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Global Axisymmetric Solutions to the 3D MHD Equations with Nonzero Swirl

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Abstract

This paper studies sufficient conditions under which axisymmetric solutions with nonzero swirl components to the Cauchy problem of the 3D incompressible magnetohydrodynamic (MHD) equations are globally well-posed. We first establish a Serrin-type regularity criterion via the swirl component of velocity for the MHD equations without magnetic diffusion. Some new estimates were introduced to overcome the difficulty caused by the absence of magnetic diffusion. Moreover, we prove the global existence of axisymmetric solutions in the presence of magnetic diffusion provided that the scaling-invariant smallness condition was prescribed only on the swirl component of initial velocity while the initial magnetic field can be arbitrarily large.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Abidi, H.: Regularity results for axisymmetric solutions of the Navier-Stokes system. Bull. Sci. Math. 132(7), 592–624 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, H., Fang, D., Zhang, T.: Regularity of 3D axisymmetric Navier-Stokes equations. Discrete Contin. Dyn. Syst. 37(4), 1923–1939 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chemin, J.Y., McCormick, D.S., Robinson, J.C., Rodrigo, J.L.: Local existence for the non-resistive MHD equations in Besov spaces. Adv. Math. 286, 1–31 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cao, C., Titi, E.S.: Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor. Arch. Ration. Mech. Anal. 202, 919–932 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chemin, J., Zhang, P., Zhang, Z.: On the critical one component regularity for 3D Navier-Stokes system: general case. Arch. Ration. Mech. Anal. 224, 871–905 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chemin, J.Y., Gallagher, I.: Wellposedness and stability results for the Navier-Stokes equations in \(\mathbb{R}^3\). Ann. Inst. H. Poincaré Anal. Non Linéaire 26(2), 599–624 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chemin, J.Y., Gallagher, I., Paicu, M.: Global regularity for some classes of large solutions to the Navier-Stokes equations. Ann. Math. 173(2), 983–1012 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, X., Guo, Z., Zhu, M.: A new regularity criterion for the 3D MHD equations involving partial components. Acta Appl. Math. 134, 161–171 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, Q., Miao, C., Zhang, Z.: On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations. Commun. Math. Phys. 284(3), 919–930 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Duvaut, G., Lions, J.L.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279 (1972)

    Article  MATH  Google Scholar 

  12. Fefferman, C.L.: Existence and smoothness of the Navier-Stokes equation, The millennium prize problems, 57–67. Clay Math. Inst, Cambridge, MA (2006)

    Google Scholar 

  13. Fefferman, C.L., McCormick, D.S., Robinson, J.C., Rodrigo, J.L.: Higher order commutator estimates and local existence for the non-resistive MHD equations and related models. J. Funct. Anal. 267(4), 1035–1056 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fefferman, C.L., McCormick, D.S., Robinson, J.C., Rodrigo, J.L.: Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces. Arch. Ration. Mech. Anal. 223(2), 677–691 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guo, Z., Wang, Y., Xie, C.: Global strong solutions to the inhomogeneous incompressible Navier-Stokes system in the exterior of a cylinder. SIAM J. Math. Anal. 53, 6804–6821 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, Z., Wang, Y., Li, Y.: Regularity criteria of axisymmetric weak solutions to the 3D MHD equations. J. Math. Phys. 62, 121502 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guo, Z., Kucera, P., Skalak, Z.: The application of anisotropic Troisi inequalities to the conditional regularity for the Navier-Stokes equations. Nonlinearity 31, 3707–3725 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guo, Z., Kucera, P., Skalak, Z.: Regularity criterion for solutions to the Navier-Stokes equations in the whole 3D space based on two vorticity components. J. Math. Anal. Appl. 458, 755–766 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guo, Z., Caggio, M., Skalak, Z.: Regularity criteria for the Navier-Stokes equations based on one component of velocity. Nonlinear Anal. Real World Appl. 35, 379–396 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guo, Z., Wittwer, P., Wang, W.: Regularity issue of the Navier-Stokes equations involving the combination of pressure and velocity field. Acta Appl. Math. 123, 99–112 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo, Z., Gala, S.: Remarks on logarithmical regularity criteria for the Navier-Stokes equations. J. Math. Phys. 52(6), 063503 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. He, C., Xin, Z.P.: Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations. J. Funct. Anal. 227, 113–152 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. He, C., Xin, Z.P.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Equations 213(2), 235–254 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jiu, Q., Liu, J.: Global regularity for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. Discrete Contin. Dyn. Syst. 35(1), 301–322 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jiu, Q., Yu, H., Zheng, X.: Global well-posedness for axisymmetric MHD system with only vertical viscosity. J. Differ. Equ. 263(5), 2954–2990 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kukavica, I., Ziane, M.: One component regularity for the Navier-Stokes equations. Nonlinearity 19, 453–469 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kubica, A., Pokorny, M., Zajączkowski, 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 

