Journal of Mathematical Sciences

, Volume 244, Issue 6, pp 1003–1009 | Cite as

Regularity of Solutions to the Navier–Stokes Equations in \( {\dot{B}}_{\infty, \infty}^{-1} \)

  • G. SereginEmail author
  • D. Zhou

It is proved that if u is a suitable weak solution to the three-dimensional Navier–Stokes equations from the space\( {L}_{\infty}\left(0,T;{\dot{B}}_{\infty, \infty}^{-1}\right) \), then all scaled energy quantities of u are bounded. As a consequence, it is shown that any axially symmetric suitable weak solution u, belonging to \( {L}_{\infty}\left(0,T;{\dot{B}}_{\infty, \infty}^{-1}\right) \), is smooth.


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  1. 1.
    A. Cheskidov and R. Shvydkoy, “On the regularity of weak solutions of the 3D Navier–Stokes equations in \( {B}_{\infty, \infty}^{-1} \),” Arch. Ration. Mech. Anal., 195, 159–169 (2010).MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. Burke and Q. Zhang, “A priori bounds for the vorticity of axially symmetric solutions to the Navier–Stokes equations,” Adv. Diff. Eq., 15, 531–560 (2010).MathSciNetzbMATHGoogle Scholar
  3. 3.
    D. Chae and J. Lee, “On the regularity of the axisymmetric solutions of the Navier–Stokes equations,” Math. Z., 239, No. 4, 645–671 (2002).MathSciNetCrossRefGoogle Scholar
  4. 4.
    C.-C. Chen, R. M. Strain, H.-T. Yau, and T.-P. Tsai, “Lower bound on the blow-up rate of the axisymmetric Navier–Stokes equations,” Int. Math. Res. Not., No 9, 016 (2008).Google Scholar
  5. 5.
    C.-C. Chen, R. M. Strain, H.-T. Yau, and T.-P. Tsai, “Lower bound on the blow-up rate of the axisymmetric Navier–Stokes equations II,” Comm. PDE’s, 34, 203–232 (2009).MathSciNetCrossRefGoogle Scholar
  6. 6.
    H. Hajaiej, L. Molinet, T. Ozawa, and B. Wang, “Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized boson equations,” in: Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS, Kyoto (2011), pp. 159–175.Google Scholar
  7. 7.
    T. Hmidi and D. Li, “Small ˙\( {\dot{B}}_{\infty, \infty}^{-1} \) implies regularity,” Dyn. PDE’s, 14, No. 1, 1–4 (2017).MathSciNetGoogle Scholar
  8. 8.
    T. Y. Hou, L. Zhen, and C. Li, “Global regularity of the 3D axi-symmetric Navier–Stokes equations with anisotropic data,” Comm. PDE’s, 33, No. 7–9, 1622–1637 (2008).MathSciNetCrossRefGoogle Scholar
  9. 9.
    T. Y. Hou and C. Li, “Dynamic stability of the three-dimensional axisymmetric Navier–Stokes equations with swirl,” Comm. Pure Appl. Math., 61, 661–697 (2008).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Q. Jiu and Z. Xin, “Some regularity criteria on suitable weak solutions of the 3-D incompressible axisymmetric Navier-Stokes equations,” in: New Studies in Advanced Mathematics, Vol. 2, International Press, Somerville (2003), pp. 119–139.Google Scholar
  11. 11.
    G. Koch, N. Nadirashvili, G. Seregin, and V. Sverak, “Liouville theorems for the Navier-Stokes equations and applications,” Acta Math., 203, 83–105 (2009).MathSciNetCrossRefGoogle Scholar
  12. 12.
    O. A. Ladyzhenskaya, “On the unique global solvability to the Cauchy problem for the Navier–Stokes equations in the presence of the axial symmetry,” Zap. Nauchn. Semin. LOMI, 7, 155–177 (1968).MathSciNetGoogle Scholar
  13. 13.
    M. Ledoux, “On improved Sobolev embedding theorems,” Math. Research Letters, 10, 659–669 (2003).MathSciNetCrossRefGoogle Scholar
  14. 14.
    Z. Lei and Q. Zhang, “A Liouville theorem for the axially-symmetric Navier–Stokes equations,” J. Funct. Anal., 261, No. 8, 2323–2345 (2011).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Z. Lei and Q. Zhang, “Structure of solutions of 3D axisymmetric Navier–Stokes equations near maximal points,” Pacific J. Math., 254, No. 2, 335–344 (2011).MathSciNetCrossRefGoogle Scholar
  16. 16.
    Z. Lei, E. Navas, and Q. Zhang, “A priori bound on the velocity in axially symmetric Navier–Stokes equations,” Comm. Math. Phys., 341, No. 1, 289–307 (2016).MathSciNetCrossRefGoogle Scholar
  17. 17.
    Z. Lei and Q. Zhang, “Criticality of the axially symmetric Navier–Stokes equations,” Pacific J. Math., 289, No. 1, 169–187 (2017).MathSciNetCrossRefGoogle Scholar
  18. 18.
    X. Pan, “Regularity of solutions to axisymmetric Navier–Stokes equations with a slightly supercritical condition,” J. Diff. Eq., 260, No. 12, 8485–8529 (2016).MathSciNetCrossRefGoogle Scholar
  19. 19.
    G. Seregin and V. Sverak, “On type I singularities of the local axi-symmetric solutions of the Navier–Stokes equations,” Comm. PDE’s, 34, 171–201 (2009).MathSciNetCrossRefGoogle Scholar
  20. 20.
    G. Seregin and W. Zajaczkowski, “A sufficient condition of regularity for axially symmetric solutions to the Navier–Stokes equations,” SIAM J. Math. Anal., 39, No. 2, 669–685 (2007).MathSciNetCrossRefGoogle Scholar
  21. 21.
    G. Seregin, “A note on bounded scale-invariant quantities for the Navier–Stokes equations,” J. Math. Sci. (N.Y.), 185, No. 5, 742–745 (2012).MathSciNetCrossRefGoogle Scholar
  22. 22.
    M. R. Ukhovskii and V. I. Yudovich, “Axially symmetric flows of ideal and viscous fluids filling the whole space,” J. Appl. Math. Mech., 32, 52–61 (1968).MathSciNetCrossRefGoogle Scholar
  23. 23.
    W. Wang and Z. Zhang, “Regularity of weak solutions for the Navier-Stokes equations in the class L(BMO1),” Comm. Contemp. Math., 14, No. 3, 1250020, (2012).MathSciNetCrossRefGoogle Scholar
  24. 24.
    D. Wei, “Regularity criterion to the axially symmetric Navier–Stokes equations,” J. Math. Anal. Appl., 435, No. 1, 402–413 (2016).MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUnited Kingdom
  2. 2.School of Mathematics and Information ScienceHenan Polytechnic UniversityJiaozuoChina

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