16 research outputs found
On selection criteria for problems with moving inhomogeneities
We study mechanical problems with multiple solutions and introduce a
thermodynamic framework to formulate two different selection criteria in terms
of macroscopic energy productions and fluxes. Studying simple examples for
lattice motion we then compare the implications for both resting and moving
inhomogeneities.Comment: revised version contains new introduction, numerical simulations of
Riemann problems, and a more detailed discussion of the causality principle;
18 pages, several figure
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Mathematical Theory of Water Waves
Water waves, that is waves on the surface of a fluid (or the interface between different fluids) are omnipresent phenomena.
However, as Feynman wrote in his lecture, water waves that are easily seen by everyone, and which are usually used as an example of waves in elementary courses, are the worst possible example; they have all the complications that waves can have. These complications make mathematical investigations particularly challenging and the physics particularly rich.
Indeed, expertise gained in modelling,
mathematical analysis and numerical simulation of water waves can be expected to lead to progress in issues of high societal impact
(renewable energies in marine environments, vorticity generation and wave breaking, macro-vortices and coastal erosion, ocean
shipping and near-shore navigation, tsunamis and hurricane-generated waves, floating airports, ice-sea interactions,
ferrofluids in high-technology applications, ...).
The workshop was mostly devoted to rigorous mathematical theory for the exact hydrodynamic
equations; numerical simulations, modelling and experimental issues were included insofar as they
had an evident synergy effect
Book of Abstracts
USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud
SĂŁo Carlos, Brasil. 3-7 february 2014
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Mini-Workshop: Multiscale and Variational Methods in Material Science and Quantum Theory of Solids
This workshop brought together 18 scientists from three different mathematical communities: (i) random SchroÌdinger operators, (ii) quantum mechanics of interacting atoms, and (iii) mathematical materials science. Several underlying themes were identified and addressed: variational principles, homogenisation techniques, thermodynamic limits, spectral theory, and dynamic and stochastic aspects
Uniform estimates for positive solutions of semilinear elliptic equations and related Liouville and one-dimensional symmetry results
We consider a semilinear elliptic equation with Dirichlet boundary conditions
in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce
a condition on the size of the domain that implies the existence of a positive
solution satisfying a uniform pointwise estimate. Here, uniform means that the
estimate is independent of the domain. The main advantage of our approach is
that it allows us to remove a restrictive monotonicity assumption that was
imposed in the recent paper. In addition, we can remove a non-degeneracy
condition that was assumed in the latter reference. Furthermore, we can
generalize an old result, concerning semilinear elliptic nonlinear eigenvalue
problems. Moreover, we study the boundary layer of global minimizers of the
corresponding singular perturbation problem. For the above applications, our
approach is based on a refinement of a result, concerning the behavior of
global minimizers of the associated energy over large balls, subject to
Dirichlet conditions. Combining this refinement with global bifurcation theory
and the sliding method, we can prove uniform estimates for solutions away from
their nodal set. Combining our approach with a-priori estimates that we obtain
by blow-up, a doubling lemma, and known Liouville type theorems, we can give a
new proof of a known Liouville type theorem without using boundary blow-up
solutions. We can also provide an alternative proof, and a useful extension, of
a Liouville theorem, involving the presence of an obstacle. Making use of the
latter extension, we consider the singular perturbation problem with mixed
boundary conditions. Moreover, we prove some new one-dimensional symmetry and
rigidity properties of certain entire solutions to Allen-Cahn type equations,
as well as in half spaces, convex cylindrical domains. In particular, we
provide a new proof of Gibbons' conjecture in two dimensions.Comment: Corrected the subsection on Gibbon's conjecture: As it is, our
Gibbons' conjecture proof works only in two dimension
Universal properties of self-organized localized structures
Ziel dieser Arbeit ist es, einen Beitrag zur AufklĂ€rung der universellen teilchenartigen Eigenschaften von selbstorganisierten lokalisierten Strukturen in unterschiedlichen rĂ€umlich ausgedehnten Systemen zu leisten. Untersucht werden zunĂ€chst Mechanismen der Bildung, Dynamik und Wechselwirkung in Prototyp-Modellsystemen wie Reaktions-Diffusions-, Ginzburg-Landau- und Swift-Hohenberg-Gleichungen. Mittels adiabatischer Eliminationsmethoden kann die Feldbeschreibung in vielen FĂ€llen auf gewöhnliche Differentialgleichungen von einheitlicher Form reduziert werden. So kann auch das Verhalten groĂer Ensembles lokalisierter Strukturen analytisch und numerisch analysiert werden. Hier ergeben sich zahlreiche neue PhĂ€nomene ohne klassisches Analogon. Zuletzt ist die Dynamik von biologischen teilchenartigen Strukturen in Form von Dictyostelium-Zellen von Interesse, die experimentell in mikrofluidischen Aufbauten erforscht wird. This work tries to contribute to explaining the universal particle-like
behavior of self-organized localized structures in different spatially
extended systems. First, mechanisms of generation, dynamics and
interaction in prototype systems like reaction-diffusion,
Ginzburg-Landau or Swift-Hohenberg equations are investigated. Using
methods of adiabatic elimination, in many cases the field description
can be reduced to ordinary differential equations of uniform structure.
In this way, one may also explore the behavior of large particle
ensembles analytically and numerically. Here, many new phenomena without
classical analogue arise. Finally, an experimental analysis of
biological particle-like structures in the form of Dictyostelium cells
is carried out using microfluidic devices