Higher-order Global Regularity of an Inviscid Voigt-regularization of the Three-dimensional Inviscid Resistive Magnetohydrodynamic Equations

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 R3 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 α → 0, 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. 1. The Inviscid Resistive MHD-Voigt Model The magnetohydrodynamic equations (MHD) are given by ∂tu + (u · ∇)u +∇(p+ 1 2 |B|) = (B · ∇)B + ν4u, (1.1a) ∂tB + (u · ∇)B − (B · ∇)u +∇q = μ4B, (1.1b) ∇ · B = ∇ · u = 0, (1.1c) with appropriate boundary and initial conditions, discussed below. Here, ν ≥ 0 is the fluid viscosity, μ ≥ 0 is the magnetic resistivity, and the unknowns are the fluid velocity field u(x, t) = (u1, u2, u3), the fluid pressure p(x, t), the magnetic field B(x, t) = (B1,B2,B3), and the magnetic pressure q(x, t), where x = (x1, x2, x3), and t ≥ 0. Note that, a posteriori, one can derive that ∇q ≡ 0. Due to the fact that these equations contain the three-dimensional Navier-Stokes equations for incompressible flows as a special case (namely, when B ≡ 0), the mathematical Date: April 3, 2011. 1991 Mathematics Subject Classification. Primary: 76W05, 76B03, 76D03, 35B44; Secondary: 76A10, 76A05.