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## Section: New Results

### Direct and Inverse problems for heterogeneous systems

Participants : Rémi Buffe, Imene Djebour, David Dos Santos Ferreira, Ludovick Gagnon, Alexandre Munnier, Julien Lequeurre, Karim Ramdani, Takéo Takahashi, Jean-Claude Vivalda.

Direct problems

In [22], Imene Djebour and Takéo Takahashi consider a fluid–structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. They use here Navier-slip boundary conditions instead of the standard no-slip boundary conditions. The main results are the local in time existence and uniqueness of strong solutions of the corresponding system and the global in time existence and uniqueness of strong solutions for small data and if one assumes the presence of frictions in the boundary conditions.

In [42], Mehdi Badra (University of Toulouse) and Takéo Takahashi analyze a bi-dimensional fluid-structure interaction system composed by a viscous incompressible fluid and a beam located at the boundary of the fluid domain. The main result is the existence and uniqueness of strong solutions for the corresponding coupled system. The proof is based on a the study of the linearized system and a fixed point procedure. In particular, they show that the linearized system can be written with a Gevrey class semigroup. The main novelty with respect to previous results is that they do not consider any approximation in the beam equation.

In [18], Muriel Boulakia (Sorbonne University), Sergio Guerrero (Sorbonne University) and Takéo Takahashi consider a system modeling the interaction between a viscous incompressible fluid and an elastic structure. The fluid motion is represented by the classical Navier–Stokes equations while the elastic displacement is described by the linearized elasticity equation. The elastic structure is immersed in the fluid and the whole system is confined into a general bounded smooth three-dimensional domain. The main result is the local in time existence and uniqueness of a strong solution of the corresponding system.

In [28], Debayan Maity (TIFR Bangalore), Jorge San Martin (University of Chile), Takéo Takahashi and Marius Tucsnak (University of Bordeaux) study the interaction of surface water waves with a floating solid constraint to move only in the vertical direction. They propose a new model for this interaction, taking into consideration the viscosity of the fluid. This is done supposing that the flow obeys a shallow water regime (modeled by the viscous Saint-Venant equations in one space dimension) and using a Hamiltonian formalism. Another contribution of this work is establishing the well-posedness of the obtained PDEs/ODEs system in function spaces similar to the standard ones for strong solutions of viscous shallow water equations. Their well-posedness results are local in time for any initial data and global in time if the initial data are close (in appropriate norms) to an equilibrium state. Moreover, they show that the linearization of the system around an equilibrium state can be described, at least for some initial data, by an integro-fractional differential equation related to the classical Cummins equation and which reduces to the Cummins equation when the viscosity vanishes and the fluid is supposed to fill the whole space. Finally, they describe some numerical tests, performed on the original nonlinear system, which illustrate the return to equilibrium and the influence of the viscosity coefficient.

In [30], Benjamin Obando and Takéo Takahashi consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. The main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). They approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body.

In [23], Alexandre Munnier and his co-authors consider the dynamics of several rigid bodies immersed in a perfect incompressible fluid. We show that this dynamics can be modelized by a second order ODE whose coefficients depend on the vorticity and the circulation of the fluid around the bodies. This formulation permits to point out the geodesic nature of the solutions, the added mass effect, the gyroscopic effects and the Kutta-Joukowski-type lift forces.

In [24], Julien Lequeurre and his co-authors study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. We prove existence of a unique local in time strong solution. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. One key point, is to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.

J.F. Scheid and M. Bouguezzi (PhD student) in collaboration with D. Hilhorst and Y. Miyamoto work on the convergence of the solution of the one-phase Stefan problem in one-space dimension to a self-similar profile. The evolutional self-similar profile is viewed as a stationary solution of a Stefan problem written in a self-similar coordinates system. The proof of the convergence relies on the construction of sub and super-solutions for which it must be proved that they both tend to the same function. It remains to show that this limiting function actually corresponds to the self-similar solution of the original Stefan problem. This work is in progress.

Rémi Buffe, Ludovick Gagnon et al. obtain the exponential decay of the solutions of coupled wave equations with a transmission condition at the interface and with a viscoelastic damping term. They prove that the exponential decay is obtained if the support of the viscoelastic term satisfies the uniform escaping geometry condition. They also deal with the case where the damping term touches the interface.

Inverse problems

In [43], the authors are interested in the homogenization of time-harmonic Maxwell's equations in a composite medium with periodically distributed small inclusions of a negative material. Here a negative material is a material modelled by negative permittivity and permeability. Due to the sign-changing coefficients in the equations, it is not straightforward to obtain uniform energy estimates to apply the usual homogenization techniques. The analysis is based on a precise study of two associated scalar problems: one involving the sign-changing permittivity with Dirichlet boundary conditions, another involving the sign-changing permeability with Neumann boundary conditions. For both problems, we obtain a criterion on the physical parameters ensuring uniform invertibility of the corresponding operators as the size of the inclusions tends to zero. Then we use the results obtained for the scalar problems to derive uniform energy estimates for Maxwell's system.

In [37], Jean-Claude Vivalda and his co-authors prove that the class of continuous-time systems who are strongly differentially observable after time sampling is everywhere dense in the set of pairs $\left(f,h\right)$ where $f$ is a (parametrized) vector field given on a compact manifold and $h$ is an observation function.

In [47], using a partial boundary measurement, Jean-Claude Vivalda and his co-authors design an observer for a system that models a desalination device; this observer being used to make an output tracking trajectory.

Rémi Buffe, David Dos Santos Ferreira and Ludovick Gagnon obtain an estimate on the magnetic Laplacian with sharp dependance on the power of the zeroth and first order potential and close to sharp norm of these potentials. This estimate is related to the observability inequality for the wave equation and to the cost of the control.