  We study the Serrin-type regularity criteria for the solutions to the
four-dimensional Navier-Stokes equations and magnetohydrodynamics system. We
show that the sufficient condition for the solution to the four-dimensional
Navier-Stokes equations to preserve its initial regularity for all time may be
reduced from a bound on the four-dimensional velocity vector field to any two
of its four components, from a bound on the gradient of the velocity vector
field to the gradient of any two of its four components, from a gradient of the
pressure scalar field to any two of its partial derivatives. Results are
further generalized to the magnetohydrodynamics system. These results may be
seen as a four-dimensional extension of many analogous results that exist in
the three-dimensional case and also component reduction results of many
classical results.
