Skip to main content
SpringerLink
Log in

Partial Boundary Regularity for the Navier-Stokes Equations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We prove two conditions of local Holder continuity for suitable weak solutions to the Navier-Stokes equations near a smooth curved part of the boundary of the domain. One of these conditions has the form of the Caffarelli-Kohn-Nirenberg condition for local boundedness of suitable weak solutions at interior points of the space-time cylinder. The corresponding results near a planar part of the boundary have been established earlier by Seregin. Bibliography: 21 titles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. O. V. Besov, V. P. Il'in, and S. M. Nikolskii, Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  2. L. Caffarelli, R. V. Kohn, and L. Nirenberg, “Partial regularity of suitable weak solutions of the Navier-Stokes equations,” Comm. Pure Appl. Math., 35, 771–831 (1982).

    MathSciNet  Google Scholar 

  3. H. J. Choe and J. L. Lewis, “On the singular set in the Navier-Stokes equations,” J. Funct. Anal., 175, 348–369 (2000).

    Article  MathSciNet  Google Scholar 

  4. L. Escauriaza, G. Seregin, and V. Sverak, “L 3,∞-solutions to the Navier-Stokes equations and backward uniqueness,” Usp. Mat. Nauk, 58, 3–44 (2003).

    MathSciNet  Google Scholar 

  5. R. Farwig and H. Sohr, “The stationary and nonstationary Stokes system in exterior domains with nonzero divergence and nonzero boundary data,” Math. Meth. Appl. Sci., 17, 269–291 (1994).

    Article  MathSciNet  Google Scholar 

  6. Y. Giga and H. Sohr, “Abstract L p-estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains,” J. Funct. Anal., 102, 72–94 (1991).

    Article  MathSciNet  Google Scholar 

  7. E. Hopf, “Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen,” Math. Nachr., 4, 213–231 (1950).

    MathSciNet  Google Scholar 

  8. O. A. Ladyzhenskaya, Mathematical Problems in the Dynamics of a Viscous Incompressible Fluid [in Russian], 2nd ed., Nauka, Moscow (1970).

    Google Scholar 

  9. O. A. Ladyzhenskaya and G. A. Seregin, “On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations,” J. Math. Fluid Mech., 1, 356–387 (1999).

    Article  MathSciNet  Google Scholar 

  10. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, 23, Amer. Math. Soc. (1967).

  11. F.-H. Lin, “A new proof of the Caffarelli-Kohn-Nirenberg theorem,” Comm. Pure Appl. Math., 51, 241–257 (1998).

    MATH  MathSciNet  Google Scholar 

  12. P. Maremonti and V. Solonnikov, “On estimates of solutions to the nonstationary Stokes problem in the anisotropic Sobolev spaces with a mixed norm,” Zap. Nauchn. Semin. POMI, 293, 124–150 (1994).

    Google Scholar 

  13. V. Scheffer, “Hausdorff measure and the Navier-Stokes equations,” Comm. Math. Phys., 55, 97–112 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  14. G. A. Seregin, “Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary,” J. Math. Fluid Mech., 4, 1–29 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  15. G. A. Seregin, “Differentiability properties of weak solutions of the Navier-Stokes equations,” St. Petersb. Math. J., 14, 147–178 (2003).

    MathSciNet  Google Scholar 

  16. G. A. Seregin, “Some estimates near the boundary for solutions to the nonstationary linearized Navier-Stokes equations,” Zap. Nauchn. Semin. POMI, 271, 204–223 (2000).

    MATH  Google Scholar 

  17. G. A. Seregin, “Remarks on regularity of weak solutions to the Navier-Stokes system near the boundary,” Zap. Nauchn. Semin. POMI, 295, 168–179 (2003).

    Google Scholar 

  18. V. A. Solonnikov, “Estimates of solutions of the nonstationary linearized system of Navier-Stokes equations,” Trudy Mat. Inst. AN SSSR, 70, 213–317 (1964).

    MATH  MathSciNet  Google Scholar 

  19. V. A. Solonnikov, “Estimates of solutions of the nonstationary Navier-Stokes system,” Zap. Nauchn. Semin. POMI, 38, 153–231 (1972).

    MathSciNet  Google Scholar 

  20. V. A. Solonnikov, “Estimates of solutions of the Stokes equations in Sobolev spaces with a mixed norm,” Zap. Nauchn. Semin. POMI, 288, 204–231 (2002).

    MATH  Google Scholar 

  21. V. A. Solonnikov, “On estimates of solutions of the nonstationary Stokes problem in anisotropic S. L. Sobolev spaces and estimate of the resolvent of the Stokes problem,” Usp. Mat. Nauk, 58, 123–156 (2003).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg Department, Steklov Mathematical Institute, St. Petersburg, Russia

    G. A. Seregin, T. N. Shilkin & V. A. Solonnikov

Authors
  1. G. A. Seregin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. N. Shilkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. A. Solonnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

To Nina Nikolaevna Uraltseva on the occasion of her 70th birthday

__________

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 158–190.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Seregin, G.A., Shilkin, T.N. & Solonnikov, V.A. Partial Boundary Regularity for the Navier-Stokes Equations. J Math Sci 132, 339–358 (2006). https://doi.org/10.1007/s10958-005-0502-7

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-005-0502-7

Keywords

Search

Navigation