Publikationen

Monografien

  • M. Liero, M. Thomas, D. Peschka, eds., Special Issue on Energy-based Mathematical Methods for Reactive Multiphase Flows, 103 of Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), Wiley-VCH Verlag, Weinheim, 2023, (Collection Published), DOI 10.1002/zamm.202302012 .

Artikel in Referierten Journalen

  • D. Abdel, A. Glitzky, M. Liero, Analysis of a drift-diffusion model for perovskite solar cells, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, published online in June 2024, DOI 10.3934/dcdsb.2024081 .
    Abstract
    This paper deals with the analysis of an instationary drift-diffusion model for perovskite solar cells including Fermi--Dirac statistics for electrons and holes and Blakemore statistics for the mobile ionic vacancies in the perovskite layer. The free energy functional is related to this choice of the statistical relations. Exemplary simulations varying the mobility of the ionic vacancy demonstrate the necessity to include the migration of ionic vacancies in the model frame. To prove the existence of weak solutions, first a problem with regularized state equations and reaction terms on any arbitrarily chosen finite time interval is considered. Its solvability follows from a time discretization argument and passage to the time-continuous limit. Applying Moser iteration techniques, a priori estimates for densities, chemical potentials and the electrostatic potential of its solutions are derived that are independent of the regularization level, which in turn ensure the existence of solutions to the original problem.

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

  • Y. Hadjimichael, Ch. Merdon, M. Liero, P. Farrell, An energy-based finite-strain model for 3D heterostructured materials and its validation by curvature analysis, International Journal for Numerical Methods in Engineering, e7508 (2024), pp. 7508/1--7508/28, DOI 10.1002/nme.7508 .
    Abstract
    This paper presents a comprehensive study of the intrinsic strain response of 3D het- erostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of these heterostructures. We propose a model for nonlinear elastic heterostructures such as bimetallic beams or nanowires that takes into account local prestrain within each distinct material region. The resulting system of partial differential equations (PDEs) in Lagrangian coordinates incorporates a nonlinear strain and a linear stress-strain relationship governed by Hooke?s law. To validate our model, we apply it to bimetallic beams and hexagonal hetero-nanowires and perform numerical simulations using finite element methods (FEM). Our simulations ex- amine how these structures undergo bending under varying material compositions and cross-sectional geometries. In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formula- tions. We derive these analytical expressions through an energy-based approach as well as a kinetic framework, adeptly accounting for the lattice constant mismatch present at each compound material of the heterostructures. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. This is particularly significant as the strain has the potential to influence the electronic band structure, piezoelectricity, and the dynamics of charge carriers.

  • 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, Physical Review Applied, 21 (2024), pp. 024052/1--024052/12, DOI 10.1103/PhysRevApplied.21.024052 .
    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.

  • L. Araujo, C. Lasser, B. Schmidt, FSSH-2: Fewest Switches Surface Hopping with robust switching probability, Journal of Chemical Theory and Computation, 20 (2024), pp. 3359--4064, DOI 10.1021/acs.jctc.4c00089 .
    Abstract
    This study introduces the FSSH-2 scheme, a redefined and numerically stable adiabatic Fewest Switches Surface Hopping (FSSH) method. It reformulates the standard FSSH hopping probability without non-adiabatic coupling vectors and allows for numerical time integration with larger step sizes. The advantages of FSSH-2 are demonstrated by numerical experiments for five different model systems in one and two spatial dimensions with up to three electronic states.

  • P. Bella, M. Kniely, Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization, Stochastic Partial Differential Equations. Analysis and Computations, published online on 27.02.2024, DOI https://doi.org/10.1007/s40072-023-00322-9 .
    Abstract
    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. Colli, G. Gilardi, A. Signori, J. Sprekels, Curvature effects in pattern formation: Well-posedness and optimal control of a sixth-order Cahn--Hilliard equation, SIAM Journal on Mathematical Analysis, 56 (2024), pp. 4928--4969, DOI 10.1137/24M1630372 .
    Abstract
    This work investigates the well-posedness and optimal control of a sixth-order Cahn--Hilliard equation, a higher-order variant of the celebrated and well-established Cahn--Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improve the understanding of the mathematical properties and control aspects of the sixth-order Cahn--Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties.

  • R. Haller, H. Meinlschmidt, J. Rehberg, Hölder regularity for domains of fractional powers of elliptic operators with mixed boundary conditions, Pure and Applied Functional Analysis, 9 (2024), pp. 169--194.
    Abstract
    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.

  • A. Mielke, S. Schindler, On self-similar patterns in coupled parabolic systems as non-equilibrium steady states, Chaos. An Interdisciplinary Journal of Nonlinear Science, 34 (2024), pp. 013150/1--013150/12, DOI 10.1063/5.0144692 .
    Abstract
    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.

  • J. Sprekels, F. Tröltzsch, Second-order sufficient conditions in the sparse optimal control of a phase field tumor growth model with logarithmic potential, ESAIM. Control, Optimisation and Calculus of Variations, 30 (2024), pp. 13/1--13/25, DOI 10.1051/cocv/2023084 .
    Abstract
    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.

  • K. Hopf, Singularities in $L^1$-supercritical Fokker--Planck equations: A qualitative analysis, Annales de l'Institut Henri Poincare. Analyse Non Lineaire, 41 (2024), pp. 357--403, DOI 10.4171/AIHPC/85 .
    Abstract
    A class of nonlinear Fokker--Planck equations with superlinear drift is investigated in the L1-supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time qualitative analysis -- the main focus being on isotropic solutions, in which case the unique minimiser of the free energy will be shown to be the global attractor. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the 3D Kaniadakis--Quarati model for Bose--Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time asymptotic behaviour for this model.

