Spectral inequality for an Oseen operator in a two dimensional channel

Rémi Buffe (1, 2) , Ludovick Gagnon (1, 2)

1 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
2 IECL - Institut Élie Cartan de Lorraine

We prove a Lebeau-Robbiano spectral inequality for the Oseen operator in a two dimensional channel, that is, the linearized Navier-Stokes operator around a laminar flow, with no-slip boundary conditions. The operator being non-self-adjoint, we place ourself into the abstract setting of [12], and prove the spectral inequaltiy through the derivation of a proper Carleman estimate. In the spirit of [4], we handle the vorticity near the boundary by using the characteristics sets of $P_ϕ$ or $Q_{ϕ0}$ in the different microlocal regions of the cotangent space. As a consequence of the spectral inequality, we derive a new estimate of the cost of the control for the small-time null-controllability.

Mots clés: Spectral inequality; Navier-Stokes equation; Null-controllability; Carleman estimate

Domaines: Equations aux dérivées partielles [math.AP]

Rémi Buffe, Ludovick Gagnon. Spectral inequality for an Oseen operator in a two dimensional channel. 2021. ⟨hal-03200711⟩