Geodesic
$L_{3,\infty}$-solutions to the MHD equations
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 37, Tome 336 (2006), pp. 112-132

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We prove that weak solutions to the MHD system are smooth providing they belong to the so-called “critical” Ladyzhenskaya–Prodi–Serrin class $L_{3,\infty}$. Besides the independent interest, this result controverts the hypothesis on the existence of collapsing self-similar solutions to the MHD equations for which the generating profile belongs to the space $L_3$. So, we extend the results that were known before for the Navier–Stokes system, for the case of the MHD equations. 
@article{ZNSL_2006_336_a5,
     author = {A. Mahalov and B. Nicolaenko and T. N. Shilkin},
     title = {$L_{3,\infty}$-solutions to the {MHD} equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {112--132},
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     volume = {336},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_336_a5/}
}
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A. Mahalov; B. Nicolaenko; T. N. Shilkin. $L_{3,\infty}$-solutions to the MHD equations. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 37, Tome 336 (2006), pp. 112-132. http://geodesic.mathdoc.fr/item/ZNSL_2006_336_a5/