Séance Séminaire

Séminaire ACSIOM

Local exact controllability for compressible Navier-Stokes equations around constant trajectories.

In this talk, I will present a recent result obtained on the local exact controllability of the 3d compressible Navier-Stokes equation around a constant trajectory with non-zero velocity, when the control is exerted on the whole boundary of the domain. The proof of this result is based on an observability inequality for the adjoint of the linearized system. The main ingredient to obtain this inequality is the tricky combination of the equations to reduce the problem to a closed subsystem easier to deal with, and obtained by the introduction of a new quantity corresponding to the effective viscous flux for the adjoint equations. This subsystem will allow to understand independently the hyperbolic and parabolic effects of the system. We will then introduce Carleman estimates with weight functions following the characteristics allowing to handle the non-linearity of the model using a fixed point argument. This result has been obtained in collaboration with Olivier Glass (Univ. Paris Dauphine) and Sergio Guerrero (Univ. Pierre et Marie Curie).