Prescribing the Motion of a Set of Particles in a Three-Dimensional Perfect Fluid

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 $\mathbb{R}^{3}$ 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.