Partial Differential Equations

Publications

Articles in Refereed Journals

• 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, New Journal of Physics, 25 (2023), pp. 023024/1--023024/16, DOI 10.1088/1367-2630/acbab6 .
  Abstract
  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. 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, Discrete and Continuous Dynamical Systems -- Series S, published online in January 2023, DOI 10.3934/dcdss.2022210 .
  Abstract
  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.

• J. Riedel, P. Gelss, R. Klein, B. Schmidt, WaveTrain: A Python package for numerical quantum mechanics of chain-like systems based on tensor trains, Journal of Chemical Physics, 158 (2023), pp. 164801/1--164801/15, DOI 10.1063/5.0147314 .
  Abstract
  WaveTrain is an open-source software for numerical simulations of chain-like quantum systems with nearest-neighbor (NN) interactions only. The Python package is centered around tensor train (TT, or matrix product) format representations of Hamiltonian operators and (stationary or time-evolving) state vectors. It builds on the Python tensor train toolbox Scikit_tt, which provides efficient construction methods and storage schemes for the TT format. Its solvers for eigenvalue problems and linear differential equations are used in WaveTrain for the time-independent and time-dependent Schrödinger equations, respectively. Employing efficient decompositions to construct low-rank representations, the tensor-train ranks of state vectors are often found to depend only marginally on the chain length N. This results in the computational effort growing only slightly more than linearly with N, thus mitigating the curse of dimensionality. As a complement to the classes for full quantum mechanics, WaveTrain also contains classes for fully classical and mixed quantum-classical (Ehrenfest or mean field) dynamics of bipartite systems. The graphical capabilities allow visualization of quantum dynamics on the fly, with a choice of several different representations based on reduced density matrices. Even though developed for treating quasi one-dimensional excitonic energy transport in molecular solids or conjugated organic polymers, including coupling to phonons, WaveTrain can be used for any kind of chain-like quantum systems, with or without periodic boundary conditions, and with NN interactions only. The present work describes version 1.0 of our WaveTrain software, based on version 1.2 of scikit_tt, both of which are freely available from the GitHub platform where they will also be further developed. Moreover, WaveTrain is mirrored at SourceForge, within the framework of the WavePacket project for numerical quantum dynamics. Worked-out demonstration examples with complete input and output, including animated graphics, are available.

• TH. Eiter, M. Kyed, Y. Shibata, Falling drop in an unbounded liquid reservoir: Steady-state solutions, Journal of Mathematical Fluid Mechanics, 25 (2023), pp. 34/1--34/34, DOI 10.1007/s00021-023-00777-9 .
  Abstract
  The equations governing the motion of a three-dimensional liquid drop moving freely in an unbounded liquid reservoir under the influence of a gravitational force are investigated. Provided the (constant) densities in the two liquids are sufficiently close, existence of a steady-state solution is shown. The proof is based on a suitable linearization of the equations. A setting of function spaces is introduced in which the corresponding linear operator acts as a homeomorphism.

• A. Mielke, R. Rossi, Balanced-viscosity solutions to infinite-dimensional multi-rate systems, Archive for Rational Mechanics and Analysis, 247 (2023), pp. 53/1--53/100, DOI 10.1007/s00205-023-01855-y .
  Abstract
  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. Mielke, On two coupled degenerate parabolic equations motivated by thermodynamics, Journal of Nonlinear Science, 33 (2023), pp. 42/1--42/55, DOI 10.1007/s00332-023-09892-3 .
  Abstract
  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 R^d for large times.

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

• 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.

• D. Bothe, W. Dreyer, P.-É. Druet, Multicomponent incompressible fluids -- An asymptotic study, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, published online on 14.01.2022, DOI 10.1002/zamm.202100174 .
  Abstract
  This paper investigates the asymptotic behavior of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy that is valid in stable subregions of the phase diagram. On the basis of this representation we classify the admissible data to construct a thermodynamically consistent constitutive model. We then analyze the incompressible limit, where the molar volume becomes independent of pressure. Here we are confronted with two problems:

  (i) Our study shows that the physical system at hand cannot remain incompressible for arbitrary large deviations from a reference pressure unless its volume is linear in the composition. (ii) As a consequence of the 2nd law of thermodynamics, the incompressible limit implies that the molar volume becomes independent of temperature as well. Most applications, however, reveal the non-appropriateness of this property. According to our mathematical treatment, the free energy as a function of temperature and partial masses tends to a limit in the sense of epi-- or Gamma--convergence. In the context of the first problem, we study the mixing of two fluids to compare the linearity with experimental observations. The second problem will be treated by considering the asymptotic behavior of both a general inequality relating thermal expansion and compressibility and a PDE-system relying on the equations of balance for partial masses, momentum and the internal energy.

