arXiv subjects

Dirk Langemann

Publications and source records attributed to Dirk Langemann.

5 recordsLinked to original sources

arXiv2025

On the asymptotic properties of solutions to a nonlinear transmission problem for a Bresse beam with thermal damping

A nonlinear transmission problem for an arch beam, which consists of two parts with different material properties is considered. One of the parts is subjected to thermal damping while another one is undamped. The thermal damping affects only the shear angle and the longitudinal equation within the damped part, which can have any positive length. We prove the well-posedness of the system in the energy space and establish the existence of a compact global attractor under certain conditions imposed solely on the coefficients of the damped part. Additionally, we study singular limits of the system as the curvature of the beam tends to zero or in case both the curvature vanishes and the shear moduli tend to infinity. Numerical simulations are also carried out to model these limiting behaviors.

math.AP↗

arXiv2024

Impact of topography and combustion functions on fire front propagation in an advection-diffusion-reaction model for wildfires

Given the recent increase in wildfires, developing a better understanding of their dynamics is crucial. For this purpose, the advection-diffusion-reaction model has been widely used to study wildfire dynamics. In this study, we introduce the previously unconsidered influence of topography through an additional advective term. Furthermore, we propose a linear term for the combustion function, comparing it with the commonly used Arrhenius law to offer a simpler model for further analysis. Our findings on the model's dynamics are supported by numerical simulations showing the differences of model extensions and approximations.

physics.flu-dyn↗

arXiv2024

Qualitative properties of solutions to a nonlinear transmission problem for an elastic Bresse beam

We consider a nonlinear transmission problem for a Bresse beam, which consists of two parts, damped and undamped. The mechanical damping in the damped part is present in the shear angle equation only, and the damped part may be of arbitrary positive length. We prove well-posedness of the corresponding PDE system in energy space and establish existence of a regular global attractor under certain conditions on nonlinearities and coefficients of the damped part only. Moreover, we study singular limits of the problem when $l\to 0$ or $l\to 0$ simultaneously with $k_i\to +\infty$ and perform numerical modelling for these processes.

math.AP↗

arXiv2023

Expand-contract plasticity on the real line

The article deals with plastic and non-plastic sub-spaces $A$ of the real line ${\mathbb{R}}$ with the usual Euclidean metric $d$. It investigates non-expansive bijections, proves properties of such maps and demonstrates their relevance by hands of examples. Finally, it is shown that the plasticity property of a sub-space $A$ contains at least two complementary questions, a purely geometric and a topological one. Both contribute essential aspects to the plasticity property and get more critical in higher dimensions and more abstract metric spaces.

math.FA↗

arXiv2022

Longterm existence of solutions of a reaction diffusion system with non-local terms modeling an immune response -- an interpretation-orientated proof

This paper shows the global existence and boundedness of solutions of a reaction diffusion system modeling liver infections. The existence proof is presented step by step and the focus lies on the interpretation of intermediate results in the context of liver infections which is modeled. Non-local effects in the dynamics between the virus and the immune system cells coming from the immune response in the lymphs lead to an integro-partial differential equation. While existence theorems for parabolic partial differential equations are textbook examples in the field, the additional integral term requires new approaches to proving the global existence of a solution. This allows to set up an existence proof with a focus on interpretation leading to more insight in the system and in the modeling perspective at the same time. We show the boundedness of the solution in the $L^1(Ω)$- and the $L^2(Ω)$-norms, and use these results to prove the global existence and boundedness of the solution. A core element of the proof is the handling of oppositely acting mechanisms in the reaction term, which occur in all population dynamics models and which results in reaction terms with opposite monotonicity behavior. In the context of modeling liver infections, the boundedness in the $L^\infty(Ω)$-norm has practical relevance: Large immune responses lead to strong inflammations of the liver tissue. Strong inflammations negatively impact the health of an infected person and lead to grave secondary diseases. The gained rough estimates are compared with numerical tests.

math.AP↗