In this presentation, we explore the local exact controllability to a constant trajectory for the compressible Navier-Stokes-Korteweg system on the torus in dimensions $din{1,2,3}$, where the control acts on an open subset. The Navier-Stokes-Korteweg system is a variant of the compressible Navier-Stokes system, which considers both viscosity and capillarity effects, relevant to describing the phase transition between liquid and vapor. In the literature, the parabolic behavior of the system is highlighted, only, in the framework of the Cauchy theory. The aim of my presentation is to explain how capture the hidden parabolic behavior of this system from the perspective of control theory. Specifically, we achieve local exact controllability to the constant state $(rho_star,0)$ for arbitrarily small positive times, without imposing any geometric condition on the control region. To this end, we study the control properties of the linearized equation and conduct a thorough analysis of the observability of the associated adjoint equations. Notably, we identify the parabolic (and potentially dispersive) structure of these adjoint equations. Using this insight, we derive observability for the adjoint system through specific Carleman estimates.

https://indico.math.cnrs.fr/event/12651/