Publications
Monographs

O. Marquardt, Simulating the electronic properties of semiconductor nanostructures using multiband $kcdot p$ models, S. Sinnott, ed., 194 of Computational Materials Science, Elsevier, Amsterdam, 2021, pp. 111, (Monograph Published), DOI 10.1016/j.commatsci.2021.110318 .
Abstract
The eightband $mathbfkcdotmathbfp$ formalism has 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 wellestablished 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 $mathbfkcdotmathbfp$ approach that allows both the application of the eightband model as well as more sophisticated approaches for novel material systems and heterostructures. 
M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, 453 pages, (Collection Published).

U.W. Pohl, A. Strittmatter, A. Schliwa, M. Lehmann, T. Niermann, T. Heindel, S. Reitzenstein, M. Kantner, U. Bandelow, Th. Koprucki, H.J. Wünsche, Chapter 3: StressorInduced Site Control of Quantum Dots for SinglePhoton Sources, in Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in SolidState Sciences, Springer, Heidelberg, 2020, pp. 5390, (Chapter Published), DOI 10.1007/9783030356569_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 selfaligned growth of single quantum dots in the center of a circular mesa. Measurements of the strain tensor applying transmissionelectron holography yield excellent agreement with the calculated strain field. Singledot spectroscopy of sitecontrolled dots proves narrow excitonic linewidth virtually free of spectral diffusion due to quantumdot growth in a defectfree matrix. Implementation of such dots in an electrically driven pin structure yields singledot electroluminescence. Singlephoton 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 cariertransport 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, Chapter 8: Deterministic Quantum Devices for Optical Quantum Communication, in: Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in SolidState Sciences, Springer, Heidelberg, 2020, pp. 285359, (Chapter Published), DOI 10.1007/9783030356569_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 nocloning theorem. Thus, all optical quantum communication networks rely crucially on the availability of suitable quantumlight sources. Such light sources with close to ideal optical and quantum optical properties can be realized by selfassembled semiconductor quantum dots. These highquality nanocrystals are predestined singlephoton emitters due to their quasi zerodimensional carrier confinement. Still, the development of practical quantumdotbased sources of single photons and entangledphoton pairs for applications in photonic quantum technology and especially for the quantumrepeater 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 photonextraction efficiency in simple planar device configurations. 
TH. Eiter, Existence and Spatial Decay of Periodic NavierStokes Flows in Exterior Domains, Th. Eiter, ed., Logos Verlag Berlin GmbH, 2020, 197 pages, (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 timeperiodic solutions to the associated NavierStokes equations when the body performs a nontrivial translation. The first part of the thesis is concerned with the question of existence of timeperiodic solutions in the case of a nonrotating 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 timeperiodic solutions. For this purpose, timeperiodic fundamental solutions to the Stokes and Oseen linearizations are introduced and investigated, and the concept of a timeperiodic fundamental solution for the vorticity field is developed. With these results, new pointwise estimates of the velocity and the vorticity field associated to a timeperiodic fluid flow are derived. 
M. Kantner, Th. Höhne, Th. Koprucki, S. Burger, H.J. Wünsche, F. Schmidt, A. Mielke, U. Bandelow, Chapter 7: MultiDimensional Modeling and Simulation of Semiconductor Nanophotonic Devices, in: Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in SolidState Sciences, Springer, Heidelberg, 2020, pp. 241283, (Chapter Published), DOI 10.1007/9783030356569_7 .
Abstract
Selfconsistent modeling and multidimensional 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 bottlenecks in new device concepts, contribute to their understanding and help to theoretically exploretheir optimization potential. The efficient implementation of multidimensional numerical simulationsfor computeraided design tasks requires sophisticated numerical methods and modeling techniques. We review recent advances in devicescale modeling of quantum dot based singlephotonsources and laser diodes by selfconsistently coupling the optical Maxwell equations with semiclassical carrier transport models using semiclassical and fully quantum mechanical descriptionsof the optically active region, respectively. For the simulation of realistic devices with complex,multidimensional geometries, we have developed a novel hpadaptive finite element approachfor the optical Maxwell equations, using mixed meshes adapted to the multiscale properties ofthe photonic structures. For electrically driven devices, we introduced novel discretization andparameterembedding techniques to solve the driftdiffusion system for strongly degenerate semiconductors at cryogenic temperature. Our methodical advances are demonstrated on variousapplications, including verticalcavity surfaceemitting lasers, grating couplers and singlephotonsources.
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 IIInitride quantum wells with varied Indium content: Connecting atomistic and continuumbased models, Journal of Applied Physics, 129 (2021), published online on 18.02.2021, DOI 10.1063/5.0031514 .

