Regularity criterion of the 4D Navier-Stokes equations involving two velocity field components

TL;DR

This paper extends regularity criteria for 4D Navier-Stokes equations, showing that controlling just two components of velocity or pressure derivatives suffices to ensure solution regularity, generalizing 3D results.

Contribution

It introduces a reduced component-based regularity criterion for 4D Navier-Stokes and magnetohydrodynamics systems, extending classical 3D results to four dimensions.

Findings

Regularity can be ensured by bounds on two velocity components.

Pressure derivative bounds can be reduced to two components.

Results generalize classical 3D regularity criteria.

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 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.