Mehrskalenmodellierung, asymptotische Analysis und Hybridmodelle

M. Kantner, Th. Höhne, Th. Koprucki, S. Burger, H.-J. Wünsche, F. Schmidt, A. Mielke, U. Bandelow, Chapter 7: Multi-Dimensional Modeling and Simulation of Semiconductor Nanophotonic Devices, in: Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in Solid-State Sciences, Springer, Heidelberg, 2020, pp. 241--283, (Chapter Published), DOI 10.1007/978-3-030-35656-9_7 .
Abstract
Self-consistent modeling and multi-dimensional simulation of semiconductor nanophotonicdevices is an important tool in the development of future integrated light sources and quantumdevices. Simulations can guide important technological decisions by revealing performance bottle-necks in new device concepts, contribute to their understanding and help to theoretically exploretheir optimization potential. The efficient implementation of multi-dimensional numerical simulationsfor computer-aided design tasks requires sophisticated numerical methods and modeling tech-niques. We review recent advances in device-scale modeling of quantum dot based single-photonsources and laser diodes by self-consistently coupling the optical Maxwell equations with semi-classical carrier transport models using semi-classical and fully quantum mechanical descriptionsof the optically active region, respectively. For the simulation of realistic devices with complex,multi-dimensional geometries, we have developed a novel hp-adaptive finite element approachfor the optical Maxwell equations, using mixed meshes adapted to the multi-scale properties ofthe photonic structures. For electrically driven devices, we introduced novel discretization andparameter-embedding techniques to solve the drift-diffusion system for strongly degenerate semi-conductors at cryogenic temperature. Our methodical advances are demonstrated on variousapplications, including vertical-cavity surface-emitting lasers, grating couplers and single-photonsources.

H.-Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., PDE 2015: Theory and Applications of Partial Differential Equations, 10 of Discrete and Continuous Dynamical Systems -- Series S, American Institute of Mathematical Science, Springfield, 2017, iv+933 pages, (Collection Published).

A. Mielke, Chapter 3: On Evolutionary $Gamma$-Convergence for Gradient Systems, in: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, A. Muntean, J.D.M. Rademacher, A. Zagaris, eds., 3 of Lecture Notes in Applied Mathematics and Mechanics, Springer International Publishing Switzerland, Cham, 2016, pp. 187--249, (Chapter Published).
Abstract
In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional E_ε and the dissipation potential R_ε or the associated dissipation distance. We assume that the functionals depend on a small parameter and the associated gradients systems have solutions u_ε. We investigate the question under which conditions the limits u of (subsequences of) the solutions u_ε are solutions of the gradient system generated by the Γ-limits E₀ and R₀. Here the choice of the right topology will be crucial as well as additional structural conditions.
We cover classical gradient systems, where R_ε is quadratic, and rate-independent systems as well as the passage from viscous to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.

A. Mielke, Chapter 5: Variational Approaches and Methods for Dissipative Material Models with Multiple Scales, in: Analysis and Computation of Microstructure in Finite Plasticity, S. Conti, K. Hackl, eds., 78 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing, Heidelberg et al., 2015, pp. 125--155, (Chapter Published).
Abstract
In a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems. These multiscale limits are formulated in the theory of evolutionary Gamma-convergence. On the one hand we consider the a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rate-independent system. On the other hand we show how the concept of Balanced-Viscosity solution arise as in the vanishing-viscosity limit.
 
As applications we discuss, first, the evolution of laminate microstructures in finite-strain elastoplasticity and, second, a two-phase model for shape-memory materials, where H-measures are used to construct the mutual recovery sequences needed in the existence theory.

A. Mielke, Chapter 21: Dissipative Quantum Mechanics Using GENERIC, in: Recent Trends in Dynamical Systems -- Proceedings of a Conference in Honor of Jürgen Scheurle, A. Johann, H.-P. Kruse, F. Rupp, S. Schmitz, eds., 35 of Springer Proceedings in Mathematics & Statistics, Springer, Basel et al., 2013, pp. 555--585, (Chapter Published).
Abstract
Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework (General Equations for Non-Equilibrium Reversible Irreversible Coupling) to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition. One of our models couples a quantum system to a finite number of heat baths each of which is described by a time-dependent temperature. The dissipation mechanism is modeled via the canonical correlation operator, which is the inverse of the Kubo-Mori metric for density matrices and which is strongly linked to the von Neumann entropy for quantum systems. Thus, one recovers the dissipative double-bracket operators of the Lindblad equations but encounters a correction term for the consistent coupling to the dissipative dynamics. For the finite-dimensional and isothermal case we provide a general existence result and discuss sufficient conditions that guarantee that all solutions converge to the unique thermal equilibrium state. Finally, we compare of our gradient flow formulation for quantum systems with the Wasserstein gradient flow formulation for the Fokker-Planck equation and the entropy gradient flow formulation for reversible Markov chains.

A. Mielke, ed., Analysis, Modeling and Simulation of Multiscale Problems, Springer, Berlin, 2006, xviii+697 pages, (Collection Published).

L. Schmeller, D. Peschka, Sharp-interface limits of Cahn--Hilliard models and mechanics with moving contact lines, , 22 (2024), pp. 869--890, DOI 10.1137/23M1546592 .
Abstract
We construct gradient structures for free boundary problems with moving capillary interfaces with nonlinear (hyper)elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phase-field models converge to certain sharp-interface models when the interface thickness tends to zero. In particular, we study the scaling of the Cahn--Hilliard mobility with certain powers of the interfacial thickness. In the presence of interfaces, it is known that the intended sharp-interface limit holds only for a particular range of powers However, in the presence of moving contact lines we show that some scalings that are valid for interfaces produce significant errors and the effective range of valid powers of the interfacial thickness in the mobility reduces.

M. Heida, M. Landstorfer, M. Liero, Homogenization of a porous intercalation electrode with phase separation, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 22 (2024), pp. 1068--1096, DOI 10.1137/21M1466189 .
Abstract
In this work, we derive a new model framework for a porous intercalation electrode with a phase separating active material upon lithium intercalation. We start from a microscopic model consisting of transport equations for lithium ions in an electrolyte phase and intercalated lithium in a solid active phase. Both are coupled through a Neumann--boundary condition modeling the lithium intercalation reaction. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a Cahn--Hilliard equation, whereas the limit model consists of a diffusion and an Allen--Cahn equation. Thus we observe a Cahn--Hilliard to Allen--Cahn transition during the upscaling process. In the sense of gradient flows, the transition goes in hand with a change in the underlying metric structure of the PDE system.

M. Heida, B. Jahnel, A.D. Vu, An ergodic and isotropic zero-conductance model with arbitrarily strong local connectivity, Electronic Communications in Probability, 29 (2024), pp. 1--13, DOI 10.1214/24-ECP633 .
Abstract
We exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all non-trivial choices of the connectivity parameter. The model is based on the so-called randomly stretched lattice where we additionally elongate layers containing few open edges.

M. Heida, Stochastic homogenization on perforated domains I: Extension operators, Networks and Heterogeneous Media, 18 (2023), pp. 1821/1--1821/78, DOI 10.3934/nhm.2023079 .
Abstract
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.

M. Heida, Stochastic homogenization on perforated domains III -- General estimates for stationary ergodic random connected Lipschitz domains, Networks and Heterogeneous Media, 18 (2023), pp. 1410--1433, DOI 10.3934/nhm.2023062 .
Abstract
This is Part III of a series on the existence of uniformly bounded extension operators on randomly perforated domains in the context of homogenization theory. Recalling that randomly perforated domains are typically not John and hence extension is possible only from W ^1,p to W ^1,r, r < p, we will show that the existence of such extension operators can be guarantied if the weighted expectations of four geometric characterizing parameters are bounded: The local Lipschitz constant M, the local Lipschitz radius Δ , the mesoscopic Voronoi diameter ∂ and the local connectivity radius R.

A.K. Barua, R. Chew, L. Shuwang, J. Lowengrub, A. Münch, B. Wagner, Sharp-interface problem of the Ohta--Kawasaki model for symmetric diblock copolymers, Journal of Computational Physics, 481 (2023), pp. 112032/1--112032/23, DOI 10.1016/j.jcp.2023.112032 .
Abstract
The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space.

A. Zafferi, K. Huber, D. Peschka, J. Vrijmoed, T. John, M. Thomas, A porous-media model for reactive fluid-rock interaction in a dehydrating rock, Journal of Mathematical Physics, 64 (2023), pp. 091504/1--091504/29, DOI 10.1063/5.0148243 .
Abstract
We study the GENERIC structure of models for reactive two-phase flows and their connection to a porous-media model for reactive fluid-rock interaction used in Geosciences. For this we discuss the equilibration of fast dissipative processes in the GENERIC framework. Mathematical properties of the porous-media model and first results on its mathematical analysis are provided. The mathematical assumptions imposed for the analysis are critically validated with the thermodynamical rock data sets.

G.L. Celora, M.G. Hennessy, A. Münch, B. Wagner, S.L. Waters, The dynamics of a collapsing polyelectrolyte gel, SIAM Journal on Applied Mathematics, 83 (2023), pp. 1146--1171, DOI 10.1137/21M1419726 .

E. Meca, A.W. Fritsch, J. Iglesias--Artola, S. Reber, B. Wagner, Predicting disordered regions driving phase separation of proteins under variable salt concentration, Frontiers in Physics, section Biophysics, 11 (2023), pp. 1213304/1--1213304/13, DOI 10.3389/fphy.2023.1213304 .
Abstract
We determine the intrinsically disordered regions (IDRs) of phase separating proteins and investigate their impact on liquid-liquid phase separation (LLPS) with a random-phase approx- imation (RPA) that accounts for variable salt concentration. We focus on two proteins, PGL-3 and FUS, known to undergo LLPS. For PGL-3 we predict that an IDR near the C-terminus pro- motes LLPS, which we validate through direct comparison with in vitro experimental results. For the structurally more complex protein FUS the role of the low complexity (LC) domain in LLPS is not as well understood. Apart from the LC domain we here identify two IDRs, one near the N-terminus and another near the C-terminus. Our RPA analysis of these domains predict that, surprisingly, the IDR at the N-terminus (aa 1-285) and not the LC domain promotes LLPS of FUS by comparison to in vitro experiments under physiological temperature and salt conditions.

A. Mielke, Non-equilibrium steady states as saddle points and EDP-convergence for slow-fast gradient systems, Journal of Mathematical Physics, 64 (2023), pp. 123502/1-- 123502/27, DOI 10.1063/5.0149910 .
Abstract
The theory of slow-fast gradient systems leads in a natural way to non-equilibrium steady states, because on the slow time scale the fast subsystem stays in steady states that are driven by the interaction with the slow system. Using the theory of convergence of gradient systems in the sense of the energy-dissipation principle shows that there is a natural characterization of these non-equilibrium steady states as saddle points of a Lagrangian where the slow variables are fixed. We give applications to slow-fast reaction-diffusion systems based on the so-called cosh-type gradient structure for reactions. It is shown that two binary reaction give rise to a ternary reaction with a state-dependent reaction coefficient. Moreover, we show that a reaction-diffusion equation with a thin membrane-like layer convergences to a transmission condition, where the formerly quadratic dissipation potential for diffusion convergences to a cosh-type dissipation potential for the transmission in the membrane limit.

M. Heida, S. Neukamm, M. Varga, Stochastic two-scale convergence and Young measures, Networks and Heterogeneous Media, 17 (2022), pp. 227--254, DOI 10.3934/nhm.2022004 .
Abstract
In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.

M. Heida, Stochastic homogenization on perforated domains II -- Application to nonlinear elasticity models, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, published online on 26.09.2022, DOI 10.1002/zamm.202100407 .
Abstract
Based on a recent work that exposed the lack of uniformly bounded W^1,p → W^1,p extension operators on randomly perforated domains, we study stochastic homogenization of nonlinear elasticity on such structures using instead the extension operators constructed in [11]. We thereby introduce two-scale convergence methods on such random domains under the intrinsic loss of regularity and prove some generally useful calculus theorems on the probability space Ω, e.g. abstract Gauss theorems.

G.L. Celora, M.G. Hennessy, A. Münch, B. Wagner, S.L. Waters, A kinetic model of a polyelectrolyte gel undergoing phase separation, Journal of the Mechanics and Physics of Solids, 160 (2022), pp. 104771/1--104771/27, DOI 10.1016/j.jmps.2021.104771 .
Abstract
In this study we use non-equilibrium thermodynamics to systematically derive a phase-field model of a polyelectrolyte gel coupled to a thermodynamically consistent model for the salt solution surrounding the gel. The governing equations for the gel account for the free energy of the internal interfaces which form upon phase separation, as well as finite elasticity and multi-component transport. The fully time-dependent model describes the evolution of small changes in the mobile ion concentrations and follows their impact on the large-scale solvent flux and the emergence of long-time pattern formation in the gel. We observe a strong acceleration of the evolution of the free surface when the volume phase transition sets in, as well as the triggering of spinodal decomposition that leads to strong inhomogeneities in the lateral stresses, potentially leading to experimentally visible patterns.

D. Bothe, W. Dreyer, P.-É. Druet, Multicomponent incompressible fluids -- An asymptotic study, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, published online on 14.01.2022, DOI 10.1002/zamm.202100174 .
Abstract
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 non-appropriateness 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 Gamma--convergence. 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 PDE-system relying on the equations of balance for partial masses, momentum and the internal energy.

M. O'Donovan, P. Farrell, T. Streckenbach, Th. Koprucki, S. Schulz, Multiscale simulations of uni-polar hole transport in (In,Ga)N quantum well systems, Optical and Quantum Electronics, 54 (2022), pp. 405/1--405/23, DOI 10.1007/s11082-022-03752-2 .
Abstract
Understanding the impact of the alloy micro-structure on carrier transport becomes important when designing III-nitride-based LED structures. In this work, we study the impact of alloy fluctuations on the hole carrier transport in (In,Ga)N single and multi-quantum well systems. To disentangle hole transport from electron transport and carrier recombination processes, we focus our attention on uni-polar (p-i-p) systems. The calculations employ our recently established multi-scale simulation framework that connects atomistic tight-binding theory with a macroscale drift-diffusion model. In addition to alloy fluctuations, we pay special attention to the impact of quantum corrections on hole transport. Our calculations indicate that results from a virtual crystal approximation present an upper limit for the hole transport in a p-i-p structure in terms of the current-voltage characteristics. Thus we find that alloy fluctuations can have a detrimental effect on hole transport in (In,Ga)N quantum well systems, in contrast to uni-polar electron transport. However, our studies also reveal that the magnitude by which the random alloy results deviate from virtual crystal approximation data depends on several factors, e.g. how quantum corrections are treated in the transport calculations.

A.F.M. TER Elst, A. Linke, J. Rehberg, On the numerical range of sectorial forms, Pure and Applied Functional Analysis, 7 (2022), pp. 1931--1940.
Abstract
We provide a sharp and optimal generic bound for the angle of the sectorial form associated to a non-symmetric second-order elliptic differential operator with various boundary conditions. Consequently this gives an, in general, sharper H^∞-angle for the H^∞-calculus on L_p for all p ∈ (1, ∞) if the coefficients are real valued.

A. Glitzky, M. Liero, G. Nika, A coarse-grained electrothermal model for organic semiconductor devices, Mathematical Methods in the Applied Sciences, 45 (2022), pp. 4809--4833, DOI 10.1002/mma.8072 .
Abstract
We derive a coarse-grained model for the electrothermal interaction of organic semiconductors. The model combines stationary drift-diffusion 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 Gauss--Fermi integral and mobility functions depending on the temperature, charge-carrier density, and field strength, which is required for a proper description of organic devices.

M. Landstorfer, M. Ohlberger, S. Rave, M. Tacke, A modelling framework for efficient reduced order simulations of parametrised lithium-ion battery cells, European Journal of Applied Mathematics, 34 (2023), pp. 554--591 (published online on 29.11.2022), DOI 10.1017/S0956792522000353 .
Abstract
In this contribution we present a new modeling and simulation framework for parametrized Lithium-ion battery cells. We first derive a new continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterized non-linear system of partial differential equations the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modeling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modeling framework.

M. Heida, S. Neukamm, M. Varga, Stochastic homogenization of Lambda-convex gradient flows, Discrete and Continuous Dynamical Systems -- Series S, 14 (2021), pp. 427--453, 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 Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen--Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established 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, Derivation of effective models from heterogenous Cosserat media via periodic unfolding, Ricerche di Matematica. A Journal of Pure and Applied Mathematics, published online on 01.07.2021, DOI 10.1007/s11587-021-00610-3 .
Abstract
We derive two different effective models from a heterogeneous Cosserat continuum taking into account the Cosserat intrinsic length of the constituents. We pass to the limit using homogenization via periodic unfolding and in doing so we provide rigorous proof to the results introduced by Forest, Pradel, and Sab (Int. J. Solids Structures 38 (26-27): 4585-4608 '01). Depending on how different characteristic lengths of the domain scale with respect to the Cosserat intrinsic length, we obtain either an effective classical Cauchy continuum or an effective Cosserat continuum. Moreover, we provide some corrector type results for each case.

A. Stephan, EDP-convergence for a linear reaction-diffusion system with fast reversible reaction, Calculus of Variations and Partial Differential Equations, 60 (2021), pp. 226/1--226/35, DOI 10.1007/s00526-021-02089-0 .
Abstract
We perform a fast-reaction limit for a linear reaction-diffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reaction-diffusion system as a gradient flow of the free energy in the space of probability measures equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and cosh-type functions for the reaction part. The fast-reaction limit is done on the level of the gradient structure by proving EDP-convergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slow-manifold. Moreover, the limit gradient system can be equivalently described by a coarse-grained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarse-grained slow variable.

D. Chaudhuri, M. O'Donovan, T. Streckenbach, O. Marquardt, P. Farrell, S.K. Patra, Th. Koprucki, S. Schulz, Multiscale simulations of the electronic structure of III-nitride quantum wells with varied indium content: Connecting atomistic and continuum-based models, Journal of Applied Physics, 129 (2021), pp. 073104/1--073104/16, DOI 10.1063/5.0031514 .

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

M. O'Donovan, D. Chaudhuri, T. Streckenbach, P. Farrell, S. Schulz, Th. Koprucki, From atomistic tight-binding theory to macroscale drift-diffusion: Multiscale modeling and numerical simulation of uni-polar charge transport in (In,Ga)N devices with random fluctuations, Journal of Applied Physics, 130 (2021), pp. 065702/1--065702/13, DOI 10.1063/5.0059014 .

A. Glitzky, M. Liero, G. Nika, Analysis of a bulk-surface thermistor model for large-area organic LEDs, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 78 (2021), pp. 187--210, DOI 10.4171/PM/2066 .
Abstract
The existence of a weak solution for an effective system of partial differential equations describing the electrothermal behavior of large-area organic light-emitting diodes (OLEDs) is proved. The effective system consists of the heat equation in the three-dimensional bulk glass substrate and two semi-linear equations for the current flow through the electrodes coupled to algebraic equations for the continuity of the electrical fluxes through the organic layers. The electrical problem is formulated on the (curvilinear) surface of the glass substrate where the OLED is mounted. The source terms in the heat equation are due to Joule heating and are hence concentrated on the part of the boundary where the current-flow equation is posed. The existence of weak solutions to the effective system is proved via Schauder's fixed-point theorem. Moreover, since the heat sources are a priori only in $L^1$, the concept of entropy solutions is used.

A. Mielke, A. Montefusco, M.A. Peletier, Exploring families of energy-dissipation landscapes via tilting: Three types of EDP convergence, Continuum Mechanics and Thermodynamics, 33 (2021), pp. 611--637, DOI 10.1007/s00161-020-00932-x .
Abstract
This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that well-known techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the Energy-Dissipation Principle that provides a variational formulation to gradient-flow equations that allows one to apply techniques from Γ-convergence of functional on states and functionals on trajectories.

A. Mielke, R.R. Netz, S. Zendehroud, A rigorous derivation and energetics of a wave equation with fractional damping, Journal of Evolution Equations, 21 (2021), pp. 3079--3102, DOI 10.1007/s00028-021-00686-2 .
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 water-air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally-damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy-dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally-damped wave equation with a time derivative of order 3/2.

A. Mielke, M.A. Peletier, A. Stephan, EDP-convergence for nonlinear fast-slow reaction systems with detailed balance, Nonlinearity, 34 (2021), pp. 5762--5798, DOI 10.1088/1361-6544/ac0a8a .
Abstract
We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.

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

M. Landstorfer, B. Prifling, V. Schmidt, Mesh generation for periodic 3D microstructure models and computation of effective properties, Journal of Computational Physics, 431 (2021), pp. 110071/1--110071/20 (published online on 23.12.2020), DOI https://doi.org/10.1016/j.jcp.2020.110071 .
Abstract
Understanding and optimizing effective properties of porous functional materials, such as permeability or conductivity, is one of the main goals of materials science research with numerous applications. For this purpose, understanding the underlying 3D microstructure is crucial since it is well known that the materials? morphology has an significant impact on their effective properties. Because tomographic imaging is expensive in time and costs, stochastic microstructure modeling is a valuable tool for virtual materials testing, where a large number of realistic 3D microstructures can be generated and used as geometry input for spatially-resolved numerical simulations. Since the vast majority of numerical simulations is based on solving differential equations, it is essential to have fast and robust methods for generating high-quality volume meshes for the geometrically complex microstructure domains. The present paper introduces a novel method for generating volume-meshes with periodic boundary conditions based on an analytical representation of the 3D microstructure using spherical harmonics. Due to its generality, the present method is applicable to many scientific areas. In particular, we present some numerical examples with applications to battery research by making use of an already existing stochastic 3D microstructure model that has been calibrated to eight differently compacted cathodes.

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

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

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

M.G. Hennessy, A. Münch, B. Wagner, Phase separation in swelling and deswelling hydrogels with a free boundary, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 101 (2020), pp. 032501/1--032501/14, DOI 10.1103/PhysRevE.101.032501 .
Abstract
We present a full kinetic model of a hydrogel that undergoes phase separation during swelling and deswelling. The model accounts for the interfacial energy of coexisting phases, finite strain of the polymer network, andsolvent transport across free boundaries. For the geometry of an initially dry layer bonded to a rigid substrate,the model predicts that forcing solvent into the gel at a fixed rate can induce a volume phase transition, whichgives rise to coexisting phases with different degrees of swelling, in systems where this cannot occur in the free-swelling case. While a nonzero shear modulus assists in the propagation of the transition front separating thesephases in the driven-swelling case, increasing it beyond a critical threshold suppresses its formation. Quenchinga swollen hydrogel induces spinodal decomposition, which produces several highly localized, highly swollenphases which coarsen and are then ejected from free boundary. The wealth of dynamic scenarios of this systemis discussed using phase-plane analysis and numerical solutions in a one-dimensional setting.

