Euler equations are not exactly controllable by a finite-dimensional external force

TL;DR

This paper demonstrates that the Euler equations cannot be exactly controlled using a finite-dimensional external force, highlighting fundamental limitations in controlling fluid dynamics systems.

Contribution

It provides a rigorous proof showing the non-controllability of the Euler system by finite-dimensional forces using entropy comparison methods.

Findings

Euler system is not exactly controllable by finite-dimensional external forces

The proof utilizes Kolmogorov epsilon-entropy comparisons

Highlights limitations in control of fluid dynamics systems

Abstract

We show that the Euler system is not exactly controllable by a finite-dimensional external force. The proof is based on the comparison of the Kolmogorov epsilon-entropy for Holder spaces and for the class of functions that can be obtained by solving the 2D Euler equations with various right-hand sides.