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

Article

First Online:

- 5 Downloads

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.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A. Cheskidov and R. Shvydkoy, “On the regularity of weak solutions of the 3
*D*Navier–Stokes equations in \( {B}_{\infty, \infty}^{-1} \),”*Arch. Ration. Mech. Anal.*,**195**, 159–169 (2010).MathSciNetCrossRefGoogle Scholar - 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.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.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.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.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.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.T. Y. Hou, L. Zhen, and C. Li, “Global regularity of the 3
*D*axi-symmetric Navier–Stokes equations with anisotropic data,”*Comm. PDE’s*,**33**, No. 7–9, 1622–1637 (2008).MathSciNetCrossRefGoogle Scholar - 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.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.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.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.M. Ledoux, “On improved Sobolev embedding theorems,”
*Math. Research Letters*,**10**, 659–669 (2003).MathSciNetCrossRefGoogle Scholar - 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.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.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.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.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.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.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.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.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.W. Wang and Z. Zhang, “Regularity of weak solutions for the Navier-Stokes equations in the class
*L*^{∞}(BMO^{−1}),”*Comm. Contemp. Math.*,**14**, No. 3, 1250020, (2012).MathSciNetCrossRefGoogle Scholar - 24.D. Wei, “Regularity criterion to the axially symmetric Navier–Stokes equations,”
*J. Math. Anal. Appl.*,**435**, No. 1, 402–413 (2016).MathSciNetCrossRefGoogle Scholar

## Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020