# On Wolf’s Regularity Criterion of Suitable Weak Solutions to the Navier–Stokes Equations

- 72 Downloads
- 1 Citations

## Abstract

*u*be a suitable weak solution in the Navier–Stokes equations. There exists an absolute positive constant \(\varepsilon \) such that if

*u*satisfies

*u*is bounded in some neighborhood of point (0, 0). This gives an improvement of previous corresponding results obtained in Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), in Guevara and Phuc (Calc Var 56:68, 2017) and Wolf (Ann Univ Ferrara 61:149–171, 2015).

## Keywords

Navier–Stokes equations Suitable weak solutions Regularity## Mathematics Subject Classification

76D03 76D05 35B33 35Q35## Notes

### Acknowledgements

We are deeply grateful to the anonymous referee and the associated editor for the invaluable comments and suggestions which helped to improve the paper significantly. Jiu was partially supported by the National Natural Science Foundation of China (No. 11671273) and by Beijing Natural Science Foundation (No. 1192001). The research of Wang was partially supported by the National Natural Science Foundation of China under Grant No. 11601492 and the Youth Core Teachers Foundation of Zhengzhou University of Light Industry. The research of Zhou is supported in part by the China Scholarship Council for one year study at Mathematical Institute of University of Oxford and Doctor Fund of Henan Polytechnic University (No. B2012-110). Part of this work was done when D. Zhou is visiting Mathematical Institute of University of Oxford. Zhou appreciates Prof. G. Seregin’s warm hospitality and support.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure. Appl. Math.
**35**, 771–831 (1982)MathSciNetCrossRefADSGoogle Scholar - 2.Chae, D., Wolf, J.: Removing discretely self-similar singularities for the 3D Navier–Stokes equations. Commun. Partial Differ. Equ.
**42**, 1359–1374 (2017)MathSciNetCrossRefGoogle Scholar - 3.Chae, D., Wolf, J.: On the Liouville type theorems for self-similar solutions to the Navier–Stokes equations. Arch. Ration. Mech. Anal.
**225**, 549–572 (2017)MathSciNetCrossRefGoogle Scholar - 4.Choe, H., Wolf, J., Yang, M.: A new local regularity criterion for suitable weak solutions of the Navier–Stokes equations in terms of the velocity gradient. Math. Ann.
**370**, 629–647 (2018)MathSciNetCrossRefGoogle Scholar - 5.Galdi, G., Simader, C., Sohr, H.: On the Stokes problem in Lipschitz domains. Annali di Mat. pura ed appl. (IV)
**167**, 147–163 (1994)MathSciNetCrossRefGoogle Scholar - 6.Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)zbMATHGoogle Scholar
- 7.Guevara, C., Phuc, N.C.: Local energy bounds and \(\varepsilon \)-regularity criteria for the 3D Navier–Stokes system. Calc. Var.
**56**, 68 (2017)MathSciNetCrossRefGoogle Scholar - 8.He, C., Wang, Y., Zhou, D.: New \(\varepsilon \)-regularity criteria and application to the box dimension of the singular set in the 3D Navier–Stokes equations. arxiv:1709.01382
- 9.Kukavica, I.: On partial regularity for the Navier–Stokes equations. Discrete Contin. Dyn. Syst.
**21**, 717–728 (2008)MathSciNetCrossRefGoogle Scholar - 10.Kukavica, I.: Regularity for the Navier–Stokes equations with a solution in a Morrey space. Indiana Univ. Math. J.
**57**, 2843–2860 (2008)MathSciNetCrossRefGoogle Scholar - 11.Ladyzenskaja, O., Seregin, G.: On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations. J. Math. Fluid Mech.
**1**, 356–387 (1999)MathSciNetCrossRefADSGoogle Scholar - 12.Lin, F.: A new proof of the Caffarelli–Kohn–Nirenberg Theorem. Commun. Pure Appl. Math.
**51**, 241–257 (1998)MathSciNetCrossRefGoogle Scholar - 13.Ren, W., Wang, Y., Wu, G.: Partial regularity of suitable weak solutions to the multi-dimensional generalized magnetohydrodynamics equations. Commun. Contemp. Math.
**18**, 1650018 (2016)MathSciNetCrossRefGoogle Scholar - 14.Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal.
**9**, 187–195 (1962)MathSciNetCrossRefGoogle Scholar - 15.Struwe, M.: Partial regularity results for the Navier–Stokes equations. Commun. Pure Appl. Math.
**41**, 437–458 (1988)MathSciNetCrossRefGoogle Scholar - 16.Tang, L., Yu, Y.: Partial regularity of suitable weak solutions to the fractional Navier–Stokes equations. Commun. Math. Phys.
**334**, 1455–1482 (2015)MathSciNetCrossRefADSGoogle Scholar - 17.Vasseur, A.: A new proof of partial regularity of solutions to Navier–Stokes equations. NoDEA Nonlinear Differ. Equ. Appl.
**14**, 753–785 (2007)MathSciNetCrossRefGoogle Scholar - 18.Wolf, J.: 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, A.S.R edn, pp. 613–630. Springer, Berlin (2010)Google Scholar
- 19.Wolf, J.: On the local regularity of suitable weak solutions to the generalized Navier–Stokes equations. Ann. Univ. Ferrara
**61**, 149–171 (2015)MathSciNetCrossRefGoogle Scholar