes in Mathematics, vol. 1607. Springer, Berlin (1995)

    Google Scholar 

  42. Solonnikov, V.A., Ščadilov, V.E.: A certain boundary value problem for the stationary system of Navier-Stokes equations. Boundary Value Problem of Mathematical Physics, 8. Trudy Mat. Inst. Steklov. 125, 196–210 (1973); translation in Proc. Steklov Inst. Math. 125, 186–199 (1973)

  43. Tang C.L., Xin Z.P.: Existence of solutions for three dimensional stationary incompressible Euler equations with non-vanishing vorticity. Chinese Ann. Math. 30(6), 801–806 (2009)

    Article  MathSciNet  Google Scholar 

  44. Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Applied Mathematical Science, vol. 68. Springer, New York (1997)

  45. Thompson P.A., Troian S.M.: A general boundary condition for liquid flow at solid surface. Nature 389, 360–362 (1997)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  47. Xiao Y.L., Xin Z.P.: Remarks on the vanishing viscosity limit for 3D Navier-Stokes equations with a slip boundary condition. Chinese Ann. Math. 32(3), 321–332 (2011)

    Article  MathSciNet  Google Scholar 

  48. Yoshida Z., Giga Y.: Remarks on spectra of operator rot. Math. Z. 204, 235–245 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, People’s Republic of China

    Yuelong Xiao

  2. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

    Yuelong Xiao & Zhouping Xin

Authors
  1. Yuelong Xiao
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  2. Zhouping Xin
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Correspondence to Yuelong Xiao.

Additional information

Communicated by H. Beirao da Veiga.

This research is supported in part by NSFC 10971174, and Zheng Ge Ru Foundation, and Hong Kong RGC Earmarked Research Grants CUHK-4041/11P, CUHK-4042/08P and a Focus Area Grant from The Chinese University of Hong Kong.

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Xiao, Y., Xin, Z. On 3D Lagrangian Navier–Stokes α Model with a Class of Vorticity-Slip Boundary Conditions. J. Math. Fluid Mech. 15, 215–247 (2013). https://doi.org/10.1007/s00021-012-0110-5

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