Stability of axially-symmetric solutions to incompressible magnetohydrodynamics with no azimuthal velocity and with only azimuthal magnetic field. (English) Zbl 1414.35179
Summary: 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.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B07 | Axially symmetric solutions to PDEs |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 35B35 | Stability in context of PDEs |
| 35D35 | Strong solutions to PDEs |
| 76E25 | Stability and instability of magnetohydrodynamic and electrohydrodynamic flows |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
incompressible viscous axially-symmetric magnetohydrodynamics; special solutions; global regular solutions located in a small neighborhood of the special solutions; stabilityReferences:
| [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). · Zbl 0352.46023 |
| [2] | C. Cao; J. Wu, Global regularity for the 2D MHD equations with mixed dissipation and magnetic diffusion, Adv. Math., 226, 1803-1822 (2011) · Zbl 1213.35159 · doi:10.1016/j.aim.2010.08.017 |
| [3] | Q. Chen; C. Miao; Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magnetohydrodynamic equations, Comm. Math. Phys., 284, 919-930 (2008) · Zbl 1168.35035 · doi:10.1007/s00220-008-0545-y |
| [4] | G. Duvaut; J. L. Lions, Inéquations en thermoélasticité et magneétohydrodynamique, Arch. Ration. Mech. Anal., 46, 241-279 (1972) · Zbl 0264.73027 · doi:10.1007/BF00250512 |
| [5] | C. He; X. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227, 113-152 (2005) · Zbl 1083.35110 · doi:10.1016/j.jfa.2005.06.009 |
| [6] | C. He; X. Xin, On the regularity of weak solutions to magnetohydrodynamic equations, J. Diff. Equas., 213, 235-254 (2005) · Zbl 1072.35154 |
| [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, 155-177 (in Russian) (1968) · Zbl 0195.10603 |
| [9] | Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Diff. Equas., 259, 3202-3215 (2015) · Zbl 1319.35195 · doi:10.1016/j.jde.2015.04.017 |
| [10] | F. Lin; P. Zhang, Global small solutions to an MHD-type system: the three-dimensional case, Comm. Pure Appl. Math., 67, 531-580 (2014) · Zbl 1298.35153 · doi:10.1002/cpa.21506 |
| [11] | Fanghua Lin; Xu Li; Zhang Ping, Global small solutions of 2-D incompressible magnetohydrodynamics system, J. Diff. Equas., 259, 5440-5485 (2015) · Zbl 1321.35138 |
| [12] | M. Sermange; R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36, 635-664 (1983) · Zbl 0524.76099 · doi:10.1002/cpa.3160360506 |
| [13] | Xu Li; Zhang Ping, Global small solutions to three-dimensional incompressible magnetohydrodynamics system, SIAM J. Math. Anal., 47, 26-65 (2015) · Zbl 1352.35099 · doi:10.1137/14095515X |
| [14] | E. Zadrzyńska; W. M. Zajączkowski, Stability of two-dimensional Navier-Stokes motions in the periodic case, J. Math. Anal. Appl., 423, 956-974 (2015) · Zbl 1308.35182 · doi:10.1016/j.jmaa.2014.10.026 |
| [15] | E. Zadrzyńska; W. M. Zajączkowski, Stability of two-dimensional heat-conducting incompressible motions in a cylinder, Nonlin. Anal. Ser. A: TMA, 125, 113-127 (2015) · Zbl 1326.35293 · 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, 84-104 (2015) |
| [17] | W. M. Zajączkowski, Stability of two-dimensional solutions to the Navier-Stokes equations in cylindrical domains under Navier boundary conditions, JMAA, 444, 275-297 (2016) · Zbl 1346.35151 · 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. · Zbl 1121.35002 |
| [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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