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 \(L^\infty \) 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.