Publications
Monographs

M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, 453 pages, (Collection Published).

H. Abels, K. Disser, H.Chr. Kaiser, A. Mielke, M. Thomas, eds., Issue on Partial Differential Equations in Fluids and Solids, 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, 292 pages, (Collection Published).

A. Mielke, M. Peletier, D. Slepcev, eds., Variational Methods for Evolution, 17 of Oberwolfach Reports, European Mathematical Society Publishing House, Zurich, 2021, 76 pages, (Collection Published), DOI 10.4171/OWR/2020/29 .
Abstract
Variational principles for evolutionary systems take advantage of the rich toolbox provided by the theory of the calculus of variations. Such principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, but also timeincremental minimization techniques for more general evolutionary problems. The new challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), new structures (systems with nonlocal transport, gradient flows on graphs, kinetic equations, systems of equations) thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, in multiagent systems, and in data science. This workshop brought together a broad spectrum of researchers from calculus of variations, partial differential equations, metric geometry, and stochastics, as well as applied and computational scientists to discuss and exchange ideas. It focused on variational tools such as minimizing movement schemes, optimal transport, gradient flows, and largedeviation principles for timecontinuous Markov processes, Gammaconvergence and homogenization.
Articles in Refereed Journals

P. Pelech, K. Tůma, M. Pavelka, M. Šípka, M. Sýkora, On compatibility of the natural configuration framework with general equation for nonequilibrium reversibleirreversible coupling (GENERIC): Derivation of anisotropic ratetype models, Journal of NonNewtonian Fluid Mechanics, 305 (2022), pp. 104808/1104808/19, DOI 10.1016/j.jnnfm.2022.104808 .
Abstract
Within the framework of natural configurations developed by Rajagopal and Srinivasa, evolution within continuum thermodynamics is formulated as evolution of a natural configuration linked with the current configuration. On the other hand, withing the General Equation for NonEquilibrium ReversibleIrreversible Coupling (GENERIC) framework, the evolution is split into Hamiltonian mechanics and (generalized) gradient dynamics. These seemingly radically different approaches have actually a lot in common and we show their compatibility on a wide range of models. Both frameworks are illustrated on isotropic and anisotropic ratetype fluid models. We propose an interpretation of the natural configurations within GENERIC and vice versa (when possible). 
P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Wellposedness and optimal control for a CahnHilliardOono system with control in the mass term, Discrete and Continuous Dynamical Systems  Series S, 15 (2022), pp. 21352172, DOI 10.3934/dcdss.2022001 .
Abstract
The paper treats the problem of optimal distributed control of a CahnHilliardOono system in R^{d}, 1 ≤ d ≤ 3 with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case d = 2. In the rest of the work, we study the necessary firstorder optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain 
P. Colli, G. Gilardi, J. Sprekels, Wellposedness for a class of phasefield systems modeling prostate cancer growth with fractional operators and general nonlinearities, Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematica e Applicazioni, 33 (2022), pp. 193228, DOI 10.4171/rlm/969 .
Abstract
This paper deals with a general system of equations and conditions arising from a mathematical model of prostate cancer growth with chemotherapy and antiangiogenic therapy that has been recently introduced and analyzed (see [P. Colli et al., Mathematical analysis and simulation study of a phasefield model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Math. Models Methods Appl. Sci. bf 30 (2020), 12531295]). The related system includes two evolutionary operator equations involving fractional powers of selfadjoint, nonnegative, unbounded linear operators having compact resolvents. Both equations contain nonlinearities and in particular the equation describing the dynamics of the tumor phase variable has the structure of a AllenCahn equation with doublewell potential and additional nonlinearity depending also on the other variable, which represents the nutrient concentration. The equation for the nutrient concentration is nonlinear as well, with a term coupling both variables. For this system we design an existence, uniqueness and continuous dependence theory by setting up a careful analysis which allows the consideration of nonsmooth potentials and the treatment of continuous nonlinearities with general growth properties. 
P. Colli, A. Signori, J. Sprekels, Optimal control problems with sparsity for tumor growth models involving variational inequalities, Journal of Optimization Theory and Applications, 194 (2022), pp. 2558 (published online on 28.02.2022), DOI 10.1007/s10957022020007 .
Abstract
This paper treats a distributed optimal control problem for a tumor growth model of CahnHilliard type including chemotaxis. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermore, the cost functional contains a nondifferentiable term like the $L^1$norm in order to include sparsity effects which is of utmost relevance, especially time sparsity, in the context of cancer therapies as applying a control to the system reflects in exposing the patient to an intensive medical treatment. To cope with the difficulties originating from the variational inequality in the state system, we employ the socalled “deep quench approximation” in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, firstorder necessary conditions of optimality can be established by invoking recent results. We use these results to derive corresponding optimality conditions also for the double obstacle case, by deducing a variational inequality in terms of the associated adjoint state variables. The resulting variational inequality can be exploited to also obtain sparsity results for the optimal controls. 
J. Fischer, K. Hopf, M. Kniely, A. Mielke, Global existence analysis of energyreactiondiffusion systems, SIAM Journal on Mathematical Analysis, 54 (2022), pp. 220267 (published online on 04.01.2022), DOI 10.1137/20M1387237 .
Abstract
We establish globalintime existence results for thermodynamically consistent reaction(cross)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model speciesdependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the nonisothermal case lies in the intrinsic presence of crossdiffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where nonintegrable diffusion fluxes or reaction terms appear. 
P. Krejčí, E. Rocca, J. Sprekels, Analysis of a tumor model as a multicomponent deformable porous medium, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 24 (2022), pp. 235262, DOI 10.4171/IFB/472 .
Abstract
We propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phasedependent elasticity coefficients. The resulting PDE system couples two CahnHilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reactiondiffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initialboundary value problem has a solution in appropriate function spaces. 
M. O'Donovan, P. Farrell, T. Streckenbach, Th. Koprucki, S. Schulz, Multiscale simulations of unipolar hole transport in (In,Ga)N quantum well systems, Optical and Quantum Electronics, 54 (2022), pp. 405/1405/23, DOI 10.1007/s11082022037522 .
Abstract
Understanding the impact of the alloy microstructure on carrier transport becomes important when designing IIInitridebased LED structures. In this work, we study the impact of alloy fluctuations on the hole carrier transport in (In,Ga)N single and multiquantum well systems. To disentangle hole transport from electron transport and carrier recombination processes, we focus our attention on unipolar (pip) systems. The calculations employ our recently established multiscale simulation framework that connects atomistic tightbinding theory with a macroscale driftdiffusion model. In addition to alloy fluctuations, we pay special attention to the impact of quantum corrections on hole transport. Our calculations indicate that results from a virtual crystal approximation present an upper limit for the hole transport in a pip structure in terms of the currentvoltage characteristics. Thus we find that alloy fluctuations can have a detrimental effect on hole transport in (In,Ga)N quantum well systems, in contrast to unipolar electron transport. However, our studies also reveal that the magnitude by which the random alloy results deviate from virtual crystal approximation data depends on several factors, e.g. how quantum corrections are treated in the transport calculations. 
A. Glitzky, M. Liero, G. Nika, A coarsegrained electrothermal model for organic semiconductor devices, Mathematical Methods in the Applied Sciences, 45 (2022), pp. 48094833 (published on 26.01.2022), DOI 10.1002/mma.8072 .
Abstract
We derive a coarsegrained model for the electrothermal interaction of organic semiconductors. The model combines stationary driftdiffusion based electrothermal models with thermistor type models on subregions of the device and suitable transmission conditions. Moreover, we prove existence of a solution using a regularization argument and Schauder's fixed point theorem. In doing so, we extend recent work by taking into account the statistical relation given by the GaussFermi integral and mobility functions depending on the temperature, chargecarrier density, and field strength, which is required for a proper description of organic devices. 
K. Hopf, M. Burger, On multispecies diffusion with size exclusion, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 224 (2022), pp. 113092/1113092/27, DOI 10.1016/j.na.2022.113092 .
Abstract
We revisit a classical continuum model for the diffusion of multiple species with sizeexclusion constraint, which leads to a degenerate nonlinear crossdiffusion system. The purpose of this article is twofold: first, it aims at a systematic study of the question of existence of weak solutions and their longtime asymptotic behaviour. Second, it provides a weakstrong stability estimate for a wide range of coefficients, which had been missing so far. In order to achieve the results mentioned above, we exploit the formal gradientflow structure of the model with respect to a logarithmic entropy, which leads to best estimates in the fullinteraction case, where all crossdiffusion coefficients are nonzero. Those are crucial to obtain the minimal Sobolev regularity needed for a weakstrong stability result. For meaningful cases when some of the coefficients vanish, we provide a novel existence result based on approximation by the fullinteraction case. 
K. Hopf, Weakstrong uniqueness for energyreactiondiffusion systems, Mathematical Models & Methods in Applied Sciences, 21 (2022), pp. 10151069 (published online on 29.04.2022), DOI 10.1142/S0218202522500233 .
Abstract
We establish weakstrong uniqueness and stability properties of renormalised solutions to a class of energyreactiondiffusion systems, which genuinely feature crossdiffusion effects. The systems considered are motivated by thermodynamically consistent models, and their formal entropy structure allows us to use as a key tool a suitably adjusted relative entropy method. Weakstrong uniqueness is obtained for general entropydissipating reactions without growth restrictions, and certain models with a nonintegrable diffusive flux. The results also apply to a class of (isoenergetic) reactioncrossdiffusion systems. 
TH. Eiter, On the Oseentype resolvent problem associated with timeperiodic flow past a rotating body, SIAM Journal on Mathematical Analysis, 54 (2022), pp. 49875012, DOI 10.1137/21M1456728 .
Abstract
Consider the timeperiodic flow of an incompressible viscous fluid past a body performing a rigid motion with nonzero translational and rotational velocity. We introduce a framework of homogeneous Sobolev spaces that renders the resolvent problem of the associated linear problem well posed on the whole imaginary axis. In contrast to the cases without translation or rotation, the resolvent estimates are merely uniform under additional restrictions, and the existence of timeperiodic solutions depends on the ratio of the rotational velocity of the body motion to the angular velocity associated with the time period. Provided that this ratio is a rational number, timeperiodic solutions to both the linear and, under suitable smallness conditions, the nonlinear problem can be established. If this ratio is irrational, a counterexample shows that in a special case there is no uniform resolvent estimate and solutions to the timeperiodic linear problem do not exist. 
TH. Eiter, On the Stokestype resolvent problem associated with timeperiodic flow around a rotating obstacle, Journal of Mathematical Fluid Mechanics, 24 (2022), pp. 52/117, DOI 10.1007/s00021021006543 .
Abstract
Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated timeperiodic linear problem. 
A. Mielke, S. Reichelt, Traveling fronts in a reactiondiffusion equation with a memory term, Journal of Dynamics and Differential Equations, (2022), published online on 23.02.2022, DOI 10.1007/s10884022101336 .
Abstract
Based on a recent work on traveling waves in spatially nonlocal reactiondiffusion equations, we investigate the existence of traveling fronts in reactiondiffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reactiondiffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that twoscale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory.The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising traveling front, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations.

