Existence and stability of spatial plane waves for the incompressible Navier-Stokes in ${\mathbb{R}}^3$

We consider the three-dimensional incompressible Navier-Stokes equation on the whole space. We observe that this system admits a $L^{\mathrm{\infty }}$ 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.