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 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… CONTINUE READING