A note on regularity conditions on ideal magnetohydrodynamic equations

A regularity criterion for ideal magnetohydrodynamical equations is known to be given in terms of the vorticity and the current density fields [Caflisch et al., Commun. Math. Phys. 184, 443 (1997)]. A simple argument which utilizes Weber transform associated with cross helicity invariant shows that if the magnetic field is smooth, then so is the velocity field, thereby suggesting some room for improving the above criterion.