Regularity criteria of the three-dimensional MHD system involving one velocity and one vorticity component

TL;DR

This paper establishes a new regularity criterion for 3D MHD systems involving only one velocity and one vorticity component, with conditions at the scaling invariant level, using a novel decomposition of nonlinear terms.

Contribution

It introduces a new regularity criterion for 3D MHD equations involving minimal components, utilizing a novel decomposition based on divergence-free conditions.

Findings

Regularity criterion involving one velocity and one vorticity component.

Norm in space and time at the scaling invariant level.

New decomposition of nonlinear terms using divergence-free conditions.

Abstract

We obtain a regularity criteria of the solution to the three-dimensional magnetohydrodynamics system to remain smooth for all time involving only one velocity and one vorticity component. Moreover, the norm in space and time with which we impose our criteria for the vorticity component is at the scaling invariant level. The proof requires a new decomposition of the four non-linear terms making use of a new identity due to the divergence-free conditions of the velocity and the magnetic vector fields.