• 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 R^d, 1 ≤ d ≤ 3 with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case d = 2. In the rest of the work, we study the necessary 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, Optimal control of a phase field system of Caginalp type with fractional operators, Pure and Applied Functional Analysis, 7 (2022), pp. 1597--1635.
  Abstract
  In their recent work “Well-posedness, regularity and asymptotic analyses for a fractional phase field system” (Asymptot. Anal. 114 (2019), 93--128), two of the present authors have studied phase field systems of Caginalp type, which model nonconserved, nonisothermal phase transitions and in which the occurring diffusional operators are given by fractional versions in the spectral sense of unbounded, monotone, selfadjoint, linear operators having compact resolvents. In this paper, we complement this analysis by investigating distributed optimal control problems for such systems. It is shown that the associated control-to-state operator is Fréchet differentiable between suitable Banach spaces, and meaningful first-order necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables.

• 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, Analysis and optimal control theory for a phase field model of Caginalp type with thermal memory, Communications in Optimization Theory, 2022 (2022), pp. 4/1--4/31, DOI 10.23952/cot.2022.4 .
  Abstract
  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.

• 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, 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, 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 O^2- 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.

• D.R.M. Renger, S. Schindler, Gradient flows for bounded linear evolution equations, Zeitschrift fur Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 41 (2022), pp. 229--238, DOI 10.4171/ZAA/1706 .
  Abstract
  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.

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

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

• 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, 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.

• TH. Eiter, On the regularity of weak solutions to time-periodic Navier--Stokes equations in exterior domains, Mathematics - Open Access Journal, 11 (2023), pp. 141/1--141/17 (published online on 27.12.2022), DOI 10.3390/math11010141 .
  Abstract
  Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities.

• TH. Koprucki, A. Maltsi, A. Mielke, Symmetries in transmission electron microscopy imaging of crystals with strain, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 478 (2022), pp. 20220317/1--20220317/23, DOI 10.1098/rspa.2022.0317 .
  Abstract
  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, S. Reichelt, Traveling fronts in a reaction-diffusion equation with a memory term, Journal of Dynamics and Differential Equations, 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. e202000019/1--e202000019/31, 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.

Contributions to Collected Editions

• Y. Hadjimichael, O. Marquardt, Ch. Merdon, P. Farrell, Band structures in highly strained 3D nanowires, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2022), Turin, Italy, 2022, J. Piprek, Bardella Paolo, eds., IEEE, 2022, pp. 119--120, DOI 10.1109/NUSOD54938.2022.9894837 .

• M. Heida, M. Thomas, GENERIC for dissipative solids with bulk-interface interaction, in: Research in the Mathematics of Materials Science, A. Schlömerkemper, ed., 31 of Association for Women in Mathematics Series, Springer, Cham, 2022, pp. 333--364, DOI 10.1007/978-3-031-04496-0_15 .
  Abstract
  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.

• O. Marquardt, M. O'Donovan, S. Schulz, O. Brandt, Th. Koprucki, Influence of random alloy fluctuations on the electronic properties of axial (In,Ga)N/GaN nanowire heterostructures, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 117--118, DOI 10.1109/NUSOD54938.2022.9894777 .
  Abstract
  Compound semiconductor heterostructures such as quantum dots, nanowires, or thin films, are commonly subject to randomly fluctuating alloy compositions if they contain ternary and quaternary alloys. These effects are obviously of an atomistic nature and thus rarely considered in heterostructure designs that require simulations on a continuum level for theory-guided design or interpretation of observations. In the following, we present a systematic approach to the treatment of alloy fluctuations in (In,Ga)N/GaN thin films and axial nanowire heterostructures. We demonstrate to what extent random alloy fluctuations can be treated in a continuum picture and discuss the impact of alloy fluctuations on the electronic properties of planar and nano wire-based (In,Ga)N/GaN heterostructures.

• M. O'Donovan, P. Farrell, T. Streckenbach, Th. Koprucki, S. Schulz, Carrier transport in (In,Ga)N quantum well systems: Connecting atomistic tight-binding electronic structure theory to drift-diffusion simulations, in: 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), Turin, Italy, 2022, J. Piprek, P. Bardella, eds., IEEE, 2022, pp. 97--98, DOI 10.1109/NUSOD54938.2022.9894745 .