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

A. Glitzky, M. Liero, G. Nika, Dimension reduction of thermistor models for large-area organic light-emitting diodes, Discrete and Continuous Dynamical Systems -- Series S, 14 (2021), pp. 3953--3971 (published online on 28.11.2020), DOI 10.3934/dcdss.2020460 .
Abstract
An effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional φ(x)-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter ε > 0 which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the current-flow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the three-dimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective current-flow equation is given by two semilinear equations in the two-dimensional cross-sections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted.

A. Caiazzo, R. Maier, D. Peterseim, Reconstruction of quasi-local numerical effective models from low-resolution measurements, Journal of Scientific Computing, 85 (2020), pp. 10/1--10/23, DOI 10.1007/s10915-020-01304-y .
Abstract
We consider the inverse problem of reconstructing an effective model for a prototypical diffusion process in strongly heterogeneous media based on low-resolution measurements. We rely on recent quasi-local numerical effective models that, in contrast to conventional homogenized models, are provably reliable beyond periodicity assumptions and scale separation. The goal of this work is to show that the identification of the matrix representation of these effective models is possible. Algorithmic aspects of the inversion procedure and its performance are illustrated in a series of numerical experiments.

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

M. Heida, S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, Asymptotic Analysis, 112 (2019), pp. 185--212, DOI 10.3233/ASY-181502 .
Abstract
In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Gamma-convergence theory.

G. Nika, B. Vernescu, Multiscale modeling of magnetorheological suspensions, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 71 (2020), pp. 14/1--14/19 (published online on 23.12.2019), DOI 10.1007/s00033-019-1238-4 .
Abstract
We develop a multiscale approach to describe the behavior of a suspension of solid magnetizable particles in a viscous non-conducting fluid in the presence of an externally applied magnetic field. By upscaling the quasi-static Maxwell equations coupled with the Stokes' equations we are able to capture the magnetorheological effect. The model we obtain generalizes the one introduced by Neuringer & Rosensweig for quasistatic phenomena. We derive the macroscopic constitutive properties explicitly in terms of the solutions of local problems. The effective coefficients have a nonlinear dependence on the volume fraction when chain structures are present. The velocity profiles computed for some simple flows, exhibit an apparent yield stress and the flowprofile resembles a Bingham fluid flow.

F. Agnelli, A. Constantinescu, G. Nika, Design and testing of 3D-printed micro-architectured polymer materials exhibiting a negative Poisson's ratio, Continuum Mechanics and Thermodynamics, 32 (2020), pp. 433--449 (published online on 20.11.2019), DOI 10.1007/s00161-019-00851-6 .
Abstract
This work proposes the complete design cycle for several auxetic materials where the cycle consists of three steps (i) the design of the micro-architecture, (ii) the manufacturing of the material and (iii) the testing of the material. We use topology optimization via a level-set method and asymptotic homogenization to obtain periodic micro-architectured materials with a prescribed effective elasticity tensor and Poisson's ratio. The space of admissible micro-architectural shapes that carries orthotropic material symmetry allows to attain shapes with an effective Poisson's ratio below -1. Moreover, the specimens were manufactured using a commercial stereolithography Ember printer and are mechanically tested. The observed displacement and strain fields during tensile testing obtained by digital image correlation match the predictions from the finite element simulations and demonstrate the efficiency of the design cycle.

A. Bacho, E. Emmrich, A. Mielke, An existence result and evolutionary Gamma-convergence for perturbed gradient systems, Journal of Evolution Equations, 19 (2019), pp. 479--522, DOI 10.1007/s00028-019-00484-x .
Abstract
We consider the initial-value problem for the perturbed gradient flows, where a differential inclusion is formulated in terms of a subdifferential of an energy functional, a subdifferential of a dissipation potential and a more general perturbation, which is assumed to be continuous and to satisfy a suitable growth condition. Under additional assumptions on the dissipation potential and the energy functional, existence of strong solutions is shown by proving convergence of a semi-implicit discretization scheme with a variational approximation technique.

P. Dondl, Th. Frenzel, A. Mielke, A gradient system with a wiggly energy and relaxed EDP-convergence, ESAIM. Control, Optimisation and Calculus of Variations, 25 (2019), pp. 68/1--68/45, DOI 10.1051/cocv/2018058 .
Abstract
If gradient systems depend on a microstructure, we want to derive a macroscopic gradient structure describing the effective behavior of the microscopic system. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. We call this notion relaxed EDP-convergence since the special structure of the dissipation functional may not be preserved under Gamma-convergence. However, by investigating the kinetic relation we derive the macroscopic dissipation potential.

V. Klika , M. Pavelka , P. Vágner, M. Grmela, Dynamic maximum entropy reduction, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 21 (2019), pp. 1--27.

M. Heida, B. Schweizer, Stochastic homogenization of plasticity equations, ESAIM. Control, Optimisation and Calculus of Variations, 24 (2018), pp. 153--176.
Abstract
In the context of infinitesimal strain plasticity with hardening, we derive a stochastic homogenization result. We assume that the coefficients of the equation are random functions: elasticity tensor, hardening parameter and flow-rule function are given through a dynamical system on a probability space. A parameter ε > 0 denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit ε → 0. The homogenization procedure is based on the fact that stochastic coefficients “allow averaging”: For one representative volume element, a strain evolution [0,T]∋ t ↦ξ(t) ∈ ℝ_s^dxd induces a stress evolution [0,T]∋ t ↦Σ (ξ) (t)∈ℝ_s^dxd. Once the hysteretic evolution law Σ is justified for averages, we obtain that the macroscopic limit equation is given by -∇ ⋅ Σ(∇^su)=f.

M. Heida, On convergences of the squareroot approximation scheme to the Fokker--Planck operator, Mathematical Models & Methods in Applied Sciences, 28 (2018), pp. 2599--2635, DOI 10.1142/S0218202518500562 .
Abstract
We study the qualitative convergence properties of a finite volume scheme that recently was proposed by Lie, Fackeldey and Weber [SIAM Journal on Matrix Analysis and Applications 2013 (34/2)] in the context of conformation dynamics. The scheme was derived from physical principles and is called the squareroot approximation (SQRA) scheme. We show that solutions to the SQRA equation converge to solutions of the Fokker-Planck equation using a discrete notion of G-convergence. Hence the squareroot approximation turns out to be a usefull approximation scheme to the Fokker-Planck equation in high dimensional spaces. As an example, in the special case of stationary Voronoi tessellations we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property. In particular, the class of tessellations for which the G-convergence result holds is not trivial.

E. Meca Álvarez, A. Münch, B. Wagner, Localized instabilities and spinodal decomposition in driven systems in the presence of elasticity, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 97 (2018), pp. 012801/1--012801/12, DOI 10.1103/PhysRevE.97.012801 .
Abstract
We study numerically and analytically the instabilities associated with phase separation in a solid layer on which an external material flux is imposed. The first instability is localized within a boundary layer at the exposed free surface by a process akin to spinodal decomposition. In the limiting static case, when there is no material flux, the coherent spinodal decomposition is recovered. In the present problem stability analysis of the time-dependent and non-uniform base states as well as numerical simulations of the full governing equations are used to establish the dependence of the wavelength and onset of the instability on parameter settings and its transient nature as the patterns eventually coarsen into a flat moving front. The second instability is related to the Mullins-Sekerka instability in the presence of elasticity and arises at the moving front between the two phases when the flux is reversed. Stability analyses of the full model and the corresponding sharp-interface model are carried out and compared. Our results demonstrate how interface and bulk instabilities can be analysed within the same framework which allows to identify and distinguish each of them clearly. The relevance for a detailed understanding of both instabilities and their interconnections in a realistic setting are demonstrated for a system of equations modelling the lithiation/delithiation processes within the context of Lithium ion batteries.

K. Disser, M. Liero, J. Zinsl, On the evolutionary Gamma-convergence of gradient systems modeling slow and fast chemical reactions, Nonlinearity, 31 (2018), pp. 3689--3706, DOI 10.1088/1361-6544/aac353 .
Abstract
We investigate the limit passage for a system of ordinary differential equations modeling slow and fast chemical reaction of mass-action type, where the rates of fast reactions tend to infinity. We give an elementary proof of convergence to a reduced dynamical system acting in the slow reaction directions on the manifold of fast reaction equilibria. Then we study the entropic gradient structure of these systems and prove an E-convergence result via Γ-convergence of the primary and dual dissipation potentials, which shows that this structure carries over to the fast reaction limit. We recover the limit dynamics as a gradient flow of the entropy with respect to a pseudo-metric.

T. Ahnert, A. Münch, B. Wagner, Models for the two-phase flow of concentrated suspensions, European Journal of Applied Mathematics, 30 (2019), pp. 585--617 (published online on 04.06.2018), DOI 10.1017/S095679251800030X .
Abstract
A new two-phase model is derived that make use of a constitutive law combining non-Brownian suspension with granular rheology, that was recently proposed by Boyer et al. [PRL, 107(18),188301 (2011)]. It is shown that for the simple channel flow geometry, the stress model naturally exhibits a Bingham type flow property with an unyielded finite-size zone in the center of the channel. As the volume fraction of the solid phase is increased, the various transitions in the flow fields are discussed using phase space methods for a boundary value problem, that is derived from the full model. The predictions of this analysis is then compared to the direct finite-element numerical solutions of the full model.

L. Donati, M. Heida, M. Weber, B. Keller, Estimation of the infinitesimal generator by square-root approximation, Journal of Physics: Condensed Matter, 30 (2018), pp. 425201/1--425201/14, DOI 10.1088/1361-648X/aadfc8 .
Abstract
For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approximated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the geometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correlation between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide.

P. Gurevich, S. Reichelt, Pulses in FitzHugh--Nagumo systems with rapidly oscillating coefficients, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 16 (2018), pp. 833--856.
Abstract
This paper is devoted to pulse solutions in FitzHugh-Nagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a two-scale FitzHugh-Nagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulse-like solutions of the original system.

A. Muntean, S. Reichelt, Corrector estimates for a thermo-diffusion model with weak thermal coupling, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 16 (2018), pp. 807--832, DOI 10.1137/16M109538X .
Abstract
The present work deals with the derivation of corrector estimates for the two-scale homogenization of a thermo-diffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged high-contrast microstructures. The terminology “weak thermal coupling” refers here to the variable scaling in terms of the small homogenization parameter ε of the heat conduction-diffusion interaction terms, while the “high-contrast” is thought particularly in terms of the heat conduction properties of the composite material. As main target, we justify the first-order terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufour-like effects. The contrasting heat conduction combined with cross coupling lead to the main mathematical difficulty in the system. Our approach relies on the method of periodic unfolding combined with ε-independent estimates for the thermal and concentration fields and for their coupled fluxes

W. Dreyer, C. Guhlke, R. Müller, Bulk-surface electro-thermodynamics and applications to electrochemistry, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 20 (2018), pp. 939/1--939/44, DOI 10.3390/e20120939 .
Abstract
We propose a modeling framework for magnetizable, polarizable, elastic, viscous, heat conducting, reactive mixtures in contact with interfaces. To this end we first introduce bulk and surface balance equations that contain several constitutive quantities. For further modeling the constitutive quantities, we formulate constitutive principles. They are based on an axiomatic introduction of the entropy principle and the postulation of Galilean symmetry. We apply the proposed formalism to derive constitutive relations in a rather abstract setting. For illustration of the developed procedure, we state an explicit isothermal material model for liquid electrolyte metal electrode interfaces in terms of free energy densities in the bulk and on the surface. Finally we give a survey of recent advancements in the understanding of electrochemical interfaces that were based on this model.

M. Liero, S. Reichelt, Homogenization of Cahn--Hilliard-type equations via evolutionary Gamma-convergence, NoDEA. Nonlinear Differential Equations and Applications, 25 (2018), pp. 6/1--6/31, DOI 10.1007/s00030-018-0495-9 .
Abstract
In this paper we discuss two approaches to evolutionary Γ-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

S. Bergmann, D.A. Barragan-Yani, E. Flegel, K. Albe, B. Wagner, Anisotropic solid-liquid interface kinetics in silicon: An atomistically informed phase-field model, Modelling and Simulation in Materials Science and Engineering, 25 (2017), pp. 065015/1--065015/20, DOI 10.1088/1361-651X/aa7862 .
Abstract
We present an atomistically informed parametrization of a phase-field model for describing the anisotropic mobility of liquid-solid interfaces in silicon. The model is derived from a consistent set of atomistic data and thus allows to directly link molecular dynamics and phase field simulations. Expressions for the free energy density, the interfacial energy and the temperature and orientation dependent interface mobility are systematically fitted to data from molecular dynamics simulations based on the Stillinger-Weber interatomic potential. The temperature-dependent interface velocity follows a Vogel-Fulcher type behavior and allows to properly account for the dynamics in the undercooled melt.

S. Reichelt, Corrector estimates for a class of imperfect transmission problems, Asymptotic Analysis, 105 (2017), pp. 3--26, DOI 10.3233/ASY-171432 .
Abstract
Based on previous homogenization results for imperfect transmission problems in two-component domains with periodic microstructure, we derive quantitative estimates for the difference between the microscopic and macroscopic solution. This difference is of order ερ, where ε > 0 describes the periodicity of the microstructure and ρ ∈ (0 , ½] depends on the transmission condition at the interface between the two components. The corrector estimates are proved without assuming additional regularity for the local correctors using the periodic unfolding method.

M. Heida, A. Mielke, Averaging of time-periodic dissipation potentials in rate-independent processes, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1303--1327.
Abstract
We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period ε. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit ε→0 and show that the effctive dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit ε→0.

M. Heida, Stochastic homogenization of rate-independent systems, Continuum Mechanics and Thermodynamics, 29 (2017), pp. 853--894, DOI 10.1007/s00161-017-0564-z .
Abstract
We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We proof some convergence results with respect to stochastic two-scale convergence, which are related to classical Gamma-convergence results. The main result is a general liminf-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rateindependent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandlt-Reuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissure.

M. Kantner, M. Mittnenzweig, Th. Koprucki, Hybrid quantum-classical modeling of quantum dot devices, Phys. Rev. B., 96 (2017), pp. 205301/1--205301/17, DOI 10.1103/PhysRevB.96.205301 .
Abstract
The design of electrically driven quantum dot devices for quantum optical applications asks for modeling approaches combining classical device physics with quantum mechanics. We connect the well-established fields of semi-classical semiconductor transport theory and the theory of open quantum systems to meet this requirement. By coupling the van Roosbroeck system with a quantum master equation in Lindblad form, we obtain a new hybrid quantum-classical modeling approach, which enables a comprehensive description of quantum dot devices on multiple scales: It allows the calculation of quantum optical figures of merit and the spatially resolved simulation of the current flow in realistic semiconductor device geometries in a unified way. We construct the interface between both theories in such a way, that the resulting hybrid system obeys the fundamental axioms of (non-)equilibrium thermodynamics. We show that our approach guarantees the conservation of charge, consistency with the thermodynamic equilibrium and the second law of thermodynamics. The feasibility of the approach is demonstrated by numerical simulations of an electrically driven single-photon source based on a single quantum dot in the stationary and transient operation regime.

M. Liero, A. Mielke, M.A. Peletier, D.R.M. Renger, On microscopic origins of generalized gradient structures, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1--35, DOI 10.3934/dcdss.2017001 .
Abstract
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

CH. Dörlemann, M. Heida, B. Schweizer, Transmission conditions for the Helmholtz equation in perforated domains, Vietnam Journal of Mathematics, 45 (2017), pp. 241--253, DOI 10.1007/s10013-016-0222-y .

M. Dziwnik, A. Münch, B. Wagner, An anisotropic phase-field model for solid-state dewetting and its sharp-interface limit, Nonlinearity, 30 (2017), pp. 1465--1496.
Abstract
We propose a phase field model for solid state dewetting in form of a Cahn-Hilliard equation with weakly anisotropic surface energy and a degenerate mobility together with a free boundary condition at the film-substrate contact line. We derive the corresponding sharp interface limit via matched asymptotic analysis involving multiple inner layers. The resulting sharp interface model is consistent with the pure surface diffusion model. In addition, we show that the natural boundary conditions, as indicated from the first variation of the total free energy, imply a contact angle condition for the dewetting front, which, in the isotropic case, is consistent with the well-known Young's equation.

A. Münch, B. Wagner, L.P. Cook, R.R. Braun, Apparent slip for an upper convected Maxwell fluid, SIAM Journal on Applied Mathematics, 77 (2017), pp. 537--564, DOI 10.1137/16M1056869 .
Abstract
In this study the flow field of a nonlocal, diffusive upper convected Maxwell (UCM) fluid with a polymer in a solvent undergoing shearing motion is investigated for pressure driven planar channel flow and the free boundary problem of a liquid layer on a solid substrate. For large ratios of the zero shear polymer viscosity to the solvent viscosity, it is shown that channel flows exhibit boundary layers at the channel walls. In addition, for increasing stress diffusion the flow field away from the boundary layers undergoes a transition from a parabolic to a plug flow. Using experimental data for the wormlike micelle solutions CTAB/NaSal and CPyCl/NaSal, it is shown that the analytic solution of the governing equations predicts these signatures of the velocity profiles. Corresponding flow structures and transitions are found for the free boundary problem of a thin layer sheared along a solid substrate. Matched asymptotic expansions are used to first derive sharp-interface models describing the bulk flow with expressions for an em apparent slip for the boundary conditions, obtained by matching to the flow in the boundary layers. For a thin film geometry several asymptotic regimes are identified in terms of the order of magnitude of the stress diffusion, and corresponding new thin film models with a slip boundary condition are derived.

A. Caiazzo, F. Caforio, G. Montecinos, L.O. Müller, P.J. Blanco, E.F. Toro, Assessment of reduced order Kalman filter for parameter identification in one-dimensional blood flow models using experimental data, International Journal of Numerical Methods in Biomedical Engineering, 33 (2017), pp. e2843/1--e2843/26, DOI 10.1002/cnm.2843 .
Abstract
This work presents a detailed investigation of a parameter estimation approach based on the reduced order unscented Kalman filter (ROUKF) in the context of one-dimensional blood flow models. In particular, the main aims of this study are (i) to investigate the effect of using real measurements vs. synthetic data (i.e., numerical results of the same in silico model, perturbed with white noise) for the estimation and (ii) to identify potential difficulties and limitations of the approach in clinically realistic applications in order to assess the applicability of the filter to such setups. For these purposes, our numerical study is based on the in vitro model of the arterial network described by [Alastruey et al. 2011, J. Biomech. bf 44], for which experimental flow and pressure measurements are available at few selected locations. In order to mimic clinically relevant situations, we focus on the estimation of terminal resistances and arterial wall parameters related to vessel mechanics (Young's modulus and thickness) using few experimental observations (at most a single pressure or flow measurement per vessel). In all cases, we first perform a theoretical identifiability analysis based on the generalized sensitivity function, comparing then the results obtained with the ROUKF, using either synthetic or experimental data, to results obtained using reference parameters and to available measurements.

W. Dreyer, C. Guhlke, M. Landstorfer, R. Müller, New insights on the interfacial tension of electrochemical interfaces and the Lippmann equation, European Journal of Applied Mathematics, 29 (2018), pp. 708--753, DOI 10.1017/S0956792517000341 .
Abstract
The Lippmann equation is considered as universal relationship between interfacial tension, double layer charge, and cell potential. Based on the framework of continuum thermo-electrodynamics we provide some crucial new insights to this relation. In a previous work we have derived a general thermodynamic consistent model for electrochemical interfaces, which showed a remarkable agreement to single crystal experimental data. Here we apply the model to a curved liquid metal electrode. If the electrode radius is large compared to the Debye length, we apply asymptotic analysis methods and obtain the Lippmann equation. We give precise definitions of the involved quantities and show that the interfacial tension of the Lippmann equation is composed of the surface tension of our general model, and contributions arising from the adjacent space charge layers. This finding is confirmed by a comparison of our model to experimental data of several mercury-electrolyte interfaces. We obtain qualitative and quantitative agreement in the 2V potential range for various salt concentrations. We also discuss the validity of our asymptotic model when the electrode curvature radius is comparable to the Debye length.

E. Meca Álvarez, A. Münch, B. Wagner, Sharp-interface formation during lithium intercalation into silicon, European Journal of Applied Mathematics, 29 (2018), pp. 118--145, DOI 10.1017/S0956792517000067 .
Abstract
In this study we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as a-Si. The governing equations couple a viscous Cahn-Hilliard-Reaction model with elasticity in the framework of the Cahn-Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface in the bulk of the layer intersects the free boundary. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

A. Mielke, C. Patz, Uniform asymptotic expansions for the infinite harmonic chain, Zeitschrift fur Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 36 (2017), pp. 437--475, DOI 10.4171/ZAA/1596 .
Abstract
We study the dispersive behavior of waves in linear oscillator chains. We show that for general general dispersions it is possible to construct an expansion such that the remainder can be estimated by $1/t$ uniformly in space. In particalur we give precise asymptotics for the transition from the $1/t^1/2$ decay of nondegenerate wave numbers to the generate $1/t^1/3$ decay of generate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.

M. Heida, B. Schweizer, Non-periodic homogenization of infinitesimal strain plasticity equations, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 96 (2016), pp. 5--23.

S. Heinz, A. Mielke, Existence, numerical convergence, and evolutionary relaxation for a rate-independent phase-transformation model, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 374 (2016), pp. 20150171/1--20150171/23, DOI 10.1098/rsta.2015.0171 .
Abstract
We revisit the two-well model for phase transformation in a linearly elastic body introduced and studied in A. Mielke, F. Theil, and V.I. Levita “A variational formulation of rate--independent phase transformations using an extremum principle", Arch. Rational Mech. Anal., 162, 137-177, 2002 ([MTL02]). This energetic rate-independent model is posed in terms of the elastic displacement and an internal variable that gives the phase portion of the second phase. We use a new approach based on mutual recovery sequences, which are adjusted to a suitable energy increment plus the associated dissipated energy and, thus, enable us to pass to the limit in the construction of energetic solutions. We give three distinct constructions of mutual recovery sequences which allow us (i) to generalize the existence result in [MTL02], (ii) to establish the convergence of suitable numerical approximations via space-time discretization, and (iii) to perform the evolutionary relaxation from the pure-state model to the relaxed mixture model. All these results rely on weak converge and involve the H-measure as an essential tool.

