Publications

Monographs

  • U.W. Pohl, A. Strittmatter, A. Schliwa, M. Lehmann, T. Niermann, T. Heindel, S. Reitzenstein, M. Kantner, U. Bandelow, Th. Koprucki, H.-J. Wünsche, Stressor-induced site control of quantum dots for single-photon sources, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., Semiconductor Nanophotonics, Springer, Heidelberg, 2020, pp. 53--90, (Chapter Published), DOI 10.1007/978-3-030-35656-9_3 .
    Abstract
    The strain field of selectively oxidized A10x current apertures in an A1GaAs/GaAs mesa is utilized to define the nucleation site of InGaAs/GaAs quantum dots. A design is developed that allows for the self-aligned growth of single quantum dots in the center of a circular mesa. Measurements of the strain tensor applying transmission-electron holography yield excellent agreement with the calculated strain field. Single-dot spectroscopy of site-controlled dots proves narrow excitonic linewidth virtually free of spectral diffusion due to quantum-dot growth in a defect-free matrix. Implementation of such dots in an electrically driven pin structure yields single-dot electroluminescence. Single-photon emission with excellent purity is provided for this device using a Hanbury Brown and Twiss setup. The injection efficiency of the initian pin design is affected by a substantial lateral current spreading close to the oxide aperture. Allpying 3D carier-transport simulation a ppn doping profile is developed achieving a substantial improvement of the current injection.

  • S. Rodt, P.-I. Schneider, L. Zschiedrich, T. Heindel, S. Bounouar, M. Kantner, Th. Koprucki, U. Bandelow, S. Burger, S. Reitzenstein, Deterministic quantum devices for optical quantum communication, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., Semiconductor Nanophotonics, Springer, Heidelberg, 2020, pp. 285--359, (Chapter Published), DOI 10.1007/978-3-030-35656-9_8 .
    Abstract
    Photonic quantum technologies are based on the exchange of information via single photons. The information is typically encoded in the polarization of the photons and security is ensured intrinsically via principles of quantum mechanics such as the no-cloning theorem. Thus, all optical quantum communication networks rely crucially on the availability of suitable quantum-light sources. Such light sources with close to ideal optical and quantum optical properties can be realized by self-assembled semiconductor quantum dots. These high-quality nanocrystals are predestined single-photon emitters due to their quasi zero-dimensional carrier confinement. Still, the development of practical quantum-dot-based sources of single photons and entangled-photon pairs for applications in photonic quantum technology and especially for the quantum-repeater scheme is very demanding and requires highly advanced device concepts and deterministic fabrication technologies. This is mainly explained by their random position and emission energy as well as by the low photon-extraction efficiency in simple planar device configurations.

  • TH. Eiter, Existence and spatial decay of periodic Navier--Stokes flows in exterior domains, Th. Eiter, ed., Logos Verlag Berlin GmbH, Berlin, 2020, (Monograph Published), DOI 10.30819/5108 .
    Abstract
    A classical problem in the field of mathematical fluid mechanics is the flow of a viscous incompressible fluid past a rigid body. In his doctoral thesis, Thomas Walter Eiter investigates time-periodic solutions to the associated Navier-Stokes equations when the body performs a non-trivial translation. The first part of the thesis is concerned with the question of existence of time-periodic solutions in the case of a non-rotating and of a rotating obstacle. Based on an investigation of the corresponding Oseen linearizations, new existence results in suitable function spaces are established. The second part deals with the study of spatially asymptotic properties of time-periodic solutions. For this purpose, time-periodic fundamental solutions to the Stokes and Oseen linearizations are introduced and investigated, and the concept of a time-periodic fundamental solution for the vorticity field is developed. With these results, new pointwise estimates of the velocity and the vorticity field associated to a time-periodic fluid flow are derived.

  • M. Kantner, Th. Höhne, Th. Koprucki, S. Burger, H.-J. Wünsche, F. Schmidt, A. Mielke, U. Bandelow, Multi-dimensional modeling and simulation of semiconductor nanophotonic devices, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., Semiconductor Nanophotonics, Springer, Heidelberg, 2020, pp. 241--283, (Chapter Published), DOI 10.1007/978-3-030-35656-9_7 .
    Abstract
    Self-consistent modeling and multi-dimensional simulation of semiconductor nanophotonicdevices is an important tool in the development of future integrated light sources and quantumdevices. Simulations can guide important technological decisions by revealing performance bottle-necks in new device concepts, contribute to their understanding and help to theoretically exploretheir optimization potential. The efficient implementation of multi-dimensional numerical simulationsfor computer-aided design tasks requires sophisticated numerical methods and modeling tech-niques. We review recent advances in device-scale modeling of quantum dot based single-photonsources and laser diodes by self-consistently coupling the optical Maxwell equations with semi-classical carrier transport models using semi-classical and fully quantum mechanical descriptionsof the optically active region, respectively. For the simulation of realistic devices with complex,multi-dimensional geometries, we have developed a novel hp-adaptive finite element approachfor the optical Maxwell equations, using mixed meshes adapted to the multi-scale properties ofthe photonic structures. For electrically driven devices, we introduced novel discretization andparameter-embedding techniques to solve the drift-diffusion system for strongly degenerate semi-conductors at cryogenic temperature. Our methodical advances are demonstrated on variousapplications, including vertical-cavity surface-emitting lasers, grating couplers and single-photonsources.

Articles in Refereed Journals

  • D. Chaudhuri, M. O'Donovan, T. Streckenbach, O. Marquart, P. Farrell, S.K. Patra, Th. Koprucki, S. Schulz, Multiscale simulations of the electronic structure of III-nitride quantum wells with varied Indium content: Connecting atomistic and continuum-based models, Journal of Applied Physics, 129 (2021), published online on 18.02.2021, DOI 10.1063/5.0031514 .

  • H. Meinlschmidt, J. Rehberg, Extrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations, Journal of Differential Equations, 280 (2021), pp. Published online on 29.01.2021 (375--404), DOI 10.1016/j.jde.2021.01.032 .
    Abstract
    In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order Xs-1,qD(Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs.

