arXiv subjects

Alexander Mielke

Publications and source records attributed to Alexander Mielke.

At least 37 records · Page 2Linked to original sources

arXiv2024

A general thermodynamical model for finitely-strained continuum with inelasticity and diffusion, its GENERIC derivation in Eulerian formulation, and some application

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.

math-ph↗

arXiv2023

Evolutionary Variational Inequalities on the Hellinger-Kantorovich and Spherical Hellinger-Kantorovich spaces

We study the minimizing movement scheme for families of geodesically semi-convex functionals defined on either the Hellinger-Kantorovich or the Spherical Hellinger-Kantorovich space. By exploiting some of the finer geometric properties of those spaces, we prove that the sequence of curves, which are produced by geodesically interpolating the points generated by the minimizing movement scheme, converges to curves that satisfy the Evolutionary Variational Inequality (EVI), when the time step goes to 0.

math.AP↗

arXiv2023

Non-equilibrium steady states as saddle points and EDP-convergence for slow-fast gradient systems

The theory of slow-fast gradient systems leads in a natural way to non-equilibrium steady states, because on the slow time scale the fast subsystem stays in steady states that are controlled by the interaction with the slow system. Using the theory of convergence of gradient systems depending on a small paramter $\varepsilon$ (here the ratio between the slow and the fast time scale) in the sense of the "energy-dissipation principle" (EDP) shows that there is a natural characterization of these non-equilibrium steady states as saddle points of a Lagrangian where the slow variables are fixed. We give applications to slow-fast reaction-diffusion systems based on the so-called cosh-type gradient structure for reactions. It is shown that two binary reaction give rise to a ternary reaction with a state-dependent reaction coefficient. Moreover, we show that a reaction-diffusion equation with a thin membrane-like layer convergences to a transmission condition, where the formerly quadratic dissipation potential for diffusion convergences to a cosh-type dissipation potential for the transmission in the membrane limit.

math.AP↗

arXiv2023

Fine properties of geodesics and geodesic $λ$-convexity for the Hellinger-Kantorovich distance

We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger--Kantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton--Jacobi equation arising in the dual dynamic formulation of HK, which are sufficiently strong to construct a characteristic transport-dilation flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of HK geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic $λ$-convexity with respect to the Hellinger--Kantorovich distance. Examples of geodesically convex functionals are provided.

math.AP↗

arXiv2023

On time-splitting methods for gradient flows with two dissipation mechanisms

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.

math.AP↗

arXiv2023

An introduction to the analysis of gradients systems

The present notes provide an extended version of a small lecture course given at the Humboldt Universität zu Berlin in the Winter Term 2022/23 (of 36 hours). The material starting in Section 5.4 was added afterwards. The aim of these notes to give an introductory overview on the analytical approaches for gradient-flow equations in Hilbert spaces, Banach spaces, and metric spaces and to show that on the first entry level these theories have a lot in common. The theories and their specific setups are illustrated by suitable examples and counterexamples.

math-ph↗

arXiv2023

Existence of similarity profiles for diffusion equations and systems

We study the existence of similarity profiles for diffusion equations and reaction diffusion systems on the real line, where the different nontrivial limits are imposed for $ x \to -\infty$ and $x \to +\infty$. Theses similarity profiles solve a coupled system of nonlinear ODEs that can be treated by monotone operator theory.

math.AP↗

arXiv2023

Convergence to self-similar profiles in reaction-diffusion systems

We study a reaction-diffusion system on the real line, where the reactions of the species are given by one reversible reaction according to the mass-action law. We describe different positive limits at both sides of infinity and investigate the long-time behavior. Rescaling space and time according to the parabolic scaling, we show that solutions converge exponentially to a constant profile. In the original variables these profiles correspond to asymptotically self-similar behavior describing the diffusive mixing or equilibration of the different states at infinity. Our method provides global exponential convergence for all initial states with finite relative entropy.

math.AP↗

arXiv2023

Self-Similar Pattern in Coupled Parabolic Systems as Non-Equilibrium Steady States

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.

math.AP↗

arXiv2022

Qualitative study of a geodynamical rate-and-state model for elastoplastic shear flows in crustal faults

The Dieterich-Ruina rate-and-state friction model is transferred to a bulk variant and the state variable (aging) influencing the dissipation mechanism is here combined also with a damage influencing standardly the elastic response. As the aging has a separate dynamics, the overall model does not have a standard variational structure. A one-dimensional model is investigated as far as the steady-state existence, localization of the ataclastic core, and its time response, too. Computational experiments with a damage-free variant show stick-slip behavior (i.e. seismic cycles of tectonic faults) as well as stable slip under very large velocities.

math.AP↗

arXiv2022

On the existence of global-in-time weak solutions and scaling laws for Kolmogorov's two-equation model of turbulence

This paper is concerned with Kolmogorov's two-equation model for the free turbulence in three dimensions. We first discuss scaling laws for slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's under space-periodic boundary conditions in a cube. To this end, we provide new a priori estimates and invoke existence result for pseudo-monotone operators.

math.AP↗

arXiv2022

Symmetries in TEM imaging of crystals with strain

TEM images of strained crystals often exhibit symmetries, the source of which is not always clear. To understand these symmetries we distinguish between symmetries that occur from the imaging process itself and symmetries of the inclusion that might affect the image. For the imaging process we prove mathematically that the intensities are invariant under specific transformations. A combination of these invariances with specific properties of the strain profile can then explain symmetries observed in TEM images. We demonstrate our approach to the study of symmetries in TEM images using selected examples in the field of semiconductor nanostructures such as quantum wells and quantum dots.

cond-mat.mtrl-sci↗

arXiv2022

On two coupled degenerate parabolic equations motivated by thermodynamics

We propose 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. Because of the degeneracies there are solutions with nontrivial support that may grow in time 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. We discuss the cross-over of these two phenomena in terms of the associated planar traveling waves. Moreover, 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.

math.AP↗

arXiv2021

Leray-Hopf solutions to a viscoelastoplastic fluid model with nonsmooth stress-strain relation

We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba-Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.

math.AP↗

arXiv2021

Balanced-Viscosity solutions to infinite-dimensional multi-rate systems

We consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable $u$ having a viscous damping with relaxation time $\varepsilon^α$ and an internal variable $z$ with relaxation time $\varepsilon$ we obtain different limits for the three cases $α\in (0,1)$, $α=1$ and $α>1$. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics.

math.AP↗

arXiv2021

Global existence analysis of energy-reaction-diffusion systems

We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.

math.AP↗

arXiv2021

Coarse-graining via EDP-convergence for linear fast-slow reaction systems

We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.

math-ph↗

arXiv2021

On the Darwin-Howie-Whelan equations for the scattering of fast electrons described by the Schrödinger equation

The Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrödinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the two-beam and the systematic-row approximation.

math-ph↗