P. Colli, G. Gilardi, J. Sprekels, An asymptotic analysis for a generalized CahnHilliard system with fractional operators, Journal of Evolution Equations, 21 (2021), pp. 2749  2778, DOI 10.1007/s00028021007061 .
Abstract
In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous CahnHilliard systems of two operator equations in which nonlinearities of doublewell 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 squareintegrable functions on a bounded and smooth threedimensional 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. 
P. Colli, A. Signori, J. Sprekels, Secondorder analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis, ESAIM. Control, Optimisation and Calculus of Variations, 27 (2021), pp. 73/173/46 (Published online on 13.07.2021), DOI 10.1051/cocv/2021072 .
Abstract
This paper concerns a distributed optimal control problem for a tumor growth model of CahnHilliard type including chemotaxis with possibly singular anpotentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak wellposedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong wellposedness 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 firstorder necessary and secondorder sufficient conditions for optimality. The mathematically challenging secondorder 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 secondorder Fréchet derivative of the controltostate operator and carry out a thorough and detailed investigation about the related properties. 
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 (375404), 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 Sobolevtype spaces of negative order X^{s1,q}_{D}(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 localintime existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs. 
J. Sprekels, F. Tröltzsch, Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth, ESAIM. Control, Optimisation and Calculus of Variations, 27 (2021), pp. S26/1S26/27, DOI 10.1051/cocv/2020088 .
Abstract
In this paper, we study an optimal control problem for a nonlinear system of reactiondiffusion 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 extracellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a doublewell 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 L^{1}norm. For such problems, we derive firstorder 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 controltostate 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 secondorder sufficient optimality conditions. 
A. Glitzky, M. Liero, G. Nika, An existence result for a class of electrothermal driftdiffusion models with GaussFermi statistics for organic semiconductors, Analysis and Applications, 19 (2021), pp. 275304, DOI 10.1142/S0219530519500246 .
Abstract
This work is concerned with the analysis of a driftdiffusion 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 GaussFermi 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 selfheating 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 driftdiffusion model for organic semiconductors on a sound analytical basis. 
TH. Eiter, On the spatially asymptotic structure of timeperiodic solutions to the NavierStokes equations, Probability Surveys, 149 (2021), pp. 34393451, DOI 10.1090/proc/15482 .
Abstract
The asymptotic behavior of weak timeperiodic solutions to the NavierStokes equations with a drift term in the threedimensional whole space is investigated. The velocity field is decomposed into a timeindependent 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. 
TH. Eiter, G.P. Galdi, Spatial decay of the vorticity field of timeperiodic viscous flow past a body, Archive for Rational Mechanics and Analysis, pp. published online on 06.07.2021 (130), DOI 10.1007/s0020502101690z .
Abstract
We study the asymptotic spatial behavior of the vorticity field associated to a timeperiodic NavierStokes flow past a body in the class of weak solutions satisfying a Serrinlike 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 timeaverage over a period and a socalled purely periodic part, we prove that inside the wake region, the timeaverage has the same algebraic decay as that known for the associated steadystate problem, whereas the purely periodic part decays even faster, uniformly in time. This implies, in particular, that “sufficiently far” from the body, the timeperiodic vorticity field behaves like the vorticity field of the corresponding steadystate problem. 
TH. Eiter, M. Kyed, Viscous flow around a rigid body performing a timeperiodic motion, Journal of Mathematical Fluid Mechanics, (2021), pp. Published online on 11.02.2021 (28/128/23), DOI 10.1007/s00021021005564 .
Abstract
The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed timeperiodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigidbody motion are compatible, and that the mean translational velocity is nonzero, existence of a timeperiodic 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 illposed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces. 
A. Mielke, Relating a rateindependent system and a gradient system for the case of onehomogeneous potentials, Journal of Dynamics and Differential Equations, pp. published online on 31.05.2021 (122), DOI 10.1007/s10884021100073 .
Abstract
We consider a nonnegative and onehomogeneous energy functional $mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradientflow equations and the energetic solutions generated via the rateinpendent system given in terms of the timedependent 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 solutiondependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the totalvariation 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 constantspeed intervals for the solutins of the gradientflow equation. As a major result we obtain a nontrivial existence and uniqueness result for the energetic rateindependent system. 
A. Mielke, A. Montefusco, M.A. Peletier, Exploring families of energydissipation landscapes via tilting: three types of EDP convergence, Continuum Mechanics and Thermodynamics, 33 (2021), pp. 611637, DOI 10.1007/s0016102000932x .
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 wellknown 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 EnergyDissipation Principle that provides a variational formulation to gradientflow equations that allows one to apply techniques from Γconvergence of functional on states and functionals on trajectories. 
A. Mielke, R.R. Netz, S. Zendehroud, A rigorous derivation and energetics of a wave equation with fractional damping, Journal of Evolution Equations, (2021), pp. published online in Apr. 2021(124).
Abstract
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 waterair 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 fractionallydamped 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 energydissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionallydamped wave equation with a time derivative of order 3/2. 
A. Mielke, M.A. Peletier, A. Stephan, EDPconvergence for nonlinear fastslow reaction systems with detailed balance, Nonlinearity, 34 (2021), pp. 57625798, DOI 10.1088/13616544/ac0a8a .
Abstract
We consider nonlinear reaction systems satisfying massaction kinetics with slow and fast reactions. It is known that the fastreactionrate 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 detailedbalance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a coshtype 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 EnergyDissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy massaction kinetics. 
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. 257/1257/11, DOI 10.1007/s1108202002356y .
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 DarwinHowieWhelan 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 modelbased geometry reconstruction of semiconductor QDs, involving machine learning techniques. 
W. Dreyer, P.É. Druet, P. Gajewski, C. Guhlke, Analysis of improved NernstPlanckPoisson 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. 119/1119/68, DOI 10.1007/s00033020013415 .
Abstract
We consider an improved NernstPlanckPoisson 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 convectiondiffusionreaction equations for the constituents of the mixture, of the NavierStokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, crossdiffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a globalintime 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 crossdiffusion systems for fluid mixtures driven by a pressure gradient, SIAM Journal on Mathematical Analysis, 52 (2020), pp. 21792197, 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 parabolichyperbolic system for the mass densities. The globalintime existence of classical and weak solutions is proved in a bounded domain with nopenetration boundary conditions. The idea is to decompose the system into a porousmediumtype 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. 294314, 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 initiobased electronic structure parameters for wurtzite GaAs, Phys. Rev. B., 101 (2020), pp. 235147/1235147/12, DOI 10.1103/PhysRevB.101.235147 .
Abstract
We develop a 16band 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 lowdiscrepancy sequences to fit k · p band structures beyond the eightband scheme to most recent ab initio data, obtained within the framework for hybridfunctional density functional theory with a screenedexchange hybrid functional. We report structural parameters, elastic constants, band structures along highsymmetry lines, and deformation potentials at the Γ point. Based on this, we compute the bulk electronic properties (Γ point energies, effective masses, Luttingerlike parameters, and optical matrix parameters) for a tenband and a sixteenband 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. 646650, 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 InAsSbP composition liquid phase on an InAs(100) substrate in the StranskiKrastanov 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 eightband k · p model taking strain and builtin electrostatic potentials into account, we have computed the hole ground state energies and charge densities for a wide range of InAs_{1xy}Sb_{x}P_{y} 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 theoryguided design for application in specific devices. 
J.A. Carrillo, K. Hopf, M.Th. Wolfram, Numerical study of BoseEinstein condensation in the KaniadakisQuarati model for bosons, Kinetic and Related Models, 13 (2020), pp. 507529, DOI 10.3934/krm.2020017 .
Abstract
Kaniadakis and Quarati (1994) proposed a FokkerPlanck 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 blowup time while having a linear diffusion term. We present a thoroughly validated timeimplicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the KaniadakisQuarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the KaniadakisQuarati 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 blowup 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 FokkerPlanck model with superlinear drift, Advances in Mathematics, 360 (2020), pp. 106883/1106883/66, DOI 10.1016/j.aim.2019.106883 .
Abstract
We consider a class of FokkerPlanck equations with linear diffusion and superlineardrift enjoying a formal Wassersteinlike gradient flow structure with convex mobility function. In the driftdominant 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 onedimensional case, is based on a reformulation of the problem in terms of the pseudoinverse distribution function. Motivated by the structure of the equation in the new variables, we establish a general framework for globalintime 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, blowup behaviour, formation of condensates (i.e. Dirac measures at zero) and longtime asymptotics are investigated. As a consequence, in the masssupercritical case,solutions will blow up in L^{∞} in finite time andunderstood in a generalised, measure sensethey 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, pp. published online on 20.10.2020, urlhttps://link.springer.com/article/10.1007/s00028020006426, DOI 10.1007/s00028020006426 .
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 L^{p} 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. 4172, DOI 10.3233/ASY191578 .
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 CahnHilliard type modelling tumor growth that has originally been proposed in HawkinsDaarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 324). The original phase field system and certain relaxed versions thereof have been studied in recent papers coauthored by the present authors and E. Rocca. The model consists of a CahnHilliard equation for the tumor cell fraction φ, coupled to a reactiondiffusion equation for a function S representing the nutrientrich 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, wellposedness 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 wellposedness 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 CahnHilliard 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. A4/1A4/18, DOI 10.1478/AAPP.98S2A4 .
Abstract
In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the CahnHilliard system, with possibly singular potentials, which we recently investigated in the paper "Wellposedness and regularity for a generalized fractional CahnHilliard system". More precisely, we give a complete characterization of the Omegalimit 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 Omegalimit is a stationary solution to the phase equation; if it is zero instead, then every element of the Omegalimit 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, Driftdiffusion simulation of Sshaped currentvoltage relations for organic semiconductor devices, Journal of Computational Electronics, 19 (2020), pp. 11641174, DOI 10.1007/s10825020015056 .
Abstract
We present an electrothermal driftdiffusion model for organic semiconductor devices with GaussFermi 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 ScharfetterGummel scheme. Using pathfollowing techniques we demonstrate that the model exhibits Sshaped currentvoltage 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. 11051134, 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 coagulationdiffusion equations with nonhomogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of βamyloid 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 nonperiodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider nonperiodic 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 EDPconvergence, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 395425 (published online in May 2020), DOI 10.3934/dcdss.2020345 .
Abstract
The notion of EnergyDissipationPrinciple convergence (EDPconvergence) 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 FokkerPlanck equation can be formulated as gradientflow equation with respect to the logarithmic relative entropy of the system and a quadratic Wassersteintype gradient structure. The EDPconvergence 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 Marcelinde Donder kinetics. 
A. Kirch, A. Fischer, M. Liero, J. Fuhrmann, A. Glitzky, S. Reineke, Experimental proof of Joule heatinginduced switchedback regions in OLEDs, Light: Science and Applications, 9 (2020), pp. 5/15/10, DOI 10.1038/s4137701902369 .

