Skip to main content
SpringerLink
Log in

A New Criterion for Partial Regularity of Suitable Weak Solutions to the Navier-Stokes Equations

  • Chapter
  • First Online:
Advances in Mathematical Fluid Mechanics

Abstract

In the present paper we study local properties of suitable weak solutions to the Navier-Stokes equation in a cylinder Q = Ω × (0, T). Using the local representation of the pressure we are able to define a positive constant ɛ⋆ such that for every parabolic subcylinder QR ⊂ Q the condition

$$R^{-2}\int_{Q_R}|u|^3dxdt\leq\varepsilon_{\ast}$$

implies \({\bf U}\in L^{\infty}(Q_{R/2})\)). As one can easily check this condition is weaker then the well known Serrin's condition as well as the condition introduced by Farwig, Kozono and Sohr in a recent paper. Since our condition can be verified for suitable weak solutions to the Navier-Stokes system it improves the known results substantially.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Caffarelli, L., Kohn, R., Nirenberg, M.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35, 771–831 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Farwig, R., Kozono, H., Sohr, H.: An L q-approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195, 21–53 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Farwig, R., Kozono, H., Sohr, H.: Energy-Based Regularity Criteria for the Navier-Stokes Equations. J. Math. Fluid Mech. 8(3), 428–442 (2008)

    MathSciNet  Google Scholar 

  4. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations—Vol. I: Linearized Steady Problems. Springer-Verlag, New York (1994)

    Google Scholar 

  5. Hopf, E.: Über die Anfangswertaufgabe für die Hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213 (1950/1951)

    Google Scholar 

  6. Ladyzhenskaya, O.A., Seregin, G.A.: On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J. Math. Fluid Mech. 1(4), 356–387 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Leray, J.: Sur le mouvements d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Scheffer, V.: Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math. 66, 535–552 (1976)

    MATH  MathSciNet  Google Scholar 

  10. Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys. 55, 97–102 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  11. Seregin, G.A.: Differentiability properties of weak soluitons of the Navier-Stokes equations (English. Russian original). St. Petersbg. Math J. 14(1), 147–178 (2003); trans. from Algebra Anal. 14(1), 194–237 (2002)

    Google Scholar 

  12. Simader, G.C.: On Dirichlet’s Boundary Value Problem. Lecture Notes Mathematics, Vol. 268. Springer, Berlin Heidelberg New York (1972)

    Google Scholar 

  13. Simader, G.C.; Sohr, H.: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Mathematics Series, Vol. 360. Addison Wesley Longman Ltd, Harlow (1996)

    Google Scholar 

  14. Sohr, H.: The Navier-Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser, Basel (2001)

    Google Scholar 

  15. Sohr, H., von Wahl, W.: On the regularity of the pressure of weak solutions of the Navier-Stokes equations. Arch. Math. (Basel) 46, 428–439 (1986)

    MATH  MathSciNet  Google Scholar 

  16. Tian, G., Xin, Z.: Gradient estimation on Navier-Stokes equations. Commun. Anal. Geom. 7(2), 221–257 (1999)

    MATH  MathSciNet  Google Scholar 

  17. Vasseur, A.F.: A new proof of partial regularity of solutions to Navier-Stokes equations. NoDEA Nonlinear Differential Equations and Appl. 14(5–6), 735–785 (2007)

    MathSciNet  Google Scholar 

  18. Wolf, J.: Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9(1), 104–138 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wolf, J.: A direct proof of the Caffarelli-Kohn-Nirenberg theorem. Banach Center Publ.: Parabolic and Navier-Stokes equation 81, 533–552 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Humboldt University Berlin, 10099, Berlin, Germany

    JÖrg Wolf

Authors
  1. JÖrg Wolf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JÖrg Wolf .

Editor information

Editors and Affiliations

  1. Interdisziplinäres Zentrum für, Universität Heidelberg, Im Neuenheimer Feld 368, Heidelberg, 69120, Germany

    Rolf Rannacher

  2. Depto. Mathemática, Istituto Superior Técnico, Av. Rovisco Pais, Lisboa, 1049-001, Portugal

    Adélia Sequeira

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Wolf, J. (2009). A New Criterion for Partial Regularity of Suitable Weak Solutions to the Navier-Stokes Equations. In: Rannacher, R., Sequeira, A. (eds) Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04068-9_34

Download citation

Publish with us

Policies and ethics

Search

Navigation