Neben Fragen der Homogenisierung von Materialien mit Mikrostruktur ist hierbei insbesondere das Verhalten von dünnen Schichten (Grenz oder Randschichten) von Interesse. Im günstigsten Fall kann die Dynamik in solchen Schichten durch dimensionsreduzierte Modelle ersetzt werden, die geeignet an die Variablen im Volumen gekoppelt werden müssen.
Dabei beschäftigt sich das WIAS mit folgenden Anwendungsgebieten:
 Herleitung von elastoplastischen Stab und Plattenmodellen
 Homogenisierung von Materialmodellen (auch mit Plastizität und Schädigung)
 Phasenfeldmodelle und deren ScharfeGrenzflächenLimites
 Epitaktisches Wachstum dünner Schichten
 Evolution von dünnen Flüssigkeitsfilmen
 Einbettung von Quantenpunkten,  drähten und filmen in Halbleitermodelle
 Aktive Grenzflächen in Halbleitern (DünnschichtPhotovoltaik)
 Kopplung von Optik und Ladungstransport in Lasern
GammaKonvergenz für Mehrskalenprobleme
Die GammaKonvergenz ist ein unverzichtbares Hilfsmittel um effektive Beschreibungen von Mehrskalenproblemen herzuleiten. Obwohl GammaKonvergenz in erster Linie für statische Minimierungsprobleme entwickelt wurde, arbeitet das WIAS an einer Erweiterung dieser Theorie für Evolutionsprobleme. Dies ist insbesondere für Systeme möglich, deren Evolution vollständig durch Funktionale beschrieben wird, zum Beispiel in verallgemeinerten Gradientensystemen. In solchen Systemen beschreiben ein Energie oder Entropiefunktional sowie ein konvexes Dissipationspotential die zeitliche Entwicklung des Systems. Mittels De Giorgi's Prinzip der Kurven maximaler Steigung kann eine äquivalente Formulierung der zugrundeliegenden Gleichung in Form einer skalaren Energie bzw. Entropiebilanz geschrieben werden. Diese Formulierung ist frei von Ableitungen, so dass es möglich ist die GammaKonvergenz der Funktionale auszunutzen um eine effektive Bilanzgleichung für ein Grenzsystem herzuleiten. Jedoch stellt sich heraus, dass die GammaKonvergenz der beteiligten Funktionale allein in vielen Fällen nicht ausreicht. Strengere (Kompatibilitäts)Bedingungen sind erforderlich, die die Konvergenz der Lösungen garantieren, siehe Mielke 2015.
Höhepunkte
Quantitative Abschätzungen für Homogenisierung von ReaktionsDiffusionsgleichungen
Für Modelle in Materialien mit periodischer Mikrostruktur, welche auch degenerierende Diffusionskoeffizienten oder Sprünge der Lösungen über interne Grenzschichten berücksichtigen, wurden neue "Wiederherstellungsoperatoren" in Reichelt 2016 entwickelt. Diese Operatoren ermöglichen es, den Fehler zwischen der mikroskopischen und der homogenisierten (makroskopischen) Lösung bezüglich der charakteristischen Periodenlänge quantitativ abzuschätzen.
Publikationen
Monografien

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. 187249, (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 Γlimits E_{0} and R_{0}. 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 rateindependent systems as well as the passage from viscous to rateindependent 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. 125155, (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 Gammaconvergence. 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 rateindependent system. On the other hand we show how the concept of BalancedViscosity solution arise as in the vanishingviscosity limit.
As applications we discuss, first, the evolution of laminate microstructures in finitestrain elastoplasticity and, second, a twophase model for shapememory materials, where Hmeasures 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. 555585, (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 NonEquilibrium Reversible Irreversible Coupling) to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradientflow contribution, which satisfy a particular noninteraction condition. One of our models couples a quantum system to a finite number of heat baths each of which is described by a timedependent temperature. The dissipation mechanism is modeled via the canonical correlation operator, which is the inverse of the KuboMori metric for density matrices and which is strongly linked to the von Neumann entropy for quantum systems. Thus, one recovers the dissipative doublebracket operators of the Lindblad equations but encounters a correction term for the consistent coupling to the dissipative dynamics. For the finitedimensional 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 FokkerPlanck 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).
Artikel in Referierten Journalen

M. Heida, S. Nesenenko, Stochastic homogenization of ratedependent models of monotone type in plasticity, Asymptotic Analysis, 112 (2019), pp. 185212, DOI 10.3233/ASY181502 .
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 ratedependent systems. The derivations of the homogenization results presented in this work are based on the stochastic twoscale convergence in Sobolev spaces. For completeness, we also present some twoscale homogenization results for convex functionals, which are related to the classical Gammaconvergence 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. 14/114/19 (published online on 23.12.2019), DOI 10.1007/s0003301912384 .
Abstract
We develop a multiscale approach to describe the behavior of a suspension of solid magnetizable particles in a viscous nonconducting fluid in the presence of an externally applied magnetic field. By upscaling the quasistatic 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 3Dprinted microarchitectured polymer materials exhibiting a negative Poisson's ratio, Continuum Mechanics and Thermodynamics, 32 (2020), pp. 433449 (published online on 20.11.2019), DOI 10.1007/s00161019008516 .
Abstract
This work proposes the complete design cycle for several auxetic materials where the cycle consists of three steps (i) the design of the microarchitecture, (ii) the manufacturing of the material and (iii) the testing of the material. We use topology optimization via a levelset method and asymptotic homogenization to obtain periodic microarchitectured materials with a prescribed effective elasticity tensor and Poisson's ratio. The space of admissible microarchitectural 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 Gammaconvergence for perturbed gradient systems, Journal of Evolution Equations, 19 (2019), pp. 479522, DOI 10.1007/s0002801900484x .
Abstract
We consider the initialvalue 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 semiimplicit discretization scheme with a variational approximation technique. 
P. Dondl, Th. Frenzel, A. Mielke, A gradient system with a wiggly energy and relaxed EDPconvergence, ESAIM. Control, Optimisation and Calculus of Variations, 25 (2019), pp. 68/168/45, 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 Gammaconvergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gammaconvergence. We call this notion relaxed EDPconvergence since the special structure of the dissipation functional may not be preserved under Gammaconvergence. 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. 127.

