Communications in Mathematical Sciences
Volume 14 (2016)
Regularity criteria of the 4D Navier–Stokes equations involving two velocity field components
Pages: 2229 – 2252
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 in the following ways: 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; and 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.
Navier–Stokes equations, magnetohydrodynamics system, scaling-invariance, regularity criteria
2010 Mathematics Subject Classification
35B65, 35Q35, 35Q86