J. Maas, A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, Journal of Statistical Physics, 181 (2020), pp. 22572303, DOI 10.1007/s10955020026634 .
Abstract
We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reactionrate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailedbalance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradientflow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailedbalance steady state. The limit of large volumes is studied in the sense of evolutionary Γconvergence 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. 443485, DOI 10.20347/WIAS.PREPRINT.2576 .
Abstract
It is wellknown 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 timedependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochnertype 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 obstacletype 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 nonautonomous Cauchy problems, Publications of the Research Institute for Mathematical Sciences, 56 (2020), pp. 83135, DOI 10.4171/PRIMS/5615 .
Abstract
In the present paper we advocate the HowlandEvans approach to solution of the abstract nonautonomous Cauchy problem (nonACP) 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 Xvalued functions on the timeinterval J. The fundamental observation is a onetoone correspondence between solution operators (propagators) for a nonACP 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 operatortheoretical methods to scrutinise the nonACP including the proof of the Trotter product approximation formulae with operatornorm 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 Kortewegtype fluids, International Journal of Engineering Science, 154 (2020), pp. 103316/1103316/28, DOI 10.1016/j.ijengsci.2020.103316 .
Abstract
We provide a derivation of several classes of boundary conditions for fluids of Kortewegtype 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 nonnegative, 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 largearea organic lightemitting diodes, Discrete and Continuous Dynamical Systems  Series S, 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 lightemitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully threedimensional 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 currentflow 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 threedimensional 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 currentflow equation is given by two semilinear equations in the twodimensional crosssections 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. 
P.É. Druet, A theory of generalised solutions for ideal gas mixtures with MaxwellStefan diffusion, Discrete and Continuous Dynamical Systems  Series S, (2020), published online in Nov. 2020, urlhttps://doi.org/10.3934/dcdss.2020458, DOI 10.3934/dcdss.2020458 .
Abstract
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 parabolichyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the BreschDesjardins 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 MaxwellStefan 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 MaxwellStefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime. 
TH. Eiter, M. Kyed, Y. Shibata, On periodic solutions for onephase and twophase problems of the NavierStokes equations, Journal of Evolution Equations, published online on 06.11.2020, DOI 10.1007/s00028020006195 .
Abstract
This paper is devoted to proving the existence of timeperiodic solutions of onephase or twophase problems for the NavierStokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in timedependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with nonhomogeneous boundary conditions or transmission conditions in fixed domains by using the socalled Hanzawa transform.We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigenvalue 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 pLq regularity theorem of the periodic solutions for the system of parabolic equations with nonhomogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of Rsolvers developed in Shibata (Diff Int Eqns 27(34):313368, 2014; On the Rbounded solution operators in the study of free boundary problem for the NavierStokes 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 203285, 2016; Comm Pure Appl Anal 17(4): 16811721. https://doi.org/10.3934/cpaa.2018081, 2018; R boundedness, maximal regularity and free boundary problems for the NavierStokes 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. (Rsolvers and their application to periodic L p estimates, Preprint in 2019) for the L p boundedness of operatorvalued 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 nonpharmaceutical interventions, Journal of Mathematics in Industry, 10 (2020), pp. 23/123/23, DOI 10.1186/s13362020000913 .
Abstract
When effective medical treatment and vaccination are not available, nonpharmaceutical interventions such as social distancing, home quarantine and farreaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptibleexposedinfectiousrecovered) model and continuoustime optimal control theory, we compute the optimal nonpharmaceutical 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 diseaserelated 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 socioeconomic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a singleintervention 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 socioeconomic 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 COVID19 pandemic in Germany. 
A. Mielke, A. Stephan, Coarsegraining via EDPconvergence for linear fastslow reaction systems, Mathematical Models & Methods in Applied Sciences, 30 (2020), pp. 17651807, 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 coarsegrained 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 coshtype gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarsegrained equation again has a coshtype gradient structure. We obtain the strongest version of convergence in the sense of the EnergyDissipation Principle (EDP), namely EDPconvergence with tilting. 
A. Mielke, T. Roubíček, Thermoviscoelasticity in KelvinVoigt rheology at large strains, Archive for Rational Mechanics and Analysis, 238 (2020), pp. 145, DOI 10.1007/s0020502001537z .
Abstract
The frameindifferent thermodynamicallyconsistent model of thermoviscoelasticity at large strain is formulated in the reference configuration with using the concept of the secondgrade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under timedependent 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

