On the three-dimensional magnetohydrodynamics system in scaling-invariant spaces

TL;DR

This paper establishes a criterion for finite-time blowup in 3D magnetohydrodynamics, linking the behavior of the magnetic field and a single velocity component in scale-invariant spaces.

Contribution

It introduces a new blowup criterion based on the magnetic field and one velocity component, utilizing the system's structure in scaling-invariant spaces.

Findings

Finite-time blowup criterion derived for 3D MHD system.

Criterion depends on magnetic field and one velocity component.

Utilizes the structure of the MHD equations in scale-invariant spaces.

Abstract

We study the criterion for the velocity and magnetic vector fields that solve the three-dimensional magnetohydrodynamics system, given any initial data sufficiently smooth, to experience a finite-time blowup. Following the work of [12] and making use of the structure of the system, we obtain a criterion that is imposed on the magnetic vector field and only one of the three components of the velocity vector field, both in scaling-invariant spaces.