Abstract
We are concerned with the problem, originated from Seregin (159–200, 2007), Seregin (J. Math. Sci. 143: 2961–2968, 2007), Seregin (Russ. Math. Surv. 62:149–168, 2007), what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3D Navier–Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled L 2 norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions.
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References
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)
Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of Navier–Stokes equation. Commun. Pure. Appl. Math. 35, 771–831 (1982)
Cao C., Wu J.: Two regularity criteria for the 3D MHD equations. J. Differ. Equ. 248, 2263–2274 (2010)
Chae D., Kang K., Lee J.: On the interior regularity of suitable weak solutions to the Navier–Stokes equations. Comm. Partial Differ. Equ. 32, 1189–1207 (2007)
Chemin J., Desjardin B., Gallagher I., Grenier E.: Fluids with anisotropic viscosity. Special issue for R. Temam’s 60th birthday. M2AN Math. Model. Numer. Anal 34, 315–335 (2000)
Dong H., Gu X.: Partial regularity of solutions to the four-dimensional Navier–Stokes equations. Dyn. Partial Differ. Equ. 11, 53–69 (2014)
Gustafson S., Kang Tsai T.: Interior regularity criteria for suitable weak solutions of the Navier–Stokes equations. Commun. Math. Phys. 273, 161–176 (2007)
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)
Lin F.: A new proof of the Caffarelli-Kohn-Nirenberg Theorem. Commun. Pure Appl. Math. 51, 241–257 (1998)
Majda A., Bertozzi A.: Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge (2002)
Miao C., Zheng X.: On the global well-posedness for the Boussinesq system with horizontal dissipation. Commun. Math. Phys 321, 33–67 (2013)
Neustupa J., Penel P.: Regularity of a suitable weak solution to the Navier–Stokes equations as a consequence of a regularity of one velocity component. In: Beiraoda Veiga, H., Sequeira, A., Videman, J. (eds.) Nonlinear applied analysis., pp. 391–402. Plenum Press, New York (1999)
Nirenberg L.: An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa 20, 733–737 (1966)
Scheffer V.: Partial regularity of solutions to the Navier–Stokes equations. Pac. J. Math. 66, 535–552 (1976)
Scheffer V.: Hausdorff measure and the Navier–Stokes equations. Commun. Math. Phys 55, 97–112 (1977)
Scheffer V.: The Navier–Stokes equations on a bounded domain. Commun. Math. Phys 73, 1–42 (1980)
Seregin, G., Šverák, V.: On smoothness of suitable weak solutions to the Navier–Stokes equations. J. Math. Sci. (N. Y.) 130, 4884–4892 (2005)
Seregin, G.: Local regularity theory of the Navier–Stokes equations. In: Friedlander S., Serre D (eds.) Handbook of mathematical fluid dynamics. Vol 4. North-Holland, Amsterdam,159–200 (2007)
Seregin, G.: Estimates of suitable weak solutions to the Navier–Stokes equations in critical Morrey spaces. J. Math. Sci. (N. Y.) 143, 2961–2968 (2007)
Seregin, G.: On the local regularity of suitable weak solutions of the Navier–Stokes equations. (Russian) Uspekhi Mat. Nauk 62 , 149–168; translation in Russian Math. Surveys 62 595–614 (2007)
Serrin J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962)
Témam, R.: Navier–Stokes equations: theory and numerical analysis. AMS Chelsea Publishing, Providence, RI, Reprint of the 1984 edition (2001)
Tian G., Xin Z.: Gradient estimation on Navier–Stokes equations. Commun. Anal. Geom. 7, 221–257 (1999)
Vasseur A.: A new proof of partial regularity of solutions to Navier–Stokes equations. Nonlinear Differ. Equ. Appl. 14, 753–785 (2007)
Wang, W., Zhang, L., Zhang Z.: On the interior regularity criteria of the 3-D Navier–Stokes equations involving two velocity components. (2014). arXiv:1410.2399
Wang W., Zhang Z.: On the interior regularity criteria for suitable weak solutions of the magnetohydrodynamics equations. SIAM J. Math. Anal. 45, 2666–2677 (2013)
Wang W., Zhang Z.: On the interior regularity criteria and the number of singular points to the Navier–Stokes equations. J. Anal. Math. 123, 139–170 (2014)
Wang, W., Zhang, Z.: Blow-up of critical norms for the 3-D Navier–Stokes equations (2015). arXiv:1510.02589
Wang Y., Wu G.: A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier–Stokes equations. J. Differ. Equ. 256, 1224–1249 (2014)
Zheng X.: A regularity criterion for the tridimensional Navier–Stokes equations in term of one velocity component. J. Differ. Equ. 256, 283–309 (2014)
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Wang, Y., Wu, G. Anisotropic Regularity Conditions for the Suitable Weak Solutions to the 3D Navier–Stokes Equations. J. Math. Fluid Mech. 18, 699–716 (2016). https://doi.org/10.1007/s00021-016-0279-0
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DOI: https://doi.org/10.1007/s00021-016-0279-0