  28. Lemarié-Rieusset, P.G.: Recent Developments in the Navier-Stokes Problem. Research Notes in Mathematics, vol. 431. Chapman & Hall/CRC, Boca Raton, FL (2002)

    MATH  Google Scholar 

  29. Leray, J.: Sur le mouvement dun liquide visqueux remplissant lespace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  Google Scholar 

  30. Lei, Z., Zhang, Q.: Criticality of the axially symmetric Navier-Stokes equations. Pac. J. Math. 289(1), 169–187 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lei, Z.: On axially symmetric incompressible magnetohydrodynamics in three dimensions. J. Differ. Equ. 259(7), 3202–3215 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu, J., Wang, W.: Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier-Stokes equation. SIAM J. Math. Anal. 41(5), 1825–1850 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ladyzhenskaya, O.A.: Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 7, 155–177 (1968). (Russian)

    MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  35. Liu, Y.: Global well-posedness of 3D axisymmetric MHD system with pure swirl magnetic field. Acta Appl. Math. 155, 21–39 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Majda, A., Bertozzi, A.: Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2002)

  37. 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 

  38. Neustupa, J., Pokorny, M.: Axisymmetric flow of Navier-Stokes fluid in the whole space with non-zero angular velocity component, Proceedings of Partial Differential Equations and Applications (Olomouc, 1999). Math. Bohem. 126(2), 469–481 (2001)

  39. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ukhovskii, M.R., Yudovich, V.I.: Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech. 32, 52–61 (1968)

    Article  MathSciNet  Google Scholar 

  41. Wang, H., Li, Y., Guo, Z., Skalak, Z.: Conditional regularity for the 3D incompressible MHD equations via partial components, Commu. Math. Sci. 17, 1025–1043 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Wang, P., Guo, Z.: Global well-posedness for axisymmetric MHD equations with vertical dissipation and vertical magnetic diffusion. Nonlinearity 35, 2147–2174 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wu, J.: Bounds and new approaches for the 3D MHD equations. J. Nonlinear Sci. 12, 395–413 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhang, P., Zhang, T.: Global axisymmetric solutions to three-dimensional Navier-Stokes system. Int. Math. Res. Not. IMRN No. 3, 610–642 (2014)

  45. Zhang, S., Guo, Z.: Regularity criteria for the 3D magnetohydrodynamics system involving only two velocity components. Math. Methods Appl. Sci. 43, 9014–9023 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhou, Y.: Remaks on regularities for the 3D MHD equations. Discrete Contin. Dyn. Syst. 12, 881–886 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhou, Y.: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(3), 491–505 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank Dr. Tianxiao Huang for his helpful discussion, and thank the referees for their valuable comments on the initial manuscript. The second author was partially supported by Natural Science Foundation of Jiangsu Province (BK20201478) and Qing Lan Project of Jiangsu Universities.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huizhou University, Huizhou, 516007, Guangdong, China

    Peng Wang

  2. School of Mathematics and Statistics, Huaiyin Normal University, Huaian, 223300, Jiangsu, China

    Zhengguang Guo

  3. Department of Mathematics, Wenzhou University, Wenzhou, 325035, Zhejiang, China

    Zhengguang Guo

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  1. Peng Wang
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Wang, P., Guo, Z. Global Axisymmetric Solutions to the 3D MHD Equations with Nonzero Swirl. J Geom Anal 32, 258 (2022). https://doi.org/10.1007/s12220-022-01006-x

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