We consider a magnetohydrodynamic-α model with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations.
We prove the existence of a global solution and a global attractor. Moreover, we provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the system. In some sense, this result provides an intermediate bound between the number of degrees of freedom for the simplified Bardina model and the Navier–Stokes-α equation.
"Global existence and finite dimensional global attractor for a 3D double viscous MHD-α model." Commun. Math. Sci. 8 (4) 1021 - 1040, December 2010.