Publikationen

Monografien

  • 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 time-incremental 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 multi-agent 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 large-deviation principles for time-continuous Markov processes, Gamma-convergence and homogenization.

Artikel in Referierten Journalen

  • M. Heida, Stochastic homogenization on perforated domains II -- Application to nonlinear elasticity models, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, published online on 26.09.2022, DOI 10.1002/zamm.202100407 .
    Abstract
    Based on a recent work that exposed the lack of uniformly bounded W1,p → W1,p extension operators on randomly perforated domains, we study stochastic homogenization of nonlinear elasticity on such structures using instead the extension operators constructed in [11]. We thereby introduce two-scale convergence methods on such random domains under the intrinsic loss of regularity and prove some generally useful calculus theorems on the probability space Ω, e.g. abstract Gauss theorems.

  • P. Pelech, K. Tůma, M. Pavelka, M. Šípka, M. Sýkora, On compatibility of the natural configuration framework with general equation for non-equilibrium reversible-irreversible coupling (GENERIC): Derivation of anisotropic rate-type models, Journal of Non-Newtonian Fluid Mechanics, 305 (2022), pp. 104808/1--104808/19, 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 Non-Equilibrium Reversible-Irreversible 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 rate-type fluid models. We propose an interpretation of the natural configurations within GENERIC and vice versa (when possible).

  • P. Vágner, M. Pavelka, J. Fuhrmann, V. Klika, A multiscale thermodynamic generalization of Maxwell--Stefan diffusion equations and of the dusty gas model, International Journal of Heat and Mass Transfer, 199 (2022), pp. 123405/1--123405/14, DOI 10.1016/j.ijheatmasstransfer.2022.123405 .
    Abstract
    Despite the fact that the theory of mixtures has been part of non-equilibrium 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 diffusion-like 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 Maxwell-Stefan diffusion equations. In a next step, we reduce that description to the mechanical equilibrium (no momentum) and derive a non-isothermal 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.

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Well-posedness and optimal control for a Cahn--Hilliard--Oono system with control in the mass term, Discrete and Continuous Dynamical Systems -- Series S, 15 (2022), pp. 2135--2172, DOI 10.3934/dcdss.2022001 .
    Abstract
    The paper treats the problem of optimal distributed control of a Cahn--Hilliard--Oono system in Rd, 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 first-order 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, Well-posedness for a class of phase-field 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. 193--228, 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 phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Math. Models Methods Appl. Sci. bf 30 (2020), 1253--1295]). 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 Allen--Cahn equation with double-well 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. 25--58 (published online on 28.02.2022), DOI 10.1007/s10957-022-02000-7 .
    Abstract
    This paper treats a distributed optimal control problem for a tumor growth model of Cahn--Hilliard 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 so-called “deep quench approximation” in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, first-order 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 energy-reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 54 (2022), pp. 220--267 (published online on 04.01.2022), DOI 10.1137/20M1387237 .
    Abstract
    We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.

  • 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. 235--262, 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 phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn--Hilliard 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 reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces.

  • V. Miloš, P. Vágner, D. Budáč, M. Carda, M. Paidar, J. Fuhrmann, K. Bouzek, Generalized Poisson--Nernst--Planck-based physical model of the O$_2$ I LSM I YSZ electrode, Journal of The Electrochemical Society, 169 (2022), pp. 044505/1--044505/17, DOI 10.1149/1945-7111/ac4a51 .
    Abstract
    The paper presents a generalized Poisson--Nernst--Planck model of an yttria-stabilized zirconia electrolyte developed from first principles of nonequilibrium thermodynamics which allows for spatial resolution of the space charge layer. It takes into account limitations in oxide ion concentrations due to the limited availability of oxygen vacancies. The electrolyte model is coupled with a reaction kinetic model describing the triple phase boundary with electron conducting lanthanum strontium manganite and gaseous phase oxygen. By comparing the outcome of numerical simulations based on different formulations of the kinetic equations with results of EIS and CV measurements we attempt to discern the existence of separate surface lattice sites for oxygen adatoms and O2- from the assumption of shared ones. Furthermore, we discern mass-action kinetics models from exponential kinetics models.

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

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

  • A. Glitzky, M. Liero, G. Nika, A coarse-grained electrothermal model for organic semiconductor devices, Mathematical Methods in the Applied Sciences, 45 (2022), pp. 4809--4833 (published on 26.01.2022), DOI 10.1002/mma.8072 .
    Abstract
    We derive a coarse-grained model for the electrothermal interaction of organic semiconductors. The model combines stationary drift-diffusion based electrothermal models with thermistor type models on subregions of the device and suitable transmission conditions. Moreover, we prove existence of a solution using a regularization argument and Schauder's fixed point theorem. In doing so, we extend recent work by taking into account the statistical relation given by the Gauss--Fermi integral and mobility functions depending on the temperature, charge-carrier density, and field strength, which is required for a proper description of organic devices.

  • K. Hopf, M. Burger, On multi-species diffusion with size exclusion, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 224 (2022), pp. 113092/1--113092/27, DOI 10.1016/j.na.2022.113092 .
    Abstract
    We revisit a classical continuum model for the diffusion of multiple species with size-exclusion constraint, which leads to a degenerate nonlinear cross-diffusion 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 long-time asymptotic behaviour. Second, it provides a weak-strong 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 gradient-flow structure of the model with respect to a logarithmic entropy, which leads to best estimates in the full-interaction case, where all cross-diffusion coefficients are non-zero. Those are crucial to obtain the minimal Sobolev regularity needed for a weak-strong stability result. For meaningful cases when some of the coefficients vanish, we provide a novel existence result based on approximation by the full-interaction case.