Preprints, Reports, Technical Reports

• J. Sprekels, F. Tröltzsch, Second-order sufficient conditions in the sparse optimal control of a phase field tumor growth model with logarithmic potential, Preprint no. 3020, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3020 .
  Abstract, PDF (344 kByte)
  his paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn--Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term in order to enhance the occurrence of sparsity effects in the optimal controls, i.e., of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions have been established for the case without sparsity, the case with sparsity has not been treated before. The results obtained in this paper also improve the known results on this phase field model for the case without sparsity.

• M. Heida, Finite volumes for simulation of large molecules, Preprint no. 3018, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3018 .
  Abstract, PDF (230 kByte)
  We study a finite volume scheme for simulating the evolution of large molecules within their reduced state space. The finite volume scheme under consideration is the SQRA scheme developed by Lie, Weber and Fackeldey. We study convergence of a more general family of FV schemes in up to 3 dimensions and provide a convergence result for the SQRA-scheme in arbitrary space dimensions.

• Y. Bokredenghel, M. Heida, Quenched homogenization of infinite range random conductance model on stationary point processes, Preprint no. 3017, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3017 .
  Abstract, PDF (388 kByte)
  We prove homogenization for elliptic long-range operators in the random conductance model on random stationary point processes in d dimensions with Dirichlet boundary conditions and with a jointly stationary coefficient field. Doing so, we identify 4 conditions on the point process and the coefficient field that have to be fulfilled at different stages of the proof in order to pass to the homogenization limit. The conditions can be clearly attributed to concentration of support, Rellich--Poincaré inequality, non-degeneracy of the homogenized matrix and ergodicity of the elliptic operator.

• A. Mielke, T. Roubíček, U. Stefanelli, A model of gravitational differentiation of compressible self-gravitating planets, Preprint no. 3015, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3015 .
  Abstract, PDF (444 kByte)
  We present a dynamic model for inhomogeneous viscoelastic media at finite strains. The model features a Kelvin--Voigt rheology, and includes a self-generated gravitational field in the actual evolving configuration. In particular, a fully Eulerian approach is adopted. We specialize the model to viscoelastic (barotropic) fluids and prove existence and a certain regularity of global weak solutions by a Faedo--Galerkin semi-discretization technique. Then, an extension to multi-component chemically reacting viscoelastic fluids based on a phenomenological approach by Eckart and Prigogine, is advanced and studied. The model is inspired by planetary geophysics. In particular, it describes gravitational differentiation of inhomogeneous planets and moons, possibly undergoing volumetric phase transitions.

• A. Glitzky, M. Liero, A drift-diffusion based electrothermal model for organic thin-film devices including electrical and thermal environment, Preprint no. 3012, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3012 .
  Abstract, PDF (463 kByte)
  We derive and investigate a stationary model for the electrothermal behavior of organic thin-film devices including their electrical and thermal environment. Whereas the electrodes are modeled by Ohm's law, the electronics of the organic device itself is described by a generalized van Roosbroeck system with temperature dependent mobilities and using Gauss--Fermi integrals for the statistical relation. The currents give rise to Joule heat which together with the heat generated by the generation/recombination of electrons and holes in the organic device occur as source terms in the heat flow equation that has to be considered on the whole domain. The crucial task is to establish that the quantities in the transfer conditions at the interfaces between electrodes and the organic semiconductor device have sufficient regularity. Therefore, we restrict the analytical treatment of the system to two spatial dimensions. We consider layered organic structures, where the physical parameters (total densities of transport states, LUMO and HOMO energies, disorder parameter, basic mobilities, activation energies, relative dielectric permittivity, heat conductivity) are piecewise constant. We prove the existence of weak solutions using Schauder's fixed point theorem and a regularity result for strongly coupled systems with nonsmooth data and mixed boundary conditions that is verified by Caccioppoli estimates and a Gehring-type lemma.

• A. Mielke, S. Schindler, Convergence to self-similar profiles in reaction-diffusion systems, Preprint no. 3008, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3008 .
  Abstract, PDF (380 kByte)
  We study a reaction-diffusion system on the real line, where the reactions of the species are given by one reversible reaction pair satisfying the mass-action law. We describe different positive limits at both sides of infinityand investigate the long-time behavior. Rescaling space and time according to the parabolic scaling, we show that solutions converge exponentially to a similarity profile when the scaled time goes to infinity. In the original variables, these profiles correspond to asymptotically self-similar behavior describing the phenomenon of diffusive mixing of the different states at infinity.Our method provides global exponential convergence for all initial states with finite relative entropy. For the case with equal stoichiometric coefficients, we can allow for self-similar profiles with arbitrary equilibrated states,while in the other case we need to assume that the two states atinfinity are sufficiently close such that the self-similar profile is relative flat.