  • TH. Eiter, R. Lasarzik, Existence of energy-variational solutions to hyperbolic conservation laws, Calculus of Variations and Partial Differential Equations, 63 (2024), pp. 103/1--103/40, DOI 10.1007/s00526-024-02713-9 .
    Abstract
    We introduce 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.

  • A. Mielke, T. Roubíček, Qualitative study of a geodynamical rate-and-state model for elastoplastic shear flows in crustal faults, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 26 (2024), pp. 245--282, DOI 10.4171/IFB/506 .
    Abstract
    The Dieterich--Ruina rate-and-state friction model is transferred to a bulk variant and the state variable (aging) influencing the dissipation mechanism is here combined also with a damage influencing standardly the elastic response. As the aging has a separate dynamics, the overall model does not have a standard variational structure. A one-dimensional model is investigated as far as the steady-state existence, localization of the 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.

  • W. van Oosterhout, M. Liero, Finite-strain poro-visco-elasticity with degenerate mobility, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, appeared online on 29.03.2024, DOI 10.1002/zamm.202300486 .
    Abstract
    A quasistatic nonlinear model for poro-visco-elastic solids at finite strains is considered in the Lagrangian frame using the concept of second-order nonsimple materials. The elastic stresses satisfy static frame-indifference, while the viscous stresses satisfy dynamic frame-indifference. The mechanical equation is coupled to a diffusion equation for a solvent or fluid content. The latter is pulled-back to the reference configuration. To treat the nonlinear dependence of the mobility tensor on the deformation gradient, the result by Healey & Krömer is used to show that the determinant of the deformation gradient is bounded away from zero. Moreover, the focus is on the physically relevant case of degenerate mobilities. The existence of weak solutions is shown using a staggered time-incremental scheme and suitable energy-dissipation inequalities.

  • M. Heida, Stochastic homogenization on perforated domains I: Extension operators, Networks and Heterogeneous Media, 18 (2023), pp. 1821/1--1821/78, DOI 10.3934/nhm.2023079 .
    Abstract
    This preprint is part of a major rewriting and substantial improvement of WIAS Preprint 2742. In this first part of a series of 3 papers, we set up a framework to study the existence of uniformly bounded extension and trace operators for W1,p-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. We drop the classical assumption of minimaly smoothness and study stationary geometries which have no global John regularity. For such geometries, uniform extension operators can be defined only from W1,p to W1,r with the strict inequality r<p. In particular, we estimate the Lr-norm of the extended gradient in terms of the Lp-norm of the original gradient. Similar relations hold for the symmetric gradients (for ℝd-valued functions) and for traces on the boundary. As a byproduct we obtain some Poincaré and Korn inequalities of the same spirit. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions: local (δ,M)-regularity to quantify statistically the local Lipschitz regularity and isotropic cone mixing to quantify the density of the geometry and the mesoscopic properties. These two properties are sufficient to reduce the problem of extension operators to the connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process, for which we can explicitly estimate the connectivity terms.

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

  • 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, Cahn--Hilliard--Brinkman model for tumor growth with possibly singular potentials, Nonlinearity, 36 (2023), pp. 4470--4500, DOI https://doi.org/10.1088/1361-6544/ace2a7 .
    Abstract
    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.

  • 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, 16 (2023), pp. 2305--2325, 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.

  • P. Colli, G. Gilardi, A. Signori, J. Sprekels, On a Cahn--Hilliard system with source term and thermal memory, Nonlinear Analysis. An International Mathematical Journal, 240 (2024), pp. 113461/1--113461/16 (published online on 14.12.2023), DOI 10.1016/j.na.2023.113461 .
    Abstract
    A nonisothermal phase field system of Cahn--Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn--Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical results under general assumptions on the source term and the double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem.

  • P. Colli, G. Gilardi, A. Signori, J. Sprekels, Optimal temperature distribution for a nonisothermal Cahn--Hilliard system in two dimensions with source term and double obstacle potential, Annals of the Academy of Romanian Scientists. Mathematics and its Applications., 15 (2023), pp. 175--204, DOI 10.56082/annalsarscimath.2023.1-2.175 .
    Abstract
    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.

  • P. Colli, G. Gilardi, A. Signori, J. Sprekels, Optimal temperature distribution for a nonisothermal Cahn--Hilliard system with source term, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 88 (2023), pp. 68/1--68/31, DOI 10.1007/s00245-023-10039-9 .

  • P.-É. Druet, K. Hopf, A. Jüngel, Hyperbolic-parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion, Communications in Partial Differential Equations, 48 (2023), pp. 863--894, DOI 10.1080/03605302.2023.2212479 .
    Abstract
    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.

  • K. Fellner, J. Fischer, M. Kniely, B.Q. Tang, Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion, Journal of Nonlinear Science, 33 (2023), pp. 66/1--66/49, DOI 10.1007/s00332-023-09926-w .
    Abstract
    The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with non-linear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract all renormalised solutions in the same compatibility class. This convergence extends even to a range of non-linear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter.

  • 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, Applied Physics Letters, 122 (2023), pp. 241104/1--241104/7, DOI 10.1063/5.0148168 .
    Abstract
    Aluminum 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, understand- ing 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 Al 0.7 Ga0.3 N/Al0.8 Ga0.2 N quantum well embedded in a p-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 sim- ulations. Moreover, we find 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 suppress such non-radiative process and poten- tially related material degradation, a careful design (wider well, multi-quantum wells) of the active region is required to improve the effi- ciency of deep UV light emitters.

