Partial Differential Equations

Publications

Monographs

• V.A. Zagrebnov, H. Neidhardt, T. Ishinose, Trotter--Kato Product Formulae, 296 of Operator Theory: Advances and Applications, Springer Nature, Cham, 2024, vii+873 pages, (Monograph Published), DOI https://doi.org/10.1007/978-3-031-56720-9 .
  Abstract
  The book captures a fascinating snapshot of the current state of results about the operator-norm convergent Trotter--Kato Product Formulae on Hilbert and Banach spaces. It also includes results on the operator-norm convergent product formulae for solution operators of the non-autonomous Cauchy problems as well as similar results on the unitary and Zeno product formulae. After the Sophus Lie product formula for matrices was established in 1875, it was generalised to Hilbert and Banach spaces for convergence in the strong operator topology by H. Trotter (1959) and then in an extended form by T. Kato (1978). In 1993 Dzh. L. Rogava discovered that convergence of the Trotter product formula takes place in the operator-norm topology. The latter is the main subject of this book, which is dedicated essentially to the operator-norm convergent Trotter--Kato Product Formulae on Hilbert and Banach spaces, but also to related results on the time-dependent, unitary and Zeno product formulae. The book yields a detailed up-to-date introduction into the subject that will appeal to any reader with a basic knowledge of functional analysis and operator theory. It also provides references to the rich literature and historical remarks.

Articles in Refereed Journals

• M. Heida, B. Jahnel, A.D. Vu, Regularized homogenization on irregularly perforated domains, Networks and Heterogeneous Media, 20 (2025), pp. 165--212, DOI 10.3934/nhm.2025010 .
  Abstract
  We study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by our approach.

• A. Esposito, G. Heinze, A. Schlichting, Graph-to-local limit for the nonlocal interaction equation, Journal de Mathématiques Pures et Appliquées, 194 (2025), pp. 103663/1--103663/50 (published online on 16.01.2025), DOI 10.1016/j.matpur.2025.103663 .
  Abstract
  We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localizing infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretization for the equation under study.

• A. Mielke, R. Rossi, A. Stephan, On time-splitting methods for gradient flows with two dissipation mechanisms, Calculus of Variations and Partial Differential Equations, 64 (2025), pp. 63/1--63/49, DOI 10.1007/s00526-024-02849-8 .
  Abstract
  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.

• TH. Eiter, A.L. Silvestre, Representation formulas and far-field behavior of time-periodic flow past a body, NoDEA. Nonlinear Differential Equations and Applications, 32 (2025), pp. 37/1--37/34, DOI 10.1007/s00030-025-01039-5 .
  Abstract
  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.

• 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, 30 (2025), pp. 99--131, 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.

• A. Erhardt, D. Peschka, Ch. Dazzi, L. Schmeller, A. Petersen, S. Checa, A. Münch, B. Wagner, Modeling cellular self-organization in strain-stiffening hydrogels, Computational Mechanics, 75 (2025), pp. 875--896 (published online on 31.08.2024), DOI 10.1007/s00466-024-02536-7 .
  Abstract
  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, 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. Heida, M. Landstorfer, M. Liero, Homogenization of a porous intercalation electrode with phase separation, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 22 (2024), pp. 1068--1096, DOI 10.1137/21M1466189 .
  Abstract
  In this work, we derive a new model framework for a porous intercalation electrode with a phase separating active material upon lithium intercalation. We start from a microscopic model consisting of transport equations for lithium ions in an electrolyte phase and intercalated lithium in a solid active phase. Both are coupled through a Neumann--boundary condition modeling the lithium intercalation reaction. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a Cahn--Hilliard equation, whereas the limit model consists of a diffusion and an Allen--Cahn equation. Thus we observe a Cahn--Hilliard to Allen--Cahn transition during the upscaling process. In the sense of gradient flows, the transition goes in hand with a change in the underlying metric structure of the PDE system.

• 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. 3413--3419, 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, 12 (2024), pp. 2246--2288, 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.

• P. Colli, J. Sprekels, F. Tröltzsch, Optimality conditions for sparse optimal control of viscous Cahn--Hilliard systems with logarithmic potential, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 90 (2024), pp. 47/1--47/48, DOI 10.1007/s00245-024-10187-6 .
  Abstract
  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.

• 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, T. Roubiček, A general thermodynamical model for finitely-strained continuum with inelasticity and diffusion, its GENERIC derivation in Eulerian formulation, and some application, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 76 (2025), pp. 1--28 (published online on 16.12.2024), DOI 10.1007/s00033-024-02391-9 .
  Abstract
  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.

• A. Mielke, S. Schindler, Convergence to self-similar profiles in reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 56 (2024), pp. 7108--7135, DOI 10.1137/23M1564298 .
  Abstract
  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, NoDEA. Nonlinear Differential Equations and Applications, 32 (2025), pp. 14/1--14/33 (published online on 11.12.2024), DOI 10.1007/s00030-024-01009-3 .
  Abstract
  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.

