Gevrey Class Regularity and Exponential Decay Property for Navier-Stokes-α Equations

Abstract

The Navier-Stokes-α equations subject to the periodic boundary conditions are considered. Analyticity in time for a class of solutions taking values in a Gevrey class of functions is proven. Exponential decay of the spatial Fourier spectrum for the analytic solutions and the lower bounds on the rate defined by the exponential decay are also obtained.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Constantin, P., Foias, C. Navier-Stokes equations. The University of Chicago Press, Chicago, 1988

  2. 2.

    Doering, C. R., Titi, E. S. Exponential decay rate of the power spectrum for solutions of the Navier-Stokes equations. Phys. Fluids, 7: 1384–1390 (1995)

    Article  MathSciNet  Google Scholar 

  3. 3.

    Foias, C., Holm, D. D. and Titi E. S. The three dimensional Viscous Camassa-Holm equations and their relation to the Navier-Stokes equations and turbulence theory. J. Dyna. Diff. Equa., 4: 1–35 (2002)

    Article  MathSciNet  Google Scholar 

  4. 4.

    Foias, C., Holm, D. D. and Titi, E. S. The Navier-Stokes-α model of fluid turbulence. Physica D., 152: 505–519 (2001)

    Article  MathSciNet  Google Scholar 

  5. 5.

    Foias, C., Temam, R. Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Func. Anal., 87: 359–369 (1989)

    Article  MathSciNet  Google Scholar 

  6. 6.

    Foias, C., Temam, R. Navier-Stokes equations and turbulence. Cambridge University Press, Cambridge, 2001

  7. 7.

    Holm, D. D., Mardsen, J. E. and Ratiu, T. S. Euler-Poincar´e equations and semidirect products with applications to continuum theories. Adv. in Math., 137: 1–81 (1998)

    Article  MathSciNet  Google Scholar 

  8. 8.

    Yu, Y.J., Li, K.T. Existence of solutions and Gevrey class regularity for Leray-alpha equations. J. Math. Anal. Appl., 306: 227–242 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yong-jiang Yu.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Yu, Yj., Li, Kt. & Huang, Ax. Gevrey Class Regularity and Exponential Decay Property for Navier-Stokes-α Equations. Acta Mathematicae Applicatae Sinica, English Series 23, 49–58 (2007). https://doi.org/10.1007/s10255-006-0348-x

Download citation

Keywords

  • Gevrey class regularity
  • Navier-Stokes-α equations
  • exponential decay

2000 MR Subject Classification

  • 35Q30
  • 35Q35