The goal of this article is to show a local exact controllability to smooth () trajectories for the density dependent incompressible Navier–Stokes equations. Our controllability result requires some geometric condition on the flow 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 flow of the target trajectory. Our result requires the proof of new Carleman estimates for heat and Stokes equations.
Cite this article
Mehdi Badra, Sylvain Ervedoza, Sergio Guerrero, Local controllability to trajectories for non-homogeneous incompressible Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 2, pp. 529–574DOI 10.1016/J.ANIHPC.2014.11.006