  The first goal of this paper is to study the large time behavior of solutions
to the Cauchy problem for the 3-dimensional incompressible Navier-Stokes
system. The Marcinkiewicz space $L^{3,\infty}$ is used to prove some asymptotic
stability results for solutions with infinite energy. Next, this approach is
applied to the analysis of two classical ``regularized'' Navier-Stokes systems.
The first one was introduced by J. Leray and consists in ``mollifying'' the
nonlinearity. The second one was proposed by J.L. Lions, who added the
artificial hyper-viscosity $(-\Delta)^{\ell/2}$, $\ell>2$, to the model. It is
shown in the present paper that, in the whole space, solutions to those
modified models converge as $t\to\infty$ toward solutions of the original
Navier-Stokes system.
