Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes system by a finite-dimensional force

摘要

In the Eulerian approach, the motion of an incompressible fluid is usuallydescribed by the velocity field which is given by the Navier--Stokes system.The velocity field generates a flow in the space of volume-preservingdiffeomorphisms. The latter plays a central role in the Lagrangian descriptionof a fluid, since it allows to identify the trajectories of individualparticles. In this paper, we show that the velocity field of the fluid and thecorresponding flow of diffeomorphisms can be simultaneously approximatelycontrolled using a finite-dimensional external force. The proof is based onsome methods from the geometric control theory introduced by Agrachev andSarychev.