  • A. Glitzky, M. Liero, G. Nika, An existence result for a class of electrothermal drift-diffusion models with Gauss--Fermi statistics for organic semiconductors, Analysis and Applications, 19 (2021), pp. 275--304, DOI 10.1142/S0219530519500246 .
    Abstract
    This work is concerned with the analysis of a drift-diffusion model for the electrothermal behavior of organic semiconductor devices. A "generalized Van Roosbroeck” system coupled to the heat equation is employed, where the former consists of continuity equations for electrons and holes and a Poisson equation for the electrostatic potential, and the latter features source terms containing Joule heat contributions and recombination heat. Special features of organic semiconductors like Gauss--Fermi statistics and mobilities functions depending on the electric field strength are taken into account. We prove the existence of solutions for the stationary problem by an iteration scheme and Schauder's fixed point theorem. The underlying solution concept is related to weak solutions of the Van Roosbroeck system and entropy solutions of the heat equation. Additionally, for data compatible with thermodynamic equilibrium, the uniqueness of the solution is verified. It was recently shown that self-heating significantly influences the electronic properties of organic semiconductor devices. Therefore, modeling the coupled electric and thermal responses of organic semiconductors is essential for predicting the effects of temperature on the overall behavior of the device. This work puts the electrothermal drift-diffusion model for organic semiconductors on a sound analytical basis.

  • TH. Eiter, M. Kyed, Viscous flow around a rigid body performing a time-periodic motion, Journal of Mathematical Fluid Mechanics, (2021), pp. Published online on 11.02.2021 (28/1--28/23), DOI 10.1007/s00021-021-00556-4 .
    Abstract
    The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

  • A. Mielke, A. Montefusco, M.A. Peletier, Exploring families of energy-dissipation landscapes via tilting: Three types of EDP convergence, Continuum Mechanics and Thermodynamics, published online on 28.01.2021, DOI 10.1007/s00161-020-00932-x .
    Abstract
    This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that well-known techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the Energy-Dissipation Principle that provides a variational formulation to gradient-flow equations that allows one to apply techniques from Γ-convergence of functional on states and functionals on trajectories.

  • A. Maltsi, T. Niermann, T. Streckenbach, K. Tabelow, Th. Koprucki, Numerical simulation of TEM images for In(Ga)As/GaAs quantum dots with various shapes, Optical and Quantum Electronics, 52 (2020), pp. 257/1--257/11 (published online on 05.05.2020), DOI 10.1007/s11082-020-02356-y .
    Abstract
    We present a mathematical model and a tool chain for the numerical simulation of TEM images of semiconductor quantum dots (QDs). This includes elasticity theory to obtain the strain profile coupled with the Darwin-Howie-Whelan equations, describing the propagation of the electron wave through the sample. We perform a simulation study on indium gallium arsenide QDs with different shapes and compare the resulting TEM images to experimental ones. This tool chain can be applied to generate a database of simulated TEM images, which is a key element of a novel concept for model-based geometry reconstruction of semiconductor QDs, involving machine learning techniques.

  • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 71 (2020), pp. 119/1--119/68, DOI 10.1007/s00033-020-01341-5 .
    Abstract
    We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a global--in--time weak solution for the full model, allowing for a general structure of the mobility tensor and for chemical reactions with highly non linear rates in the bulk and on the active boundary. We characterise the singular states of the system, showing that the chemical species can vanish only globally in space, and that this phenomenon must be concentrated in a compact set of measure zero in time. With respect to our former study [DDGG16], we also essentially improve the a priori estimates, in particular concerning the relative chemical potentials.

  • P.-É. Druet, A. Jüngel, Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient, SIAM Journal on Mathematical Analysis, 52 (2020), pp. 2179--2197, DOI 10.1137/19M1301473 .
    Abstract
    The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy?s law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.

  • M. Heida, R. Kornhuber, J. Podlesny, Fractal homogenization of a multiscale interface problem, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 18 (2020), pp. 294--314, DOI 10.1137/18M1204759 .
    Abstract
    Inspired from geological problems, we introduce a new geometrical setting for homogenization of a well known and well studied problem of an elliptic second order differential operator with jump condition on a multiscale network of interfaces. The geometrical setting is fractal and hence neither periodic nor stochastic methods can be applied to the study of such kind of multiscale interface problem. Instead, we use the fractal nature of the geometric structure to introduce smoothed problems and apply methods from a posteriori theory to derive an estimate for the order of convergence. Computational experiments utilizing an iterative homogenization approach illustrate that the theoretically derived order of convergence of the approximate problems is close to optimal.

  • O. Marquardt, M.A. Caro, Th. Koprucki, P. Mathé, M. Willatzen, Multiband k $cdot$ p model and fitting scheme for ab initio-based electronic structure parameters for wurtzite GaAs, Phys. Rev. B., 101 (2020), pp. 235147/1--235147/12, DOI 10.1103/PhysRevB.101.235147 .
    Abstract
    We develop a 16-band k · p model for the description of wurtzite GaAs, together with a novel scheme to determine electronic structure parameters for multiband k · p models. Our approach uses low-discrepancy sequences to fit k · p band structures beyond the eight-band scheme to most recent ab initio data, obtained within the framework for hybrid-functional density functional theory with a screened-exchange hybrid functional. We report structural parameters, elastic constants, band structures along high-symmetry lines, and deformation potentials at the Γ point. Based on this, we compute the bulk electronic properties (Γ point energies, effective masses, Luttinger-like parameters, and optical matrix parameters) for a ten-band and a sixteen-band k · p model for wurtzite GaAs. Our fitting scheme can assign priorities to both selected bands and k points that are of particular interest for specific applications. Finally, ellipticity conditions can be taken into account within our fitting scheme in order to make the resulting parameter sets robust against spurious solutions.

  • K.M. Gambaryan, T. Boeck, A. Trampert, O. Marquardt, Nucleation chronology and electronic properties of In(As,Sb,P) graded composition quantum dots grown on InAs(100) substrate, ACS Applied Electronic Materials, 2 (2020), pp. 646--650, DOI https://doi.org/10.1021/acsaelm.9b00739 .
    Abstract
    We provide a detailed study of nucleation process, characterization, electronic and optical properties of graded composition quantum dots (GCQDs) grown from In-As-Sb-P composition liquid phase on an InAs(100) substrate in the Stranski-Krastanov growth mode. Our GCQDs exhibit diameters from 10 to 120 nm and heights from 2 to 20 nm with segregation profiles having a maximum Sb content of approximately 20% at the top and a maximum P content of approximately 15% at the bottom of the GCQDs so that hole confinement is expected in the upper parts of the GCQDs. Using an eight-band k · p model taking strain and built-in electrostatic potentials into account, we have computed the hole ground state energies and charge densities for a wide range of InAs1-x-ySbxPy GCQDs as close as possible to the systems observed in experiment. Finally, we have obtained an absorption spectrum for an ensemble of GCQDs by combining data from both experiment and theory. Excellent agreement between measured and simulated absorption spectra indicates that such GCQDs can be grown following a theory-guided design for application in specific devices.

