Section: New Results

Analysis, control and stabilization of heterogeneous systems

Participant : Takéo Takahashi.

In [12], T. Takahashi (with D. Maity and M. Tucsnak, both from Institut de Mathématiques de Bordeaux, France) has considered a free boundary problem modeling the motion of a piston in a viscous gas. The gas-piston system fills a cylinder with fixed extremities, which possibly allow gas from the exterior to penetrate inside the cylinder. The gas is modeled by the 1D compressible Navier-Stokes system and the piston motion is described by the second Newton law. They prove the existence and uniqueness of global in time strong solutions. The main novelty brought in is that the case of nonhomogeneous boundary conditions is considered. Moreover, even for homogeneous boundary conditions, their results require less regularity of the initial data than those obtained in previous works.

In [32], T. Takahashi (with C. Lacave from Institut Fourier, Grenoble, France) has studied the motion of a single disk moving under the influence of a 2D viscous fluid. They deal with the asymptotic as the size of the solid tends to zero. If the density of the solid is independent of the size of the solid, the energy equality is not sufficient to obtain a uniform estimate for the solid velocity. This will be achieved thanks to the optimal $L^p-L^q$ decay estimates of the semigroup associated to the fluid-rigid body system and to a fixed point argument. Next, they deduce the convergence to the solution of the Navier-Stokes equations in ${ℝ}^2$.

In [7], T. Takahashi (with C. Bianchini (Dimai, Florence, Italy) and A. Henrot (IECL, Nancy, France)) has tackled a model for the shape of vesicles. In order to do this, they consider a shape optimization problem associated with a Willmore type energy in the plane. More precisely, they study a Blaschke-Santaló diagram involving the area, the perimeter and the elastic energy of planar convex bodies. Existence, regularity and geometric properties of solutions to this shape optimization problem are shown.

We have studied the self-propelled motions of a rigid body immersed in a viscous incompressible fluid which fills the exterior domain of the rigid body. The mechanism used by the body to reach the desired motion is modeled through a distribution of velocities at its boundary.

T. Takahashi (with J. San Martín (DIM, Santiago, Chile) and M. Tucsnak (Institut de Mathématiques de Bordeaux, France)) considers in [16] a class of swimmers of low Reynolds number, of prolate spheroidal shape, which can be seen as simplified models of ciliated microorganisms. Within this model, the form of the swimmer does not change, the propelling mechanism consisting in tangential displacements of the material points of swimmer's boundary. Using explicit formulas for the solution of the Stokes equations at the exterior of a translating prolate spheroid the governing equations reduce to a system of ODE's with the control acting in some of its coefficients (bilinear control system). The main theoretical result asserts the exact controllability of the prolate spheroidal swimmer. In the same geometrical situation, they define a concept of efficiency which reduces to the classical one in the case of a spherical swimmer and they consider the optimal control problem of maximizing this efficiency during a stroke. Moreover, they analyse the sensitivity of this efficiency with respect to the eccentricity of the considered spheroid. They provide semi-explicit formulas for the Stokes equations at the exterior of a prolate spheroid, with an arbitrary tangential velocity imposed on the fluid-solid interface. Finally, they use numerical optimization tools to investigate the dependence of the efficiency on the number of inputs and on the eccentricity of the spheroid. The “best” numerical result obtained yields an efficiency of 30.66% with 13 scalar inputs. In the limiting case of a sphere their best numerically obtained efficiency is of 30.4%, whereas the best computed efficiency previously reported in the literature was of 22%.

In [10], T. Takahashi (with T. Hishida (Nagoya University, Japan) and A.L. Silvestre (IST, Lisboa, Portugal)) tackles the stationary case. The fluid motion is modeled by the stationary Navier-Stokes system coupled with two relations for the balance of forces and torques. They prove that there exists a control allowing the rigid body to move with a prescribed rigid velocity provided the velocity is small enough. They also show that since the net force exerted by the fluid to the rigid body vanishes, we have a better summability of the fluid velocity than the classical summability result for the solutions of the stationary Navier-Stokes system in exterior domains.