| Summary: | 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 $ {{\boldsymbol x}}_{N} $ and $ {{\boldsymbol 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 $ {{\boldsymbol x}}_{N} $ implies the regularity of the weak solution $ ({{\boldsymbol u}}, {{\boldsymbol h}}) $; thus, no restriction on the magnetic field is needed.
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