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PROC

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.

PROC

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.

DCDS

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

EECT

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.

PROC

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

EECT

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

EECT

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

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

EECT

This volume collects a number of contributions in the fields of partial differential equations and control theory, following the Special Session

For more information please click the “Full Text” above.

*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.For more information please click the “Full Text” above.

keywords:

EECT

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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