GAMM2025

Session

S14: Applied analysis

S14

8 Apr 2025, 16:30

Lecture and Conference Centre

Conveners

S14: Applied analysis: S14.01

• Tomasz Dębiec
• Katharina Hopf

S14: Applied analysis: S14.02

• Tomasz Dębiec
• Aneta Wróblewska-Kamińska

S14: Applied analysis: S14.03

• Manuel Friedrich
• Marita Thomas

S14: Applied analysis: S14.04

• Sebastian Throm
• Katharina Hopf

S14: Applied analysis: S14.05

• Florian Oschmann
• Markus Schmidtchen

S14: Applied analysis: S14.06

• Michael Kniely
• Thomas Eiter

S14: Applied analysis: S14.07

• Francesco De Anna
• Theresa Simon

Description

This session is devoted to the mathematical analysis of natural phenomena and engineering problems. In this area PDEs play a basic role. Therefore lectures discussing analytical aspects of PDE problems as well as problems in the Calculus of Variations are welcome.

306. From compressible to incompressible, MHD with non-conservative boundary condition

Aneta Wróblewska-Kamińska

08/04/2025, 16:30

We consider a general compressible viscous, heat and magnetic conducting fluid described bycompressible Naiver–Stokes–Fourier system coupled with induction equation. In particular we do not assume conservative boundary condition for temperature and allow heating or cooling on the surface of the domain. We are interested in mathematical analysis when Mach, Froude, and Alfvén number are small -...

307. Existence and weak-strong uniqueness of suitable weak solutions to an anisotropic electrokinetic flow model

Luisa Plato

08/04/2025, 16:50

In this presentation, we discuss the existence proof of weak solutions in three spatial dimensions to an anisotropic Navier--Stokes--Nernst--Planck--Poisson system that satisfy an energy inequality. This system models the electrokinetic flow generated by charged particles dissolved in a liquid crystal with a constant director field. The existence proof is based on an approximation scheme and...

308. Long-time asymptotics of the damped Euler equations by parabolic scaling

Thomas Eiter

08/04/2025, 17:10

We study the compressible Euler equations on the real line subject to frictional damping. We assume that the velocity vanishes at spatial infinity, and for the density we prescribe two positive limit values. If these values coincide, solutions convergence to a constant state in the long-time limit. However, in the case of different limit values, there is no convergence towards a steady-state,...

309. Analysis of a viscoplastic Burgers equation

Marita Thomas

08/04/2025, 17:30

We study a viscous Burgers equation, where the viscosity is governed by a positively 1-homogeneous potential. This problem is motivated by the so-called Hibler’s sea ice model, which treats sea ice as a Non-Newtonian fluid, where the stress tensor includes such a term in order to account for the plastic response of the ice. For this simplified model given by the viscoplastic Burgers equation...

310. Darcy's law for inhomogeneous incompressible flows

Florian Oschmann

08/04/2025, 17:50

We investigate the evolutionary incompressible inhomogeneous Navier-Stokes equations in a perforated domain. Provided the inclusions are large enough, we show that, as their number tends to infinity, the limiting system is given by Darcy's law. This result is already known for 1) purely incompressible fluids with constant density, and 2) compressible fluids when the inclusions are the same...

311. Variational modelling of porosity waves

Andrea Zafferi

09/04/2025, 08:30

In this presentation, we address the application of variational methods to partial differential equation (PDE) systems, particularly within the context of geological phenomena. Our work leverages the GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling) framework and its variants, such as damped Hamiltonian or extended gradient systems, to describe porosity waves...

312. Energy-variational structure in evolution equations

Robert Lasarzik

09/04/2025, 08:50

We consider different measure-valued solvability concepts from the literature and show that they could be simplified by using the energy-variational structure of the underlying system of partial differential equations. In the considered examples, we prove that a measure-valued solution can be equivalently expressed as an energy-variational solution. The first concept represents the solution...

314. On some explicit solutions of the linearised Prandtl equations via hypergeometric functions

Francesco De Anna

09/04/2025, 09:10

The Prandtl boundary layer theory has influenced many scientific disciplines, from aerodynamics to rheology. Analytically, it is nowadays widely recognized that the Prandtl equations are ill-posed in Sobolev spaces, even near certain shear flows, unless additional structural assumptions of monotonicity are imposed. However, they are well-posed in Gevrey class 2 spaces and whether...

313. On the connection of the Prandtl equations and the harmonic oscillator

Joshua Kortum

09/04/2025, 09:30

In this talk, we investigate the linearized Prandtl equations around a shear flow with non-degenerate critical point. Under a separation of variable ansatz, we completely classify all separated solutions of the equation and discuss qualitative properties of the latter. The solutions of the corresponding (algebraic) eigenvalue problem can be computed by the well-known harmonic oscillator. As a...

