## On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models.(English)Zbl 1202.35172

The initial boundary value problem to the Euler-Voigt equations is studied.
$-\alpha^2\frac{\partial \Delta u}{\partial t}+\frac{\partial u}{\partial t}+(u\cdot\nabla)u+\nabla p=0,\quad \text{div}\,u=0,\quad x\in\mathbb T^3,\;t\in(-\infty,\infty),$
$u(x,0)=u_0(x),\quad x\in\mathbb T^3.$
Here $$u(x,t)$$ is the velocity of the fluid, $$p$$ is the pressure, $$\mathbb T^3=[0,1]^3$$ is the unit torus, $$\alpha$$ is the nonnegative parameter, $$u$$ satisfies periodic boundary conditions.
The following results are obtained in the paper.
(1)
If $$u_0\in H^m$$ then $$u\in H^m$$, $$m=1,2,\dots$$.
(2)
If $$u_0$$ is analytic then $$u$$ enjoys spatial analyticity.
(3)
The solution to the problem converges to sufficiently regular solution of the Euler equations, as $$\alpha\rightarrow 0$$, on any closed interval of time where the solution to the Euler equation exists. A criterion for blow-up of the Euler equations is proved.
(4)
$$\alpha$$-regularization procedure is applied to the hydrodynamic models involving magnetism. Global regularity of the MHD system is established.

### MSC:

 35Q35 PDEs in connection with fluid mechanics 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 76F20 Dynamical systems approach to turbulence 35B44 Blow-up in context of PDEs 76W05 Magnetohydrodynamics and electrohydrodynamics
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