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The edoc server is the Open Access publication server and institutional repository of the Humboldt-Universität zu Berlin. It is available to university members for the publication of academic work as a first or second publication. In addition to text publications, research data can also be published via the edoc server. These publications are available to the general public worldwide, free of charge and for the long term.

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Publication
2023-03-08Zeitschriftenartikel
On Two Coupled Degenerate Parabolic Equations Motivated by Thermodynamics
(Mathematisch-Naturwissenschaftliche Fakultät) Mielke, Alexander
We discuss a system of two coupled parabolic equations that have degenerate diffusion constants depending on the energy-like variable. The dissipation of the velocity-like variable is fed as a source term into the energy equation leading to conservation of the total energy. The motivation of studying this system comes from Prandtl’s and Kolmogorov’s one- and two-equation models for turbulence, where the energy-like variable is the mean turbulent kinetic energy. Because of the degeneracies, there are solutions with time-dependent support like in the porous medium equation, which is contained in our system as a special case. The motion of the free boundary may be driven by either self-diffusion of the energy-like variable or by dissipation of the velocity-like variable. The crossover of these two phenomena is exemplified for the associated planar traveling fronts. We provide existence of suitably defined weak and very weak solutions. After providing a thermodynamically motivated gradient structure, we also establish convergence into steady state for bounded domains and provide a conjecture on the asymptotically self-similar behavior of the solutions in Rdfor large times.
Publication
2023-05-16Zeitschriftenartikel
Balanced-Viscosity Solutions to Infinite-Dimensional Multi-Rate Systems
(Mathematisch-Naturwissenschaftliche Fakultät) Mielke, Alexander; Rossi, Riccarda
We consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable u having a viscous damping with relaxation time εαand an internal variable z with relaxation time εwe obtain different limits for the three cases α∈(0,1), α=1and α>1. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics.
Publication
2022-06-22Zeitschriftenartikel
Political Theory of the Digital Constellation
(Kultur-, Sozial- und Bildungswissenschaftliche Fakultät) Berg, Sebastian; Staemmler, Daniel; Thiel, Thorsten
The introductory contribution to this special issue on “Political Theory of the Digital Constellation” addresses the conditions and possibilities of political theory’s engagement with digital developments. The motivation for this inquiry is the growing interest in questions of political theory arising from the digital transformation, as well as the acknowledgement that digitalisation not only changes politics, but conversely that politics also shapes digitalisation. The article identifies three pitfalls of previous engagement: The narrowing of the subject of “digitalisation” to the topic of the “internet” and, thereby, to the aspect of communication, the disregard for the technicality of the digital, and the insufficient recognition that (digital) technology is political. To avoid these pitfalls, the research perspective of the digital constellation is presented. The digital constellation serves as an epistemological guide that helps to structure theoretical reflection on the interrelationship between digitalisation and political questions. Ultimately, the outlines of the political theory in the digital constellation become clear in the fourteen contributions of the special volume, which are presented in conclusion.
Publication
2022-02-23Zeitschriftenartikel
Traveling Fronts in a Reaction–Diffusion Equation with a Memory Term
(Mathematisch-Naturwissenschaftliche Fakultät) Mielke, Alexander; Reichelt, Sina
Based on a recent work on traveling waves in spatially nonlocal reaction–diffusion equations, we investigate the existence of traveling fronts in reaction–diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction–diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory. The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising traveling front, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations.
Publication
2023-12-07Zeitschriftenartikel
Time-domain modeling of interband transitions in plasmonic systems
(Mathematisch-Naturwissenschaftliche Fakultät) Pfeifer, Max; Huynh, Dan-Nha; Wegner, Gino; Intravaia, Francesco; Peschel, Ulf; Busch, Kurt
Efficient modeling of dispersive materials via time-domain simulations of the Maxwell equations relies on the technique of auxiliary differential equations. In this approach, a material’s frequency-dependent permittivity is represented via a sum of rational functions, e.g., Lorentz poles, and the associated free parameters are determined by fitting to experimental data. In the present work, we present a modified approach for plasmonic materials that requires considerably fewer fit parameters than traditional approaches. Specifically, we consider the underlying microscopic theory and, in the frequency domain, separate the hydrodynamic contributions of the quasi-free electrons in partially filled bands from the interband transitions. As an illustration, we apply our approach to gold and demonstrate how to treat the interband transitions within the effective model via connecting to the underlying electronic band structure, thereby assigning physical meaning to the remaining fit parameters. Finally, we show how to utilize this approach within the technique of auxiliary differential equations. Our approach can be extended to other plasmonic materials and leads to efficient time-domain simulations of plasmonic structures for frequency ranges where interband transitions have to be considered.