M. Khodayari, P. Reinsberg, A.A. Abd-El-Latif, Ch. Merdon, J. Fuhrmann, H. Baltruschat, Determining solubility and diffusivity by using a flow cell coupled to a mass spectrometer, ChemPhysChem, 17 (2016), pp. 1647--1655.

M. Korzec, A. Münch, E. Süli, B. Wagner, Anisotropy in wavelet based phase field models, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 21 (2016), pp. 1167--1187.
Abstract
Anisotropy is an essential feature of phase-field models, in particular when describing the evolution of microstructures in solids. The symmetries of the crystalline phases are reflected in the interfacial energy by introducing corresponding directional dependencies in the gradient energy coefficients, which multiply the highest order derivative in the phase-field model. This paper instead considers an alternative approach, where the anisotropic gradient energy terms are replaced by a wavelet analogue that is intrinsically anisotropic and linear. In our studies we focus on the classical coupled temperature - Ginzburg-Landau type phase-field model for dendritic growth. For the resulting derivative-free wavelet analogue existence, uniqueness and continuous dependence on initial data for weak solutions is proved. The ability to capture dendritic growth similar to the results obtained from classical models is investigated numerically.

S. Reichelt, Error estimates for elliptic equations with not exactly periodic coefficients, Advances in Mathematical Sciences and Applications, 25 (2016), pp. 117--131.
Abstract
This note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly ε-periodic and the ellipticity constant may degenerate for vanishing ε. Here ε>0 denotes the ratio between the microscopic and the macroscopic length scale. It is shown that for degenerating and non-degenerating coefficients the error between the original solution and the effective solution is of order √ε. Therefore suitable test functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions on the given data and the effective solution with respect to the macroscopic scale.

W. Dreyer, C. Guhlke, R. Müller, A new perspective on the electron transfer: Recovering the Butler--Volmer equation in non-equilibrium thermodynamics, Physical Chemistry Chemical Physics, 18 (2016), pp. 24966--24983, DOI 10.1039/C6CP04142F .
Abstract
Understanding and correct mathematical description of electron transfer reaction is a central question in electrochemistry. Typically the electron transfer reactions are described by the Butler-Volmer equation which has its origin in kinetic theories. The Butler-Volmer equation relates interfacial reaction rates to bulk quantities like the electrostatic potential and electrolyte concentrations. Since in the classical form, the validity of the Butler-Volmer equation is limited to some simple electrochemical systems, many attempts have been made to generalize the Butler-Volmer equation. Based on non-equilibrium thermodynamics we have recently derived a reduced model for the electrode-electrolyte interface. This reduced model includes surface reactions but does not resolve the charge layer at the interface. Instead it is locally electroneutral and consistently incorporates all features of the double layer into a set of interface conditions. In the context of this reduced model we are able to derive a general Butler-Volmer equation. We discuss the application of the new Butler-Volmer equations to different scenarios like electron transfer reactions at metal electrodes, the intercalation process in lithium-iron-phosphate electrodes and adsorption processes. We illustrate the theory by an example of electroplating.

A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Journal of Non-Equilibrium Thermodynamics, 41 (2016), pp. 141--149.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

K. Disser, M. Liero, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, Networks and Heterogeneous Media, 10 (2015), pp. 233-253.
Abstract
We study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then, we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order.

D. Peschka, Thin-film free boundary problems for partial wetting, Journal of Computational Physics, 295 (2015), pp. 770--778.
Abstract
We present a novel framework to solve thin-film equations with an explicit non-zero contact angle, where the support of the solution is treated as an unknown. The algorithm uses a finite element method based on a gradient formulation of the thin-film equations coupled to an arbitrary Lagrangian-Eulerian method for the motion of the support. Features of this algorithm are its simplicity and robustness. We apply this algorithm in 1D and 2D to problems with surface tension, contact angles and with gravity.

S. Yanchuk, L. Lücken, M. Wolfrum, A. Mielke, Spectrum and amplitude equations for scalar delay-differential equations with large delay, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 537--553.
Abstract
The subject of the paper are scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold.

P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Vanishing viscosities and error estimate for a Cahn--Hilliard type phase field system related to tumor growth, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 26 (2015), pp. 93--108.
Abstract
In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn--Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [Colli-Gilardi-Hilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.

M.G. Hennessy, V.M. Burlakov, A. Münch, B. Wagner, A. Goriely, Controlled topological transitions in thin-film phase separation, SIAM Journal on Applied Mathematics, 75 (2015), pp. 38--60.
Abstract
In this paper the evolution of a binary mixture in a thin-film geometry with a wall at the top and bottom is considered. Bringing the mixture into its miscibility gap so that no spinodal decomposition occurs in the bulk, a slight energetic bias of the walls towards each one of the constituents ensures the nucleation of thin boundary layers that grow until the constituents have moved into one of the two layers. These layers are separated by an interfacial region, where the composition changes rapidly. Conditions that ensure the separation into two layers with a thin interfacial region are investigated based on a phase-field model and using matched asymptotic expansions a corresponding sharp-interface problem for the location of the interface is established. It is then argued that a thus created two-layer system is not at its energetic minimum but destabilizes into a controlled self-replicating pattern of trapezoidal vertical stripes by minimizing the interfacial energy between the phases while conserving their area. A quantitative analysis of this mechanism is carried out via a new thin-film model for the free interfaces, which is derived asymptotically from the sharp-interface model.

R. Huth, S. Jachalski, G. Kitavtsev, D. Peschka, Gradient flow perspective on thin-film bilayer flows, Journal of Engineering Mathematics, 94 (2015), pp. 43--61.
Abstract
We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thin-film model and formally identify solutions with those of the corresponding free-boundary problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using Gamma-convergence. For time-dependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions.

CH. Bayer, H.A. Hoel, A. Kadir, P. Plechac, M. Sandberg, A. Szepessy, Computational error estimates for Born--Oppenheimer molecular dynamics with nearly crossing potential surfaces, Applied Mathematics Research Express, 2015 (2015), pp. 329--417.

A. Caiazzo, I. Ramis-Conde, Multiscale modeling of palisade formation in glioblastoma multiforme, Journal of Theoretical Biology, 383 (2015), pp. 145--156.
Abstract
Palisades are characteristic tissue aberrations that arise in glioblastomas. Observation of palisades is considered as a clinical indicator of the transition from a noninvasive to an invasive tumour. In this article we propose a computational model to study the influence of genotypic and phenotypic heterogeneity in palisade formation. For this we produced three dimensional realistic simulations, based on a multiscale hybrid model, coupling the evolution of tumour cells and the oxygen diffusion in tissue, that depict the shape of palisades during its formation. Our results can be summarized as the following: (1) we show that cell heterogeneity is a crucial factor in palisade formation and tumour growth; (2) we present results that can explain the observed fact that recursive tumours are more malignant than primary tumours; and (3) the presented simulations can provide to clinicians and biologists for a better understanding of palisades 3D structure as well as glioblastomas growth dynamics

W. Dreyer, C. Guhlke, R. Müller, Modeling of electrochemical double layers in thermodynamic non-equilibrium, Physical Chemistry Chemical Physics, 17 (2015), pp. 27176--27194, DOI 10.1039/C5CP03836G .
Abstract
We consider the contact between an electrolyte and a solid electrode. At first we formulate a thermodynamic consistent model that resolves boundary layers at interfaces. The model includes charge transport, diffusion, chemical reactions, viscosity, elasticity and polarization under isothermal conditions. There is a coupling between these phenomena that particularly involves the local pressure in the electrolyte. Therefore the momentum balance is of major importance for the correct description of the layers.

The width of the boundary layers is typically very small compared to the macroscopic dimensions of the system. In a second step we thus apply the method of asymptotic analysis to derive a simpler reduced model that does not resolve the boundary layers but instead incorporates the electrochemical properties of the layers into a set of new boundary conditions. For a metal-electrolyte interface, we derive a qualitative description of the double layer capacitance without the need to resolve space charge layers.

H. Hanke, D. Knees, Homogenization of elliptic systems with non-periodic, state dependent coefficients, Asymptotic Analysis, 92 (2015), pp. 203--234.
Abstract
In this paper, a homogenization problem for an elliptic system with non-periodic, state dependent coefficients representing microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by ε > 0. The aim is the derivation of an effective model by investigating the limit process ε → 0 of the state functions rigorously. The effective model is independent of the parameter ε > 0 but preserves the microscopic structure of the state functions (ε > 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the non-periodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to time-dependent damage models (discussed in a future paper), we provide a Γ-convergence result of sequences of functionals being related to the previous microscopic models with state dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided.

A. Mielke, Deriving amplitude equations via evolutionary Gamma convergence, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 2679--2700.
Abstract
We discuss the justification of the Ginzburg-Landau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional Swift-Hohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary Gamma convergence by reformulating both equations as gradient systems. Using a suitable linear transformation we show Gamma convergence of the associated energies in suitable function spaces. The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savaré 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in L², while for the case of a quadratic nonlinearity we need to impose weak convergence in H¹. However, we do not need wellpreparedness of the initial conditions.

C. Kreisbeck, L. Mascarenhas, Asymptotic spectral analysis in semiconductor nanowire heterostructures, Applicable Analysis. An International Journal, (published online on June 2, 2014), DOI 10.1080/00036811.2014.919052 .

S. Jachalski, G. Kitavtsev, R. Taranets, Weak solutions to lubrication systems describing the evolution of bilayer thin films, Communications in Mathematical Sciences, 12 (2014), pp. 527--544.
Abstract
The existence of global nonnegative weak solutions is proved for coupled one-dimen- sional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown.

G. Aki, W. Dreyer, J. Giesselmann, Ch. Kraus, A quasi-incompressible diffuse interface model with phase transition, Mathematical Models & Methods in Applied Sciences, 24 (2014), pp. 827--861.
Abstract
This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier-Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier-Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.

P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A vanishing diffusion limit in a nonstandard system of phase field equations, Evolution Equations and Control Theory, 3 (2014), pp. 257--275.
Abstract
We are concerned with a nonstandard phase field model of Cahn--Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011, and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient $sigma$ in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.

P. Hornung, S. Neukamm, I. Velcic, Derivation of a homogenized nonlinear plate theory from 3D elasticity, Calculus of Variations and Partial Differential Equations, 51 (2014), pp. 677--699.

A. Caiazzo, J. Mura, Multiscale modeling of weakly compressible elastic materials in harmonic regime and application to microscale structure estimation, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12 (2014), pp. 514--537.
Abstract
This article is devoted to the modeling of elastic materials composed by an incompressible elastic matrix and small compressible gaseous inclusions, under a time harmonic excitation. In a biomedical context, this model describes the dynamics of a biological tissue (e.g. lung or liver) when wave analysis methods (such as Magnetic Resonance Elastography) are used to estimate tissue properties. Due to the multiscale nature of the problem, direct numerical simulations are prohibitive. We extend the homogenized model introduced in [Baffico, Grandmont, Maday, Osses, SIAM J. Mult. Mod. Sim., 7(1), 2008] to a time harmonic regime to describe the solid-gas mixture from a macroscopic point of view in terms of an effective elasticity tensor. Furthermore, we derive and validate numerically analytical approximations for the effective elastic coefficients in terms of macroscopic parameters. This simplified description is used to to set up an efficient variational approach for the estimation of the tissue porosity, using the mechanical response to external harmonic excitations.

W. Dreyer, J. Giesselmann, Ch. Kraus, A compressible mixture model with phase transition, Physica D. Nonlinear Phenomena, 273--274 (2014), pp. 1--13.
Abstract
We introduce a new thermodynamically consistent diffuse interface model of Allen-Cahn/Navier-Stokes type for multi-component flows with phase transitions and chemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized Allen-Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a Young-Laplace and a Stefan type law.

A. Mielke, S. Reichelt, M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Networks Heterogeneous Media, 9 (2014), pp. 353--382.
Abstract
We derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.

S. Neukamm, H. Olbermann, Homogenization of the nonlinear bending theory for plates, Calculus of Variations and Partial Differential Equations, (published online on Sept. 14, 2014), DOI 10.1007/s00526-014-0765-2 .
Abstract
We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gamma-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions, using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.

S. Neukamm, I. Velcic, Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 2701--2748.
Abstract
We rigorously derive a homogenized von-Kármán plate theory as a Gamma-limit from nonlinear three-dimensional elasticity by combining homogenization and dimension reduction. Our starting point is an energy functional that describes a nonlinear elastic, three-dimensional plate with spatially periodic material properties. The functional features two small length scales: the period $epsilon$ of the elastic composite material, and the thickness h of the slender plate. We study the behavior as $epsilon$ and h simultaneously converge to zero in the von-Kármán scaling regime. The obtained limit is a homogenized von-Kármán plate model. Its effective material properties are determined by a relaxation formula that exposes a non-trivial coupling of the behavior of the out-of-plane displacement with the oscillatory behavior in the in-plane directions. In particular, the homogenized coefficients depend on the relative scaling between h and $epsilon$, and different values arise for h<<$epsilon$, $epsilon$   h and $epsilon$ << h.

P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, An asymptotic analysis for a nonstandard Cahn--Hilliard system with viscosity, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013), pp. 353--368.
Abstract
This paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter $rho$ and the chemical potential $mu$; each equation includes a viscosity term -- respectively, $varepsilon,partial_tmu$ and $delta,partial_trho$ -- with $varepsilon$ and $delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is well-posed and investigated the long-time behavior of its $(varepsilon,delta)-$solutions. Here we discuss the asymptotic limit of the system as $eps$ tends to 0. We prove convergence of $(varepsilon,delta)-$solutions to the corresponding solutions for the case $eps$ =0, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.

P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn--Hilliard system with viscosity, Journal of Differential Equations, 254 (2013), pp. 4217--4244.
Abstract
Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic [19]; in the balance equations of microforces and microenergy, the two unknowns are the order parameter $rho$ and the chemical potential $mu$. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phase-field equation for $rho$ is coupled with a genuinely nonlinear diffusion equation for $mu$. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of costant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables.

M.A. Peletier, M. Renger, M. Veneroni, Variational formulation of the Fokker--Planck equation with decay: A particle approach, Communications in Contemporary Mathematics, 15 (2013), pp. 1350017/1--1350017/43.

A. Mielke, E. Rohan, Homogenization of elastic waves in fluid-saturated porous media using the Biot model, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 873--916.
Abstract
We consider periodically heterogeneous fluid-saturated poroelastic media described by the Biot model with inertia effects. The weak and semistrong formulations for displacement, seepage and pressure fields involve three equations expressing the momentum and mass balance and the Darcy law. Using the two-scale homogenization method we obtain the limit two-scale problem and prove the existence and uniqueness of its weak solutions. The Laplace transformation in time is used to decouple the macroscopic and microscopic scales. It is shown that the seepage velocity is eliminated form the macroscopic equations involving strain and pressure fields only. The plane harmonic wave propagation is studied using an example of layered medium. Illustrations show some influence of the orthotropy on the dispersion phenomena.

A. Mielke, U. Stefanelli, Linearized plasticity is the evolutionary Gamma limit of finite plasticity, Journal of the European Mathematical Society (JEMS), 15 (2013), pp. 923--948.
Abstract
We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via Gamma-convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity

M. Liero, Th. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via $Gamma$-convergence, NoDEA. Nonlinear Differential Equations and Applications, 19 (2012), pp. 437--457.
Abstract
This paper deals with dimension reduction in linearized elastoplasticity in the rate-independent case. The reference configuration of the elastoplastic body is given by a two-dimensional middle surface and a small but positive thickness. We derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations which are coupled via plastic strains. The convergence analysis is based on an abstract Gamma convergence theory for rate-independent evolution formulated in the framework of energetic solutions. This concept is based on an energy-storage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance.

S. Arnrich, A. Mielke, M.A. Peletier, G. Savaré, M. Veneroni, Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction, Calculus of Variations and Partial Differential Equations, 44 (2012), pp. 419--454.
Abstract
We study a singular-limit problem arising in the modelling of chemical reactions. At finite $e>0$, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by $1/e$, and in the limit $eto0$, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savaré, and Veneroni, em SIAM Journal on Mathematical Analysis, 42(4):1805--1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular, we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the propety of being a emphcurve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the $e$-problem converge to a solution of the limiting problem.

P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Continuous dependence for a nonstandard Cahn--Hilliard system with nonlinear atom mobility, Rendiconti del Seminario Matematico. Universita e Politecnico Torino, 70 (2012), pp. 27--52.
Abstract
This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice [Podio-Guidugli 2006]; it consists of the balance equations of microforces and microenergy; the two unknowns are the order parameter $rho$ and the chemical potential $mu$. Some recent results obtained for this class of problems is reviewed and, in the case of a nonconstant and nonlinear atom mobility, uniqueness and continuous dependence on the initial data are shown with the help of a new line of argumentation developed in Colli/Gilardi/Podio-Guidugli/Sprekels 2012.

P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence for a strongly coupled Cahn--Hilliard system with viscosity, Bollettino della Unione Matematica Italiana. Serie 9, 5 (2012), pp. 495--513.
Abstract
An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in [CGPS11]. Both systems conform to the general theory developed in [Pod06]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter $rho$ and the chemical potential $mu$. In the system studied in this note, a phase-field equation in $rho$ fairly more general than in [CGPS11] is coupled with a highly nonlinear diffusion equation for $mu$, in which the conductivity coefficient is allowed to depend nonlinearly on both variables.

W. Dreyer, F. Duderstadt, M. Hantke, G. Warnecke, Bubbles in liquids with phase transition, Continuum Mechanics and Thermodynamics, 24 (2012), pp. 461--483.
Abstract
We consider a bubble of vapor and inert gas surrounded by the corresponding liquid phase. We study the behavior of the bubble due to phase change, i.e. condensation and evaporation, at the interface. Special attention is given to the effects of surface tension and heat production on the bubble dynamics as well as the propagation of acoustic elastic waves by including slight compressibility of the liquid phase. Separately we study the influence of the three phenomena heat conduction, elastic waves, and phase transition on the evolution of the bubble. The objective is to derive relations including the mass, momentum, and energy transfer between the phases. We find ordinary differential equations, in the cases of heat transfer and the emission of acoustic waves partial differential equations, that describe the bubble dynamics. From numerical evidence we deduce that the effect of phase transition and heat transfer on the behavior of the radius of the bubble is negligible. It turns out that the elastic waves in the liquid are of greatest importance to the dynamics of the bubble radius. The phase transition has a strong influence on the evolution of the temperature, in particular at the interface. Furthermore the phase transition leads to a drastic change of the water content in the bubble, so that a rebounding bubble is only possible, if it contains in addition an inert gas. In a forthcoming paper the equations derived are sought in order to close equations for multi-phase mixture balance laws for dispersed bubbles in liquids involving phase change. Also the model is used to make comparisons with experimental data on the oscillation of a laser induced bubble. For this case it was necessary to include the effect of an inert gas in the thermodynamic modeling of the phase transition.

G. Aki, J. Daube, W. Dreyer, J. Giesselmann, M. Kränkel, Ch. Kraus, A diffuse interface model for quasi-incompressible flows: Sharp interface limits and numerics, ESAIM Proceedings, 38 (2012), pp. 54--77.
Abstract
In this contribution, we investigate a diffuse interface model for quasi-incompressible flows. We determine corresponding sharp interface limits of two different scalings. The sharp interface limit is deduced by matched asymptotic expansions of the fields in powers of the interface. In particular, we study solutions of the derived system of inner equations and discuss the results within the general setting of jump conditions for sharp interface models. Furthermore, we treat, as a subproblem, the convective Cahn-Hilliard equation numerically by a Local Discontinuous Galerkin scheme.

W. Dreyer, J. Giesselmann, Ch. Kraus, Ch. Rohde, Asymptotic analysis for Korteweg models, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 14 (2012), pp. 105--143.
Abstract
This paper deals with a sharp interface limit of the isothermal Navier-Stokes-Korteweg system. The sharp interface limit is performed by matched asymptotic expansions of the fields in powers of the interface width. These expansions are considered in the interfacial region (inner expansions) and in the bulk (outer expansion) and are matched order by order. Particularly we consider the first orders of the corresponding inner equations obtained by a change of coordinates in an interfacial layer. For a specific scaling we establish solvability criteria for these inner equations and recover the results within the general setting of jump conditions for sharp interface models.

A. Mielke, T. Roubíček, M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, Journal of Elasticity. The Physical and Mathematical Science of Solids, 109 (2012), pp. 235--273.
Abstract
Brittle Griffith-type delamination of compounds is deduced by means of Gamma-convergence from partial, isotropic damage of three-specimen-sandwich-structures by flattening the middle component to the thickness 0. The models used here allow for nonlinearly elastic materials at small strains and consider the processes to be unidirectional and rate-independent. The limit passage is performed via a double limit: first, we gain a delamination model involving the gradient of the delamination variable, which is essential to overcome the lack of a uniform coercivity arising from the passage from partial damage to delamination. Second, the delamination-gradient is supressed. Noninterpenetration- and transmission-conditions along the interface are obtained.

A. Mielke, L. Truskinovsky, From discrete visco-elasticity to continuum rate-independent plasticity: Rigorous results, Archive for Rational Mechanics and Analysis, 203 (2012), pp. 577--619.
Abstract
We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a visco-elastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete to continuum limit. In the course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent dissipation of visco-elastic solid transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize ideas we employ in our proofs the simplest prototypical system describing transformational plasticity of shape-memory alloys. The approach, however, is sufficiently general and can be used for similar reductions in the cases of more general plasticity and damage models.

M. Liero, A. Mielke, An evolutionary elastoplastic plate model derived via $Gamma$-convergence, Mathematical Models & Methods in Applied Sciences, 21 (2011), pp. 1961--1986.
Abstract
This paper is devoted to dimension reduction for linearized elastoplasticity in the rate-independent case. The reference configuration of the three-dimensional elastoplastic body has a two-dimensional middle surface and a positive but small thickness. Under suitable scalings we derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations (linear Kirchhoff--Love plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Gamma-convergence theory for rate-independent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energy-storage functional and a dissipation functional, such that energetic solutions are defined in terms of a stability condition and an energy balance. The Mosco convergence of the quadratic energy-storage functional follows the arguments of the elastic case. To handle the evolutionary situation the interplay with the dissipation functional is controlled by cancellation properties for Mosco-convergent quadratic energies.