M. Heida, B. Schweizer, Stochastic homogenization of plasticity equations, ESAIM. Control, Optimisation and Calculus of Variations, 24 (2018), pp. 153176.
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 flowrule 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(∇^{s}u)=f. 
P. Gurevich, S. Reichelt, Pulses in FitzHughNagumo systems with rapidly oscillating coefficients, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 16 (2018), pp. 833856.
Abstract
This paper is devoted to pulse solutions in FitzHughNagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a twoscale FitzHughNagumo 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 pulselike solutions of the original system. 
A. Muntean, S. Reichelt, Corrector estimates for a thermodiffusion model with weak thermal coupling, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 16 (2018), pp. 807832, DOI 10.1137/16M109538X .
Abstract
The present work deals with the derivation of corrector estimates for the twoscale homogenization of a thermodiffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged highcontrast microstructures. The terminology “weak thermal coupling” refers here to the variable scaling in terms of the small homogenization parameter ε of the heat conductiondiffusion interaction terms, while the “highcontrast” is thought particularly in terms of the heat conduction properties of the composite material. As main target, we justify the firstorder terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufourlike 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 εindependent estimates for the thermal and concentration fields and for their coupled fluxes 
M. Liero, S. Reichelt, Homogenization of CahnHilliardtype equations via evolutionary Gammaconvergence, NoDEA. Nonlinear Differential Equations and Applications, 25 (2018), pp. 6/16/31, DOI 10.1007/s0003001804959 .
Abstract
In this paper we discuss two approaches to evolutionary Γconvergence 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 Γconvergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the timedependent 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 energydissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for CahnHilliardtype equations. Using the method of weak and strong twoscale convergence via periodic unfolding, we show that the energy and dissipation functionals Γconverge. In conclusion, we will give specific examples for the applicability of each of the two approaches. 
S. Reichelt, Corrector estimates for a class of imperfect transmission problems, Asymptotic Analysis, 105 (2017), pp. 326, DOI 10.3233/ASY171432 .
Abstract
Based on previous homogenization results for imperfect transmission problems in twocomponent 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. Kantner, M. Mittnenzweig, Th. Koprucki, Hybrid quantumclassical modeling of quantum dot devices, Phys. Rev. B., 96 (2017), pp. 205301/1205301/17, 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 wellestablished fields of semiclassical 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 quantumclassical 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 singlephoton source based on a single quantum dot in the stationary and transient operation regime. 
CH. Dörlemann, M. Heida, B. Schweizer, Transmission conditions for the Helmholtz equation in perforated domains, Vietnam Journal of Mathematics, 45 (2017), pp. 241253, DOI 10.1007/s100130160222y .

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 onedimensional blood flow models using experimental data, International Journal of Numerical Methods in Biomedical Engineering, 33 (2017), pp. e2843/1e2843/26, 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 onedimensional 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. 
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. 437475, 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. 
S. Heinz, A. Mielke, Existence, numerical convergence, and evolutionary relaxation for a rateindependent phasetransformation model, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 374 (2016), pp. 20150171/120150171/23, DOI 10.1098/rsta.2015.0171 .
Abstract
We revisit the twowell 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 rateindependent phase transformations using an extremum principle", Arch. Rational Mech. Anal., 162, 137177, 2002 ([MTL02]). This energetic rateindependent 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 spacetime discretization, and (iii) to perform the evolutionary relaxation from the purestate model to the relaxed mixture model. All these results rely on weak converge and involve the Hmeasure as an essential tool. 
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. 11671187.
Abstract
Anisotropy is an essential feature of phasefield 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 phasefield 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  GinzburgLandau type phasefield model for dendritic growth. For the resulting derivativefree 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. 117131.
Abstract
This note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly εperiodic 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 nondegenerating 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. 
A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Journal of NonEquilibrium Thermodynamics, 41 (2016), pp. 141149.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic timereversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradientflow, steepestascent, or maximalentropyproduction equation. Onsager's original theorem is limited to closetoequilibrium 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 timereversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows. 
S. Yanchuk, L. Lücken, M. Wolfrum, A. Mielke, Spectrum and amplitude equations for scalar delaydifferential equations with large delay, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 537553.
Abstract
The subject of the paper are scalar delaydifferential 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 delaydifferential equations close to the destabilization threshold. 
A. Mielke, Deriving amplitude equations via evolutionary Gamma convergence, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 26792700.
Abstract
We discuss the justification of the GinzburgLandau equation with real coefficients as an amplitude equation for the weakly unstable onedimensional SwiftHohenberg 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 timedependent 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 L^{2}, while for the case of a quadratic nonlinearity we need to impose weak convergence in H^{1}. 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 .

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. 677699.

