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We obtain a regularity criteria of the solution to the three-dimensionalmagnetohydrodynamics system to remain smooth for all time involving only onevelocity and one vorticity component. Moreover, the norm in space and time withwhich we impose our criteria for the vorticity component is at the scalinginvariant level. The proof requires a new decomposition of the four non-linearterms making use of a new identity due to the divergence-free conditions of thevelocity and the magnetic vector fields.

Regularity criteria of the three-dimensional MHD system involving one velocity and one vorticity component