Skip to main content
Log in

Some regularity criteria of a weak solution to the 3D Navier–Stokes equations in a domain

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We give some regularity criterion (of a weak\(-L^{p}\) Serrin type) of a weak solution to the 3D Navier–Stokes equations in a bounded domain \(\Omega \subset \mathbb {R}^3\) with a smooth boundary. In particular, in case of the half space, we give a regularity condition of a weak solution with respect to a tangential component of the velocity flow vector.

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

Access this article

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. Bae, H.-O., Choe, H.J.: A regularity criterion for the Navier–Stokes equations. Comm. Partial Differ. Equ. 32, 1173–1187 (2007)

    Article  MathSciNet  Google Scholar 

  2. Bae, H.-O., Wolf, J.: A local regularity condition involving two velocity components of Serrin-type for the Navier–Stokes equations. C. R. Math. Acad. Sci. Paris 354, 167–174 (2016)

    Article  MathSciNet  Google Scholar 

  3. Beirão da Veiga, H., Berselli, L.C.: Navier–Stokes equations: Green’s matrices, vorticity direction, and regularity up to the boundary. J. Differ. Equ. 246, 597–628 (2009)

  4. Beirão da Veiga, H., Crispo, F.: Sharp inviscid limit results under Navier type boundary conditions. An \(L^p\) theory. J. Math. Fluid Mech. 12, 397–411 (2010)

  5. Berselli, L.C., Spirito, S.: On the vanishing viscosity limit of 3D Navier–Stokes equations under slip boundary conditions in general domains. Comm. Math. Phys. 316, 171–198 (2012)

    Article  MathSciNet  Google Scholar 

  6. Bosia, S., Pata, V., Robinson, J.C.: A weak-\(L^p\) Prodi–Serrin type regularity criterion for the Navier–Stokes equations. J. Math. Fluid Mech. 16, 721–725 (2014)

    Article  MathSciNet  Google Scholar 

  7. Chae, D.: On the regularity conditions of suitable weak solutions of the 3D Navier–Stokes equations. J. Math. Fluid Mech. 12, 171–180 (2010)

    Article  MathSciNet  Google Scholar 

  8. Chae, D., Choe, H.J.: Regularity of solutions to the Navier–Stokes equation. Electron. J. Differ. Equ. 5, 1–7 (1999)

    Article  MathSciNet  Google Scholar 

  9. Guo, Z., Kučera, P., Skalák, Z.: Regularity criterion for solutions to the Navier–Stokes equations in the whole 3D space based on two vorticity components. J. Math. Anal. Appl. 458, 755–766 (2018)

    Article  MathSciNet  Google Scholar 

  10. Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundglei-chungen. Math. Nachr. 4, 213–231 (1951)

    Article  MathSciNet  Google Scholar 

  11. Ji, X., Wang, Y., Wei, W.: New regularity criteria based on pressure or gradient of velocity in Lorentz spaces for the 3D Navier–Stokes equations. J. Math. Fluid Mech. 22(1), Paper No. 13, 8 pp. (2020)

  12. Kang, K., Kim, J.-M.: Regularity criteria of the magenetohydrodynamic equations in bounded domains or a half space. J. Differ. Equ. 253, 764–794 (2012)

    Article  Google Scholar 

  13. Kukavica, I., Ziane, M.: Navier–Stokes equations with regularity in one direction. J. Math. Phys. 48, 065203, 10 pp (2007)

  14. Ladyženskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI (1967)

  15. Lee, J.: Notes on the geometric regularity criterion of 3D Navier–Stokes system. J. Math. Phys. 53, (2012)

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

    Article  MathSciNet  Google Scholar 

  17. O’Neil, R.: Convolution operators and \(L(p, q)\) spaces. Duke Math. J. 30, 129–142 (1963)

    Article  MathSciNet  Google Scholar 

  18. Prodi, G.: Un teorema di unicita per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)

    Article  MathSciNet  Google Scholar 

  19. Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 9, 187–195 (1962)

    Article  MathSciNet  Google Scholar 

  20. Skalák, Z.: Commentary on local and boundary regularity of weak solutions to Navier–Stokes equations. Electron. J. Differ. Equ., No. 9, 14 pp. (2004)

  21. Suzuki, T.: A remark on the regularity of weak solutions to the Navier–Stokes equations in terms of the pressure in Lorentz spaces. Nonlinear Anal. 75, 3849–3853 (2012)

    Article  MathSciNet  Google Scholar 

  22. Triebel, H.: Theory of Function Spaces. Birkhäuser Verlag, Basel (1983)

    Book  Google Scholar 

  23. von Wahl, W.: Estimating \(\nabla u\) by \(\text{div} u\) and \({\rm {curl}}\,u\). Math. Methods Appl. Sci. 15, 123–143 (1992)

  24. Wang, W., Zhang, L., Zhang, Z.: On the interior regularity criteria of the 3-D Navier–Stokes equations involving two velocity components. Discrete Contin. Dyn. Syst. 38, 2609–2627 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks the knowledgeable referee for his/her valuable comments and helpful suggestions. Jae-Myoung Kim was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2020R1C1C1A01006521).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jae-Myoung Kim.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, JM. Some regularity criteria of a weak solution to the 3D Navier–Stokes equations in a domain. Arch. Math. 117, 215–225 (2021). https://doi.org/10.1007/s00013-021-01613-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-021-01613-0

Keywords

Mathematics Subject Classification

Navigation