• 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.

• 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.

• TH. Eiter, Y. Shibata, Viscous flow past a translating body with oscillating boundary, Journal of the Mathematical Society of Japan, 77 (2025), pp. 103--134 (published online on 22.07.2024), DOI 10.2969/jmsj/91649164 .
  Abstract
  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.

• M. Heida, B. Jahnel, A.D. Vu, An ergodic and isotropic zero-conductance model with arbitrarily strong local connectivity, Electronic Communications in Probability, 29 (2024), pp. 1--13, DOI 10.1214/24-ECP633 .
  Abstract
  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.

• 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, 104 (2024), pp. e202300486/1--e202300486/22, 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.

Contributions to Collected Editions

• B. Schembera, F. Wübbeling, H. Kleikamp, B. Schmidt, A. Shehu, M. Reidelbach, Ch. Biedinger, J. Fiedler, Th. Koprucki, D. Iglezakis, D. Göddeke, Towards a knowledge graph for models and algorithms in applied mathematics, in: Metadata and Semantics Research, M. Sfakakis, E. Garoufallou, M. Damigos, A. Salaba, Ch. Papatheodorou, eds., Communications in Computer and Information Science, Springer, Cham, 2025, pp. 95--109, DOI 10.1007/978-3-031-81974-2_8 .
  Abstract
  Mathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. To make this research data FAIR, we present how two previously distinct ontologies, MathAlgoDB for algorithms and MathModDB for models, were merged and extended into a living knowledge graph as the key outcome. This was achieved by connecting the ontologies through computational tasks that correspond to algorithmic tasks. Moreover, we show how models and algorithms can be enriched with subject-specific metadata, such as matrix symmetry or model linearity, essential for defining workflows and determining suitable algorithms. Additionally, we propose controlled vocabularies to be added, along with a new class that differentiates base quantities from specific use case quantities. We illustrate the capabilities of the developed knowledge graph using two detailed examples from different application areas of applied mathematics, having already integrated over 250 research assets into the knowledge graph.

• E. Gladin, P. Dvurechensky, A. Mielke , J.-J. Zhu, Interaction-force transport gradient flows, in: Advances in Neural Information Processing Systems, Proceedings of NeurIPS 2024, the Thirty-Eighth Annual Conference on Neural Information Processing Systems, 2024, A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, C. Zhang, eds., 37 of Proceedings of NeurIPS, Curran Associates, Inc., 2024, pp. 14484--14508.

• A. Mielke, On EVI flows in the (spherical) Hellinger-Kantorovich space, in: Report 7: Applications of Optimal Transportation, G. Carlier, M. Colombo, V. Ehrlacher, M. Matthes, eds., 21 of Oberwolfach Reports, European Mathematical Society Publishing House, Zürich, 2024, pp. 309--388, DOI 10.4171/OWR/2024/7 .

• B. Schembera, F. Wübbeling, H. Kleikamp, Ch. Biedinger, 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, in: Metadata and Semantics Research, E. Garoufallou, F. Sartori, eds., Communications in Computer and Information Science, Springer, Cham, 2024, pp. 161--168, DOI 10.1007/978-3-031-65990-4_14 .
  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.

• A. Glitzky, On a drift-diffusion model for perovskite solar cells, in: 94th Annual Meeting 2024 of the International Association of Applied Mathematics and Mechanics (GAMM), 24 of Proc. Appl. Math. Mech. (Special Issue), Wiley-VCH Verlag, Weinheim, 2024, pp. 17/1--e202400017/8, DOI 10.1002/pamm.202400017 .
  Abstract
  We introduce a vacancy-assisted charge transport model for perovskite solar cells. This instationary drift-diffusion system describes the motion of electrons, holes, and ionic vacancies and takes into account Fermi--Dirac statistics for electrons and holes and the Fermi--Dirac integral of order -1 for the mobile ionic vacancies in the perovskite. The free energy functional we work with corresponds to that choice of the statistical relations. To verify the existence of weak solutions, we consider a problem with regularized state equations and reaction terms on any arbitrarily chosen finite time interval. We motivate its solvability by time discretization and passage to the time-continuous limit. A priori estimates for the chemical potentials that are independent of the regularization level ensure the existence of solutions to the original problem. These types of estimates rely on Moser iteration techniques and can also be obtained for solutions to the original problem.

Preprints, Reports, Technical Reports

• G. Heinze, A. Mielke, A. Stephan, Discrete-to-continuum limit for nonlinear reaction-diffusion systems via EDP convergence for gradient systems, Preprint no. 3194, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3194 .
  Abstract, PDF (460 kByte)
  We investigate the convergence of spatial discretizations for reaction-diffusion systems with mass-action law satisfying a detailed balance condition. Considering systems on the d-dimensional torus, we construct appropriate space-discrete processes and show convergence not only on the level of solutions, but also on the level of the gradient systems governing the evolutions. As an important step, we prove chain rule inequalities for the reaction-diffusion systems as well as their discretizations, featuring a non-convex dissipation functional. The convergence is then obtained with variational methods by building on the recently introduced notion of gradient systems in continuity equation format.