  • G. Gilardi, A. Signori, J. Sprekels, Nutrient control for a viscous Cahn--Hilliard--Keller--Segel model with logistic source describing tumor growth, Discrete and Continuous Dynamical Systems -- Series S, 16 (2023), pp. 3552--3572, DOI 10.3934/dcdss.2023123 .
    Abstract
    In this paper, we address a distributed control problem for a system of partial differential equations describing the evolution of a tumor that takes the biological mechanism of chemotaxis into account. The system describing the evolution is obtained as a nontrivial combination of a Cahn--Hilliard type system accounting for the segregation between tumor cells and healthy cells, with a Keller--Segel type equation accounting for the evolution of a nutrient species and modeling the chemotaxis phenomenon. First, we develop a robust mathematical background that allows us to analyze an associated optimal control problem. This analysis forced us to select a source term of logistic type in the nutrient equation and to restrict the analysis to the case of two space dimensions. Then, the existence of an optimal control and first-order necessary conditions for optimality are established.

  • 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.

  • J. Sprekels, F. Tröltzsch, Second-order sufficient conditions for sparse optimal control of singular Allen--Cahn systems with dynamic boundary conditions, Discrete and Continuous Dynamical Systems -- Series S, 16 (2023), pp. 3784--3812, DOI 10.3934/dcdss.2023163 .
    Abstract
    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.

  • A. Glitzky, M. Liero, A drift-diffusion based electrothermal model for organic thin-film devices including electrical and thermal environment, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, published online on 27.11.2023, DOI 10.1002/zamm.202300376 .
    Abstract
    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.

  • TH. Eiter, M. Kyed, Y. Shibata, Periodic Lp estimates by R-boundedness: Applications to the Navier--Stokes equations, Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications, 188 (2023), pp. 1/1--1/43, DOI 10.1007/s10440-023-00612-3 .
    Abstract
    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.

  • 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.

  • P. Farrell, J. Moatti, M. O'Donovan, S. Schulz, Th. Koprucki, Importance of satisfying thermodynamic consistency in optoelectronic device simulations for high carrier densities, Optical and Quantum Electronics, 55 (2023), pp. 978/1--978/12, DOI 10.1007/s11082-023-05234-5 .
    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.

  • M. Liero, A. Mielke, G. Savaré, Fine properties of geodesics and geodesic $lambda$-convexity for the Hellinger--Kantorovich distance, Archive for Rational Mechanics and Analysis, 247 (2023), pp. 112/1--112/73, DOI 10.1007/s00205-023-01941-1 .
    Abstract
    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.

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

  • 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 Rd for large times.

Beiträge zu Sammelwerken

  • R. Danabalan, M. Hintermüller, Th. Koprucki, K. Tabelow, MaRDI: Building research data infrastructures for mathematics and the mathematical sciences, in: 1st Conference on Research Data Infrastructure (CoRDI) -- Connecting Communities, Y. Sure-Vetter, C. Goble, eds., 1 of Proceedings of the Conference on Research Data Infrastructure, TIB Open Publishing, Hannover, 2023, pp. 69/1--69/4, DOI 10.52825/cordi.v1i.397 .
    Abstract
    MaRDI is building a research data infrastructure for mathematics and beyond based on semantic technologies (metadata, ontologies, knowledge graphs) and data repositories. Focusing on the algorithms, models and workflows, the MaRDI infrastructure will connect with other disciplines and NFDI consortia on data processing methods, solving real world problems and support mathematicians on research datamanagement

  • A. Maltsi, A. Mielke, Th. Koprucki, Symmetries in Transmission Electron Microscopy images of semiconductor nanostructures with strain, in: 23nd International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2023), P. Bardella, A. Tibaldi, eds., IEEE, 2023, pp. 111-112, DOI 10.1109/NUSOD59562.2023.10273568 .
    Abstract
    Transmission electron microscopy is often used to image semiconductor nanostructures with strain. The resulting images exhibit symmetries, the source of which is not always known. We prove mathematically that the intensities are invariant under specific transformations, which allows us to distinguish between symmetries of the imaging process itself and symmetries of the inclusion.

  • L. Ermoneit, B. Schmidt, J. Fuhrmann, Th. Koprucki, L.R. Schreiber, M. Kantner, Simulation of single-electron shuttling for spin-qubit transport in a SiGe quantum bus, in: Book of Abstracts of the International Workshop on Computational Nanotechnology 2023 (IWCN 2023), X. Orios Plaedvall, G. Abadal Berini, X. Cartoixà Soler, A. Cummings, C.F. Destefani, D. Jiménez Jiménez, J. Mart'in Mart'inez, R. Rodr'iguez Mart'inez, A. Benali, eds., pp. 88-89.

  • M. Heida, Finite volumes for simulation of large molecules, in: Finite Volumes for Complex Applications X -- Volume 1, Elliptic and Parabolic Problems: FVCA10, Strasbourg, France, October 30, 2023 -- November 03, 2023, Invited Contributions, E. Franck, J. Fuhrmann, V. Michel-Dansac, L. Navoret, eds., 432 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham, 2023, pp. 305--313, DOI 10.1007/978-3-031-40864-9_25 .
    Abstract
    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.

  • M. O'Donovan, R. Finn, S. Schulz, Th. Koprucki, Atomistic study of Urbach tail energies in (Al,Ga)N quantum well systems, in: 23nd International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2023), P. Bardella, A. Tibaldi, eds., IEEE, 2023, pp. 79--80, DOI 10.1109/NUSOD59562.2023.10273479 .
    Abstract
    Aluminium gallium nitride is a system of interest for developing ultraviolet (UV) optoelectronic devices. Here Urbach tails induced by carrier localization effects play a key role in determining device behaviour. We study the electronic structure of Al x Ga 1-x N/Al y Ga 1-y N single quantum wells using an atomistic framework. Results show that the density of states exhibits a tail at low energies due to disorder in the alloy microstructure. Our analysis allows for insight into the orbital character of the states forming the Urbach tails, which can affect light polarization characteristics, and important quantity for deep UV light emitters.