A. Mielke, S. Reichelt, M. Thomas, Twoscale homogenization of nonlinear reactiondiffusion systems with slow diffusion, Networks Heterogeneous Media, 9 (2014), pp. 353382.
Abstract
We derive a twoscale homogenization limit for reactiondiffusion 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 twoscale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit. 
S. Neukamm, I. Velcic, Derivation of a homogenized vonKármán plate theory from 3D nonlinear elasticity, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 27012748.
Abstract
We rigorously derive a homogenized vonKármán plate theory as a Gammalimit from nonlinear threedimensional elasticity by combining homogenization and dimension reduction. Our starting point is an energy functional that describes a nonlinear elastic, threedimensional 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 vonKármán scaling regime. The obtained limit is a homogenized vonKármán plate model. Its effective material properties are determined by a relaxation formula that exposes a nontrivial coupling of the behavior of the outofplane displacement with the oscillatory behavior in the inplane 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. 
M.A. Peletier, M. Renger, M. Veneroni, Variational formulation of the FokkerPlanck equation with decay: A particle approach, Communications in Contemporary Mathematics, 15 (2013), pp. 1350017/11350017/43.

A. Mielke, E. Rohan, Homogenization of elastic waves in fluidsaturated porous media using the Biot model, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 873916.
Abstract
We consider periodically heterogeneous fluidsaturated 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 twoscale homogenization method we obtain the limit twoscale 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. 923948.
Abstract
We provide a rigorous justification of the classical linearization approach in plasticity. By taking the smalldeformations limit, we prove via Gammaconvergence for rateindependent processes that energetic solutions of the quasistatic finitestrain 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. 437457.
Abstract
This paper deals with dimension reduction in linearized elastoplasticity in the rateindependent case. The reference configuration of the elastoplastic body is given by a twodimensional 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 rateindependent evolution formulated in the framework of energetic solutions. This concept is based on an energystorage 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. 419454.
Abstract
We study a singularlimit problem arising in the modelling of chemical reactions. At finite $e>0$, the system is described by a FokkerPlanck convectiondiffusion equation with a doublewell 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):18051825, 2010, using the linear structure of the equation. In this paper we reprove the result by using solely the Wasserstein gradientflow structure of the system. In particular, we make no use of the linearity, nor of the fact that it is a secondorder 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 gradientflow structure, we prove that the sequence of rescaled solutions is precompact in an appropriate topology. We then prove a Gammaconvergence 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. 
W. Dreyer, F. Duderstadt, M. Hantke, G. Warnecke, Bubbles in liquids with phase transition, Continuum Mechanics and Thermodynamics, 24 (2012), pp. 461483.
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 multiphase 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 quasiincompressible flows: Sharp interface limits and numerics, ESAIM Proceedings, 38 (2012), pp. 5477.
Abstract
In this contribution, we investigate a diffuse interface model for quasiincompressible 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 CahnHilliard 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. 105143.
Abstract
This paper deals with a sharp interface limit of the isothermal NavierStokesKorteweg 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. 235273.
Abstract
Brittle Griffithtype delamination of compounds is deduced by means of Gammaconvergence from partial, isotropic damage of threespecimensandwichstructures 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 rateindependent. 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 delaminationgradient is supressed. Noninterpenetration and transmissionconditions along the interface are obtained. 
A. Mielke, L. Truskinovsky, From discrete viscoelasticity to continuum rateindependent plasticity: Rigorous results, Archive for Rational Mechanics and Analysis, 203 (2012), pp. 577619.
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 viscoelastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishingviscosity limit and the discrete to continuum limit. In the course of these limits a nonconvex elastic problem transforms into a convex elastic problem while the quadratic ratedependent dissipation of viscoelastic solid transforms into a singular rateindependent dissipation of an ideally plastic solid. In order to emphasize ideas we employ in our proofs the simplest prototypical system describing transformational plasticity of shapememory 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. 19611986.
Abstract
This paper is devoted to dimension reduction for linearized elastoplasticity in the rateindependent case. The reference configuration of the threedimensional elastoplastic body has a twodimensional 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 KirchhoffLove plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Gammaconvergence theory for rateindependent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energystorage 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 energystorage 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 Moscoconvergent quadratic energies. 
K. Hermsdörfer, Ch. Kraus, D. Kröner, Interface conditions for limits of the NavierStokesKorteweg model, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 13 (2011), pp. 239254.
Abstract
In this contribution we will study the behaviour of the pressure across phase boundaries in liquidvapour flows. As mathematical model we will consider the static version of the NavierStokesKorteweg 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. 
H. Hanke, Homogenization in gradient plasticity, Mathematical Models & Methods in Applied Sciences, 21 (2011), pp. 16511684.
Abstract
This paper yields a twoscale homogenization result for a rateindependent 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 twoscale 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, GAMMMitteilungen, 34 (2011), pp. 102106.
Abstract
This paper yields a twoscale homogenization result for a rateindependent 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 twoscale 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, Completedamage evolution based on energies and stresses, Discrete and Continuous Dynamical Systems  Series S, 4 (2011), pp. 423439.
Abstract
The rateindependent damage model recently developed in Bouchitté, Mielke, Roubíček “A completedamage problem at small strains" allows for complete damage, such that the deformation is no longer welldefined. 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 nonquadratic elastic energies by providing Gamma convergence of energetic solutions from partial to complete damage under rather general conditions. 
H. Garcke, Ch. Kraus, An anisotropic, inhomogeneous, elastically modified GibbsThomson law as singular limit of a diffuse interface model, Advances in Mathematical Sciences and Applications, 20 (2010), pp. 511545.
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 GibbsThomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the EulerLagrange equation of the diffuse phase field energy. 
J. Giannoulis, A. Mielke, Ch. Sparber, Highfrequency averaging in semiclassical Hartreetype equations, Asymptotic Analysis, 70 (2010), pp. 87100.
Abstract
We investigate the asymptotic behavior of solutions to semiclassical 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 highfrequency 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 WaalsCahnHilliard phasefield model and its equilibria conditions in the sharp interface limit, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 140 A (2010), pp. 11611186.
Abstract
We study the equilibria of liquidvapor 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 WaalsCahnHilliard 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 twophase 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. 14931512.
Abstract
We derive dispersive stability results for oscillator chains like the FPU chain or the discrete KleinGordon 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. 12421253.
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 rateindependent. The stored energy involves the gradient of the damage variable, which determines an internal lengthscale. Quasistatic fully rateindependent evolution is considered as well as ratedependent evolution including viscous/inertial effects. Illustrative 2dimensional computer simulations are presented, too. 
S. Zelik, A. Mielke, Multipulse evolution and spacetime chaos in dissipative systems, Memoirs of the American Mathematical Society, 198 (2009), pp. 197.