• P. Colli, J. Sprekels, On the optimal control of viscous Cahn--Hilliard systems with hyperbolic relaxation of the chemical potential, Preprint no. 3190, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3190 .
  Abstract, PDF (296 kByte)
  In this paper, we study an optimal control problem for a viscous Cahn--Hilliard system with zero Neumann boundary conditions in which a hyperbolic relaxation term involving the second time derivative of the chemical potential has been added to the first equation of the system. For the initial-boundary value problem of this system, results concerning well-posedness, continuous dependence and regularity are known. We show Fréchet differentiability of the associated control-to-state operator, study the associated adjoint state system, and derive first-order necessary optimality conditions. Concerning the nonlinearities driving the system, we can include the case of logarithmic potentials. In addition, we perform an asymptotic analysis of the optimal control problem as the relaxation coefficient approaches zero.

• P. Colli, J. Sprekels, F. Tröltzsch, First- and second-order optimality conditions in the sparse optimal control of Cahn--Hilliard systems, Preprint no. 3184, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3184 .
  Abstract, PDF (262 kByte)
  This paper deals with the sparse distributed control of viscous and nonviscous Cahn--Hilliard systems. We report on results concerning first-order necessary and second-order sufficient optimality conditions that have recently established by the authors. The analysis covers both the cases when the nonlinear double well potential governing the evolution is of either regular or logarithmic type. A major difficulty originates from the sparsity-enhancing term in the cost functional which typically is nondifferentiable.

• A. Mielke, An Eulerian formulation for dissipative materials using Lie derivatives and GENERIC, Preprint no. 3180, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3180 .
  Abstract, PDF (342 kByte)
  We recall the systematic formulation of Eulerian mechanics in terms of Lie derivatives along the vector field of the material points. Using the abstract properties of Lie derivatives we show that the transport via Lie derivatives generates in a natural way a Poisson structure on the chosen phase space. The evolution equations for thermo-viscoelastic-viscoplastic materials in the Eulerian setting is formulated in the abstract framework of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). The equations may not be new, but the systematic splitting between reversible Hamiltonian and dissipative effects allows us to see the equations in a new light that is especially useful for future generalizing of the system, e.g.for adding new effects like reactive species.

• D. Peschka, M. Thomas, A. Zafferi, Reference map approach to Eulerian thermomechanics using GENERIC, Preprint no. 3178, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3178 .
  Abstract, PDF (7829 kByte)
  An Eulerian GENERIC model for thermo-viscoelastic materials with diffusive components is derived based on a transformation framework that maps a Lagrangian formulation to corresponding Eulerian coordi- nates. The key quantity describing the deformation in Eulerian coordinates is the inverse of the deformation, i.e. the reference map. The Eulerian model is formally constructed, and by reducing the GENERIC system to a damped Hamiltonian system, the isothermal limit is derived. A structure-preserving weak formulation is developed. As an example, the coupling of finite strain viscoelasticity and diffusion in a multiphase system governed by Lagrangian indicator functions is demonstrated.

• A. Mielke, J.-J. Zhu, Hellinger--Kantorovich gradient flows: Global exponential decay of entropy functionals, Preprint no. 3176, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3176 .
  Abstract, PDF (508 kByte)
  We investigate a family of gradient flows of positive and probability measures, focusing on the Hellinger--Kantorovich (HK) geometry, which unifies transport mechanism of Otto--Wasserstein, and the birth-death mechanism of Hellinger (or Fisher--Rao). A central contribution is a complete characterization of global exponential decay behaviors of entropy functionals under Otto--Wasserstein and Hellinger-type gradient flows. In particular, for the more challenging analysis of HK gradient flows on positive measures---where the typical log-Sobolev arguments fail---we develop a specialized shape-mass decomposition that enables new analysis results. Our approach also leverages the Polyak--Łojasiewicz-type functional inequalities and a careful extension of classical dissipation estimates. These findings provide a unified and complete theoretical framework for gradient flows and underpin applications in computational algorithms for statistical inference, optimization, and machine learning.

• K. Kunisch, J. Rehberg, On non-autonomous parabolic equations with measure-valued right hand sides and applications to optimal control, Preprint no. 3165, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3165 .
  Abstract, PDF (413 kByte)
  The main aim of this paper is to develop a theory for non-autonomous parabolic equations with time-dependent measures on the spatial domain appearing as right hand sides. Restricting these measures to ones which have their supports on 'curves' or 'surfaces' -- the latter understood in the sense of geometric measure theory -- we succeed in interpreting them as distributional objects from a (negatively indexed) Sobolev--Slobodetskii space with differentiability index close to minus one. For these indices a tailor suited parabolic theory is established, based on previous results. It is also demonstrated that the proposed frame work is well-suited for optimal control with controls acting on sub-manifolds.