  • 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, 130 of EMS Magazine, European Mathematical Society, 2023, pp. 40--47, DOI 10.4171/MAG/152 .

  • R. Finn, M. O'Donovan, P. Farrell, T. Streckenbach, J. Moatti, Th. Koprucki, S. Schulz, Theoretical investigation of carrier transport and recombination processes for deep UV (Al,Ga)N light emitters, in: 23nd International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2023), P. Bardella, A. Tibaldi, eds., IEEE, 2023, pp. 83--84, DOI 10.1109/NUSOD59562.2023.10273485 .
    Abstract
    We present a theoretical study on the impact of alloy disorder on carrier transport and recombination rates in an (Al,Ga)N single quantum well based LED operating in the deep UV spectral range. Our calculations indicate that alloy fluctuations enable percolative pathways which can result in improved carrier injection into the well, but may also increase carrier leakage from the well. Additionally, we find that alloy disorder induces carrier localization effects, a feature particularly noticeable for the holes. These localization effects can lead to locally increased carrier densities when compared to a virtual crystal approximation which neglects alloy disorder. We observe that both radiative and non-radiative recombination rates are increased. Our calculations also indicate that Auger--Meitner recombination increases faster than the radiative rate, based on a comparison with a virtual crystal approximation.

  • B. Schembera, F. Wübbeling, Th. Koprucki, Ch. Biedinger, M. Reidelbach, B. Schmidt, D. Göddeke, J. Fiedler, Building ontologies and knowledge graphs for mathematics and its applications, in: 1st Conference on Research Data Infrastructure (CoRDI) -- Connecting Communities, Y. Sure-Vetter, C. Goble, eds., 1 of Proceedings of the Conference on Research Data Infrastructure, TIB Open Publishing, Hannover, 2023, pp. 29/1--29/5, DOI 10.52825/cordi.v1i.255 .
    Abstract
    Ontologies and knowledge graphs for mathematical algorithms and modelsare presented, that have been developed by the Mathematical Research Data Initiative.This enables FAIR data handling in mathematics and the applied disciplines. Moreover, challenges of harmonization during the ontology development are discussed.

  • A. Glitzky, M. Liero, A bulk-surface model for the electrothermal feedback in large-area organic light-emitting diodes, in: 93rd Annual Meeting 2023 of the International Association of Applied Mathematics and Mechanics (GAMM), 23 of Proc. Appl. Math. Mech. (Special Issue), Wiley-VCH Verlag, Weinheim, 2023, pp. e202300018/1--e202300018/8, DOI 10.1002/pamm.202300018 .
    Abstract
    This work deals with an effective bulk-surface thermistor model describing the electrothermal behavior of large-area thin-film organic light-emitting diodes (OLEDs). This model was rigorously derived from a -Laplace thermistor model by dimension reduction and consists of the heat equation in the three-dimensional glass substrate and two semi-linear equations describing the current flow through the electrodes coupled to algebraic equations that express the continuity of the electrical fluxes through the organic layers. The electrical problem lives on the surface of the glass substrate where the OLED is mounted. The source terms in the heat equation result from Joule heating and are concentrated on the part of the boundary where the current-flow problem is formulated. Schauder's fixed-point theorem is used to establish the existence of weak solutions to this effective system. Since the heat source terms at the surface are a priori only in L1, the concept of entropy solutions for the heat equation is worked with.

  • M. Landstorfer, M. Heida, Energie effizienter speichern, Spektrum der Wissenschaft, Spektrum der Wissenschaft Verlagsgesellschaft mbH, Heidelberg, 2023, pp. 72--79.

Preprints, Reports, Technical Reports

  • T. Böhnlein, M. Egert, J. Rehberg, Bounded functional calculus for divergence form operators with dynamical boundary conditions, Preprint no. 3115, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3115 .
    Abstract, PDF (417 kByte)
    We consider divergence form operators with complex coefficients on an open subset of Euclidean space. Boundary conditions in the corresponding parabolic problem are dynamical, that is, the time derivative appears on the boundary. As a matter of fact, the elliptic operator and its semigroup act simultaneously in the interior and on the boundary. We show that the elliptic operator has a bounded holomorphic calculus in Lebesgue spaces if the coefficients satisfy a p-adapted ellipticity condition. A major challenge in the proof is that different parts of the spatial domain of the operator have different dimensions. Our strategy relies on extending a contractivity criterion due to Nittka and a non-linear heat flow method recently popularized by Carbonaro--Dragicevic to our setting.

  • P. Colli, J. Sprekels, Second-order optimality conditions for the sparse optimal control of nonviscous Cahn--Hilliard systems, Preprint no. 3114, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3114 .
    Abstract, PDF (363 kByte)
    In this paper we study the optimal control of an initial-boundary value problem for the classical nonviscous Cahn--Hilliard system with zero Neumann boundary conditions. Phase field systems of this type govern the evolution of diffusive phase transition processes with conserved order parameter. 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 sparsity-enhancing nondifferentiable term like the $L^1$-norm. For such cases, we establish first-order necessary and second-order sufficient optimality conditions for locally optimal controls, where in the approach to second-order sufficient conditions we employ a technique introduced by E. Casas, C. Ryll and F. Tröltzsch in the paper SIAM J. Control Optim. 53 (2015), 2168--2202. The main novelty of this paper is that this method, which has recently been successfully applied to systems of viscous Cahn--Hilliard type, can be adapted also to the classical nonviscous case. Since in the case without viscosity the solutions to the state and adjoint systems turn out to be considerably less regular than in the viscous case, numerous additional technical difficulties have to be overcome, and additional conditions have to be imposed. In particular, we have to restrict ourselves to the case when the nonlinearity driving the phase separation is regular, while in the presence of a viscosity term also nonlinearities of logarithmic type turn could be admitted. In addition, the implicit function theorem, which was employed to establish the needed differentiability properties of the control-to-state operator in the viscous case, does not apply in our situation and has to be substituted by other arguments.