• A. Mielke, S. Schindler, Existence of similarity profiles for diffusion equations and systems, Preprint no. 3007, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3007 .
  Abstract, PDF (403 kByte)
  We study the existence of self-similar profiles for diffusion equations and reaction-diffusion systems on the real line, where different nontrivial limits are imposed at both sides of infinity. The theses profiles solve a coupled system of nonlinear ODEs that can be treated by monotone operator theory.

• K. Hopf, A. Jüngel, Convergence of a finite volume scheme and dissipative measure-valued--strong stability for a hyperbolic-parabolic cross-diffusion system, Preprint no. 3006, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3006 .
  Abstract, PDF (444 kByte)
  This article is concerned with the approximation of hyperbolic-parabolic cross-diffusion systems modeling segregation phenomena for populations by a fully discrete finite-volume scheme. It is proved that the numerical scheme converges to a dissipative measure-valued solution of the PDE system and that, whenever the latter possesses a strong solution, the convergence holds in the strong sense. Furthermore, the “parabolic density part” of the limiting measure-valued solution is atomic and converges to its constant state for long times. The results are based on Young measure theory and a weak-strong stability estimate combining Shannon and Rao entropies. The convergence of the numerical scheme is achieved by means of discrete entropy dissipation inequalities and an artificial diffusion, which vanishes in the continuum limit.

• J. Sprekels, F. Tröltzsch, Second-order sufficient conditions for sparse optimal control of singular Allen--Cahn systems with dynamic boundary conditions, Preprint no. 3005, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3005 .
  Abstract, PDF (385 kByte)
  In this paper we study the optimal control of a parabolic initial-boundary value problem of Allen--Cahn type with dynamic boundary conditions. Phase field systems of this type govern the evolution of coupled diffusive phase transition processes with nonconserved order parameters that occur in a container and on its surface, respectively. It is assumed that the nonlinear functions driving the physical processes within the bulk and on the surface are double well potentials of logarithmic type whose derivatives become singular at the boundary of their respective domains of definition. For such systems, optimal control problems have been studied in the past. We focus here on the situation when the cost functional of the optimal control problem contains a nondifferentiable term like the L¹-norm leading to sparsity of optimal controls. For such cases, we derive second-order sufficient conditions for locally optimal controls.

• P. Colli, G. Gilardi, A. Signori, J. Sprekels, Optimal temperature distribution for a nonisothermal Cahn--Hilliard system in two dimensions with source term and double obstacle potential, Preprint no. 3003, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3003 .
  Abstract, PDF (312 kByte)
  In this note, we study the optimal control of a nonisothermal phase field system of Cahn--Hilliard type that 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 further mathematical difficulties because the mass conservation of the order parameter is no longer satisfied. In this paper, we study the case that the double-well potential driving the evolution of the phase transition is given by the nondifferentiable double obstacle potential, thereby complementing recent results obtained for the differentiable cases of regular and logarithmic potentials. Besides existence results, we derive first-order necessary optimality conditions for the control problem. The analysis is carried out by employing the so-called deep quench approximation in which the nondifferentiable double obstacle potential is approximated by a family of potentials of logarithmic structure for which meaningful first-order necessary optimality conditions in terms of suitable adjoint systems and variational inequalities are available. Since the results for the logarithmic potentials crucially depend on the validity of the so-called strict separation property which is only available in the spatially two-dimensional situation, our whole analysis is restricted to the two-dimensional case.

• TH. Eiter, Y. Shibata, Viscous flow past a translating body with oscillating boundary, Preprint no. 3000, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3000 .
  Abstract, PDF (326 kByte)
  We study an incompressible viscous flow around an obstacle with an oscillating boundary that moves by a translational periodic motion, and we show existence of strong time-periodic solutions for small data in different configurations: If the mean velocity of the body is zero, existence of time-periodic solutions is provided within a framework of Sobolev functions with isotropic pointwise decay. If the mean velocity is non-zero, this framework can be adapted, but the spatial behavior of flow requires a setting of anisotropically weighted spaces. In the latter case, we also establish existence of solutions within an alternative framework of homogeneous Sobolev spaces. These results are based on the time-periodic maximal regularity of the associated linearizations, which is derived from suitable R-bounds for the Stokes and Oseen resolvent problems. The pointwise estimates are deduced from the associated time-periodic fundamental solutions.

