On a Partial Voigt Regularization of the 3D Magnetohydrodynamic Equations in Velocity-Vorticity Form

TL;DR

This paper extends Voigt regularization to the 3D magnetohydrodynamics equations in velocity-vorticity form, proving global well-posedness and convergence, and establishing a blow-up criterion for solutions.

Contribution

It introduces a partial Voigt regularization for 3D MHD equations that preserves the structure and proves key properties like well-posedness and convergence.

Findings

Global well-posedness of the regularized system is established.

The regularized system converges to the original system up to blow-up time.

A blow-up criterion for the original 3D MHD system is derived.

Abstract

The Velocity-Vorticity (VV) formulation of the incompressible Navier-Stokes equations has become popular in recent years, especially in numerical studies, due to its structural advantages. Recently, with L. Rebholz, we introduced a Voigt regularization to the momentum equation in this formulation, establishing global well-posedness of the regularized system in 3D, along with convergence results and a blow-up criterion. In the present work, we extend these ideas to the 3D magnetohydrodynamics (MHD) equations. While it may seem that a ``VV-type'' split on the magnetic equation is required, we show that no such modification is necessary, and global well-posedness holds with a Voigt regularization only on the momentum equation, preserving the structure of both the vorticity and magnetic equations. We also prove that the regularized system converges to the original system, up to a possible blow-up time, and we establish a blow-up criterion for solutions to the original 3D MHD system.