Low Regularity Global Solutions for a generalized MHD-α system

The Magneto-Hydrodynamic (MHD) system of equations governs the motion of viscous fluids subject to a magnetic field. Due to the difficulty of obtaining global solutions to the MHD system, it has become common to study modified versions of the system. In this paper, we prove the existence of a unique global solution to the incompressible MHD-α system with diffusion terms which are Fourier multipliers with symbols of the form m(ξ)=|ξ|^γ/g(|ξ|) for γ>0 and g (essentially) a logarithm. Letting γ₁ and γ₂ be the regularity of the diffusion terms, we obtain global existence when γ₁ and γ₂ satisfy γ₁,γ₂>1, γ₁≥n/3, and γ₁+γ₂≥n in Rⁿ for n≥3.