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 ${\mathit{x}}_N$ and ${\mathit{y}}_N$ 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 ${\mathit{x}}_N$ implies the regularity of the weak solution $(\mathit{u},\mathit{h})$; thus, no restriction on the magnetic field is needed.