• G. Heinze, J.-F. Pietschmann, A. Schlichting, Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits, Preprint no. 3161, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3161 .
  Abstract, PDF (1430 kByte)
  We study evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on the case where the dynamics are driven by an entropy functional defined both on the metric edges and vertices, we provide a rigorous understanding of such systems of coupled ordinary and partial differential equations as (generalized) gradient flows in continuity equation format. Approximating the edges by a sequence of vertices, which yields a fully discrete system, we are able to establish existence of solutions in this formalism. Furthermore, we study several scaling limits using the recently developed framework of EDP convergence with embeddings to rigorously show convergence to gradient flows on reduced metric and combinatorial graphs. Finally, numerical studies confirm our theoretical findings and provide additional insights into the dynamics under rescaling.

• K. Remini, L. Schmeller, D. Peschka, B. Wagner, R. Seemann, Shape of polystyrene droplets on soft PDMS: Exploring the gap between theory and experiment at the three-phase contact line, Preprint no. 3160, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3160 .
  Abstract, PDF (6173 kByte)
  The shapes of liquid polystyrene (PS) droplets on viscoelastic polydimethylsiloxane (PDMS) substrates are investigated experimentally using atomic force microscopy for a range of droplet sizes and substrate elasticities. These shapes, which comprise the PS-air, PS-PDMS, and PDMS-air interfaces as well as the three-phase contact line, are compared to theoretical predictions using axisymmetric sharp-interface models derived through energy minimization. We find that the polystyrene droplets are cloaked by a thin layer of uncrosslinked molecules migrating from the PDMS substrate. By incorporating the effects of cloaking into the surface energies in our theoretical model, we show that the global features of the experimental droplet shapes are in excellent quantitative agreement for all droplet sizes and substrate elasticities. However, our comparisons also reveal systematic discrepancies between the experimental results and the theoretical predictions in the vicinity of the three-phase contact line. Moreover, the relative importance of these discrepancies systematically increases for softer substrates and smaller droplets. We demonstrate that global variations in system parameters, such as surface tension and elastic shear moduli, cannot explain these differences but instead point to a locally larger elastocapillary length, whose possible origin is discussed thoroughly.

• K. Hopf, M. Kniely, A. Mielke, On the equilibrium solutions of electro-energy-reaction-diffusion systems, Preprint no. 3157, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3157 .
  Abstract, PDF (447 kByte)
  Electro-energy-reaction-diffusion systems are thermodynamically consistent continuum models for reaction-diffusion processes that account for temperature and electrostatic effects in a way that total charge and energy are conserved. The question of the long-time asymptotic behavior in electro-energy-reaction-diffusion systems and the characterization of their equilibrium solutions leads to a maximization problem of the entropy on the manifold of states with fixed values for the linear charge and the nonlinear convex energy functional. As the main result, we establish the existence, uniqueness, and regularity of solutions to this constrained optimization problem. We give two conceptually different proofs, which are related to different perspectives on the constrained maximization problem. The first one is based on the Lagrangian approach, whereas the second one employs the direct method of the calculus of variations.

• M. Tsopanopoulos, Spectral bounds for the operator pencil of an elliptic system in an angle, Preprint no. 3155, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3155 .
  Abstract, PDF (709 kByte)
  The model problem of a plane angle for a second-order elliptic system with Dirichlet, mixed, and Neumann boundary conditions is analyzed. The existence of solutions is, for each boundary condition, reduced to solving a matrix equation. Leveraging these matrix equations and focusing on Dirichlet and mixed boundary conditions, optimal bounds on these solutions are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results in linear elasticity.

• J. Rehberg, Regularity for non-autonomous parabolic equations with right-hand side singular measures involved, Preprint no. 3150, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3150 .
  Abstract, PDF (345 kByte)
  This article provides a theory for non-autonomous parabolic equations the right hand side of which includes singular measures - depending on the time parameter - on the spatial domain. In two space dimensions all bounded Radon measures are admissable as such. In higher dimensions the focus is on measures whose support is concentrated on l-sets in the sense of Jonsson and Wallin. It is shown that they may interpreted as elements from a Sobolev space W. So the right hand side is considered as an element from a W-valued Lebesgue space on the time interval. Having this at hand, previous results on maximal (non-autonomous) maximal parabolic regularity apply and show that the solution lies in the corresponding space of maximal parabolic regularity. In contrast to other work in this field we only require absolute minimal smothness for the data of the problem: the domain, the coefficients - and mixed boundary conditions are allowed. Under minimally stronger assumptions we even show the Hölder property in space and time. Overall, this work contains an interplay of geometric measure theory with advanced parabolic theory which delivers as much parabolic regularity for the solution as one can maximally expect.

