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A Regularity Condition of 3d Axisymmetric Navier-Stokes Equations

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Abstract

In this paper, we study the regularity of 3d axisymmetric Navier-Stokes equations under a prior point assumption on \(v^{r}\) or \(v^{z}\). That is, the weak solution of the 3d axisymmetric Navier-Stokes equations \(v\) is smooth if

$$ rv^{r}\geq-1; \quad\mbox{or}\quad r\bigl|v^{r}(t,x)\bigr|\leq Cr^{\alpha}, \ \alpha\in(0,1];\quad\mbox{or} \quad r\bigl|v^{z}(t,x)\bigr| \leq Cr^{ \beta},\ \beta\in[0,1]; $$

where \(r\) is the distance from the point \(x\) to the symmetric axis.

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References

  1. Burke Loftus, J., Zhang, Q.S.: A priori bounds for the vorticity of axially symmetric solutions to the Navier-Stokes equations. Adv. Differ. Equ. 15(5–6), 531–560 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Chae, D., Lee, J.: On the regularity of the axisymmetric solutions of the Navier-Stokes equations. Math. Z. 239(4), 645–671 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, C.-C., Strain, R.M., Yau, H.-T., Tsai, T.-P.: Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations. Int. Math. Res. Not. 2008(9), rnn016, 31 pp. (2008)

    MathSciNet  MATH  Google Scholar 

  4. Chen, C.-C., Strain, R.M., Tsai, T.-P., Yau, H.-T.: Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations. II. Commun. Partial Differ. Equ. 34(1–3), 203–232 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, H., Fang, D., Zhang, T.: Regularity of 3D axisymmetric Navier-Stokes equations. arXiv:1505.00905

  6. Hou, T.Y., Li, C.: Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl. Commun. Pure Appl. Math. 61(5), 661–697 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hou, T.Y., Lei, Z., Li, C.: Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data. Commun. Partial Differ. Equ. 33(7–9), 1622–1637 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jiu, Q., Xin, Z.: Some regularity criteria on suitable weak solutions of the 3-D incompressible axisymmetric Navier-Stokes equations. In: Lectures on Partial Differential Equations. New Stud. Adv. Math., vol. 2, pp. 119–139. Int. Press, Somerville (2003)

    Google Scholar 

  9. Jiu, Q., Xin, Z.: On Liouville type of theorems to the 3-D incompressible axisymmetric Navier-Stokes equations. arXiv:1501.02412

  10. Koch, G., Nadirashvili, N., Seregin, G.A., Sverak, V.: Liouville theorems for the Navier-Stokes equations and applications. Acta Math. 203(1), 83–105 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kubica, A., Pokorný, M., Zajaczkowski, W.: Remarks on regularity criteria for axially symmetric weak solutions to the Navier-Stokes equations. Math. Methods Appl. Sci. 35(3), 360–371 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ladyzenskaja, O.A.: 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, 155–177 (1968) (Russian)

    MathSciNet  Google Scholar 

  13. Lei, Z., Zhang, Q.S.: A Liouville theorem for the axially-symmetric Navier-Stokes equations. J. Funct. Anal. 261(8), 2323–2345 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lei, Z., Zhang, Q.S.: Criticality of the axially symmetric Navier-Stokes equations. Pac. J. Math. (2017, to appear). arXiv:1505.02628

  15. Lei, Z., Navas, E.A., Zhang, Q.S.: A priori bound on the velocity in axially symmetric Navier-Stokes equations. Commun. Math. Phys. 341(1), 289–307 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Leonardi, S., Málek, J., Nečas, J., Pokorný, M.: On axially symmetric flows in \({\mathbb{R}}^{3}\). Z. Anal. Anwend. 18(3), 639–649 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Neustupa, J., Pokorny, M.: An interior regularity criterion for an axially symmetric suitable weak solution to the Navier-Stokes equations. J. Math. Fluid Mech. 2(4), 381–399 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ukhovskii, M.R., Iudovich, V.I.: Axially symmetric flows of ideal and viscous fluids filling the whole space. Prikl. Mat. Meh. 32, 59–69 (1968) (Russian); translated as J. Appl. Math. Mech. 32, 52–61

    MathSciNet  Google Scholar 

  19. Wei, D.: Regularity criterion to the axially symmetric Navier-Stokes equations. J. Math. Anal. Appl. 435(1), 402–413 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, P., Zhang, T.: Global axisymmetric solutions to three-dimensional Navier-Stokes system. Int. Math. Res. Not. 3, 610–642 (2014)

    MathSciNet  Google Scholar 

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Acknowledgements

I want to express my gratitude to the anonymous referees for their ideas and suggestions in the proof of this work.

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  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

    Xinghong Pan

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  1. Xinghong Pan
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Pan, X. A Regularity Condition of 3d Axisymmetric Navier-Stokes Equations. Acta Appl Math 150, 103–109 (2017). https://doi.org/10.1007/s10440-017-0096-3

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