  • J.A. Carrillo, K. Hopf, M.-Th. Wolfram, Numerical study of Bose--Einstein condensation in the Kaniadakis--Quarati model for bosons, Kinetic and Related Models, 13 (2020), pp. 507--529, DOI 10.3934/krm.2020017 .
    Abstract
    Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting. It is an open question whether this equation has solutions exhibiting condensates in finite time. The main analytical challenge lies in the continuation of exploding solutions beyond their first blow-up time while having a linear diffusion term. We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the Kaniadakis--Quarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model in 3D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional at an exponential rate. Our simulations further indicate that the spatial blow-up profile near the origin follows a universal power law and that transient condensates can occur for sufficiently concentrated initial data.

  • J.A. Carrillo, K. Hopf, J.L. Rodrigo, On the singularity formation and relaxation to equilibrium in 1D Fokker--Planck model with superlinear drift, Advances in Mathematics, 360 (2020), pp. 106883/1--106883/66, DOI 10.1016/j.aim.2019.106883 .
    Abstract
    We consider a class of Fokker--Planck equations with linear diffusion and superlineardrift enjoying a formal Wasserstein-like gradient flow structure with convex mobility function. In the drift-dominant regime, the equations have a finite critical mass above which the measure minimising the associated entropy functional displays a singular component. Our approach, which addresses the one-dimensional case, is based on a reformulation of the problem in terms of the pseudo-inverse distribution function. Motivated by the structure of the equation in the new variables, we establish a general framework for global-in-time existence, uniqueness and regularity of monotonic viscosity solutions to a class of nonlinear degenerate (resp. singular) parabolic equations, using as a key tool comparison principles and maximum arguments. We then focus on a specific equation and study in more detail the regularity and dynamics of solutions. In particular, blow-up behaviour, formation of condensates (i.e. Dirac measures at zero) and long-time asymptotics are investigated. As a consequence, in the mass-supercritical case,solutions will blow up in L in finite time and--understood in a generalised, measure sense--they will eventually have condensate. We further show that the singular part of the measure solution does in general interact with the density and that condensates can be transient. The equations considered are motivated by a model for bosons introduced by Kaniadakis and Quarati (1994), which has a similar entropy structure and a critical mass if d≥3.

  • R. Chill, H. Meinlschmidt, J. Rehberg, On the numerical range of second order elliptic operators with mixed boundary conditions in L$^p$, Journal of Evolution Equations, (2020), pp. published online on 20.10.2020 (urlhttps://link.springer.com/article/10.1007/s00028-020-00642-6), DOI 10.1007/s00028-020-00642-6 .
    Abstract
    We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on Lp in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in [7]. Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin- instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterization of elements of the form domains inducing mixed boundary conditions.

  • P. Colli, G. Gilardi, J. Sprekels, Asymptotic analysis of a tumor growth model with fractional operators, Asymptotic Analysis, 120 (2020), pp. 41--72, DOI 10.3233/ASY-191578 .
    Abstract
    In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn--Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases.

  • P. Colli, G. Gilardi, J. Sprekels, Longtime behavior for a generalized Cahn--Hilliard system with fractional operators, Atti della Accademia Peloritana dei Pericolanti. Classe di Scienze, Fisiche, Matematiche e Naturali. AAPP. Physical, Mathematical, and Natural Sciences, 98 (2020), pp. A4/1--A4/18, DOI 10.1478/AAPP.98S2A4 .
    Abstract
    In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn--Hilliard system, with possibly singular potentials, which we recently investigated in the paper "Well-posedness and regularity for a generalized fractional CahnHilliard system". More precisely, we give a complete characterization of the Omega-limit of the phase parameter. The characterization depends on the first eigenvalue of one of the involved operators: if this eigenvalue is positive, then the chemical potential vanishes at infinity, and every element of the Omega-limit is a stationary solution to the phase equation; if it is zero instead, then every element of the Omega-limit solves a problem containing a real function which is related to the chemical potential. Such a function is nonunique and time dependent, in general, as we show by means of an example; however, we give sufficient conditions for it to be uniquely determined and constant.

  • D.H. Doan, A. Fischer, J. Fuhrmann, A. Glitzky, M. Liero, Drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices, Journal of Computational Electronics, 19 (2020), pp. 1164--1174, DOI 10.1007/s10825-020-01505-6 .
    Abstract
    We present an electrothermal drift-diffusion model for organic semiconductor devices with Gauss-Fermi statistics and positive temperature feedback for the charge carrier mobilities. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and discretize the system by a finite volume based generalized Scharfetter-Gummel scheme. Using path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, which were only recently observed experimentally.

  • B. Franchi, M. Heida, S. Lorenzani, A mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation, Communications in Mathematical Sciences, 18 (2020), pp. 1105--1134, DOI 10.4310/CMS.2020.v18.n4.a10 .
    Abstract
    In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes.

  • TH. Frenzel, M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDP-convergence, Discrete and Continuous Dynamical Systems -- Series S, 14 (2021), pp. 395--425 (published online in May 2020), DOI 10.3934/dcdss.2020345 .
    Abstract
    The notion of Energy-Dissipation-Principle convergence (EDP-convergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker--Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure. The EDP-convergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelin--de Donder kinetics.

  • A. Kirch, A. Fischer, M. Liero, J. Fuhrmann, A. Glitzky, S. Reineke, Experimental proof of Joule heating-induced switched-back regions in OLEDs, Light: Science and Applications, 9 (2020), pp. 5/1--5/10, DOI 10.1038/s41377-019-0236-9 .

  • J. Maas, A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, Journal of Statistical Physics, 181 (2020), pp. 2257--2303, DOI 10.1007/s10955-020-02663-4 .
    Abstract
    We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary Γ-convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.

  • H. Meinlschmidt, Ch. Meyer, J. Rehberg, Regularization for optimal control problems associated to nonlinear evolution equations, Journal of Convex Analysis, 27 (2020), pp. 443--485, DOI 10.20347/WIAS.PREPRINT.2576 .
    Abstract
    It is well-known that in the case of a sufficiently nonlinear general optimal control problem there is very frequently the necessity for a compactness argument in order to pass to the limit in the state equation in the standard “calculus of variations” proof for the existence of optimal controls. For time-dependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochner-type spaces. In this paper, we propose an abstract function space and a suitable regularization- or Tychonov term for the objective functional which allows for the usual standard reasoning in the proof of existence of optimal controls and which admits a reasonably favorable structure in the characterization of optimal solutions via first order necessary conditions in, generally, the form of a variational inequality of obstacle-type in time. We establish the necessary properties of the function space and the Tychonov term and derive the aforementioned variational inequality. The variational inequality can then be reformulated as a projection identity for the optimal control under additional assumptions. We give sufficient conditions on when these are satisfied. The considerations are complemented with a series of practical examples of possible constellations and choices in dependence on the varying control spaces required for the evolution equations at hand.

