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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 8

### Regularity criteria of the 4D Navier–Stokes equations involving two velocity field components

Pages: 2229 – 2252

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n8.a6

#### Author

#### Abstract

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.

#### Keywords

Navier–Stokes equations, magnetohydrodynamics system, scaling-invariance, regularity criteria

#### 2010 Mathematics Subject Classification

35B65, 35Q35, 35Q86

Published 26 October 2016