• A. Glitzky, M. Liero, Uniqueness and regularity of weak solutions of a drift-diffusion system for perovskite solar cells, Preprint no. 3142, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3142 .
  Abstract, PDF (290 kByte)
  We establish a novel uniqueness result for an instationary drift-diffusion model for perovskite solar cells. This model for vacancy-assisted charge transport uses Fermi--Dirac statistics for electrons and holes and Blakemore statistics for the mobile ionic vacancies in the perovskite. Existence of weak solutions and their boundedness was proven in a previous work. For the uniqueness proof, we establish improved integrability of the gradients of the charge-carrier densities. Based on estimates obtained in the previous paper, we consider suitably regularized continuity equations with partly frozen arguments and apply the regularity results for scalar quasilinear elliptic equations by Meinlschmidt & Rehberg, Evolution Equations and Control Theory, 2016, 5(1):147-184.

• J. Ginster, A. Pešić, B. Zwicknagl, Nonlinear interpolation inequalities with fractional Sobolev norms and pattern formation in biomembranes, Preprint no. 3131, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3131 .
  Abstract, PDF (343 kByte)
  We consider a one-dimensional version of a variational model for pattern formation in biological membranes. The driving term in the energy is a coupling between the order parameter and the local curvature of the membrane. We derive scaling laws for the minimal energy. As a main tool we present new nonlinear interpolation inequalities that bound fractional Sobolev seminorms in terms of a Cahn--Hillard/Modica--Mortola energy.

• TH. Eiter, L. Schmeller, Weak solutions to a model for phase separation coupled with finite-strain viscoelasticity subject to external distortion, Preprint no. 3130, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3130 .
  Abstract, PDF (376 kByte)
  We study the coupling of a viscoelastic deformation governed by a Kelvin--Voigt model at equilibrium, based on the concept of second-grade nonsimple materials, with a plastic deformation due to volumetric swelling, described via a phase-field variable subject to a Cahn--Hilliard model expressed in a Lagrangian frame. Such models can be used to describe the time evolution of hydrogels in terms of phase separation within a deformable substrate. The equations are mainly coupled via a multiplicative decomposition of the deformation gradient into both contributions and via a Korteweg term in the Eulerian frame. To treat time-dependent Dirichlet conditions for the deformation, an auxiliary variable with fixed boundary values is introduced, which results in another multiplicative structure. Imposing suitable growth conditions on the elastic and viscous potentials, we construct weak solutions to this quasistatic model as the limit of time-discrete solutions to incremental minimization problems. The limit passage is possible due to additional regularity induced by the hyperelastic and viscous stresses.

• P. Colli, J. Sprekels, Hyperbolic relaxation of the chemical potential in the viscous Cahn--Hilliard equation, Preprint no. 3128, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3128 .
  Abstract, PDF (300 kByte)
  In this paper, we study a hyperbolic relaxation of the viscous Cahn--Hilliard system with zero Neumann boundary conditions. In fact, we consider a relaxation term involving the second time derivative of the chemical potential in the first equation of the system. We develop a well-posedness, continuous dependence and regularity theory for the initial-boundary value problem. Moreover, we investigate the asymptotic behavior of the system as the relaxation parameter tends to 0 and prove the convergence to the viscous Cahn--Hilliard system.

• A. Mielke, R. Rossi, On De Giorgi's lemma for variational interpolants in metric and Banach spaces, Preprint no. 3127, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3127 .
  Abstract, PDF (349 kByte)
  Variational interpolants are an indispensable tool for the construction of gradient-flow solutions via the Minimizing Movement Scheme. De Giorgi's lemma provides the associated discrete energy-dissipation inequality. It was originally developed for metric gradient systems. Drawing from this theory we study the case of generalized gradient systems in Banach spaces, where a refined theory allows us to extend the validity of the discrete energy-dissipation inequality and to establish it as an equality. For the latter we have to impose the condition of radial differentiability of the dissipation potential. Several examples are discussed to show how sharp the results are.

• W. van Oosterhout, Linearization of finite-strain poro-visco-elasticity with degenerate mobility, Preprint no. 3123, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3123 .
  Abstract, PDF (344 kByte)
  A quasistatic nonlinear model for finite-strain poro-visco-elasticity is considered in the Lagrangian frame using Kelvin--Voigt rheology. The model consists of a mechanical equation which is coupled to a diffusion equation with a degenerate mobility. Having shown existence of weak solutions in a previous work, the focus is first on showing boundedness of the concentration using Moser iteration. Afterwards, it is assumed that the external loading is small, and it is rigorously shown that solutions of the nonlinear, finite-strain system converge to solutions of the linear, small-strain system.

• P. Colli, G. Gilardi, A. Signori, J. Sprekels, Solvability and optimal control of a multi-species Cahn--Hilliard--Keller--Segel tumor growth model, Preprint no. 3118, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3118 .
  PDF (367 kByte)

• 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.