J. Giannoulis, M. Herrmann, A. Mielke, Lagrangian and Hamiltonian twoscale reduction, Journal of Mathematical Physics, 49 (2008), pp. 103505/1103505/42.
Abstract
Studying highdimensional 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 noncanonical 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 twoscale 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 wavelength 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. 939963.
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 elasticplastic systems with hardening, Journal of Mathematical Analysis and Applications, 343 (2008), pp. 10071021.
Abstract
This paper discusses the stability of quasistatic paths for a continuous elasticplastic system with hardening in a onedimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasistatic problems involving elasticplastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasistatic paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasistatic 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 quasistatic paths of these elasticplastic systems is the main result proved in the paper. 
W. Dreyer, M. Herrmann, Numerical experiments on the modulation theory for the nonlinear atomic chain, Physica D. Nonlinear Phenomena, 237 (2008), pp. 255282.
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. 494516.

A. Mielke, T. Roubíček, U. Stefanelli, $Gamma$limits and relaxations for rateindependent evolutionary problems, Calculus of Variations and Partial Differential Equations, 31 (2008), pp. 387416.
Abstract
This work uses the energetic formulation of rateindependent systems that is based on the storedenergy 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 semicontinuity 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 timeincremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finiteelement 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, Weakconvergence methods for Hamiltonian multiscale problems, Discrete and Continuous Dynamical Systems, 20 (2008), pp. 5379.
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 welldeveloped 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 rateindependent quasistatic limit, Journal of Mathematical Analysis and Applications, 348 (2008), pp. 10121020.
Abstract
This paper discusses the convergence of kinetic variational inequalities to rateindependent quasistatic variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rateindependent quasistatic problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rateindependent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to threedimensional elasticplastic systems with hardening is given. 
J.A.C. Martins, M.D.P. Monteiro, A. Petrov, On the stability of quasistatic paths for finite dimensional elasticplastic systems with hardening, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 87 (2007), pp. 303313.

A. Mielke, A. Timofte, Twoscale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM Journal on Mathematical Analysis, 39 (2007), pp. 642668.
Abstract
This paper is devoted to the twoscale homogenization for a class of rateindependent 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 energystorage functional and a dissipation functional. Using the recently developed method of weak and strong twoscale convergence via periodic unfolding, we show that these two functionals have a suitable twoscale 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 socalled 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, Infinitedimensional hyperbolic sets and spatiotemporal chaos in reactiondiffusion systems in $R^n$, Journal of Dynamics and Differential Equations, 19 (2007), pp. 333389.

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. 493523.

M.A. Efendiev, A. Mielke, On the rateindependent limit of systems with dry friction and small viscosity, Journal of Convex Analysis, 13 (2006), pp. 151167.

W. Dreyer, M. Herrmann, A. Mielke, Micromacro transitions in the atomic chain via Whitham's modulation equation, Nonlinearity, 19 (2006), pp. 471500.

A. Mielke, Macroscopic behavior of microscopic oscillations in harmonic lattices via WignerHusimi transforms, Archive for Rational Mechanics and Analysis, 181 (2006), pp. 401448.

M. Baro, H. Neidhardt, J. Rehberg, Current coupling of driftdiffusion models and dissipative SchrödingerPoisson systems: Dissipative hybrid models, SIAM Journal on Mathematical Analysis, 37 (2005), pp. 941981.

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. 181911/1181911/3.

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. 379409.
Beiträge zu Sammelwerken

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. 269293, DOI 10.1007/9783030331160 .
Abstract
We discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantumclassical model that selfconsistently couples van Roosbroeck's driftdiffusion system for classical charge transport with a Lindbladtype quantum master equation. The coupling is shown to obey fundamental principles of nonequilibrium 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: Selfconsistent coupling of driftdiffusion, SchrödingerPoisson, 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/110912OU/8, DOI 10.1117/12.2515209 .
Abstract
The devicescale simulation of electrically driven solid state quantum light emitters, such as singlephoton 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 selfconsistently coupled the semiclassical driftdiffusion 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 lightmatter interaction in the coupled quantum dotphoton system out of one box. In this paper, we extend our hybrid quantumclassical 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 singlephoton emitting diode. 
M. Kantner, Simulation of quantum light sources using the selfconsistently coupled SchrödingerPoissonDriftDiffusionLindblad 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. 1516, DOI 10.1109/NUSOD.2019.8806839 .
Abstract
The devicescale 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 quantumclassical modelsystem ? where we have coupled the driftdiffusion system witha Lindbladtype quantum master equation ? by including a selfconsistent Schrödinger?Poisson problem. The latter describes the(quasi)bound states of the quantum dot carriers. The extendedmodel allows to describe the biasdependency of the emissionspectrum due to the quantum confined Stark effect 
M. Kantner, M. Mittnenzweig, Th. Koprucki, A hybrid quantumclassical 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. 1052603/11052603/6, 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 wellestablished fields of semiclassical 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 quantumclassical system obeys the fundamental laws of nonequilibrium 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 2327, 2018, Berlin, Germany (Conference Program), 2018, pp. 135.

