Abstract
We give a new proof of the zero limit to the solution of the two-dimensional Navier-Stokes equations, as time goes to infinity. This proof is done in the frequency space; it is simpler and shorter compared to the existing proofs. Based on this limit, we derive some analytic properties of the solution. Mainly, it becomes infinitely differentiable with respect to time and has value in all Sobolev spaces. Moreover, its regularity grows in an exponential way and its L-2(R-2) norm decays exponentially fast, as time tends to infinity. Especially, we obtain, for any time t >= 0, that
integral(xi) e((1/2)root vt\xi\)vertical bar F(u)(t,xi)vertical bar(2) d xi <= parallel to u(t/2)parallel to(2)(L2(R2)).
We describe the long time behaviour of its homogeneous Sobolev norm for any positive, real exponent, by comparing to usual functions and we ameliorate some existing results. We establish that the Leray solution is stable as time increases.