Lorena Bociu

Wave equations with super-critical interior and boundary nonlinearities

Abstract: This article presents a unified overview of the latest, to date, results on boundary value problems for wave equations with super-critical nonlinear sources on both the interior and the boundary of a bounded domain @W@?R^n. The presented theorems include Hadamard local wellposedness, global existence, blow-up and non-existence theorems, as well as estimates on the uniform energy dissipation rates for the appropriate classes of solutions.

Linearization of a Coupled System of Nonlinear Elasticity and Viscous Fluid

We model the coupled system formed by an incompressible fluid and a nonlinear elastic body.We work with large displacement, small deformation elasticity (or St. Venant elasticity), which makes the problem very interesting from the physical point of view.The elastic body is three-dimensional Ω \(\epsilon\, \mathbf{R}^3\), and thus it can not be reduced to its boundary Γ (like in the case of a membrane or a shell). In this paper, we study the static problem, and in view of the stability analysis we derive the linearization of the system, which turns out to be different from the usual coupling of classical linear models. New extra terms (for example those involving the boundary curvatures) play an important role in the final linearized system around some equilibrium.

Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

We study the well-posedness of a total linearization, with respect to a perturbation of the external forcing, of a free-boundary nonlinear elasticity--incompressible fluid interaction. The total linearization for the coupling modeled by the Navier--Stokes equations and the nonlinear equations of elastodynamics was obtained recently in [L. Bociu and J.-P. Zolesio, Evol. Equ. Control Theory, 2 (2013), pp. 55--79]. The equations and the free boundary were linearized together, and the result turned out to be quite different from the usual coupling of classical linear models. New additional terms are present on the common interface, some of them involving boundary curvatures and boundary acceleration. These terms play an important role in the final linearized system and cannot be neglected; their presence also introduces new challenges in the well-posedness analysis, which proceeds to establish that the evolution operator associated to the linearized system can be represented as a bounded perturbation of a max...

Strong Shape Derivative for the Wave Equation with Neumann Boundary Condition

The paper provides shape derivative analysis for the wave equation with mixed boundary conditions on a moving domain Ω s in the case of non smooth neumann boundary datum. The key ideas in the paper are (i) bypassing the classical sensitivity analysis of the state by using parameter differentiability of a functional expressed in the form of Min-Max of a convex-concave Lagrangian with saddle point, and (ii) using a new regularity result on the solution of the wave problem (where the Dirichlet condition on the fixed part of the boundary is essential) to analyze the strong derivative.

The role of structural viscoelasticity in deformable porous media with incompressibleconstituents: Applications in biomechanics

The main goal of this work is to clarify and quantify, by means of mathematical analysis, the role of structural viscoelasticity in the biomechanical response of deformable porous media with incompressible constituents to sudden changes in external applied loads. Models of deformable porous media with incompressible constituents are often utilized to describe the behavior of biological tissues, such as cartilages, bones and engineered tissue scaffolds, where viscoelastic properties may change with age, disease or by design. Here, for the first time, we show that the fluid velocity within the medium could increase tremendously, even up to infinity, should the external applied load experience sudden changes in time and the structural viscoelasticity be too small. In particular, we consider a one-dimensional poro-visco-elastic model for which we derive explicit solutions in the cases where the external applied load is characterized by a step pulse or a trapezoidal pulse in time. By means of dimensional analysis, we identify some dimensionless parameters that can aid the design of structural properties and/or experimental conditions as to ensure that the fluid velocity within the medium remains bounded below a certain given threshold, thereby preventing potential tissue damage. The application to confined compression tests for biological tissues is discussed in detail. Interestingly, the loss of viscoelastic tissue properties has been associated with various disease conditions, such as atherosclerosis, Alzheimers disease and glaucoma. Thus, the findings of this work may be relevant to many applications in biology and medicine.

System Modeling and Optimization

This paper presents the modelling of the evolution of plasma equilibrium in the presence of external poloidal field circuits and passive structures. The optimization of plasma scenarios is formulated as an optimal control problem where the equations for the evolution of the plasma equilibrium are the constraints. The procedure determines the voltages applied to the external circuits that minimize a certain costfunction representing the distance to a desired plasma augmented by an energetic cost of the electrical system. A sequential quadratic programming method is used to solve the minimization of the cost-function and an application to the optimization of a discharge for ITER is shown.