Uniform Boundedness and Long-Time Asymptotics for the Two-Dimensional Navier-Stokes Equations in an Infinite Cylinder

The incompressible Navier-Stokes equations are considered in the two-dimensional strip R x [0,L], with periodic boundary conditions and no exterior forcing. If the initial velocity is bounded, it is shown that the solution remains uniformly bounded for all time, and that the vorticity distribution converges to zero as t -> infinity. This implies, after a transient period, the emergence of a laminar regime in which the solution rapidly converges to a shear flow described by the one-dimensional heat equation in an appropriate Galilean frame. The approach is constructive and provides explicit estimates on the size of the solution and the lifetime of the turbulent period in terms of the initial Reynolds number.