M. Heida, S. Neukamm, M. Varga, Stochastic homogenization of Lambdaconvex gradient flows, in: Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, pp. 427453, DOI 10.3934/dcdss.2020328 .
Abstract
In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λconvex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are AllenCahn type equations and evolutionary equations driven by the pLaplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic twoscale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the wellestablished notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ)convex functionals. 
G. Nika, B. Vernescu, Microgeometry effects on the nonlinear effective yield strength response of magnetorheological fluids, in: Emerging Problems in the Homogenization of Partial Differential Equations, P. Donato, M. LunaLaynez, eds., 10 of SEMA SIMAI Springer Series, Springer, Cham, 2021, pp. 116, DOI 9783030620301_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. 
P. Colli, G. Gilardi, J. Sprekels, Deep quench approximation and optimal control of general CahnHilliard systems with fractional operators and double obstacle potentials, in: Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, pp. 243271, DOI 10.3934/dcdss.2020213 .
Abstract
Recently, the authors derived wellposedness and regularity results for general evolutionary operator equations having the structure of a CahnHilliard system. The involved operators were fractional versions in the spectral sense of general linear operators that have compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions. The class of admissible doublewell potentials driving the phase separation process modeled by the CahnHilliard system included polynomial, logarithmic, and double obstacle nonlinearities. In a subsequent paper, distributed optimal control problems for such systems were investigated, where only differentiable polynomial and logarithmic potentials were admitted. Existence of optimizers and firstorder optimality conditions were derived. In this paper, these results are complemented for nondifferentiable double obstacle nonlinearities. It is well known that for such nonlinearities standard constraint qualifications to construct Lagrange multipliers cannot be applied. To overcome this difficulty, we follow the socalled “deep quench” method, which has proved to be a powerful tool in optimal control problems with double obstacle potentials. We give a general convergence analysis of the deep quench approximation, including an error estimate, and demonstrate that its use leads to meaningful firstorder necessary optimality conditions. 
TH. Frenzel, M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDPconvergence, in: Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, pp. 395425, DOI 10.3934/dcdss.2020345 .
Abstract
The notion of EnergyDissipationPrinciple convergence (EDPconvergence) 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 FokkerPlanck equation can be formulated as gradientflow equation with respect to the logarithmic relative entropy of the system and a quadratic Wassersteintype gradient structure. The EDPconvergence 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 Marcelinde Donder kinetics. 
S. Schulz, D. Chaudhuri, M. O'Donovan, S. Patra, T. Streckenbach, P. Farrell, O. Marquardt, Th. Koprucki, Multiscale modeling of electronic, optical, and transport properties of IIIN alloys and heterostructures, in: Physics and Simulation of Optoelectronic Devices XXVIII, B. Witzigmann, M. Osiński, Y. Arakawa, eds., 11274 of Proceedings of SPIE, San Francisco, 2020, pp. 1127416/11127416/10, DOI 10.1117/12.2551055 .

