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. 
Sim\~ao Correia, M\'ario Figueira
math.AP