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Regularity Issue of the Navier-Stokes Equations Involving the Combination of Pressure and Velocity Field

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Abstract

We establish some regularity criteria for the incompressible Navier-Stokes equations in a bounded three-dimensional domain concerning the quotients of the pressure, the velocity field and the pressure gradient.

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References

  1. Beirão da Veiga, H.: A new regularity class for the Navier-Stokes equations in ℝn. Chin. Ann. Math., Ser. B 16, 407–412 (1995)

    MATH  Google Scholar 

  2. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chae, D., Choe, H.-J.: On the regularity criterion for the solutions of the Navier-Stokes equations. Electron. J. Differ. Equ. 1999, 1–7 (1999)

    Article  MathSciNet  Google Scholar 

  6. Cao, C., Titi, E.S.: Regularity criteria for the three dimensional Navier-Stokes equations. Indiana Univ. Math. J. 57, 2643–2661 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cao, C., Qin, J.L., Titi, E.S.: Regularity criterion for solutions of three-dimensional turbulent channel flows. Commun. Partial Differ. Equ. 33, 419–428 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fan, J., Jiang, S., Nakamura, G., Zhou, Y.: Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations. J. Math. Fluid Mech. 13, 557–571 (2011)

    Article  MathSciNet  Google Scholar 

  9. Giga, Y.: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Differ. Equ. 62, 186–212 (1986)

    Article  MathSciNet  Google Scholar 

  10. Guo, Z., Gala, S.: A note on the regularity criteria for the Navier-Stokes equations. Appl. Math. Lett. 25, 305–309 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, Z., Gala, S.: Remarks on logarithmical regularity criteria for the Navier-Stokes equations. J. Math. Phys. 52, 063503 (2011)

    Article  MathSciNet  Google Scholar 

  12. Hopf, E.: Über die Anfangwertaufgaben für die hydromischen Grundgleichungen. Math. Nachr. 4, 213–321 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  13. Iwashita, H.: L q-L r estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in L q spaces. Math. Ann. 285, 265–288 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kyungkeun, K., Lee, J.: On regularity criteria in conjunction with the pressure of Navier-Stokes equations. Int. Math. Res. Not., 80762 (2006)

  15. Kyungkeun, K., Lee, J.: Erratum: on regularity criteria in conjunction with the pressure of the Navier-Stokes equations. Int. Math. Res. Not. 9, 1772–1774 (2010). doi:10.1093/imrn/rnq073

    Google Scholar 

  16. Kato, T.: Strong L p-solutions to the Navier-Stokes equations in ℝm, with applications to weak solutions. Math. Z. 187, 471–480 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kozono, H., Sohr, H.: Regularity criterion on weak solutions to the Navier-Stokes equations. Adv. Differ. Equ. 2, 535–554 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Kozono, H., Taniuchi, Y.: Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235, 173–194 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kozono, H.: Global L n-solution and its decay property for the Navier-Stokes equations in half space \(R_{+}^{n}\). J. Differ. Equ. 79, 79–88 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Leray, J.: Étude de divers équations intégrales nonlinearies et de quelques problemes que posent lhydrodinamique. J. Math. Pures Appl. 12, 1–82 (1931)

    MathSciNet  Google Scholar 

  21. Núñez, M.: Regularity criteria for the Navier-Stokes equations involving the ratio pressure-gradient of velocity. Math. Methods Appl. Sci. 33, 323–331 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Penel, P., Pokorný, M.: Some new regularity criteria for the Navier-Stokes equations containing gradient of velocity. Appl. Math. 5, 483–493 (2004)

    Article  Google Scholar 

  23. Scheffer, V.: Partial regularity of solutions to the Navier-Stokes equations. Pac. J. Math. 66, 535–552 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Tian, G., Xin, Z.: Gradient estimation on Navier-Stokes equations. Commun. Anal. Geom. 7, 221–257 (1999)

    MathSciNet  MATH  Google Scholar 

  27. von Wahl, W.: Regularity of weak solutions of the Navier-Stokes equations. In: Proceedings of the 1983 Summer Institute on Nonlinear Functional Analysis and Applications. Proc. Symposia in Pure Mathematics, vol. 45, pp. 497–503. Am. Math. Soc., Providence (1989)

    Google Scholar 

  28. Zhou, Y.: On regularity criteria in terms of pressure for the Navier-Stokes equations in ℝ3. Proc. Am. Math. Soc. 134, 149–156 (2006)

    Article  MATH  Google Scholar 

  29. Zhou, Y.: A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component. Methods Appl. Anal. 9, 563–578 (2002)

    MathSciNet  MATH  Google Scholar 

  30. Zhou, Y.: A new regularity criterion for weak solutions to the Navier-Stokes equations. J. Math. Pures Appl. 84, 1496–1514 (2005)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhou, Y.: On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in ℝN. Z. Angew. Math. Phys. 57, 384–392 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhou, Y., Pokorný, M.: On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component. J. Math. Phys. 50, 123514 (2009)

    Article  MathSciNet  Google Scholar 

  34. Zhou, Y., Pokorný, M.: On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity 23, 1097–1107 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the referees for their comments on the initial version of this paper. Guo is partially support by Zhejiang Provincial Natural Science Foundation of China under Grant No. Q12A010047, the SSSTC Project (No. EG19-032009), the Swiss National Science Foundation under Grant No. 200021-124403 and National Natural Science Foundation of China, No. 11171257, Wittwer is supported by the Swiss National Science Foundation under Grant No. 200021-124403, Wang is partially supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. R1110261 and Y12A010041.

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Guo, Z., Wittwer, P. & Wang, W. Regularity Issue of the Navier-Stokes Equations Involving the Combination of Pressure and Velocity Field. Acta Appl Math 123, 99–112 (2013). https://doi.org/10.1007/s10440-012-9717-z

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  • DOI: https://doi.org/10.1007/s10440-012-9717-z

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