315. On an inhomogeneous coagulation model describing sedimentation

Iulia Cristian

09/04/2025, 09:50

Coagulation equations describe the evolution in time of a system of particles that are characterized by their volume. Multi-dimensional coagulation equations have been used in recent years in order to include information about the system of particles which cannot be otherwise incorporated. Depending on the model, we can describe the evolution of the shape, chemical composition or position in...

316. Magnetic skyrmions

Theresa Simon

09/04/2025, 16:30

In extremely thin ferromagnetic films, an additional interaction, the so-called Dzyaloshinskii-Moriya interaction (DMI), arises in the micromagnetic energy along with a magnetocrystalline anisotropy favoring the magnetization to be out-of-plane. In such materials, topologically nontrivial, point-like configurations of the magnetization called magnetic skyrmions are observed, which are of great...

317. Amplitude equations for the fractional Swift-Hohenberg equation

Sebastian Throm

09/04/2025, 17:10

The Swift-Hohenberg equation arises as a basic pattern forming model. We consider the dynamics of a space fractional version of this model near instability using amplitude equations. More precisely, we prove that there exists an approximation by a Ginzburg-Landau equation near the first bifurcation point.

318. Stress-Modulated Growth in the Presence of Nutrients

Julian Blawid

09/04/2025, 17:30

In this talk, we discuss a quasi-stationary model for stress-modulated growth of elastic materials. Inspired by models for crystal plasticity, the model features a multiplicative decomposition of the total deformation gradient into an elastic part and the growth tensor. The growth tensor is given by the solution to an ordinary differential equation on a suitable Banach space that depends,...

319. On the Derivation of the Timoshenko Beam Model from Nonlinear Elasticity by Gamma-Convergence

Tamara Fastovska

09/04/2025, 17:50

A hierarchy of classical plate models such as nonlinear bending theory, von Karman theory and linearized von Karman theory was derived from three-dimensional nonlinear elasticity by Gamma-convergence (see [2], [3]). The justification of the linear Timoshenko beam model and the Reissner-Mindlin plate model via Gamma-limit of the linear-elasticity model for a three-dimensional body was obtained...

383. Legendre-Hadamard conditions in the nonlinear theory of fiber-reinforced elastic solids and shells

Mircea Birsan

09/04/2025, 18:10

Pointwise convexity conditions of the Legendre-Hadamard type for energy minimizers are derived in the context of a special Cosserat elasticity model for fiber-reinforced solids [1]. In this approach, the Cosserat rotation tensor accounts for the kinematics of embedded fibers, regarded as continuously distributed Kirchhoff rods. Firstly, we obtain new Legendre-Hadamard inequalities for...

321. Localisation Limits and Degenerate Cross-Diffusion Systems

Markus Schmidtchen

10/04/2025, 08:30

In recent years, there has been a spike in interest in multiphase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through the nonlocal Brinkman law or the local Darcy law. While both velocity-pressure relations have been studied in the literature separately, little emphasis has been placed on the fine relationship between them. In this talk, we...

322. Advection and enhanced diffusion in some active scalar problems

Konstantin Kalinin

10/04/2025, 08:50

We start by demonstrating that the interplay between advection and diffusion in the incompressible porous media equation with diffusion -- a classical active scalar transport equation -- can lead to the enhanced dissipation. Subsequently, we derive a scaling limit that perfectly balances these two physical mechanisms. The high degeneracy of the limiting equation prevents us from proving...

323. Discrete-to-continuum limit for reaction-diffusion systems via variational convergence of gradient systems

Georg Heinze

10/04/2025, 09:10

This talk is about a convergence result for the spatial discretization of a reaction-diffusion system. The approximation is based on a homogeneous lattice, where in each node a reaction-ODE system describes the evolution of the concentrations, and where the transport between the different lattice nodes is given by additional exchange reactions. Assuming detailed balance, this large coupled...

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

Artur Stephan

10/04/2025, 09:30

A gradient system (X, ℰ, ℛ) consists of a state space X (a separable, reflexive Banach space), an energy functional ℰ : dom(ℰ) ⊂ X → ℝ ∪ {∞}, and a dissipation potential ℛ : X → [0, ∞), which is convex, lower semicontinuous, and satisfies ℛ(0) = 0. The associated gradient-flow equation is given by

0 ∈ ∂ ℛ(u'(t)) + ∇ ℰ(u(t))   or equivalently   u'(t) ∈ ∂ ℛ*(−∇ ℰ(u(t))).