  • K. Hopf, Weak-strong uniqueness for energy-reaction-diffusion systems, Mathematical Models & Methods in Applied Sciences, 21 (2022), pp. 1015--1069 (published online on 29.04.2022), DOI 10.1142/S0218202522500233 .
    Abstract
    We establish weak-strong uniqueness and stability properties of renormalised solutions to a class of energy-reaction-diffusion systems, which genuinely feature cross-diffusion effects. The systems considered are motivated by thermodynamically consistent models, and their formal entropy structure allows us to use as a key tool a suitably adjusted relative entropy method. Weak-strong uniqueness is obtained for general entropy-dissipating reactions without growth restrictions, and certain models with a non-integrable diffusive flux. The results also apply to a class of (isoenergetic) reaction-cross-diffusion systems.

  • P.-É. Druet, Maximal mixed parabolic-hyperbolic regularity for the full equations of multicomponent fluid dynamics, Nonlinearity, 35 (2022), pp. 3812--3882, DOI 10.1088/1361-6544/ac5679 .
    Abstract
    We consider a Navier--Stokes--Fick--Onsager--Fourier system of PDEs describing mass, energy and momentum balance in a Newtonian fluid with composite molecular structure. For the resulting parabolic-hyperbolic system, we introduce the notion of optimal regularity of mixed type, and we prove the short-time existence of strong solutions for a typical initial boundary-value-problem. By means of a partial maximum principle, we moreover show that such a solution cannot degenerate in finite time due to blow-up 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 volume-additive mixtures.

  • TH. Eiter, K. Hopf, R. Lasarzik, Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models, Advances in Nonlinear Analysis, 12 (2023), pp. 20220274/1--20220274/31 (published online on 03.10.2022), DOI 10.1515_anona-2022-0274 .
    Abstract
    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 Zaremba--Jaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show short-time existence of strong solutions as well as their uniqueness in a class of Leray--Hopf type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The global-in-time 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 energy-variational solutions, which is based on an inequality for the relative energy. We derive general properties of energy-variational solutions and show their existence by passing to the non-diffusive limit in the relative energy inequality satisfied by generalized solutions for non-zero stress diffusion.

  • TH. Eiter, On the Oseen-type resolvent problem associated with time-periodic flow past a rotating body, SIAM Journal on Mathematical Analysis, 54 (2022), pp. 4987--5012, DOI 10.1137/21M1456728 .
    Abstract
    Consider the time-periodic flow of an incompressible viscous fluid past a body performing a rigid motion with non-zero 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 time-periodic 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, time-periodic 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 time-periodic linear problem do not exist.

  • TH. Eiter, On the Stokes-type resolvent problem associated with time-periodic flow around a rotating obstacle, Journal of Mathematical Fluid Mechanics, 24 (2022), pp. 52/1--17, DOI 10.1007/s00021-021-00654-3 .
    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 time-periodic linear problem.

  • A. Mielke, S. Reichelt, Traveling fronts in a reaction-diffusion equation with a memory term, Journal of Dynamics and Differential Equations, (2022), published online on 23.02.2022, DOI 10.1007/s10884-022-10133-6 .
    Abstract
    Based on a recent work on traveling waves in spatially nonlocal reaction-diffusion equations, we investigate the existence of traveling fronts in reaction-diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction-diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale 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 global-in-time weak solutions and scaling laws for Kolmogorov's two-equation model of turbulence, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 102 (2022), pp. 1--31 (published online on 01.07.2022), DOI 10.1002/zamm.202000019 .
    Abstract
    This paper is concerned with Kolmogorov's two-equation 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 two-equation 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 two-equation model under space-periodic boundary conditions in cubes with positive side length l. To this end, we provide new a priori estimates and invoke existence result for pseudo-monotone operators.

  • P.-É. Druet, Global-in-time existence for liquid mixtures subject to a generalised incompressibility constraint, Journal of Mathematical Analysis and Applications, 499 (2021), pp. 125059/1--125059/56, 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 non-solenoidal velocity field in the Navier-Stokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDE-system 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 L1. 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 Fokker--Planck operator, ESAIM: Mathematical Modelling and Numerical Analysis, 55 (2021), pp. 3017--3042, DOI 10.1051/m2an/2021078 .
    Abstract
    We introduce a family of various finite volume discretization schemes for the Fokker--Planck operator, which are characterized by different weight functions on the edges. This family particularly includes the well-established Scharfetter--Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the quality of the weights. We show by numerical experiments that for small gradients the choice of the optimal representative of the discretization family is highly non-trivial while for large gradients the Scharfetter--Gummel scheme stands out compared to the others.

  • M. Heida, S. Neukamm, M. Varga, Stochastic homogenization of Lambda-convex gradient flows, Discrete and Continuous Dynamical Systems -- Series S, 14 (2021), pp. 427--453, DOI 10.3934/dcdss.2020328 .
    Abstract
    In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen--Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals.

