Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations

TL;DR

This paper proves the global regularity and convergence of a Voigt-regularized inviscid resistive MHD model, providing a stable numerical approach and a new blow-up criterion for the original system.

Contribution

It introduces a new inviscid Voigt-regularization for 3D resistive MHD equations, establishing global regularity, convergence, and a blow-up criterion, extending applicability to various hydrodynamic models.

Findings

Existence and uniqueness of strong solutions for the regularized model.

Global-in-time solutions for weak solutions of the regularized system.

Convergence of regularized solutions to original MHD solutions as regularization parameter tends to zero.

Abstract

We prove existence, uniqueness, and higher-order global regularity of strong solutions to a particular Voigt-regularization of the three-dimensional inviscid resistive Magnetohydrodynamic (MHD) equations. Specifically, the coupling of a resistive magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid resistive MHD system. The results hold in both the whole space $\nR^3$ and in the context of periodic boundary conditions. Weak solutions for this regularized model are also considered, and proven to exist globally in time, but the question of uniqueness for weak solutions is still open. Since the main purpose of this line of research is to introduce a reliable and stable inviscid numerical regularization of the underlying model we, in particular, show that the solutions of the Voigt regularized system converge, as the regularization parameter $\alpha\maps0$, to strong solutions of the original inviscid resistive MHD, on the corresponding time interval of existence of the latter. Moreover, we also establish a new criterion for blow-up of solutions to the original MHD system inspired by this Voigt regularization. This type of regularization, and the corresponding results, are valid for, and can also be applied to, a wide class of hydrodynamic models.