Local controllability to trajectories for non-homogeneous 2-d incompressible Navier-Stokes equations

TL;DR

This paper establishes local exact controllability to smooth trajectories for 2D density-dependent incompressible Navier-Stokes equations, using novel Carleman estimates and geometric conditions related to the flow.

Contribution

It introduces new Carleman estimates and a fixed point approach to achieve controllability under geometric conditions on the flow.

Findings

Controllability to smooth trajectories is achieved under specific geometric conditions.

New Carleman estimates for heat and Stokes equations are developed.

The method applies to 2D density-dependent incompressible Navier-Stokes equations.

Abstract

The goal of this article is to show a local exact controllability to smooth (C2) trajectories for the 2-d density dependent incompressible Navier-Stokes equations. Our controllability result requires some geometric condition on the ow of the target trajectory, which is remanent from the transport equation satisfied by the density. The proof of this result uses a fixed point argument in suitable spaces adapted to a Carleman weight function that follows the ow of the target trajectory. Our result requires the proof of new Carleman estimates for heat and Stokes equations.