Regularity criteria for 3D MHD flows in terms of spectral components

We extend the spectral regularity criteria of the Prodi-Serrin kind for the Navier-Stokes equations in a torus to the MHD equations. More precisely, the following is established: for any N > 0, let xN and y_N be the sum of all spectral components of the velocity and magnetic field whose wave numbers possess absolute value greater that N; then, it is possible to show that for any N the finiteness of the Prodi-Serrin norm of xN implies the regularity of the weak solution (u, h); thus, no restriction on the magnetic field is needed.