  We consider the three-dimensional incompressible Navier-Stokes equation on
the whole space. We observe that this system admits a $L^\infty$ family of
global spatial plane wave solutions, which are connected with the
two-dimensional equation. We then proceed to prove local well-posedness over a
space which includes $L^3(\mathbb{R}^3)$ and these solutions. Finally, we prove
$L^3$-stability of spatial plane waves, with no condition on their size.