  • O. Marquardt, Simulating the electronic properties of semiconductor nanostructures using multiband $kcdot p$ models, Computational Materials Science, 194 (2021), pp. 110318/1--110318/11, DOI 10.1016/j.commatsci.2021.110318 .
    Abstract
    The eight-band $kcdot p$ formalism been successfully applied to compute the electronic properties of a wide range of semiconductor nanostructures in the past and can be considered the backbone of modern semiconductor heterostructure modelling. However, emerging novel material systems and heterostructure fabrication techniques raise questions that cannot be answered using this well-established formalism, due to its intrinsic limitations. The present article reviews recent studies on the calculation of electronic properties of semiconductor nanostructures using a generalized multiband $kcdot p$ approach that allows both the application of the eight-band model as well as more sophisticated approaches for novel material systems and heterostructures.

  • G. Nika, Derivation of effective models from heterogenous Cosserat media via periodic unfolding, Ricerche di Matematica. A Journal of Pure and Applied Mathematics, published online on 01.07.2021, DOI 10.1007/s11587-021-00610-3 .
    Abstract
    We derive two different effective models from a heterogeneous Cosserat continuum taking into account the Cosserat intrinsic length of the constituents. We pass to the limit using homogenization via periodic unfolding and in doing so we provide rigorous proof to the results introduced by Forest, Pradel, and Sab (Int. J. Solids Structures 38 (26-27): 4585-4608 '01). Depending on how different characteristic lengths of the domain scale with respect to the Cosserat intrinsic length, we obtain either an effective classical Cauchy continuum or an effective Cosserat continuum. Moreover, we provide some corrector type results for each case.

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

  • E. Davoli, M. Kružík, P. Pelech, Separately global solutions to rate-independent processes in large-strain inelasticity, Nonlinear Analysis. An International Mathematical Journal, 215 (2022), pp. 112668/1--112668/37 (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 large-strain rate-independent systems, and we provide an existence result for a model describing bulk damage. Our analysis covers non-convex 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 well-posedness in classes of strong solutions for general class-one models, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 210 (2021), pp. 112389/1--112389/53, 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 Fick-Onsager or Maxwell- Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the Navier-Stokes equations. The thermodynamic pressure is defined by the Gibbs-Duhem 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 parabolic-hyperbolic type. We prove two theoretical results concerning the well-posedness of the model in classes of strong solutions: 1. The solution always exists and is unique for short-times and 2. If the initial data are sufficiently near to an equilibrium solution, the well-posedness 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, Well-posedness analysis of multicomponent incompressible flow models, Journal of Evolution Equations, 21 (2021), pp. 4039--4093, DOI 10.1007/s00028-021-00712-3 .
    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 Navier--Stokes equations and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.

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

  • P. Colli, G. Gilardi, J. Sprekels, An asymptotic analysis for a generalized Cahn--Hilliard system with fractional operators, Journal of Evolution Equations, 21 (2021), pp. 2749--2778, DOI 10.1007/s00028-021-00706-1 .
    Abstract
    In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous Cahn--Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers in the spectral sense of general linear operators, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space of square-integrable functions on a bounded and smooth three-dimensional domain, and have compact resolvents. Here, for the case of the viscous system, we analyze the asymptotic behavior of the solution as the fractional power coefficient of the second operator tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of the second operator appears.

  • P. Colli, A. Signori, J. Sprekels, Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis, ESAIM. Control, Optimisation and Calculus of Variations, 27 (2021), pp. 73/1--73/46, DOI 10.1051/cocv/2021072 .
    Abstract
    This paper concerns a distributed optimal control problem for a tumor growth model of Cahn--Hilliard type including chemotaxis with possibly singular anpotentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

  • P. Colli, G. Gilardi, J. Sprekels, Deep quench approximation and optimal control of general Cahn--Hilliard systems with fractional operators and double obstacle potentials, Discrete and Continuous Dynamical Systems -- Series S, 14 (2021), pp. 243--271, DOI 10.3934/dcdss.2020213 .
    Abstract
    Recently, the authors derived well-posedness and regularity results for general evolutionary operator equations having the structure of a Cahn--Hilliard 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 double-well potentials driving the phase separation process modeled by the Cahn--Hilliard 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 first-order 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 so-called “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 first-order necessary optimality conditions.

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

  • A. Kirch, A. Fischer, M. Liero, J. Fuhrmann, A. Glitzky, S. Reineke, Electrothermal tristability causes sudden burn-in 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. 375--404, DOI 10.1016/j.jde.2021.01.032 .
    Abstract
    In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order Xs-1,qD(Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs.

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

  • 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. S26/1--S26/27, DOI 10.1051/cocv/2020088 .
    Abstract
    In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L1-norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.

  • A.F.M. TER Elst, R. Haller-Dintelmann, J. Rehberg, P. Tolksdorf, On the $L^p$-theory for second-order elliptic operators in divergence form with complex coefficients, Journal of Evolution Equations, 21 (2021), pp. 3963--4003, DOI 10.1007/s00028-021-00711-4 .
    Abstract
    Given a complex, elliptic coefficient function we investigate for which values of p the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on Lp(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. 125815/1--125815/26 (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 electro-thermal behavior of semiconductor heterostructures. This hybrid model combines an electro-thermal model based on drift-diffusion 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 drift-diffusion models with Gauss--Fermi statistics for organic semiconductors, Analysis and Applications, 19 (2021), pp. 275--304, DOI 10.1142/S0219530519500246 .
    Abstract
    This work is concerned with the analysis of a drift-diffusion model for the electrothermal behavior of organic semiconductor devices. A "generalized Van Roosbroeck” system coupled to the heat equation is employed, where the former consists of continuity equations for electrons and holes and a Poisson equation for the electrostatic potential, and the latter features source terms containing Joule heat contributions and recombination heat. Special features of organic semiconductors like Gauss--Fermi statistics and mobilities functions depending on the electric field strength are taken into account. We prove the existence of solutions for the stationary problem by an iteration scheme and Schauder's fixed point theorem. The underlying solution concept is related to weak solutions of the Van Roosbroeck system and entropy solutions of the heat equation. Additionally, for data compatible with thermodynamic equilibrium, the uniqueness of the solution is verified. It was recently shown that self-heating significantly influences the electronic properties of organic semiconductor devices. Therefore, modeling the coupled electric and thermal responses of organic semiconductors is essential for predicting the effects of temperature on the overall behavior of the device. This work puts the electrothermal drift-diffusion model for organic semiconductors on a sound analytical basis.

