A geometric measure-type regularity criterion for solutions to the magnetohydrodynamical system

TL;DR

This paper introduces a geometric regularity criterion for the 3D magnetohydrodynamical system, linking local geometric conditions to preventing finite-time singularities in solutions.

Contribution

It develops a new geometric measure-type condition on super-level sets that ensures regularity and analyticity of solutions, advancing understanding of singularity prevention in MHD.

Findings

Established existence, uniqueness, and analyticity of solutions.

Provided a sharp lower bound on the radius of analyticity.

Linked geometric conditions to singularity prevention.

Abstract

Several formulations of a local geometric measure-type condition are imposed on super-level sets of mild solutions to the homogeneous incompressible 3D magnetohydrodynamical system with bounded initial data to prevent finite-time singularity formation. Supporting this, results regarding the existence, uniqueness, and real analyticity of mild solutions are established as is a sharp lower bound on the radius of analyticity.