A. Mielke, J. Naumann, On the existence of globalintime weak solutions and scaling laws for Kolmogorov's twoequation model of turbulence, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 102 (2022), pp. 131 (published online on 01.07.2022), DOI 10.1002/zamm.202000019 .
Abstract
This paper is concerned with Kolmogorov's twoequation model for free turbulence in space dimension 3, involving the mean velocity u, the pressure p, an average frequency omega, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general twoequation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's twoequation model under spaceperiodic boundary conditions in cubes with positive side length l. To this end, we provide new a priori estimates and invoke existence result for pseudomonotone operators. 
P.É. Druet, Globalintime existence for liquid mixtures subject to a generalised incompressibility constraint, Journal of Mathematical Analysis and Applications, 499 (2021), pp. 125059/1125059/56, DOI 10.1016/j.jmaa.2021.125059 .
Abstract
We consider a system of partial differential equations describing diffusive and convective mass transport in a fluid mixture of N > 1 chemical species. A weighted sum of the partial mass densities of the chemical species is assumed to be constant, which expresses the incompressibility of the fluid, while accounting for different reference sizes of the involved molecules. This condition is different from the usual assumption of a constant total mass density, and it leads in particular to a nonsolenoidal velocity field in the NavierStokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDEsystem of mass transport equations and momentum balance is fully coupled. Another striking feature of such incompressible mixtures is the algebraic formula connecting the pressure and the densities, which can be exploited to prove a pressure bound in L^{1}. In this paper, we consider incompressible initial states with bounded energy and show the global existence of weak solutions with defect measure. 
M. Heida, M. Kantner, A. Stephan, Consistency and convergence for a family of finite volume discretizations of the FokkerPlanck operator, ESAIM: Mathematical Modelling and Numerical Analysis, 55 (2021), pp. 30173042, DOI 10.1051/m2an/2021078 .
Abstract
We introduce a family of various finite volume discretization schemes for the FokkerPlanck operator, which are characterized by different weight functions on the edges. This family particularly includes the wellestablished ScharfetterGummel discretization as well as the recently developed squareroot approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the quality of the weights. We show by numerical experiments that for small gradients the choice of the optimal representative of the discretization family is highly nontrivial while for large gradients the ScharfetterGummel scheme stands out compared to the others. 
M. Heida, S. Neukamm, M. Varga, Stochastic homogenization of Lambdaconvex gradient flows, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 427453, DOI 10.3934/dcdss.2020328 .
Abstract
In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λconvex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are AllenCahn type equations and evolutionary equations driven by the pLaplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic twoscale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the wellestablished notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ)convex functionals. 
O. Marquardt, Simulating the electronic properties of semiconductor nanostructures using multiband $kcdot p$ models, Computational Materials Science, 194 (2021), pp. 110318/1110318/11, DOI 10.1016/j.commatsci.2021.110318 .
Abstract
The eightband $kcdot p$ formalism been successfully applied to compute the electronic properties of a wide range of semiconductor nanostructures in the past and can be considered the backbone of modern semiconductor heterostructure modelling. However, emerging novel material systems and heterostructure fabrication techniques raise questions that cannot be answered using this wellestablished formalism, due to its intrinsic limitations. The present article reviews recent studies on the calculation of electronic properties of semiconductor nanostructures using a generalized multiband $kcdot p$ approach that allows both the application of the eightband model as well as more sophisticated approaches for novel material systems and heterostructures. 
G. Nika, Derivation of effective models from heterogenous Cosserat media via periodic unfolding, Ricerche di Matematica. A Journal of Pure and Applied Mathematics, published online on 01.07.2021, DOI 10.1007/s11587021006103 .
Abstract
We derive two different effective models from a heterogeneous Cosserat continuum taking into account the Cosserat intrinsic length of the constituents. We pass to the limit using homogenization via periodic unfolding and in doing so we provide rigorous proof to the results introduced by Forest, Pradel, and Sab (Int. J. Solids Structures 38 (2627): 45854608 '01). Depending on how different characteristic lengths of the domain scale with respect to the Cosserat intrinsic length, we obtain either an effective classical Cauchy continuum or an effective Cosserat continuum. Moreover, we provide some corrector type results for each case. 
A. Stephan, EDPconvergence for a linear reactiondiffusion system with fast reversible reaction, Calculus of Variations and Partial Differential Equations, 60 (2021), pp. 226/1226/35, DOI 10.1007/s00526021020890 .
Abstract
We perform a fastreaction limit for a linear reactiondiffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reactiondiffusion system as a gradient flow of the free energy in the space of probability measures equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and coshtype functions for the reaction part. The fastreaction limit is done on the level of the gradient structure by proving EDPconvergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slowmanifold. Moreover, the limit gradient system can be equivalently described by a coarsegrained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarsegrained slow variable. 
E. Davoli, M. Kružík, P. Pelech, Separately global solutions to rateindependent processes in largestrain inelasticity, Nonlinear Analysis. An International Mathematical Journal, 215 (2022), pp. 112668/1112668/37 (published online on 17.11.2021), DOI 10.1016/j.na.2021.112668 .
Abstract
In this paper, we introduce the notion of separately global solutions for largestrain rateindependent systems, and we provide an existence result for a model describing bulk damage. Our analysis covers nonconvex energies blowing up for extreme compressions, yields solutions excluding interpenetration of matter, and allows to handle nonlinear couplings of the deformation and the internal variable featuring both Eulerian and Lagrangian terms. In particular, motivated by the theory developed in Roubíček (2015) in the small strain setting, and for separately convex energies we provide a solution concept suitable for large strain inelasticity. 
D. Bothe, P.É. Druet, Mass transport in multicomponent compressible fluids: Local and global wellposedness in classes of strong solutions for general classone models, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 210 (2021), pp. 112389/1112389/53, DOI 10.1016/j.na.2021.112389 .
Abstract
We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the FickOnsager or Maxwell Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the NavierStokes equations. The thermodynamic pressure is defined by the GibbsDuhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents. The resulting PDEs are of mixed parabolichyperbolic type. We prove two theoretical results concerning the wellposedness of the model in classes of strong solutions: 1. The solution always exists and is unique for shorttimes and 2. If the initial data are sufficiently near to an equilibrium solution, the wellposedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity. 
D. Bothe, P.É. Druet, Wellposedness analysis of multicomponent incompressible flow models, Journal of Evolution Equations, 21 (2021), pp. 40394093, DOI 10.1007/s00028021007123 .
Abstract
In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities of the species stays constant. In this type of models, non solenoidal effects affect the velocity field in the NavierStokes equations and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the localintime wellposedness in classes of strong solutions, and the globalintime existence of solutions for initial data sufficiently close to a smooth equilibrium solution. 
D. Chaudhuri, M. O'Donovan, T. Streckenbach, O. Marquardt, P. Farrell, S.K. Patra, Th. Koprucki, S. Schulz, Multiscale simulations of the electronic structure of IIInitride quantum wells with varied indium content: Connecting atomistic and continuumbased models, Journal of Applied Physics, 129 (2021), pp. 073104/1073104/16, DOI 10.1063/5.0031514 .

