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Journal of Nonlinear Science

, Volume 19, Issue 2, pp 133–152 | Cite as

Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations

  • Varga K. Kalantarov
  • Boris Levant
  • Edriss S. TitiEmail author
Article

Abstract

Recently, the Navier–Stokes–Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier–Stokes equations for the purpose of direct numerical simulations. In this work, we prove that the global attractor of the 3D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one. This result provides additional evidence that the 3D NSV with the small regularization parameter enjoys similar statistical properties as the 3D Navier–Stokes equations. Finally, we calculate a lower bound for the exponential decaying scale—the scale at which the spectrum of the solution start to decay exponentially, and establish a similar bound for the steady state solutions of the 3D NSV and 3D Navier–Stokes equations. Our estimate coincides with the known bounds for the smallest length scale of the solutions of the 3D Navier–Stokes equations, established earlier by Doering and Titi.

Keywords

Navier–Stokes–Voight equations Navier–Stokes equations Global attractor Regularization of the Navier–Stokes equations Turbulence models Viscoelastic models Gevrey regularity 

Mathematics Subject Classification (2000)

35Q30 35Q35 35B40 35B41 76F20 76F55 

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References

  1. Bardina, J., Ferziger, J., Reynolds, W.: Improved subgrid scale models for large Eddy simulation. Am. Inst. Aeron. Astronaut. 80, 80–1357 (1980) Google Scholar
  2. Berselli, L.C., Iliescu, T., Layton, W.J.: Mathematics of large Eddy simulation of turbulent flows. In: Scientific Computation. Springer, New York (2006) Google Scholar
  3. Cao, Y., Lunasin, E.M., Titi, E.S.: Global well-posedness of the three dimensional viscous and inviscid simplified Bardina turbulence models. Commun. Math. Sci. 4, 823–884 (2006) zbMATHMathSciNetGoogle Scholar
  4. Chueshov, I., Polat, M., Siegmund, S.: Gevrey regularity of global attractor for generalized Benjamin–Bona–Mahony equation. Mat. Fiz. Anal. Geom. 11(2), 226–242 (2004) zbMATHMathSciNetGoogle Scholar
  5. Constantin, P., Foias, C.: Navier–Stokes Equations. The University of Chicago Press, Chicago (1988) zbMATHGoogle Scholar
  6. Constantin, P., Foias, C., Manley, O.P., Temam, R.: Determining modes and fractal dimension of turbulent flows. J. Fluid Mech. 150, 427–440 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  7. Doering, C., Titi, E.S.: Exponential decay rate of the power spectrum for the solutions of the Navier–Stokes equations. Phys. Fluids 7(6), 1384–1390 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  8. Ferrari, A.B., Titi, E.S.: Gevrey regularity for nonlinear analytic parabolic equations. Commun. Part. Differ. Equ. 23(1–2), 1–16 (1998) zbMATHMathSciNetGoogle Scholar
  9. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87, 359–369 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  10. Foias, C., Manley, O., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. Cambridge University Press, Cambridge (2001) zbMATHGoogle Scholar
  11. Goubet, O.: Regularity of the attractor for a weakly damped nonlinear Schrödinger equation. Appl. Anal. 60, 99–119 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  12. Hale, J.K., Raugel, G.: Regularity, determining modes and Galerkin methods. J. Math. Pure Appl. 82, 1075–1136 (2003) zbMATHMathSciNetGoogle Scholar
  13. Henshaw, W.D., Kreiss, H.O., Reyna, L.G.: On the smallest scale for the incompressible Navier–Stokes equations. Theor. Comput. Fluid Dyn. 1, 65–95 (1989) zbMATHGoogle Scholar
  14. Henshaw, W.D., Kreiss, H.O., Reyna, L.G.: Smallest scale estimates for the Navier–Stokes equations for incompressible fluids. Arch. Ration. Mech. Anal. 112, 21–44 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  15. Jones, D.A., Titi, E.S.: Determining finite volume elements for the 2D Navier–Stokes equations. Physica D 60, 165–174 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  16. Ilyin, A.A., Titi, E.S.: On the domain of analyticity and small scales for the solutions of the damped-driven 2D Navier–Stokes equations. Dyn. Part. Differ. Equ. 4(2), 111–127 (2007) zbMATHMathSciNetGoogle Scholar
  17. Kalantarov, V.K.: Attractors for some nonlinear problems of mathematical physics. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 152, 50–54 (1986) zbMATHGoogle Scholar
  18. Kalantarov, V.K., Titi, E.S.: Global attractors and estimates of the number of degrees of determining modes for the 3D Navier–Stokes–Voight equations. arXiv:0705.3972v1 (2007)
  19. Khouider, B., Titi, E.S.: An inviscid regularization for the surface quasi-geostrophic equation. Commun. Pure Appl. Math. 61(10), 1331–1346 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  20. Kukavica, I.: On the dissipative scale for the Navier–Stokes equations. Indiana Univ. Math. J. 48, 1057–1081 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  21. Layton, R., Lewandowski, R.: On a well-posed turbulence model. Discrete Contin. Dyn. Syst. B 6, 111–128 (2006) zbMATHMathSciNetGoogle Scholar
  22. Levermore, C.D., Oliver, M.: Analyticity of solutions for a generalized Euler equation. J. Differ. Equ. 133, 321–339 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  23. Métivier, G.: Valeurs propres d’opérateurs définis par la restriction de systèmes variationnels à des sous-espaces. J. Math. Pure Appl. 57(2), 133–156 (1978) zbMATHGoogle Scholar
  24. Oliver, M., Titi, E.S.: Analyticity of the attractor and the number of determining nodes for a weakly damped driven nonlinear Schrödinger equation. Indiana Univ. Math. J. 47(1), 49–73 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  25. Oliver, M., Titi, E.S.: Gevrey regularity for the attractor of a partially dissipative model of Bénard convection in a porous medium. J. Differ. Equ. 163, 292–311 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  26. Oliver, M., Titi, E.S.: On the domain of analyticity for solutions of second order analytic nonlinear differential equations. J. Differ. Equ. 174, 55–74 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  27. Olson, E., Titi, E.S.: Determining modes for continuous data assimilation in 2-D turbulence. J. Stat. Phys. 113, 799–840 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  28. Oskolkov, A.P.: The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 38, 98–136 (1973) zbMATHMathSciNetGoogle Scholar
  29. Oskolkov, A.P.: On the theory of Voight fluids. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 96, 233–236 (1980) zbMATHMathSciNetGoogle Scholar
  30. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988) zbMATHGoogle Scholar
  31. Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis, 3rd edn. North-Holland, Amsterdam (2001) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Varga K. Kalantarov
    • 1
  • Boris Levant
    • 2
  • Edriss S. Titi
    • 2
    • 3
    Email author
  1. 1.Department of MathematicsKoc UniversityIstanbulTurkey
  2. 2.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael
  3. 3.Department of Mathematics and Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaIrvineUSA

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