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Mathematische Annalen

, Volume 332, Issue 1, pp 219–238 | Cite as

On smoothness of L3,∞-solutions to the Navier–Stokes equations up to boundary

  • G. Seregin
Article

Abstract.

We show that L3,∞-solutions to the three-dimensional Navier-Stokes equations near a flat part of the boundary are smooth.

Keywords

Flat Part 
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References

  1. 1.
    Caffarelli, L., Kohn, R.-V., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. Vol. XXXV, 771–831 (1982)Google Scholar
  2. 2.
    Escauriaza, L., Seregin, G., Šverák, V.: On backward uniqueness for parabolic equations. Arch. Rational Mech. Anal. 169(2), 147–157 (2003)Google Scholar
  3. 3.
    Escauriaza, L., Seregin, G., Šverák, V.: Backward uniqueness for the heat operator in half space. Algebra and Analiz 15(1), 201–214 (2003)Google Scholar
  4. 4.
    Escauriaza, L., Seregin, G., Šverák, V.: L3,∞-Solutions to the Navier-Stokes Equations and Backward Uniqueness. Russian Mathematical Surveys 58(2), 211–250 (2003)Google Scholar
  5. 5.
    Giga, Y., Sohr, H.: Abstract Lp-estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Ladyzhenskaya, O. A.: Mathematical problems of the dynamics of viscous incompressible fluids. Fizmatgiz, Moscow 1961; English transltion, Gordon and Breach, New York-London, 1969Google Scholar
  7. 7.
    Ladyzhenskaya, O. A.: Mathematical problems of the dynamics of viscous incompressible fluids. 2nd edition, Nauka, Moscow, 1970Google Scholar
  8. 8.
    Lin, F.-H.: A new proof of the Caffarelly-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51(3), 241–257 (1998)Google Scholar
  9. 9.
    Maremonti, P., Solonnikov, V. A.: On the estimate of solutions of evolution Stokes problem in anisotropic Sobolev spaces with a mixed norm. Zap. Nauchn. Sem. LOMI 223, 124–150 (1994)Google Scholar
  10. 10.
    Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys. 55, 97–112 (1977)Google Scholar
  11. 11.
    Seregin, G. A.: Some estimates near the boundary for solutions to the non-stationary linearized Navier-Stokes equations. Zapiski Nauchn. Seminar. POMI 271, 204–223 (2000)Google Scholar
  12. 12.
    Seregin, G. A.: On the number of singular points of weak solutions to the Navier-Stokes equations. Commun. Pure Appl. Math. 54(8), 1019–1028 (2001)Google Scholar
  13. 13.
    Seregin, G. A.: Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary. J. Math. fluid Mech. 4(1), 1–29 (2002)Google Scholar
  14. 14.
    Seregin, G. A.: Remarks on regularity of weak solutions to the Navier-Stokes equations near the boundary. Zapiski Nauchn. Seminar. POMI 295, 168–179 (2003)Google Scholar
  15. 15.
    Seregin, G., Šverák, V.: The Navier-Stokes equations and backward uniqueness. Nonlinear Problems in Mathematical Physics II, In Honor of Professor O.A. Ladyzhenskaya, International Mathematical Series II, 2002, pp. 359–370Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • G. Seregin
    • 1
  1. 1.Steklov Institute of Mathematics at St.PetersburgSt.PetersburgRussia

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