## Abstract

We present generalizations of results concerning conditional global regularity of weak Leray–Hopf solutions to incompressible Navier–Stokes equations presented by Zhou and Pokorný in articles (Pokorný, Electron J Differ Equ (11):1–8, 2003; Zhou, Methods Appl Anal 9(4):563–578, 2002; Zhou, J Math Pure Appl 84(11):1496–1514, 2005); see also Neustupa et al. (Quaderni di Matematica, vol. 10. Topics in Mathematical Fluid Mechanics, 2002, pp. 163–183) We are able to replace the condition on one velocity component (or its gradient) by a corresponding condition imposed on a projection of the velocity (or its gradient) onto a more general vector field. Comparing to our other recent results from Axmann and Pokorný (A note on regularity criteria for the solutions to Navier-Stokes equations involving one velocity component, in preparation), the conditions imposed on the projection are more restrictive here, however due to the technique used in Axmann and Pokorný (A note on regularity criteria for the solutions to Navier-Stokes equations involving one velocity component, in preparation), there appeared a specific additional restriction on geometrical properties of the reference field, which could be omitted here.

To Yoshihiro Shibata

## Access this chapter

Tax calculation will be finalised at checkout

Purchases are for personal use only

### Similar content being viewed by others

## References

Š. Axmann, M. Pokorný, A note on regularity criteria for the solutions to Navier-Stokes equations involving one velocity component (in preparation)

C. Cao, E.S. Titi, Regularity criteria for the three-dimensional Navier–Stokes equations. Indiana U. Math. J.

**57**(6), 2643–2660 (2008)H. El-Owaidy et al., On some new integral inequalities of Gronwall–Bellman type. Appl. Math. Comput.

**106**, 289–303 (1999)G.P. Galdi, An introduction to the Navier–Stokes initial–boundary value problem, in

*Fundamental Directions in Mathematical Fluid Mechanics*. Advances in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2000), pp. 1–70C. He, Regularity for solutions to the Navier–Stokes equations with one velocity component regular. Electron. J. Differ. Equ.

**2002**(29), 1–13 (2002)E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr.

**4**(1), 213–231 (1951)A. Kufner, O. John, S. Fučík,

*Function Spaces*, 1st edn. (Academia, Prague, 1977)I. Kukavica, M. Ziane, One component regularity for the Navier–Stokes equations. Nonlinearity

**19**(2), 453–469 (2006)O.A. Ladyzhenskaya,

*The Mathematical Theory of Viscous Incompressible Flow*, 2nd edn. (Gordon and Breach, New York/London/Paris, 1969)O.A. Ladyzhenskaya, Sixth problem of the millenium: Navier–Stokes equations, existence and smoothness. Russ. Math. Surv.

**58**(2), 251–286 (2003)J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pure Appl.

**12**(2), 1–82 (1933)J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math.

**63**(1), 193–248 (1934)J. Neustupa, P. Penel, Regularity of a suitable weak solution to the Navier–Stokes equations as a consequence of regularity of one velocity component, in

*Applied Nonlinear Analysis*, ed. by A. Sequeira et al. (Kluwer Academic/Plenum Pulishers, New York, 1999), pp. 391–402J. Neustupa, A. Novotný, P. Penel, An interior regularity of weak solution to the Navier–Stokes equations in dependence on one component of velocity, in

*Quaderni di Matematica*, ed. by A.O. Eden. Topics in Mathematical Fluid Mechanics, vol. 10 (Dept. Math., Seconda Univ. Napoli, Caserta, 2002), pp. 163–183M. Pokorný, On the result of He concerning the smoothness of solutions to the Navier–Stokes equations. Electron. J. Differ. Equ.

**2003**(11), 1–8 (2003)R. Temam, Some developments on Navier–Stokes equations in the second half of the 20th century, in

*Development of Mathematics 1950–2000*(Birkhäuser, Basel, 2000), pp. 1049–1106Y. Zhou, A new regularity criterion for the Navier–Stokes equations in terms of the gradient of one velocity component. Methods Appl. Anal.

**9**(4), 563–578 (2002)Y. Zhou, A new regularity criterion for weak solutions to the Navier–Stokes equations. J. Math. Pure Appl.

**84**(11), 1496–1514 (2005)Y. Zhou, M. Pokorný, On a regularity criterion for the Navier–Stokes equations involving gradient of one velocity component. J. Math. Phys.

**50**(12), 123514 (2009)Y. Zhou, M. Pokorný, On the regularity criterion of the Navier–Stokes equations via one velocity component. Nonlinearity

**23**(5), 1097–1107 (2010)

## Acknowledgements

The work of the first author was supported by the grant SVV-2015-260226. The work of the second author was partially supported by the grant No. 201/09/0917 of the Grant Agency of the Czech Republic.

## Author information

### Authors and Affiliations

### Corresponding author

## Editor information

### Editors and Affiliations

## Rights and permissions

## Copyright information

© 2016 Springer Basel

## About this chapter

### Cite this chapter

Axmann, Š., Pokorný, M. (2016). A Generalization of Some Regularity Criteria to the Navier–Stokes Equations Involving One Velocity Component. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds) Recent Developments of Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0939-9_5

### Download citation

DOI: https://doi.org/10.1007/978-3-0348-0939-9_5

Publisher Name: Birkhäuser, Basel

Print ISBN: 978-3-0348-0938-2

Online ISBN: 978-3-0348-0939-9

eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)