P. Colli, G. Gilardi, J. Sprekels, An asymptotic analysis for a generalized CahnHilliard system with fractional operators, Journal of Evolution Equations, 21 (2021), pp. 27492778, DOI 10.1007/s00028021007061 .
Abstract
In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous CahnHilliard systems of two operator equations in which nonlinearities of doublewell type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers in the spectral sense of general linear operators, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space of squareintegrable functions on a bounded and smooth threedimensional domain, and have compact resolvents. Here, for the case of the viscous system, we analyze the asymptotic behavior of the solution as the fractional power coefficient of the second operator tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of the second operator appears. 
P. Colli, A. Signori, J. Sprekels, Secondorder analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis, ESAIM. Control, Optimisation and Calculus of Variations, 27 (2021), pp. 73/173/46, DOI 10.1051/cocv/2021072 .
Abstract
This paper concerns a distributed optimal control problem for a tumor growth model of CahnHilliard type including chemotaxis with possibly singular anpotentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak wellposedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong wellposedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both firstorder necessary and secondorder sufficient conditions for optimality. The mathematically challenging secondorder analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the secondorder Fréchet derivative of the controltostate operator and carry out a thorough and detailed investigation about the related properties. 
P. Colli, G. Gilardi, J. Sprekels, Deep quench approximation and optimal control of general CahnHilliard systems with fractional operators and double obstacle potentials, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 243271, DOI 10.3934/dcdss.2020213 .
Abstract
Recently, the authors derived wellposedness and regularity results for general evolutionary operator equations having the structure of a CahnHilliard system. The involved operators were fractional versions in the spectral sense of general linear operators that have compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions. The class of admissible doublewell potentials driving the phase separation process modeled by the CahnHilliard system included polynomial, logarithmic, and double obstacle nonlinearities. In a subsequent paper, distributed optimal control problems for such systems were investigated, where only differentiable polynomial and logarithmic potentials were admitted. Existence of optimizers and firstorder optimality conditions were derived. In this paper, these results are complemented for nondifferentiable double obstacle nonlinearities. It is well known that for such nonlinearities standard constraint qualifications to construct Lagrange multipliers cannot be applied. To overcome this difficulty, we follow the socalled “deep quench” method, which has proved to be a powerful tool in optimal control problems with double obstacle potentials. We give a general convergence analysis of the deep quench approximation, including an error estimate, and demonstrate that its use leads to meaningful firstorder necessary optimality conditions. 
TH. Frenzel, M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDPconvergence, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 395425, DOI 10.3934/dcdss.2020345 .
Abstract
The notion of EnergyDissipationPrinciple convergence (EDPconvergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The FokkerPlanck equation can be formulated as gradientflow equation with respect to the logarithmic relative entropy of the system and a quadratic Wassersteintype gradient structure. The EDPconvergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelinde Donder kinetics. 
A. Kirch, A. Fischer, M. Liero, J. Fuhrmann, A. Glitzky, S. Reineke, Electrothermal tristability causes sudden burnin phenomena in organic LEDs, Advanced Functional Materials, published online in September 2021, DOI 10.1002/adfm.202106716 .

H. Meinlschmidt, J. Rehberg, Extrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations, Journal of Differential Equations, 280 (2021), pp. 375404, DOI 10.1016/j.jde.2021.01.032 .
Abstract
In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolevtype spaces of negative order X^{s1,q}_{D}(Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of localintime existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs. 
M. O'Donovan, D. Chaudhuri, T. Streckenbach, P. Farrell, S. Schulz, Th. Koprucki, From atomistic tightbinding theory to macroscale driftdiffusion: Multiscale modeling and numerical simulation of unipolar charge transport in (In,Ga)N devices with random fluctuations, Journal of Applied Physics, 130 (2021), pp. 065702/1065702/13, DOI 10.1063/5.0059014 .

J. Sprekels, F. Tröltzsch, Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth, ESAIM. Control, Optimisation and Calculus of Variations, 27 (2021), pp. S26/1S26/27, DOI 10.1051/cocv/2020088 .
Abstract
In this paper, we study an optimal control problem for a nonlinear system of reactiondiffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extracellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a doublewell potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L^{1}norm. For such problems, we derive firstorder necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding controltostate operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of secondorder sufficient optimality conditions. 
A.F.M. TER Elst, R. HallerDintelmann, J. Rehberg, P. Tolksdorf, On the $L^p$theory for secondorder elliptic operators in divergence form with complex coefficients, Journal of Evolution Equations, 21 (2021), pp. 39634003, DOI 10.1007/s00028021007114 .
Abstract
Given a complex, elliptic coefficient function we investigate for which values of p the corresponding secondorder divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on L^{p}(Ω). Additional properties like analyticity of the semigroup, H^{∞}calculus and maximal regularity arealso discussed. Finally we prove a perturbation result for real coefficients that gives the whole range of ^{p}'s for small imaginary parts of the coefficients. Our results are based on the recent notion of ^{p}ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients. 
A. Glitzky, M. Liero, G. Nika, Analysis of a hybrid model for the electrothermal behavior of semiconductor heterostructures, Journal of Mathematical Analysis and Applications, 507 (2022), pp. 125815/1125815/26 (published online on 16.11.2021), DOI 10.1016/j.jmaa.2021.125815 .
Abstract
We prove existence of a weak solution for a hybrid model for the electrothermal behavior of semiconductor heterostructures. This hybrid model combines an electrothermal model based on driftdiffusion with thermistor type models in different subregions of the semiconductor heterostructure. The proof uses a regularization method and Schauder's fixed point theorem. For boundary data compatible with thermodynamic equilibrium we verify, additionally, uniqueness. Moreover, we derive bounds and higher integrability properties for the electrostatic potential and the quasi Fermi potentials as well as the temperature. 
A. Glitzky, M. Liero, G. Nika, An existence result for a class of electrothermal driftdiffusion models with GaussFermi statistics for organic semiconductors, Analysis and Applications, 19 (2021), pp. 275304, DOI 10.1142/S0219530519500246 .
Abstract
This work is concerned with the analysis of a driftdiffusion model for the electrothermal behavior of organic semiconductor devices. A "generalized Van Roosbroeck” system coupled to the heat equation is employed, where the former consists of continuity equations for electrons and holes and a Poisson equation for the electrostatic potential, and the latter features source terms containing Joule heat contributions and recombination heat. Special features of organic semiconductors like GaussFermi statistics and mobilities functions depending on the electric field strength are taken into account. We prove the existence of solutions for the stationary problem by an iteration scheme and Schauder's fixed point theorem. The underlying solution concept is related to weak solutions of the Van Roosbroeck system and entropy solutions of the heat equation. Additionally, for data compatible with thermodynamic equilibrium, the uniqueness of the solution is verified. It was recently shown that selfheating significantly influences the electronic properties of organic semiconductor devices. Therefore, modeling the coupled electric and thermal responses of organic semiconductors is essential for predicting the effects of temperature on the overall behavior of the device. This work puts the electrothermal driftdiffusion model for organic semiconductors on a sound analytical basis. 
A. Glitzky, M. Liero, G. Nika, Analysis of a bulksurface thermistor model for largearea organic LEDs, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 78 (2021), pp. 187210, DOI 10.4171/PM/2066 .
Abstract
The existence of a weak solution for an effective system of partial differential equations describing the electrothermal behavior of largearea organic lightemitting diodes (OLEDs) is proved. The effective system consists of the heat equation in the threedimensional bulk glass substrate and two semilinear equations for the current flow through the electrodes coupled to algebraic equations for the continuity of the electrical fluxes through the organic layers. The electrical problem is formulated on the (curvilinear) surface of the glass substrate where the OLED is mounted. The source terms in the heat equation are due to Joule heating and are hence concentrated on the part of the boundary where the currentflow equation is posed. The existence of weak solutions to the effective system is proved via Schauder's fixedpoint theorem. Moreover, since the heat sources are a priori only in $L^1$, the concept of entropy solutions is used. 
TH. Eiter, On the spatially asymptotic structure of timeperiodic solutions to the NavierStokes equations, Proceedings of the American Mathematical Society, 149 (2021), pp. 34393451, DOI 10.1090/proc/15482 .
Abstract
The asymptotic behavior of weak timeperiodic solutions to the NavierStokes equations with a drift term in the threedimensional whole space is investigated. The velocity field is decomposed into a timeindependent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions. 
TH. Eiter, G.P. Galdi, Spatial decay of the vorticity field of timeperiodic viscous flow past a body, Archive for Rational Mechanics and Analysis, 242 (2021), pp. 149178, DOI 10.1007/s0020502101690z .
Abstract
We study the asymptotic spatial behavior of the vorticity field associated to a timeperiodic NavierStokes flow past a body in the class of weak solutions satisfying a Serrinlike condition. We show that outside the wake region the vorticity field decays pointwise at an exponential rate, uniformly in time. Moreover, decomposing it into its timeaverage over a period and a socalled purely periodic part, we prove that inside the wake region, the timeaverage has the same algebraic decay as that known for the associated steadystate problem, whereas the purely periodic part decays even faster, uniformly in time. This implies, in particular, that “sufficiently far” from the body, the timeperiodic vorticity field behaves like the vorticity field of the corresponding steadystate problem. 
TH. Eiter, M. Kyed, Viscous flow around a rigid body performing a timeperiodic motion, Journal of Mathematical Fluid Mechanics, 23 (2021), pp. 28/128/23, DOI 10.1007/s00021021005564 .
Abstract
The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed timeperiodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigidbody motion are compatible, and that the mean translational velocity is nonzero, existence of a timeperiodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is illposed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces. 
TH. Eiter, K. Hopf, A. Mielke, LerayHopf solutions to a viscoelastic fluid model with nonsmooth stressstrain relation, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 65 (2022), pp. 103491/1103491/30 (published online on 20.12.2021), DOI 10.1016/j.nonrwa.2021.103491 .
Abstract
We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the ZarembaJaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of globalintime weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor. 
TH. Koprucki, A. Maltsi, A. Mielke, On the DarwinHowieWhelan equations for the scattering of fast electrons described by the Schrödinger equation, SIAM Journal on Applied Mathematics, 81 (2021), pp. 15521578, DOI 10.1137/21M139164X .
Abstract
The DarwinHowieWhelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrödinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the twobeam and the systematicrow approximation. 
A. Mielke, Relating a rateindependent system and a gradient system for the case of onehomogeneous potentials, Journal of Dynamics and Differential Equations, published online on 31.05.2021, DOI 10.1007/s10884021100073 .
Abstract
We consider a nonnegative and onehomogeneous energy functional $mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradientflow equations and the energetic solutions generated via the rateinpendent system given in terms of the timedependent functional $mathcal E(t,u)=t mathcal J(u)$ and the norm as a dissipation distance. The relation between the two flows is given via a solutiondependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the totalvariation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constantspeed intervals for the solutins of the gradientflow equation. As a major result we obtain a nontrivial existence and uniqueness result for the energetic rateindependent system. 
A. Mielke, A. Montefusco, M.A. Peletier, Exploring families of energydissipation landscapes via tilting: Three types of EDP convergence, Continuum Mechanics and Thermodynamics, 33 (2021), pp. 611637, DOI 10.1007/s0016102000932x .
Abstract
This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that wellknown techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the EnergyDissipation Principle that provides a variational formulation to gradientflow equations that allows one to apply techniques from Γconvergence of functional on states and functionals on trajectories. 
A. Mielke, R.R. Netz, S. Zendehroud, A rigorous derivation and energetics of a wave equation with fractional damping, Journal of Evolution Equations, 21 (2021), pp. 30793102, DOI 10.1007/s00028021006862 .
Abstract
We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the waterair interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionallydamped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energydissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionallydamped wave equation with a time derivative of order 3/2. 
A. Mielke, M.A. Peletier, A. Stephan, EDPconvergence for nonlinear fastslow reaction systems with detailed balance, Nonlinearity, 34 (2021), pp. 57625798, DOI 10.1088/13616544/ac0a8a .
Abstract
We consider nonlinear reaction systems satisfying massaction kinetics with slow and fast reactions. It is known that the fastreactionrate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailedbalance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a coshtype dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the EnergyDissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy massaction kinetics.
Contributions to Collected Editions