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

  • TH. Eiter, On the spatially asymptotic structure of time-periodic solutions to the Navier--Stokes equations, Proceedings of the American Mathematical Society, 149 (2021), pp. 3439--3451, DOI 10.1090/proc/15482 .
    Abstract
    The asymptotic behavior of weak time-periodic solutions to the Navier--Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.

  • TH. Eiter, G.P. Galdi, Spatial decay of the vorticity field of time-periodic viscous flow past a body, Archive for Rational Mechanics and Analysis, 242 (2021), pp. 149--178, DOI 10.1007/s00205-021-01690-z .
    Abstract
    We study the asymptotic spatial behavior of the vorticity field associated to a time-periodic Navier-Stokes flow past a body in the class of weak solutions satisfying a Serrin-like condition. We show that outside the wake region the vorticity field decays pointwise at an exponential rate, uniformly in time. Moreover, decomposing it into its time-average over a period and a so-called purely periodic part, we prove that inside the wake region, the time-average has the same algebraic decay as that known for the associated steady-state problem, whereas the purely periodic part decays even faster, uniformly in time. This implies, in particular, that “sufficiently far” from the body, the time-periodic vorticity field behaves like the vorticity field of the corresponding steady-state problem.

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

  • TH. Eiter, K. Hopf, A. Mielke, Leray--Hopf solutions to a viscoelastic fluid model with nonsmooth stress-strain relation, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 65 (2022), pp. 103491/1--103491/30 (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 Zaremba--Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time 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 Darwin--Howie--Whelan equations for the scattering of fast electrons described by the Schrödinger equation, SIAM Journal on Applied Mathematics, 81 (2021), pp. 1552--1578, DOI 10.1137/21M139164X .
    Abstract
    The Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrödinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the two-beam and the systematic-row approximation.

  • A. Mielke, Relating a rate-independent system and a gradient system for the case of one-homogeneous potentials, Journal of Dynamics and Differential Equations, 34 (2022), pp. 3143--3164 (published online on 31.05.2021), DOI 10.1007/s10884-021-10007-3 .
    Abstract
    We consider a non-negative and one-homogeneous energy functional $mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-inpendent system given in terms of the time-dependent functional $mathcal E(t,u)=t mathcal J(u)$ and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutins of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

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

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

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

Beiträge zu Sammelwerken

  • O. Marquardt, L. Geelhaar, O. Brandt, Wave-function engineering in In0.53Ga0.47As/InxAl1-xAs core/shell nanowires, in: 2021 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), IEEE Conference Publications Management Group, 2021, pp. 15--16, DOI 10.1109/NUSOD52207.2021.9541519 .
    Abstract
    We study the electronic properties of In 0.53 Ga 0.47 As/In x Al 1-x 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 1-x 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, Micro-geometry effects on the nonlinear effective yield strength response of magnetorheological fluids, in: Emerging Problems in the Homogenization of Partial Differential Equations, P. Donato, M. Luna-Laynez, eds., 10 of SEMA SIMAI Springer Series, Springer, Cham, 2021, pp. 1--16, DOI 10.1007/978-3-030-62030-1_1 .
    Abstract
    We use the novel constitutive model in [15], derived using the homogenization method, to investigate the effect particle chain microstructures have on the properties of the magnetorheological fluid. The model allows to compute the constitutive coefficients for different geometries. Different geometrical realizations of chains can significantly change the magnetorheological effect of the suspension. Numerical simulations suggest that particle size is also important as the increase of the overall particle surface area can lead to a decrease of the overall magnetorheological effect while keeping the volume fraction constant.

  • A. Stephan, EDP convergence for nonlinear fast-slow 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. 1456--1459, 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. 133--156, DOI 10.1007/978-3-030-90051-9_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 gradient-polyconvex 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 orientation-preserving and globally injective everywhere in the domain representing the specimen. Theoretical results are supported by three-dimensional 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 tight-binding 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. 135--136, DOI 10.1109/NUSOD52207.2021.9541461 .
    Abstract
    We present a multi-scale framework for calculating electronic and transport properties of nitride-based devices. Here, an atomistic tight-binding model is connected with continuum-based 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 continuum-based 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, uni-polar 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 energy-reaction-diffusion 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. 1418--1421, DOI 10.4171/OWR/2020/29 .

Preprints, Reports, Technical Reports

  • TH. Eiter, R. Lasarzik, Existence of energy-variational solutions to hyperbolic conservation laws, Preprint no. 2974, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2974 .
    Abstract, PDF (546 kByte)
    oduce the concept of energy-variational solutions for hyperbolic conservation laws. Intrinsically, these energy-variational solutions fulfill the weak-strong uniqueness principle and the semi-flow property, and the set of solutions is convex and weakly-star closed. The existence of energy-variational solutions is proven via a suitable time-discretization scheme under certain assumptions. This general result yields existence of energy-variational solutions to the magnetohydrodynamical equations for ideal incompressible fluids and to the Euler equations in both the incompressible and the compressible case. Moreover, we show that energy-variational solutions to the Euler equations coincide with dissipative weak solutions.

