Skip to main content
SpringerLink
Log in

On the Truth, and Limits, of a Full Equivalence \({\mathbf{p \cong \,v^2 }}\) in the Regularity Theory of the Navier–Stokes Equations: A Point of View

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

The motivation at the origin of this note is the well known sufficient condition for regularity of solutions to the evolution Navier–Stokes equations, sometimes referred to in the literature as Ladyzhenskaya–Prodi–Serrin’s condition. Such a condition requires that the velocity field \(\,v\,\), alone, satisfies sufficiently strong integrability requirements in space–time. On the other hand, a relation like \(\,{p \cong \,|v|^2}\,\), with p pressure field, is loosely suggested by the Navier–Stokes equations themselves. In three papers published nearly 20 years ago we have considered this problem. The results obtained there immediately suggest new interesting questions. In this paper, we propose, and solve, some of them, while many other related problems remain still open.

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. Beirão da Veiga, H.: Existence and asymptotic behaviour for strong solutions of the Navier–Stokes equations in the whole space. Indiana Univ. Math. J. 36, 149–166 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beirão da Veiga, H.: Concerning the regularity of the solutions to the Navier–Stokes equations via the truncation method; Part I. Differ. Integral Equ. 10, 1149–1156 (1997)

    MATH  Google Scholar 

  3. Beirão da Veiga, H.: Concerning the regularity of the solutions to the Navier–Stokes equations via the truncation method. Part II. Équations aux Dérivées Partielles et Applications; Articles dédiés à J.L. Lions à l’occasion de son 70. Anniversaire. Gauthier-Villars, Paris, pp. 127–138 (1998)

  4. Beirão da Veiga, H.: A sufficient condition on the pressure for the regularity of weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 2, 99–106 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Berselli, L.C., Galdi, G.P.: Regularity criteria involving the pressure for weak solutions to the Navier–Stokes equations. Proc. Am. Math. Soc. 130, 3585–3595 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berselli, L.C., Manfrin, R.: On a theorem of Sohr for the Navier–Stokes equations. J. Evol. Equ. 4, 193–211 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kaniel, S.: A sufficient condition for smoothness of solutions of Navier–Stokes equations. Isr. J. Math. 6, 354–358 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rionero, S., Galdi, G.P.: The weight function approach to uniquiness of viscous flows in unbounded domains. Arch. Ration. Mech. Anal. 69, 37–52 (1979)

    Article  MATH  Google Scholar 

  9. Seregin, G., Sverak, V.: Navier–Stokes equations with lower bounds on the pressure. Arch. Ration. Mech. Anal. 163, 65–86 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sohr, H.: Zur Regularitätstheorie der instationären Gleichungen von Navier–Stokes. Math. Z. 184, 359–375 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhou, Y.: Regularity criteria in terms of pressure for the $3-D$ Navier–Stokes equations. Math. Ann. 328, 173–192 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Pisa University, Pisa, Italy

    H. Beirão da Veiga

Authors
  1. H. Beirão da Veiga
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Beirão da Veiga.

Additional information

Partially supported by FCT (Portugal) under Grant UID/MAT/04561/3013.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beirão da Veiga, H. On the Truth, and Limits, of a Full Equivalence \({\mathbf{p \cong \,v^2 }}\) in the Regularity Theory of the Navier–Stokes Equations: A Point of View. J. Math. Fluid Mech. 20, 889–898 (2018). https://doi.org/10.1007/s00021-018-0377-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00021-018-0377-2

Search

Navigation