On the Higher-Order Global Regularity of the Inviscid Voigt-Regularization of Three-Dimensional Hydrodynamic Models

TL;DR

This paper proves higher-order and analytic regularity of solutions to the Euler-Voigt inviscid regularization of 3D Euler equations, establishes convergence to true solutions, and introduces a regularized MHD system with proven global regularity.

Contribution

It provides new regularity results for the Euler-Voigt model, convergence to Euler solutions, and extends regularization techniques to MHD systems.

Findings

Higher-order regularity of Euler-Voigt solutions

Convergence of Euler-Voigt to Euler solutions

Global regularity of the regularized MHD system

Abstract

We prove higher-order and a Gevrey class (spatial analytic) regularity of solutions to the Euler-Voigt inviscid $\alpha$-regularization of the three-dimensional Euler equations of ideal incompressible fluids. Moreover, we establish the convergence of strong solutions of the Euler-Voigt model to the corresponding solution of the three-dimensional Euler equations for inviscid flow on the interval of existence of the latter. Furthermore, we derive a criterion for finite-time blow-up of the Euler equations based on this inviscid regularization. The coupling of a magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid irresistive magneto-hydrodynamic (MHD) system. Global regularity of the regularized MHD system is also established.