  • H. Neidhardt, A. Stephan, V.A. Zagrebnov, Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems, Publications of the Research Institute for Mathematical Sciences, 56 (2020), pp. 83--135, DOI 10.4171/PRIMS/56-1-5 .
    Abstract
    In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space Lp(J,X), consisting of X-valued functions on the time-interval J. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in Lp(J,X). We show that the latter also allows to apply a full power of the operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces.

  • O. Souček, M. Heida, J. Málek, On a thermodynamic framework for developing boundary conditions for Korteweg-type fluids, International Journal of Engineering Science, 154 (2020), pp. 103316/1--103316/28, DOI 10.1016/j.ijengsci.2020.103316 .
    Abstract
    We provide a derivation of several classes of boundary conditions for fluids of Korteweg-type using a simple and transparent thermodynamic approach that automatically guarentees that the derived boundary conditions are compatible with the second law of thermodynamics. The starting assumption of our approach is to describe the boundary of the domain as the membrane separating two different continua, one inside the domain, and the other outside the domain. With this viewpoint one may employ the framework of continuum thermodynamics involving singular surfaces. This approach allows us to identify, for various classes of surface Helmholtz free energies, the corresponding surface entropy production mechanisms. By establishing the constitutive relations that guarantee that the surface entropy production is non-negative, we identify a new class of boundary conditions, which on one hand generalizes in a nontrivial manner the Navier's slip boundary conditions, and on the other hand describes dynamic and static contact angle conditions. We explore the general model in detail for a particular case of Korteweg fluid where the Helmholtz free energy in the bulk is that of a van der Waals fluid. We perform a series of numerical experiments to document the basic qualitative features of the novel boundary conditions and their practical applicability to model phenomena such as the contact angle hysteresis.

  • A. Glitzky, M. Liero, G. Nika, Dimension reduction of thermistor models for large-area organic light-emitting diodes, Discrete and Continuous Dynamical Systems -- Series S, (2020), Published online on 28.11.2020, DOI 10.3934/dcdss.2020460 .
    Abstract
    An effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional φ(x)-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter ε > 0 which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the current-flow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the three-dimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective current-flow equation is given by two semilinear equations in the two-dimensional cross-sections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted.

  • TH. Eiter, M. Kyed, Y. Shibata, On periodic solutions for one-phase and two-phase problems of the Navier--Stokes equations, Journal of Evolution Equations, published online on 06.11.2020, DOI 10.1007/s00028-020-00619-5 .
    Abstract
    This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier--Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform.We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal L p-Lq regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of R-solvers developed in Shibata (Diff Int Eqns 27(3-4):313-368, 2014; On the R-bounded solution operators in the study of free boundary problem for the Navier--Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203-285, 2016; Comm Pure Appl Anal 17(4): 1681-1721. https://doi.org/10.3934/cpaa.2018081, 2018; R boundedness, maximal regularity and free boundary problems for the Navier--Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. (R-solvers and their application to periodic L p estimates, Preprint in 2019) for the L p boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.

  • M. Kantner, Th. Koprucki, Beyond just ``flattening the curve'': Optimal control of epidemics with purely non-pharmaceutical interventions, Journal of Mathematics in Industry, 10 (2020), pp. 23/1--23/23, DOI 10.1186/s13362-020-00091-3 .
    Abstract
    When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple ?flattening of the curve?. Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.

  • A. Mielke, A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction systems, Mathematical Models & Methods in Applied Sciences, 30 (2020), pp. 1765--1807, DOI 10.1142/S0218202520500360 .
    Abstract
    We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.

  • A. Mielke, T. Roubíček, Thermoviscoelasticity in Kelvin--Voigt rheology at large strains, Archive for Rational Mechanics and Analysis, 238 (2020), pp. 1--45, DOI 10.1007/s00205-020-01537-z .
    Abstract
    The frame-indifferent thermodynamically-consistent model of thermoviscoelasticity at large strain is formulated in the reference configuration with using the concept of the second-grade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under time-dependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the determinant of the deformation gradient. The heat transfer is governed by the Fourier law in the actual deformed configuration, which leads to a nontrivial description when pulled back into the reference configuration. Existence of weak solutions in the quasistatic setting, i.e. inertial forces are ignored, is shown by time discretization.

Contributions to Collected Editions

  • G. Nika, B. Vernescu, Micro-geometry effects on the nonlinear effective yield strength response of magnetorheological fluids, in: Emerging Problems in the Homogenization of Partial Differential Equations, P. Donato, M. Luna-Laynez, eds., 10 of SEMA SIMAI Springer Series, Springer, Cham, 2021, pp. 1--16, DOI 978-3-030-62030-1_1 .
    Abstract
    We use the novel constitutive model in [15], derived using the homogenization method, to investigate the effect particle chain microstructures have on the properties of the magnetorheological fluid. The model allows to compute the constitutive coefficients for different geometries. Different geometrical realizations of chains can significantly change the magnetorheological effect of the suspension. Numerical simulations suggest that particle size is also important as the increase of the overall particle surface area can lead to a decrease of the overall magnetorheological effect while keeping the volume fraction constant.

  • C. Cancès, C. Chainais-Hillairet, J. Fuhrmann, B. Gaudeul, On four numerical schemes for a unipolar degenerate drift-diffusion model, in: Proceedings of Finite Volumes for Complex Applications IX, Bergen, Norway, June 2020, R. Klöfkorn, F. Radu, E. Keijgavlen, J. Fuhrmann, eds., Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2020, pp. 163--171, DOI 10.1007/978-3-030-43651-3_13 .

  • S. Schulz, D. Chaudhuri, M. O'Donovan, S. Patra, T. Streckenbach, P. Farrell, O. Marquardt, Th. Koprucki, Multi-scale modeling of electronic, optical, and transport properties of III-N alloys and heterostructures, in: Proceedings Physics and Simulation of Optoelectronic Devices XXVIII, 11274, San Francisco, California, USA, 2020, pp. 416--426, DOI 10.1117/12.2551055 .

