Existence and Convergence of an MHD Approximate Deconvolution Model

We consider a Large Eddy Simulation (LES) model for the equations of Magnetohydrodynamics (MHD). We study an -model that is obtained by adapting to the MHD the approach by Stolz and Adams with van Cittert approximate deconvolution operators. We work with periodic boundary conditions and use the Helmholtz filter. We prove existence and uniqueness of a regular weak solution for a system with filtering and deconvolution in both equations. We show that when the deconvolution parameter goes to infinity, then the solution converges — in an appropriate sense — to the solution of the filtered MHD equations. These results can be extended to the problem with filtering acting only on the velocity.