S. Reichelt, Error estimates for nonlinear reactiondiffusion systems involving different diffusion length scales, in: MURPHYSHSFS2014: 7th MUltiRate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and SlowFast 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. 012013/1012013/15.
Abstract
We derive quantitative error estimates for coupled reactiondiffusion systems, whose coefficient functions are quasiperiodically 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 microstructurelength scale. We consider an effective system, which is rigorously obtained via twoscale convergence, and we derive quantitative error estimates. 
M. Kantner, U. Bandelow, Th. Koprucki, H.J. Wünsche, Multiscale modelling and simulation of singlephoton sources on a device level, in: EuroTMCS 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, BalancedViscosity solutions for multirate systems, in: MURPHYSHSFS2014: 7th MUltiRate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and SlowFast 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. 012010/1012010/26.
Abstract
Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rateindependent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finitedimensional setting, and regularize both the static equation and the rateindependent 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 vanishingviscosity 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 rateindependent 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 SelfOrganizing Nonlinear Systems, E. Schöll, S. Klapp, P. Hövel, eds., Understanding Complex Systems, Springer International Publishing AG Switzerland, Cham, 2016, pp. 235251.

A. Mielke, Relaxation of a rateindependent phase transformation model for the evolution of microstructure, in: Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure, Workshop, March 1418, 2016, R. Kienzler, D.L. Mcdowell, S. Müller, E.A. Werner, eds., 13 of Oberwolfach Reports, European Mathematical Society, 2016, pp. 840842.

M. Kantner, U. Bandelow, Th. Koprucki, H.J. Wünsche, Modeling and numerical simulation of electrically pumped singlephoton emitters, in: Proceedings of the 15th International Conference on Numerical Simulation of Optoelectronic Devices 2015, J. Piprek, W. YuhRenn, eds., IEEE Conference Publications Management Group, Piscataway, 2015, pp. 151152.

A. Mielke, Evolutionary relaxation of a twophase model, in: Scales in Plasticity, MiniWorkshop, November 814, 2015, G.A. Francfort, S. Luckhaus, eds., 12 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2015, pp. 30273030.

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. 331348.
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 gradientflow contribution, which satisfy a particular noninteraction condition:
We give three applications of the theory. First, we consider a finitedimensional 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 onedimensional Schrödinger operator connected to a onedimensional heat equation on the left and on the right. Finally, we consider thermooptoelectronics, where the MaxwellBloch system of optics is coupled to the energydriftdiffusion system for semiconductor electronics. 
A. Mielke, Multiscale gradient systems and their amplitude equations, in: Dynamics of Pattern, Workshop, Dezember 1622, 2012, 9 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2012, pp. 35883591.

M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, in: Microstructures in Solids: From Quantum Models to Continua, Workshop, March 1420, 2010, 7 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2010, pp. 783785.

A. Petrov, J.A.C. Martins, M.D.P. Monteiro Marques, Mathematical results on the stability of quasistatic paths of elasticplastic systems with hardening, in: Topics on Mathematics for Smart Systems, B. Miara, G. Stavroulakis, V. Valente, eds., World Scientific, Singapore, 2007, pp. 167182.

A. Mielke, Gamma convergence for rateindependent processes with applications to damage, in: Phase Transition, Workshop, June 4  8, 2007, 4 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2007, pp. 16171620.

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. 435466.

C. Patz, A. Mielke, Dispersive and longtime 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), WileyVCH Verlag, Weinheim, 2006, pp. 503504.

A. Timofte, A. Mielke, Twoscale homogenization for rateindependent systems, in: GAMM Annual Meeting 2006  Berlin, Special Issue (Vol. 6, Issue 1) of PAMM (Proceedings of Applied Mathematics and Mechanics), WileyVCH Verlag, Weinheim, 2006, pp. 513514.

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. 467500.

A. Mielke, Deriving modulation equations via Lagrangian and Hamiltonian reduction, in: Mathematical Theory of Water Waves, Workshop, November 1218, 2006, 3 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2006, pp. 30323034.

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. 16971700.

A. Mielke, Twoscale modelling for Hamiltonian systems: Formal and rigorous results, in: PDE and Materials, Workshop, September 2430, 2006, 3 of Oberwolfach Reports, Mathematisches Forschunsinstitut Oberwolfach, 2006, pp. 26562659.

A. Mielke, Energy transport in periodic lattices, in: Thermodynamische Materialtheorien, Workshop, December 1218, 2004, 1 (4) of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2004, pp. 30193022.

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. 355364.

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. 377390.

U. Bandelow, H. Gajewski, H.Chr. Kaiser, Modeling combined effects of carrier injection, photon dynamics and heating in Strained MultiQuantumWell 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. 301310.

