# American Institute of Mathematical Sciences

May  2019, 18(3): 1333-1350. doi: 10.3934/cpaa.2019064

## Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations

 1 Department of Mathematics and IMS, Nanjing University, Nanjing, 210093, China 2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

* Corresponding author

Received  June 2018 Revised  September 2018 Published  November 2018

Fund Project: The first author is supported by China Scholar Council (NO. 201606190089). The second author is supported by the Research Foundation of Nanjing University of Aeronautics and Astronautics (NO. 1008-YAH17070)

Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical component $u^z$, accompanied by a $log$ subcritical assumption on the horizontally angular component $u^θ$ of the velocity. Second, the precise Green function for the operator $-(Δ-\frac{1}{r^2})$ under the axially symmetric situation, where $r$ is the distance to the symmetric axis, and some weighted $L^p$ estimates of it will be given. This will serve as a tool for the study of axially symmetric Navier-Stokes equations. As an application, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a critical (or a subcritical) assumption on the angular component $w^θ$ of the vorticity.

Citation: Zijin Li, Xinghong Pan. Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1333-1350. doi: 10.3934/cpaa.2019064
##### References:
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Pan, Regularity of solutions to axisymmetric Navier-Stokes equations with a slightly supercritical condition, J. Differential Equations, 260 (2016), 8485-8529. doi: 10.1016/j.jde.2016.02.026. Google Scholar [26] X. Pan, A regularity condition of 3d axisymmetric Navier-Stokes equations, Acta Appl. Math., 150 (2017), 103-109. doi: 10.1007/s10440-017-0096-3. Google Scholar [27] G. Tian and Z. Xin, One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal., 11 (1998), 135-145. doi: 10.12775/TMNA.1998.008. Google Scholar [28] M. R. Ukhovskii and V. I Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Meh., 32, 59-69 (Russian); translated as J. Appl. Math. Mech., 32 (1968), 52-61. doi: 10.1016/0021-8928(68)90147-0. Google Scholar [29] G. N. Watson, A Treatise on the Theory of Bessel Functions, Reprint of the second edition, 1994, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. 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##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. Google Scholar [2] H. Abidi and P. Zhang, Global smooth axisymmetric solutions of 3-D inhomogeneous incompressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 54 (2015), 3251-3276. doi: 10.1007/s00526-015-0902-6. Google Scholar [3] L. J. Burke and Q. S. Zhang, A priori bounds for the vorticity of axially symmetric solutions to the Navier-Stokes equations, Adv. Differential Equations, 15 (2010), 531-560. Google Scholar [4] D. Chae and J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671. doi: 10.1007/s002090100317. Google Scholar [5] E. A. Carlen and M. Loss, Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier-Stokes equation, Duke Math. J., 81 (1995), 135-157. doi: 10.1215/S0012-7094-95-08110-1. Google Scholar [6] C. C. Chen, R. M. Strain, H. Z. Yau and T. P. Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations, Int. Math. Res. Not. (IMRN), (2008), Art. ID rnn016, 31 pp. doi: 10.1093/imrn/rnn016. Google Scholar [7] C. C. Chen, R. M. Strain, T. P. Tsai and H. Z. Yau, Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations Ⅱ, Comm. Partial Differential Equations, 34 (2009), 203-232. doi: 10.1080/03605300902793956. Google Scholar [8] H. Chen, D. Fang and T. Zhang, Regularity of 3D axisymmetric Navier-Stokes equations, Discrete and Continuous Dynamical Systems, 37 (2017), 1923-1939. doi: 10.3934/dcds.2017081. Google Scholar [9] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. Google Scholar [10] C. L. Fefferman, Existence and Smoothness of the Navier-Stokes Equation. The Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, (2006), 57-67. Google Scholar [11] A. Grigor'yan, Heat kernels on weighted manifolds and applications, The Ubiquitous Heat Kernel, Amer. Math. Soc., Providence, RI, Contemp. Math., 398 (2006), 93-191. doi: 10.1090/conm/398/07486. Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar [13] T. Gallay and V. Sverak, Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations, preprint, arXiv: 1510.01036.Google Scholar [14] G. Seregin and D. Zhou, Regularity of solutions to the Navier-Stokes equations in $\dot{B}^{-1}_{\infty, \infty }$ preprint, arXiv: 1802.03600. doi: 10.1007/s00021-002-8533-z. Google Scholar [15] T.Y. Hou, Z. Lei and C. Li, Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data, Comm. Partial Differential Equations, 33 (2008), 1622-1637. doi: 10.1080/03605300802108057. Google Scholar [16] T. Y. Hou and C. Li, Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (2008), 661-697. doi: 10.1002/cpa.20212. Google Scholar [17] Q. Jiu and Z. Xin, Some regularity criteria on suitable weak solutions of the 3-D incompressible axisymmetric Navier-Stokes equations, New Stud. Adv. Math., 2 (2003), 119-139. Google Scholar [18] G. Koch, N. Nadirashvili, G. A. Seregin and V. Sverak, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105. doi: 10.1007/s11511-009-0039-6. Google Scholar [19] O. A. Ladyzenskaja, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968) 155-177 (Russian). Google Scholar [20] Z. Lei and Q. S. Zhang, A Liouville theorem for the axially-symmetric Navier-Stokes equations, J. Funct. Anal., 261 (2011), 2323-2345. doi: 10.1016/j.jfa.2011.06.016. Google Scholar [21] Z. Lei and Q. S. Zhang, Structure of solutions of 3D axisymmetric Navier-Stokes equations near maximal points, Pacific J. Math., 254 (2011), 335-344. doi: 10.2140/pjm.2011.254.335. Google Scholar [22] Z. Lei and Q. S. Zhang, Notes on axially symmetric Navier-Stokes equations, (2014) (private communications).Google Scholar [23] Z. Lei and Q. S. Zhang, Criticality of the axially symmetric Navier-Stokes equations, Pacific J. Math., 289 (2017), 169-187. doi: 10.2140/pjm.2017.289.169. Google Scholar [24] J. Neustupa and M. Pokorny, An interior regularity criterion for an axially symmetric suitable weak solution to the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 381-399. doi: 10.1007/PL00000960. Google Scholar [25] X. Pan, Regularity of solutions to axisymmetric Navier-Stokes equations with a slightly supercritical condition, J. Differential Equations, 260 (2016), 8485-8529. doi: 10.1016/j.jde.2016.02.026. Google Scholar [26] X. Pan, A regularity condition of 3d axisymmetric Navier-Stokes equations, Acta Appl. Math., 150 (2017), 103-109. doi: 10.1007/s10440-017-0096-3. Google Scholar [27] G. Tian and Z. Xin, One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal., 11 (1998), 135-145. doi: 10.12775/TMNA.1998.008. Google Scholar [28] M. R. Ukhovskii and V. I Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Meh., 32, 59-69 (Russian); translated as J. Appl. Math. Mech., 32 (1968), 52-61. doi: 10.1016/0021-8928(68)90147-0. Google Scholar [29] G. N. Watson, A Treatise on the Theory of Bessel Functions, Reprint of the second edition, 1994, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. Google Scholar [30] D. Wei, Regularity criterion to the axially symmetric Navier-Stokes equations, J. Math. Anal. Appl., 435 (2016), 402-413. doi: 10.1016/j.jmaa.2015.09.088. Google Scholar
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