Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping
Lorena Bociu Petronela Radu
In this paper we show existence of finite energy solutions for the Cauchy problem associated with a semilinear wave equation with interior damping and supercritical source terms. The main contribution consists in dealing with super-supercritical source terms (terms of the order of $|u|^p$ with $p\geq 5$ in $n=3$ dimensions), an open and highly recognized problem in the literature on nonlinear wave equations.
keywords: damping and source terms wave equations energy identity weak solutions
Optimal control in a free boundary fluid-elasticity interaction
Lorena Bociu Lucas Castle Kristina Martin Daniel Toundykov
We establish existence of an optimal control for the problem of minimizing flow turbulence in the case of a nonlinear fluid-structure interaction model in the framework of the known local well-posedness theory. If the initial configuration is regular, in an appropriate sense, then a class of sufficiently smooth control inputs contains an element that minimizes, within the control class, the vorticity of the fluid flow around a moving and deforming elastic solid.
keywords: fluid-structure Optimal control nonlinear elasticity Navier-Stokes moving boundary.
Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping
Lorena Bociu Irena Lasiecka
We consider finite energy solutions of a wave equation with supercritical nonlinear sources and nonlinear damping. A distinct feature of the model under consideration is the presence of nonlinear sources on the boundary driven by Neumann boundary conditions. Since Lopatinski condition fails to hold (unless the $\text{dim} (\Omega) = 1$), the analysis of the nonlinearities supported on the boundary, within the framework of weak solutions, is a rather subtle issue and involves the strong interaction between the source and the damping. Thus, it is not surprising that existence theory for this class of problems has been established only recently. However, the uniqueness of weak solutions was declared an open problem. The main result in this work is uniqueness of weak solutions. This result is proved for the same (even larger) class of data for which existence theory holds. In addition, we prove that weak solutions are continuously depending on initial data and that the flow corresponding to weak and global solutions is a dynamical system on the finite energy space.
keywords: boundary source critical exponents nonlinear damping uniqueness interior source wave equation
Sensitivity analysis for a free boundary fluid-elasticity interaction
Lorena Bociu Jean-Paul Zolésio
In this paper a total linearization is derived for the free boundary nonlinear elasticity - incompressible fluid interaction. The equations and the free boundary are linearized together and the new linearization turns out to be different from the usual coupling of classical linear models. New extra terms are present on the common interface, some of them involving the boundary curvatures. These terms play an important role in the final linearized system and can not be neglected.
keywords: Navier-Stokes Free boundary linearization coupled system nonlinear elasticity shape derivative.
Existence for the linearization of a steady state fluid/nonlinear elasticity interaction
Lorena Bociu Jean-Paul Zolésio
A linearized steady state three-dimensional fluid-structure interaction is considered and its solvability is studied. The linearization (obtained in a previous work by these authors) that we deal with has new features, including the presence of the curvature terms on the common interface. These new extra terms, coming from the geometrical aspect of the problem, are critical for a correct physical interpretation of the fluid/structure coupling. We prove that the linearization has unique solution.
keywords: Coupled System Viscous Fluid Existence Nonlinear Elasticity Navier-Stokes Linearization
The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics
Maurizio Verri Giovanna Guidoboni Lorena Bociu Riccardo Sacco

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, Alzheimer’s disease and glaucoma. Thus, the findings of this work may be relevant to many applications in biology and medicine.

keywords: Deformable porous media flow incompressible constituents viscoelasticity explicit solution velocity blow-up confined compression
Errata: Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping
Lorena Bociu Petronela Radu Daniel Toundykov
This note is an errata for the paper [2] which discusses regular solutions to wave equations with super-critical source terms. The purpose of this note is to address the gap in the proof of uniqueness of such solutions.
keywords: nonlinear damping. super-critical critical exponent regular solutions Wave equation
Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping
Lorena Bociu Petronela Radu Daniel Toundykov
We study regular solutions to wave equations with super-critical source terms, e.g., of exponent $p>5$ in 3D. Such sources have been a major challenge in the investigation of finite-energy ($H^1 \times L^2$) solutions to wave PDEs for many years. The wellposedness has been settled in part, but even the local existence, for instance, in 3 dimensions requires the relation $p\leq 6m/(m+1)$ between the exponents $p$ of the source and $m$ of the viscous damping.
    We prove that smooth initial data ($H^2 \times H^1$) yields regular solutions that do not depend on the above correlation. Local existence is demonstrated for any source exponent $p\geq 1$ and any monotone damping including feedbacks growing exponentially or logarithmically at infinity, or with no damping at all. The result holds in dimensions 3 and 4, and with some restrictions on $p$ in dimensions $n\geq 5$. Furthermore, if we assert the classical condition that the damping grows as fast as the source, then these regular solutions are global.
keywords: critical exponent regular solutions nonlinear damping. Wave equation super-critical
Preface: Introduction to the Special Volume on Nonlinear PDEs and Control Theory with Applications
Lorena Bociu Barbara Kaltenbacher Petronela Radu
This volume collects a number of contributions in the fields of partial differential equations and control theory, following the Special Session Nonlinear PDEs and Control Theory with Applications held at the 9th AIMS conference on Dynamical Systems, Differential Equations and Applications in Orlando, July 1--5, 2012.

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Linearized hydro-elasticity: A numerical study
Lorena Bociu Steven Derochers Daniel Toundykov
In view of control and stability theory, a recently obtained linearization around a steady state of a fluid-structure interaction is considered. The linearization was performed with respect to an external forcing term and was derived in an earlier paper via shape optimization techniques. In contrast to other approaches, like transporting to a fixed reference configuration, or using transpiration techniques, the shape optimization route is most suited to incorporating the geometry of the problem into the analysis. This refined description brings up new terms---missing in the classical coupling of linear Stokes flow and linear elasticity---in the matching of the normal stresses and the velocities on the interface. Later, it was demonstrated that this linear PDE system generates a $C_0$ semigroup, however, unlike in the standard Stokes-elasticity coupling, the wellposedness result depended on the fluid's viscosity and the new boundary terms which, among other things, involve the curvature of the interface. Here, we implement a finite element scheme for approximating solutions of this fluid-elasticity dynamics and numerically investigate the dependence of the discretized model on the ``new" terms present therein, in contrast with the classical Stokes-linear elasticity system.
keywords: sensitivity. moving boundary free boundary linearization elasticity Stokes hydro-elasticity finite elements Fluid-structure

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