Approximate controllability for Navier--Stokes equations in $\mathrm{3D}$ rectangles under Lions boundary conditions

TL;DR

This paper proves approximate controllability of the 3D Navier-Stokes equations in rectangles under Lions boundary conditions, by constructing an explicit saturating set.

Contribution

It introduces an explicit saturating set for 3D rectangles, advancing control theory for fluid dynamics systems.

Findings

Approximate controllability is established under certain conditions.

An explicit saturating set for 3D rectangles is provided.

The results extend control methods to 3D Navier-Stokes systems.

Abstract

The $\mathrm{3D}$ Navier--Stokes system, under Lions boundary conditions, is proven to be approximately controllable provided a suitable saturating set does exist. An explicit saturating set for $\mathrm{3D}$ rectangles is given.