  • J. Fuhrmann, D.H. Doan, A. Glitzky, M. Liero, G. Nika, Unipolar drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices, in: Proceedings of ``Finite Volumes for Complex Applications IX'', R. Klöfkorn, E. Keilegavlen, F.A. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2020, pp. 625--633, DOI 10.1007/978-3-030-43651-3_59 .
    Abstract
    We discretize a unipolar electrothermal drift-diffusion model for organic semiconductor devices with Gauss--Fermi statistics and charge carrier mobilities having positive temperature feedback. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and use a finite volume based generalized Scharfetter-Gummel scheme. Applying path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, only recently observed experimentally.

  • M. Kantner, Th. Koprucki, Non-isothermal Scharfetter--Gummel scheme for electro-thermal transport simulation in degenerate semiconductors, in: Proceedings of ``Finite Volumes for Complex Applications IX'', R. Klöfkorn, E. Keilegavlen, F.A. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2020, pp. 173--182, DOI 10.1007/978-3-030-43651-3_14 .
    Abstract
    Electro-thermal transport phenomena in semiconductors are described by the non-isothermal drift-diffusion system. The equations take a remarkably simple form when assuming the Kelvin formula for the thermopower. We present a novel, non-isothermal generalization of the Scharfetter--Gummel finite volume discretization for degenerate semiconductors obeying Fermi--Dirac statistics, which preserves numerous structural properties of the continuous model on the discrete level. The approach is demonstrated by 2D simulations of a heterojunction bipolar transistor.

Preprints, Reports, Technical Reports

  • G. Nika, Derivation of effective models from heterogenous Cosserat media via periodic unfolding, Preprint no. 2817, WIAS, Berlin, 2021.
    Abstract, PDF (285 kByte)
    We derive two different effective models from a heterogeneous Cosserat continuum taking into account the Cosserat intrinsic length of the constituents. We pass to the limit using homogenization via periodic unfolding and in doing so we provide rigorous proof to the results introduced by Forest, Pradel, and Sab (Int. J. Solids Structures 38 (26-27): 4585-4608 '01). Depending on how different characteristic lengths of the domain scale with respect to the Cosserat intrinsic length, we obtain either an effective classical Cauchy continuum or an effective Cosserat continuum. Moreover, we provide some corrector type results for each case.

  • K. Hopf, Weak-strong uniqueness for energy-reaction-diffusion systems, Preprint no. 2808, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2808 .
    Abstract, PDF (444 kByte)
    We establish weak-strong uniqueness and stability properties of renormalised solutions to a class of energy-reaction-diffusion systems, which genuinely feature cross-diffusion effects. The systems considered are motivated by thermodynamically consistent models, and their formal entropy structure allows us to use as a key tool a suitably adjusted relative entropy method. Weak-strong uniqueness is obtained for general entropy-dissipating reactions without growth restrictions, and certain models with a non-integrable diffusive flux. The results also apply to a class of (isoenergetic) reaction-cross-diffusion systems.

  • J. Fischer, K. Hopf, M. Kniely, A. Mielke, Global existence analysis of energy-reaction-diffusion systems, Preprint no. 2807, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2807 .
    Abstract, PDF (489 kByte)
    We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.

  • G. Nika, An existence result for a class of nonlinear magnetorheological composites, Preprint no. 2804, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2804 .
    Abstract, PDF (257 kByte)
    We prove existence of a weak solution for a nonlinear, multi-physics, multi-scale problem of magnetorheological suspensions introduced in Nika & Vernescu (Z. Angew. Math. Phys., 71(1):1--19, '20). The hybrid model couples the Stokes' equation with the quasi-static Maxwell's equations through the Lorentz force and the Maxwell stress tensor. The proof of existence is based on: i) the augmented variational formulation of Maxwell's equations, ii) the definition of a new function space for the magnetic induction and the proof of a Poincaré type inequality, iii) the Altman--Shinbrot fixed point theorem when the magnetic Reynold's number, Rm, is small.

  • TH. Koprucki, A. Maltsi, A. Mielke, On the Darwin--Howie--Whelan equations for the scattering of fast electrons described by the Schrödinger equation, Preprint no. 2801, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2801 .
    Abstract, PDF (4221 kByte)
    The Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrödinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the two-beam and the systematic-row approximation.

  • A. Stephan, EDP-convergence for a linear reaction-diffusion system with fast reversible reaction, Preprint no. 2793, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2793 .
    Abstract, PDF (422 kByte)
    We perform a fast-reaction limit for a linear reaction-diffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reaction-diffusion system as a gradient flow of the free energy in the space of probability measures equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and cosh-type functions for the reaction part. The fast-reaction limit is done on the level of the gradient structure by proving EDP-convergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slow-manifold. Moreover, the limit gradient system can be equivalently described by a coarse-grained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarse-grained slow variable.

  • TH. Eiter, G.P. Galdi, Spatial decay of the vorticity field of time-periodic viscous flow past a body, Preprint no. 2791, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2791 .
    Abstract, PDF (297 kByte)
    We study the asymptotic spatial behavior of the vorticity field associated to a time-periodic Navier-Stokes flow past a body in the class of weak solutions satisfying a Serrin-like condition. We show that outside the wake region the vorticity field decays pointwise at an exponential rate, uniformly in time. Moreover, decomposing it into its time-average over a period and a so-called purely periodic part, we prove that inside the wake region, the time-average has the same algebraic decay as that known for the associated steady-state problem, whereas the purely periodic part decays even faster, uniformly in time. This implies, in particular, that “sufficiently far” from the body, the time-periodic vorticity field behaves like the vorticity field of the corresponding steady-state problem.

  • A. Mielke, M.A. Peletier, A. Stephan, EDP-convergence for nonlinear fast-slow reaction systems with detailed balance, Preprint no. 2781, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2781 .
    Abstract, PDF (897 kByte)
    We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.

  • K.M. Gambaryan, O. Marquardt, T. Boeck, A. Trampert, Micro- and nano-scale engineering and structures shape architecture at nucleation from In-As-Sb-P composition liquid phase on an InAs(100) surface, Preprint no. 2775, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2775 .
    Abstract, PDF (4287 kByte)
    In this review paper we present results of the growth, characterization and electronic properties of In(As,Sb,P) composition strain-induced micro- and nanostructures. Nucleation is performed from In-As-Sb-P quaternary composition liquid phase in Stranski--Krastanow growth mode using steady-state liquid phase epitaxy. Growth features and the shape transformation of pyramidal islands, lens-shape and ellipsoidal type-II quantum dots (QDs), quantum rings and QD-molecules are under consideration. It is shown that the application of a quaternary In(As,Sb,P) composition wetting layer allows not only more flexible control of lattice-mismatch between the wetting layer and an InAs(100) substrate, but also opens up new possibilities for nanoscale engineering and nanoarchitecture of several types of nanostructures. HR-SEM, AFM, TEM and STM are used for nanostructure characterization. Optoelectronic properties of the grown structures are investigated by FTIR and photoresponse spectra measurements. Using an eight-band $mathbfkcdotmathbfp$ model taking strain and built-in electrostatic potentials into account, the electronic properties of a wide range of InAs$_1-x-y$Sb$_x$P$_y$ QDs and QD-molecules are computed. Two types of QDs mid-infrared photodetectors are fabricated and investigated. It is shown that the incorporation of QDs allows to improve some output device characteristics, in particularly sensitivity, and to broaden the spectral range.

