We study the following inverse problem: an inaccessible rigid body D is immersed in a viscous fluid, in such a way that D plays the role of an obstacle around which the fluid is flowing in a greater bounded domain , and we wish to determine D (i.e., its form and location) via boundary measurement on the boundary ∂. Both for the stationary and the evolution problem, we show that under reasonable smoothness assumptions on and D, one can identify D via the measurement of the velocity of the fluid and the Cauchy forces on some part of the boundary ∂. We also show that the dependence of the Cauchy forces on deformations of D is analytic, and give some stability results for the inverse problem.
Abstract. In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in R d . We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency ξ, are not continuous at the origin. Nevertheless, when ξ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.
In this paper we introduce the functional framework and the necessary conditions for the well-posedness of an inverse problem arising in the mathematical modeling of disease transmission. The direct problem is given by an initial boundary value problem for a reaction diffusion system. The inverse problem consists in the determination of the disease and recovery transmission rates from observed measurement of the direct problem solution at the end time. The unknowns of the inverse problem are coefficients of the reaction term. We formulate the inverse problem as an optimization problem for an appropriate cost functional. Then, the existence of solutions of the inverse problem is deduced by proving the existence of a minimizer for the cost functional. Moreover, we establish the uniqueness up an additive constant of identification problem. The uniqueness is a consequence of the first order necessary optimality condition and a stability of the inverse problem unknowns with respect to the observations.
We study the homogenization and localization of high-frequency waves in a locally periodic media with period ε. We consider initial data that are localized Bloch-wave packets, i.e. that are the product of a fast oscillating Bloch wave at a given frequency ξ and of a smooth envelope function whose support is concentrated at a point x with length scale √ ε. We assume that (ξ, x) is a stationary point in the phase space of the Hamiltonian λ(ξ, x), i.e. of the corresponding Bloch eigenvalue. Upon rescaling at size √ ε, we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian λ(ξ, x) is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger equation is made of a countable sequence of finite multiplicity eigenvalues with exponentially decaying eigenfunctions.
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