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

  • A. Mielke, T. Roubiček, A general thermodynamical model for finitely-strained continuum with inelasticity and diffusion, its GENERIC derivation in Eulerian formulation, and some application, Preprint no. 3107, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3107 .
    Abstract, PDF (484 kByte)
    A thermodynamically consistent visco-elastodynamical model at finite strains is derived that also allows for inelasticity (like plasticity or creep), thermal coupling, and poroelasticity with diffusion. The theory is developed in the Eulerian framework and is shown to be consistent with the thermodynamic framework given by General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). For the latter we use that the transport terms are given in terms of Lie derivatives. Application is illustrated by two examples, namely volumetric phase transitions with dehydration in rocks and martensitic phase transitions in shape-memory alloys. A strategy towards a rigorous mathematical analysis is only very briefly outlined.

  • M. Heida, B. Jahnel, A.D. Vu, An ergodic and isotropic zero-conductance model with arbitrarily strong local connectivity, Preprint no. 3095, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3095 .
    Abstract, PDF (377 kByte)
    We exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all non-trivial choices of the connectivity parameter. The model is based on the so-called randomly stretched lattice where we additionally elongate layers containing few open edges.

  • P. Colli, J. Sprekels, F. Tröltzsch, Optimality conditions for sparse optimal control of viscous Cahn--Hilliard systems with logarithmic potential, Preprint no. 3094, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3094 .
    Abstract, PDF (407 kByte)
    In this paper we study the optimal control of a parabolic initial-boundary value problem of viscous Cahn--Hilliard type with zero Neumann boundary conditions. Phase field systems of this type govern the evolution of diffusive phase transition processes with conserved order parameter. It is assumed that the nonlinear function driving the physical processes within the spatial domain 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 $L^1$-norm, which leads to sparsity of optimal controls. For such cases, we establish first-order necessary and second-order sufficient optimality conditions for locally optimal controls. In the approach to second-order sufficient conditions, the main novelty of this paper, we adapt a technique introduced by E. Casas, C. Ryll and F. Tröltzsch in the paper [em SIAM J. Control Optim. bf 53 (2015), 2168--2202]. In this paper, we show that this method can also be successfully applied to systems of viscous Cahn--Hilliard type with logarithmic nonlinearity. Since the Cahn--Hilliard system corresponds to a fourth-order partial differential equation in contrast to the second-order systems investigated before, additional technical difficulties have to be overcome.

  • TH. Eiter, A.L. Silvestre, Representation formulas and far-field behavior of time-periodic flow past a body, Preprint no. 3091, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3091 .
    Abstract, PDF (325 kByte)
    This paper is concerned with integral representations and asymptotic expansions of solutions to the time-periodic incompressible Navier-Stokes equations for fluid flow in the exterior of a rigid body that moves with constant velocity. Using the time-periodic Oseen fundamental solution, we derive representation formulas for solutions with suitable regularity. From these formulas, the decomposition of the velocity component of the fundamental solution into steady-state and purely periodic parts and their detailed decay rate in space, we deduce complete information on the asymptotic structure of the velocity and pressure fields.

  • J. Li, X. Liu, D. Peschka, Local well-posedness and global stability of one-dimensional shallow water equations with surface tension and constant contact angle, Preprint no. 3084, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3084 .
    Abstract, PDF (405 kByte)
    We consider the one-dimensional shallow water problem with capillary surfaces and moving contact lines. An energy-based model is derived from the two-dimensional water wave equations, where we explicitly discuss the case of a stationary force balance at a moving contact line and highlight necessary changes to consider dynamic contact angles. The moving contact line becomes our free boundary at the level of shallow water equations, and the depth of the shallow water degenerates near the free boundary, which causes singularities for the derivatives and degeneracy for the viscosity. This is similar to the physical vacuum of compressible flows in the literature. The equilibrium, the global stability of the equilibrium, and the local well-posedness theory are established in this paper.

  • L. Ermoneit, B. Schmidt, Th. Koprucki, J. Fuhrmann, T. Breiten, A. Sala, N. Ciroth, R. Xue, L.R. Schreiber, M. Kantner, Optimal control of conveyor-mode spin-qubit shuttling in a Si/SiGe quantum bus in the presence of charged defects, Preprint no. 3082, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3082 .
    PDF (9473 kByte)

  • A. Erhardt, D. Peschka, Ch. Dazzi, L. Schmeller, A. Petersen, S. Checa, A. Münch, B. Wagner, Modeling cellular self-organization in strain-stiffening hydrogels, Preprint no. 3076, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3076 .
    Abstract, PDF (14 MByte)
    We develop a three-dimensional mathematical model framework for the collective evolution of cell populations by an agent-based model (ABM) that mechanically interacts with the surrounding extracellular matrix (ECM) modeled as a hydrogel. We derive effective two-dimensional models for the geometrical set-up of a thin hydrogel sheet to study cell-cell and cell-hydrogel mechanical interactions for a range of external conditions and intrinsic material properties. We show that without any stretching of the hydrogel sheets, cells show the well-known tendency to form long chains with varying orientations. Our results further show that external stretching of the sheet produces the expected nonlinear strain-softening or stiffening response, with, however, little qualitative variation of the overall cell dynamics for all the materials considered. The behavior is remarkably different when solvent is entering or leaving from strain softening or stiffening hydrogels, respectively.