  • O. Marquardt, Simulating the electronic properties of semiconductor nanostructures using multiband $kcdot p$ models, Preprint no. 2773, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2773 .
    Abstract, PDF (706 kByte)
    The eight-band $kcdot p$ formalism been successfully applied to compute the electronic properties of a wide range of semiconductor nanostructures in the past and can be considered the backbone of modern semiconductor heterostructure modelling. However, emerging novel material systems and heterostructure fabrication techniques raise questions that cannot be answered using this well-established formalism, due to its intrinsic limitations. The present article reviews recent studies on the calculation of electronic properties of semiconductor nanostructures using a generalized multiband $kcdot p$ approach that allows both the application of the eight-band model as well as more sophisticated approaches for novel material systems and heterostructures.

  • A. Mielke, Relating a rate-independent system and a gradient system for the case of one-homogeneous potentials, Preprint no. 2771, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2771 .
    Abstract, PDF (342 kByte)
    We consider a non-negative and one-homogeneous energy functional $mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-inpendent system given in terms of the time-dependent functional $mathcal E(t,u)=t mathcal J(u)$ and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutins of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

  • P. Colli, A. Signori, J. Sprekels, Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis, Preprint no. 2770, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2770 .
    Abstract, PDF (451 kByte)
    This paper concerns a distributed optimal control problem for a tumor growth model of Cahn--Hilliard type including chemotaxis with possibly singular anpotentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

  • A. Glitzky, M. Liero, G. Nika, An effective bulk-surface thermistor model for large-area organic light-emitting diodes, Preprint no. 2757, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2757 .
    Abstract, PDF (315 kByte)
    The existence of a weak solution for an effective system of partial differential equations describing the electrothermal behavior of large-area organic light-emitting diodes (OLEDs) is proved. The effective system consists of the heat equation in the three-dimensional bulk glass substrate and two semi-linear equations for the current flow through the electrodes coupled to algebraic equations for the continuity of the electrical fluxes through the organic layers. The electrical problem is formulated on the (curvilinear) surface of the glass substrate where the OLED is mounted. The source terms in the heat equation are due to Joule heating and are hence concentrated on the part of the boundary where the current-flow equation is posed. The existence of weak solutions to the effective system is proved via Schauder's fixed-point theorem. Moreover, since the heat sources are a priori only in $L^1$, the concept of entropy solutions is used.

  • D. Bothe, P.-É. Druet, On the structure of continuum thermodynamical diffusion fluxes -- A novel closure scheme and its relation to the Maxwell--Stefan and the Fick--Onsager approach, Preprint no. 2749, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2749 .
    Abstract, PDF (439 kByte)
    This paper revisits the modeling of multicomponent diffusion within the framework of thermodynamics of irreversible processes. We briefly review the two well-known main approaches, leading to the generalized Fick--Onsager multicomponent diffusion fluxes or to the generalized Maxwell--Stefan equations. The latter approach has the advantage that the resulting fluxes are consistent with non-negativity of the partial mass densities for non-singular and non-degenerate Maxwell--Stefan diffusivities. On the other hand, this approach requires computationally expensive matrix inversions since the fluxes are only implicitly given. We propose and discuss a novel and more direct closure which avoids the inversion of the Maxwell--Stefan equations. It is shown that all three closures are actually equivalent under the natural requirement of positivity for the concentrations, thus revealing the general structure of continuum thermodynamical diffusion fluxes.

  • M. Kantner, Th. Koprucki, Beyond just "flattening the curve": Optimal control of epidemics with purely non-pharmaceutical interventions, Preprint no. 2748, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2748 .
    Abstract, PDF (3116 kByte)
    When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple "flattening of the curve". Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.

  • M. Heida, Stochastic homogenization on randomly perforated domains, Preprint no. 2742, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2742 .
    Abstract, PDF (1175 kByte)
    We study the existence of uniformly bounded extension and trace operators for W1,p-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions on the geometry which we call local (δ, M)-regularity, isotropic cone mixing and bounded average connectivity. The first concept measures local Lipschitz regularity of the domain while the second measures the mesoscopic distribution of void space. The third is the most tricky part and measures the ”mesoscopic” connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process. We finally introduce suitable Sobolev spaces on Rd and Ω in order to construct a stochastic two-scale convergence method and apply the resulting theory to the homogenization of a p-Laplace problem on a randomly perforated domain.

  • P. Colli, G. Gilardi, J. Sprekels, An asymptotic analysis for a generalized Cahn--Hilliard system with fractional operators, Preprint no. 2741, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2741 .
    Abstract, PDF (311 kByte)
    In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous Cahn--Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers in the spectral sense of general linear operators, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space of square-integrable functions on a bounded and smooth three-dimensional domain, and have compact resolvents. Here, for the case of the viscous system, we analyze the asymptotic behavior of the solution as the fractional power coefficient of the second operator tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of the second operator appears.

  • TH. Eiter, On the spatially asymptotic structure of time-periodic solutions to the Navier--Stokes equations, Preprint no. 2727, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2727 .
    Abstract, PDF (258 kByte)
    The asymptotic behavior of weak time-periodic solutions to the Navier--Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.

  • P. Colli, G. Gilardi, J. Sprekels, Optimal control of a phase field system of Caginalp type with fractional operators, Preprint no. 2725, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2725 .
    Abstract, PDF (360 kByte)
    In their recent work “Well-posedness, regularity and asymptotic analyses for a fractional phase field system” (Asymptot. Anal. 114 (2019), 93--128), two of the present authors have studied phase field systems of Caginalp type, which model nonconserved, nonisothermal phase transitions and in which the occurring diffusional operators are given by fractional versions in the spectral sense of unbounded, monotone, selfadjoint, linear operators having compact resolvents. In this paper, we complement this analysis by investigating distributed optimal control problems for such systems. It is shown that the associated control-to-state operator is Fréchet differentiable between suitable Banach spaces, and meaningful first-order necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables.