• A. Mielke, Non-equilibrium steady states as saddle points and EDP-convergence for slow-fast gradient systems, Preprint no. 2998, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2998 .
  Abstract, PDF (444 kByte)
  The theory of slow-fast gradient systems leads in a natural way to non-equilibrium steady states, because on the slow time scale the fast subsystem stays in steady states that are driven by the interaction with the slow system. Using the theory of convergence of gradient systems in the sense of the energy-dissipation principle shows that there is a natural characterization of these non-equilibrium steady states as saddle points of a Lagrangian where the slow variables are fixed. We give applications to slow-fast reaction-diffusion systems based on the so-called cosh-type gradient structure for reactions. It is shown that two binary reaction give rise to a ternary reaction with a state-dependent reaction coefficient. Moreover, we show that a reaction-diffusion equation with a thin membrane-like layer convergences to a transmission condition, where the formerly quadratic dissipation potential for diffusion convergences to a cosh-type dissipation potential for the transmission in the membrane limit.

• P. Colli, G. Gilardi, A. Signori, J. Sprekels, Optimal temperature distribution for a nonisothermal Cahn--Hilliard system with source term, Preprint no. 2997, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2997 .
  Abstract, PDF (336 kByte)
  In this note, we study the optimal control of a nonisothermal phase field system of Cahn--Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system 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, typical of the classic Cahn--Hilliard equation, is no longer satisfied. In this paper, we analyze the case that the double-well potential driving the evolution of the phase transition is differentiable, either (in the regular case) on the whole set of reals or (in the singular logarithmic case) on a finite open interval; nondifferentiable cases like the double obstacle potential are excluded from the analysis. We prove the Fréchet differentiability of the control-to-state operator between suitable Banach spaces for both the regular and the logarithmic cases and establish the solvability of the corresponding adjoint systems in order to derive the associated first-order necessary optimality conditions for the optimal control problem. Crucial for the whole analysis to work is the so-called “strict separation property”, which states that the order parameter attains its values in a compact subset of the interior of the effective domain of the nonlinearity. While this separation property turns out to be generally valid for regular potentials in three dimensions of space, it can be shown for the logarithmic case only in two dimensions.

• P. Gelss, S. Matera, R. Klein, B. Schmidt, Quantum dynamics of coupled excitons and phonons in chain-like systems: Tensor train approaches and higher-order propagators, Preprint no. 2995, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2995 .
  Abstract, PDF (546 kByte)
  We investigate the use of tensor-train approaches to the solution of the time-dependent Schrödinger equation for chain-like quantum systems with on-site and nearest-neighbor interactions only. Using the efficient SLIM representation for low-rank tensor train representations of quantum-mechanical Hamiltonians, we aim at reducing the memory consumption as well as the computation costs, in order to mitigate the curse of dimensionality as much as possible. As an example, coupled excitons and phonons modeled in terms of Fröhlich--Holstein type Hamiltonians are studied here. By comparing our tensor-train based results with semi-analytical results, we demonstrate the key role of the ranks of tensor-train representations for quantum state vectors. Both the computational effort of the propagations and the accuracy that can be reached crucially depend on the maximum number of ranks chosen. Typically, an excellent quality of the solutions is found only when the ranks exceeds a certain value. That threshold, however, is very different for excitons, phonons, and coupled systems. One class of propagation schemes used in the present work builds on splitting the Hamiltonian into two groups of interleaved nearest-neighbor interactions which commutate within each of the groups. In addition to the first order Lie--Trotter and the second order Strang--Marchuk splitting schemes, we have also implemented the 4-th order Yoshida--Neri and the 8-th order Kahan--Li symplectic compositions. Especially the latter two are demonstrated to yield very accurate results, close to machine precision. However, due to the computational costs, currently their use is restricted to rather short chains. Another class of propagators involves explicit, time-symmetrized Euler integrators. Building on the traditional second order differencing method, we have also implemented higher order methods. Especially the 4-th order variant is recommended for quantum simulations of longer chains, even though the high precision of the splitting schemes cannot be reached. Moreover, the scaling of the computational effort with the dimensions of the local Hilbert spaces is much more favorable for the differencing than for the splitting schemes.

• A. Mielke, S. Schindler, Self-similar pattern in coupled parabolic systems as non-equilibrium steady states, Preprint no. 2992, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2992 .
  Abstract, PDF (370 kByte)
  We consider reaction-diffusion systems and other related dissipative systems on unbounded domains which would have a Liapunov function (and gradient structure) when posed on a finite domain. In this situation, the system may reach local equilibrium on a rather fast time scale but the infinite amount of mass or energy leads to persistent mass or energy flow for all times. In suitably rescaled variables the system converges to a steady state that corresponds to asymptotically self-similar behavior in the original system.

• 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.

• 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.

• 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.

• 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.

Talks, Poster

• A. Maltsi, Symmetries in TEM imaging of semiconductor nanostructures with strain, 15th Annual Meeting Photonic Devices, March 29 - 31, 2023, Zuse-Institut Berlin, March 31, 2023.

