On the regularity of weak solutions of the MHD equations in BMO−1 and Ḃ∞,∞−1

In this paper, we are concerned with the regularity of weak solutions to three-dimensional magnetohydrodynamics equations in critical spaces BMO−1 and B∞,∞−1. It is shown that weak solutions to this system are smooth if the velocity field belongs to C((0, T]; BMO−1) or the velocity field is in C((0,T];B∞,∞−1) and the magnetic field is bounded in L∞(0,T;B∞,∞−1). In addition, regularity criteria just via ∇u in homogeneous Besov spaces of negative regular indices are established. Our results improve previous corresponding criteria due to Chen, Miao, and Zhang [Commun. Math. Phys. 284, 919–930 (2008)]; He and Xin [J. Differ. Equations 213, 235–254 (2005)]; and Liu [Acta Math. Sci. 30, 335–343 (2010)].