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- ItemExploring families of energy-dissipation landscapes via tilting -- Three types of EDP convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Mielke, Alexander; Montefusco, Alberto; Peletier, Mark A.This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that well-known techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the Energy-Dissipation Principle that provides a variational formulation to gradient-flow equations that allows one to apply techniques from Γ-convergence of functional on states and functionals on trajectories.
- ItemRelating a rate-independent system and a gradient system for the case of one-homogeneous potentials(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Mielke, AlexanderWe consider a non-negative and one-homogeneous energy functional $mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-inpendent system given in terms of the time-dependent functional $mathcal E(t,u)=t mathcal J(u)$ and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutins of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.
- ItemA rigorous derivation and energetics of a wave equation with fractional damping(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Mielke, Alexander; Netz, Roland R.; Zendehroud, SinaWe consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water-air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally-damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy-dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally-damped wave equation with a time derivative of order 3/2.
- ItemThermoviscoelasticity in Kelvin--Voigt rheology at large strains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Mielke, Alexander; Roubíček, TomášThe frame-indifferent thermodynamically-consistent model of thermoviscoelasticity at large strain is formulated in the reference configuration with using the concept of the second-grade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under time-dependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the determinant of the deformation gradient. The heat transfer is governed by the Fourier law in the actual deformed configuration, which leads to a nontrivial description when pulled back into the reference configuration. Existence of weak solutions in the quasistatic setting, i.e. inertial forces are ignored, is shown by time discretization.
- ItemOn two coupled degenerate parabolic equations motivated by thermodynamics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Mielke, AlexanderWe 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.
- ItemBalanced-Viscosity solutions to infinite-dimensional multi-rate systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Mielke, Alexander; Rossi, RiccardaWe 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.
- ItemGlobal existence analysis of energy-reaction-diffusion systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Fischer, Julian; Hopf, Katharina; Kniely, Michael; Mielke, AlexanderWe establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.
- ItemMulti-dimensional modeling and simulation of semiconductor nanophotonic devices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Kantner, Markus; Höhne, Theresa; Koprucki, Thomas; Burger, Sven; Wünsche, Hans-Jürgen; Schmidt, Frank; Mielke, Alexander; Bandelow, UweSelf-consistent modeling and multi-dimensional simulation of semiconductor nanophotonic devices is an important tool in the development of future integrated light sources and quantum devices. Simulations can guide important technological decisions by revealing performance bottlenecks in new device concepts, contribute to their understanding and help to theoretically explore their optimization potential. The efficient implementation of multi-dimensional numerical simulations for computer-aided design tasks requires sophisticated numerical methods and modeling techniques. We review recent advances in device-scale modeling of quantum dot based single-photon sources and laser diodes by self-consistently coupling the optical Maxwell equations with semiclassical carrier transport models using semi-classical and fully quantum mechanical descriptions of the optically active region, respectively. For the simulation of realistic devices with complex, multi-dimensional geometries, we have developed a novel hp-adaptive finite element approach for the optical Maxwell equations, using mixed meshes adapted to the multi-scale properties of the photonic structures. For electrically driven devices, we introduced novel discretization and parameter-embedding techniques to solve the drift-diffusion system for strongly degenerate semiconductors at cryogenic temperature. Our methodical advances are demonstrated on various applications, including vertical-cavity surface-emitting lasers, grating couplers and single-photon sources.
- ItemModeling of chemical reaction systems with detailed balance using gradient structures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Maas, Jan; Mielke, AlexanderWe consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary Γ-convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.
- ItemOn the Darwin--Howie--Whelan equations for the scattering of fast electrons described by the Schrödinger equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Koprucki, Thomas; Maltsi, Anieza; Mielke, AlexanderThe Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrödinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the two-beam and the systematic-row approximation.