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Anja Schlömerkemper

Publications and source records attributed to Anja Schlömerkemper.

At least 19 recordsLinked to original sources
arXiv2026

Existence of homeomorphic minimizers via mappings of finite distortion in compressible magnetoelasticity

We establish the existence of an energy minimizer for a variational model of compressible magnetoelastic solids. The analysis is carried out in a new admissible class of deformations consisting of mappings of finite distortion, which extends previously available existence frameworks. A key ingredient is a compactness result under the critical integrability assumption on the outer distortion coefficient, which significantly weakens the regularity requirements imposed in earlier works. To obtain this result, we prove a diameter estimate for finite-distortion mappings satisfying the Ciarlet-Nečas condition and derive an open mapping theorem under the optimal integrability assumption, that is, the outer distortion is in $L^{n-1}$. This provides a partial positive result in the direction of the Iwaniec-Šverák conjecture and implies that admissible deformations are homeomorphisms. These topological and compactness properties allow us to apply the direct method of the calculus of variations and establish the existence of minimizers for compressible magnetoelastic solids within the admissible class of deformations consisting of mappings of finite distortions.

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arXiv2025

A study of ferronematic thin films including a stray field energy

Ferronematic materials are colloidal suspensions of magnetic particles in liquid crystals. They are complex materials with potential applications in display technologies, sensors, microfluidics devices, etc. We consider a model for ferronematics in a 2D domain with a variational approach. The proposed free energy of the ferronematic system depends on the Landau-de Gennes (LdG) order parameter $\mathbf{Q}$ and the magnetization $\mathbf{M}$, and incorporates the complex interaction between the liquid crystal molecules and the magnetic particles in the presence of an external magnetic field $\mathbf{H}_{ext}$. The energy functional combines the Landau-de Gennes nematic energy density and energy densities from the theory of micromagnetics including (an approximation of) the stray field energy and energetic contributions from an external magnetic field. For the proposed ferronematic energy, we first prove the existence of an energy minimizer and then the uniqueness of the minimizer in certain parameter regimes. Secondly, we numerically compute stable ferronematic equilibria by solving the gradient flow equations associated with the proposed ferronematic energy. The numerical results show that the stray field influences the localization of the interior nematic defects and magnetic vortices.

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arXiv2024

A variational approach to the modeling of compressible magnetoelastic materials

We analyze a model of the evolution of a (solid) magnetoelastic material. More specifically, the model we consider describes the evolution of a compressible magnetoelastic material with a non-convex energy and coupled to a gradient flow equation for the magnetization in the quasi-static setting. The viscous dissipation considered in this model induces an extended material derivative in the magnetic force balance. We prove existence of weak solutions based on De Giorgi's minimizing movements scheme, which allows us to deal with the non-convex energy as well as the non-convex state space for the deformation. In the application of this method we rely on the fact that the magnetic force balance in the model can be expressed in terms of the same energy and dissipation potentials as the equation of motion, allowing us to model the functional for the discrete minimization problem based on these potentials.

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arXiv2024

Modeling of fluid-rigid body interaction in an electrically conducting fluid

We derive a mathematical model for the motion of several insulating rigid bodies through an electrically conducting fluid. Starting from a universal model describing this phenomenon in generality, we elaborate (simplifying) physical assumptions under which a mathematical analysis of the model becomes feasible. Our main focus lies on the derivation of the boundary and interface conditions for the electromagnetic fields as well as the derivation of the magnetohydrodynamic approximation carried out via a nondimensionalization of the system.

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arXiv2023

Fluid-rigid body interaction in an incompressible electrically conducting fluid

We analyze a mathematical model that describes the interaction between an insulating rigid body and an incompressible electrically conducting fluid surrounding it. The model as well as the mathematical analysis involve the fields of fluid-structure interaction and of magnetohydrodynamics. Our main result shows the existence of a weak solution to the corresponding system of partial differential equations. The proof relies on a use of a time discretization via the Rothe method to decouple the system. This allows us to deal with test functions, depending on the position of the moving body and therefore on the solution of the system, in the weak formulation of the induction equation. The proof moreover makes use of the Brinkman penalization in order to cope with the mechanical part of the problem.