• 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.

Talks, Poster

• S. Schindler, On asymptotically self-similar behavior in reaction-diffusion systems, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2025), Section 14.04 ``Applied Analysis'', April 7 - 11, 2025, Poznan University of Technology, Poland, April 10, 2025.

• M. Heida, Upscaling battery models: Connecting microscopic dynamics with macroscopic behavior, 24th GAMM Seminar on Microstructures, January 30 - 31, 2025, Humboldt-Universität zu Berlin, January 31, 2025.

• D. Peschka, Fluid-structure interaction and wetting of soft polymer substrates, Leibniz MMS Days 2025, March 26 - 28, 2025, Leibniz-Institut für Ostseeforschung Warnemünde, March 28, 2025.

• A. Pešić, Nonlinear interpolation inequalities with fractional Sobolev norms and pattern formation in biomembranes, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2025), Poland, April 7 - 11, 2025.

• A. Pešić, Pattern formation in biomembranes: From interpolation inequalities to a scaling law result, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2025), Young Researchers' Minisymposium 4 ``Mulit-scale Phenomena in Magnetic and Elastic Materials'', April 7 - 11, 2025, Poznan University of Technology, Poland, April 7, 2025.

• S. Schindler, On asymptotically self-similar patterns in coupled systems, Mathematics of Wave Phenomena, Minisymposium 01 ``Dissipative Patterns and Waves'', February 24 - 28, 2025, Karlsruhe Institute of Technology (KIT), February 26, 2025.

• M. Thomas, Analysis of a dynamic phase-field fracture model and how spaghettis break, Colloquium in Memory of Prof. Anna-Margarete Sändig, March 27, 2025, Universität Stuttgart.

• M. Thomas, Analysis of non-smooth PDEs in materials modeling - fractures, faults and sea ice, BMS Student Conference 2025, February 19 - 21, 2025, Berlin Mathematical School, February 20, 2025.

• M. Thomas, First-order formulation for dynamic phase-field fracture in viscoelastic materials, Seminar on Nonlinear Partial Differential Equations, Texas A&M University, Department of Mathematics, College Station, USA, March 25, 2025.

• TH. Eiter, Existence results for fluid flow around a periodically deforming body, Frankenstein 25 - Maximal Regularity Methods in Mathematical Fluid Mechanics, March 31 - April 4, 2025, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, Frankenstein, April 1, 2025.

• TH. Eiter, Long-time asymptotics of the damped Euler equations by parabolic scaling, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2025), Section 14.01 ``Applied Analysis'', April 7 - 11, 2025, Poznan University of Technology, Poland, April 8, 2025.

• J. Ginster, A scaling law for a model of epitaxially strained elastic films with dislocations, 24th GAMM Seminar on Microstructures, January 30 - 31, 2025, Humboldt-Universität zu Berlin, January 31, 2025.

• M. Heida, Modeling intercalation processes in batteries, Mixtures: Modeling, Analysis and Computing, February 5 - 7, 2025, Charles University, Prague, Czech Republic, February 6, 2025.

• G. Heinze, Discrete-to-continuum limit for reaction-diffusion systems via variational convergence of gradient systems, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2025), Section 14.04 ``Applied Analysis'', April 7 - 11, 2025, Poznan University of Technology, Poland, April 10, 2025.

• G. Heinze, Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits, Research Seminar ``Numerical Analysis of Stochastic and Deterministic Partial Differential Equations'', Freie Universität Berlin, January 16, 2025.

• M. Liero, EDP convergence for evolutionary systems with gradient flow structure, Calculus of Variations & Applications, May 16, 2025, Università degli Studi di Firenze, Italy.

• T. Dörffel, M. Landstorfer, M. Liero, Modeling battery electrodes with mechanical interactions and multiple phase transistions upon ion insertion, MATH+ Day, Urania Berlin, October 18, 2024.

• L. Ermoneit, B. Schmidt, J. Fuhrmann, A. Sala, N. Ciroth, L. Schreiber, T. Breiten, Th. Koprucki, M. Kantner, Optimal control of a Si/SiGe quantum bus for scalable quantum computing architectures, Quantum Optimal Control: From Mathematical Foundations to Quantum Technologies, MATH+, Berlin, May 21, 2024.

• L. Ermoneit, B. Schmidt, Th. Koprucki, T. Breiten, M. Kantner, Coherent transport of semiconductor spin-qubits: Modeling, simulation and optimal control, MATH+ Day 2024, Urania Berlin, October 2, 2024.

• M. Heida, Materials with discontinuities on many scales, SCCS Days 2024 of the Collaborative Research Center - CRC 1114 ``Scaling Cascades in Complex Systems'', October 28 - 29, 2024, Freie Universität Berlin, October 28, 2024.