O. Marquardt, L. Geelhaar, O. Brandt, Wavefunction engineering in In0.53Ga0.47As/InxAl1xAs core/shell nanowires, in: 2021 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), IEEE Conference Publications Management Group, 2021, pp. 1516, DOI 10.1109/NUSOD52207.2021.9541519 .
Abstract
We study the electronic properties of In 0.53 Ga 0.47 As/In x Al 1x As core/shell nanowires for light emission in the telecommunication range. In particular, we systematically investigate the influence of the In content x of the In x Al 1x As shell and the diameter d of the In 0.53 Ga 0.47 As core on strain distribution, transition energies, and the character of the hole wave function. We show that the character of the hole state, and thus the polarization of the light emitted by such core/shell nanowires, can be easily tuned via these two experimentally accessible parameters. 
G. Nika, B. Vernescu, Microgeometry effects on the nonlinear effective yield strength response of magnetorheological fluids, in: Emerging Problems in the Homogenization of Partial Differential Equations, P. Donato, M. LunaLaynez, eds., 10 of SEMA SIMAI Springer Series, Springer, Cham, 2021, pp. 116, DOI 10.1007/9783030620301_1 .
Abstract
We use the novel constitutive model in [15], derived using the homogenization method, to investigate the effect particle chain microstructures have on the properties of the magnetorheological fluid. The model allows to compute the constitutive coefficients for different geometries. Different geometrical realizations of chains can significantly change the magnetorheological effect of the suspension. Numerical simulations suggest that particle size is also important as the increase of the overall particle surface area can lead to a decrease of the overall magnetorheological effect while keeping the volume fraction constant. 
A. Stephan, EDP convergence for nonlinear fastslow reaction systems, in: Report 29: Variational Methods for Evolution (hybrid meeting), A. Mielke, M. Peletier, D. Slepcev, eds., 17 of Oberwolfach Reports, European Mathematical Society Publishing House, Zurich, 2021, pp. 14561459, DOI 10.4171/OWR/2020/29 .

M. Horák, M. Kružík, P. Pelech, A. Schlömerkemper, Gradient polyconvexity and modeling of shape memory alloys, in: Variational Views in Mechanics, P.M. Mariano, ed., 46 of Advances in Mechanics and Mathematics, Birkhäuser, Cham, 2021, pp. 133156, DOI 10.1007/9783030900519_5 .
Abstract
We show existence of an energetic solution to a model of shape memory alloys in which the elastic energy is described by means of a gradientpolyconvex functional. This allows us to show existence of a solution based on weak continuity of nonlinear minors of deformation gradients in Sobolev spaces. Admissible deformations do not necessarily have integrable second derivatives. Under suitable assumptions, our model allows for solutions which are orientationpreserving and globally injective everywhere in the domain representing the specimen. Theoretical results are supported by threedimensional computational examples. This work is an extended version of [36]. 
S. Schulz, M. O'Donovan, D. Chaudhuri, S.K. Patra, P. Farrell, O. Marquardt, T. Streckenbach, Th. Koprucki, Connecting atomistic and continuum models for (In,Ga)N quantum wells: From tightbinding energy landscapes to electronic structure and carrier transport, in: 2021 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), IEEE Conference Publications Management Group, 2021, pp. 135136, DOI 10.1109/NUSOD52207.2021.9541461 .
Abstract
We present a multiscale framework for calculating electronic and transport properties of nitridebased devices. Here, an atomistic tightbinding model is connected with continuumbased electronic structure and transport models. In a first step, the electronic structure of (In,Ga)N quantum wells is analyzed and compared between atomistic and continuumbased approaches, showing that even though the two models operate on the same energy landscape, the obtained results differ noticeably; we briefly discuss approaches to improve the agreement between the two methods. Equipped with this information, unipolar carrier transport is investigated. Our calculations reveal that both random alloy fluctuations and quantum corrections significantly impact the transport, consistent with previous literature results. 
K. Hopf, Global existence analysis of energyreactiondiffusion systems, in: Report 29: Variational Methods for Evolution (hybrid meeting), A. Mielke, M. Peletier, D. Slepcev, eds., 17 of Oberwolfach Reports, European Mathematical Society Publishing House, Zurich, 2021, pp. 14181421, DOI 10.4171/OWR/2020/29 .
Preprints, Reports, Technical Reports