K. Hermsdörfer, Ch. Kraus, D. Kröner, Interface conditions for limits of the Navier--Stokes--Korteweg model, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 13 (2011), pp. 239--254.
Abstract
In this contribution we will study the behaviour of the pressure across phase boundaries in liquid-vapour flows. As mathematical model we will consider the static version of the Navier-Stokes-Korteweg model which belongs to the class of diffuse interface models. From this static equation a formula for the pressure jump across the phase interface can be derived. If we perform then the sharp interface limit we see that the resulting interface condition for the pressure seems to be inconsistent with classical results of hydrodynamics. Therefore we will present two approaches to recover the results of hydrodynamics in the sharp interface limit at least for special situations.

G. Kitavtsev, L. Recke, B. Wagner, Center manifold reduction approach for the lubrication equation, Nonlinearity, 24 (2011), pp. 2347--2369.
Abstract
The goal of this study is the reduction of the lubrication equation, modelling thin film dynamics, onto an approximate invariant manifold. The reduction is derived for the physical situation of the late phase evolution of a dewetting thin liquid film, where arrays of droplets connected by an ultrathin film of thickness $eps$ undergo a slow-time coarsening dynamics. With this situation in mind, we construct an asymptotic approximation of the corresponding invariant manifold, that is parametrized by a family of droplet pressures and positions, in the limit when $epsto 0$. The approach is inspired by the paper by Mielke and Zelik [Mem. Amer. Math. Soc., Vol. 198, 2009], where the center manifold reduction was carried out for a class of semilinear systems. In this study this approach is considered for quasilinear degenerate parabolic PDE's such as lubrication equations. While it has previously been shown by Glasner and Witelski [Phys. Rev. E, Vol. 67, 2003], that the system of ODEs governing the coarsening dynamics, can be obtained via formal asymptotic methods, the center manifold reduction approach presented here, pursues the rigorous justification of this asymptotic limit.

A. Münch, B. Wagner, Impact of slippage on the morphology and stability of a dewetting rim, Journal of Physics: Condensed Matter, 23 (2011), pp. 184101/1--184101/12.
Abstract
In this study lubrication theory is used to describe the stability and morphology of the rim that forms as a thin polymer film dewets from a hydrophobized silicon wafer. Thin film equations are derived from the governing hydrodynamic equations for the polymer to enable the systematic mathematical and numerical analysis of the properties of the solutions for different regimes of slippage and for a range of time scales. Dewetting rates and the cross sectional profiles of the evolving rims are derived for these models and compared to experimental results. Experiments also show that the rim is typically unstable in the spanwise direction and develops thicker and thinner parts that may grow into “fingers”. Linear stability analysis as well as nonlinear numerical solutions are presented to investigate shape and growth rate of the rim instability. It is demonstrated that the difference in morphology and the rate at which the instability develops can be directly attributed to the magnitude of slippage. Finally, a derivation is given for the dominant wavelength of the bulges along the unstable rim.

A. Münch, C.P. Please, B. Wagner, Spin coating of an evaporating polymer solution, Physics of Fluids, 23 (2011), pp. 102101/1--102101/12.
Abstract
We consider a mathematical model of spin coating of a single polymer blended in a solvent. The model describes the one-dimensional development of the thin layer of the mixture as the layer thins due to flow created by a balance of viscous forces and centrifugal forces and due to evaporation of the solvent. In the model both the diffusivity of the solvent in the polymer and the viscosity of the mixture are very rapidly varying functions of the solvent volume fraction. Guided by numerical solutions an asymptotic analysis reveals a number of different possible behaviours of the thinning layer dependent on the nondimensional parameters describing the system. The main practical interest is in controlling the appearance and development of a “skin” on the polymer where the solvent concentration reduces rapidly on the outer surface leaving the bulk of the layer still with high concentrations of solvent. The critical parameters controlling this behaviour are found to be $eps$ the ratio of the diffusion to advection time scales, $delta$ the ratio of the evaporation to advection time scales and $exp(-gamma)$, the ratio of the diffusivity of the initial mixture and the pure polymer. In particular, our analysis shows that for very small evaporation with $delta ll exp(-3/(4gamma)) eps^3/4$ skin formation can be prevented.

H. Hanke, Homogenization in gradient plasticity, Mathematical Models & Methods in Applied Sciences, 21 (2011), pp. 1651--1684.
Abstract
This paper yields a two-scale homogenization result for a rate-independent elastoplastic system. The presented model is a generalization of the classical model of linearized elastoplacticity with hardening, which is extended by a gradient term of the plastic variables. The associated stored elastic energy density has periodically oscillating coefficients, where the period is scaled by ε > 0 . The additional gradient term of the plastic variables z is contained in the elastic energy with a prefactor ε^γ (γ ≥ 0) . We derive different limiting models for ε → 0 in dependence of &gamma ;. For γ > 1 the limiting model is the two-scale model derived in [MielkeTimofte07], where no gradient term was present. For γ = 1 the gradient term of the plastic variable survives on the microscopic cell poblem, while for γ ∈ [0,1) the limit model is defined in terms of a plastic variable without microscopic fluctuation. The latter model can be simplified to a purely macroscopic elastoplasticity model by homogenisation of the elastic part.

H. Hanke, Homogenization in gradient plasticity, GAMM-Mitteilungen, 34 (2011), pp. 102--106.
Abstract
This paper yields a two-scale homogenization result for a rate-independent elastoplastic system. The presented model is a generalization of the classical model of linearized elastoplacticity with hardening, which is extended by a gradient term of the plastic variables. The associated stored elastic energy density has periodically oscillating coefficients, where the period is scaled by ε > 0 . The additional gradient term of the plastic variables z is contained in the elastic energy with a prefactor ε^γ (γ ≥ 0) . We derive different limiting models for ε → 0 in dependence of &gamma ;. For γ > 1 the limiting model is the two-scale model derived in [MielkeTimofte07], where no gradient term was present. For γ = 1 the gradient term of the plastic variable survives on the microscopic cell poblem, while for γ ∈ [0,1) the limit model is defined in terms of a plastic variable without microscopic fluctuation. The latter model can be simplified to a purely macroscopic elastoplasticity model by homogenisation of the elastic part.

A. Mielke, Complete-damage evolution based on energies and stresses, Discrete and Continuous Dynamical Systems -- Series S, 4 (2011), pp. 423--439.
Abstract
The rate-independent damage model recently developed in Bouchitté, Mielke, Roubíček “A complete-damage problem at small strains" allows for complete damage, such that the deformation is no longer well-defined. The evolution can be described in terms of energy densities and stresses. Using concepts of parametrized Gamma convergence, we generalize the theory to convex, but non-quadratic elastic energies by providing Gamma convergence of energetic solutions from partial to complete damage under rather general conditions.

G. Kitavtsev, B. Wagner, Coarsening dynamics of slipping droplets, Journal of Engineering Mathematics, 66 (2010), pp. 271--292.
Abstract
This paper studies the late phase dewetting process of nanoscopic thin polymer films on hydrophobized substrates using some recently derived lubrication models that take account of large slippage at the polymer-substrate interface. The late phase of this process is characterized by the slow-time coarsening dynamics of arrays of droplets that remain after rupture and the initial dewetting phases. For this situation a reduced system of ordinary differential equations is derived from the lubrication model for large slippage using asymptotic analysis. This extends known results for the no-slip case. On the basis of the reduced model, the role of the slippage as a control parameter for droplet migration is analysed and several new qualitative effects for the coarsening process are identified.

D. Peschka, A. Münch, B. Niethammer, Self-similar rupture of viscous thin films in the strong-slip regime, Nonlinearity, 23 (2010), pp. 409--427.

D. Peschka, A. Münch, B. Niethammer, Thin film rupture for large slip, Journal of Engineering Mathematics, 66 (2010), pp. 33--51.
Abstract
This paper studies the rupture of thin liquid films on hydrophobic substrates, assuming large slip at the liquidsolid interface. Using a recently developed em strong slip lubrication model, it is shown that the rupture passes through up to three self-similar regimes with different dominant balances and different scaling exponents. For one of these regimes the similarity is of second kind, and the similarity exponent is determined by solving a boundary value problem for a nonlinear ODE. For this regime we also prove finite-time rupture.

H. Garcke, Ch. Kraus, An anisotropic, inhomogeneous, elastically modified Gibbs--Thomson law as singular limit of a diffuse interface model, Advances in Mathematical Sciences and Applications, 20 (2010), pp. 511--545.
Abstract
We consider the sharp interface limit of a diffuse phase field model with prescribed total mass taking into account a spatially inhomogeneous anisotropic interfacial energy and an elastic energy. The main aim is the derivation of a weak formulation of an anisotropic, inhomogeneous, elastically modified Gibbs-Thomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the Euler-Lagrange equation of the diffuse phase field energy.

J. Giannoulis, A. Mielke, Ch. Sparber, High-frequency averaging in semi-classical Hartree-type equations, Asymptotic Analysis, 70 (2010), pp. 87--100.
Abstract
We investigate the asymptotic behavior of solutions to semi-classical Schröodinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.

W. Dreyer, Ch. Kraus, On the van der Waals--Cahn--Hilliard phase-field model and its equilibria conditions in the sharp interface limit, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 140 A (2010), pp. 1161--1186.
Abstract
We study the equilibria of liquid--vapor phase transitions of a single substance at constant temperature and relate the sharp interface model of classical thermodynamics to a phase field model that determines the equilibria by the stationary van der Waals--Cahn--Hilliard theory.
For two reasons we reconsider this old problem. 1. Equilibria in a two phase system can be established either under fixed total volume of the system or under fixed external pressure. The latter case implies that the domain of the two--phase system varies. However, in the mathematical literature rigorous sharp interface limits of phase transitions are usually considered under fixed volume. This brings the necessity to extend the existing tools for rigorous sharp interface limits to changing domains since in nature most processes involving phase transitions run at constant pressure. 2. Thermodynamics provides for a single substance two jump conditions at the sharp interface, viz. the continuity of the specific Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. The existing estimates for rigorous sharp interface limits show only the first condition. We identify the cause of this phenomenon and develop a strategy that yields both conditions up to the first order.
The necessary information on the equilibrium conditions are achieved by an asymptotic expansion of the density which is valid for an arbitrary double well potential. We establish this expansion by means of local energy estimates, uniform convergence results of the density and estimates on the Laplacian of the density.

A. Mielke, C. Patz, Dispersive stability of infinite dimensional Hamiltonian systems on lattices, Applicable Analysis. An International Journal, 89 (2010), pp. 1493--1512.
Abstract
We derive dispersive stability results for oscillator chains like the FPU chain or the discrete Klein-Gordon chain. If the nonlinearity is of degree higher than 4, then small localized initial data decay like in the linear case. For this, we provide sharp decay estimates for the linearized problem using oscillatory integrals and avoiding the nonoptimal interpolation between different $ell^p$ spaces.

A. Mielke, T. Roubíček, J. Zeman, Complete damage in elastic and viscoelastic media and its energetics, Computer Methods in Applied Mechanics and Engineering, 199 (2010), pp. 1242--1253.
Abstract
A model for the evolution of damage that allows for complete disintegration is addressed. Small strains and a linear response function are assumed. The “flow rule” for the damage parameter is rate-independent. The stored energy involves the gradient of the damage variable, which determines an internal length-scale. Quasi-static fully rate-independent evolution is considered as well as rate-dependent evolution including viscous/inertial effects. Illustrative 2-dimensional computer simulations are presented, too.

D. Peschka, A. Münch, B. Niethammer, Self-similar rupture of viscous thin films in the strong-slip regime, Nonlinearity, 23 (2010), pp. 409--427.
Abstract
We consider rupture of thin viscous films in the strong-slip regime with small Reynolds numbers. Numerical simulations indicate that near the rupture point viscosity and van-der-Waals forces are dominant and that there are self-similar solutions of the second kind. For a corresponding simplified model we rigorously analyse self-similar behaviour. There exists a one-parameter family of self-similar solutions and we establish necessary and sufficient conditions for convergence to any self-similar solution in a certain parameter regime. We also present a conjecture on the domains of attraction of all self-similar solutions which is supported by numerical simulations.

J.R. King, A. Münch, B. Wagner, Linear stability analysis of a sharp-interface model for dewetting thin films, Journal of Engineering Mathematics, 63 (2009), pp. 177--195.
Abstract
The topic of this study concerns the stability of the three-phase contact-line of a dewetting thin liquid film on a hydrophobised substrate driven by van der Waals forces. The role of slippage in the emerging instability at the three-phase contact-line is studied by deriving a sharp-interface model for the dewetting thin film via matched asymptotic expansions. This allows for a derivation of travelling waves and their linear stability via eigenmode analysis. In contrast to the dispersion relations typically encountered for the finger-instabilty, where the dependence of the growth rate on the wave number is quadratic, here it is linear. Using the separation of time scales of the slowly growing rim of the dewetting film and time scale on which the contact line destabilises, the sharp-interface results are compared to earlier results for the full lubrication model and good agreement for the most unstable modes is obtained.

S. Zelik, A. Mielke, Multi-pulse evolution and space-time chaos in dissipative systems, Memoirs of the American Mathematical Society, 198 (2009), pp. 1--97.

K. Afanasiev, A. Münch, B. Wagner, Thin film dynamics on a vertically rotating disk partially immersed in a liquid bath, Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems. Elsevier Science Inc., New York, NY. English, English abstracts., 32 (2008), pp. 1894-1911.
Abstract
The axisymmetric flow of a thin liquid film subject to surface tension, gravity and centrifugal forces is considered for the problem of a vertically rotating disk that is partially immersed in a liquid bath. This problem constitutes a generalization of the classic Landau-Levich drag-out problem to axisymmetric flow. A generalized lubrication model that includes the meniscus region connecting the thin film to the bath is derived. The resulting nonlinear fourth-order partial differential equation is solved numerically using a finite element scheme. For a range of parameters steady states are found. While the solutions for the height profile of the film near the drag-out region show excellent agreement with the asymptotic solutions to the corresponding classic Landau-Levich problem, they show novel patterns away from the meniscus region. The implications for possible industrial applications are discussed.

J. Giannoulis, M. Herrmann, A. Mielke, Lagrangian and Hamiltonian two-scale reduction, Journal of Mathematical Physics, 49 (2008), pp. 103505/1--103505/42.
Abstract
Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.

J. Giannoulis, A. Mielke, Ch. Sparber, Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential, Journal of Differential Equations, 245 (2008), pp. 939--963.
Abstract
We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.

J.A.C. Martins, M.D.P. Monteiro Marques, A. Petrov, On the stability of elastic-plastic systems with hardening, Journal of Mathematical Analysis and Applications, 343 (2008), pp. 1007--1021.
Abstract
This paper discusses the stability of quasi-static paths for a continuous elastic-plastic system with hardening in a one-dimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasi-static problems involving elastic-plastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasi-static paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasi-static ones, when (as in Lyapunov stability) the size of initial perturbations is decreased and the rate of application of the forces (which plays the role of the small parameter in singular perturbation problems) is also decreased to zero. The stability of the quasi-static paths of these elastic-plastic systems is the main result proved in the paper.

A. Münch, B. Wagner, Galerkin method for feedback controlled Rayleigh--Bénard convection, Nonlinearity, 21 (2008), pp. 2625-2651.

M. Rauscher, R. Blossey, A. Münch, B. Wagner, Spinodal dewetting of thin films with large interfacial slip: Implications from the dispersion relation, Langmuir, 24 (2008), pp. 12290-12294.

W. Dreyer, M. Herrmann, Numerical experiments on the modulation theory for the nonlinear atomic chain, Physica D. Nonlinear Phenomena, 237 (2008), pp. 255-282.
Abstract
Modulation theory with periodic traveling waves is a powerful, but not rigorous tool to derive a thermodynamic description for the atomic chain. We investigate the validity of this theory by means of several numerical experiments.

A. Mielke, M. Ortiz, A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems, ESAIM. Control, Optimisation and Calculus of Variations, 14 (2008), pp. 494--516.

A. Mielke, T. Roubíček, U. Stefanelli, $Gamma$-limits and relaxations for rate-independent evolutionary problems, Calculus of Variations and Partial Differential Equations, 31 (2008), pp. 387--416.
Abstract
This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals $calE$ and the dissipation distance $calD$. For sequences $(calE_k)_kin N$ and $(calD_k)_kin N$ we address the question under which conditions the limits $q_infty$ of solutions $q_k:[0,T]to calQ$ satisfy a suitable limit problem with limit functionals $calE_infty$ and $calD_infty$, which are the corresponding $Gamma$-limits. We derive a sufficient condition, called emphconditional upper semi-continuity of the stable sets, which is essential to guarantee that $q_infty$ solves the limit problem. In particular, this condition holds if certain emphjoint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator convergece if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit $kto infty$, which in the limit can be described by an effective macroscopic model.

A. Mielke, Weak-convergence methods for Hamiltonian multiscale problems, Discrete and Continuous Dynamical Systems, 20 (2008), pp. 53--79.
Abstract
We consider Hamiltonian problems depending on a small parameter like in wave equations with rapidly oscillating coefficients or the embedding of an infinite atomic chain into a continuum by letting the atomic distance tend to $0$. For general semilinear Hamiltonian systems we provide abstract convergence results in terms of the existence of a family of joint recovery operators which guarantee that the effective equation is obtained by taking the $Gamma$-limit of the Hamiltonian. The convergence is in the weak sense with respect to the energy norm. Exploiting the well-developed theory of $Gamma$-convergence, we are able to generalize the admissible coefficients for homogenization in the wave equations. Moreover, we treat the passage from a discrete oscillator chain to a wave equation with general $rmL^infty$ coefficients

A. Mielke, A. Petrov, J.A.C. Martins, Convergence of solutions of kinetic variational inequalities in the rate-independent quasi-static limit, Journal of Mathematical Analysis and Applications, 348 (2008), pp. 1012--1020.
Abstract
This paper discusses the convergence of kinetic variational inequalities to rate-independent quasi-static variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rate-independent quasi-static problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rate-independent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to three-dimensional elastic-plastic systems with hardening is given.

K. Afanasiev, A. Münch, B. Wagner, On the Landau--Levich problem for non-Newtonian liquids, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 76 (2007), pp. 036307/1--036307/12.
Abstract
In this paper the drag-out problem for shear-thinning liquids at variable inclination angle is considered. For this free boundary problem dimension-reduced lubrication equations are derived for the most commonly used viscosity models, namely, the power-law, Ellis and Carreau model. For the resulting lubrication models a system of ordinary differential equation governing the steady state solutions is obtained. Phase plane analysis is used to characterize the type of possible steady state solutions and their dependence on the rheological parameters.

J.A.C. Martins, M.D.P. Monteiro, A. Petrov, On the stability of quasi-static paths for finite dimensional elastic-plastic systems with hardening, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 87 (2007), pp. 303--313.

P. Krejčí, J. Sprekels, Elastic-ideally plastic beams and Prandtl--Ishlinskii hysteresis operators, Mathematical Methods in the Applied Sciences, 30 (2007), pp. 2371--2393.

A. Mielke, A. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM Journal on Mathematical Analysis, 39 (2007), pp. 642--668.
Abstract
This paper is devoted to the two-scale homogenization for a class of rate-independent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case that the period tends to 0. Our approach is based on the notion of energetic solutions which is phrased in terms of a stability condition and an energy balance of an energy-storage functional and a dissipation functional. Using the recently developed method of weak and strong two-scale convergence via periodic unfolding, we show that these two functionals have a suitable two-scale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of Gamma convergence for the energetic formulation using so-called joint recovery sequences it is possible to show that the solutions of the problem with periodicity converge to the energetic solution associated with the limit functionals.

A. Mielke, S. Zelik, Infinite-dimensional hyperbolic sets and spatio-temporal chaos in reaction-diffusion systems in $R^n$, Journal of Dynamics and Differential Equations, 19 (2007), pp. 333--389.

J. Giannoulis, A. Mielke, Dispersive evolution of pulses in oscillator chains with general interaction potentials, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 6 (2006), pp. 493--523.

M.A. Efendiev, A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, Journal of Convex Analysis, 13 (2006), pp. 151--167.

P. Evans, A. Münch, Interaction of advancing fronts and meniscus profiles formed by surface-tension-on-gradient-driven liquid films, SIAM Journal on Applied Mathematics, 66 (2006), pp. 1610--1631.

P.L. Evans, J.R. King, A. Münch, Intermediate-asymptotic structure of a dewetting rim with strong slip, Applied Mathematics Research Express, (2006), pp. 25262/1--25262/25.
Abstract
When a thin viscous liquid film dewets, it typically forms a rim which spreads outwards, leaving behind a growing dry region. We consider the dewetting behaviour of a film, when there is strong slip at a liquid-substrate interface. The film can be modelled by two coupled partial differential equations (PDEs) describing the film thickness and velocity. Using asymptotic methods, we describe the structure of the rim as it evolves in time, and the rate of dewetting, in the limit of large slip lengths. An inner region emerges, closest to the dewetted region, where surface tension is important; in an outer region, three subregions develop. This asymptotic description is compared with numerical solutions of the full system of PDEs.

J.R. King, A. Münch, B. Wagner, Linear stability of a ridge, Nonlinearity, 19 (2006), pp. 2813-2831.
Abstract
We investigate the stability of the three-phase contact-line of a thin liquid ridge on a hydrophobic substrate for flow driven by surface tension and van der Waals forces. We study the role of slippage in the emerging instability at the three-phase contact-line by comparing the lubrication models for no-slip and slip-dominated conditions at the liquid/substrate interface. For both cases we derive a sharp-interface model via matched asymptotic expansions and derive the eigenvalues from a linear stability analysis of the respective reduced models. We compare our asymptotic results with the eigenvalues obtained numerically for the full lubrication models.

A. Münch, B. Wagner, M. Rauscher, R. Blossey, A thin film model for corotational Jeffreys fluids under strong slip, The European Physical Journal. E. Soft Matter, 20 (2006), pp. 365-368.
Abstract
We derive a thin film model for viscoelastic liquids under strong slip which obey the stress tensor dynamics of corotational Jeffreys fluids.

W. Dreyer, M. Herrmann, A. Mielke, Micro-macro transitions in the atomic chain via Whitham's modulation equation, Nonlinearity, 19 (2006), pp. 471--500.

A. Mielke, Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner--Husimi transforms, Archive for Rational Mechanics and Analysis, 181 (2006), pp. 401--448.

M. Baro, H. Neidhardt, J. Rehberg, Current coupling of drift-diffusion models and dissipative Schrödinger--Poisson systems: Dissipative hybrid models, SIAM Journal on Mathematical Analysis, 37 (2005), pp. 941--981.

