Sur la contrôlabilité de l'équation de Navier-Stokes dans un rectangle avec l'aide d'une force fantôme distribuée

International audienceThis note echoes the talk given by the second author during the Journées EDP 2018 in Obernai. Its aim is to provide an overview and a sketch of proof of the result obtained by the authors, concerning the controllability of the Navier-Stokes equation. We refer the interested readers to the original paper for the full technical details of the proof, which will be omitted here, to focus on the main underlying ideas

Exact controllability of incompressible ideal magnetohydrodynamics in 22D

This work examines the controllability of planar incompressible ideal magnetohydrodynamics (MHD). Interior controls are obtained for problems posed in doubly-connected regions; simply-connected configurations are driven by boundary controls. Up to now, only straight channels regulated at opposing walls have been studied. Hence, the present program adds to the literature an exploration of interior controllability, extends the known boundary controllability results, and contributes ideas for treating general domains. To transship obstacles stemming from the MHD coupling and the magnetic field topology, a divide-and-control strategy is proposed. This leads to a family of nonlinear velocity-controlled sub-problems which are solved using J.-M. Coron's return method. The latter is here developed based on a reference trajectory in the domain's first cohomology space.Comment: 41 pages, 11 figure

On the hydrostatic approximation of the Navier-Stokes equations in a thin strip

In this paper, we first prove the global well-posedness of a scaled anisotropic Navier-Stokes system and the hydrostatic Navier-Stokes system in a 2-D striped domain with small analytic data in the tangential variable. Then we justify the limit from the anisotropic Navier-Stokes system to the hydrostatic Navier-Stokes system with analytic data

Spectral inequality for an Oseen operator in a two dimensional channel

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

An inverse problem in Fluid Mechanics applied in Biomedicine

In this thesis, new advances are presented in inverse problems of Fluid Mechanics in steady state, with direct applications in the recovery of domain deformations and obstacles, and whose purpose is to contribute to the detection of aortic valve conditions (such as insufficiency or stenosis). The first main result of this thesis is an asymptotic approximation result between the obstacle detection problems and the recovery of a non-negative permeability parameter that assumes significantly large values in the regions with obstacles or the value 0 in other parts. This result is supported by numerical tests that confirm the approximation result. The second result of this thesis presents a logarithmic inequality for the identification problem of the permeability parameter on Navier-Stokes equations from local measurements of fluid velocity. Numerical tests on the recovery of smooth and non-smooth parameters by a minimization problem and adaptive refinement algorithms are also included. Finally, a parameter identification problem for the Oseen and Navier-Stokes equations is studied in order to recover a permeability parameter from local or global measurements of the fluid velocity. Several numerical experiments using Navier-Stokes flow illustrate the applicability of the method, for the localization of a simulated 2D cardiac valve from synthetic MRI and also recovering of the permeability parameter from 3D synthetic MRI