M. Liero, A. Mielke, G. Savaré, Fine properties of geodesics and geodesic lambdaconvexity for the HellingerKantorovich distance, Preprint no. 2956, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2956 .
Abstract, PDF (691 kByte)
We study the fine regularity properties of optimal potentials for the dual formulation of the HellingerKantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the HamiltonJacobi equation arising in the dual dynamic formulation of HK, which are sufficiently strong to construct a characteristic transportdilation flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of HK geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic lambdaconvexity with respect to the HellingerKantorovich distance. 
R. Bazaes, A. Mielke, Ch. Mukherjee, Stochastic homogenization of HamiltonJacobiBellman equations on continuum percolation clusters, Preprint no. 2955, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2955 .
Abstract, PDF (598 kByte)
We prove homogenization properties of random HamiltonJacobiBellman (HJB) equations on continuum percolation clusters, almost surely w.r.t. the law of the environment when the origin belongs to the unbounded component in the continuum. Here, the viscosity term carries a degenerate matrix, the Hamiltonian is convex and coercive w.r.t. the degenerate matrix and the underlying environment is nonelliptic and its law is nonstationary w.r.t. the translation group. We do not assume uniform ellipticity inside the percolation cluster, nor any finiterange dependence (i.i.d.) assumption on the percolation models and the effective Hamiltonian admits a variational formula which reflects some key properties of percolation. The proof is inspired by a method of KosyginaRezakhanlouVaradhan developed for the case of HJB equations with constant viscosity and uniformly coercive Hamiltonian in a stationary, ergodic and elliptic random environment. In the nonstationary and nonelliptic set up, we leverage the coercivity property of the underlying Hamiltonian as well as a relative entropy structure (both being intrinsic properties of HJB, in any framework) and make use of the random geometry of continuum percolation. 
A. Mielke, T. Roubíček, Qualitative study of a geodynamical rateandstate model for elastoplastic shear flows in crustal faults, Preprint no. 2954, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2954 .
Abstract, PDF (3349 kByte)
The DieterichRuina rateandstate friction model is transferred to a bulk variant and the state variable (aging) influencing the dissipation mechanism is here combined also with a damage influencing standardly the elastic response. As the aging has a separate dynamics, the overall model does not have a standard variational structure. A onedimensional model is investigated as far as the steadystate existence, localization of the cataclastic core, and its time response, too. Computational experiments with a damagefree variant show stickslip behavior (i.e. seismic cycles of tectonic faults) as well as stable slip under very large velocities. 
P. Colli, G. Gilardi, A. Signori, J. Sprekels, On a CahnHilliard system with source term and thermal memory, Preprint no. 2950, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2950 .
Abstract, PDF (322 kByte)
A nonisothermal phase field system of CahnHilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a CahnHilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a secondorder in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical results under general assumptions on the source term and the doublewell nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initialboundary value problem. 
P. Colli, G. Gilardi, A. Signori, J. Sprekels, Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential, Preprint no. 2949, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2949 .
Abstract, PDF (339 kByte)
In this paper, we investigate optimal control problems for a nonlinear state system which constitutes a version of the Caginalp phase field system modeling nonisothermal phase transitions with a nonconserved order parameter that takes thermal memory into account. The state system, which is a firstorder approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. It consists of two nonlinearly coupled partial differential equations that govern the phase dynamics and the universal balance law for internal energy, written in terms of the phase variable and the socalled thermal displacement, i.e., a primitive with respect to time of temperature. We extend recent results obtained for optimal control problems in which the free energy governing the phase transition was differentiable (i.e., of regular or logarithmic type) to the nonsmooth case of a double obstacle potential. As is well known, in this nondifferentiable case standard methods to establish the existence of appropriate Lagrange multipliers fail. This difficulty is overcome utilizing of the socalled deep quench approach. Namely, the double obstacle potential is approximated by a family of (differentiable) logarithmic ones for which the existence of optimal controls and firstorder necessary conditions of optimality in terms of the adjoint state variables and a variational inequality are known. By proving appropriate bounds for the adjoint states of the approximating systems, we can pass to the limit in the corresponding firstorder necessary conditions, thereby establishing meaningful firstorder necessary optimality conditions also for the case of the double obstacle potential. 
P. Vágner, M. Pavelka, J. Fuhrmann, V. Klika, A multiscale thermodynamic generalization of MaxwellStefan diffusion equations and of the dusty gas model, Preprint no. 2947, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2947 .
Abstract, PDF (1288 kByte)
Despite the fact that the theory of mixtures has been part of nonequilibrium thermodynamics and engineering for a long time, it is far from complete. While it is well formulated and tested in the case of mechanical equilibrium (where only diffusionlike processes take place), the question how to properly describe homogeneous mixtures that flow with multiple independent velocities that still possess some inertia (before mechanical equilibrium is reached) is still open. Moreover, the mixtures can have several temperatures before they relax to a common value. In this paper, we derive a theory of mixtures from Hamiltonian mechanics in interaction with electromagnetic fields. The resulting evolution equations are then reduced to the case with only one momentum (classical irreversible thermodynamics), providing a generalization of the MaxwellStefan diffusion equations. In a next step, we reduce that description to the mechanical equilibrium (no momentum) and derive a nonisothermal variant of the dusty gas model. These reduced equations are solved numerically, and we illustrate the results on effciency analysis, showing where in a concentration cell effciency is lost. Finally, the theory of mixtures identifies the temperature difference between constituents as a possible new source of the Soret coeffcient. For the sake of clarity, we restrict the presentation to the case of binary mixtures; the generalization is straightforward. 
M. Heida, On quenched homogenization of longrange random conductance models on stationary ergodic point processes, Preprint no. 2942, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2942 .
Abstract, PDF (359 kByte)
We study the homogenization limit on bounded domains for the longrange random conductance model on stationary ergodic point processes on the integer grid. We assume that the conductance between neares neighbors in the point process are always positive and satisfy certain weight conditions. For our proof we use longrange twoscale convergence as well as methods from numerical analysis of finite volume methods. 
P. Colli, G. Gilardi, A. Signori, J. Sprekels, CahnHilliardBrinkman model for tumor growth with possibly singular potentials, Preprint no. 2939, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2939 .
Abstract, PDF (350 kByte)
We analyze a phase field model for tumor growth consisting of a CahnHilliardBrinkman system, ruling the evolution of the tumor mass, coupled with an advectionreactiondiffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular, the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical noflux condition, a Dirichlet boundary condition for the chemical potential appearing in the CahnHilliardtype equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated. 
TH. Koprucki, A. Maltsi, A. Mielke, Symmetries in TEM imaging of crystals with strain, Preprint no. 2938, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2938 .
Abstract, PDF (6008 kByte)
TEM images of strained crystals often exhibit symmetries, the source of which is not always clear. To understand these symmetries we distinguish between symmetries that occur from the imaging process itself and symmetries of the inclusion that might affect the image. For the imaging process we prove mathematically that the intensities are invariant under specific transformations. A combination of these invariances with specific properties of the strain profile can then explain symmetries observed in TEM images. We demonstrate our approach to the study of symmetries in TEM images using selected examples in the field of semiconductor nanostructures such as quantum wells and quantum dots. 
A. Mielke, On two coupled degenerate parabolic equations motivated by thermodynamics, Preprint no. 2937, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2937 .
Abstract, PDF (3772 kByte)
We discuss a system of two coupled parabolic equations that have degenerate diffusion constants depending on the energylike variable. The dissipation of the velocitylike variable is fed as a source term into the energy equation leading to conservation of the total energy. The motivation of studying this system comes from Prandtl's and Kolmogorov's one and twoequation models for turbulence, where the energylike variable is the mean turbulent kinetic energy. Because of the degeneracies there are solutions with timedependent support like in the porous medium equation, which is contained in our system as a special case. The motion of the free boundary may be driven by either selfdiffusion of the energylike variable or by dissipation of the velocitylike variable. The crossover of these two phenomena is exemplified for the associated planar traveling fronts. We provide existence of suitably defined weak and very weak solutions. After providing a thermodynamically motivated gradient structure we also establish convergence into steady state for bounded domains and provide a conjecture on the asymptotically selfsimilar behavior of the solutions in R^{d} for large times. 
M. Heida, Stochastic homogenization on perforated domains III  General estimates for stationary ergodic random connected Lipschitz domains, Preprint no. 2932, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2932 .
Abstract, PDF (394 kByte)
This is Part III of a series on the existence of uniformly bounded extension operators on randomly perforated domains in the context of homogenization theory. Recalling that randomly perforated domains are typically not John and hence extension is possible only from W ^{1,p} to W ^{1,r}, r < p, we will show that the existence of such extension operators can be guarantied if the weighted expectations of four geometric characterizing parameters are bounded: The local Lipschitz constant M, the local Lipschitz radius Δ , the mesoscopic Voronoi diameter ∂ and the local connectivity radius R. 
TH. Eiter, M. Kyed, Y. Shibata, Periodic Lp estimates by Rboundedness: Applications to the NavierStokes equations, Preprint no. 2931, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2931 .
Abstract, PDF (400 kByte)
General evolution equations in Banach spaces are investigated. Based on an operatorvalued version of de Leeuw's transference principle, timeperiodic Lp estimates of maximal regularity type are established from Rbounds of the family of solution operators (Rsolvers) to the corresponding resolvent problems. With this method, existence of timeperiodic solutions to the NavierStokes equations is shown for two configurations: in a periodically moving bounded domain and in an exterior domain, subject to prescribed timeperiodic forcing and boundary data. 
P.É. Druet, Incompressible limit for a fluid mixture, Preprint no. 2930, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2930 .
Abstract, PDF (627 kByte)
In this paper we discuss the incompressible limit for multicomponent fluids in the isothermal ideal case. Both a direct limitpassage in the equation of state and the low Machnumber limit in rescaled PDEs are investigated. Using the relative energy inequality, we obtain convergence results for the densities and the velocityfield under the condition that the incompressible model possesses a sufficiently smooth solution, which is granted at least for a short time. Moreover, in comparison to singlecomponent flows, uniform estimates and the convergence of the pressure are needed in the multicomponent case because the incompressible velocity field is not divergencefree. We show that certain constellations of the mobility tensor allow to control gradients of the entropic variables and yield the convergence of the pressure in L1. 
M. Heida, A. Sikorski, M. Weber, Consistency and order 1 convergence of cellcentered finite volume discretizations of degenerate elliptic problems in any space dimension, Preprint no. 2913, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2913 .
Abstract, PDF (601 kByte)
We study consistency of cellcentered finite difference methods for elliptic equations with degenerate coefficients in any space dimension $dgeq2$. This results in order of convergence estimates in the natural weighted energy norm and in the weighted discrete $L^2$norm on admissible meshes. The cells of meshes under consideration may be very irregular in size. We particularly allow the size of certain cells to remain bounded from below even in the asymptotic limit. For uniform meshes we show that the order of convergence is at least 1 in the energy seminorm, provided the discrete and continuous solutions exist and the continuous solution has $H^2$ regularity. 
M. Heida, M. Landstorfer, M. Liero, Homogenization of a porous intercalation electrode with phase separation, Preprint no. 2905, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2905 .
Abstract, PDF (1704 kByte)
In this work, we derive a new model framework for a porous intercalation electrode with a phase separating active material upon lithium intercalation. We start from a microscopic model consisting of transport equations for lithium ions in an electrolyte phase and intercalated lithium in a solid active phase. Both are coupled through a Neumannboundary condition modeling the lithium intercalation reaction. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a CahnHilliard equation, whereas the limit model consists of a diffusion and an AllenCahn equation. Thus we observe a CahnHilliard to AllenCahn transition during the upscaling process. In the sense of gradient flows, the transition goes in hand with a change in the underlying metric structure of the PDE system. 