  • J. Sprekels, F. Tröltzsch, Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth, Preprint no. 2721, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2721 .
    Abstract, PDF (326 kByte)
    In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L1-norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.

  • D. Bothe, P.-É. Druet, Well-posedness analysis of multicomponent incompressible flow models, Preprint no. 2720, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2720 .
    Abstract, PDF (465 kByte)
    In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities of the species stays constant. In this type of models, non solenoidal effects affect the velocity field in the Navier--Stokes equations and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.

  • A. Glitzky, M. Liero, G. Nika, Dimension reduction of thermistor models for large-area organic light-emitting diodes, Preprint no. 2719, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2719 .
    Abstract, PDF (328 kByte)
    An effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional φ(x)-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter ε > 0 which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the current-flow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the three-dimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective current-flow equation is given by two semilinear equations in the two-dimensional cross-sections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted.

  • A. Mielke, R.R. Netz, S. Zendehroud, A rigorous derivation and energetics of a wave equation with fractional damping, Preprint no. 2718, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2718 .
    Abstract, PDF (312 kByte)
    We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water-air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally-damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy-dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally-damped wave equation with a time derivative of order 3/2.

  • P.-É. Druet, A theory of generalised solutions for ideal gas mixtures with Maxwell--Stefan diffusion, Preprint no. 2700, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2700 .
    Abstract, PDF (363 kByte)
    After the pioneering work by Giovangigli on mathematics of multicomponent flows, several attempts were made to introduce global weak solutions for the PDEs describing the dynamics of fluid mixtures. While the incompressible case with constant density was enlighted well enough due to results by Chen and Jüngel (isothermal case), or Marion and Temam, some open questions remain for the weak solution theory of gas mixtures with their corresponding equations of mixed parabolic-hyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the Bresch-Desjardins technique and a regularisation of pressure preventing vacuum. The result by Dreyer, Druet, Gajewski and Guhlke avoids emphex machina stabilisations, but the mathematical assumption that the Onsager matrix is uniformly positive on certain subspaces leads, in the dilute limit, to infinite diffusion velocities which are not compatible with the Maxwell-Stefan form of diffusion fluxes. In this paper, we prove the existence of global weak solutions for isothermal and ideal compressible mixtures with natural diffusion. The main new tool is an asymptotic condition imposed at low pressure on the binary Maxwell-Stefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime.

  • M. Heida, M. Kantner, A. Stephan, Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck operator, Preprint no. 2684, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2684 .
    Abstract, PDF (2719 kByte)
    We introduce a family of various finite volume discretization schemes for the Fokker--Planck operator, which are characterized by different weight functions on the edges. This family particularly includes the well-established Scharfetter--Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the quality of the weights. We show by numerical experiments that for small gradients the choice of the optimal representative of the discretization family is highly non-trivial while for large gradients the Scharfetter--Gummel scheme stands out compared to the others.

Talks, Poster

  • A. Maltsi, Quantum dots and TEM images from a mathematician's perspective, Women in Mathematics Webinar (online participation), UK, February 11 - 12, 2021.

  • P. Pelech, Separately global solutions to rate-independent systems - applications to large-strain deformations of damageable solids (online talk), 20th GAMM Seminar on Microstructures (Online Event), Technische Universität Wien, Austria, January 29, 2021.

  • P.-É. Druet, The free energy of incompressible fluid mixtures: An asymptotic study (online talk), TES-Seminar on Energy-based Mathematical Methods and Thermodynamics, Technische Universität Berlin WIAS Berlin, January 21, 2021.

  • M. Heida, A. Mielke, A. Stephan, Effective models for materials and interfaces with multiple scales, SCCS Days 2020 of the Collaborative Research Center - CRC 1114 "Scaling Cascades in Complex Systems" (Online event), December 2 - 4, 2020.

  • M. Heida, Stochastic homogenization in randomly perforated domains (online talk), Annual Workshop of the GAMM Activity Group on Analysis of PDEs (Online Event), September 30 - October 2, 2020, Institute of Science and Technology Austria (IST Austria), Klosterneuburg, October 2, 2020.

  • M. Heida, Stochastic homogenization on perforated domains, Friedrich-Alexander Universität Erlangen-Nürnberg, Department Mathematik, July 2, 2020.

  • M. Heida, Stochastic homogenization on perforated domains (online talk), MA4M: Mathematical Analysis for Mechanics (Online Event), November 23 - 25, 2020, WIAS Berlin, November 24, 2020.

  • O. Marquardt, Data-driven electronic structure calculations for semiconductor nanostructures, Efficient algorithms for numerical problems - Workshop on the occasion of the retirement of Peter Mathé, January 17, 2020, WIAS Berlin, January 17, 2020.

  • O. Marquardt, Electronic properties of semiconductor heterostructures using SPHInX (online tutorial), NUSOD 2020: 20th International Conference on Numerical Simulation of Optoelectronic Devices (Online Event), September 14 - 25, 2020, Politecnico di Torino, September 14, 2020.

  • O. Marquardt, Nucleation chronology and electronic properties of In(As,Sb,P) graded-composition quantum dots (online talk), NUSOD 2020: 20th International Conference on Numerical Simulation of Optoelectronic Devices (Online Event), September 14 - 25, 2020, Politecnico di Torino, September 14, 2020.

  • O. Marquardt, Nucleation chronology and electronic properties of In(As,Sb,P) graded-composition quantum dots (online talk), CMD2020GEFES (Online Event), August 31 - September 4, 2020, European Physical Society & La Real Sociedad Española de Física, September 4, 2020.

  • O. Marquardt, Th. Koprucki, A. Mielke, DESCANT - Data-driven electronic structure calculations for semiconductor nanostructures (online poster session), MATH+ Day 2020 (Online Event), November 6, 2020.

  • O. Marquardt, Semiconductor nanostructures (online talk), IKZ-WIAS Workshop (Online Event), October 30, 2020, WIAS Berlin, IKZ Berlin, October 30, 2020.

  • G. Nika, An existence result for a class of electrothermal drift-diffusion models with Fermi--Gauss statistics for organic semiconductors, Joint Mathematics Meeting, January 15 - 18, 2020, American Mathematical Society/ Mathematical Association of America, Denver, USA, January 15, 2020.

  • P. Pelech, Separately global solutions to rate-independent systems - applications to large-strain deformations of damageable solids (online talk), MA4M: Mathematical Analysis for Mechanics (Online Event), November 23 - 25, 2020, WIAS Berlin, November 23, 2020.

