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

Mehdi Badra (1) , Sylvain Ervedoza (2) , Sergio Guerrero (3)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 LJLL - Laboratoire Jacques-Louis Lions

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.

Mots clés: Non-homogeneous Navier-Stokes equations; Local exact controllability to trajectories

Domaines: Equations aux dérivées partielles [math.AP]

Mehdi Badra, Sylvain Ervedoza, Sergio Guerrero. Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2016, 33 (2), pp.529--574. ⟨10.1016/j.anihpc.2014.11.006⟩. ⟨hal-01018132⟩