Long-time asymptotics of Navier-Stokes and vorticity equations in a three-dimensional layer

We study the long-time behavior of solutions of the Navier-Stokes equation in R-2 X (0, 1). After introducing self-similar variables, we compute the long-time asymptotics of the vorticity up to second order, assuming that the initial vorticity is sufficiently small and has polynomial decay at infinity. Afterwards, we relax this smallness assumption and we prove again that the long-time behavior of global bounded solutions is governed by the two-dimensional Navier-Stokes equation. In particular, we show that solutions converge towards Oseen vortices.