TH. Koprucki, M. Baro, U. Bandelow, Th. Tien, F. Weik, J.W. Tomm, M. Grau, M.-Ch. Amann, Electronic structure and optoelectronic properties of strained InAsSb/GaSb multiple quantum wells, Applied Physics Letters, 87 (2005), pp. 181911/1--181911/3.

B. Jin, A. Acrivos, A. Münch, The drag-out problem in film coating, Physics of Fluids, 17 (2005), pp. 103603/1-103603/12.

A. Münch, P.L. Evans, Marangoni-driven liquid films rising out of a meniscus onto a nearly-horizontal substrate, Physica D. Nonlinear Phenomena, 209 (2005), pp. 164--177.

A. Münch, B. Wagner, Th.P. Witelski, Lubrication models for small to large slip lengths, Journal of Engineering Mathematics, 53 (2005), pp. 359-383.

A. Münch, B. Wagner, Contact-line instability for dewetting thin films, Physica D. Nonlinear Phenomena, 209 (2005), pp. 178--190.

U. Bandelow, H.-Chr. Kaiser, Th. Koprucki, J. Rehberg, Spectral properties of $k cdot p$ Schrödinger operators in one space dimension, Numerical Functional Analysis and Optimization. An International Journal, 21 (2000), pp. 379--409.

S. Bartels, M. Milicevic, M. Thomas, S. Tornquist, N. Weber, Approximation schemes for materials with discontinuities, in: Non-standard Discretisation Methods in Solid Mechanics, J. Schröder, P. Wriggers, eds., 98 of Lecture Notes in Applied and Computational Mechanics, Springer, Cham, 2022, pp. 505--565, DOI 10.1007/978-3-030-92672-4_17 .
Abstract
Damage and fracture phenomena are related to the evolution of discontinuities both in space and in time. This contribution deals with methods from mathematical and numerical analysis to handle these: Suitable mathematical formulations and time-discrete schemes for problems with discontinuities in time are presented. For the treatment of problems with discontinuities in space, the focus lies on FE-methods for minimization problems in the space of functions of bounded variation. The developed methods are used to introduce fully discrete schemes for a rate-independent damage model and for the viscous approximation of a model for dynamic phase-field fracture. Convergence of the schemes is discussed.

G. Nika, B. Vernescu, Micro-geometry effects on the nonlinear effective yield strength response of magnetorheological fluids, in: Emerging Problems in the Homogenization of Partial Differential Equations, P. Donato, M. Luna-Laynez, eds., 10 of SEMA SIMAI Springer Series, Springer, Cham, 2021, pp. 1--16, DOI 10.1007/978-3-030-62030-1_1 .
Abstract
We use the novel constitutive model in [15], derived using the homogenization method, to investigate the effect particle chain microstructures have on the properties of the magnetorheological fluid. The model allows to compute the constitutive coefficients for different geometries. Different geometrical realizations of chains can significantly change the magnetorheological effect of the suspension. Numerical simulations suggest that particle size is also important as the increase of the overall particle surface area can lead to a decrease of the overall magnetorheological effect while keeping the volume fraction constant.

M. Kantner, A. Mielke, M. Mittnenzweig, N. Rotundo, Mathematical modeling of semiconductors: From quantum mechanics to devices, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 269--293, DOI 10.1007/978-3-030-33116-0 .
Abstract
We discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantum-classical model that self-consistently couples van Roosbroeck's drift-diffusion system for classical charge transport with a Lindblad-type quantum master equation. The coupling is shown to obey fundamental principles of non-equilibrium thermodynamics. The appealing thermodynamic properties are shown to arise from the underlying mathematical structure of a damped Hamitlonian system, which is an isothermal version of socalled GENERIC systems. The evolution is governed by a Hamiltonian part and a gradient part involving a Poisson operator and an Onsager operator as geoemtric structures, respectively. Both parts are driven by the conjugate forces given in terms of the derivatives of a suitable free energy.

M. Kantner, Hybrid modeling of quantum light emitting diodes: Self-consistent coupling of drift-diffusion, Schrödinger--Poisson, and quantum master equations, in: Proc. SPIE 10912, B. Witzigmann, M. Osiński, Y. Arakawa, eds., Physics and Simulation of Optoelectronic Devices XXVII, SPIE Digital Library, Bellingham, 2019, pp. 10912OU/1--10912OU/8, DOI 10.1117/12.2515209 .
Abstract
The device-scale simulation of electrically driven solid state quantum light emitters, such as single-photon sources and nanolasers based on semiconductor quantum dots, requires a comprehensive modeling approach, that combines classical device physics with cavity quantum electrodynamics. In a previous work, we have self-consistently coupled the semi-classical drift-diffusion system with a Markovian quantum master equation in Lindblad form to describe (i) the spatially resolved current injection into a quantum dot embedded within a semiconductor device and (ii) the fully quantum mechanical light-matter interaction in the coupled quantum dot-photon system out of one box. In this paper, we extend our hybrid quantum-classical modeling approach by including a Schroedinger?Poisson problem to account for energy shifts of the quantum dot carriers in response to modifications of its macroscopic environment (e.g., quantum confined Stark effect due to the diode's internal electric field and plasma screening). The approach is demonstrated by simulations of a single-photon emitting diode.

M. Kantner, Simulation of quantum light sources using the self-consistently coupled Schrödinger--Poisson-Drift-Diffusion-Lindblad system, in: Proceedings of the 19th International Conference on Numerical Simulation of Optoelectronic Devices -- NUSOD 2019, J. Piprek, K. Hinzer, eds., IEEE Conference Publications Management Group, Piscataway, 2019, pp. 15--16, DOI 10.1109/NUSOD.2019.8806839 .
Abstract
The device-scale simulation of electrically drivenquantum light sources based on semiconductor quantum dotsrequires a combination of the (classical) semiconductor deviceequations with cavity quantum electrodynamics. In this paper, weextend our previously developed hybrid quantum-classical modelsystem ? where we have coupled the drift-diffusion system witha Lindblad-type quantum master equation ? by including a self-consistent Schrödinger?Poisson problem. The latter describes the(quasi-)bound states of the quantum dot carriers. The extendedmodel allows to describe the bias-dependency of the emissionspectrum due to the quantum confined Stark effect

M. Kantner, M. Mittnenzweig, Th. Koprucki, A hybrid quantum-classical modeling approach for electrically driven quantum dot devices, in: Proc. SPIE 10526, Physics and Simulation of Optoelectronic Devices XXVI, B. Witzigmann, M. Osiński, Y. Arakawa, eds., SPIE Digital Library, 2018, pp. 1052603/1--1052603/6, DOI 10.1117/12.2289185 .
Abstract
The design of electrically driven quantum light sources based on semiconductor quantum dots, such as singlephoton emitters and nanolasers, asks for modeling approaches combining classical device physics with cavity quantum electrodynamics. In particular, one has to connect the well-established fields of semi-classical semiconductor transport theory and the theory of open quantum systems. We present a first step in this direction by coupling the van Roosbroeck system with a Markovian quantum master equation in Lindblad form. The resulting hybrid quantum-classical system obeys the fundamental laws of non-equilibrium thermodynamics and provides a comprehensive description of quantum dot devices on multiple scales: It enables the calculation of quantum optical figures of merit (e.g. the second order intensity correlation function) together with the spatially resolved simulation of the current flow in realistic semiconductor device geometries in a unified way.

M. Kantner, M. Mittnenzweig, Th. Koprucki, Modeling and simulation of electrically driven quantum light sources: From classical device physics to open quantum systems, in: 14th International Conference on Nonlinear Optics and Excitation Kinetics in Semiconductors, September 23--27, 2018, Berlin, Germany (Conference Program), 2018, pp. 135.

S. Reichelt, Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales, in: MURPHYS-HSFS-2014: 7th MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012013/1--012013/15.
Abstract
We derive quantitative error estimates for coupled reaction-diffusion systems, whose coefficient functions are quasi-periodically oscillating modeling microstructure of the underlying macroscopic domain. The coupling arises via nonlinear reaction terms, and we allow for different diffusion length scales, i.e. whereas some species have characteristic diffusion length of order 1, other species may diffuse much slower, namely, with order of the characteristic microstructure-length scale. We consider an effective system, which is rigorously obtained via two-scale convergence, and we derive quantitative error estimates.

M. Kantner, U. Bandelow, Th. Koprucki, H.-J. Wünsche, Multi-scale modelling and simulation of single-photon sources on a device level, in: Euro-TMCS II -- Theory, Modelling & Computational Methods for Semiconductors, 7th -- 9th December 2016, Tyndall National Institute, University College Cork, Ireland, E. O'Reilly, S. Schulz, S. Tomic, eds., Tyndall National Institute, 2016, pp. 65.

A. Mielke, R. Rossi, G. Savaré, Balanced-Viscosity solutions for multi-rate systems, in: MURPHYS-HSFS-2014: 7th MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012010/1--012010/26.
Abstract
Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients ε^α and ε, where 0<ε<1 and α>0 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishing-viscosity analysis as ε tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0<α<1.

A. Mielke, Deriving effective models for multiscale systems via evolutionary $Gamma$-convergence, in: Control of Self-Organizing Nonlinear Systems, E. Schöll, S. Klapp, P. Hövel, eds., Understanding Complex Systems, Springer International Publishing AG Switzerland, Cham, 2016, pp. 235--251.

A. Mielke, Relaxation of a rate-independent phase transformation model for the evolution of microstructure, in: Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure, Workshop, March 14--18, 2016, R. Kienzler, D.L. Mcdowell, S. Müller, E.A. Werner, eds., 13 of Oberwolfach Reports, European Mathematical Society, 2016, pp. 840---842.

M. Kantner, U. Bandelow, Th. Koprucki, H.-J. Wünsche, Modeling and numerical simulation of electrically pumped single-photon emitters, in: Proceedings of the 15th International Conference on Numerical Simulation of Optoelectronic Devices 2015, J. Piprek, W. Yuh-Renn, eds., IEEE Conference Publications Management Group, Piscataway, 2015, pp. 151--152.

A. Mielke, Evolutionary relaxation of a two-phase model, in: Scales in Plasticity, Mini-Workshop, November 8--14, 2015, G.A. Francfort, S. Luckhaus, eds., 12 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2015, pp. 3027--3030.

A. Mielke, On thermodynamical couplings of quantum mechanics and macroscopic systems, in: Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, P. Exner, W. König, H. Neidhardt, eds., World Scientific Publishing, Singapore, 2015, pp. 331--348.
Abstract
Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the Liouville equation for the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition:

We give three applications of the theory. First, we consider a finite-dimensional quantum system that is coupled to a finite number of simple heat baths, each of which is described by a scalar temperature variable. Second, we model quantum system given by a one-dimensional Schrödinger operator connected to a one-dimensional heat equation on the left and on the right. Finally, we consider thermo-opto-electronics, where the Maxwell-Bloch system of optics is coupled to the energy-drift-diffusion system for semiconductor electronics.

A. Mielke, Multiscale gradient systems and their amplitude equations, in: Dynamics of Pattern, Workshop, Dezember 16--22, 2012, 9 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2012, pp. 3588--3591.

M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, in: Microstructures in Solids: From Quantum Models to Continua, Workshop, March 14--20, 2010, 7 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2010, pp. 783--785.

A. Petrov, J.A.C. Martins, M.D.P. Monteiro Marques, Mathematical results on the stability of quasi-static paths of elastic-plastic systems with hardening, in: Topics on Mathematics for Smart Systems, B. Miara, G. Stavroulakis, V. Valente, eds., World Scientific, Singapore, 2007, pp. 167--182.

A. Mielke, Gamma convergence for rate-independent processes with applications to damage, in: Phase Transition, Workshop, June 4 -- 8, 2007, 4 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2007, pp. 1617--1620.

J. Giannoulis, M. Herrmann, A. Mielke, Continuum descriptions for the dynamics in discrete lattices: Derivation and justification, in: Analysis, Modeling and Simulation of Multiscale Problems, A. Mielke, ed., Springer, Heidelberg, 2006, pp. 435--466.

C. Patz, A. Mielke, Dispersive and long-time behavior of oscillations in lattices, in: GAMM Annual Meeting 2006 -- Berlin, Special Issue (Vol. 6, Issue 1) of PAMM (Proceedings of Applied Mathematics and Mechanics), Wiley-VCH Verlag, Weinheim, 2006, pp. 503--504.

A. Timofte, A. Mielke, Two-scale homogenization for rate-independent systems, in: GAMM Annual Meeting 2006 -- Berlin, Special Issue (Vol. 6, Issue 1) of PAMM (Proceedings of Applied Mathematics and Mechanics), Wiley-VCH Verlag, Weinheim, 2006, pp. 513--514.

W. Dreyer, M. Herrmann, J.D.M. Rademacher, Wave trains, solitons and modulation theory in FPU chains, in: Analysis, Modeling and Simulation of Multiscale Problems, A. Mielke, ed., Springer, Heidelberg, 2006, pp. 467--500.

A. Mielke, Deriving modulation equations via Lagrangian and Hamiltonian reduction, in: Mathematical Theory of Water Waves, Workshop, November 12--18, 2006, 3 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2006, pp. 3032--3034.

A. Mielke, Gamma convergence for evolutionary problems, in: Applications of Asymptotic Analysis, Workshop, June 18 -- 24, 2006, 2 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2006, pp. 1697--1700.

A. Mielke, Two-scale modelling for Hamiltonian systems: Formal and rigorous results, in: PDE and Materials, Workshop, September 24--30, 2006, 3 of Oberwolfach Reports, Mathematisches Forschunsinstitut Oberwolfach, 2006, pp. 2656--2659.

A. Mielke, Energy transport in periodic lattices, in: Thermodynamische Materialtheorien, Workshop, December 12--18, 2004, 1 (4) of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2004, pp. 3019-3022.

H. Gajewski, H.-Chr. Kaiser, H. Langmach, R. Nürnberg, R.H. Richter, Mathematical modelling and numerical simulation of semiconductor detectors, in: Mathematics --- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.-J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 355--364.

H.-Chr. Kaiser, U. Bandelow, Th. Koprucki, J. Rehberg, Modelling and simulation of strained quantum wells in semiconductor lasers, in: Mathematics --- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.-J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 377--390.

U. Bandelow, H. Gajewski, H.-Chr. Kaiser, Modeling combined effects of carrier injection, photon dynamics and heating in Strained Multi-Quantum-Well Laser, in: Physics and Simulation of Optoelectronic Devices VIII, R.H. Binder, P. Blood, M. Osinski, eds., 3944 of Proceedings of SPIE, SPIE, Bellingham, WA, 2000, pp. 301--310.

H.-Chr. Kaiser, J. Rehberg, About some mathematical questions concerning the embedding of Schrödinger-Poisson systems into the drift-diffusion model of semiconductor devices, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., 2, World Scientific, Singapore [u. a.], 2000, pp. 1328--1333.

A. Mielke, M.A. Peletier, J. Zimmer, Deriving a GENERIC system from a Hamiltonian system, Preprint no. 3108, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3108 .
Abstract, PDF (651 kByte)
We reconsider the fundamental problem of coarse-graining infinite-dimensional Hamiltonian dynamics to obtain a macroscopic system which includes dissipative mechanisms. In particular, we study the thermodynamical implications concerning Hamiltonians, energy, and entropy and the induced geometric structures such as Poisson and Onsager brackets (symplectic and dissipative brackets). We start from a general finite-dimensional Hamiltonian system that is coupled linearly to an infinite-dimensional heat bath with linear dynamics. The latter is assumed to admit a compression to a finite-dimensional dissipative semigroup (i.e., the heat bath is a dilation of the semigroup) describing the dissipative evolution of new macroscopic variables. Already in the finite-energy case (zero-temperature heat bath) we obtain the so-called GENERIC structure (General Equations for Non-Equilibrium Reversible Irreversibe Coupling), with conserved energy, nondecreasing entropy, a new Poisson structure, and an Onsager operator describing the dissipation. However, their origin is not obvious at this stage. After extending the system in a natural way to the case of positive temperature, giving a heat bath with infinite energy, the compression property leads to an exact multivariate Ornstein--Uhlenbeck process that drives the rest of the system. Thus, we are able to identify a conserved energy, an entropy, and an Onsager operator (involving the Green-Kubo formalism) which indeed provide a GENERIC structure for the macroscopic system.

M. Hintermüller, D. Korolev, A hybrid physics-informed neural network based multiscale solver as a partial differential equation constrained optimization problem, Preprint no. 3052, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3052 .
Abstract, PDF (1045 kByte)
In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating multiscale PDEs. From a continuous perspective, our formulation corresponds to a non-standard PDE-constrained optimization problem with a PINN-type objective. From a discrete standpoint, the formulation represents a hybrid numerical solver that utilizes both neural networks and finite elements. We propose a function space framework for the problem and develop an algorithm for its numerical solution, combining an adjoint-based technique from optimal control with automatic differentiation. The multiscale solver is applied to a heat transfer problem with oscillating coefficients, where the neural network approximates a fine-scale problem, and a coarse-scale problem constrains the learning process. We show that incorporating coarse-scale information into the neural network training process through our modelling framework acts as a preconditioner for the low-frequency component of the fine-scale PDE, resulting in improved convergence properties and accuracy of the PINN method. The relevance of the hybrid solver to numerical homogenization is discussed.

Y. Bokredenghel, M. Heida, Quenched homogenization of infinite range random conductance model on stationary point processes, Preprint no. 3017, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3017 .
Abstract, PDF (391 kByte)
We prove homogenization for elliptic long-range operators in the random conductance model on random stationary point processes in d dimensions with Dirichlet boundary conditions and with a jointly stationary coefficient field. Doing so, we identify 4 conditions on the point process and the coefficient field that have to be fulfilled at different stages of the proof in order to pass to the homogenization limit. The conditions can be clearly attributed to concentration of support, Rellich--Poincaré inequality, non-degeneracy of the homogenized matrix and ergodicity of the elliptic operator.

R. Bazaes, A. Mielke, Ch. Mukherjee, Stochastic homogenization of Hamilton--Jacobi--Bellman equations on continuum percolation clusters, Preprint no. 2955, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2955 .
Abstract, PDF (598 kByte)
We prove homogenization properties of random Hamilton--Jacobi--Bellman (HJB) equations on continuum percolation clusters, almost surely w.r.t. the law of the environment when the origin belongs to the unbounded component in the continuum. Here, the viscosity term carries a degenerate matrix, the Hamiltonian is convex and coercive w.r.t. the degenerate matrix and the underlying environment is non-elliptic and its law is non-stationary w.r.t. the translation group. We do not assume uniform ellipticity inside the percolation cluster, nor any finite-range dependence (i.i.d.) assumption on the percolation models and the effective Hamiltonian admits a variational formula which reflects some key properties of percolation. The proof is inspired by a method of Kosygina--Rezakhanlou--Varadhan developed for the case of HJB equations with constant viscosity and uniformly coercive Hamiltonian in a stationary, ergodic and elliptic random environment. In the non-stationary and non-elliptic set up, we leverage the coercivity property of the underlying Hamiltonian as well as a relative entropy structure (both being intrinsic properties of HJB, in any framework) and make use of the random geometry of continuum percolation.

M. Heida, On quenched homogenization of long-range random conductance models on stationary ergodic point processes, Preprint no. 2942, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2942 .
Abstract, PDF (359 kByte)
We study the homogenization limit on bounded domains for the long-range random conductance model on stationary ergodic point processes on the integer grid. We assume that the conductance between neares neighbors in the point process are always positive and satisfy certain weight conditions. For our proof we use long-range two-scale convergence as well as methods from numerical analysis of finite volume methods.

A. Stephan, Coarse-graining and reconstruction for Markov matrices, Preprint no. 2891, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2891 .
Abstract, PDF (248 kByte)
We present a coarse-graining (or model order reduction) procedure for stochastic matrices by clustering. The method is consistent with the natural structure of Markov theory, preserving positivity and mass, and does not rely on any tools from Hilbert space theory. The reconstruction is provided by a generalized Penrose-Moore inverse of the coarse-graining operator incorporating the inhomogeneous invariant measure of the Markov matrix. As we show, the method provides coarse-graining and reconstruction also on the level of tensor spaces, which is consistent with the notion of an incidence matrix and quotient graphs, and, moreover, allows to coarse-grain and reconstruct fluxes. Furthermore, we investigate the connection with functional inequalities and Poincaré-type constants.

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 two-scale 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.

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

M.G. Hennessy, G.L. Celora, A. Münch, S.L. Waters, B. Wagner, Asymptotic study of the electric double layer at the interface of a polyelectrolyte gel and solvent bath, Preprint no. 2751, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2751 .
Abstract, PDF (2265 kByte)
An asymptotic framework is developed to study electric double layers that form at the inter-face between a solvent bath and a polyelectrolyte gel that can undergo phase separation. The kinetic model for the gel accounts for the finite strain of polyelectrolyte chains, free energy ofinternal interfaces, and Stefan?Maxwell diffusion. By assuming that the thickness of the doublelayer is small compared to the typical size of the gel, matched asymptotic expansions are used toderive electroneutral models with consistent jump conditions across the gel-bath interface in two-dimensional plane-strain as well as fully three-dimensional settings. The asymptotic frameworkis then applied to cylindrical gels that undergo volume phase transitions. The analysis indicatesthat Maxwell stresses are responsible for generating large compressive hoop stresses in the double layer of the gel when it is in the collapsed state, potentially leading to localised mechanicalinstabilities that cannot occur when the gel is in the swollen state. When the energy cost of in-ternal interfaces is sufficiently weak, a sharp transition between electrically neutral and chargedregions of the gel can occur. This transition truncates the double layer and causes it to have finitethickness. Moreover, phase separation within the double layer can occur. Both of these featuresare suppressed if the energy cost of internal interfaces is sufficiently high. Thus, interfacial freeenergy plays a critical role in controlling the structure of the double layer in the gel.

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 two-scale convergence method and apply the resulting theory to the homogenization of a p-Laplace problem on a randomly perforated domain.

G.L. Celora, M.G. Hennessy, A. Münch, S.L. Waters, B. Wagner, Spinodal decomposition and collapse of a polyelectrolyte gel, Preprint no. 2731, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2731 .
Abstract, PDF (2259 kByte)
The collapse of a polyelectrolyte gel in a (monovalent) salt solution is analysed using a new model that includes interfacial gradient energy to account for phase separation in the gel, finite elasticity and multicomponent transport. We carry out a linear stability analysis to determine the stable and unstable spatially homogeneous equilibrium states and how they phase separate into localized regions that eventually coarsen to a new stable state. We then investigate the problem of a collapsing gel as a response to increasing the salt concentration in the bath. A phase space analysis reveals that the collapse is obtained by a front moving through the gel that eventually ends in a new stable equilibrium. For some parameter ranges, these two routes to gel shrinking occur together.