• M. O'Donovan, Multi-scale simulation of electronic and transport properties in (Al,Ga)N quantum well systems for UV-C emission, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), September 10 - 13, 2024, WIAS Berlin, September 11, 2024.

• M. O'Donovan, Simulation of the alloy fluctuations on luminescence and transport in AIGaN-based UV-LEDs, XXXV. Heimbach Workshop, September 23 - 27, 2024, Technische Universität Berlin, Mansfeld, September 26, 2024.

• M. O'Donovan, Theoretical investigations on different scales towards novel III-N materials and devices, Roundtable discussion of the SPP 2477 ``Nitrides4Future'', Magdeburg, September 24 - 25, 2024.

• D. Peschka, Dissipative processes in thin film flows, Liquid Thin Films, August 26 - 30, 2024, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, August 27, 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.

• N. Ciroth, A. Sala, L. Ermoneit, Th. Koprucki, M. Kantner, L. Schreiber, Numerical simulation of coherent spin-shuttling in silicon devices across dilute charge defects, Silicon Quantum Electronics Workshop 2024, Davos, Switzerland, September 4 - 6, 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, Asymptotic self-similar behavior in reaction-diffusion systems on $R^n$, Applied Analysis Complex Systems & Dynamics Seminar, Universität Graz, Institut für Mathematik und Wissenschaftliches Rechnen, Austria, November 13, 2024.

• A. Mielke, Asymptotic self-similar behaviour in reaction-diffusion systems on Rd, Dynamical Systems Approaches towards Nonlinear PDEs, August 28 - 30, 2024, Universität Stuttgart, August 29, 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, Balanced-viscosity solutions for generalized gradient systems in mechanics, Frontiers of the Calculus of Variations, September 16 - 20, 2024, University of the Aegean, Karlovasi, Greece, September 17, 2024.

• A. Mielke, Hellinger--Kantorovich (aka WFR) spaces and gradient flows, Optimal Transport from Theory to Applications: Interfacing Dynamical Systems, Optimization, and Machine Learning, March 11 - 15, 2024, WAS Berlin, March 11, 2024.

• A. Mielke, Non-equilibrium steady states for gradient systems with ports, Analysis and Applied Mathematics Seminar, Università Commerciale Luigi Bocconi, Milano, Italy, May 15, 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. Mielke, On EVI flows for the spherical Hellinger distance and the spherical Hellinger--Kantorovich distance, Optimal Transportation and Applications, December 2 - 6, 2024, Scuola Normale Superiore di Pisa, Centro di Ricerca Matematica Ennio De Giorgi, Italy, December 6, 2024.

• A. Mielke, On the stability of NESS in gradient systems with ports, Gradient Flows Face-to-Face 4, September 9 - 12, 2024, Technische Universität München, Raitenhaslach, September 10, 2024.

• J. Sprekels, Optimality conditions in the sparse optimal control of viscous Cahn--Hilliard systems, Seminari di Matematica Applicata, Università degli Studi di Pavia, Dipartimento di Matematica, Italy, April 23, 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. Glitzky, Electrothermal models for organic semiconductor devices, Applied Mathematics and Simulation for Semiconductor Devices (AMaSiS 2024), Berlin, September 10 - 13, 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, 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.

• 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.

• TH. Eiter, Artificial boundary conditions for oscillatory flow past a body, Mathematical Fluid Mechanics, Czech Academy of Sciences, Prague, Czech Republic, August 22, 2024.

• TH. Eiter, Energy-variational solutions for a model for rock deformation, SCCS Days 2024 of the Collaborative Research Center - CRC 1114 ``Scaling Cascades in Complex Systems'', October 28 - 29, 2024, Freie Universität Berlin, October 28, 2024.

• TH. Eiter, Existence of time-periodic flow past a rotating body by uniform resolvent estimates, Seminar on Mathematics of Fluids, Ochanomizu University, Tokyo, Japan, December 5, 2024.

• TH. Eiter, Far-field behavior of oscillatory viscous flow past an obstacle, Oberseminar Analysis und Angewandte Mathematik, Universität Kassel, July 15, 2024.

• TH. Eiter, Far-field behavior of viscous flow past a body driven by time-periodic boundary flux, Mathematical Analysis of Viscous Incompressible Fluid, December 2 - 4, 2024, Kyoto University, Japan, December 3, 2024.

• TH. Eiter, On energy-variational solutions for hyperbolic conservation laws, Mathematics of Fluids in Motion: Recent Results and Trends, November 11 - 15, 2024, Centre International de Rencontres Mathématiques (CIRM), Marseille, France, November 14, 2024.

• TH. Eiter, On the spatially asymptotic structure of oscillating flow past a body, Seminar ``Funktionalanalysis und Stochastische Analysis'', Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, December 19, 2024.

