Ana L. Silvestre

Solving Inverse Source Problems Using Observability. Applications to the Euler-Bernoulli Plate Equation

The aim of this paper is to provide a general framework for solving a class of inverse source problems by using exact observability of infinite dimensional systems. More precisely, we show that if a system is exactly observable, then a source term in this system can be identified by knowing its intensity and appropriate observations which often correspond to measurements of some boundary traces. This abstract theory is then applied to obtain new identifiability results for a system governed by the Euler-Bernoulli plate equation. Using a different methodology, we show that exact observability can be used to identify both the locations and the intensities of combinations of point sources in the plate equation.

Strong Solutions to the Problem of Motion of a Rigid Body in a Navier—Stokes Liquid under the Action of Prescribed Forces and Torques

This paper is devoted to the motion of a rigid body in an infinite Navier-Stokes liquid under the action of external forces and torques. For sufficiently regular data, we prove the existence of a local strong solution to the corresponding initial-boundary-value problem for the system body-liquid.

On the slow motion of a self-propelled rigid body in a viscous incompressible fluid

Abstract In this paper we study the Stokes approximation of the self-propelled motion of a rigid body in a viscous liquid that fills all the three-dimensional space exterior to the body. We prove the existence and uniqueness of strong solution to the coupled systems of equations describing the motion of the system body–liquid, for any time and any regular distribution of velocity on the boundary of the body. For the corresponding stationary problem we derive L p -estimates for the solution in terms of the data. Finally, we prove that every steady solution is attainable as the limit, when t →∞, of an unsteady self-propelled solution which starts from rest.

On the determination of point-forces on a Stokes system

We consider the steady Stokes equations in a bounded domain Ω ⊂ Rn, n = 2, 3 describing the motion of an incompressible viscous fluid under the action of point-forces located inside Ω. The objective of this work is to solve the inverse problem of determining the number, location and strength of the these point-forces.The detection of the unknown point-forces is made by means of the prescribed fluid velocity at the boundary ∂Ω and by a single measurement of the stress exerted by the fluid on ∂Ω. The main tool is the use of a reciprocity gap function, which is obtained by applying a reciprocity gap functional to Stokeslets located outside Ω. This procedure leads to the resolution of a nonlinear minimization problem.

On the motion of a rigid body with a cavity filled with a viscous liquid

We study the motion of a rigid body with a cavity filled with a viscous liquid. The main objective is to investigate the well-posedness of the coupled system formed by the Navier-Stokes equations describing the motion of the fluid and the ordinary differential equations for the motion of the rigid part. To this end, appropriate function spaces and operators are introduced and analysed by considering a completely general three-dimensional bounded domain. We prove the existence of weak solutions using the Galerkin method. In particular, we show that if the initial velocity is orthogonal, in a certain sense, to all rigid velocities, then the velocity of the system decays exponentially to zero as time goes to infinity. Then, following a functional analytic approach inspired by Katos scheme, we prove the existence and uniqueness of mild solutions. Finally, the functional analytic approach is extended to obtain the existence and uniqueness of strong solutions for regular data.

Existence and uniqueness of time-periodic solutions with finite kinetic energy for the Navier–Stokes equations in {\mathbb R}^3

Our aim is to prove existence and uniqueness of time-periodic strong solutions with finite kinetic energy for the Navier–Stokes equations in . For this, appropriate conditions are imposed on the external force, together with a smallness condition involving the viscosity of the fluid and the period of motion. We extend the method we have recently used to construct steady states with finite kinetic energy to the time-periodic case. First, existence and uniqueness of strong solutions with finite kinetic energy are established for a linearized version of the problem, using the Galerkin method and the Fourier transform in the space variables. Then, a strong solution with finite kinetic energy for the nonlinear problem is obtained by means of the contraction mapping principle. We also show that such a solution satisfies the energy equality and is unique within a class of weak solutions.

Integral equations for an inverse boundary value problem for the two-dimensional Stokes equations

Abstract The problem to determine the location and shape of a perfect conductor within a conducting homogeneous host medium from measured current and voltages on the accessible exterior boundary of the host medium can be modelled by an inverse Dirichlet boundary value problem for the Laplace equation. For this, recently Kress and Rundell suggested a novel inverse algorithm based on nonlinear integral equations arising from the reciprocity gap principle. The present paper extends this approach to the problem to recover the location and shape of a rigid body immersed in a fluid from the measured velocity and traction at the exterior boundary of the fluid, that is, to an inverse boundary value problem for the Stokes equation. The mathematical foundation of this extension is provided and numerical examples illustrate the feasibility of the method.