TH. Eiter, K. Hopf, R. Lasarzik, Weakstrong uniqueness and energyvariational solutions for a class of viscoelastoplastic fluid models, Preprint no. 2904, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2904 .
Abstract, PDF (345 kByte)
We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and an internal stress. This stress tensor is transported via the ZarembaJaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show shorttime existence of strong solutions as well as their uniqueness in a class of LerayHopf type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The globalintime existence of such generalized solutions has been established in a previous work. We further study the limit when stress diffusion vanishes. In this case, the above notion of generalized solutions is no longer suitable, and we introduce the concept of energyvariational solutions, which is based on an inequality for the relative energy. We derive general properties of energyvariational solutions and show their existence by passing to the nondiffusive limit in the relative energy inequality satisfied by generalized solutions for nonzero stress diffusion. 
A. Mielke, R. Rossi, BalancedViscosity solutions to infinitedimensional multirate systems, Preprint no. 2902, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2902 .
Abstract, PDF (778 kByte)
We consider generalized gradient systems with rateindependent and ratedependent dissipation potentials. We provide a general framework for performing a vanishingviscosity limit leading to the notion of parametrized and true BalancedViscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable $u$ having a viscous damping with relaxation time $eps^alpha$ and an internal variable $z$ with relaxation time $eps$ we obtain different limits for the three cases $alpha in (0,1)$, $alpha=1$ and $alpha>1$. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics. 
A. Stephan, H. Stephan, Positivity and polynomial decay of energies for squarefield operators, Preprint no. 2901, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2901 .
Abstract, PDF (328 kByte)
We show that for a general Markov generator the associated squarefield (or carré du champs) operator and all their iterations are positive. The proof is based on an interpolation between the operators involving the generator and their semigroups, and an interplay between positivity and convexity on Banach lattices. Positivity of the squarefield operators allows to define a hierarchy of quadratic and positive energy functionals which decay to zero along solutions of the corresponding evolution equation. Assuming that the Markov generator satisfies an operatortheoretic normality condition, the sequence of energies is logconvex. In particular, this implies polynomial decay in time for the energy functionals along solutions. 
A. Stephan, Coarsegraining and reconstruction for Markov matrices, Preprint no. 2891, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2891 .
Abstract, PDF (248 kByte)
We present a coarsegraining (or model order reduction) procedure for stochastic matrices by clustering. The method is consistent with the natural structure of Markov theory, preserving positivity and mass, and does not rely on any tools from Hilbert space theory. The reconstruction is provided by a generalized PenroseMoore inverse of the coarsegraining operator incorporating the inhomogeneous invariant measure of the Markov matrix. As we show, the method provides coarsegraining and reconstruction also on the level of tensor spaces, which is consistent with the notion of an incidence matrix and quotient graphs, and, moreover, allows to coarsegrain and reconstruct fluxes. Furthermore, we investigate the connection with functional inequalities and Poincarétype constants. 
M. Heida, S. Neukamm, M. Varga, Stochastic twoscale convergence and Young measures, Preprint no. 2885, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2885 .
Abstract, PDF (354 kByte)
In this paper we compare the notion of stochastic twoscale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic twoscale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic twoscale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic twoscale convergence. 
D.R.M. Renger, S. Schindler, Gradient flows for bounded linear evolution equations, Preprint no. 2881, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2881 .
Abstract, PDF (198 kByte)
We study linear evolution equations in separable Hilbert spaces defined by a bounded linear operator. We answer the question which of these equations can be written as a gradient flow, namely those for which the operator is real diagonalisable. The proof is constructive, from which we also derive geodesic lambdaconvexity. 
M. Heida, B. Jahnel, A.D. Vu, Stochastic homogenization on irregularly perforated domains, Preprint no. 2880, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2880 .
Abstract, PDF (668 kByte)
We study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by our approach. 
F. Agnelli, G. Nika, A. Constantinescu, Design of thin microarchitectured panels with extensionbending coupling effects using topology optimization, Preprint no. 2873, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2873 .
Abstract, PDF (3850 kByte)
We design thin microarchitectured panels with programmable macroscopic behaviour using inverse homogenization, the Hadamard shape derivative, and a level set method in the diffuse interface context. The optimally designed microstructures take into account the extensionbending effect in addition to inplane stiffness and outofplane bending stiffness. Furthermore, we present numerical examples of optimal microstructures that attain different targets for different volume fractions and interpret the physical significance of the extensionbending coupling. The simultaneous control of the inplane, outofplane and their coupled behaviour enables to shift a flat panel into a dome or saddle shaped structure under the action of an inplane loading. Moreover, the obtained unit cells are elementary blocks to create threedimensional objects with shapemorphing capabilities. 
M. Heida, M. Thomas, GENERIC for dissipative solids with bulkinterface interaction, Preprint no. 2872, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2872 .
Abstract, PDF (322 kByte)
The modeling framework of GENERIC was originally introduced by Grmela and Öttinger for thermodynamically closed systems. It is phrased with the aid of the energy and entropy as driving functionals for reversible and dissipative processes and suitable geometric structures. Based on the definition functional derivatives we propose a GENERIC framework for systems with bulkinterface interaction and apply it to discuss the GENERIC structure of models for delamination processes. 
P.É. Druet, Maximal mixed parabolichyperbolic regularity for the full equations of multicomponent fluid dynamics, Preprint no. 2869, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2869 .
Abstract, PDF (561 kByte)
We consider a NavierStokesFickOnsagerFourier system of PDEs describing mass, energy and momentum balance in a Newtonian fluid with composite molecular structure. For the resulting parabolichyperbolic system, we introduce the notion of optimal regularity of mixed type, and we prove the shorttime existence of strong solutions for a typical initial boundaryvalueproblem. By means of a partial maximum principle, we moreover show that such a solution cannot degenerate in finite time due to blowup or vanishing of the temperature or the partial mass densities. This second result is however only valid under certain growth conditions on the phenomenological coefficients. In order to obtain some illustration of the theory, we set up a special constitutive model for volumeadditive mixtures. 
M. Heida, Stochastic homogenization on perforated domains II  Application to nonlinear elasticity models, Preprint no. 2865, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2865 .
Abstract, PDF (314 kByte)
Based on a recent work that exposed the lack of uniformly bounded W^{1,p} → W^{1,p} extension operators on randomly perforated domains, we study stochastic homogenization of nonlinear elasticity on such structures using instead the extension operators constructed in [11]. We thereby introduce twoscale convergence methods on such random domains under the intrinsic loss of regularity and prove some generally useful calculus theorems on the probability space Ω, e.g. abstract Gauss theorems. 
P. Colli, A. Signori, J. Sprekels, Analysis and optimal control theory for a phase field model of Caginalp type with thermal memory, Preprint no. 2863, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2863 .
Abstract, PDF (369 kByte)
A nonlinear extension of the Caginalp phase field system is considered that takes thermal memory into account. The resulting model, which is a firstorder approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. Two equations, resulting from phase dynamics and the universal balance law for internal energy, are written in terms of the phase variable (representing a nonconserved order parameter) and the socalled thermal displacement, i.e., a primitive with respect to time of temperature. Existence and continuous dependence results are shown for weak and strong solutions to the corresponding initialboundary value problem. Then, an optimal control problem is investigated for a suitable cost functional, in which two data act as controls, namely, the distributed heat source and the initial temperature. Fréchet differentiability between suitable Banach spaces is shown for the controltostate operator, and meaningful firstorder necessary optimality conditions are derived in terms of variational inequalities involving the adjoint variables. Eventually, characterizations of the optimal controls are given. 
K. Hopf, Singularities in $L^1$supercritical FokkerPlanck equations: A qualitative analysis, Preprint no. 2860, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2860 .
Abstract, PDF (402 kByte)
A class of nonlinear FokkerPlanck equations with superlinear drift is investigated in the L^{1}supercritical regime, which exhibits a finite critical mass. The equations have a formal Wassersteinlike gradientflow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finitetime appearance constitutes a primary technical difficulty. This paper aims at a globalintime qualitative analysis  the main focus being on isotropic solutions, in which case the unique minimiser of the free energy will be shown to be the global attractor. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the 3D KaniadakisQuarati model for BoseEinstein particles, and thus provides a first rigorous result on the continuation beyond blowup and longtime asymptotic behaviour for this model. 
M. Heida, Precompact probability spaces in applied stochastic homogenization, Preprint no. 2852, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2852 .
Abstract, PDF (346 kByte)
We provide precompactness and metrizability of the probability space Ω for random measures and random coefficients such as they widely appear in stochastic homogenization and are typically given from data. We show that these properties are enough to implement the convenient twoscale formalism by Zhikov and Piatnitsky (2006). To further demonstrate the benefits of our approach we provide some useful trace and extension operators for Sobolev functions on Ω, which seem not known in literature. On the way we close some minor gaps in the Sobolev theory on Ω which seemingly have not been proven up to date. 
M. Heida, Stochastic homogenization on perforated domains I: Extension operators, Preprint no. 2849, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2849 .
Abstract, PDF (750 kByte)
This preprint is part of a major rewriting and substantial improvement of WIAS Preprint 2742. In this first part of a series of 3 papers, we set up a framework to study the existence of uniformly bounded extension and trace operators for W^{1,p}functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. We drop the classical assumption of minimaly smoothness and study stationary geometries which have no global John regularity. For such geometries, uniform extension operators can be defined only from W^{1,p} to W^{1,r} with the strict inequality r<p. In particular, we estimate the L^{r}norm of the extended gradient in terms of the L^{p}norm of the original gradient. Similar relations hold for the symmetric gradients (for ℝ^{d}valued functions) and for traces on the boundary. As a byproduct we obtain some Poincaré and Korn inequalities of the same spirit. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions: local (δ,M)regularity to quantify statistically the local Lipschitz regularity and isotropic cone mixing to quantify the density of the geometry and the mesoscopic properties. These two properties are sufficient to reduce the problem of extension operators to the connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process, for which we can explicitly estimate the connectivity terms. 
D. Bothe, W. Dreyer, P.É. Druet, Multicomponent incompressible fluids  An asymptotic study, Preprint no. 2825, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2825 .
Abstract, PDF (519 kByte)
This paper investigates the asymptotic behavior of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy that is valid in stable subregions of the phase diagram. On the basis of this representation we classify the admissible data to construct a thermodynamically consistent constitutive model. We then analyze the incompressible limit, where the molar volume becomes independent of pressure. Here we are confronted with two problems:(i) Our study shows that the physical system at hand cannot remain incompressible for arbitrary large deviations from a reference pressure unless its volume is linear in the composition. (ii) As a consequence of the 2nd law of thermodynamics, the incompressible limit implies that the molar volume becomes independent of temperature as well. Most applications, however, reveal the nonappropriateness of this property. According to our mathematical treatment, the free energy as a function of temperature and partial masses tends to a limit in the sense of epi or Gammaconvergence. In the context of the first problem, we study the mixing of two fluids to compare the linearity with experimental observations. The second problem will be treated by considering the asymptotic behavior of both a general inequality relating thermal expansion and compressibility and a PDEsystem relying on the equations of balance for partial masses, momentum and the internal energy.