In this talk,...

325. On asymptotically self-similar behavior in reaction-diffusion systems

Stefanie Schindler

10/04/2025, 09:50

Self-similar behavior is a well-studied phenomenon in extended systems. However, the consideration is often restricted to scalar problems having exact self-similar solutions (e.g. the porous medium equation) or to problems with a trivial behavior at infinity. In this talk, we study a reaction-diffusion system and other related dissipative systems on the whole real line with prescribed...

326. Conditional Exponential Equilibration of Electro-Energy-Reaction-Diffusion Systems

Michael Kniely

10/04/2025, 10:10

We present a thermodynamically consistent framework for reaction-diffusion systems modeling the evolution of a finite number of charged species in a temperature-dependent setting. Thermodynamical consistency guarantees, in particular, that the fundamental laws of charge and energy conservation hold, and that the total entropy is monotone as time evolves. This is achieved by starting from an...

327. On a non-isothermal Allen-Cahn model for tumor growth

Erica Ipocoana

10/04/2025, 14:00

We present a new diffuse interface model describing the growth of a tumor, whose evolution is assumed to be governed by biological mechanisms such as proliferation of cells via nutrient consumption and apoptosis. In this context, the tumor is seen as an expanding mass surrounded by healthy tissues, while the interface in between contains a mixture of both healthy and tumor cells. More...

328. Viscoelastic Phase Separation: Well-posedness and Singular Limit to Viscous Cahn–Hilliard Equation

Moritz Gau

10/04/2025, 14:20

Viscoelastic phase separation plays an important role in biological cells, for instance, RNA or proteins can undergo phase separation to form membrane-less condensates, which is crucial for biological functions. In our contribution, we consider a model for phase separation in polymer solutions consistent with the second law of thermodynamics, introduced by Zhou et al. 2006. For the full model...

329. Analysis of a Cahn-Hilliard model for viscoelastoplastic two phase flows in geodynamics

Fan Cheng

10/04/2025, 14:40

A model describing both viscoelastic and viscoplastic behavior of two-phase flows has been developed. This model is used in geodynamics, e.g., to describe the evolution of fault systems in the lithosphere over geological time scales. The system consists of a momentum balance, an evolution law for an extra stress tensor, and a Cahn-Hilliard-type evolution law that governs phase separation. The...

330. Sharp Interface Reduction of a Mesoscale Model for Two-Species Surfactant Films

Jakob Fuchs

10/04/2025, 15:00

We propose a variational model for two-phase surfactant films separating aqueous and oily fluids. Considering two species of surfactant molecules we describe a phase seperation within the film. The analysis builds on a model for single species lipid biomembranes proposed by Peletier and Röger [ARMA 2009]. We prove a Gamma-convergence result in the limit of vanishing surfactant length and show...

331. A fully coupled Stokes-transport system modeling thermoregulation in human skin

Kilian Hacker

10/04/2025, 15:20

We consider a Stokes flow coupled with advective-diffusive transport in an evolving domain with boundary conditions allowing for inflow and outflow. The domain evolution is induced by the transport process, leading to a fully coupled problem. We prove existence of weak solutions to the system using a fixed-point method which allows us to treat the nonlinear coupling. Our approach aims to model...

334. Γ-Convergence and Stochastic Homogenization of Second-Order Singular Perturbation Models for Phase Transitions

Antonio Flavio Donnarumma

10/04/2025, 15:40

We study the effective behavior of random, heterogeneous, anisotropic, second order phase transitions energies that arise in the study of pattern formations in physical-chemical systems. Specifically, we study the asymptotic behavior, as $\varepsilon$ goes to zero, of random heterogeneous anisotropic functionals in which the second order perturbation competes not only with a double well...

339. Polarization filter as a homogenisation limit for Maxwell's equations

David Wiedemann

10/04/2025, 16:30

We consider the time harmonic Maxwell equations in a complex geometry. We are interested in complex geometries that model polarization filters or Faraday cages. We study the situation that the domain of interest contains inclusions with large or infinite conductivity, the inclusions are distributed in a periodic fashion along a surface. The periodicity is h\>0 and the shape of the inclusion...

341. Time-harmonic Maxwell’s equations in half-waveguides

Tim Schubert

10/04/2025, 16:50

Maxwell’s equations are considered in a half-waveguide Ω₊ := ℝ₊ × S where S ⊂ ℝ² is a bounded Lipschitz domain and ℝ₊ = (0,∞). The electric permittivity ε and the magnetic permeability μ are assumed to be strictly positive and periodic outside a compact set. Our Maxwell system is accompanied by a radiation condition that was introduced and investigated in [1]. We give a result on existence and...

