Prescribing the motion of a set of particles in a 3D perfect fluid

TL;DR

This paper proves that in a 3D perfect fluid, it is possible to control the motion of fluid particles from one configuration to another by boundary manipulation within a finite time, demonstrating a form of Lagrangian controllability.

Contribution

It establishes the first controllability result for the Euler equation in 3D, showing boundary control can steer fluid particles between configurations.

Findings

Boundary control can approximately move particle sets to desired positions.

The result applies to smooth, contractible particle sets in 3D fluids.

The proof involves new techniques in fluid controllability theory.

Abstract

We establish a result concerning the so-called Lagrangian controllability of the Euler equation for incompressible perfect fluids in dimension 3. More precisely we consider a connected bounded domain of R^3 and two smooth contractible sets of fluid particles, surrounding the same volume. We prove that given any initial velocity field, one can find a boundary control and a time interval such that the corresponding solution of the Euler equation makes the first of the two sets approximately reach the second one.