J. Fischer, K. Hopf, M. Kniely, A. Mielke, Global existence analysis of energyreactiondiffusion systems, Preprint no. 2807, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2807 .
Abstract, PDF (489 kByte)
We establish globalintime existence results for thermodynamically consistent reaction(cross)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model speciesdependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the nonisothermal case lies in the intrinsic presence of crossdiffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where nonintegrable diffusion fluxes or reaction terms appear. 
G. Nika, An existence result for a class of nonlinear magnetorheological composites, Preprint no. 2804, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2804 .
Abstract, PDF (257 kByte)
We prove existence of a weak solution for a nonlinear, multiphysics, multiscale problem of magnetorheological suspensions introduced in Nika & Vernescu (Z. Angew. Math. Phys., 71(1):119, '20). The hybrid model couples the Stokes' equation with the quasistatic Maxwell's equations through the Lorentz force and the Maxwell stress tensor. The proof of existence is based on: i) the augmented variational formulation of Maxwell's equations, ii) the definition of a new function space for the magnetic induction and the proof of a Poincaré type inequality, iii) the AltmanShinbrot fixed point theorem when the magnetic Reynold's number, R_{m}, is small.
Talks, Poster

A. Maltsi, Symmetries in TEM images of strained crystals, ``European Women in Mathematics'' General Meeting 2022, Aalto, Finland, August 22  26, 2022.

A. Maltsi, Symmetries in TEM images of strained crystals, BMSBGSMath Junior Meeting, September 5  7, 2022, Universidad de Barcelona, Spain, September 6, 2022.

S. Schindler, Convergence to selfsimilar profiles for a coupled reactiondiffusion system on the real line, CRC 910: Workshop on Control of SelfOrganizing Nonlinear Systems, September 26  28, 2022.

S. Schindler, Energy approach for a coupled reactiondiffusion system on the real line (online talk), SFB 910 Symposium ``Pattern formation and coherent structure in dissipative systems'' (Online Event), Technische Universität Berlin, January 14, 2022.

Y. Hadjimichael, O. Marquardt, Ch. Merdon, P. Farrell, Band structures in highly strained 3D nanowires, 22th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) (Online Event, Italy, September 12  16, 2022.

M. Heida, Convergence of the infinite range SQRA operator, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), Session 14 ``Applied Analysis'', August 15  19, 2022, RheinischWestfälische Technische Hochschule Aachen, August 17, 2022.

M. Heida, Homogenization on locally Lipschitz random domains (online talk), SIAM Conference on Analysis of Partial Differential Equations, Minisymposium ``Disordered Media and Homogenization" (Online Event), March 14  18, 2022, Society for Industrial and Applied Mathematics, March 15, 2022.

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

O. Marquardt, Simulating the electronic properties of semiconductor nanostructures, 5th Leibniz MMS Days, April 25  27, 2022, PotsdamInstitut für Klimafolgenforschung (PIK), April 26, 2022.

O. Marquardt, SPHInXTutorial 2022 (Hybrid Event), March 14  April 11, 2022, WIAS Berlin.

P. Pelech, PenroseFife model as a gradient flow  interplay between signed measures and functionals on Sobolev spaces, DMV Annual Meeting 2022, Section 09 ``Applied Analysis and Partial Differential Equations", September 12  16, 2022, Freie Universität Berlin, September 13, 2022.

P. Pelech, PenroseFife model with activated phase transformation  existence and effective model for slowloading regimes, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), Session 08 ``Multiscales and Homogenization'', August 15  19, 2022, RheinischWestfälische Technische Hochschule Aachen, August 18, 2022.

A. Stephan, EDPconvergence for a linear reactiondiffusion systems with fast reversible reaction (online talk), SIAM Conference on Analysis of Partial Differential Equations (Online Event), March 14  18, 2022, Society for Industrial and Applied Mathematics, March 14, 2022.

P. Vágner, Capacitance of the blocking YSZ I Au electrode, 18th Symposium on Modeling and Experimental Validation of Electrochemical Energy Technologies, March 14  16, 2022, DLR Institut für Technische Thermodynamik, Hohenkammer, March 16, 2022.

A. Glitzky, A driftdiffusion based electrothermal model for organic thinfilm devices including electrical and thermal environment, DMV Annual Meeting 2022, Section 09 ``Applied Analysis and Partial Differential Equations", September 12  16, 2022, Freie Universität Berlin, September 14, 2022.

K. Hopf, A variational approach to stability in strongly coupled parabolic systems (online talk), SIAM Conference on Analysis of Partial Differential Equations, Minisymposium ``Variational Evolution: Analysis and MultiScale Aspects" (Online Event), March 14  18, 2022, Society for Industrial and Applied Mathematics, March 16, 2022.

TH. Eiter, Energyvariational solutions for a viscoelastoplastic fluid model (online talk), SIAM Conference on Analysis of Partial Differential Equations, Minisymposium ``Generalized Solvability Concepts for Evolutionary PDEs and their Properties" (Online Event), March 14  18, 2022, Society for Industrial and Applied Mathematics, March 16, 2022.

TH. Eiter, Junior Richard von Mises Lecture: On timeperiodic viscous flow around a moving body, Richard von Mises Lecture 2022, HumboldtUniversität zu Berlin, June 17, 2022.

TH. Eiter, On the resolvent problem associated with flow outside a rotating body, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), Session 14 ``Applied Analysis'', August 15  19, 2022, RheinischWestfälische Technische Hochschule Aachen, August 16, 2022.

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

TH. Eiter, On the timeperiodic Viscous flow outside a rotating body (online talk), SIAM Conference on Analysis of Partial Differential Equations, Minisymposium ``Recent Developments in the Mathematical Analysis of Viscous Fluids" (Online Event), March 14  18, 2022, Society for Industrial and Applied Mathematics, March 15, 2022.

TH. Eiter, On timeperiodic NavierStokes flow around a rotating body (online talk), EDP non linéaires en dynamique des fluides (Hybrid Event), May 9  13, 2022, Centre International de Rencontres Mathématiques, Marseille, France, May 9, 2022.

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

TH. Eiter, On uniformity of the resolvent estimates associated with timeperiodic flow past a rotating body, GermanyJapan Workshop on Free and Singular Boundaries in Fluid Dynamics and Related Topics (Hybrid Event), August 10  12, 2022, HeinrichHeineUniversität Düsseldorf, August 10, 2022.