  • Y. Hadjimichael, Ch. Merdon, M. Liero, P. Farrell, An energy-based finite-strain model for 3D heterostructured materials and its validation by curvature analysis, Preprint no. 3064, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3064 .
    Abstract, PDF (6517 kByte)
    This paper presents a comprehensive study of the intrinsic strain response of 3D het- erostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of these heterostructures. We propose a model for nonlinear elastic heterostructures such as bimetallic beams or nanowires that takes into account local prestrain within each distinct material region. The resulting system of partial differential equations (PDEs) in Lagrangian coordinates incorporates a nonlinear strain and a linear stress-strain relationship governed by Hooke?s law. To validate our model, we apply it to bimetallic beams and hexagonal hetero-nanowires and perform numerical simulations using finite element methods (FEM). Our simulations ex- amine how these structures undergo bending under varying material compositions and cross-sectional geometries. In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formula- tions. We derive these analytical expressions through an energy-based approach as well as a kinetic framework, adeptly accounting for the lattice constant mismatch present at each compound material of the heterostructures. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. This is particularly significant as the strain has the potential to influence the electronic band structure, piezoelectricity, and the dynamics of charge carriers.

  • M. Heida, On the computation of high dimensional Voronoi diagrams, Preprint no. 3041, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3041 .
    Abstract, PDF (553 kByte)
    We investigate a recently implemented new algorithm for the computation of a Voronoi diagram in high dimensions and generalize it to N nodes in general or non-general position using a geometric characterization of edges merging in a given vertex. We provide a mathematical proof that the algorithm is exact, convergent and has computational costs of O(E*nn(N)), where E is the number of edges and nn(N) is the computational cost to calculate the nearest neighbor among N points. We also provide data from performance tests in the recently developed Julia package ,,HighVoronoi.jl”.

  • K.M. Gambaryan, O. Ernst, T. Boeck, O. Marquardt, Energy level alignment of confined hole states in $InAs_1-x-ySb_xP_y$ double quantum dots, Preprint no. 3040, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3040 .
    Abstract, PDF (310 kByte)
    We present a combined experimental and theoretical study of uncapped In(As,Sb,P) double quantum dots (DQD), suited for application in novel resonant tunneling nanodiods or single-photon nanooptical up- and down-converters in the mid-infrared spectral range. We provide details on the growth process using liquid-phase epitaxy (LPE), as well as on the characterization using atomic-force microscopy (AFM) and scanning electron microscopy (SEM). We find that most DQDs exhibit an asymmetry such that the two QDs of each pair have different dimensions, giving rise to correspondingly different quantum confinement of hole states localized in each QD. Based on these data, we have performed systematic simulations based on an eight-band $mathbfkcdotmathbfp$ model to identify the relationship between QD dimensions and the energy difference between corresponding confined hole states in the two QDs. Finally, we have determined the strength of an applied electric field required to energetically align the hole ground states of two QDs of different dimensions in order to facilitate hole tunneling.

  • A. Mielke, R. Rossi, A. Stephan, On time-splitting methods for gradient flows with two dissipation mechanisms, Preprint no. 3033, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3033 .
    Abstract, PDF (530 kByte)
    We consider generalized gradient systems in Banach spaces whose evolutions are generated by the interplay between an energy functional and a dissipation potential. We focus on the case in which the dual dissipation potential is given by a sum of two functionals and show that solutions of the associated gradient-flow evolution equation with combined dissipation can be constructed by a split-step method, i.e. by solving alternately the gradient systems featuring only one of the dissipation potentials and concatenating the corresponding trajectories. Thereby the construction of solutions is provided either by semiflows, on the time-continuous level, or by using Alternating Minimizing Movements in the time-discrete setting. In both cases the convergence analysis relies on the energy-dissipation principle for gradient systems.

  • A. Stephan, Trotter-type formula for operator semigroups on product spaces, Preprint no. 3030, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3030 .
    Abstract, PDF (252 kByte)
    We consider a Trotter-type-product formula for approximating the solution of a linear abstract Cauchy problem (given by a strongly continuous semigroup), where the underlying Banach space is a product of two spaces. In contrast to the classical Trotter-product formula, the approximation is given by freezing subsequently the components of each subspace. After deriving necessary stability estimates for the approximation, which immediately provide convergence in the natural strong topology, we investigate convergence in the operator norm. The main result shows that an almost optimal convergence rate can be established if the dominant operator generates a holomorphic semigroup and the off-diagonal coupling operators are bounded.

  • A. Mielke, An introduction to the analysis of gradients systems, Preprint no. 3022, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3022 .
    Abstract, PDF (862 kByte)
    The present notes provide an extended version of a small lecture course (of 36 hours) given at the Humboldt-Universität zu Berlin in the Winter Term 2022/23. The material starting in Section 5.4 was added afterwards. The aim of these notes to give an introductory overview on the analytical approaches for gradient-flow equations in Hilbert spaces, Banach spaces, and metric spaces and to show that on the first entry level these theories have a lot in common. The theories and their specific setups are illustrated by suitable examples and counterexamples.

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

  • 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. 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.

  • 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.

  • 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.

Vorträge, Poster

  • L. Ermoneit, M. Kantner, Th. Koprucki, J. Fuhrmann, B. Schmidt, Optimal control of a Si/SiGe quantum bus for scalable quantum computing architectures, QUANTUM OPTIMAL CONTROL From Mathematical Foundations to Quantum Technologies, Berlin, May 21, 2024.

  • D. Peschka, Wetting of soft deformable substrate -- Phase fields for fluid structure interaction with moving contact lines, Colloquium on ``Interfaces, Complex Structures, and Singular Limits in Continuum Mechanics -- Analysis and Numerics'', Universität Regensburg, Fakultät für Mathematik, May 24, 2024.