  • P. Pelech, Separately global solutions to rate-independent systems: Applications to large-strain deformations of damageable solids, Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (Online Event), October 12 - 23, 2020.

  • P. Pelech, Separately global solutions to rate-independent systems: Applications to large-strain deformations of damageable solids (online talk), Thematic Einstein Semester: Kick-off Conference (Online Event), October 26 - 30, 2020, WIAS Berlin, October 29, 2020.

  • A. Stephan, On gradient flows and gradient systems (online talk), CRC 1114 PhD Seminar (Online Event), Freie Universität Berlin, November 11, 2020.

  • A. Stephan, On gradient systems and applications to interacting particle systems (online talk), CRC 1114 PhD Seminar (Online Event), Freie Universität Berlin, November 25, 2020.

  • A. Stephan, Coarse-graining for gradient systems with applications to reaction systems (online talk), Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (Online Event), October 12 - 23, 2020, WIAS Berlin, October 15, 2020.

  • A. Stephan, EDP-convergence for nonlinear fast-slow reaction systems (online talk), Annual Workshop of the GAMM Activity Group on Analysis of PDEs (Online Event), September 30 - October 2, 2020, Institute of Science and Technology Austria (IST Austria), Klosterneuburg, October 1, 2020.

  • A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction systems, Seminar ``Applied Analysis'', Eindhoven University of Technology, Centre for Analysis, Scientific Computing, and Applications - Mathematics and Computer Science, Netherlands, January 20, 2020.

  • A. Stephan, EDP-convergence for nonlinear fast-slow reactions, Variational Methods for Evolution, September 13 - 19, 2020, Mathematisches Forschungszentrum Oberwolfach, September 18, 2020.

  • A. Stephan, On mathematical coarse-graining for linear reaction systems, 8th BMS Student Conference, February 19 - 21, 2020, Technische Universität Berlin, February 21, 2020.

  • A. Glitzky, A hybrid model for the electrothermal behaviour of semiconductor devices (online talk), Annual Workshop of the GAMM Activity Group on Analysis of PDEs (Online Event), September 30 - October 2, 2020, Institute of Science and Technology Austria (IST Austria), Klosterneuburg, October 1, 2020.

  • K. Hopf, Global existence analysis of energy-reaction-diffusion systems, Variational Methods for Evolution, September 13 - 19, 2020, Mathematisches Forschungszentrum Oberwolfach, September 15, 2020.

  • TH. Eiter, Spatially asymptotic structure of time-periodic Navier--Stokes flows (online talk), MA4M: Mathematical Analysis for Mechanics (Online Event), November 23 - 25, 2020, WIAS Berlin, November 24, 2020.

  • TH. Eiter , Spatially asymptotic structure of time-periodic Navier--Stokes flows (online talk), Annual Workshop of the GAMM Activity Group on Analysis of PDEs (Online Event), September 30 - October 2, 2020, Institute of Science and Technology Austria (IST Austria), Klosterneuburg, October 1, 2020.

  • J. Fuhrmann, D.H. Doan, A. Glitzky, M. Liero, G. Nika, Unipolar drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices, Finite Volumes for Complex Applications IX (Online Event), Bergen, Norway, June 15 - 19, 2020.

  • TH. Koprucki, The Mathematical Research Data Initiative (MaRDI) for the National Research Data Infrastructure (online talk), DMV Jahrestagung 2020 (Online Event), minisymposium on ``A Research Infrastructure Tailored for Mathematics in the Digital Age", September 14 - 17, 2020, Technische Universität Chemnitz, September 15, 2020.

  • TH. Koprucki, K. Tabelow, T. Streckenbach, T. Niermann, A. Maltsi, Model-based geometry reconstruction of TEM images (online poster session), MATH+ Day 2020 (Online Event), November 6, 2020.

  • M. Liero, A. Mielke, Analysis for thermo-mechanical models with internal variables, Vorstellung der Projektanträge im SPP 2256, Bad Honnef, January 30, 2020.

  • M. Liero, Drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices (online talk), SimOEP 2020: International Conference on Simulation of Organic Electronics and Photovoltaics (Online Event), August 31 - September 2, 2020, Zürcher Hochschule für Angewandte Wissenschaften, Switzerland, September 1, 2020.

  • M. Liero, Evolutionary Gamma-convergence for multiscale problems (online talks), Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (Online Event), October 12 - 23, 2020, WIAS Berlin, October 15, 2020.

  • A. Mielke, Finite-strain viscoelasticity with temperature coupling, Calculus of Variations and Applications, January 27 - February 1, 2020, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, January 28, 2020.

  • A. Mielke, Gradient systems and evolutionary Gamma-convergence (online talk), Oberseminar ``Mathematik in den Naturwissenschaften'' (Online Event), Julius-Maximilians-Universität Würzburg, June 5, 2020.

  • A. Mielke, On finite-strain thermo-viscoelasticity, Mechanics of Materials: Towards Predictive Methods for Kinetics in Plasticity, Fracture, and Damage, March 8 - 14, 2020, Mathematisches Forschungszentrum Oberwolfach, March 12, 2020.

  • A. Mielke, Differential equations as gradient flows, with applications in mechanics, stochastics, and chemistry (online talk), Würzburger Mathematisches Kolloquium (Online Event), Julius-Maximilians-Universität Würzburg, November 9, 2020.

  • A. Mielke, EDP-convergence for multiscale gradient systems with applications to fast-slow reaction systems (online talk), One World Dynamics Seminar (Online Event), Technische Universität München, November 13, 2020.

  • A. Mielke, Global existence for finite-strain viscoelasticity with temperature coupling (online talk), One World Dynamics Seminar (Online Event), University of Bath, UK, December 1, 2020.

  • A. Mielke, Similarity solutions for Kolmogorov's two-equation model for turbulence, Workshop on Control of Self-Organizing Nonlinear Systems, September 2 - 3, 2020, Technische Universität Berlin, September 2, 2020.

  • A. Mielke, Variational structures for the analysis of PDE systems (online talks), Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (Online Event), October 12 - 23, 2020, WIAS Berlin, October 13, 2020.

  • J. Rehberg, Explicit and uniform estimates for second order divergence operators on Lp spaces, Oberseminar ``Analysis und Theoretische Physik'', Leibniz Universität Hannover, Institut für Angewandte Mathematik, January 28, 2020.

External Preprints

  • J. Fischer , K. Hopf , M. Kniely, A. Mielke, Global existence analysis of energy-reaction-diffusion systems, Preprint no. arXiv:2012.03792, Cornell University, 2020.
    Abstract
    We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.