• A. Maltsi, Symmetries in TEM imaging of semiconductor nanostructures with strain, Leibniz MMS Days 2023, April 17 - 19, 2023, Leibniz-Institut für Agrartechnik und Bioökonomie (ATB), Potsdam, April 18, 2023.

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

• M. O'Donovan, Modeling random alloy fluctuations in carrier transport simulations of III-N based light emitting diodes - Connecting atomistic tight-binding to drift-diffusion, 15th Annual Meeting Photonic Devices, March 29 - 31, 2023, Zuse-Institut Berlin, March 31, 2023.

• M. O'Donovan, Tight binding simulations of localization in alloy fluctuations in nitride based LEDs, Seminar zu Physik der Gruppe III-Nitrid-Halbleiter und nanophotonischer Bauelemente und Advanced III-Nitride Materials and Photonic Devices (IIIN-MPD), Technische Universität Berlin, AG Experimentelle Nanophysik und Photonik, May 17, 2023.

• L. Schütz, An existence theory for solitary waves on a ferrofluid jet, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session 14 ``Applied Analysis'', May 30 - June 2, 2023, Technische Universität Dresden, May 30, 2023.

• M. Kniely, On a thermodynamically consistent electro-energy-reaction-diffusion system, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session 14 ``Applied Analysis'', May 30 - June 2, 2023, Technische Universität Dresden, June 1, 2023.

• J. Rehberg, Maximal parabolic regularity for the treatment of real world problems, Oberseminar für Optimale Steuerung und Inverse Probleme, Universität Duisburg-Essen, Fakultät für Mathematik, May 4, 2023.

• J. Sprekels, Sparse optimal control of singular Allen--Cahn systems with dynamic boundary conditions, Kolloquium, Università di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, April 18, 2023.

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

• K. Hopf, The Cauchy problem for multi-component systems with strong cross-diffusion, Johannes Gutenberg-Universität Mainz, Fachbereich Physik, Mathematik und Informatik, January 11, 2023.

• TH. Eiter, R. Lasarzik, Analysis of energy-variational solutions for hyperbolic conservation laws, Presentation of project proposals in SPP 2410 ``Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness'', Bad Honnef, April 28, 2023.

• TH. Eiter, Energy-variational solutions for a class of hyperbolic conservation laws, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session 14 ``Applied Analysis'', May 30 - June 2, 2023, Technische Universität Dresden, June 2, 2023.

• TH. Eiter, The concept of energy-variational solutions for hyperbolic conservation laws, Seminar on Partial Differential Equations, Czech Academy of Sciences, Institute of Mathematics, Prague, Czech Republic, March 28, 2023.

• TH. Koprucki, Building research data services for the community, Leibniz MMS Days 2023, April 17 - 19, 2023, Leibniz-Institut für Agrartechnik und Bioökonomie (ATB), Potsdam, April 17, 2023.

• TH. Koprucki, MaRDI - The Mathematical Research Data Initiative within the German National Research Data Infrastructure (NFDI), SIAM Conference on Computational Science and Engineering, Minisymposium 301 ``Interfaces, Workflows, and Knowledge Graphs for FAIR CSE'', February 27 - March 3, 2023, Society for Industrial and Applied Mathematics, Amsterdam, Netherlands, March 2, 2023.

• TH. Koprucki, MaRDI - The Mathematical Research Data Initiative within the German National Research Data Infrastructure (NFDI), Kolloquium der AG ``Modellierung, Numerik, Differentialgleichungen'', Technische Universität Berlin, January 31, 2023.

• M. Liero, Analysis for thermo-mechanical models with internal variables, Presentation of project proposals in SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, March 27, 2023.

• M. Liero, Balanced-viscosity solutions for a Penrose--Fife phasefield model with friction, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', May 30 - June 2, 2023, Technische Universität Dresden, June 1, 2023.

• M. Liero, Luminance inhomogeneities in large-area OLEDs due to electrothermal feedback, Hybride Optoelektronische Materialsysteme (HYD Seminar), Integrative Research Institute for the Sciences (IRIS Adlershof), Hybrid Devices Group, Berlin, April 20, 2023.

• M. Liero, On the geometry of the Hellinger--Kantorovich space (hybrid talk), Seminar ``Modern Methods in Applied Stochastics and Nonparametric Statistics'', WIAS Berlin, January 31, 2023.

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

• A. Mielke, EDP-convergence for gradient systems and Non-Equilibrium Steady States, Nonlinear Diffusion and nonlocal Interaction Models -- Entropies, Complexity, and Multi-Scale Structures, May 28 - June 2, 2023, Universidad de Granada, Spain, May 30, 2023.

