**Davide Catania** and **Paolo Secchi**

*Global regularity for some MHD-α systems *

**Pages:** 25-39

**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.

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