  • B. Schmidt, J.-P. Thiele, Code and perish?! How about publishing your software?, Leibniz MMS Days 2024, Mini Workshop, April 10 - 12, 2024, Leibniz-Institut für Verbundwerkstoffe (IVW), Kaiserslautern, April 10, 2024.

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

  • A. Mielke, Asymptotic self-similar behavior in reaction-diffusion systems, Analysis Seminars, Heriot-Watt University, Mathematical and Computer Sciences, Edinburgh, UK, March 20, 2024.

  • A. Mielke, Balanced-viscosity solutions for generalized gradient systems and a delamination problem, Measures and Materials, March 25 - 28, 2024, University of Warwick, Coventry, UK, March 25, 2024.

  • A. Mielke, Non-equilibrium steady states for port gradient systems, 23rd Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2024), April 3 - 5, 2024, Julius-Maximilians-Universität Würzburg, April 4, 2024.

  • A. Mielke, On EVI flows for gradient systems on the (spherical) Hellinger--Kantorovich space, Workshop ``Applications of Optimal Transportation'', February 5 - 9, 2024, Mathematisches Forschungsinstitut Oberwolfach, February 5, 2024.

  • A. Glitzky, Analysis of a drift-diffusion model for perovskite solar cells, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 14.07 ``Various topics in Applied Analysis'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 21, 2024.

  • A. Maltsi, Introduction to photoacoustic imaging, Women in Math - Introduction of the Iris Runge Program, Weierstraß-Institut für Angewandte Analysis und Stochastik, March 18, 2024.

  • A. Maltsi, The mathematics behind imaging, WINS School 2024: Cross Sections and Interfaces in Science and its Environment, May 31 - June 3, 2024, Humboldt-Universität zu Berlin, Blossin, May 31, 2024.

  • M. Thomas, Analysis of a model for visco-elastoplastic two-phase flows in geodynamics, 23rd Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2024), April 3 - 5, 2024, Julius-Maximilians-Universität Würzburg, April 5, 2024.

  • M. Thomas, Analysis of a model for visco-elastoplastic two-phase flows in geodynamics, Seminar on Nonlinear Partial Differential Equations, Texas A&M University, Department of Mathematics, College Station, USA, March 19, 2024.

  • M. Thomas, Analysis of a model for visco-elastoplastic two-phase flows in geodynamics, 9th European Congress of Mathematics (9ECM), Minisymposium 27 ``New Trends in Calculus of Variations", July 15 - 19, 2024, Universidad de Sevilla, Spain, July 16, 2024.

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

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

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

  • M. Heida, Voronoi diagrams and finite volume methods in any dimension, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 18.01 ``Discontinuous Galerkin and Software'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 19, 2024.

  • G. Heinze, Fast-slow limits for gradient flows on metric graphs, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 14.01 ``Various Topics in Applied Analysis'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 19, 2024.

  • G. Heinze, Graph-based nonlocal gradient systems and their local limits, Aggregation-Diffusion Equations & Collective Behavior: Analysis, Numerics and Applications, Marseille, France, April 8 - 12, 2024.

  • W. van Oosterhout, Finite-strain poro-visco-elasticity with degenerate mobility, Spring School 2024 ``Mathematical Advances for Complex Materials with Microstructures'', April 8 - 12, 2024.

  • A. Maltsi, Symmetries in Transmission Electron Microscopy images of semiconductor nanostructures with strain, 23rd International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2023), September 18 - 21, 2023, Politecnico di Torino, Italy, September 21, 2023.

  • 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.

  • A. Maltsi, Symmetries in transmission electron microscopy imaging of crystals with strain, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Tokyo, Japan, August 20 - 25, 2023.

  • L. Ermoneit, B. Schmidt, J. Fuhrmann, Th. Koprucki, M. Kantner, Coherent spin-qubit shuttling in a SiGe quantum bus: Device-scale modeling, simulation and optimal control, Leibniz MMS Days 2023, Potsdam, April 17 - 19, 2023.

  • L. Ermoneit, M. Kantner, Th. Koprucki, B. Schmidt, Coherent spin-qubit shuttling for scalable quantum processors: Modeling, simulation and optimal control, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

  • M. Heida, Finite volumes for simulation of large molecules, Finite Volumes for Complex Applications (FVCA10), Strasbourg, France, October 30 - November 3, 2023.

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

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

  • M. O'Donovan, Atomistic study of Urbach tail energies in (Al,Ga)N quantum well systems, 23rd International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2023), September 18 - 21, 2023, Politecnico di Torino, Italy, September 21, 2023.

  • M. O'Donovan, Impact of alloy disorder on carrier transport and recombination in (Al,Ga)N-based UV-C emitters, The 6th International Workshop on Ultraviolet Materials and Devices (IWUMD 2023), June 5 - 8, 2023, Metz Congrès Robert Schuman, France, June 7, 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, M. Heida, M. Thomas, Materials with discontinuities on many scales, SCCS Days 2023 of the Collaborative Research Center -- CRC 1114 ``Scaling Cascades in Complex Systems'', November 13 - 15, 2023.

  • L. Schütz, 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.

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

  • M. Kniely, A thermodynamically correct framework for electro-energy-reaction-diffusion systems, 22nd ECMI Conference on Industrial and Applied Mathematics, June 26 - 30, 2023, Wrocław University of Science and Technology, Poland, June 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. Rehberg, A view on the Kohn-Sham system from the perspective of functional analysists, Technische Universität Braunschweig Institut für Analysis und Algebra, November 13, 2023.

  • J. Rehberg, Nonsmooth elliptic and parabolic regularity, Technische Universität Darmstadt, Fachbereich Mathematik, November 27, 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, Normal form and the Cauchy problem for cross-diffusive mixtures, Workshop ``Variational Methods for Evolution'', December 3 - 8, 2023, Mathematisches Forschungsinstitut Oberwolfach, December 4, 2023.