• A. Mielke, On time-splitting methods for gradient flows with two dissipation mechanisms, Non-equilibrium steady states and EDP-convergence for slow-fast gradient systems, April 24 - 28, 2023, Westfälische Wilhelms-Universität Münster, April 25, 2023.

• A. Mielke, Viscoelastic fluid models for geodynamic processes in the lithosphere, ``SPP Meets TP'' Workshop: Variational Methods for Complex Phenomena in Solids, February 21 - 24, 2023, Universität Bonn, Hausdorff Institute for Mathematics, February 24, 2023.

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

• A. Stephan, On time-splitting methods for gradient flows with two dissipation mechanisms, In search of model structures for non-equilibrium systems, April 24 - 28, 2023, Westfälische Wilhelms-Universität Münster, April 28, 2023.

• W. van Oosterhout, Analysis of poro-visco-elastic solids at finite strains, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session 14 ``Applied Analysis'', May 30 - June 2, 2023, Technische Universität Dresden, June 2, 2023.

• 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 (PD22) (Online Event), Minisymposium ``Disordered Media and Homogenization'', March 14 - 18, 2022, March 15, 2022.

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

• M. Heida, Measure theoretic aspects of stochastic homogenization, Seminar Interacting Random Systems (Hybrid Event), WIAS Berlin, April 20, 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, Balanced-viscosity solutions for a Penrose--Fife model with rate-independent friction (hybrid talk), Oberseminar ``Mathematik in den Naturwissenschaften'', Julius-Maximilians-Universität Würzburg, December 8, 2022.

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

• 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 (PD22) (Online Event), Minisymposium ``Bridging Gradient Flows, Hypocoercivity and Reaction-Diffusion Systems'', March 14 - 18, 2022, 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'', July 11 - 15, 2022, Masaryk University, Brno, Czech Republic, July 12, 2022.

• J. Sprekels, Deep quench approach and sparsity in the optimal control of a phase field model for tumor growth, PHAse field MEthods in applied sciences (PHAME 2022), May 23 - 27, 2022, Istituto Nazionale di Alta Matematica, Rome, Italy, May 27, 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, Relative entropies and stability in strongly coupled parabolic systems (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium ``Variational Evolution: Analysis and Multi-Scale Aspects'', March 14 - 18, 2022, 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 (PD22) (Online Event), Minisymposium ``Generalized Solvability Concepts for Evolutionary PDEs and their Properties'', March 14 - 18, 2022, March 16, 2022.

• TH. Eiter, Existence of time-periodic flows in domains with oscillating boundaries, International Workshop on Multiphase Flows: Analysis, Modelling and Numerics, December 5 - 9, 2022, Waseda University, Tokyo, Japan, December 6, 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, 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. Eiter, Time-periodic maximal Lp regularity by R-boundedness in the context of incompressible viscous flows, Research Seminar Function Spaces, Friedrich-Schiller-Universität Jena, November 4, 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. Landstorfer, A. Selahi, M. Heida, M. Eigel, Recovery of battery ageing dynamics with multiple timescales, MATH+-Day 2022, Technische Universität Berlin, November 18, 2022.

• M. Liero, 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, Rome, 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 (PD22) (Online Event), Minisymposium ``Bridging Gradient Flows, Hypocoercivity and Reaction-Diffusion Systems'', March 14 - 18, 2022, March 14, 2022.

• M. Liero, Modeling, analysis, and simulation of electrothermal feedback in organic devices, Audit 2022, September 22 - 23, 2022, Weierstraß-Institut Berlin, September 22, 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 (PD22) (Online Event), Minisymposium MS43: ``Nonlinear Parabolic Equations and Systems'', March 14 - 18, 2022, 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.

• J. Rehberg, On non-autonomous and quasilinear parabolic equations, Oberseminar AG Analysis, Technische Universität Darmstadt, December 8, 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 gradient systems and applications to fast-slow chemical reaction systems, Block Course ``Multiscale Problems and Homogenization'' at Freie Universität Berlin from Nov. 10 to Dec. 15, 2022, Berlin Mathematical School & Berlin Mathematics Research Center MATH+, November 24, 2022.