H.Chr. Kaiser, J. Rehberg, About some mathematical questions concerning the embedding of SchrödingerPoisson systems into the driftdiffusion 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. 13281333.
Preprints, Reports, Technical Reports

A. Glitzky, M. Liero, G. Nika, Dimension reduction of thermistor models for largearea organic lightemitting diodes, Preprint no. 2719, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2719 .
Abstract, PDF (328 kByte)
An effective system of partial differential equations describing the heat and current flow through a thin organic lightemitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully threedimensional φ(x)Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter ε > 0 which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the currentflow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the threedimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective currentflow equation is given by two semilinear equations in the twodimensional crosssections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted. 
A. Mielke, R.R. Netz, S. Zendehroud, A rigorous derivation and energetics of a wave equation with fractional damping, Preprint no. 2718, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2718 .
Abstract, PDF (312 kByte)
We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the waterair interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionallydamped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energydissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionallydamped wave equation with a time derivative of order 3/2. 
J. Maas, A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, Preprint no. 2712, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2712 .
Abstract, PDF (552 kByte)
We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reactionrate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailedbalance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradientflow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailedbalance steady state. The limit of large volumes is studied in the sense of evolutionary Γconvergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels. 
G. Nika, B. Vernescu, Microgeometry effects on the nonlinear effective yield strength response of magnetorheological fluids, Preprint no. 2673, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2673 .
Abstract, PDF (1608 kByte)
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. 
A. Mielke, A. Montefusco, M.A. Peletier, Exploring families of energydissipation landscapes via tilting  Three types of EDP convergence, Preprint no. 2668, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2668 .
Abstract, PDF (372 kByte)
This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that wellknown techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the EnergyDissipation Principle that provides a variational formulation to gradientflow equations that allows one to apply techniques from Γconvergence of functional on states and functionals on trajectories. 
M. Kantner, Th. Höhne, Th. Koprucki, S. Burger, H.J. Wünsche, F. Schmidt, A. Mielke, U. Bandelow, Multidimensional modeling and simulation of semiconductor nanophotonic devices, Preprint no. 2653, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2653 .
Abstract, PDF (9836 kByte)
Selfconsistent modeling and multidimensional simulation of semiconductor nanophotonic devices is an important tool in the development of future integrated light sources and quantum devices. Simulations can guide important technological decisions by revealing performance bottlenecks in new device concepts, contribute to their understanding and help to theoretically explore their optimization potential. The efficient implementation of multidimensional numerical simulations for computeraided design tasks requires sophisticated numerical methods and modeling techniques. We review recent advances in devicescale modeling of quantum dot based singlephoton sources and laser diodes by selfconsistently coupling the optical Maxwell equations with semiclassical carrier transport models using semiclassical and fully quantum mechanical descriptions of the optically active region, respectively. For the simulation of realistic devices with complex, multidimensional geometries, we have developed a novel hpadaptive finite element approach for the optical Maxwell equations, using mixed meshes adapted to the multiscale properties of the photonic structures. For electrically driven devices, we introduced novel discretization and parameterembedding techniques to solve the driftdiffusion system for strongly degenerate semiconductors at cryogenic temperature. Our methodical advances are demonstrated on various applications, including verticalcavity surfaceemitting lasers, grating couplers and singlephoton sources. 
A. Mielke, A. Stephan, Coarsegraining via EDPconvergence for linear fastslow reaction systems, Preprint no. 2643, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2643 .
Abstract, PDF (426 kByte)
We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarsegrained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant coshtype gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarsegrained equation again has a coshtype gradient structure. We obtain the strongest version of convergence in the sense of the EnergyDissipation Principle (EDP), namely EDPconvergence with tilting. 
TH. Frenzel, M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDPconvergence, Preprint no. 2601, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2601 .
Abstract, PDF (359 kByte)
The notion of EnergyDissipationPrinciple convergence (EDPconvergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The FokkerPlanck equation can be formulated as gradientflow equation with respect to the logarithmic relative entropy of the system and a quadratic Wassersteintype gradient structure. The EDPconvergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelinde Donder kinetics. 
G. Nika, B. Vernescu, Multiscale modeling of magnetorheological suspensions, Preprint no. 2598, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2598 .
Abstract, PDF (3159 kByte)
We develop a multiscale approach to describe the behavior of a suspension of solid magnetizable particles in a viscous nonconducting fluid in the presence of an externally applied magnetic field. By upscaling the quasistatic 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. 
M. Heida, S. Neukamm, M. Varga, Stochastic homogenization of Lambdaconvex gradient flows, Preprint no. 2594, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2594 .
Abstract, PDF (429 kByte)
In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λconvex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are AllenCahn type equations and evolutionary equations driven by the pLaplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic twoscale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the wellestablished notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ)convex functionals.
Vorträge, Poster

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

A. Mielke, Gradient systems and evolutionary Gammaconvergence (online talk), Oberseminar ``Mathematik in den Naturwissenschaften'', JuliusMaximiliansUniversität Würzburg, June 5, 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, The fractional pLaplacian 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: Selfconsistent coupling of driftdiffusion, SchrödingerPoisson, 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 singlephoton sources using the SchrödingerPoissonDriftDiffusionLindblad 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 selfconsistently coupled SchrödingerPoissonDriftDiffusionLindblad 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 multiscale model of magnetorheological suspension, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Minisymposium MS ME13 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 nonisothermal CahnHilliard 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.

TH. Koprucki, Towards multiscale modeling of IIINbased 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 EDPconvergence, 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, COPDESCWorkshop ``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 Gammaconvergence for gradient systems, Mathematisches Kolloquium, AlbertLudwigsUniversität 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 WilhelmsUniversität Münster, September 17, 2019.

A. Mielke, Gradient systems and evolutionary Gammaconvergence, DMVJahrestagung 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 ME13 9 ``Entropy Methods for Multidimensional 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, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, June 5, 2019.