J. Fuhrmann, D.H. Doan, A. Glitzky, M. Liero, G. Nika, Unipolar driftdiffusion simulation of Sshaped currentvoltage relations for organic semiconductor devices, in: Finite Volumes for Complex Applications IX  Methods, Theoretical Aspects, Examples  FVCA 9, Bergen, June 2020, R. Klöfkorn, E. Keilegavlen, F.A. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2020, pp. 625633, DOI 10.1007/9783030436513_59 .
Abstract
We discretize a unipolar electrothermal driftdiffusion model for organic semiconductor devices with GaussFermi 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 ScharfetterGummel scheme. Applying pathfollowing techniques we demonstrate that the model exhibits Sshaped currentvoltage curves with regions of negative differential resistance, only recently observed experimentally. 
M. Kantner, Th. Koprucki, Nonisothermal ScharfetterGummel scheme for electrothermal transport simulation in degenerate semiconductors, in: Finite Volumes for Complex Applications IX  Methods, Theoretical Aspects, Examples  FVCA 9, Bergen, June 2020, R. Klöfkorn, E. Keilegavlen, F.A. Radu, J. Fuhrmann, eds., 323 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2020, pp. 173182, DOI 10.1007/9783030436513_14 .
Abstract
Electrothermal transport phenomena in semiconductors are described by the nonisothermal driftdiffusion system. The equations take a remarkably simple form when assuming the Kelvin formula for the thermopower. We present a novel, nonisothermal generalization of the ScharfetterGummel finite volume discretization for degenerate semiconductors obeying FermiDirac 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

K. Hopf, Singularities in $L^1$supercritical FokkerPlanck equations: A qualitative analysis, Preprint no. 2860, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2860 .
Abstract, PDF (402 kByte)
A class of nonlinear FokkerPlanck equations with superlinear drift is investigated in the L^{1}supercritical regime, which exhibits a finite critical mass. The equations have a formal Wassersteinlike gradientflow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finitetime appearance constitutes a primary technical difficulty. This paper aims at a globalintime qualitative analysis  the main focus being on isotropic solutions, in which case the unique minimiser of the free energy will be shown to be the global attractor. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the 3D KaniadakisQuarati model for BoseEinstein particles, and thus provides a first rigorous result on the continuation beyond blowup and longtime asymptotic behaviour for this model. 
P. Pelech, K. Tůma, M. Pavelka, M. Šípka, M. Sýkora, On compatibility of the natural configuration framework with general equation for nonequilibrium reversibleirreversible coupling (GENERIC): Derivation of anisotropic ratetype models, Preprint no. 2856, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2856 .
Abstract, PDF (485 kByte)
Within the framework of natural configurations developed by Rajagopal and Srinivasa, evolution within continuum thermodynamics is formulated as evolution of a natural configuration linked with the current configuration. On the other hand, withing the General Equation for NonEquilibrium ReversibleIrreversible Coupling (GENERIC) framework, the evolution is split into Hamiltonian mechanics and (generalized) gradient dynamics. These seemingly radically different approaches have actually a lot in common and we show their compatibility on a wide range of models. Both frameworks are illustrated on isotropic and anisotropic ratetype fluid models. We propose an interpretation of the natural configurations within GENERIC and vice versa (when possible). 
M. Heida, Precompact probability spaces in applied stochastic homogenization, Preprint no. 2852, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2852 .
Abstract, PDF (346 kByte)
We provide precompactness and metrizability of the probability space Ω for random measures and random coefficients such as they widely appear in stochastic homogenization and are typically given from data. We show that these properties are enough to implement the convenient twoscale formalism by Zhikov and Piatnitsky (2006). To further demonstrate the benefits of our approach we provide some useful trace and extension operators for Sobolev functions on Ω, which seem not known in literature. On the way we close some minor gaps in the Sobolev theory on Ω which seemingly have not been proven up to date. 
M. Horák, M. Kružík, P. Pelech, A. Schlömerkemper, Gradient polyconvexity and modeling of shape memory alloys, Preprint no. 2851, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2851 .
Abstract, PDF (526 kByte)
We show existence of an energetic solution to a model of shape memory alloys in which the elastic energy is described by means of a gradientpolyconvex functional. This allows us to show existence of a solution based on weak continuity of nonlinear minors of deformation gradients in Sobolev spaces. Admissible deformations do not necessarily have integrable second derivatives. Under suitable assumptions, our model allows for solutions which are orientationpreserving and globally injective everywhere in the domain representing the specimen. Theoretical results are supported by threedimensional computational examples. This work is an extended version of [36]. 
M. Heida, Stochastic homogenization on perforated domains I: Extension operators, Preprint no. 2849, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2849 .
Abstract, PDF (750 kByte)
This preprint is part of a major rewriting and substantial improvement of WIAS Preprint 2742. In this first part of a series of 3 papers, we set up a framework to study the existence of uniformly bounded extension and trace operators for W^{1,p}functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. We drop the classical assumption of minimaly smoothness and study stationary geometries which have no global John regularity. For such geometries, uniform extension operators can be defined only from W^{1,p} to W^{1,r} with the strict inequality r<p. In particular, we estimate the L^{r}norm of the extended gradient in terms of the L^{p}norm of the original gradient. Similar relations hold for the symmetric gradients (for ℝ^{d}valued functions) and for traces on the boundary. As a byproduct we obtain some Poincaré and Korn inequalities of the same spirit. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions: local (δ,M)regularity to quantify statistically the local Lipschitz regularity and isotropic cone mixing to quantify the density of the geometry and the mesoscopic properties. These two properties are sufficient to reduce the problem of extension operators to the 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, for which we can explicitly estimate the connectivity terms. 
P. Colli, A. Signori, J. Sprekels, Optimal control problems with sparsity for phase field tumor growth models involving variational inequalities, Preprint no. 2845, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2845 .
Abstract, PDF (376 kByte)
This paper treats a distributed optimal control problem for a tumor growth model of CahnHilliard type including chemotaxis. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermore, the cost functional contains a nondifferentiable term like the $L^1$norm in order to include sparsity effects which is of utmost relevance, especially time sparsity, in the context of cancer therapies as applying a control to the system reflects in exposing the patient to an intensive medical treatment. To cope with the difficulties originating from the variational inequality in the state system, we employ the socalled “deep quench approximation” in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, firstorder necessary conditions of optimality can be established by invoking recent results. We use these results to derive corresponding optimality conditions also for the double obstacle case, by deducing a variational inequality in terms of the associated adjoint state variables. The resulting variational inequality can be exploited to also obtain sparsity results for the optimal controls. 
P. Krejčí, E. Rocca, J. Sprekels, Analysis of a tumor model as a multicomponent deformable porous medium, Preprint no. 2842, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2842 .
Abstract, PDF (270 kByte)
We propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phasedependent elasticity coefficients. The resulting PDE system couples two CahnHilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reactiondiffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initialboundary value problem has a solution in appropriate function spaces. 
A. Mielke, S. Reichelt, Traveling fronts in a reactiondiffusion equation with a memory term, Preprint no. 2836, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2836 .
Abstract, PDF (1093 kByte)
Based on a recent work on traveling waves in spatially nonlocal reactiondiffusion equations, we investigate the existence of traveling fronts in reactiondiffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reactiondiffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that twoscale 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.

