Continuous Dependence Estimates Research Articles

Effect of anisotropic permeability on Darcy's law

Abstract The concept of structural stability is discussed and briefly reviewed within the context of fluid flow in porous media. Then the equations for non‐Boussinesq (penetrative) convection in a porous medium with anisotropic permeability are analysed. It is shown, by deriving truly a priori estimates, that the solution exists in the natural functional framework and depends continuously on changes in the permeability. Since under appropriate circumstances the onset of thermal convection may be via Hopf bifurcation rather than stationary convection, such a priori continuous dependence estimates are of much value. Copyright © 2001 John Wiley & Sons, Ltd.

Boundary Integral Formulae for the Reconstruction of Imperfections of Small Diameter in an Elastic Medium

We consider an isotropic Lame system for an elastic medium consisting of finitely many imperfections of small diameter, embedded in a homogeneous reference medium. The Lame coefficients of the imperfections are different from those of the background medium. First, we establish an asymptotic formula for the displacement vector in terms of the reference Lame coefficients, the location of the imperfections, and their geometry. Second, we use this asymptotic expansion to establish continuous dependence estimates for certain characteristics of the imperfections in terms of the boundary data and to derive integral boundary formulae that yield to a very effective identification procedure.

Abstract stochastic integrodifferential delay equations

We investigate a class of abstract stochastic integrodifferential delay equations dependent upon a family of probability measures in a separable Hilbert space. We establish the existence and uniqueness of a mild solution, along with various continuous dependence estimates and Markov (weak and strong) properties of this solution. This is followed by a convergence result concerning the strong solutions of the Yosida approximations of our equation, from which we deduce the weak convergence of the measures induced by these strong solutions to the measure induced by the mild solution of the primary problem under investigation. Next, we establish the pth moment and almost sure exponential stability of the mild solution. Finally, an analysis of two examples, namely a generalized stochastic heat equation and a Sobolev‐type evolution equation, is provided to illustrate the applicability of the general theory.

Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction

We derive an asymptotic formula for the steady-state voltage potential in the presence of a finite number of diametrically small inhomogeneities with conductivity different from the background conductivity. We use this formula to establish continuous dependence estimates and to design an effective computational identification procedure.

Well-posedness of the weak formulation for the phase-field model with memory

A phase-field model based on the Gurtin-Pipkin heat flux law is considered. This model consists in a Volterra integrodifferential equation of hyperbolic type coupled with a nonlinear parabolic equation. The system is then associated with a set of initial and Neumann boundary conditions. The resulting problem was already studied by the authors who proved existence and uniqueness of a smooth solution. A~careful and detailed investigation on weak solutions is the goal of this paper, going from the aspects of the approximation to the proof of continuous dependence estimates. In addition, a sufficient condition for the boundedness of the phase variable is given.

Structural Stability for the Brinkman Equations of Porous Media

Explicit a priori continuous dependence estimates are derived for the Brinkman equations for non-isothermal flow in porous media. Continuous dependence on the cooling coefficient is shown when a boundary condition of Newton cooling type is employed. Continuous dependence on the model itself is proved when the Boussinesq model is allowed to change to one appropriate to penetrative convection. The final result derives an a priori continuous dependence estimate for the heat supply and body force.

Sensitivity with respect to the underlying information in stochastic programs

Quantitative estimates of the continuous dependence on the underlying probability measure, are established for the value of stochastic programming problems. Lipschitz dependence is derived as a particular case. The class of problems is a general one; examples show how the conditions can be verified for more structured cases. The general result is used to establish the continuous dependence estimates for the value of sensors for the stochastic problems.