$L_{3,\infty}$-solutions to the MHD equations

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.