P. Colli, G. Gilardi, J. Sprekels, Wellposedness for a class of phasefield systems modeling prostate cancer growth with fractional operators and general nonlinearities, Preprint no. 2832, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2832 .
Abstract, PDF (310 kByte)
This paper deals with a general system of equations and conditions arising from a mathematical model of prostate cancer growth with chemotherapy and antiangiogenic therapy that has been recently introduced and analyzed (see [P. Colli et al., Mathematical analysis and simulation study of a phasefield model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Math. Models Methods Appl. Sci. bf 30 (2020), 12531295]). The related system includes two evolutionary operator equations involving fractional powers of selfadjoint, nonnegative, unbounded linear operators having compact resolvents. Both equations contain nonlinearities and in particular the equation describing the dynamics of the tumor phase variable has the structure of a AllenCahn equation with doublewell potential and additional nonlinearity depending also on the other variable, which represents the nutrient concentration. The equation for the nutrient concentration is nonlinear as well, with a term coupling both variables. For this system we design an existence, uniqueness and continuous dependence theory by setting up a careful analysis which allows the consideration of nonsmooth potentials and the treatment of continuous nonlinearities with general growth properties. 
TH. Eiter, K. Hopf, A. Mielke, LerayHopf solutions to a viscoelastic fluid model with nonsmooth stressstrain relation, Preprint no. 2829, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2829 .
Abstract, PDF (342 kByte)
We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the ZarembaJaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of globalintime weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor. 
D. Bothe, W. Dreyer, P.É. Druet, Multicomponent incompressible fluids  An asymptotic study, Preprint no. 2825, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2825 .
Abstract, PDF (519 kByte)
This paper investigates the asymptotic behavior of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy that is valid in stable subregions of the phase diagram. On the basis of this representation we classify the admissible data to construct a thermodynamically consistent constitutive model. We then analyze the incompressible limit, where the molar volume becomes independent of pressure. Here we are confronted with two problems:(i) Our study shows that the physical system at hand cannot remain incompressible for arbitrary large deviations from a reference pressure unless its volume is linear in the composition. (ii) As a consequence of the 2nd law of thermodynamics, the incompressible limit implies that the molar volume becomes independent of temperature as well. Most applications, however, reveal the nonappropriateness of this property. According to our mathematical treatment, the free energy as a function of temperature and partial masses tends to a limit in the sense of epi or Gammaconvergence. In the context of the first problem, we study the mixing of two fluids to compare the linearity with experimental observations. The second problem will be treated by considering the asymptotic behavior of both a general inequality relating thermal expansion and compressibility and a PDEsystem relying on the equations of balance for partial masses, momentum and the internal energy.

