Vanishing Dissipation Limits For a Leray-alpha Magneto-Hydrodynamic Equation
The Magnetohydrodynamic (MHD) system of equations governs kinematic fluids that are subjected to a magnetic field. The equation is a combination of the Navier-Stokes equations and Maxwell's equations. Due to the difficulty in solving the MHD system, it has become common to study approximating modifications of the equations, including the MHD-alpha system, which regularizes the velocity field in exchange for the addition of non-linear terms. Both the kinematic and magnetic parts of the MHD-alpha system have diffusive terms which dissipate the initial energy of the system. Setting those terms equal to zero returns the Ideal MHD-alpha system, and the goal of this project is to show that solutions to the MHD-alpha system with diffusion will converge to the Ideal MHD-alpha system as the diffusion parameters are sent to zero by adapting known results for the analogous problem of determining when solutions to the Navier-Stokes equations will converge to a solution of the Euler equation.