Navier-Stokes Equation on the Rectangle

TL;DR

This paper investigates the controllability of the 2D Navier-Stokes equations on a rectangle with Lions boundary conditions, using harmonic basis and control theory to achieve approximate controllability with low modes forcing.

Contribution

It introduces a novel approach combining harmonic basis and geometric control theory to establish controllability results for the Navier-Stokes system on a rectangle.

Findings

Proves controllability of the system using finite-dimensional Galerkin approximations.

Establishes the continuity of the control-to-solution map in the relaxation metric.

Demonstrates both solid and approximate controllability with low modes forcing.

Abstract

We study controllability issues for the Navier-Stokes Equation on a two dimensional rectangle with so-called Lions boundary conditions. Rewriting the Equation using a basis of harmonic functions we arrive to an infinite-dimensional system of ODEs. Methods of Geometric/Lie Algebraic Control Theory are used to prove controllability by means of low modes forcing of finite-dimensional Galerkin approximations of that system. Proving the continuity of the ``control $\mapsto$ solution'' map in the so-called relaxation metric we use it to prove both solid controllability on observed component and $\mathbf L^2$-approximate controllability of the Equation (full system) by low modes forcing.