A. Glitzky, M. Liero, G. Nika, A coarsegrained electrothermal model for organic semiconductor devices, Preprint no. 2822, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2822 .
Abstract, PDF (374 kByte)
We derive a coarsegrained model for the electrothermal interaction of organic semiconductors. The model combines stationary driftdiffusion based electrothermal models with thermistor type models on subregions of the device and suitable transmission conditions. Moreover, we prove existence of a solution using a regularization argument and Schauder's fixed point theorem. In doing so, we extend recent work by taking into account the statistical relation given by the GaussFermi integral and mobility functions depending on the temperature, chargecarrier density, and field strength, which is required for a proper description of organic devices. 
G. Nika, Derivation of effective models from heterogenous Cosserat media via periodic unfolding, Preprint no. 2817, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2817 .
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 (2627): 45854608 '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, Weakstrong uniqueness for energyreactiondiffusion systems, Preprint no. 2808, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2808 .
Abstract, PDF (444 kByte)
We establish weakstrong uniqueness and stability properties of renormalised solutions to a class of energyreactiondiffusion systems, which genuinely feature crossdiffusion 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. Weakstrong uniqueness is obtained for general entropydissipating reactions without growth restrictions, and certain models with a nonintegrable diffusive flux. The results also apply to a class of (isoenergetic) reactioncrossdiffusion systems. 
J. Fischer, K. Hopf, M. Kniely, A. Mielke, Global existence analysis of energyreactiondiffusion systems, Preprint no. 2807, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2807 .
Abstract, PDF (489 kByte)
We establish globalintime existence results for thermodynamically consistent reaction(cross)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model speciesdependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the nonisothermal case lies in the intrinsic presence of crossdiffusion 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 nonintegrable 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, multiphysics, multiscale problem of magnetorheological suspensions introduced in Nika & Vernescu (Z. Angew. Math. Phys., 71(1):119, '20). The hybrid model couples the Stokes' equation with the quasistatic 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 AltmanShinbrot fixed point theorem when the magnetic Reynold's number, R_{m}, is small. 
TH. Koprucki, A. Maltsi, A. Mielke, On the DarwinHowieWhelan 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 (993 kByte)
The DarwinHowieWhelan 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 twobeam and the systematicrow approximation. 
A. Stephan, EDPconvergence for a linear reactiondiffusion system with fast reversible reaction, Preprint no. 2793, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2793 .
Abstract, PDF (422 kByte)
We perform a fastreaction limit for a linear reactiondiffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reactiondiffusion 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 coshtype functions for the reaction part. The fastreaction limit is done on the level of the gradient structure by proving EDPconvergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slowmanifold. Moreover, the limit gradient system can be equivalently described by a coarsegrained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarsegrained slow variable. 
K.M. Gambaryan, O. Marquardt, T. Boeck, A. Trampert, Micro and nanoscale engineering and structures shape architecture at nucleation from InAsSbP 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 straininduced micro and nanostructures. Nucleation is performed from InAsSbP quaternary composition liquid phase in StranskiKrastanow growth mode using steadystate liquid phase epitaxy. Growth features and the shape transformation of pyramidal islands, lensshape and ellipsoidal typeII quantum dots (QDs), quantum rings and QDmolecules 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 latticemismatch 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. HRSEM, 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 eightband $mathbfkcdotmathbfp$ model taking strain and builtin electrostatic potentials into account, the electronic properties of a wide range of InAs$_1xy$Sb$_x$P$_y$ QDs and QDmolecules are computed. Two types of QDs midinfrared 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. 
A. Glitzky, M. Liero, G. Nika, An effective bulksurface thermistor model for largearea organic lightemitting 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 largearea organic lightemitting diodes (OLEDs) is proved. The effective system consists of the heat equation in the threedimensional bulk glass substrate and two semilinear 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 currentflow equation is posed. The existence of weak solutions to the effective system is proved via Schauder's fixedpoint 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 MaxwellStefan and the FickOnsager 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 wellknown main approaches, leading to the generalized FickOnsager multicomponent diffusion fluxes or to the generalized MaxwellStefan equations. The latter approach has the advantage that the resulting fluxes are consistent with nonnegativity of the partial mass densities for nonsingular and nondegenerate MaxwellStefan 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 MaxwellStefan 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 nonpharmaceutical 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, nonpharmaceutical interventions such as social distancing, home quarantine and farreaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptibleexposedinfectiousrecovered) model and continuoustime optimal control theory, we compute the optimal nonpharmaceutical 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 diseaserelated 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 socioeconomic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a singleintervention 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 socioeconomic 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 COVID19 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 W^{1,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 R^{d} and Ω in order to construct a stochastic twoscale convergence method and apply the resulting theory to the homogenization of a pLaplace problem on a randomly perforated domain. 
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 “Wellposedness, regularity and asymptotic analyses for a fractional phase field system” (Asymptot. Anal. 114 (2019), 93128), 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 controltostate operator is Fréchet differentiable between suitable Banach spaces, and meaningful firstorder necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables. 
D. Bothe, P.É. Druet, Wellposedness 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 NavierStokes 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 localintime wellposedness in classes of strong solutions, and the globalintime 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 largearea organic lightemitting 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 lightemitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully threedimensional 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 currentflow 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 threedimensional 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 currentflow equation is given by two semilinear equations in the twodimensional crosssections 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. 
M. Heida, M. Kantner, A. Stephan, Consistency and convergence for a family of finite volume discretizations of the FokkerPlanck 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 FokkerPlanck operator, which are characterized by different weight functions on the edges. This family particularly includes the wellestablished ScharfetterGummel discretization as well as the recently developed squareroot 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 nontrivial while for large gradients the ScharfetterGummel scheme stands out compared to the others.
Talks, Poster

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

M. Heida, Stochastic homogenization on random geometries (online talk), Applied Analysis Seminar, Universität Heidelberg, June 17, 2021.

P. Pelech, On compatibility of the natural configuration framework with general equation of nonequilibrium reversibleirreversible coupling (GENERIC), 16thJoint European Thermodynamics Conference (Hybrid Event), June 14  18, 2021, Charles University Prague, Czech Republic, June 14, 2021.

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

P. Pelech, Separately global solutions to rateindependent systems  applications to largestrain deformations of damageable solids (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S14 ``Applied Analysis'', March 15  19, 2021, Universität Kassel, March 19, 2021.

P. Pelech , Separately global solutions to rateindependent systems  Applications to largestrain deformations of damageable solids (online talk), MS21: SIAM Conference on Mathematical Aspects of Materials Science, Minisymposium 33 ``Asymptotic Analysis of Variational Models in Solid Mechanics'' (Online Event), May 17  28, 2021, Society for Industrial and Applied Mathematics, May 24, 2021.