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

External Preprints

• R. Finn, M. O'Donovan, P. Farrell, J. Moatti, T. Streckenbach, Th. Koprucki, S. Schulz, Theoretical study of the impact of alloy disorder on carrier transport and recombination processes in deep UV (Al, Ga)N light emitters, Preprint no. hal-04037215, Hyper Articles en Ligne (HAL), 2023.
  Abstract
  Aluminium gallium nitride ((Al,Ga)N) has gained significant attention in recent years due to its potential for highly efficient light emitters operating in the deep ultra-violet (UV) range (< 280 nm). However, given that current devices exhibit extremely low efficiencies, understanding the fundamental properties of (Al,Ga)N-based systems is of key importance. Here, using a multi-scale simulation framework, we study the impact of alloy disorder on carrier transport, radiative and non-radiative recombination processes in a c-plane Al0.7Ga0.3N/Al0.8Ga0.2N quantum well embedded in a p-i-n junction. Our calculations reveal that alloy fluctuations can open "percolative" pathways that promote transport for the electrons and holes into the quantum well region. Such an effect is neglected in conventional, and widely used transport simulations. Moreover, we find also that the resulting increased carrier density and alloy induced carrier localization effects significantly increase non-radiative Auger-Meitner recombination in comparison to the radiative process. Thus, to avoid such non-radiative process and potentially related material degradation, a careful design (wider well, multi quantum wells) of the active region is required to improve the efficiency of deep UV light emitters.

• P. Farrell, J. Moatti, M. O'Donovan, S. Schulz, Th. Koprucki, Importance of satisfying thermodynamic consistency in light emitting diode simulations, Preprint no. hal-04012467, Hyper Articles en Ligne (HAL), 2023.
  Abstract
  We show the importance of using a thermodynamically consistent flux discretization when describing drift-diffusion processes within light emitting diode simulations. Using the classical Scharfetter-Gummel scheme with Fermi-Dirac statistics is an example of such an inconsistent scheme. In this case, for an (In,Ga)N multi quantum well device, the Fermi levels show steep gradients on one side of the quantum wells which are not to be expected. This result originates from neglecting diffusion enhancement associated with Fermi-Dirac statistics in the numerical flux approximation. For a thermodynamically consistent scheme, such as the SEDAN scheme, the spikes in the Fermi levels disappear. We will show that thermodynamic inconsistency has far reaching implications on the current-voltage curves and recombination rates.

• T. Boege, R. Fritze, Ch. Görgen, J. Hanselman, D. Iglezakis, L. Kastner, Th. Koprucki, T. Krause, Ch. Lehrenfeld, S. Polla, M. Reidelbach, Ch. Riedel, J. Saak, B. Schembera, K. Tabelow, M. Weber, Research-data management planning in the German mathematical community, Preprint no. arXiv:2211.12071, Cornell University, 2022, DOI 10.48550/arXiv.2211.12071 .
  Abstract
  In this paper we discuss the notion of research data for the field of mathematics and report on the status quo of research-data management and planning. A number of decentralized approaches are presented and compared to needs and challenges faced in three use cases from different mathematical subdisciplines. We highlight the importance of tailoring research-data management plans to mathematicians' research processes and discuss their usage all along the data life cycle.

• 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, 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.

• M. Oliva, T. Flissikowsky, M. Góra, J. Lähnemann, J. Herranz, R. Lewis, O. Marquardt, M. Ramsteiner, L. Geelhaar, O. Brandt, Carrier recombination in highly uniform and phase-pure GaAs/(Al,Ga)As core/shell nanowire arrays on Si(111): Mott transition and internal quantum efficiency, Preprint no. arXiv:2211.17167, Cornell University, 2022, DOI 10.48550/arXiv.2211.17167 .
  Abstract
  GaAs-based nanowires are among the most promising candidates for realizing a monolithical integration of III-V optoelectronics on the Si platform. To realize their full potential for applications as light absorbers and emitters, it is crucial to understand their interaction with light governing the absorption and extraction efficiency, as well as the carrier recombination dynamics determining the radiative efficiency. Here, we study the spontaneous emission of zincblende GaAs/(Al,Ga)As core/shell nanowire arrays by μ -photoluminescence spectroscopy. These ordered arrays are synthesized on patterned Si(111) substrates using molecular beam epitaxy, and exhibit an exceptionally low degree of polytypism for interwire separations exceeding a critical value. We record emission spectra over more than five orders of excitation density for both steady-state and pulsed excitation to identify the nature of the recombination channels. An abrupt Mott transition from excitonic to electron-hole-plasma recombination is observed, and the corresponding Mott density is derived. Combining these experiments with simulations and additional direct measurements of the external quantum efficiency using a perfect diffuse reflector as reference, we are able to extract the internal quantum efficiency as a function of carrier density and temperature as well as the extraction efficiency of the nanowire array. The results vividly document the high potential of GaAs/(Al,Ga)As core/shell nanowires for efficient light emitters integrated on the Si platform. Furthermore, the methodology established in this work can be applied to nanowires of any other materials system of interest for optoelectronic applications.