A. Münch, B. Wagner, Self-consistent field theory for a polymer brush. Part II: The effective chemical potential, Preprint no. 2649, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2649 .
Abstract, PDF (318 kByte)
The most successful mean-field model to describe the collective behaviour of the large class of macromolecular polymers is the self-consistent field theory (SCFT). Still, even for the simple system of a grafted dry polymer brush, the mean-field equations have to be solved numerically. As one of very few alternatives that offer some analytical tractability the strong-stretching theory (SST) has led to explicit expressions for the effective chemical potential and consequently the free energy to promote an understanding of the underlying physics. Yet, a direct derivation of these analytical results from the SCFT model is still outstanding. In this study we present a systematic asymptotic theory based on matched asymtptotic expansions to obtain the effective chemical potential from the SCFT model for a dry polymer brush for large but finite stretching.

A. Münch, B. Wagner, Self-consistent field theory for a polymer brush. Part I: Asymptotic analysis in the strong-stretching limit, Preprint no. 2648, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2648 .
Abstract, PDF (854 kByte)
In this study we consider the self-consistent field theory for a dry, in- compressible polymer brush, densely grafted on a substrate, describing the average segment density of a polymer in terms of an effective chemical potential for the interaction between the segments of the polymer chain. We present a systematic singular perturbation analysis of the self-consistent field theory in the strong-stretching limit, when the length scale of the ratio of the radius of gyration of the polymer chain to the extension of the brush from the substrate vanishes. Our analysis yields, for the first time, an approximation for the average segment density that is correct to leading order in the outer scaling and resolves the boundary layer singularity at the end of the polymer brush in the strong-stretching limit. We also show that in this limit our analytical results agree increasingly well with our numerical solutions to the full model equations comprising the self-consistent field theory.

M. Heida, S. Neukamm, M. Varga, Stochastic unfolding and homogenization, Preprint no. 2460, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2460 .
Abstract, PDF (596 kByte)
The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogenization result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method introduced in this paper extends discrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition.

J. Ben-Artzi, D. Marahrens, S. Neukamm, Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations, Preprint no. 1985, WIAS, Berlin, 2014, DOI 10.20347/WIAS.PREPRINT.1985 .
Abstract, PDF (413 kByte)
We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions d ≥ 2. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green's function in weighted spaces.

W. Dreyer, J. Giesselmann, Ch. Kraus, Modeling of compressible electrolytes with phase transition, Preprint no. 1955, WIAS, Berlin, 2014, DOI 10.20347/WIAS.PREPRINT.1955 .
Abstract, PDF (457 kByte)
A novel thermodynamically consistent diffuse interface model is derived for compressible electrolytes with phase transitions. The fluid mixtures may consist of N constituents with the phases liquid and vapor, where both phases may coexist. In addition, all constituents may consist of polarizable and magnetizable matter. Our introduced thermodynamically consistent diffuse interface model may be regarded as a generalized model of Allen-Cahn/Navier-Stokes/Poisson type for multi-component flows with phase transitions and electrochemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-coupled and a coupled regime, where the coupling takes place between the smallness parameter in the Poisson equation and the width of the interface. We recover in the sharp interface limit a generalized Allen-Cahn/Euler/Poisson system for mixtures with electrochemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satisfy, for instance, a generalized Gibbs-Thomson law and a dynamic Young-Laplace law.

W. Dreyer, C. Guhlke, R. Müller, Rational modeling of electrochemical double-layers and derivation of Butler--Volmer equations, Preprint no. 1860, WIAS, Berlin, 2013, DOI 10.20347/WIAS.PREPRINT.1860 .
Abstract, PDF (443 kByte)
We derive the boundary conditions for the contact between an electrolyte and a solid electrode. At first we revisit the thermodynamic consistent complete model that resolves the actual electrode--electrolyte interface and its adjacent boundary layers. The width of these layers is controlled by the Debye length that is typically very small, leading to strongly different length scales in the system. We apply the method of asymptotic analysis to derive a simpler reduced model that does not resolve the boundary layers but instead incorporates the electrochemical properties of the layers into a set of new boundary conditions. This approach fully determines the relation of bulk quantities to the boundary conditions of the reduced model. In particular, the Butler-Volmer equations for electrochemical reactions, which are still under discussion in the literature, are rational consequences of our approach. For illustration and to compare with the literature, we consider a simple generic reaction.

A. Lamacz, S. Neukamm, F. Otto, Moment bounds for the corrector in stochastic homogenization of a percolation model, Preprint no. 1836, WIAS, Berlin, 2013, DOI 10.20347/WIAS.PREPRINT.1836 .
Abstract, PDF (472 kByte)
We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on Z^d, d > 2. The model is obtained from the classical Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. This extends a previous result by Gloria and the third author, where uniformly elliptic conductances are treated, to the degenerate case. Our argument is based on estimates on the gradient of the elliptic Green's function.

H. Hanke, D. Knees, Derivation of an effective damage model with evolving micro-structure, Preprint no. 1749, WIAS, Berlin, 2012, DOI 10.20347/WIAS.PREPRINT.1749 .
Abstract, PDF (554 kByte)
In this paper rate-independent damage models for elastic materials are considered. The aim is the derivation of an effective damage model by investigating the limit process of damage models with evolving micro-defects. In all presented models the damage is modeled via a unidirectional change of the material tensor. With progressing time this tensor is only allowed to decrease in the sense of quadratic forms. The magnitude of the damage is given by comparing the actual material tensor with two reference configurations, denoting completely undamaged material and maximally damaged material (no complete damage). The starting point is a microscopic model, where the underlying micro-defects, describing the distribution of either undamaged material or maximally damaged material (but nothing in between), are of a given time-dependent shape but of different sizes. Scaling the microstructure of this microscopic model by a parameter ε>0 the limit passage ε→0 is preformed via two-scale convergence techniques. Therefore, a regularization approach for piecewise constant functions is introduced to guaranty enough regularity for identifying the limit model. In the limit model the material tensor depends on a damage variable z:[0,T]→ W^1,p(Ω) taking values between 0 and 1 such that, in contrast to the microscopic model, some kind of intermediate damage for a material point x∈Ω is possible. Moreover, this damage variable is connected to the material tensor via an explicit formula, namely, a unit cell formula known from classical homogenization results.

W. Dreyer, M. Gaberšček, J. Jamnik, Phase transition and hysteresis in a rechargeable lithium battery, Preprint no. 1284, WIAS, Berlin, 2007, DOI 10.20347/WIAS.PREPRINT.1284 .
Abstract, Postscript (3236 kByte), PDF (740 kByte)
We represent a model which describes the evolution of a phase transition that occurs in some part of a rechargeable lithium battery during the process of charging/discharging. The model is capable to simulate the hysteretic behavior of the voltage - charge characteristics. During discharging of the battery, the interstitial lattice sites of a small crystalline host system are filled up with lithium atoms and these are released again during charging. We show within the context of a sharp interface model that two mechanical phenomena go along with a phase transition that appears in the host system during supply and removal of lithium. At first the lithium atoms need more space than it is available by the interstitial lattice sites, which leads to a maximal relative change of the crystal volume of about $6%$. Furthermore there is an interface between two adjacent phases that has very large curvature of the order of magnitude 100 m, which evoke here a discontinuity of the normal component of the stress. In order to simulate the dynamics of the phase transitions and in particular the observed hysteresis we establish a new initial and boundary value problem for a nonlinear PDE system that can be reduced in some limiting case to an ODE system.

W. Dreyer, Ch. Kraus, The equilibria of vapour-liquid systems revisited, Preprint no. 1238, WIAS, Berlin, 2007, DOI 10.20347/WIAS.PREPRINT.1238 .
Abstract, Postscript (3894 kByte), PDF (624 kByte)
We study equilibrium conditions of liquid-vapour phase transitions for a single substance at constant temperature. The phase transitions are modelled by a classical sharp interface model with boundary contact energy. We revisit this old problem mainly for the following reasons. Equilibria in a two-phase system can be established either under fixed external pressure or under fixed total volume. These two different settings lead to distinct equilibria, a fact that is usually ignored in the literature. In nature and in most technical processes, the approach of a two-phase system to equilibrium runs at constant pressure, whereas mathematicians prefer to study processes in constant domains, i.e. at constant volume. Furthermore, in the literature the sharp interface of the liquid and the vapour phase is usually described by a surface with high symmetry like a plane interface or a radially symmetric interface which has the shape of the boundary of a ball. In this paper we establish equilibrium conditions for pressure control as well as for volume control with arbitrary shapes of the interface. The results are derived by methods of differential geometry. Further, the common features and differences of pressure and volume control are worked out for some simple cases.

M. Heida, Materials with discontinuities on many scales, SCCS Days 2024 of the Collaborative Research Center - CRC 1114 ``Scaling Cascades in Complex Systems'', October 28 - 29, 2024, Freie Universität Berlin, October 28, 2024.

A. Mielke, Analysis of (fast-slow) reaction-diffusion systems using gradient structures, Conference on Differential Equations and their Applications (EQUADIFF 24), June 10 - 14, 2024, Karlstad University, Sweden, June 14, 2024.

TH. Eiter, The effect of time-periodic boundary flux on the decay of viscous flow past, Conference on Differential Equations and their Applications (EQUADIFF 24), Minisymposium 12 ``Fluid-structure Interactions'', June 10 - 14, 2024, Karlstad University, Sweden, June 11, 2024.

TH. Eiter, Time-periodic flow past a body: Approximation by problems on bounded domains, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 14.05 ``PDEs Related to Fluid Mechanics'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 20, 2024.

M. Heida, Permissible random geometries for homogenization, Leibniz MMS Days 2024, Parallel Session ``Computational Material Science'', April 10 - 12, 2024, Leibniz-Institut für Verbundwerkstoffe (IVW), Kaiserslautern, April 11, 2024.

M. Heida, Diskrete Operatoren in Modellbildung und Numerik, Universität der Bundeswehr München, Institut für Mathematik und Informatik, July 13, 2023.

M. Heida, Perspectives for homogenization on randomly perforated domains, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', May 30 - June 2, 2023, Technische Universität Dresden, June 1, 2023.

D. Peschka, Multiscale limits of thin-film models with moving support, In Search of Model Structures for Non-equilibrium Systems, April 24 - 28, 2023, Westfälische Wilhelms-Universität Münster, April 27, 2023.

L. Schütz, M. Heida, M. Thomas, Materials with discontinuities on many scales, SCCS Days 2023 of the Collaborative Research Center -- CRC 1114 ``Scaling Cascades in Complex Systems'', November 13 - 15, 2023.

L. Schütz, Towards stochastic homogenization of a rate-independent delamination model, Hausdorff School ``Analysis of PDEs: Variational and Geometric Perspectives'', Bonn, July 10 - 14, 2023.

A. Glitzky, An effective bulk-surface thermistor model for large-area organic light-emitting diodes, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session 14 ``Applied Analysis'', May 30 - June 2, 2023, Technische Universität Dresden, May 30, 2023.

M. Thomas, Damage in viscoelastic materials at finite strains, Workshop ``Variational Methods for Evolution'', December 3 - 8, 2023, Mathematisches Forschungsinstitut Oberwolfach, December 7, 2023.

M. Thomas, Some aspects of damage in nonlinearly elastic materials: From damage to delamination in nonlinearly elastic materials, Variational and Geometric Structures for Evolution, October 9 - 13, 2023, Università Commerciale Luigi Bocconi, Levico Terme, Italy, October 10, 2023.

M. Liero, Variational modeling of biomechanical systems, 10th International Conference on Computational Bioengineering (ICCB 2023), Minisymposium 22-3 ``Continuum Biomechanics of Active Biological Systems'', September 20 - 22, 2023, Technische Universität Wien, Austria, September 22, 2023.

M. Liero, EDP-convergence for evolutionary systems with gradient flow structure, 29th Nordic Congress of Mathematicians with EMS, July 3 - 7, 2023, Aalborg University, Department of Mathematical Sciences, Denmark, July 4, 2023.

A. Mielke, Asymptotic self-similar behavior in reaction-diffusion systems on the real line, Minisymposium ``Interacting Particle Systems and Variational Methods'', Einhoven University of Technology, Department of Mathematics and Computer Science, Netherlands, February 3, 2023.

A. Mielke, EDP-convergence for gradient systems and non-equilibrium steady states, Nonlinear Diffusion and Nonlocal Interaction Models -- Entropies, Complexity, and Multi-Scale Structures, May 28 - June 2, 2023, Universidad de Granada, Spain, May 30, 2023.

A. Stephan, Fast-slow chemical reaction systems: Gradient systems and EDP-convergence, Oberseminar Dynamics, Technische Universität München, Department of Mathematics, April 17, 2023.

M. Heida, Elasticity on randomly perforated domains, Jahrestreffen des SPP 2256, September 28 - 30, 2022, Universität Regensburg, September 29, 2022.

M. Heida, Homogenization on locally Lipschitz random domains (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium ``Disordered Media and Homogenization'', March 14 - 18, 2022, March 15, 2022.

M. Heida, Homogenization on randomly perforated domains, Block Course ``Multiscale Problems and Homogenization'' at Freie Universität Berlin from Nov. 10 to Dec. 15, 2022, Berlin Mathematical School & Berlin Mathematics Research Center MATH+, November 17, 2022.

M. Heida, Upscaling of intercalation electrodes featuring Cahn--Hilliard to Allen--Cahn transitions (online talk), 21st GAMM Seminar on Microstructures (Online Event), Technische Universität Wien, Austria, January 28, 2022.

A. Selahi, Homogenization in PDE-based battery models and recovery of ageing dynamics via Bayesian inference (online talk), Second Conference of Young Applied Mathematicians (Hybrid Event), September 18 - 22, 2022, Arenzano, Italy, September 21, 2022.

A. Stephan, EDP-convergence for a linear reaction-diffusion systems with fast reversible reaction (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium MS11: ``Bridging Gradient Flows, Hypocoercivity and Reaction-Diffusion Systems'', March 14 - 18, 2022, March 14, 2022.

TH. Eiter, On the resolvent problems associated with rotating viscous flow, DMV Annual Meeting 2022, Section 09 ``Applied Analysis and Partial Differential Equations", September 12 - 16, 2022, Freie Universität Berlin, September 14, 2022.

TH. Eiter, On uniform resolvent estimates associated with time-periodic rotating viscous flow, Mathematical Fluid Mechanics in 2022 (Hybrid Event), August 22 - 26, 2022, Czech Academy of Sciences, Prague, Czech Republic, August 24, 2022.

M. Landstorfer, A. Selahi, M. Heida, M. Eigel, Recovery of battery ageing dynamics with multiple timescales, MATH+-Day 2022, Technische Universität Berlin, November 18, 2022.

M. Liero, EDP-convergence for evolutionary systems with gradient flow structure, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), Minisymposium 4 ``Evolution Equations with Gradient Flow Structure'', August 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, August 16, 2022.

M. Liero, From diffusion to reaction-diffusion in thin structures via EDP-convergence (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium ``Bridging Gradient Flows, Hypocoercivity and Reaction-Diffusion Systems'', March 14 - 18, 2022, March 14, 2022.

A. Mielke, Gamma convergence for evolutionary problems: Using EDP convergence for deriving nontrivial kinetic relations, Calculus of Variations. Back to Carthage, May 16 - 20, 2022, Carthage, Tunisia, May 18, 2022.

A. Mielke, Gradient flows: Existence and Gamma-convergence via the energy-dissipation principle, Horizons in Non-linear PDEs, September 26 - 30, 2022, Universität Ulm.

A. Stephan, EDP-convergence for a linear reaction-diffusion system with fast reversible reaction, Mathematical Models for Biological Multiscale Systems (Hybrid Event), September 12 - 14, 2022, WIAS Berlin, September 12, 2022.

A. Stephan, EDP-convergence for gradient systems and applications to fast-slow chemical reaction systems, Block Course ``Multiscale Problems and Homogenization'' at Freie Universität Berlin from Nov. 10 to Dec. 15, 2022, Berlin Mathematical School & Berlin Mathematics Research Center MATH+, November 24, 2022.

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

M. Heida, Stochastic homogenization on randomly perforated domains (online talk), SIAM Conference on Mathematical Aspects of Materials Science (MS21, Online Event), Minisymposium ``Stochastic Homogenization and Related Topics'', May 17 - 28, 2021, Basque Center for Applied Mathematics, Bilbao, Spain, May 24, 2021.

G. Nika, Derivation of an effective bulk-surface thermistor model for OLEDs, AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6 - 9, 2021.

A. Stephan, Gradient systems and mulit-scale 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, Coarse-graining via EDP-convergence for linear fast-slow reaction-diffusion 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.

A. Glitzky, A coarse-grained electrothermal model for organic semiconductor devices (online talk), DMV-ÖMG Jahrestagung 2021 (Online Event), September 27 - October 1, 2021, Universität Passau, September 29, 2021.

P.-E. Druet, Modeling and analysis for multicomponent incompressible fluids (online talk), 8th European Congress of Mathematics (8ECM), Minisymposium ID 51 ``Partial Differential Equations describing Far-from-Equilibrium Open Systems'' (Online Event), June 20 - 26, 2021, Portorož, Slovenia, June 23, 2021.

A. Mielke, A. Stephan, Effective models for materials and interfaces with multiple scales, CRC 1114 Conference 2021 (Online Event), March 1 - 3, 2021.

A. Mielke, Gradient structures and EDP convergence for reaction and diffusion (online talk), Recent Advances in Gradient Flows, Kinetic Theory, and Reaction-Diffusion Equations (Online Event), July 13 - 16, 2021, Universität Wien, July 15, 2021.

A. Mielke, On a rigorous derivation of a wave equation with fractional damping from a system with fluid-structure interaction (online talk), Tbilisi Analysis and PDE Seminar (Online Event), The University of Georgia, School of Science and Technology, December 20, 2021.

O. Marquardt, Data-driven electronic structure calculations for semiconductor nanostructures, Efficient Algorithms for Numerical Problems, January 17, 2020, WIAS Berlin, January 17, 2020.

D.R.M. Renger, Fast reaction limits via Γ-convergence of the Flux Rate Functional, Variational Methods for Evolution, September 13 - 19, 2020, Mathematisches Forschungszentrum Oberwolfach, September 18, 2020.

A. Stephan, EDP-convergence for nonlinear fast-slow reactions, Workshop ``Variational Methods for Evolution'', September 13 - 19, 2020, Mathematisches Forschungsinstitut Oberwolfach, September 18, 2020.

A. Stephan, Coarse-graining for gradient systems with applications to reaction systems (online talk), Thematic Einstein Semester on Energy-based 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, EDP-convergence for nonlinear fast-slow reaction systems (online talk), Annual Workshop of the GAMM Activity Group on Analysis of PDEs (Online Event), September 30 - October 2, 2020, Institute of Science and Technology Austria (IST Austria), Klosterneuburg, October 1, 2020.

M. Thomas, Weierstraß-Gruppe "Volumen-Grenzschicht-Prozesse", Sitzung des Wissenschaftlichen Beirats, WIAS Berlin, September 18, 2020.

B. Wagner, Pattern formation in dewetting films (online talk), Workshop ``Mathematical Modeling and Scientific Computing: Focus on Complex Processes and Systems'' (Online Event), November 19 - 20, 2020, Technische Universität München, November 19, 2020.

B. Wagner, Phase-field models of the lithiation/delithiation cycle of thin-film electrodes (online talk), Oxford Battery Modelling Symposium (Online Event), March 16 - 17, 2020, University of Oxford, UK, March 16, 2020.

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

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

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

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

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

M. Heida, A. Mielke, A. Stephan, Effective models for materials and interfaces with multiple scales, SCCS Days 2019 of the Collaborative Research Center - CRC 1114, Zeuthen, May 20 - 22, 2019.

M. Heida, Stochastic homogenization of PDE on non-uniformly Lipschitz and percolating structures, DMV-Jahrestagung 2019, September 23 - 26, 2019, KIT - Karlsruher Institut für Technologie, September 24, 2019.

M. Heida, The SQRA operator: Convergence behaviour and applications, Universität Wien, Fakultät für Mathematik, Lehrstuhl Analysis, Austria, March 19, 2019.

M. Heida, The fractional p-Laplacian emerging from discrete homogenization of the random conductance model with degenerate ergodic weights, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18 - 22, 2019, Universität Wien, Technische Universität Wien, Austria, February 19, 2019.

M. Kantner, Hybrid modeling of quantum light emitting diodes: Self-consistent coupling of drift-diffusion, Schrödinger--Poisson, and quantum master equations, SPIE Photonics West, February 5 - 7, 2019, San Francisco, USA, February 6, 2019, DOI 10.1117/12.2515209 .

M. Kantner, Simulation of quantum dot based single-photon sources using the Schrödinger--Poisson-Drift-Diffusion-Lindblad system, International Conference on Simulation of Semiconductor Processes and Devices (SISPAD 2019), September 4 - 6, 2019, Università degli Studi di Udine, Italy, September 6, 2019.

M. Kantner, Simulation of quantum light sources using the self-consistently coupled Schrödinger--Poisson-Drift-Diffusion-Lindblad system, 19th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), July 8 - 12, 2019, University of Ottawa, Canada, July 8, 2019.

G. Nika, Homogenization for a multi-scale model of magnetorheological suspension, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Minisymposium MS ME-1-3 1 ``Emerging Problems in the Homogenization of Partial Differential Equations'', July 15 - 19, 2019, Valencia, Spain, July 15, 2019.

A. Stephan, Rigorous derivation of the effective equation of a linear reaction system with different time scales, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18 - 22, 2019, Universität Wien, Technische Universität Wien, Austria, February 21, 2019.

A. Zafferi, Dynamics of rock dehydration on multiple scales, SCCS Days 2019 of the Collaborative Research Center - CRC 1114, Zeuthen, May 20 - 22, 2019.

A. Zafferi, Some regularity results for a non-isothermal Cahn-Hilliard model, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18 - 22, 2019, Technische Universität Wien, Austria, February 20, 2019.

B. Wagner, Free boundary problems of active and driven hydrogels, PIMS-Germany Workshop on Modelling, Analysis and Numerical Analysis of PDEs for Applications, June 24 - 26, 2019, Universität Heidelberg, Interdisciplinary Center for Scientific Computing and BIOQUANT Center, June 24, 2019.

