Localized and degenerate controls for the incompressible Navier-Stokes system

TL;DR

This paper demonstrates the approximate controllability of the 2D incompressible Navier-Stokes system using localized, degenerate controls by reducing the problem to a linear transport equation through innovative geometric control techniques.

Contribution

It introduces a novel method combining Coron's return method with geometric control to achieve controllability with localized controls in the Navier-Stokes system.

Findings

Successfully reduces nonlinear control problem to linear transport equation

Achieves controllability with controls supported in a small subdomain

Provides explicit construction methods for control implementation

Abstract

We consider the global approximate controllability of the two-dimensional incompressible Navier-Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls into a small region, essentially by tracking their actions along the characteristic curves of a linearized vorticity equation. In this way, through explicit constructions and by connecting Coron's return method with recent concepts from geometric control, the original problem for the nonlinear Navier-Stokes system is reduced to one for a linear transport equation steered by a global force. This article can be viewed as an attempt to tackle a well-known open problem due to Agrachev.