Local null controllability of the two-dimensional Navier–Stokes system in the torus with a control force having a vanishing component

Abstract

In this paper we deal with the two-dimensional Navier–Stokes system in a torus. The main result establishes the local null controllability with internal controls having one vanishing component. The linearized control system around 0 is not null controllable: the nonlinear term is essential to get this null controllability. Our proof uses the return method together with previous results by Fursikov and Imanuvilov.