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

Larios, Adam; Titi, Edriss S.

Discrete Contin. Dyn. Syst., Ser. B 14, No. 2, 603-627 (2010).

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.

Reviewer: Il’ya Sh. Mogilevskij (Tver’)

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

Keywords:

Euler-Voigt equations; inviscid regularization; turbulence models; MHD equations