• TH. Eiter, The effect of time-periodic boundary flux on the decay of viscous flow past a body, 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, Discrete-to-continuum limit for reaction-diffusion systems via variational convergence of gradient systems, Oberseminar Angewandte Analysis, Universität Ulm, Institut für Angewandte Analysis, November 25, 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, Fast-slow limits for gradient flows on metric graphs, Annual Workshop of the GAMM Activity Group ``Analysis of PDEs'' 2024, September 16 - 19, 2024, Czech Academy of Sciences, Prague, Czech Republic, September 17, 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.

• M. Kantner, L. Ermoneit, B. Schmidt, A. Sala, N. Ciroth, L. Schreiber, Th. Koprucki, Optimal control of conveyor-mode electron shuttling in a Si/SiGe quantum bus in the presence of charged defects, Silicon Quantum Electronics Workshop 2024, Davos, Switzerland, September 4 - 6, 2024.

• J. Rehberg, Estimates for operator functions, Oberseminar Analysis und Theoretische Physik, Leibniz Universität Hannover, Institut für Analysis, October 15, 2024.

• J. Rehberg, Parabolic equations with measure-valued right hand sides, Forschungsseminar Analysis, Universität Graz, Institut für Mathematik und Wissenschaftliches Rechnen, Austria, October 23, 2024.

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

• W. van Oosterhout, Linearization of finite-strain poro-viscoelasticity with degenerate mobility, Annual Workshop of the GAMM Activity Group ``Analysis of PDEs'' 2024, September 16 - 19, 2024, Czech Academy of Sciences, Prague, Czech Republic, September 16, 2024.

External Preprints

• R. Finn, M. O'Donovan, Th. Koprucki, S. Schulz, Theoretical study of the impact of carrier density screening on Urbach tail energies and optical polarization in (Al,Ga)N quantum well systems, Preprint no. arXiv:2501.16808, Cornell University, 2025, DOI 10.48550/arXiv.2501.16808 .
  Abstract
  Aluminium Gallium Nitride ((Al,Ga)N) presents an ideal platform for designing ultra-violet (UV) light emitters across the entire UV spectral range. However, in the deep-UV spectral range (<280 nm) these emitters exhibit very low quantum efficiencies, which in part is linked to the light polarization characteristics of (Al,Ga)N quantum wells (QWs). In this study we provide insight into the degree of optical polarization of (Al,Ga)N QW systems operating across the UV-C spectral range by means of an atomistic, multi-band electronic structure model. Our model not only captures the difference in valence band ordering in AlN and GaN, it accounts also for alloy disorder induced band mixing effects originating from random alloy fluctuations in (Al,Ga)N QWs. The latter aspect is not captured in widely employed continuum based models. The impact of alloy disorder on the electronic structure is studied in terms of Urbach tail energies, which reflect the broadening of the valence band density of states due to carrier localization effects. We find that especially in wider wells, Urbach tail energies are reduced with increasing carrier densities in the well, highlighting that alloy disorder induced carrier localization effects in (Al,Ga)N QWs are also tightly linked to electrostatic built-in fields. Our calculations show that for QWs designed to emit at the longer wavelength end of the UV-C spectrum, carrier density and well width are of secondary importance for their light emission properties, meaning that one observes mainly transverse electrical polarization. However, for (Al,Ga)N QWs with high Al contents, we find that both well width and carrier density will impact the degree of optical polarization. Our calculations suggest that wider wells will increase the degree of optical polarization and may therefore be a viable option to improve the light extraction efficiency in deep UV light emitters.

• E. Gladin, P. Dvurechensky, A. Mielke , J.-J. Zhu, Interaction-force transport gradient flows, Preprint no. arXiv:2405.17075, Cornell University, 2024, 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.

• B. Schembera, F. Wübbeling, H. Kleikamp, B. Schmidt, A. Shehu, M. Reidelbach, Ch. Biedinger, J. Fiedler, Th. Koprucki, D. Iglezakis, D. Göddeke, Towards a knowledge graph for models and algorithms in applied mathematics, Preprint no. arXiv:2408.10003, Cornell University, 2024, DOI 10.48550/arXiv.2408.10003 .
  Abstract
  Mathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. In order to represent models and algorithms as well as their relationship semantically to make this research data FAIR, two previously distinct ontologies were merged and extended, becoming a living knowledge graph. The link between the two ontologies is established by introducing computational tasks, as they occur in modeling, corresponding to algorithmic tasks. Moreover, controlled vocabularies are incorporated and a new class, distinguishing base quantities from specific use case quantities, was introduced. Also, both models and algorithms can now be enriched with metadata. Subject-specific metadata is particularly relevant here, such as the symmetry of a matrix or the linearity of a mathematical model. This is the only way to express specific workflows with concrete models and algorithms, as the feasible solution algorithm can only be determined if the mathematical properties of a model are known. We demonstrate this using two examples from different application areas of applied mathematics. In addition, we have already integrated over 250 research assets from applied mathematics into our knowledge graph.