May 2019, 18(3): 1447-1482. doi: 10.3934/cpaa.2019070

Stability of axially-symmetric solutions to incompressible magnetohydrodynamics with no azimuthal velocity and with only azimuthal magnetic field

1. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw, Poland

2. 

Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw, Poland

The author thanks to professor Yoshihiro Shibata for the essential correction of the proof of Proposition 2

Received  December 2017 Revised  September 2018 Published  November 2018

Incompressible viscous axially-symmetric magnetohydrodynamics is considered in a bounded axially-symmetric cylinder. Vanishing of the normal components, azimuthal components and also azimuthal components of rotation of the velocity and the magnetic field is assumed on the boundary. First, global existence of regular special solutions, such that the velocity is without the swirl but the magnetic field has only the swirl component, is proved. Next, the existence of global regular axially-symmetric solutions which are initially close to the special solutions and remain close to them for all time is proved. The result is shown under sufficiently small differences of the external forces. All considerations are performed step by step in time and are made by the energy method. In view of complicated calculations estimates are only derived so existence should follow from the Faedo-Galerkin method.

Citation: Wojciech M. Zajączkowski. Stability of axially-symmetric solutions to incompressible magnetohydrodynamics with no azimuthal velocity and with only azimuthal magnetic field. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1447-1482. doi: 10.3934/cpaa.2019070
References:
[1]

O. V. Besov, V. P. Il'in and S. M. Nikol'skii, Integral representations of functions and imbedding theorems, Nauka, Moscow 1975 (in Russian).

[2]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.

[3]

Q. ChenC. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magnetohydrodynamic equations, Comm. Math. Phys., 284 (2008), 919-930. doi: 10.1007/s00220-008-0545-y.

[4]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magneétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.

[5]

C. He and X. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152. doi: 10.1016/j.jfa.2005.06.009.

[6]

C. He and X. Xin, On the regularity of weak solutions to magnetohydrodynamic equations, J. Diff. Equas., 213 (2005), 235-254.

[7]

X. Hu and F. H. Lin, Global existence of two-dimensional incompressible magnetohydrodynamic flows with zero magnetic diffusivity, arXiv: 1405.0082.

[8]

O. A. Ladyzhenskaya, On unique solvability of three-dimensional Cauchy problem for the Navier-Stokes equations under the axial symmetry, Zap. Nauchn. Sem LOMI, 7 (1968), 155-177 (in Russian).

[9]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Diff. Equas., 259 (2015), 3202-3215. doi: 10.1016/j.jde.2015.04.017.

[10]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: the three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580. doi: 10.1002/cpa.21506.

[11]

Fanghua LinXu Li and Zhang Ping, Global small solutions of 2-D incompressible magnetohydrodynamics system, J. Diff. Equas., 259 (2015), 5440-5485.

[12]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.

[13]

Xu Li and Zhang Ping, Global small solutions to three-dimensional incompressible magnetohydrodynamics system, SIAM J. Math. Anal., 47 (2015), 26-65. doi: 10.1137/14095515X.

[14]

E. Zadrzyńska and W. M. Zajączkowski, Stability of two-dimensional Navier-Stokes motions in the periodic case, J. Math. Anal. Appl., 423 (2015), 956-974. doi: 10.1016/j.jmaa.2014.10.026.

[15]

E. Zadrzyńska and W. M. Zajączkowski, Stability of two-dimensional heat-conducting incompressible motions in a cylinder, Nonlin. Anal. Ser. A: TMA, 125 (2015), 113-127. doi: 10.1016/j.na.2015.05.014.

[16]

W. M. Zajączkowski, Stability of nonswirl axisymmetric solution to the Navier-Stokes equations, Research Inst. Math. Sc. Kyoto Univ., 2009 (2015), 84-104.

[17]

W. M. Zajączkowski, Stability of two-dimensional solutions to the Navier-Stokes equations in cylindrical domains under Navier boundary conditions, JMAA, 444 (2016), 275-297. doi: 10.1016/j.jmaa.2016.05.059.

[18]

W. M. Zajączkowski, Global special regular solutions to the Navier-Stokes equations in a cylindrical domain under boundary slip conditions, Gakuto International Series, 21 (2004), pp. 188.

[19]

T. Zhang, An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system, arXiv: 1404.5681.

show all references

References:
[1]

O. V. Besov, V. P. Il'in and S. M. Nikol'skii, Integral representations of functions and imbedding theorems, Nauka, Moscow 1975 (in Russian).

[2]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.

[3]

Q. ChenC. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magnetohydrodynamic equations, Comm. Math. Phys., 284 (2008), 919-930. doi: 10.1007/s00220-008-0545-y.

[4]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magneétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.

[5]

C. He and X. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152. doi: 10.1016/j.jfa.2005.06.009.

[6]

C. He and X. Xin, On the regularity of weak solutions to magnetohydrodynamic equations, J. Diff. Equas., 213 (2005), 235-254.

[7]

X. Hu and F. H. Lin, Global existence of two-dimensional incompressible magnetohydrodynamic flows with zero magnetic diffusivity, arXiv: 1405.0082.

[8]

O. A. Ladyzhenskaya, On unique solvability of three-dimensional Cauchy problem for the Navier-Stokes equations under the axial symmetry, Zap. Nauchn. Sem LOMI, 7 (1968), 155-177 (in Russian).

[9]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Diff. Equas., 259 (2015), 3202-3215. doi: 10.1016/j.jde.2015.04.017.

[10]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: the three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580. doi: 10.1002/cpa.21506.

[11]

Fanghua LinXu Li and Zhang Ping, Global small solutions of 2-D incompressible magnetohydrodynamics system, J. Diff. Equas., 259 (2015), 5440-5485.

[12]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.

[13]

Xu Li and Zhang Ping, Global small solutions to three-dimensional incompressible magnetohydrodynamics system, SIAM J. Math. Anal., 47 (2015), 26-65. doi: 10.1137/14095515X.

[14]

E. Zadrzyńska and W. M. Zajączkowski, Stability of two-dimensional Navier-Stokes motions in the periodic case, J. Math. Anal. Appl., 423 (2015), 956-974. doi: 10.1016/j.jmaa.2014.10.026.

[15]

E. Zadrzyńska and W. M. Zajączkowski, Stability of two-dimensional heat-conducting incompressible motions in a cylinder, Nonlin. Anal. Ser. A: TMA, 125 (2015), 113-127. doi: 10.1016/j.na.2015.05.014.

[16]

W. M. Zajączkowski, Stability of nonswirl axisymmetric solution to the Navier-Stokes equations, Research Inst. Math. Sc. Kyoto Univ., 2009 (2015), 84-104.

[17]

W. M. Zajączkowski, Stability of two-dimensional solutions to the Navier-Stokes equations in cylindrical domains under Navier boundary conditions, JMAA, 444 (2016), 275-297. doi: 10.1016/j.jmaa.2016.05.059.

[18]

W. M. Zajączkowski, Global special regular solutions to the Navier-Stokes equations in a cylindrical domain under boundary slip conditions, Gakuto International Series, 21 (2004), pp. 188.

[19]

T. Zhang, An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system, arXiv: 1404.5681.

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