  • K. Hopf, On cross-diffusive coupling of hyperbolic-parabolic type, Variational and Geometric Structures for Evolution, October 9 - 13, 2023, Centro Internazionale per la Ricerca Matematica (CIRM), Levico Terme, Italy, October 13, 2023.

  • K. Hopf, Structure and approximation of cross-diffusive mixtures with incomplete diffusion, Universität Kassel, Fachbereich Mathematik und Naturwissenschaften, September 21, 2023.

  • K. Hopf, Structure, dynamics, and approximation of cross-diffusive mixtures with incomplete diffusion, Universität Hamburg, Fachbereich Mathematik, May 10, 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.

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

  • M. Eigel, M. Heida, M. Landstorfer, A. Selahi, Recovery of battery ageing dynamics with multiple timescales, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 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, Artificial boundary conditions for time-periodic flow past a body, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00558 ``Bifurcations, Periodicity and Stability in Fluid-structure Interactions'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 21, 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.

  • M. Kantner, L. Ermoneit, B. Schmidt, J. Fuhrmann, A. Sala, L.R. Schreiber, Th. Koprucki, Optimal control of a SiGe-quantum bus for coherent electron shuttling in scalable quantum computing architectures, Silicon Quantum Electronics Workshop 2023, Kyoto, Japan, October 31 - November 2, 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 (CSE23), Minisymposium MS301 ``Interfaces, Workflows, and Knowledge Graphs for FAIR CSE'', February 27 - March 3, 2023, 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, Variational modeling of biomechanical systems, 10th International Conference on Computational Bioengineering (ICCB 2023), Minisymposium 22-3 ``Continuum Biomechanics of Active Biological Systems'', September 20 - 22, 2023, Technische Universität Wien, Austria, September 22, 2023.

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

  • 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, Variational and geometric structures for thermomechanical systems, Variational and Geometric Structures for Evolution, October 9 - 13, 2023, Centro Internazionale per la Ricerca Matematica (CIRM), Levico Terme, Italy, October 11, 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, Balanced-viscosity solutions as limits in generalized gradient systems under slow loading, Hausdorff School ``Analysis of PDEs: Variational and Geometric Perspectives'', July 10 - 14, 2023, Universität Bonn, Hausdorff School for Advanced Studies in Mathematics.

  • 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, Non-equilibrium steady states and EDP-convergence for slow-fast gradient systems, In Search of Model Structures for Non-equilibrium 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, Gradient systems and time-splitting methods (online talk), PDE & Applied Mathematics Seminar, University of California, Riverside, Department of Mathematics, USA, November 8, 2023.

  • A. Stephan, On time-splitting methods for gradient flows with two dissipation mechanisms, Gradient Flows face-to-face 3, September 11 - 14, 2023, Université Claude Bernard Lyon 1, France, September 11, 2023.

  • A. Stephan, On time-splitting methods for gradient flows with two dissipation mechanisms, PDE Afternoon, Technische Universität Wien, Austria, December 13, 2023.

  • A. Stephan, Positivity and polynomial decay of energies for square-field operators, Variational and Geometric Structures for Evolution, October 9 - 13, 2023, Centro Internazionale per la Ricerca Matematica (CIRM), Levico Terme, Italy, October 13, 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.

  • A. Stephan, On time-splitting methods for gradient flows with two dissipation mechanisms, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 01181 ``Variational Methods for Multi-scale Dynamics'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 24, 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.

  • W. van Oosterhout, Poro-visco-elastic solids at finite strains with degenerate mobilities, Nonlinear PDEs: Recent Trends in the Analysis of Continuum Mechanics, July 17 - 21, 2023, Universität Bonn, Hausdorff School for Advanced Studies in Mathematics, July 19, 2023.

Preprints im Fremdverlag

  • E. Gladin, P. Dvurechensky, A. Mielke , J.-J. Zhu, Interaction-force transport gradient flows, Preprint no. arXiv:2405.17075, Cornell University, , DOI 10.48550/arXiv.2405.17075 .
    Abstract
    This paper presents a new type of gradient flow geometries over non-negative and probability measures motivated via a principled construction that combines the optimal transport and interaction forces modeled by reproducing kernels. Concretely, we propose the interaction-force transport (IFT) gradient flows and its spherical variant via an infimal convolution of the Wasserstein and spherical MMD Riemannian metric tensors. We then develop a particle-based optimization algorithm based on the JKO-splitting scheme of the mass-preserving spherical IFT gradient flows. Finally, we provide both theoretical global exponential convergence guarantees and empirical simulation results for applying the IFT gradient flows to the sampling task of MMD-minimization studied by Arbel et al. [2019]. Furthermore, we prove that the spherical IFT gradient flow enjoys the best of both worlds by providing the global exponential convergence guarantee for both the MMD and KL energy.

  • 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.

  • B. Schembera, F. Wübbeling, H. Kleikamp, Ch. Bledinger, J. Fiedler, M. Reidelbach, A. Shehu, B. Schmidt, Th. Koprucki, D. Iglezakis, D. Göddeke, Ontologies for models and algorithms in applied mathematics and related disciplines, Preprint no. arXiv:2310.20443, Cornell University, 2023, DOI 10.48550/arXiv.2310.20443 .
    Abstract
    In applied mathematics and related disciplines, the modeling-simulationoptimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data, the Mathematical Research Data Initiative has developed, merged and implemented ontologies and knowledge graphs. This contributes to making mathematical research data FAIR by introducing semantic technology and documenting the mathematical foundations accordingly. Using the concrete example of microfracture analysis of porous media, it is shown how the knowledge of the underlying mathematical model and the corresponding numerical algorithms for its solution can be represented by the ontologies.

  • 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.