  • V. Laschos, A. Mielke, Evolutionary variational inequalities on the Hellinger--Kantorovich and spherical Hellinger--Kantorovich spaces, Preprint no. 2973, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2973 .
    Abstract, PDF (491 kByte)
    We study the minimizing movement scheme for families of geodesically semiconvex functionals defined on either the Hellinger--Kantorovich or the Spherical Hellinger--Kantorovich space. By exploiting some of the finer geometric properties of those spaces, we prove that the sequence of curves, which are produced by geodesically interpolating the points generated by the minimizing movement scheme, converges to curves that satisfy the Evolutionary Variational Inequality (EVI), when the time step goes to 0.

  • M. Mirahmadi, B. Friedrich, B. Schmidt, J. Pérez-Ríos, Mapping atomic trapping in an optical superlattice onto the libration of a planar rotor in electric fields, Preprint no. 2972, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2972 .
    Abstract, PDF (4889 kByte)
    We show that two seemingly unrelated problems -- the trapping of an atom in a one-dimensional optical superlattice (OSL) formed by the interference of optical lattices whose spatial periods differ by a factor of two, and the libration of a polar polarizable planar rotor (PR) in combined electric and optical fields -- have isomorphic Hamiltonians. Since the OSL gives rise to a periodic potential that acts on atomic translation via the AC Stark effect, it is possible to establish a map between the translations of atoms in this system and the rotations of the PR due to the coupling of the rotor's permanent and induced electric dipole moments with the external fields. The latter system belongs to the class of conditionally quasi-exactly solvable (C-QES) problems in quantum mechanics and shows intriguing spectral properties, such as avoided and genuine crossings, studied in details in our previous works [our works]. We make use of both the spectral characteristics and the quasi-exact solvability to treat ultracold atoms in an optical superlattice as a semifinite-gap system. The band structure of this system follows from the eigenenergies and their genuine and avoided crossings obtained as solutions of the Whittaker--Hill equation. Furthermore, the mapping makes it possible to establish correspondence between concepts developed for the two eigenproblems individually, such as localization on the one hand and orientation/alignment on the other. This correspondence may pave the way to unraveling the dynamics of the OSL system in analytic form.

  • P. Bella, M. Kniely, Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization, Preprint no. 2971, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2971 .
    Abstract, PDF (354 kByte)
    We consider degenerate elliptic equations of second order in divergence form with a symmetric random coefficient field a. Extending the work of the first author, Fehrman, and Otto [Ann. Appl. Probab. 28 (2018), no. 3, 1379-1422], who established the large-scale regularity of a-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius describing the minimal scale for this regularity. As an application to stochastic homogenization, we partially generalize results by Gloria, Neukamm, and Otto [Anal. PDE 14 (2021), no. 8, 2497-2537] on the growth of the corrector, the decay of its gradient, and a quantitative two-scale expansion to the degenerate setting. On a technical level, we demand the ensemble of coefficient fields to be stationary and subject to a spectral gap inequality, and we impose moment bounds on the coefficient field and its inverse. We also introduce the ellipticity radius, which encodes the minimal scale where these moments are close to their positive expectation value.

  • P.-É. Druet, K. Hopf, A. Jüngel, Hyperbolic-parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion, Preprint no. 2967, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2967 .
    Abstract, PDF (380 kByte)
    We investigate degenerate cross-diffusion equations with a rank-deficient diffusion matrix that are considered to model populations which move as to avoid spatial crowding and have recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of such equations which reveals their structure of a symmetric hyperbolic-parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in H^s(mathbbT^d) for s>d/2+1.

  • R. Haller, H. Meinlschmidt, J. Rehberg, Hölder regularity for domains of fractional powers of elliptic operators with mixed boundary conditions, Preprint no. 2959, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2959 .
    Abstract, PDF (330 kByte)
    This work is about global Hölder regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the realization of an elliptic differential operator in a negative Sobolev space with integrability q > d embeds into a space of Hölder continuous functions, then so do the domains of suitable fractional powers of this operator. The second main result then establishes that the premise of the first is indeed satisfied. The proof goes along the classical techniques of localization, transformation and reflection which allows to fall back to the classical results of Ladyzhenskaya or Kinderlehrer. One of the main features of our approach is that we do not require Lipschitz charts for the Dirichlet boundary part, but only an intriguing metric/measure-theoretic condition on the interface of Dirichlet- and Neumann boundary parts. A similar condition was posed in a related work by ter Elst and Rehberg in 2015 [10], but the present proof is much simpler, if only restricted to space dimension up to 4.

  • M. Liero, A. Mielke, G. Savaré, Fine properties of geodesics and geodesic lambda-convexity for the Hellinger--Kantorovich 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 Hellinger--Kantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton--Jacobi equation arising in the dual dynamic formulation of HK, which are sufficiently strong to construct a characteristic transport-dilation 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 lambda-convexity with respect to the Hellinger--Kantorovich distance.

