Forced Rapidly Dissipative Navier--Stokes Flows

TL;DR

This paper demonstrates how a finite parameter control force can significantly increase energy dissipation in small Navier--Stokes solutions, with the force's magnitude bounded by the initial velocity's Sobolev norm and support arbitrarily localized.

Contribution

It introduces a method to enhance energy dissipation in Navier--Stokes flows using localized control forces based on initial data norms.

Findings

Energy dissipation rate can be increased via finite parameter control.

Control force support can be arbitrarily small in space or time.

Magnitude of control force is bounded by initial velocity Sobolev norm.

Abstract

We show that, by acting on a finite number of parameters of a compactly supported control force, we can increase the energy dissipation rate of any small solution of the Navier--Stokes equations in $\mathbb{R}^n$ . The magnitude of the control force is bounded by a negative Sobolev norm of the initial velocity. Its support can be chosen to be contained in an arbitrarily small region, in time or in space.