Remarks on the global solutions of 3-D Navier-Stokes system with one slow variable

By applying Wiegner' method in \cite{Wiegner}, we first prove the large time decay estimate for the global solutions of a 2.5 dimensional Navier-Stokes system, which is a sort of singular perturbed 2-D Navier-Stokes system in three space dimension. As an application of this decay estimate, we give a simplified proof for the global wellposedness result in \cite{cg3} for 3-D Navier-Stokes system with one slow variable. Let us also mention that compared with the assumptions for the initial data in \cite{cg3}, here the assumptions in Theorem \ref{slowvarsimplifie} are weaker.