Geodesic
On the local smoothness of some class of axi-symmetric solutions to the MHD equations
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 46, Tome 459 (2017), pp. 127-148

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In this paper we consider a special class of weak axi-symmetric solutions to the MHD equations for which the velocity field has only poloidal component and the magnetic field is toroidal. We prove local regularity for such solutions. The global strong solvability of the initial-boundary value problem for the corresponding system in a cylindrical domain with non-slip boundary conditions for the velocity on the cylindrical surface is established as well. 
@article{ZNSL_2017_459_a7,
     author = {T. Shilkin},
     title = {On the local smoothness of some class of axi-symmetric solutions to the {MHD} equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {127--148},
     publisher = {mathdoc},
     volume = {459},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_459_a7/}
}
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T. Shilkin. On the local smoothness of some class of axi-symmetric solutions to the MHD equations. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 46, Tome 459 (2017), pp. 127-148. http://geodesic.mathdoc.fr/item/ZNSL_2017_459_a7/