Abstract
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 L3,∞ 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 (−Δ)l/ 2, l > 2 to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as t → ∞ toward solutions of the original Navier–Stokes system.
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