A. Stephan, Gradient systems and EDPconvergence with applications to nonlinear fastslow reaction systems (online talk), DS21: SIAM Conference on Applications of Dynamical Systems, Minisymposium 19 ``Applications of Stochastic Reaction Networks'' (Online Event), May 23  27, 2021, Society for Industrial and Applied Mathematics, May 23, 2021.

A. Stephan, Gradient systems and mulitscale reaction networks (online talk), Limits and control of stochastic reaction networks (Online Event), July 26  30, 2021, American Institute of Mathematics, San Jose, USA, July 29, 2021.

A. Stephan, Coarsegraining via EDPconvergence for linear fastslow reactiondiffusion systems (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S14 ``Applied Analysis'', March 15  19, 2021, Universität Kassel, March 17, 2021.

P.E. Druet, Modeling and analysis for multicomponent incompressible fluids (online talk), 8th European Congress of Mathematics, Minisymposium ID 51 ``Partial differential equations describing farfromequilibrium open systems'' (Online Event), June 20  26, 2021, European Mathematical Society, June 23, 2021.

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

P.E. Druet, Wellposedness results for mixedtype systems modelling pressuredriven multicomponent fluid flows (online talk), 8th European Congress of Mathematics, Minisymposium ID 42 ``Multicomponent diffusion in porous media'' (Online Event), June 20  26, 2021, European Mathematical Society, June 22, 2021.

M. Liero, Machine learning and PDEs with Julia (online talk), FUHRI2021: Finite Volume Methoods for Realworld AppIications (Online Event), April 29, 2021, WIAS Berlin, April 29, 2021.

A. Mielke, Effective models for materials and interfaces with multiple scales (online poster session), CRC 1114 Conference 2021 (online participation), March 1  3, 2021.

A. Mielke, Thermoviscoelasticity at finite strain (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S14 ``Applied Analysis'', March 15  19, 2021, Universität Kassel, March 19, 2021.

M. Heida, A. Mielke, A. Stephan, Effective models for materials and interfaces with multiple scales, CRC 1114: Scaling Cascades in Complex Systems (SCCS Days) (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, FriedrichAlexander Universität ErlangenNürnberg, 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, Datadriven electronic structure calculations for semiconductor nanostructures, Efficient Algorithms for Numerical Problems, 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) gradedcomposition 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) gradedcomposition 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  Datadriven electronic structure calculations for semiconductor nanostructures, MATH+ Day 2020 (Online Event), Berlin, November 6, 2020.

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

G. Nika, An existence result for a class of electrothermal driftdiffusion models with FermiGauss 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 rateindependent systems  Applications to largestrain 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 rateindependent systems: Applications to largestrain 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 rateindependent systems: Applications to largestrain deformations of damageable solids (online talk), Thematic Einstein Semester: Kickoff 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, Coarsegraining for gradient systems with applications to reaction systems (online talk), Thematic Einstein Semester on Energybased Mathematical Methods for Reactive Multiphase Flows: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12  23, 2020, WIAS Berlin, October 15, 2020.

A. Stephan, EDPconvergence for nonlinear fastslow 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, Coarsegraining via EDPconvergence for linear fastslow 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, EDPconvergence for nonlinear fastslow reactions, Workshop ``Variational Methods for Evolution'', September 13  19, 2020, Mathematisches Forschungsinstitut Oberwolfach, September 18, 2020.

A. Stephan, On mathematical coarsegraining 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 energyreactiondiffusion systems, Workshop ``Variational Methods for Evolution'', September 13  19, 2020, Mathematisches Forschungsinstitut Oberwolfach, September 15, 2020.

TH. Eiter, Spatially asymptotic structure of timeperiodic NavierStokes 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 timeperiodic NavierStokes 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 driftdiffusion simulation of Sshaped currentvoltage 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, Modelbased geometry reconstruction of TEM images, MATH+ Day 2020 (Online Event), Berlin, November 6, 2020.

M. Liero, A. Mielke, Analysis for thermomechanical models with internal variables, Presentation of project proposals in DFG SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, January 30, 2020.

M. Liero, Driftdiffusion simulation of Sshaped currentvoltage 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 Gammaconvergence 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, Finitestrain 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 Gammaconvergence (online talk), Oberseminar ``Mathematik in den Naturwissenschaften'' (Online Event), JuliusMaximiliansUniversität Würzburg, June 5, 2020.

A. Mielke, On finitestrain thermoviscoelasticity, 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), JuliusMaximiliansUniversität Würzburg, November 9, 2020.

A. Mielke, EDPconvergence for multiscale gradient systems with applications to fastslow reaction systems (online talk), One World Dynamics Seminar (Online Event), Technische Universität München, November 13, 2020.

A. Mielke, Global existence for finitestrain 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 twoequation model for turbulence, Workshop on Control of SelfOrganizing Nonlinear Systems, September 2  3, 2020, CRC 910, Technische Universität Berlin, September 2, 2020.

A. Mielke, Variational structures for the analysis of PDE systems (online talks), Thematic Einstein Semester on Energybased Mathematical Methods for Reactive Multiphase Flows: 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 $L^p$ spaces, Oberseminar ``Analysis und Theoretische Physik'', Leibniz Universität Hannover, Institut für Angewandte Mathematik, January 28, 2020.
Research Groups
 Partial Differential Equations
 Laser Dynamics
 Numerical Mathematics and Scientific Computing
 Nonlinear Optimization and Inverse Problems
 Interacting Random Systems
 Stochastic Algorithms and Nonparametric Statistics
 Thermodynamic Modeling and Analysis of Phase Transitions
 Nonsmooth Variational Problems and Operator Equations