58 research outputs found
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Mini-Workshop: Wellposedness and Controllability of Evolution Equations
This mini-workshop brought together mathematicians engaged in partial differential equations, operator theory, functional analysis and harmonic analysis in order to address a number of current problems in the wellposedness and controllability of infinite-dimensional systems
Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus
We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional
torus T = R/(2{\pi}Z). We prove a Unique Continuation Property (UCP) for small
data in H^1(T) with nonnegative zero means. Next we extend the UCP to certain
BBM-like equations, including the equal width wave equation and the KdV-BBM
equation. Applications to the stabilization of the above equations are given.
In particular, we show that when an internal control acting on a moving
interval is applied in BBM equation, then a semiglobal exponential
stabilization can be derived in H^s(T) for any s \geq 1. Furthermore, we prove
that the BBM equation with a moving control is also locally exactly
controllable in H^s(T) for any s \geq 0 and globally exactly controllable in H
s (T) for any s \geq 1
Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer
In this paper dynamic von Karman equations with localized interior damping
supported in a boundary collar are considered. Hadamard well-posedness for von
Karman plates with various types of nonlinear damping are well-known, and the
long-time behavior of nonlinear plates has been a topic of recent interest.
Since the von Karman plate system is of "hyperbolic type" with critical
nonlinearity (noncompact with respect to the phase space), this latter topic is
particularly challenging in the case of geometrically constrained and nonlinear
damping. In this paper we first show the existence of a compact global
attractor for finite-energy solutions, and we then prove that the attractor is
both smooth and finite dimensional. Thus, the hyperbolic-like flow is
stabilized asymptotically to a smooth and finite dimensional set.
Key terms: dynamical systems, long-time behavior, global attractors,
nonlinear plates, nonlinear damping, localized dampin
Landau-Lifshitz-Gilbert equations: Controllability by Low Modes Forcing for deterministic version and Support Theorems for Stochastic version
In this article, we study the controllability issues of the
Landau-Lifshitz-Gilbert Equations (LLGEs), accompanied with non-zero exchange
energy only, in an interval in one spatial dimension with Neumann boundary
conditions. The paper is of twofold. In the first part of the paper, we study
the controllability issues of the LLGEs. The control force acting here is
degenerate i.e., it acts through a few numbers of low mode frequencies. We
exploit the Fourier series expansion of the solution. We borrow methods of
differential geometric control theory (Lie bracket generating property) to
establish the global controllability of the finite-dimensional Galerkin
approximations of LLGEs. We show approximate controllability of the full
system. In the second part, we consider the LLGEs with lower-dimensional
degenerate random forcing (finite-dimensional Brownian motions) and study
support theorems.Comment: 2
Semigroup Well-posedness of A Linearized, Compressible Fluid with An Elastic Boundary
We address semigroup well-posedness of the fluid-structure interaction of a
linearized compressible, viscous fluid and an elastic plate (in the absence of
rotational inertia). Unlike existing work in the literature, we linearize the
compressible Navier-Stokes equations about an arbitrary state (assuming the
fluid is barotropic), and so the fluid PDE component of the interaction will
generally include a nontrivial ambient flow profile . The
appearance of this term introduces new challenges at the level of the
stationary problem. In addition, the boundary of the fluid domain is
unavoidably Lipschitz, and so the well-posedness argument takes into account
the technical issues associated with obtaining necessary boundary trace and
elliptic regularity estimates. Much of the previous work on flow-plate models
was done via Galerkin-type constructions after obtaining good a priori
estimates on solutions (specifically \cite {Chu2013-comp}---the work most
pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach,
with a view of associating solutions of the fluid-structure dynamics with a
-semigroup on the natural
finite energy space of initial data. So, given this approach, the major
challenge in our work becomes establishing of the maximality of the operator
which models the fluid-structure dynamics. In sum: our main
result is semigroup well-posedness for the fully coupled fluid-structure
dynamics, under the assumption that the ambient flow field has zero normal component trace on the boundary (a
standard assumption with respect to the literature). In the final sections we
address well-posedness of the system in the presence of the von Karman plate
nonlinearity, as well as the stationary problem associated with the dynamics.Comment: 1 figur
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Small Collaboration: Numerical Analysis of Electromagnetic Problems (hybrid meeting)
The classical theory of electromagnetism describes the interaction of electrically charged particles through electromagnetic forces, which are carried by the electric and magnetic fields. The propagation of the electromagnetic fields can be described by Maxwell's equations. Solving Maxwell's equations numerically is a challenging problem which appears in many different technical applications. Difficulties arise for instance from material interfaces or if the geometrical features are much larger than or much smaller than a typical wavelength. The spatial discretization needs to combine good geometrical flexibility with a relatively high order of accuracy.
The aim of this small-scale, week-long interactive mini-workshop jointly organized by the University of Duisburg-Essen and the University of Twente, and kindly hosted at the MFO, is to bring together experts in non-standard and mixed finite elements methods with experts in the field of electromagnetism
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