349. Boundary-field formulation for transient electromagnetic scattering by dielectric scatterers and coated conductors

Wolfgang Wendland

10/04/2025, 17:10

In this lecture, the authors analyse the regularity properties of the solution to time domain transmission problems with respect to the data in electromagnetism using the Laplace approach. The model problems are considered: (i) the analysis is conducted by considering first the scattered and transmitted fields at a dielectric surface Gamma. (ii) the slight changes to be made in the case of a...

354. Existence and Uniqueness of Fractional Integro-Differential Equations with Singular Kernel

Pratibha Verma

10/04/2025, 17:30

This work examines the solution of fractional integro-differential equations using the Adomian decomposition method, utilizing a unique kernel operator. The fractional term is specified in the Caputo framework, and the solution is derived by an approximate and analytical method. The fixed-point theorems are employed to establish the existence and uniqueness of the solution. Further, the...

355. Global Solver based on the Sperner-Lemma and Mazurkewicz-Knaster-Kuratowski-Lemma based proof of the Brouwer Fixed-Point theorem

Thilo Moshagen

10/04/2025, 17:50

A fixed-point solver for mappings from a simplex into itself that is gradient-free, global, and requires d function evaluations for halving the error is presented, where d is the dimension. It is based on topological arguments and uses the constructive proof of the Mazurkewicz-Knaster-Kuratowski lemma when used as part of the proof for Brouwer's Fixed-Point theorem. Its explorative placing of...

363. On regularity for systems of elliptic equations with mixed boundary conditions

Michael Tsopanopoulos

10/04/2025, 18:10

In this talk, we present new insights into the regularity of elliptic systems within certain three-dimensional polyhedral domains under mixed boundary conditions.

Our approach begins by analyzing geometric model problems, focusing on a \linebreak two-dimensional angle that naturally extends to edges in three dimensions.

Utilizing algebraic properties of the elliptic system, we construct...

369. On the passage from nonlinear to linearized viscoelastodynamics

Malte Kampschulte

11/04/2025, 08:30

In the spirit of preceeding results for the quasistatic case, we study the limiting process from nonlinear to linearized viscoelastodynamics for a general frame-indifferent model of Kelvin-Voigt rheology. We prove existence of weak solutions to both cases using a recently developed variational time-delayed approach. We then prove that solutions to the nonlinear model indeed converge to the...

378. Positive temperature in nonlinear thermoviscoelasticity and the derivation of linearized models

Lennart Machill

11/04/2025, 08:50

According to the Nernst theorem or, equivalently, the third law of thermodynamics, the absolute zero temperature is not attainable. Starting with an initial positive temperature, we show that there exist solutions to a Kelvin-Voigt model for quasi-static nonlinear thermoviscoelasticity at a finite-strain setting [Mielke-Roubíček '20], obeying an exponential-in-time lower bound on the...

320. Dynamic Optimal Transport with Optimal Preferential Paths

Juliane Krautz

11/04/2025, 09:10

In this talk we study a dynamic optimal transport type problem on a compact set (bulk) coupled with a non-intersecting and sufficiently regular curve. On each of them, a Benamou-Brenier type dynamic optimal transport problem is considered, yet with an additional mechanism that allows the exchange (at a cost) of mass between bulk and curve. In the respective actions, we allow for non-linear...

386. Balanced viscosity solutions for rate-independent systems with state-dependent dissipation and applications in non-associated plasticity

Samira Boddin

11/04/2025, 09:30

Various solution concepts for rate-independent systems have been developed, generally leading to different solutions. If the underlying energy functional is not convex or the dissipation potential depends not only on the rate but also on the state of the solution, a discontinuous evolution may be necessary. The concept of balanced viscosity (BV) solutions allows such temporal jumps and shows...

387. Linearization of quasistatic evolution in fracture

Manuel Friedrich

11/04/2025, 09:50

In this talk, I present a linearization result for quasistatic fracture evolution in nonlinear elasticity. As the stiffness of the material tends to infinity, we show that rescaled displacement fields and their associated crack sets converge to a solution of quasistatic crack growth in linear elasticity without any a priori assumptions on the geometry of the crack set. The proof relies on a...

388. Characterizing BV- and BD-ellipticity for a class of positively 1-homogeneous surface energy densities

Dominik Engl

11/04/2025, 10:10

Lower semicontinuity of surface energies in integral form is known to be equivalent to BV-ellipticity of the surface density. In this talk, we prove that BV-ellipticity coincides with the simpler notion of biconvexity for a class of densities that depend only on the jump height and jump normal and are positively 1-homogeneous in the first argument. The second main result is the analogous...