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arXiv2022

Temperature dependent extensions of the Cahn-Hilliard equation

The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In this paper we focus on the dynamics of these binary media, when the underlying temperature is not constant. The aim of this paper is twofold. We first derive two distinct models that extend the classical Cahn-Hilliard equation with an evolutionary equation for the absolute temperature. Secondly, we analyse the local well-posedness of classical solutions for one of these systems. Our modelling introduces the systems of PDEs by means of a general and unified formalism. This formalism couples standard principles of mechanics together with the main laws of thermodynamics. Our work highlights how certain assumptions on the transport of the temperature effect the overall physics of the systems. The variety of these thermodynamically consistent models opens the question of which one should be more appropriate. Our analysis shows that one of the derived models might be more desirable to the well-posedness theory of classical solutions, a property that might be natural as a selection criteria.

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arXiv2021

Onset of fracture in random heterogeneous particle chains

In mechanical systems it is of interest to know the onset of fracture in dependence of the boundary conditions. Here we study a one-dimensional model which allows for an underlying heterogeneous structure in the discrete setting. Such models have recently been studied in the passage to the continuum by means of variational convergence ($Γ$-convergence). The $Γ$-limit results determine thresholds of the boundary condition, which mark a transition from purely elastic behaviour to the occurrence of a crack. In this article we provide a notion of fracture in the discrete setting and show that its continuum limit yields the same threshold as that obtained from the $Γ$-limit. Since the calculation of the fracture threshold is much easier with the new method, we see a good chance that this new approach will turn out useful in applications.

math.AP↗
arXiv2021

Strong well-posedness, stability and optimal control theory for a mathematical model for magneto-viscoelastic fluids

In this article, we study the strong well-posedness, stability and optimal control of an incompressible magneto-viscoelastic fluid model in two dimensions. The model consists of an incompressible Navier--Stokes equation for the velocity field, an evolution equation for the deformation tensor, and a gradient flow equation for the magnetization vector. First, we prove that the model under consideration posseses a global strong solution in a suitable functional framework. Second, we derive stability estimates with respect to an external magnetic field. Based on the stability estimates we use the external magnetic field as the control to minimize a cost functional of tracking-type. We prove existence of an optimal control and derive first-order necessary optimality conditions. Finally, we consider a second optimal control problem, where the external magnetic field, which represents the control, is generated by a finite number of fixed magnetic field coils.

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arXiv2021

Existence of weak solutions to a diffuse interface model involving magnetic fluids with unmatched densities

In this article we prove the global existence of weak solutions for a diffuse interface model in a bounded domain (both in 2D and 3D) involving incompressible magnetic fluids with unmatched densities. The model couples the incompressible Navier-Stokes equations, gradient flow of the magnetization vector and the Cahn-Hilliard dynamics describing the partial mixing of two fluids. The density of the mixture depends on an order parameter and the modelling, specifically the density dependence, is inspired from Abels, Garcke and Grün 2011.

math.AP↗
arXiv2021

Derivation of a variational model for brittle fracture from a random heterogeneous particle chain

A mathematical continuum limit of the interaction energy of a random particle chain is shown to yield new insight into the effect of microscopic heterogeneities on macroscopic fracture laws in brittle materials. We derive a formula which yields that either elastic behaviour or a crack is energetically preferred. The formula explicitly shows the dependence on the boundary condition and the microstructure of the chain. The mathematical analysis is based on a variational convergence $Γ$-convergence of convex-concave potentials together with ergodic theorems which are common tools in stochastic homogenization.

