Davide Catania and Paolo Secchi
Global regularity for some MHD-α systems
Received: 23 September 2010
Accepted in revised form: 13 april 2011
Mathematics Subject Classification (2000): 35Q35, 76D03.
Keywords:Magnetohydrodynamics, MHD-α, Simplified Bardina, MHD-Voight, regularizing MHD, turbulence, incompressible fluid, global existence, hyperbolic system, parabolic system.
Abstract: The global existence of strong solutions (for arbitrarily large initial data) to the incompressible Euler equations is a major open problem. This problem is open as well for the ideal MHD system, that is to say in the inviscid irresistive case, for both space dimension n=2 or n=3. We review some results, appeared in previous papers, concerning the global existence of regularized models for incompressible magnetofluids. In particular, we observe that a partial viscous (i.e., with positive kinematic viscosity and no magnetic resistivity) α-regularization (which yields a hyperbolic-parabolic system) is capable to provide strong global in time solvability for the ideal MHD system of equations in the 2D framework. In the more complex 3D case, we have strong global existence also for an ideal purely hyperbolic system, known as MHD-Voight model, when both the velocity and the magnetic fields are α-regularized, and when we regularize only the velocity, but the magnetic resistivity is strictly positive. If, in the latter case, we consider a double viscous model, we can get as well the existence of a unique compact global attractor and give estimates for its Hausdorff and fractal dimension. We will introduce the four different regularized magnetohydrodynamic models and motivate such a choice. In all cases, we will state the strong global existence and uniqueness result that we have obtained for the solution to the respective systems. Finally, we will give an idea of some proofs, referring to the original related papers for more details.