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

  • A. Mielke, T. Roubíček, Qualitative study of a geodynamical rate-and-state 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 Dieterich--Ruina rate-and-state 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 one-dimensional model is investigated as far as the steady-state existence, localization of the cataclastic core, and its time response, too. Computational experiments with a damage-free variant show stick-slip 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 Cahn--Hilliard 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 Cahn--Hilliard 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 Cahn--Hilliard 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 second-order 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 double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary 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 first-order 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 so-called 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 so-called 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 first-order 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 first-order necessary conditions, thereby establishing meaningful first-order necessary optimality conditions also for the case of the double obstacle potential.

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

  • P. Colli, G. Gilardi, A. Signori, J. Sprekels, Cahn--Hilliard--Brinkman 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 Cahn--Hilliard--Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion 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 no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn--Hilliard-type 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 energy-like variable. The dissipation of the velocity-like 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 two-equation models for turbulence, where the energy-like variable is the mean turbulent kinetic energy. Because of the degeneracies there are solutions with time-dependent 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 self-diffusion of the energy-like variable or by dissipation of the velocity-like variable. The cross-over 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 self-similar behavior of the solutions in Rd 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 R-boundedness: Applications to the Navier--Stokes 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 operator-valued version of de Leeuw's transference principle, time-periodic Lp estimates of maximal regularity type are established from R-bounds of the family of solution operators (R-solvers) to the corresponding resolvent problems. With this method, existence of time-periodic solutions to the Navier--Stokes equations is shown for two configurations: in a periodically moving bounded domain and in an exterior domain, subject to prescribed time-periodic 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 limit-passage in the equation of state and the low Mach-number limit in rescaled PDEs are investigated. Using the relative energy inequality, we obtain convergence results for the densities and the velocity-field 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 single-component flows, uniform estimates and the convergence of the pressure are needed in the multicomponent case because the incompressible velocity field is not divergence-free. 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 cell-centered 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 cell-centered 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 semi-norm, 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 Neumann--boundary condition modeling the lithium intercalation reaction. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a Cahn--Hilliard equation, whereas the limit model consists of a diffusion and an Allen--Cahn equation. Thus we observe a Cahn--Hilliard to Allen--Cahn transition during the upscaling process. In the sense of gradient flows, the transition goes in hand with a change in the underlying metric structure of the PDE system.

  • A. Mielke, R. Rossi, Balanced-Viscosity solutions to infinite-dimensional multi-rate systems, Preprint no. 2902, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2902 .
    Abstract, PDF (778 kByte)
    We consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity 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 square-field 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 square-field (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 square-field 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 operator-theoretic normality condition, the sequence of energies is log-convex. In particular, this implies polynomial decay in time for the energy functionals along solutions.

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

  • M. Heida, S. Neukamm, M. Varga, Stochastic two-scale 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 two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.

  • 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 lambda-convexity.

  • 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.

  • M. Heida, M. Thomas, GENERIC for dissipative solids with bulk-interface 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 bulk-interface interaction and apply it to discuss the GENERIC structure of models for delamination processes.

  • 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 first-order 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 non-conserved order parameter) and the so-called 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 initial-boundary 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 control-to-state operator, and meaningful first-order 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 Fokker--Planck equations: A qualitative analysis, Preprint no. 2860, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2860 .
    Abstract, PDF (402 kByte)
    A class of nonlinear Fokker--Planck equations with superlinear drift is investigated in the L1-supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow 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 finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time 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 Kaniadakis--Quarati model for Bose--Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time 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 two-scale formalism by Zhikov and Piatnitsky (2006). To further demonstrate the benefits of our approach we provide some useful trace and extension operators for Sobolev functions on Ω, which seem not known in literature. On the way we close some minor gaps in the Sobolev theory on Ω which seemingly have not been proven up to date.

  • 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 W1,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 W1,p to W1,r with the strict inequality r<p. In particular, we estimate the Lr-norm of the extended gradient in terms of the Lp-norm of the original gradient. Similar relations hold for the symmetric gradients (for ℝd-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 non-appropriateness of this property. According to our mathematical treatment, the free energy as a function of temperature and partial masses tends to a limit in the sense of epi-- or Gamma--convergence. In the context of the first problem, we study the mixing of two fluids to compare the linearity with experimental observations. The second problem will be treated by considering the asymptotic behavior of both a general inequality relating thermal expansion and compressibility and a PDE-system relying on the equations of balance for partial masses, momentum and the internal energy.

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

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

Vorträge, Poster

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

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

  • S. Schindler, Convergence to self-similar profiles for a coupled reaction-diffusion system on the real line, CRC 910: Workshop on Control of Self-Organizing Nonlinear Systems, September 26 - 28, 2022.

  • S. Schindler, Energy approach for a coupled reaction-diffusion 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.

  • S. Schindler, Entropy method for a coupled reaction-diffusion system on the real line, Annual Workshop of the GAMM Activity Group ``Analysis of PDEs" 2022, October 5 - 7, 2022, Institute of Science and Technology Austria (ISTA), Klosterneuburg, October 5, 2022.

  • S. Schindler, On asymptotic self-similar behavior of solutions to parabolic systems (hybrid talk), SFB910: International Conference on Control of Self-Organizing Nonlinear Systems (Hybrid Event), November 23 - 26, 2022, Technische Universität Berlin, Potsdam, November 25, 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, Rheinisch-Westfälische Technische Hochschule Aachen, August 17, 2022.

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

  • M. Heida, Homogenization on locally Lipschitz random domains (online talk), SIAM Conference on Analysis of Partial Differential Equations, 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 Cahn--Hilliard to Allen--Cahn 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, Potsdam-Institut für Klimafolgenforschung (PIK), April 26, 2022.

