Journal of Statistical Physics

, Volume 135, Issue 4, pp 737–750 | Cite as

A New Boundary Problem for the Two Dimensional Navier-Stokes System

  • Efim Dinaburg
  • Dong Li
  • Yakov G. Sinai


We formulate a new boundary value problem for the 2D Navier-Stokes system on the unit square. Under some suitable assumptions on the initial velocity, we obtain quantitative decay estimates of the Fourier modes of both the vorticity and the velocity. It is found that in one direction the Fourier modes decay exponentially and along the other direction their decay is only power like.


Navier-Stokes equations 


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  1. 1.
    Cao, C., Rammaha, M., Titi, E.S.: Gevrey regularity for nonlinear analytic parabolic equations on the sphere. J. Dyn. Differ. Equ. 12, 411–433 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chernov, N.: Numerical studies of a two-dimensional Navier-Stokes system with new boundary conditions. J. Stat. Phys. (to appear) Google Scholar
  3. 3.
    Constantin, P., Foias, C.: Navier-Stokes Equations. University of Chicago Press, Chicago (1988) zbMATHGoogle Scholar
  4. 4.
    Dinaburg, E.I., Li, D., Sinai, Ya.G.: Navier-Stokes system on the flat cylinder and unit square with slip boundary conditions. Commun. Contemp. Math. (submitted) Google Scholar
  5. 5.
    Doering, C.R., Titi, E.S.: Exponential decay rate of the power spectrum for solutions of the Navier-Stokes equations. Phys. Fluids 7(6), 1384–1390 (1995) zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Ferrari, A., Titi, E.S.: Gevrey regularity for nonlinear analytic parabolic equations. Commun. Partial Differ. Equ. 23, 1–16 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal. 87(2), 359–369 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow. Gordon and Breach, New York (1969) zbMATHGoogle Scholar
  10. 10.
    Leray, J.: Essai sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Mattingly, J.C., Sinai, Ya.G.: An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations. Commun. Contemp. Math. 1(4), 497–516 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2. North-Holland Publishing Co., Amsterdam (1979), revised edn. zbMATHGoogle Scholar
  13. 13.
    Temam, R.: Navier-Stokes equations and nonlinear functional analysis. CBMS-NSF Regional Conferences Series in Applied Mathematics, vol. 66, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1995) zbMATHGoogle Scholar
  14. 14.
    Yudovich, V.I.: The Linearization Method in Hydrodynamical Stability Theory. American Mathematical Society, Providence (1989). Translated from Russian by J.R. Schunlenberger zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute of Physics of EarthRussia Academy of SciencesMoscowRussia
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  3. 3.Mathematics DepartmentPrinceton UniversityPrincetonUSA

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