  We consider the incompressible Navier-Stokes equations in the cylinder $\R
\times \T$, with no exterior forcing, and we investigate the long-time behavior
of solutions arising from merely bounded initial data. Although we do not know
if such solutions stay uniformly bounded for all times, we prove that they
converge in an appropriate sense to the family of spatially homogeneous
equilibria as $t \to \infty$. Convergence is uniform on compact subdomains, and
holds for all times except on a sparse subset of the positive real axis. We
also improve the known upper bound on the $L^\infty$ norm of the solutions,
although our results in this direction are not optimal. Our approach is based
on a detailed study of the local energy dissipation in the system, in the
spirit of a recent work devoted to a class of dissipative partial differential
equations with a formal gradient structure (arXiv:1212.1573).