S. Reichelt, Pulses in FitzHughNagumo 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 quantumclassical 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 quantumclassical modeling of quantum dot based singlephoton 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, FriedrichAlexanderUniversität ErlangenNürnberg, 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 FitzHughNagumo systems with rapidly oscillating coefficients, Workshop ``Control of Selforganizing Nonlinear Systems'', August 29  31, 2017, Collaborative Research Center 910: Control of selforganizing 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 timeperiodic dissipation potentials in rateindependent 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 Gconvergence and stochastic twoscale convergences of the squareroot approximation scheme to the FokkerPlanck operator, GAMMWorkshop 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 quantumclassical 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 semiclassical 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 bulkinterfaceinteraction, HumboldtUniversität zu Berlin, Institut für Mathematik, October 20, 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 energyreactiondiffusion 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, Econvergence to the quasisteadystate 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, 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 CahnHilliardtype 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 CahnHilliardtype equations via evolutionary Gammaconvergence, Workshop ``Patterns of Dynamics'', Freie Universität Berlin, Fachbereich Mathematik und Informatik, July 25  29, 2016.

S. Reichelt, Homogenization of CahnHilliardtype 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. HarzSeminar zur Strukturbildung in Chemie und Biophysik, February 21  22, 2016, PhysikalischTechnische Bundesanstalt, Hahnenklee, February 22, 2016.

TH. Frenzel, Evolutionary Gammaconvergence 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 rateindependent systems, sc Matheon Multiscale Seminar, Technische Universität Berlin, February 17, 2016.

M. Heida, Stochastic homogenization of 1homogeneous 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 rateindependent systems, Berlin Dresden Prague Würzburg Workshop ``Homogenization and Related Topics'', Technische Universität Dresden, Fachbereich Mathematik, June 22, 2016.

M. Kantner, Multiscale modeling and numerical simulation of singlephoton emitters, Matheon Workshop9th Annual Meeting ``Photonic Devices", Zuse Institut, Berlin, March 3, 2016.

M. Kantner, Multiscale modelling and simulation of singlephoton sources on a device level, EuroTMCS II Theory, Modelling & Computational Methods for Semiconductors, Tyndall National Institute and University College Cork, Cork, Ireland, December 9, 2016.

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

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

A. Mielke, Evolutionary Gammaconvergence, 2nd CENTRAL School on Analysis and Numerics for Partial Differential Equations, August 29  September 2, 2016, HumboldtUniversität zu Berlin, Institut für Mathematik.

A. Mielke, Evolutionary relaxation for a rateindependent phasetransformation 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 rateindependent elastic twophase 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, KTGUIMU Mathematics Colloquia, March 30  31, 2016, Kyoto University, Department of Mathematics, Japan, March 31, 2016.

A. Mielke, Rateindependent microstructure evolution via relaxation of a twophase 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 CahnHilliard equations, Workshop on Control of SelfOrganizing Nonlinear Systems, September 14  16, 2015, SFB 910 ``Control of selforganizing nonlinear systems: Theoretical methods and concepts of application'', Lutherstadt Wittenberg, September 15, 2015.

S. Reichelt, Twoscale homogenization of reactiondiffusion 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 PrandtlReuss 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, Multiscale modeling and simulation of electrically pumped singlephoton sources, International NanoOptoelectronics Workshop (iNOW 2015), Tokio, Japan, August 3  7, 2015.

TH. Koprucki, On device concepts for CMOScompatible edgeemitters based on strained germanium, Symposium ``Alternative Semiconductor Integration in Si Microelectronics: Materials, Techniques and Applications'' of the EMRS 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 viscoelastodynamics, 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 edgeemitting 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, EDPconvergence 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 twophase model, MiniWorkshop ``Scales in Plasticity'', November 8  14, 2015, Mathematisches Forschungsinstitut Oberwolfach, November 11, 2015.

A. Mielke, Homogenizing the PenroseFife 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 FokkerPlanck 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 FokkerPlanck and Liouville equations for chemical reactions as largevolume 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, Thinfilm limits of functionals on Afree vector fields and applications, Workshop on Trends in NonLinear Analysis 2014, July 31  August 1, 2014, Instituto Superior Técnico, Departamento de Matemática, Lisbon, Portugal, August 1, 2014.

C. Kreisbeck, Thinfilm limits of functionals on Afree 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, Thinfilm limits of functionals on Afree vector fields and applications, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, July 16, 2014.

S. Reichelt, Effective equations for reactiondiffusion systems in strongly heterogeneous media, 7th International Workshop on MultiRate Processes & Hysteresis, 2nd International Workshop on Hysteresis and SlowFast Systems (MURPHYSHSFS2014), April 7  11, 2014, WIAS Berlin, April 10, 2014.

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

S. Reichelt, Twoscale homogenization of reactiondiffusion systems with small diffusion, 13th GAMM Seminar on Microstructures, January 17  18, 2014, RuhrUniversität Bochum, Lehrstuhl für Mechanik  Materialtheorie, January 18, 2014.

S. Reichelt, Twoscale homogenization of reactiondiffusion 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 twoscale expansion in stochastic homogenization, 13th GAMM Seminar on Microstructures, January 17  18, 2014, RuhrUniversität 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, BMSWIAS Summer School ``Applied Analysis for Materials'', August 25  September 5, 2014, Berlin Mathematical School, Technische Universität Berlin.