B. Wagner, Ill-posedness of two-phase flow models of concentrated suspensions, 9th International Congress on Industrial and Applied Mathematics ICIAM2019, Minisymposium MS ME-0-7 6 ``Recent Advances in Understanding Suspensions and Granular Media Flow -- Part 2'', July 15 - 19, 2019, Valencia, Spain, July 17, 2019.

B. Wagner, Mathematical modeling of real world processes, CERN Academic Training Programme 2018--2019, March 14 - 15, 2019, CERN, Geneva, Switzerland.

M. Heida, The SQRA operator: Convergence behaviour and applications, Politechnico di Milano, Dipartimento di Matematica, Italy, March 13, 2019.

TH. Koprucki, Towards multiscale modeling of III-N-based LEDs, 19th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2019) , Session ``Postdeadline Session and Outlook", July 8 - 12, 2019, University of Ottawa, Canada, July 12, 2019.

M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDP-convergence, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18 - 22, 2019, Universität Wien, Technische Universität Wien, Austria, February 20, 2019.

A. Mielke, Effective kinetic relations and EDP convergence, COPDESC-Workshop ``Calculus of Variation and Nonlinear Partial Differential Equations'', March 25 - 28, 2019, Universität Regensburg, March 28, 2019.

A. Mielke, Effective kinetic relations and EDP convergence for gradient systems, Necas Seminar on Continuum Mechanics, Charles University, Prague, Czech Republic, March 18, 2019.

A. Mielke, Evolutionary Gamma-convergence for gradient systems, Mathematisches Kolloquium, Albert-Ludwigs-Universität Freiburg, January 24, 2019.

A. Mielke, Gamma convergence of dissipation functionals and EDP convergence for gradient systems, 6th Applied Mathematics Symposium Münster: Recent Advances in the Calculus of Variations, September 16 - 19, 2019, Westfälische Wilhelms-Universität Münster, September 17, 2019.

A. Mielke, Gradient systems and evolutionary Gamma-convergence, DMV-Jahrestagung 2019, September 23 - 26, 2019, KIT -- Karlsruher Institut für Technologie, September 24, 2019.

A. Mielke, Pattern formation in coupled parabolic systems on extended domains, Fundamentals and Methods of Design and Control of Complex Systems -- Introductory Lectures 2019/20 of CRC 910, Technische Universität Berlin, November 25, 2019.

A. Mielke, EDP convergence for the membrane limit in the porous medium equation, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS ME-1-3 9 ``Entropy Methods for Multi-dimensional Systems in Mechanics'', July 15 - 19, 2019, Valencia, Spain, July 19, 2019.

A. Mielke, Effective models for materials and interfaces with multiple scales, SCCS Days 2019 of the Collaborative Research Center - CRC 1114, May 20 - 22, 2019, SFB 1114, Freie Universität Berlin, Zeuthen, May 21, 2019.

A. Mielke, Gradient systems and the derivation of effective kinetic relations via EDP convergence, Material Theories, Statistical Mechanics, and Geometric Analysis: A Conference in Honor of Stephan Luckhaus' 66th Birthday, June 3 - 6, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, June 5, 2019.

S. Reichelt, Pulses in FitzHugh--Nagumo systems with rapidly oscillating coefficients, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), Section S14 ``Applied Analysis'', March 19 - 23, 2018, Technische Universität München, March 23, 2018.

M. Heida, Mathematische Mehrskalenmethoden in Natur und Technik, Seminar ``Angewandte Analysis'', Universität Konstanz, Institut für Mathematik, October 31, 2018.

M. Kantner, M. Mittnenzweig, Th. Koprucki, A hybrid quantum-classical modeling approach for electrically driven quantum dot devices, SPIE Photonics West 2018: Physics and Simulation of Optoelectronic Devices XXVI, January 29 - February 1, 2018, The Moscone Center, San Francisco, USA, January 29, 2018.

M. Kantner, Hybrid quantum-classical modeling of quantum dot based single-photon emitting diodes, Workshop Applied Mathematics and Simulation for Semiconductors, WIAS Berlin, October 10, 2018.

M. Kantner, Modeling and simulation of electrically driven quantum light emitters, Leibniz MMS Days, Leibniz Institut für Oberflächenmodifizierung (IOM), Leipzig, March 2, 2018.

M. Kantner, Thermodynamically consistent modeling of electrically driven quantum dot based light emitters on a device scale, Workshop ,,Nonlinear Dynamics in Semiconductor Lasers (NDSL2018)'', June 18 - 20, 2018, WIAS, Berlin, June 18, 2018.

D. Peschka, Steering pattern formation during dewetting with interface and contact lines properties, The 20th European Conference on Mathematics for Industry (ECMI 2018), Minisymposium 38 ``ECMI Special Interest Group: Material Design and Performance in Sustainable Energies'', June 18 - 22, 2018, Budapest, Hungary, June 21, 2018.

M. Thomas, Optimization of the radiative emission for mechanically strained optoelectronic semiconductor devices, 9th International Conference ``Inverse Problems: Modeling and Simulation'' (IPMS 2018), Minisymposium M16 ``Inverse and Control Problems in Mechanics'', May 21 - 25, 2018, The Eurasian Association on Inverse Problems, Malta, May 24, 2018.

A. Mielke, EDP convergence and optimal transport, Workshop ``Optimal Transportation and Applications'', November 12 - 15, 2018, Scuola Normale Superiore, Università di Pisa, Università di Pavia, Pisa, Italy, November 13, 2018.

A. Mielke, Energy, dissipation, and evolutionary Gamma convergence for gradient systems, Kolloquium ``Applied Analysis'', Universität Bremen, December 18, 2018.

A. Mielke, Entropy and gradient structures for quantum Markov semigroups and couplings to macroscopic thermodynamical systems, Nonlinear Mechanics Seminar, University of Bath, Mathematical Sciences, UK, May 22, 2018.

A. Mielke, On notions of evolutionary Gamma convergence for gradient systems, Workshop ``Gradient Flows: Challenges and New Directions'', September 10 - 14, 2018, International Centre for Mathematical Sciences (ICMS), Edinburgh, UK, September 13, 2018.

S. Reichelt, Corrector estimates for elliptic and parabolic equations with periodic coefficients, Analysis Seminar, Friedrich-Alexander-Universität Erlangen-Nürnberg, Institut für Angewandte Mathematik, Erlangen, May 18, 2017.

S. Reichelt, Corrector estimates for elliptic and parabolic equations with periodic coefficients, Analysis Seminar, Universität Augsburg, Institut für Mathematik, May 23, 2017.

S. Reichelt, Corrector estimates for imperfect transmission problems, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 7, 2017.

S. Reichelt, Traveling waves in FitzHugh--Nagumo systems with rapidly oscillating coefficients, Workshop ``Control of Self-organizing Nonlinear Systems'', August 29 - 31, 2017, Collaborative Research Center 910: Control of self-organizing nonlinear systems: Theoretical methods and concepts of application, Lutherstadt Wittenberg, August 30, 2017.

M. Heida, A. Mielke, Effective models for interfaces with many scales, CRC 1114 Conference ''Scaling Cascades in Complex Systems 2017'', Berlin, March 27 - 29, 2017.

M. Heida, Averaging of time-periodic dissipation potentials in rate-independent processes, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 9, 2017.

M. Heida, On G-convergence and stochastic two-scale convergences of the squareroot approximation scheme to the Fokker--Planck operator, GAMM-Workshop on Analysis of Partial Differential Equations, September 27 - 29, 2017, Eindhoven University of Technology, Mathematics and Computer Science Department, Netherlands, September 28, 2017.

M. Kantner, Hybrid quantum-classical modeling of electrically driven quantum light sources, Meeting of the MATHEON Scientific Advisory Board 2017, TU Berlin, Institut für Mathematik, November 13, 2017.

M. Kantner, Simulations of quantum dot devices by coupling of quantum master equations and semi-classical transport theory, 17th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD2017), July 24 - 28, 2017, Technical University of Denmark, Copenhagen, July 27, 2017.

M. Mittnenzweig, An entropic gradient structure Lindblad equations, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 9, 2017.

M. Mittnenzweig, An entropic gradient structure for quantum Markov semigroups, Workshop ``Applications of Optimal Transportation in the Natural Sciences'', January 30 - February 3, 2017, Mathematisches Forschungsinstitut Oberwolfach, January 31, 2017.

D. Peschka, Modelling and simulation of suspension flow, Graduate Seminar PDE in the Sciences, Universität Bonn, Institut für Angewandte Mathematik, January 20, 2017.

M. Thomas, Mathematical modeling and analysis of evolution processes in solids and the influence of bulk-interface-interaction, Humboldt-Universität zu Berlin, Institut für Mathematik, October 20, 2017.

A. Caiazzo, Homogenization methods for weakly compressible elastic materials forward and inverse problem, Workshop on Numerical Inverse and Stochastic Homogenization, February 13 - 17, 2017, Universität Bonn, Hausdorff Research Institute for Mathematics, February 17, 2017.

A. Mielke, Interfaces with many scales, Second CRC 1114 Days ''Scaling Cascades in Complex Systems'', November 8 - 10, 2017, Freie Universität Berlin, Schmöckwitz, November 10, 2017.

A. Mielke, Uniform exponential decay for energy-reaction-diffusion systems, Analysis Seminar, University of Pavia, Department of Mathematics, Italy, March 21, 2017.

K. Disser, Convergence for gradient systems of slow and fast chemical reactions, Joint Annual Meeting of DMV and GAMM, Session ``Applied Analysis'', March 7 - 11, 2016, Technische Universität Braunschweig, Braunschweig, March 11, 2016.

K. Disser, E-convergence to the quasi-steady-state approximation in systems of chemical reactions, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 25, 2016.

S. Reichelt, Competing patterns in anti-symmetrically coupled Swift--Hohenberg equations, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of Self-Organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4 - 8, 2016.

S. Reichelt, Error estimates for elliptic and parabolic equations with oscillating coefficients, Karlstad Applied Analysis Seminar, Karlstad University, Department of Mathematics and Computer Science, Sweden, April 13, 2016.

S. Reichelt, Error estimates for elliptic equations with not exactly periodic coefficients, Berlin Dresden Prague Würzburg Workshop ``Homogenization and Related Topics'', Technische Universität Dresden, Fachbereich Mathematik, June 22, 2016.

S. Reichelt, Homogenization of Cahn--Hilliard-type equations via evolutionary $Gamma$-convergence, Joint Annual Meeting of DMV and GAMM, Young Researchers' Minisymposium ``Multiscale Evolutionary Problems'', March 7 - 11, 2016, Technische Universität Braunschweig, March 7, 2016.

S. Reichelt, Homogenization of Cahn--Hilliard-type equations via evolutionary Gamma-convergence, Workshop ``Patterns of Dynamics'', Freie Universität Berlin, Fachbereich Mathematik und Informatik, July 25 - 29, 2016.

S. Reichelt, Homogenization of Cahn--Hilliard-type equations via gradient structures, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 2 ``Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equation'', July 1 - 5, 2016, The American Institute of Mathematical Sciences, Orlando (Florida), USA, July 3, 2016.

S. Reichelt, On periodic homogenization, 20. Harz-Seminar zur Strukturbildung in Chemie und Biophysik, February 21 - 22, 2016, Physikalisch-Technische Bundesanstalt, Hahnenklee, February 22, 2016.

TH. Frenzel, Evolutionary Gamma-convergence for amplitude equations and for wiggly energy models, Winter School 2016: Calculus of Variations in Physics and Materials Science, Würzburg, February 15 - 19, 2016.

M. Heida, A. Mielke, Ch. Kraus, M. Thomas, Effective models for interfaces with many scales, SCCS Days, CRC 1114 ``Complex Processes involving Cascades of Scales'', Ketzin, October 10 - 12, 2016.

M. Heida, Homogenization of the random conductance model, 7th European Congress of Mathematics (ECM), session ``Probability, Statistics and Financial Mathematics'', July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 20, 2016.

M. Heida, Homogenization of the random conductance model, Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 26 - 28, 2016, Technische Universität Dortmund, Fachbereich Mathematik, Dortmund, September 26, 2016.

M. Heida, Interfaces with many scales, SCCS Days, October 10 - 12, 2016, CRC ``Complex processes involving cascades of scales'', Ketzin, October 11, 2016.

M. Heida, On homogenization of rate-independent systems, sc Matheon Multiscale Seminar, Technische Universität Berlin, February 17, 2016.

M. Heida, Stochastic homogenization of 1-homogeneous functionals, 7th European Congress of Mathematics (7ECM), Minisymposium 29 ``Nonsmooth PDEs in the Modeling Damage, Delamination, and Fracture'', July 18 - 22, 2016, Technische Universität Berlin, July 22, 2016.

M. Heida, Stochastic homogenization of rate-independent systems, Berlin Dresden Prague Würzburg Workshop ``Homogenization and Related Topics'', Technische Universität Dresden, Fachbereich Mathematik, June 22, 2016.

M. Heida, Stochastic homogenization of rate-independent systems, Joint Annual Meeting of DMV and GAMM, Session ``Multiscales and Homogenization'', March 7 - 11, 2016, Technische Universität Braunschweig, Braunschweig, March 10, 2016.

M. Kantner, Multi-scale modeling and numerical simulation of single-photon emitters, Matheon Workshop--9th Annual Meeting ``Photonic Devices", Zuse Institut, Berlin, March 3, 2016.

M. Kantner, Multi-scale modelling and simulation of single-photon sources on a device level, Euro--TMCS II Theory, Modelling & Computational Methods for Semiconductors, Tyndall National Institute and University College Cork, Cork, Ireland, December 9, 2016.

R. Müller, W. Dreyer, J. Fuhrmann, C. Guhlke, New insights into Butler--Volmer kinetics from thermodynamic modeling, The 67th Annual Meeting of the International Society of Electrochemistry, Den Haag, Netherlands, August 21 - 26, 2016.

M. Becker, Th. Frenzel, Th. Niedermayer, S. Reichelt, M. Bär, A. Mielke, Competing patterns in anti-symmetrically coupled Swift--Hohenberg equations, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of Self-Organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4 - 8, 2016.

J. Fuhrmann, Ch. Merdon, A thermodynamically consistent numerical approach to Nernst--Planck--Poisson systems with volume constraints, The 67th Annual Meeting of the International Society of Electrochemistry, Den Haag, Netherlands, August 21 - 26, 2016.

J. Fuhrmann, W. Dreyer, C. Guhlke, M. Landstorfer, R. Müller, A. Linke, Ch. Merdon, Modeling and numerics for electrochemical systems, Micro Battery and Capacitive Energy Harvesting Materials -- Results of the MatFlexEnd Project, Universität Wien, Austria, September 19, 2016.

C. Guhlke, J. Fuhrmann, W. Dreyer, R. Müller, M. Landstorfer, Modeling of batteries, Batterieforum Deutschland 2016, Berlin, April 6 - 8, 2016.

A. Mielke, Evolution driven by energy and entropy, SFB1114 Kolloquium, Freie Universität Berlin, Berlin, June 30, 2016.

A. Mielke, Evolutionary Gamma-convergence, 2nd CENTRAL School on Analysis and Numerics for Partial Differential Equations, August 29 - September 2, 2016, Humboldt-Universität zu Berlin, Institut für Mathematik.

A. Mielke, Evolutionary relaxation for a rate-independent phase-transformation model, Workshop ``Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure'', March 14 - 18, 2016, Mathematisches Forschungszentrum Oberwolfach, March 14, 2016.

A. Mielke, Microstructure evolution via relaxation for a rate-independent elastic two-phase model, Joint Annual Meeting of DMV and GAMM, Session ``Applied Analysis'', March 7 - 11, 2016, Technische Universität Braunschweig, Braunschweig, March 10, 2016.

A. Mielke, Multiscale modeling via evolutionary Gamma convergence, 3rd PhD Workshop, May 30 - 31, 2016, International Research Training Group of the Collaborative Research Center (SFB) 1114 ``Scaling Cascades in Complex Systems'', Güstrow, May 30, 2016.

A. Mielke, On a model for the evolution of microstructures in solids -- Evolutionary relaxation, KTGU-IMU Mathematics Colloquia, March 30 - 31, 2016, Kyoto University, Department of Mathematics, Japan, March 31, 2016.

A. Mielke, Rate-independent microstructure evolution via relaxation of a two-phase model, Workshop ``Advances in the Mathematical Analysis of Material Defects in Elastic Solids'', June 6 - 10, 2016, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, June 10, 2016.

S. Reichelt, Homogenization of Cahn--Hilliard equations, Workshop on Control of Self-Organizing Nonlinear Systems, September 14 - 16, 2015, SFB 910 ``Control of self-organizing nonlinear systems: Theoretical methods and concepts of application'', Lutherstadt Wittenberg, September 15, 2015.

S. Reichelt, Two-scale homogenization of reaction-diffusion systems involving different diffusion length scales, CASA Colloquium, Eindhoven University of Technology, Centre for Analysis, Scientific Computing and Applications (CASA), Eindhoven, Netherlands, March 11, 2015.

M. Heida, Stochastic homogenization of Prandtl--Reuss plasticity, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30 - October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, October 1, 2015.

M. Kantner, Multi-scale modeling and simulation of electrically pumped single-photon sources, International Nano-Optoelectronics Workshop (iNOW 2015), Tokio, Japan, August 3 - 7, 2015.

TH. Koprucki, On device concepts for CMOS-compatible edge-emitters based on strained germanium, Symposium ``Alternative Semiconductor Integration in Si Microelectronics: Materials, Techniques and Applications'' of the E-MRS Fall Meeting 2015, September 15 - 18, 2015, Warsaw University of Technology, Krakow, Poland, September 18, 2015.

M. Thomas, Evolutionary Gamma convergence with application to damage and delamination, Seminar DICATAM, Università di Brescia, Dipartimento di Matematica, Brescia, Italy, June 3, 2015.

M. Thomas, From adhesive contact to brittle delamination in visco-elastodynamics, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30 - October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, October 2, 2015.

M. Thomas, Modeling of edge-emitting lasers based on tensile strained germanium microstripes, Applied Mathematics and Simulation for Semiconductors (AMaSiS 2015), March 11 - 13, 2015, WIAS Berlin, Berlin, March 11, 2015.

A. Mielke, Chemical Master Equation: Coarse graining via gradient structures, Kolloquium des SFB 1114 ``Scaling Cascades in Complex Systems'', Freie Universität Berlin, Fachbereich Mathematik, Berlin, June 4, 2015.

A. Mielke, EDP-convergence and the limit from diffusion to reaction, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30 - October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, October 2, 2015.

A. Mielke, Evolutionary $Gamma$-convergence for gradient systems explained via applications, Symposium ``Variational Methods for Stationary and Evolutionary Problems'', University of Warwick, Mathematics Institute, Warwick, UK, May 12, 2015.

A. Mielke, Evolutionary relaxation of a two-phase model, Mini-Workshop ``Scales in Plasticity'', November 8 - 14, 2015, Mathematisches Forschungsinstitut Oberwolfach, November 11, 2015.

A. Mielke, Homogenizing the Penrose--Fife system via evolutionary $Gamma$-convergence, INdAM Workshop ``Special Materials in Complex Systems -- SMaCS 2015'', May 18 - 20, 2015, Rome, Italy, May 19, 2015.

A. Mielke, The Chemical Master Equation as a discretization of the Fokker--Planck and Liouville equation for chemical reactions, Colloquium of Collaborative Research Center/Transregio ``Discretization in Geometry and Dynamics'', Technische Universität Berlin, Institut für Mathematik, Berlin, February 10, 2015.

A. Mielke, The Chemical Master Equation as entropic gradient flow, Conference ``New Trends in Optimal Transport'', March 2 - 6, 2015, Universität Bonn, Institut für Angewandte Mathematik, March 2, 2015.

A. Mielke, The Fokker--Planck and Liouville equations for chemical reactions as large-volume approximations of the Chemical Master Equation, Workshop ``Stochastic Limit Analysis for Reacting Particle Systems'', December 16 - 18, 2015, WIAS Berlin, Berlin, December 18, 2015.

A. Mielke, Variational approaches and methods for dissipative material models with multiple scales, Workshop ``Analysis and Computation of Microstructure in Finite Plasticity'', May 4 - 5, 2015, Hausdorff Center for Mathematics, Bonn, May 4, 2015.

C. Kreisbeck, Thin-film limits of functionals on A-free vector fields and applications, Workshop on Trends in Non-Linear Analysis 2014, July 31 - August 1, 2014, Instituto Superior Técnico, Departamento de Matemática, Lisbon, Portugal, August 1, 2014.

C. Kreisbeck, Thin-film limits of functionals on A-free vector fields and applications, XIX International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2014), September 8 - 11, 2014, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 9, 2014.

C. Kreisbeck, Thin-film limits of functionals on A-free vector fields and applications, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, July 16, 2014.

S. Reichelt, Effective equations for reaction-diffusion systems in strongly heterogeneous media, 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems (MURPHYS-HSFS-2014), April 7 - 11, 2014, WIAS Berlin, April 10, 2014.

S. Reichelt, Two-scale homogenization of nonlinear reaction-diffusion systems involving different diffusion length scales, scshape Matheon Multiscale Seminar, Technische Universität Berlin, Institut für Mathematik, December 3, 2014.

S. Reichelt, Two-scale homogenization of reaction-diffusion systems with small diffusion, 13th GAMM Seminar on Microstructures, January 17 - 18, 2014, Ruhr-Universität Bochum, Lehrstuhl für Mechanik - Materialtheorie, January 18, 2014.

S. Reichelt, Two-scale homogenization of reaction-diffusion systems with small diffusion, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 8: Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equations from Mathematical Science, July 7 - 11, 2014, Madrid, Spain, July 8, 2014.

S. Neukamm, Optimal quantitative two-scale expansion in stochastic homogenization, 13th GAMM Seminar on Microstructures, January 17 - 18, 2014, Ruhr-Universität Bochum, Lehrstuhl für Mechanik - Materialtheorie, January 18, 2014.

M. Thomas, Thermomechanical modeling of dissipative processes in elastic media via energy and entropy, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 8: Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equations from Mathematical Science, July 7 - 11, 2014, Madrid, Spain, July 8, 2014.

B. Wagner, Asymptotic analysis of interfacial evolution, BMS-WIAS Summer School ``Applied Analysis for Materials'', August 25 - September 5, 2014, Berlin Mathematical School, Technische Universität Berlin.