  • O. Marquardt, SPHInX-Tutorial 2022 (Hybrid Event), March 14 - April 11, 2022, WIAS Berlin.

  • P. Pelech, Penrose--Fife 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, Penrose--Fife model as a gradient flow - interplay between signed measures and functionals on Sobolev spaces, Annual Workshop of the GAMM Activity Group ``Analysis of PDEs" 2022, October 5 - 7, 2022, Institute of Science and Technology Austria (ISTA), Klosterneuburg, October 7, 2022.

  • P. Pelech, Penrose--Fife 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, Rheinisch-Westfälische Technische Hochschule Aachen, August 18, 2022.

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

  • A. Stephan, EDP-convergence for a linear reaction-diffusion 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.

  • M. Kniely, Degenerate random elliptic operators: Regularity aspects and stochastic homogenization, Annual Workshop of the GAMM Activity Group ``Analysis of PDEs" 2022, October 5 - 7, 2022, Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria, October 6, 2022.

  • M. Kniely, Global renormalized solutions and equilibration of reaction-diffusion systems with nonlinear diffusion (online talk), SIAM Conference on Analysis of Partial Differential Equations, Minisymposium ``Bridging Gradient Flows, Hypocoercivity and Reaction-Diffusion Systems" (Online Event), March 14 - 18, 2022, Society for Industrial and Applied Mathematics, March 14, 2022.

  • M. Kniely, Global solutions to a class of energy-reaction-diffusion systems, Conference on Differential Equations and Their Applications (EQUADIFF 15), Minisymposium NAA-03 ``Evolution Differential Equations with Application to Physics and Biology II'', July 11 - 15, 2022, Masaryk University, Brno, Czech Republic, July 12, 2022.

  • A. Glitzky, A drift-diffusion based electrothermal model for organic thin-film 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 Multi-Scale Aspects" (Online Event), March 14 - 18, 2022, Society for Industrial and Applied Mathematics, March 16, 2022.

  • K. Hopf, The Cauchy problem for a cross-diffusion system with incomplete diffusion, Annual Workshop of the GAMM Activity Group ``Analysis of PDEs" 2022, October 5 - 7, 2022, Institute of Science and Technology Austria (ISTA), Klosterneuburg, October 5, 2022.

  • P.-É. Druet, Global existence and weak-strong uniqueness for isothermal ideal multicomponent flows, Against the flow, October 18 - 22, 2022, Polish Academy of Sciences, Będlewo, Poland, October 19, 2022.

  • TH. Eiter, Energy-variational 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 time-periodic viscous flow around a moving body, Richard von Mises Lecture 2022, Humboldt-Universität 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, Rheinisch-Westfälische 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 time-periodic 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 time-periodic Navier--Stokes 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 time-periodic rotating viscous flow, Mathematical Fluid Mechanics in 2022 (Hybrid Event), August 22 - 26, 2022, Czech Academy of Sciences, Prague, Czech Republic, August 24, 2022.

  • TH. Eiter, On uniformity of the resolvent estimates associated with time-periodic flow past a rotating body, Germany-Japan Workshop on Free and Singular Boundaries in Fluid Dynamics and Related Topics (Hybrid Event), August 10 - 12, 2022, Heinrich-Heine-Universität Düsseldorf, August 10, 2022.

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

  • TH. Eiter, The Navier--Stokes equations in domains with oscillating boundaries, Against the flow, October 18 - 22, 2022, Polish Academy of Sciences, Będlewo, Poland, October 20, 2022.

  • TH. Koprucki, K. Tabelow, HackMD (online talk), E-Coffee-Lecture (Online Event), WIAS Berlin, March 25, 2022.

  • TH. Koprucki, MaRDI - The Mathematical Research Data Initiative within the German National Research Data Infrastructure (NFDI), 1st MaRDI Workshop on Scientific Computing, October 26 - 28, 2022, Westfälische Wilhelms-Universität Münster, October 26, 2022.

  • M. Liero, Analysis of an electrothermal drift-diffusion 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), E-Coffee-Lecture (Online Event), WIAS Berlin, January 21, 2022.

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

  • M. Liero, From diffusion to reaction-diffusion in thin structures via EDP-convergence (online talk), SIAM Conference on Analysis of Partial Differential Equations, Minisymposium ``Bridging Gradient Flows, Hypocoercivity and Reaction-Diffusion 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.

  • M. Liero, Viscoelastodynamics of solids at large strains coupled to diffusion processes, Jahrestreffen des SPP 2256, September 28 - 30, 2022, Universität Regensburg, September 29, 2022.

  • A. Mielke, Convergence of a split-step scheme for gradient flows with a sum of two dual dissipation potentials, Nonlinear evolutionary equations and applications 2022, September 6 - 9, 2022, Technische Universität Chemnitz, September 8, 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 Hellinger--Kantorovich 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, Rheinisch-Westfälische Technische Hochschule Aachen, August 16, 2022.

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

  • A. 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.

  • A. Mielke, On time-splitting methods for gradient flows with two dissipation mechanisms, Annual Workshop of the GAMM Activity Group ``Analysis of PDEs" 2022, October 5 - 7, 2022, Institute of Science and Technology Austria (ISTA), Klosterneuburg, October 7, 2022.

  • J. Rehberg, Explicit Lp-estimates for second-order divergence operators, Oberseminar Analysis und Angewandte Mathematik, Universität Kassel, June 9, 2022.