A. Mielke, S. Reichelt, M. Thomas, Pattern formation in systems with multiple scales, Evaluation of the DFG Collaborative Research Center 910 ``Control of Selforganizing 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, FriedrichAlexander Universität ErlangenNürnberg, 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 rateindependent systems using balancedviscosity solutions, 7th International Workshop on MultiRate Processes & Hysteresis, 2nd International Workshop on Hysteresis and SlowFast Systems (MURPHYSHSFS2014), April 7  11, 2014, WIAS Berlin, April 8, 2014.

A. Mielke, Multiscale modeling and evolutionary Gammaconvergence for gradient flows, BMSWIAS 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 reactiondiffusion 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 BritishGerman Frontiers of Science Symposium, Potsdam, March 6  9, 2014.

S. Neukamm, Characterization and approximation of macroscopic properties with homogenization, 4th BritishGerman Frontiers of Science Symposium, March 6  9, 2014, Alexander von HumboldtStiftung, Potsdam, March 7, 2014.

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

S. Neukamm, Homogenization of slender structures in smallstrain 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 FokkerPlanck equation, Oberseminar Analysis, MartinLutherUniversität HalleWittenberg, Institut für Mathematik, Halle, November 20, 2013.

S. Reichelt, Homogenization of degenerated reactiondiffusion equations, Doktorandenforum der LeibnizGemeinschaft, Sektion D, Berlin, June 6  7, 2013.

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

S. Reichelt, Twoscale homogenization of nonlinear reactiondiffusion systems with small diffusion, Workshop on Control of SelfOrganizing Nonlinear Systems, August 28  30, 2013, SFB 910 ``Control of selforganizing nonlinear systems: Theoretical methods and concepts of application'', Lutherstadt Wittenberg, August 30, 2013.

S. Reichelt, Twoscale homogenization in nonlinear reactiondiffusion 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.

CH. Kraus, Sharp interface limit of a diffuse interface model of NavierStokesAllenCahn 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 microstructure, Oberseminar zur Analysis, Universität DuisburgEssen, Fachbereich Mathematik, Essen, January 29, 2013.

H. Hanke, Derivation of an effective damage model with nonperiodic evolving microstructure, 12th GAMM Seminar on Microstructures, February 8  9, 2013, HumboldtUniversität zu Berlin, Institut für Mathematik, February 9, 2013.

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

A. Mielke, Deriving the GinzburgLandau 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 entropydriven dissipative quantum mechanical systems, Analysis and Stochastics in Complex Physical Systems, March 20  22, 2013, Universität Leipzig, Mathematisches Institut, March 21, 2013.

A. Mielke, Rateindependent plasticity as vanishingviscosity 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 MathematikerVereinigung (DMV), Studierendenkonferenz, September 17  20, 2012, Universität des Saarlandes, Fakultät für Mathematik und Informatik, Saarbrücken, September 20, 2012.

S. Heinz, Quasiconvexity equals rankone convexity for isotropic sets of 2x2 matrices, 11th GAMM Seminar on Microstructures, January 20  21, 2012, Universität DuisburgEssen, January 20, 2012.

S. Heinz, Regularization and relaxation of timecontinuous problems in plasticity, 11th GAMM Seminar on Microstructures, Universität DuisburgEssen, 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 reactiondiffusion systems, Seminar ``Dünne Schichten'', Technische Universität Berlin, Institut für Mathematik, February 9, 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 twoscale 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 smallstrain to finitestrain 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 (IMATICNR), 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, Smallstrain elastoplasticity is the evolutionary Gamma limit of finitestrain 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 reactiondiffusion problems, SFB 910 Symposium ``Pattern Formation and Instabilities in Systems with Multiple Scales'', Technische Universität Berlin, Institut für Theoretische Physik, November 25, 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, ISAMTopMath 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.

CH. Kraus, An inhomogeneous, anisotropic and elastically modified GibbsThomson 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 phasefield 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, Rateindependent plasticity as Gamma limit of a slow viscous gradient flow for wiggly discrete energy, ZweiStädteKolloquium zur Analysis, FriedrichAlexanderUniversität ErlangenNürnberg, Fachbereich Mathematik, November 26, 2010.

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

H. Hanke, Gammalimits in rateindependent evolutionary problems and homogenization in gradient plasticity, DKSeminar ``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 Gammaconvergence 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 energydriven 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.

J. Rehberg, On SchrödingerPoisson 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 SchrödingerPoissonSysteme, Chemnitzer Mathematisches Colloquium, Technische Universität Chemnitz, Fakultät für Mathematik, May 24, 2007.

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

J. Rehberg, Some analytical ideas concerning the quantumdriftdiffusion systems, Workshop ``Problèmes spectraux nonlinéaires 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 semiclassical 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, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, December 12, 2003.
Preprints im Fremdverlag

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 timereversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradientflow, steepestascent, or maximalentropyproduction equation. Onsager's original theorem is limited to closetoequilibrium 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 timereversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.
Ansprechpartner
Anwendungen
 Dynamik von Halbleiterlasern
 Modellierung dünner Filme und Nanostrukturen auf Substraten
 Modellierung von Phasenseparation und Schädigung in modernen Materialien
 Nichtlineare Materialmodelle, multifunktionale Materialien und Hysterese im Zusammenhang mit elastoplastischen Prozessen
 Pulse in optisch nichtlinearen Medien
 Quantenmechanische Modelle für Halbleiter
Projekte/Drittmittel
 Analysis für thermomechanische Modelle mit internen Variablen
 Datengetriebene Berechnung der elektronischen Struktur von Nanostrukturen
 Hybride Modelle für das elektrothermische Verhalten organischer Halbleiterbauelemente
 MultiMaterial Electrocatalysis (MULTECAT)
 Räumliche Stochastische Teilchenprozesse für Koagulation