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

TL;DR

This paper investigates the existence, stability, and well-posedness of spatial plane wave solutions for the 3D incompressible Navier-Stokes equations, revealing their stability in the $L^3$ space without size restrictions.

Contribution

It establishes the existence of a family of global spatial plane wave solutions and proves their $L^3$-stability, extending understanding of solution behavior in three dimensions.

Findings

Existence of a family of global spatial plane wave solutions.

Proved local well-posedness in spaces including $L^3(R^3)$.

Demonstrated $L^3$-stability of these solutions without size constraints.

Abstract

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.