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Pavel Exner, Chezh Technical University, EMS.

Quantum Hamiltonians Exhibiting a Spectral Transition

Abstract: The aim of this talk is to discuss several classes of Schrődinger operators with potentials that are below unbounded but their negative part is localized in narrow channels. A prototype of such a behavior can be found in Smilansky-Solomyak model devised to illustrate that an an irreversible behavior is possible even if the heat bath to which the systems is coupled has a finite number of degrees of freedom. We review its properties and analyze several modifications of this model, with regular potentials or a magnetic field, as well as another system in which x^py^p potential is amended by a negative radially symmetric term. All of them have the common property that they exhibit an abrupt parameter-dependent spectral transition: if the coupling constant exceeds a critical value the spectrum changes from a below bounded, partly or fully discrete, to the continuous one covering the whole real axis. We also discuss resonance effects in such models.
The results come from a common work with Diana Barseghyan, Andrii Khrabustovskyi, Jiří Lipovský, Vladimir Lotoreichik, and Miloš Tater.

Dietmar Kroener, Freiburg Albert-Ludwig University.

Numerical Treatment of Interfaces

Abstract: In this contribution we will consider moving interfaces and partial differential equations on moving interfaces in different contexts. First we will present the existence, uniqueness and numerical experiments for solutions of nonlinear conservation laws on moving surfaces. In addition to the “hydrodynamical” shocks, geometrically induced shocks will appear. In the second part we study the compressible two phase flow with phase transition on the basis of the Navier-Stokes-Korteweg- and a phasefield model. It turns out that it is extremely important for the numerical schemes of both models that they satisfy a discrete energy inequality to satisfy the second law of thermodynamics. Different numerical experiments will be presented. In the third part we will report on recent research on our experience of the application of the volume of fluid method (VOF) for the resolution of interfaces. The main advantage compared to level set methods is , that the VOF method is mass conserving. We will show different numerical experiments for the movement of droplets on solid walls.
These results have been obtained together with S. Burbulla, D. Diehl, J. Gerstenberger, M. Kränkel, T. Malkmus, T. Müller, M. Nolte, C. Rohde.

Holm Altenbach, Otto-von-Guericke-Universitat Magdeburg.

Modeling of Plastics and Composite Materials

Abstract: There are several approaches to model the material behavior under mechanical loading:
- the deductive approach based on Continuum Mechanics and Material Theory,
- the inductive approach based on some experimental observations and a step-by-step generalization, and
- the method of rheological modeling
After some general statements with the help of the inductive approach combined with rheological modeling the mechanical behavior of plastics and composite materials will be presented.

Reinhold Kienzler, University of Bremen.

A Beam - Just a Beam in Plane Bending

Abstract: We derive one-dimensional beam theories from the three-dimensional theory of linear elasticity by a power-law expansion of the displacements in height and width direction. The strain energy and the potential of external forces are calculated and integrated over the cross-sectional area. Both appear as power laws of the small beam parameters c^2=h^2/(12\ell^2) and d^2=b^2/(12\ell^2), where \ell is the characteristic dimension in length direction of the beam and h and b are height and width of the rectangular cross-section, respectively. Hierarchical beam theories arise from the consistent truncation of the elastic energy after a specific power 2N=2n+2m of the fast decaying factors c^{2n}d^{2m}. It turns out that the first-order (N=1) approximation delivers the classical Euler-Bernoulli beam theory whereas the second-order approximation (N=2) leads to a Timoshenko-type theory. A special feature of the derivation is that no a priori assumptions are invoked.
These results have been obtained together with P.Schneider.

Flavia Lanzara, Rome University La Sapienza.

Recent Developments in the Computation of High-Dimensional Volume Potentials Based on Approximate Approximations

Alberto Cialdea, University Basilicata.

The L^p-Dissipativity of First Order Partial Differential Operators

Abstract: In 2005 we proved, together with Vladimir Mazya, that the algebraic condition |p-2|\, |\langle ImA \xi,\xi\rangle| \leq 2 \sqrt{p-1}\, \langle ReA\xi,\xi\rangle (for any \xi\in{\mathbb{R}}^{n}) is necessary and sufficient for the L^p-dissipativity of the Dirichlet problem for the differential operator \nabla^{t}(A\nabla), where A is a matrix whose entries are complex measures and whose imaginary part is symmetric. This condition characterizes the L^p-dissipativity individually, for each p, while usually the results in the literature concern the L^p-dissipativity for all p's simultaneously.
Later on we have determined necessary and sufficient conditions for the L^p-dissipativity of other partial differential operators of the second order, including some systems.
The aim of the present talk is to present recent results in this direction concerning first order partial differential operators. Also these results have been obtained together with Vladimir Mazya.

Lucian Beznea, Simion Stoilow Institute of Mathematics of the Romanian Academy.

Stochastic Equation of Fragmentation and Branching Processes Related to Avalanches

Abstract: We develop a method for the construction of continuous time fragmentation-branching processes on the space of all fragmentation sizes, induced either by continuous fragmen- tation kernels or by discontinuous ones. This construction leads to a stochastic model for the fragmentation phase of an avalanche. We introduce an approximation scheme for the process which solves the corresponding stochastic differential equations of fragmentation. Finally, we present numerical results that confirm the validity of a fractal property which is emphasized by our model for an avalanche.
The talk is based on joint works with Madalina Deaconu and Oana Lupaşcu.