TH. Eiter, Resolvent estimates for the flow past a rotating body and existence of timeperiodic solutions, CEMAT Seminar, July 24  30, 2022, University of Lisbon, Center for Computational and Stochastic Mathematics, Portugal, July 27, 2022.

TH. Koprucki, K. Tabelow, HackMD (online talk), ECoffeeLecture (Online Event), WIAS Berlin, March 25, 2022.

M. Liero, Analysis of an electrothermal driftdiffusion model for organic semiconductor devices, PHAse field MEthods in applied sciences (PHAME 2022), May 23  27, 2022, Istituto Nazionale di Alta Matematica, Roma, Italy, May 24, 2022.

M. Liero, Automated building and testing of software projects using the WIAS Gitlab server (online talk), ECoffeeLecture (Online Event), WIAS Berlin, January 21, 2022.

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

M. Liero, From diffusion to reactiondiffusion in thin structures via EDPconvergence (online talk), SIAM Conference on Analysis of Partial Differential Equations, Minisymposium ``Bridging Gradient Flows, Hypocoercivity and ReactionDiffusion Systems" (Online Event), March 14  18, 2022, Society for Industrial and Applied Mathematics, March 14, 2022.

M. Liero, The impact of modeling, analysis, and simulation on organic semiconductor development (online talk), ERCOM Meeting 2022 (Hybrid Event), March 25  26, 2022, European Research Centers on Mathematics, Bilbao, Spain, March 26, 2022.

A. Mielke, Convergence to thermodynamic equilibrium in a degenerate parabolic system, DMV Annual Meeting 2022, Section 09 ``Applied Analysis and Partial Differential Equations", September 12  16, 2022, Freie Universität Berlin, September 13, 2022.

A. Mielke, Existence and longtime behavior of solutions to a degenerate parabolic system, Journées Équations aux Dérivées Partielles 2022, May 30  June 3, 2022, Centre national de la recherche scientifique, Obernai, France, May 31, 2022.

A. Mielke, Gamma convergence for evolutionary problems: using EDP convergence for deriving nontrivial kinetic relations, Calculus of Variations, May 16  20, 2022, Università degli Studi di Roma ``Tor Vergata'', Tunis, Tunisia, May 18, 2022.

A. Mielke, Gradient flows in the HellingerKantorovich space, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), Minisymposium 4 ``Evolution Equations with Gradient Flow Structure'', August 15  19, 2022, RheinischWestfälische Technische Hochschule Aachen, August 16, 2022.

A. Mielke, On the existence and longtime behavior of solutions to a degenerate parabolic system (online talk), SIAM Conference on Analysis of Partial Differential Equations (Online Event), March 14  18, 2022, Society for Industrial and Applied Mathematics, March 16, 2022.

A. Mielke, On the longtime behavior of solutions to a coupled degenerate parabolic system motivated by thermodynamics (online talk), Nonlinear Waves and Coherent Structures Webinar Series (Online Event), University of Massachusetts, Amherst, USA, January 25, 2022.

J. Rehberg, Explicit Lpestimates for secondorder divergence operators, Oberseminar Analysis und Angewandte Mathematik, Universität Kassel, June 9, 2022.

W. van Oosterhout, Analysis of a poroviscoelastic material model, Mathematical models for biomedical sciences, Como, Italy, June 20  24, 2022.

A. Maltsi, Modelbased geometry reconstruction of TEM images, MATH+ Day 2021 (Online Event), Technische Universität Berlin, November 5, 2021.

A. Maltsi, On the DarwinHowieWhelan equations in TEM imaging (online talk), Young Women in PDEs and Applications (Online Event), September 20  22, 2021, Universität Bonn, September 22, 2021.

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

S. Schindler, Selfsimilar diffusive equilibration for a coupled reactiondiffusion system with massaction kinetics, SFB910: International Conference on Control of SelfOrganizing Nonlinear Systems (Hybrid Event), August 29  September 2, 2021, Technische Universität Berlin, Potsdam, September 1, 2021.

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

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

O. Marquardt, Modelling the electronic properties of semiconductor nanowire heterostructures (online talk), PDISeminar, PaulDrudeInstitut für Festkörperelektronik, August 23, 2021.

O. Marquart, Wavefunction engineering in (In,Ga)As/(InAl)/As core/shell nanowires (online talk), 2021 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) (Online Event), September 13  17, 2021, Politecnico die Torino, September 13, 2021.

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

P. Pelech, Balancedviscosity solution for a PenroseFife model with rateindependent friction (online talk), Jahrestreffen des SPP 2256 (Online Event), September 22  24, 2021, Universität Regensburg, September 22, 2021.

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

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

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

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

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

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

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

A. Glitzky, A coarsegrained electrothermal model for organic semiconductor devices (online talk), DMVÖMG Jahrestagung 2021 (Online Event), September 27  October 1, 2021, Universität Passau, September 29, 2021.

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

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

P.E. Druet, Wellposedness results for mixedtype systems modelling pressuredriven multicomponent fluid flows (online talk), 8th European Congress of Mathematics (8ECM), Minisymposium ID 42 ``Multicomponent Diffusion in Porous Media'' (Online Event), June 20  26, 2021, Portorož, Slovenia, June 22, 2021.

TH. Eiter, Viscoelastic fluid flows with nonsmooth stress dissipation (online talk), DMVÖMG Jahrestagung 2021 (Online Event), September 27  October 1, 2021, Universität Passau, September 29, 2021.

TH. Koprucki, MaRDI, the mathematical research data iniative within the NFDI (online talk), DMVÖMG Jahrestagung 2021 (Online Event), September 27  October 1, 2021, Universität Passau, September 27, 2021.

M. Landstorfer, M. Eigel, M. Heida, A. Selahi, Recovery of battery ageing dynamics with multiple timescales (online poster), MATH+ Day 2021 (Online Event), Technische Universität Berlin, November 5, 2021.

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

M. Liero, Heat and carrier flow in organic semiconductor devices  Modeling, analysis, and simulation (online talk), AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6  9, 2021, WIAS Berlin, September 6, 2021.

M. Liero, Mathematical research data in Applied Analysis (online talk), MaRDI Kickoff Workshop, November 2  4, 2021, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, November 2, 2021.

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

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

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

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

V. Laschos, A. Mielke, Evolutionary variational inequalities on the HellingerKantorovich and spherical HellingerKantorovich spaces, Preprint no. arXiv:2207.09815, Cornell University Library, arXiv.org, 2022, DOI 10.48550/arXiv.2207.09815 .
Abstract
We study the minimizing movement scheme for some families of geodesically semiconvex functionals defined on either the HellingerKantorovich or the spherical HellingerKantorovich space. By exploiting some of the finer geometric properties of those spaces, we prove that the sequence of curves produced by geodesically interpolating the points generated by the scheme, parameterized by the step size, converges to curves that satisfy the Evolutionary Variational Inequality (EVI), when the step size goes to zero. 
A. Mielke, T. Roubíček, Qualitative study of a geodynamical rateandstate model for elastoplastic shear flows in crustal faults, Preprint no. arXiv:2207.11074, Cornell University Library, arXiv.org, 2022, DOI 10.48550/arXiv.2207.11074 .
Abstract
The DieterichRuina rateandstate friction model is transferred to a bulk variant and the state variable (aging) influencing the dissipation mechanism is here combined also with a damage influencing standardly the elastic response. As the aging has a separate dynamics, the overall model does not have a standard variational structure. A onedimensional model is investigated as far as the steadystate existence, localization of the ataclastic core, and its time response, too. Computational experiments with a damagefree variant show stickslip behavior (i.e. seismic cycles of tectonic faults) as well as stable slip under very large velocities. 
M. O'Donovan, P. Farrell, T. Streckenbach, Th. Koprucki, S. Schulz, Multiscale simulations of unipolar hole transport in (In,Ga)N quantum well systems, Preprint no. arXiv:2111.01644, Cornell University Library, arXiv.org, 2021.
Abstract
Understanding the impact of the alloy microstructure on carrier transport becomes important when designing IIInitridebased LED structures. In this work, we study the impact of alloy fluctuations on the hole carrier transport in (In,Ga)N single and multiquantum well systems. To disentangle hole transport from electron transport and carrier recombination processes, we focus our attention on unipolar (pip) systems. The calculations employ our recently established multiscale simulation framework that connects atomistic tightbinding theory with a macroscale driftdiffusion model. In addition to alloy fluctuations, we pay special attention to the impact of quantum corrections on hole transport. Our calculations indicate that results from a virtual crystal approximation present an upper limit for the hole transport in a pip structure in terms of the currentvoltage characteristics. Thus we find that alloy fluctuations can have a detrimental effect on hole transport in (In,Ga)N quantum well systems, in contrast to unipolar electron transport. However, our studies also reveal that the magnitude by which the random alloy results deviate from virtual crystal approximation data depends on several factors, e.g. how quantum corrections are treated in the transport calculations.
Research Groups
 Partial Differential Equations
 Laser Dynamics
 Numerical Mathematics and Scientific Computing
 Nonlinear Optimization and Inverse Problems
 Interacting Random Systems
 Stochastic Algorithms and Nonparametric Statistics
 Thermodynamic Modeling and Analysis of Phase Transitions
 Nonsmooth Variational Problems and Operator Equations