A. Caiazzo, Multiscale modeling of weakly compressible elastic materials in harmonic regime, Université de Franche-Comté, Laboratoire de Mathématiques de Besançon, France, July 1, 2014.

A. Mielke, S. Reichelt, M. Thomas, Pattern formation in systems with multiple scales, Evaluation of the DFG Collaborative Research Center 910 ``Control of Self-organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Berlin, June 30 - July 1, 2014.

A. Mielke, Evolutionary Gamma convergence and amplitude equations, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Analysis, March 10 - 14, 2014, Friedrich-Alexander Universität Erlangen-Nürnberg, March 13, 2014.

A. Mielke, Homogenization of parabolic gradient systems via evolutionary $Gamma$-convergence, Second Workshop of the GAMM Activity Group on ``Analysis of Partial Differential Equations'', September 29 - October 1, 2014, Universität Stuttgart, Institut für Analysis, Dynamik und Modellierung, September 30, 2014.

A. Mielke, Modeling jumps in rate-independent systems using balanced-viscosity solutions, 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems (MURPHYS-HSFS-2014), April 7 - 11, 2014, WIAS Berlin, April 8, 2014.

A. Mielke, Multiscale modeling and evolutionary Gamma-convergence for gradient flows, BMS-WIAS Summer School ``Applied Analysis for Materials'', August 25 - September 5, 2014, Berlin Mathematical School, Technische Universität Berlin.

A. Mielke, On the microscopic origin of generalized gradient structures for reaction-diffusion systems, XIX International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2014), September 8 - 11, 2014, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 11, 2014.

S. Neukamm, Characterization and approximation of macroscopic properties in elasticity with homogenization, 4th British-German Frontiers of Science Symposium, Potsdam, March 6 - 9, 2014.

S. Neukamm, Characterization and approximation of macroscopic properties with homogenization, 4th British-German Frontiers of Science Symposium, March 6 - 9, 2014, Alexander von Humboldt-Stiftung, Potsdam, March 7, 2014.

S. Neukamm, Homogenization of nonlinear bending plates, Workshop ``Relaxation, Homogenization, and Dimensional Reduction in Hyperelasticity'', March 25 - 27, 2014, Université Paris-Nord, France, March 26, 2014.

S. Neukamm, Homogenization of slender structures in small-strain regimes, 14th Dresden Polymer Discussion, Meißen, May 25 - 28, 2014.

K. Disser, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, BMS Intensive Course on Evolution Equations and their Applications, November 27 - 29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 28, 2013.

K. Disser, Passage to the limit of the entropic gradient structure of reversible Markov processes to the Wasserstein Fokker--Planck equation, Oberseminar Analysis, Martin-Luther-Universität Halle-Wittenberg, Institut für Mathematik, Halle, November 20, 2013.

S. Reichelt, Homogenization of degenerated reaction-diffusion equations, Doktorandenforum der Leibniz-Gemeinschaft, Sektion D, Berlin, June 6 - 7, 2013.

S. Reichelt, Introduction to homogenization concepts, Freie Universität Berlin, Institut für Mathematik, April 11, 2013.

S. Reichelt, Two-scale homogenization of nonlinear reaction-diffusion systems with small diffusion, Workshop on Control of Self-Organizing Nonlinear Systems, August 28 - 30, 2013, SFB 910 ``Control of self-organizing nonlinear systems: Theoretical methods and concepts of application'', Lutherstadt Wittenberg, August 30, 2013.

S. Reichelt, Two-scale homogenization in nonlinear reaction-diffusion systems with small diffusion, BMS Intensive Course on Evolution Equations and their Applications, November 27 - 29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 28, 2013.

P. Gussmann, Linearized elasticity as $Gamma$-limit of finite elasticity in the case of cracks, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Section ``Applied Analysis'', March 18 - 22, 2013, University of Novi Sad, Serbia, March 20, 2013.

S. Neukamm, Optimal decay estimate on the semigroup associated with a random walk among random conductances, Dirichlet Forms and Applications, German-Japanese Meeting on Stochastic Analysis, September 9 - 13, 2013, Universität Leipzig, Mathematisches Institut, September 9, 2013.

H. Abels, J. Daube, Ch. Kraus, D. Kröner, Sharp interface limit for the Navier--Stokes--Korteweg model, DIMO2013 -- Diffuse Interface Models, Levico Terme, Italy, September 10 - 13, 2013.

CH. Kraus, Damage and phase separation processes: Modeling and analysis of nonlinear PDE systems, DIMO2013 -- Diffuse Interface Models, September 10 - 13, 2013, Levico Terme, Italy, September 11, 2013.

CH. Kraus, Modeling and analysis of a nonlinear PDE system for phase separation and damage, Università di Pavia, Dipartimento di Matematica, Italy, January 22, 2013.

CH. Kraus, Sharp interface limit of a diffuse interface model of Navier--Stokes--Allen--Cahn type for mixtures, Workshop ``Hyperbolic Techniques for Phase Dynamics'', June 10 - 14, 2013, Mathematisches Forschungsinstitut Oberwolfach, June 11, 2013.

M. Thomas, Mathematical modeling, analysis and optimization of strained germanium microbridges, sc Matheon Center Days, Technische Universität Berlin, November 5, 2013.

H. Hanke, Derivation of an effective damage model with evolving micro-structure, Oberseminar zur Analysis, Universität Duisburg-Essen, Fachbereich Mathematik, Essen, January 29, 2013.

H. Hanke, Derivation of an effective damage model with non-periodic evolving micro-structure, 12th GAMM Seminar on Microstructures, February 8 - 9, 2013, Humboldt-Universität zu Berlin, Institut für Mathematik, February 9, 2013.

A. Mielke, Introduction to evolutionary Gamma convergence for gradient systems, School ``Multi-scale and Multi-field Representations of Condensed Matter Behavior'', November 25 - 29, 2013, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy.

A. Mielke, Deriving the Ginzburg--Landau equation as amplitude equation via evolutionary Gamma convergence, ERC Workshop on Variational Views on Mechanics and Materials, June 24 - 26, 2013, University of Pavia, Department of Mathematics, Italy, June 26, 2013.

A. Mielke, Emergence of rate independence in gradient flows with wiggly energies, SIAM Conference on Mathematical Aspects of Materials Science (MS13), Minisymposium ``The Origins of Hysteresis in Materials'' (MS56), June 9 - 12, 2013, Philadelphia, USA, June 12, 2013.

A. Mielke, Evolutionary Gamma convergence and amplitude equations, sc Matheon Multiscale Seminar, Technische Universität Berlin, Institut für Mathematik, April 8, 2013.

A. Mielke, On entropy-driven dissipative quantum mechanical systems, Analysis and Stochastics in Complex Physical Systems, March 20 - 22, 2013, Universität Leipzig, Mathematisches Institut, March 21, 2013.

A. Mielke, Rate-independent plasticity as vanishing-viscosity limit for wiggly energy landscape, Workshop on Evolution Problems for Material Defects: Dislocations, Plasticity, and Fracture, September 30 - October 4, 2013, International School of Advanced Studies (SISSA), Trieste, Italy, September 30, 2013.

P. Gussmann, Linearisierte Elastizität als Grenzwert finiter Elastizität im Falle von Schlitzgebieten, Jahrestagung der Deutsche Mathematiker-Vereinigung (DMV), Studierendenkonferenz, September 17 - 20, 2012, Universität des Saarlandes, Fakultät für Mathematik und Informatik, Saarbrücken, September 20, 2012.

S. Heinz, Quasiconvexity equals rank-one convexity for isotropic sets of 2x2 matrices, 11th GAMM Seminar on Microstructures, January 20 - 21, 2012, Universität Duisburg-Essen, January 20, 2012.

S. Heinz, Regularization and relaxation of time-continuous problems in plasticity, 11th GAMM Seminar on Microstructures, Universität Duisburg-Essen, January 20 - 21, 2012.

S. Heinz, Rigorous derivation of a dissipation for laminate microstructures, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Material Modelling in Solid Mechanics, March 26 - 30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 27, 2012.

M. Liero, Interfaces in reaction-diffusion systems, Seminar ``Dünne Schichten'', Technische Universität Berlin, Institut für Mathematik, February 9, 2012.

CH. Kraus, A nonlinear PDE system for phase separation and damage, Universität Freiburg, Abteilung Angewandte Mathematik, November 13, 2012.

CH. Kraus, Cahn--Larché systems coupled with damage, Università degli Studi di Milano, Dipartimento di Matematica, Italy, November 28, 2012.

CH. Kraus, Phase field systems for phase separation and damage processes, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11 - 15, 2012, Frauenchiemsee, June 12, 2012.

CH. Kraus, Phasenfeldsysteme für Entmischungs- und Schädigungsprozesse, Mathematisches Kolloquium, Universität Stuttgart, Fachbereich Mathematik, May 15, 2012.

H. Hanke, Derivation of an effective damage evolution model, ``A sc Matheon Multiscale Workshop'', Technische Universität Berlin, Institut für Mathematik, April 20, 2012.

H. Hanke, Derivation of an effective damage evolution model using two-scale convergence techniques, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Damage Processes and Contact Problems, March 26 - 30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 29, 2012.

A. Mielke, From small-strain to finite-strain elastoplasticity via evolutionary Gamma convergence, Variational Models and Methods for Evolution, September 10 - 12, 2012, Centro Internazionale per la Ricerca Matematica (CIRM) and Istituto di Matematica Applicata e Tecnologie Informatiche/Consiglio Nazionale delle Ricerche (IMATI-CNR), Levico, Italy, September 11, 2012.

A. Mielke, Gamma convergence and evolution, International Conference ``Trends in Mathematical Analysis'', March 1 - 3, 2012, Politecnico di Milano, Dipartimento di Matematica ``F. Brioschi'', Italy, March 1, 2012.

A. Mielke, Multiscale gradient systems and their amplitude equations, Workshop ``Dynamics of Patterns'', December 17 - 21, 2012, Mathematisches Forschungsinstitut Oberwolfach, December 18, 2012.

A. Mielke, Multiscale modeling for evolutionary systems via Gamma convergence, NDNS$^+$ Summer School in Applied Analysis, June 18 - 20, 2012, University of Twente, Applied Analysis & Mathematical Physics, Enschede, Netherlands.

A. Mielke, Small-strain elastoplasticity is the evolutionary Gamma limit of finite-strain elastoplasticity, International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2012), September 3 - 6, 2012, Israel Institute of Technology (Technion), Faculty of Aerospace Engineering, Haifa, September 4, 2012.

S. Reichelt, Homogenization of reaction-diffusion problems, SFB 910 Symposium ``Pattern Formation and Instabilities in Systems with Multiple Scales'', Technische Universität Berlin, Institut für Theoretische Physik, November 25, 2011.

CH. Kraus, Diffuse interface systems for phase separation and damage, Seminar on Partial Differential Equations, Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, May 3, 2011.

CH. Kraus, Phase separation systems coupled with elasticity and damage, ICIAM 2011, July 18 - 22, 2011, Vancouver, Canada, July 18, 2011.

A. Mielke, An evolutionary elastoplastic plate model obtained via Mosco convergence, 10th GAMM Seminar on Microstructures, January 20 - 22, 2011, Technische Universität Darmstadt, Fachbereich Mathematik, January 22, 2011.

A. Mielke, Geometry and thermodynamics for the coupling of quantum mechanics and dissipative systems, Workshop ``Applied Dynamics and Geometric Mechanics'', August 15 - 19, 2011, Mathematisches Forschungsinstitut Oberwolfach, August 16, 2011.

A. Mielke, Multiscale problems in systems driven by functionals, ISAM-TopMath Summer School 2011 on Variational Methods, September 12 - 16, 2011, Technische Universität München, Fakultät für Mathematik.

A. Mielke, Remarks on evolutionary multiscale systems driven by functionals, Intellectual Challenges in Multiscale Modeling of Solids, July 4 - 5, 2011, Oxford University, Mathematical Institute, UK, July 4, 2011.

A. Mielke, Mathematical approaches to thermodynamic modeling, Autumn School on Mathematical Principles for and Advances in Continuum Mechanics, November 7 - 12, 2011, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy.

M. Thomas, From damage to delamination in nonlinearly elastic materials, 6th Singular Days on Asymptotic Methods for PDEs, April 29 - May 1, 2010, WIAS, May 1, 2010.

M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, Workshop ``Microstructures in Solids: From Quantum Models to Continua'', March 14 - 20, 2010, Mathematisches Forschungsinstitut Oberwolfach, March 18, 2010.

W. Dreyer, On a paradox within the phase field modeling of storage systems and its resolution, 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, May 25 - 28, 2010, Technische Universität Dresden, May 26, 2010.

CH. Kraus, An inhomogeneous, anisotropic and elastically modified Gibbs-Thomson law as singular limit of a diffuse interface model, 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), March 22 - 26, 2010, Karlsruhe, March 23, 2010.

CH. Kraus, Inhomogeneous and anisotropic phase-field quantities in the sharp interface limit, 6th Singular Days 2010, April 29 - May 1, 2010, WIAS, Berlin, April 30, 2010.

A. Mielke, A mathematical model for the evolution of microstructures in elastoplasticity, Fifth International Conference on Multiscale Materials Modeling, Symposium on Mathematical Methods, October 4 - 8, 2010, Fraunhofer Institut für Werkstoffmechanik (IWM), Freiburg, October 4, 2010.

A. Mielke, Rate-independent plasticity as Gamma limit of a slow viscous gradient flow for wiggly discrete energy, Zwei-Städte-Kolloquium zur Analysis, Friedrich-Alexander-Universität Erlangen-Nürnberg, Fachbereich Mathematik, November 26, 2010.

A. Mielke, Rate-independent plasticity as Gamma limit of a slow viscous gradient flow for wiggly energies, Nečas Seminar on Continuum Mechanics, Jindrich Nečas Center for Mathematical Modeling, Prague, Czech Republic, November 8, 2010.

M. Thomas, Rate-independent damage and delamination processes, Workshop ``Rate-independent Systems: Modeling, Analysis, and Computations'', August 30 - September 3, 2010, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, August 31, 2010.

G. Kitavtsev, L. Recke, B. Wagner, Derivation, analysis and numerics of reduced ODE models describing coarsening dynamics, 3textsuperscriptrd European Postgraduate Fluid Dynamics Conference, Nottingham, UK, July 13 - 16, 2009.

G. Kitavtsev, Derivation, analysis and numerics of reduced ODE models describing coarsening dynamics, 3$^rm rd$ European Postgraduate Fluid Dynamics Conference, July 13 - 16, 2009, University of Nottingham, UK, July 15, 2009.

G. Kitavtsev, Reduced ODE models describing coarsening dynamics of slipping droplets and a geometrical approach for their derivation, Oberseminar, Universität Bonn, Institut für Angewandte Mathematik, July 23, 2009.

W. Dreyer, On a paradox within the phase field modeling of storage systems and its resolution, PF09 -- 2nd Symposium on Phase-Field Modelling in Materials Science, August 30 - September 2, 2009, Universität Aachen, Kerkrade, Netherlands, August 31, 2009.

W. Dreyer, Phase field models and their sharp limits in the context of hydrogen storage and lithium-ion batteries, 1textsuperscriptst International Conference on Material Modelling (1textsuperscriptst ICMM), September 15 - 18, 2009, Technische Universität Dortmund, September 16, 2009.

H. Hanke, Gamma-limits in rate-independent evolutionary problems and homogenization in gradient plasticity, DK-Seminar ``Numerical Simulations in Technical Sciences'', Graz University of Technology, Faculty of Technical Mathematics and Technical Physics, Austria, November 18, 2009.

H. Hanke, Homogenization in gradient plasticity, ISIMM Workshop on Mathematical Problems of Solid Mechanics, October 8 - 9, 2009, Technische Universität Darmstadt, Fachbereich Mathematik, October 9, 2009.

A. Mielke, Homogenization via Gamma-convergence for hyperbolic problems, Workshop ``The Future for Complexity Sciences'', September 15 - 16, 2009, University of Bath, Department of Mathematical Sciences, UK, September 15, 2009.

A. Mielke, Multiscale modeling for energy-driven systems, Workshop on Scale Transitions in Space and Time for Materials, October 19 - 23, 2009, Lorentz Center, Leiden, Netherlands, October 19, 2009.

J. Rehberg, Functional analytic properties of the quantum mechanical particle density operator, International Workshop on Quantum Systems and Semiconductor Devices: Analysis, Simulations, Applications, April 20 - 24, 2009, Peking University, School of Mathematical Sciences, Beijing, China, April 21, 2009.

J. Rehberg, Quasilinear parabolic equations in distribution spaces, International Conference on Nonlinear Parabolic Problems in Honor of Herbert Amann, May 10 - 16, 2009, Stefan Banach International Mathematical Center, Bedlewo, Poland, May 12, 2009.

CH. Kraus, A phase-field model with anisotropic surface tension in the sharp interface limit, Second GAMM-Seminar on Multiscale Material Modelling, July 10 - 12, 2008, Universität Stuttgart, Institut für Mechanik (Bauwesen), July 12, 2008.

CH. Kraus, Ein Phasenfeldmodell vom Cahn-Hilliard-Typ im singulären Grenzwert, Oberseminar Analysis, Universität Regensburg, Fakultät für Mathematik, April 25, 2008.

CH. Kraus, Phase field models and corresponding Gibbs--Thomson laws. Part II, SIMTECH Seminar Multiscale Modelling in Fluid Mechanics, Universität Stuttgart, Institut für Angewandte Analysis und Numerische Simulation, November 5, 2008.

CH. Kraus, On jump conditions at phase interfaces, Oberseminar über Angewandte Mathematik, December 10 - 15, 2007, Universität Freiburg, Abteilung für Angewandte Mathematik, December 11, 2007.

J. Rehberg, On Schrödinger--Poisson systems, International Conference ``Nonlinear Partial Differential Equations'' (NPDE 2007), September 10 - 15, 2007, Institute of Applied Mathematics and Mechanics of NASU, Yalta, Ukraine, September 13, 2007.

J. Rehberg, Über Schrödinger-Poisson-Systeme, Chemnitzer Mathematisches Colloquium, Technische Universität Chemnitz, Fakultät für Mathematik, May 24, 2007.

CH. Kraus, Equilibrium conditions for liquid-vapor system in the sharp interface limit, Seminar Thermodynamische Modellierung und Analyse von Phasenübergängen, WIAS, Berlin, July 18, 2006.

CH. Kraus, Equilibria conditions in the sharp interface limit of the van der Waals-Cahn-Hilliard phase model, Recent Advances in Free Boundary Problems and Related Topics (FBP2006), September 14 - 16, 2006, Levico, Italy, September 14, 2006.

CH. Kraus, The sharp interface limit of the van der Waals--Cahn--Hilliard model, Polish-German Workshop ``Modeling Structure Formation'', Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, Poland, September 8, 2006.

J. Rehberg, The Schrödinger--Poisson system, Colloquium in Honor of Prof. Demuth, September 10 - 11, 2006, Universität Clausthal, September 10, 2006.

CH. Kraus, On the sharp limit of the Van der Waals-Cahn-Hilliard model, WIAS Workshop ``Dynamic of Phase Transitions'', November 30 - December 3, 2005, Berlin, December 2, 2005.

CH. Kraus, On the sharp limit of the Van der Waals-Cahn-Hilliard model, Workshop ``Micro-Macro Modeling and Simulation of Liquid-Vapor Flows'', November 16 - 18, 2005, Universität Freiburg, Mathematisches Institut, Kirchzarten, November 17, 2005.

CH. Kraus, Maximale Konvergenz in höheren Dimensionen, Seminar Thermodynamische Modellierung und Analyse von Phasenübergängen, WIAS, Berlin, May 24, 2005.

J. Rehberg, Some analytical ideas concerning the quantum-drift-diffusion systems, Workshop ``Problèmes spectraux non-linéaires et modèles de champs moyens'', April 4 - 8, 2005, Institut Henri Poincaré, Paris, France, April 5, 2005.

J. Rehberg, Analysis of macroscopic and quantum mechanical semiconductor models, International Visitor Program ``Nonlinear Parabolic Problems'', August 8 - November 18, 2005, Finnish Mathematical Society (FMS), University of Helsinki, and Helsinki University of Technology, Finland, November 1, 2005.

J. Fuhrmann, H.-Chr. Kaiser, Th. Koprucki, G. Schmidt, Electronic states in semiconductor nanostructures and upscaling to semi-classical models, Evaluation Colloquium of the DFG Priority Program ``Analysis, Modeling and Simulation of Multiscale Problems'', Bad Honnef, May 20 - 21, 2004.

M. Baro, H. Gajewski, R. Hünlich, H.-Chr. Kaiser, Optoelektronische Bauelemente: mikroskopische & makroskopische Modelle, MathInside --- Überall ist Mathematik, event of the DFG Research Center ``Mathematics for Key Technologies'' on the occasion of the Open Day of Urania, Berlin, September 13, 2003 - December 3, 2004.

J. Rehberg, A combined quantum mechanical and macroscopic model for semiconductors, Workshop on Multiscale problems in quantum mechanics and averaging techniques, December 11 - 12, 2003, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, December 12, 2003.

M. O'Donovan, P. Farrell, T. Streckenbach, Th. Koprucki, S. Schulz, Multiscale simulations of uni-polar hole transport in (In,Ga)N quantum well systems, Preprint no. arXiv:2111.01644, Cornell University Library, arXiv.org, 2021.
Abstract
Understanding the impact of the alloy micro-structure on carrier transport becomes important when designing III-nitride-based LED structures. In this work, we study the impact of alloy fluctuations on the hole carrier transport in (In,Ga)N single and multi-quantum well systems. To disentangle hole transport from electron transport and carrier recombination processes, we focus our attention on uni-polar (p-i-p) systems. The calculations employ our recently established multi-scale simulation framework that connects atomistic tight-binding theory with a macroscale drift-diffusion model. In addition to alloy fluctuations, we pay special attention to the impact of quantum corrections on hole transport. Our calculations indicate that results from a virtual crystal approximation present an upper limit for the hole transport in a p-i-p structure in terms of the current-voltage characteristics. Thus we find that alloy fluctuations can have a detrimental effect on hole transport in (In,Ga)N quantum well systems, in contrast to uni-polar electron transport. However, our studies also reveal that the magnitude by which the random alloy results deviate from virtual crystal approximation data depends on several factors, e.g. how quantum corrections are treated in the transport calculations.

A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Preprint no. arXiv:1510.06219, Cornell University Library, arXiv.org, 2015.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.