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Some Interior Regularity Criteria Involving Two Components for Weak Solutions to the 3D Navier–Stokes Equations

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Abstract

In this paper, we prove new local criteria in terms of the horizontal gradient of the horizontal velocity \( \nabla _{h}u_{h}\) for suitable weak solutions of the 3D Navier–Stokes equations. In addition, we present an alternative proof of some known interior criteria involving two components of suitable weak solutions to this system.

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References

  1. Bae, H., Choe, H.: A regularity criterion for the Navier–Stokes equations. Commun. Part. Differ. Equ. 32, 1173–1187 (2007)

    Article  MathSciNet  Google Scholar 

  2. Bae, H., Wolf, J.: Boundary regularity for the steady Stokes type flow with shear thickening viscosity. J. Differ. Equ. 258, 3811–3850 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  3. Bae, H., Wolf, J.: Existence of strong solutions to the equations of unsteady motion of shear thickening incompressible fluids. Nonlinear Anal. Real World Appl. 23, 160–182 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bae, H., 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 

  5. Bae, H., Wolf, J.: Sufficient conditions for local regularity to the generalized Newtonian fluid with shear thinning viscosity. Z. Angew. Math. Phys. 68, Art. 23 (2017)

  6. Beale, J., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  7. Beirao da Veiga, H.: A new regularity class for the Navier–Stokes equations in \(R^{n}\). Chin. Ann. Math. Ser. B 16, 407–412 (1995)

    MATH  Google Scholar 

  8. Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of Navier–Stokes equation. Commun. Pure Appl. Math. 35, 771–831 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  9. 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  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  11. Chae, D., Kang, K., Lee, J.: On the interior regularity of suitable weak solutions to the Navier–Stokes equations. Commun. Part. Differ. Equ. 32, 1189–1207 (2007)

    Article  MathSciNet  Google Scholar 

  12. Chae, D., Lee, J.: On the geometric regularity conditions for the 3D Navier–Stokes equations. Nonlinear Anal. 151, 265–273 (2017)

    Article  MathSciNet  Google Scholar 

  13. Dong, B., Chen, Z.: Regularity criterion of weak solutions to the 3D Navier–Stokes equations via two velocity components. J. Math. Anal. Appl. 338, 1–10 (2008)

    Article  MathSciNet  Google Scholar 

  14. Escauriaza, L., Seregin, G., Šverák, V.: On \(L^{\infty }L^{3}\) -solutions to the Navier–Stokes equations and backward uniqueness. Russ. Math. Surv. 58, 211–250 (2003)

    Article  Google Scholar 

  15. Fabes, E.B., Jones, B.F., Riviere, N.M.: The initial value problem for the Navier–Stokes equations with datain \(L^p\). Arch. Ration. Mech. Anal. 45, 222–240 (1972)

    Article  Google Scholar 

  16. Galdi, G.: An Introduction to the Navier–Stokes Initial-Boundary Value Problem, Fundamental Directions in Mathematical Fluid Mechanics, pp. 1–70. Birkhäuser, Basel (2000)

    MATH  Google Scholar 

  17. Giga, Y., Sohr, H.: Abstract \(L^p\)-estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)

    Article  MathSciNet  Google Scholar 

  18. Grujić, Z.: Localization and geometric depletion of vortex-stretching in the 3D NSE. Commun. Math. Phys. 290, 861–870 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  19. He, C.: Regularity for solutions to the Navier–Stokes equations with one velocity component regular. Electron. J. Differ. Equ. 29, 1–13 (2002)

    MathSciNet  Google Scholar 

  20. Hopf, E.: Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1950) (German)

    Article  Google Scholar 

  21. Ignatova, M., Kukavica, I., Ryzhik, L.: The Harnack inequality for second-order parabolic equations with divergence-free drifts of low regularity. Commun. Part. Differ. Equ. 41, 208–226 (2016)

    Article  MathSciNet  Google Scholar 

  22. Jia, X., Zhou, Y.: Ladyzhenskaya–Prodi–Serrin type regularity criteria for the 3D incompressible MHD equations in terms of \(3\times 3\) mixture matrices. Nonlinearity 28, 3289–3307 (2015)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  24. Kukavica, I., Rusin, W., Ziane, M.: Localized anisotropic regularity conditions for the Navier–Stokes equations. J. Nonlinear Sci. 27, 1725–1742 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  25. Leray, J.: Sur le mouvement déun liquide visqueux emplissant léspace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  Google Scholar 

  26. Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  27. Neustupa, J., Penel, P.: Regularity of a suitable weak solution to the Navier-Stokes Equations as a Consequence of Regularity of One Velocity Component, Applied Nonlinear Analysis. Kluwer, New York (1999)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Solonnikov, V.A.: Estimates for solutions of nonstationary Navier–Stokes equations. J. Sov. Math. 8, 467–523 (1977)

    Article  Google Scholar 

  30. Struwe, M.: On partial regularity results for the Navier–Stokes equations. Commun. Pure Appl. Math. 41, 437–458 (1988)

    Article  MathSciNet  Google Scholar 

  31. 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 

  32. Wolf, J.: A regularity criterion of Serrin-type for the Navier–Stokes equations involving the gradient of one velocity component. Analysis (Berlin) 35, 259–292 (2015)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the anonymous referee and the associated editor for the invaluable comments and suggestions which helped to improve the paper greatly. 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. Wu was partially supported by the National Natural Science Foundation of China under Grant Nos. 11771423 and 11671378. 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).

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Authors and Affiliations

  1. Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450002, Henan, People’s Republic of China

    Yanqing Wang

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Gang Wu

  3. School of Mathematics and Informatics Science, Henan Polytechnic University, Jiaozuo, 454000, Henan, People’s Republic of China

    Daoguo Zhou

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  1. Yanqing Wang
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  2. Gang Wu
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  3. Daoguo Zhou
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Correspondence to Yanqing Wang.

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Communicated by D. Chae

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Wang, Y., Wu, G. & Zhou, D. Some Interior Regularity Criteria Involving Two Components for Weak Solutions to the 3D Navier–Stokes Equations. J. Math. Fluid Mech. 20, 2147–2159 (2018). https://doi.org/10.1007/s00021-018-0402-5

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