A proof of approximate controllability of the 3D Navier-Stokes system via a linear test

TL;DR

This paper demonstrates that the approximate controllability of the 3D Navier-Stokes equations can be established by leveraging the controllability of a linearized Euler system, offering a shorter proof and new insights into control structure.

Contribution

It introduces a novel, shorter proof of approximate controllability for the 3D Navier-Stokes system based on linearized Euler system controllability, expanding control space insights.

Findings

Shorter proof of controllability using linearized Euler system

Control space dimension is larger but viscosity-independent

Provides new structural information about the control mechanism

Abstract

We consider the 3D Navier-Stokes system driven by an additive finite-dimensional control force. The purpose of this paper is to show how the approximate controllability of this system can be derived from the approximate controllability of the Euler system linearised around some suitable trajectory. The proof presented here is shorter than the previous ones obtained by Lie algebraic methods and gives some new information about the structure of the control. The dimension of the control space provided by this approach is larger, but it is still uniform with respect to the viscosity.