Analytical Study of Certain Magnetohydrodynamic-α Models

In this paper we present an analytical study of a subgrid scale turbulence model of the threedimensional magnetohydrodynamic (MHD) equations, inspired by the Navier-Stokes-α (also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes-α model. Specifically, we show the global well-posedness and regularity of solutions of a certain MHD-α model (which is a particular case of the Lagrangian averaged magnetohydrodynamic-α model without enhancing the viscosity for the magnetic field). We also introduce other subgrid scale turbulence models, inspired by the Leray-α and the modified Leray-α models of turbulence. Finally, we discuss the relation of the MHD-α model to the MHD equations by proving a convergence theorem, that is, as the length scale α tends to zero, a subsequence of solutions of the MHD-α equations converges to a certain solution (a Leray-Hopf solution) of the three-dimensional MHD equations.