  • W. van Oosterhout, Analysis of a poro-visco-elastic material model, Mathematical models for bio-medical sciences, Como, Italy, June 20 - 24, 2022.

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

  • A. Maltsi, On the Darwin--Howie--Whelan 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, Self-similar diffusive equilibration for a coupled reaction-diffusion system with mass-action kinetics, SFB910: International Conference on Control of Self-Organizing 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), PDI-Seminar, Paul-Drude-Institut für Festkörperelektronik, August 23, 2021.

  • O. Marquart, Wave-function 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 bulk-surface thermistor model for OLEDs, AMaSiS 2021: Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems (Online Event), September 6 - 9, 2021.

  • P. Pelech, Balanced-viscosity solution for a Penrose--Fife model with rate-independent 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 non-equilibrium reversible-irreversible 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 rate-independent systems - applications to large-strain deformations of damageable solids (online talk), 20th GAMM Seminar on Microstructures (Online Event), Technische Universität Wien, Austria, January 29, 2021.

  • P. Pelech, Separately global solutions to rate-independent systems - applications to large-strain 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 rate-independent systems -- Applications to large-strain 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 EDP-convergence with applications to nonlinear fast-slow 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 mulit-scale reaction networks (online talk), Limits and Control of Stochastic Reaction Networks (Online Event), July 26 - 30, 2021, American Institute of Mathematics, San Jose, USA, July 29, 2021.

  • A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction-diffusion systems (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S14 ``Applied Analysis'', March 15 - 19, 2021, Universität Kassel, March 17, 2021.

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

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

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

  • P.-E. Druet, Well-posedness results for mixed-type systems modelling pressure-driven 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), DMV-ÖMG 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), DMV-ÖMG 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 Real-world 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, Max-Planck-Institut 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 Reaction-Diffusion Equations (Online Event), July 13 - 16, 2021, Universität Wien, July 15, 2021.

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

  • A. Mielke, Thermo-visco-elasticity 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.

Preprints im Fremdverlag

  • V. Laschos, A. Mielke, Evolutionary variational inequalities on the Hellinger--Kantorovich and spherical Hellinger--Kantorovich 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 Hellinger--Kantorovich or the spherical Hellinger--Kantorovich 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.

  • M. O'Donovan, P. Farrell, J. Moatti, T. Streckenbach, Th. Koprucki, S. Schulz, Impact of random alloy fluctuations on the carrier distribution in multi-color (In,Ga)N/GaN quantum well systems, Preprint no. arXiv.2209.11657, Cornell University Library, arXiv.org, 2022, DOI 10.48550/arXiv.2209.11657 .
    Abstract
    In this work, we study the impact that random alloy fluctuations have on the distribution of electrons and holes across the active region of a (In,Ga)N/GaN multi-quantum well based light emitting diode (LED). To do so, an atomistic tight-binding model is employed to account for alloy fluctuations on a microscopic level and the resulting tight-binding energy landscape forms input to a drift-diffusion model. Here, quantum corrections are introduced via localization landscape theory and we show that when neglecting alloy disorder our theoretical framework yields results similar to commercial software packages that employ a self-consistent Schroedinger-Poisson-drift-diffusion solver. Similar to experimental studies in the literature, we have focused on a multi-quantum well system where two of the three wells have the same In content while the third well differs in In content. By changing the order of wells in this multicolor quantum well structure and looking at the relative radiative recombination rates of the different emitted wavelengths, we (i) gain insight into the distribution of carriers in such a system and (ii) can compare our findings to trends observed in experiment. Our results indicate that the distribution of carriers depends significantly on the treatment of the quantum well microstructure. When including random alloy fluctuations and quantum corrections in the simulations, the calculated trends in the relative radiative recombination rates as a function of the well ordering are consistent with previous experimental studies. The results from the widely employed virtual crystal approximation contradict the experimental data. Overall, our work highlights the importance of a careful and detailed theoretical description of the carrier transport in an (In,Ga)N/GaN multi-quantum well system to ultimately guide the design of the active region of III-N-based LED structures.

  • A. Mielke, T. Roubíček, Qualitative study of a geodynamical rate-and-state 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 Dieterich--Ruina rate-and-state 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 one-dimensional model is investigated as far as the steady-state existence, localization of the ataclastic core, and its time response, too. Computational experiments with a damage-free variant show stick-slip 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 uni-polar hole transport in (In,Ga)N quantum well systems, Preprint no. arXiv:2111.01644, Cornell University Library, arXiv.org, 2021.
    Abstract
    Understanding the impact of the alloy micro-structure on carrier transport becomes important when designing III-nitride-based LED structures. In this work, we study the impact of alloy fluctuations on the hole carrier transport in (In,Ga)N single and multi-quantum well systems. To disentangle hole transport from electron transport and carrier recombination processes, we focus our attention on uni-polar (p-i-p) systems. The calculations employ our recently established multi-scale simulation framework that connects atomistic tight-binding theory with a macroscale drift-diffusion model. In addition to alloy fluctuations, we pay special attention to the impact of quantum corrections on hole transport. Our calculations indicate that results from a virtual crystal approximation present an upper limit for the hole transport in a p-i-p structure in terms of the current-voltage characteristics. Thus we find that alloy fluctuations can have a detrimental effect on hole transport in (In,Ga)N quantum well systems, in contrast to uni-polar electron transport. However, our studies also reveal that the magnitude by which the random alloy results deviate from virtual crystal approximation data depends on several factors, e.g. how quantum corrections are treated in the transport calculations.