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

Wojciech M. Zajączkowski

doi:10.3934/cpaa.2019070

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2019, Volume 18, Issue 3: 1447-1482. Doi: 10.3934/cpaa.2019070

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

Wojciech M. Zajączkowski^1,2,

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: May 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 35A01, 35B07, 76W05; Secondary: 35B35, 35D35, 76E25.

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