math.AP↗
arXiv2021

Struwe-like solutions for an evolutionary model of magnetoviscoelastic fluids

In this work we investigate the existence and uniqueness of Struwe-like solutions for a system of partial differential equations modeling the dynamics of magnetoviscoelastic fluids. The considered system couples a Navier-Stokes type equation with a dissipative equation for the deformation tensor and a Landau-Lifshitz-Gilbert type equation for the magnetization field. The main purpose is to establish a well-posedness theory in a two-dimensional periodic domain under standard assumption of critical regularity for the (possibly large) initial data. We prove that the considered weak solutions are everywhere smooth, except for a discrete set of time values. The proof of the uniqueness is based on suitable energy estimates for the solutions within a functional framework which is less regular than the one of the Struwe energy level. These estimates rely on several techniques of harmonic analysis and paradifferential calculus.

math.AP↗
arXiv2020

Global existence of weak solutions to a diffuse interface model for magnetic fluids

This article is devoted to the derivation and analysis of a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids. The system consists of the incompressible Navier-Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn-Hilliard equations. We show global in time existence of weak solutions to the system using the time discretization method.

math.AP↗
arXiv2019

Dissipative solutions to a system for the flow of magnetoviscoelastic materials

We address the question of global in time existence of solutions to a magnetoviscoelastic system with general initial data. We show that the notion of dissipative solutions allows to prove such an existence in two and three dimensions. This extends an earlier result for the viscoelastic subsystem to the setting which includes the magnetization vector and its evolution in terms of a Landau-Lifshitz-Gilbert equation.

math.AP↗
arXiv2019

Mechanical behaviour of heterogeneous nanochains in the $Γ$-limit of stochastic particle systems

Nanochains of atoms, molecules and polymers have gained recent interest in the experimental sciences. This article contributes to an advanced mathematical modeling of the mechanical properties of nanochains that allow for heterogenities, which may be impurities or a deliberately chosen composition of different kind of atoms. We consider one-dimensional systems of particles which interact through a large class of convex-concave potentials, which includes the classical Lennard-Jones potentials. We allow for a stochastic distribution of the material parameters and investigate the effective behaviour of the system as the distance between the particles tends to zero. The mathematical methods are based on $Γ$-convergence, which is a suitable notion of convergence for variational problems, and on ergodic theorems as is usual in the framework of stochastic homogenization. The allowed singular structure of the interaction potentials causes mathematical difficulties that we overcome by an approximation. We consider the case of $K$ interacting neighbours with $K\in \mathbb{N}$ arbitrary, i.e., interactions of finite range.

math.AP↗
arXiv2019

External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a $Γ$-development

This paper is concerned with equilibrium configurations of one-dimensional particle system with non-convex nearest-neighbour and next-to-nearest-neighbour interactions and its passage to the continuum. The goal is to derive compactness results for a $Γ$-development of the energy with the novelty that external forces are allowed. In particular, the forces may depend on Lagrangian or Eulerian coordinates. Our result is based on a new technique for deriving compactness results which are required for calculating the first-order $Γ$-limit: instead of comparing a configuration of $n$ atoms to a global minimizer of the $Γ$-limit, we compare the configuration to a minimizer in some subclass of functions which in some sense are "close to" the configuration. This new technique is required due to the additional presence of forces with non-convex potentials. The paper is complemented with the study of the minimizers of the $Γ$-limit.

math.AP↗
arXiv2019

Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity

The paper is concerned with the analysis of an evolutionary model for magnetoviscoelastic materials in two dimensions. The model consists of a Navier-Stokes system featuring a dependence of the stress tensor on elastic and magnetic terms, a regularized system for the evolution of the deformation gradient and the Landau-Lifshitz-Gilbert system for the dynamics of the magnetization. First, we show that our model possesses global in time weak solutions, thus extending work by Benešová et al. 2018. Compared to that work, we include the stray field energy and relax the assumptions on the elastic energy density. Second, we prove the local in time existence of strong solutions. Both existence results are based on the Galerkin method. Finally, we show a weak-strong uniqueness property.

math.AP↗
arXiv2018

Shape memory alloys as gradient-polyconvex materials

We show existence of an energetic solution to a model of shape memory alloys in which the elastic energy is described by means of a gradient-polyconvex functional. This allows us to show existence of a solution based on weak continuity of nonlinear minors of deformation gradients in Sobolev spaces. Resulting deformations are orientation-preserving and injective everywhere in a domain representing the specimen.

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