Uniform boundedness and long-time asymptotics for the two-dimensional Navier-Stokes equations in an infinite cylinder

We study the incompressible Navier-Stokes equations in the two-dimensional strip $\mathbb{R} \times [0,L]$, with periodic boundary conditions and no exterior forcing. If the initial velocity is bounded, we prove that the solution remains uniformly bounded for all times, and that the vorticity distribution converges to zero as $t \to \infty$. We deduce that, after a transient period, a laminar regime emerges in which the solution rapidly converges to a shear flow governed by the one-dimensional heat equation. Our approach is constructive and gives explicit estimates on the size of the solution and the lifetime of the turbulent period in terms of the initial Reynolds number.