arXiv subjects

Klemens Fellner

Publications and source records attributed to Klemens Fellner.

30 records · Page 2Linked to original sources

arXiv2016

Growth behaviour of periodic tame friezes

We examine the growth behaviour of the entries occurring in $n$-periodic tame friezes of real numbers. Extending \cite{T}, we prove that generalised recursive relations exist between all entries of such friezes. These recursions are parametrised by a sequence of so-called growth coefficients, which are shown to satisfy itself a recursive relation. Thus, all growth coefficients are determined by a \emph{principle growth coefficients}, which can be read off directly from the frieze. We place special emphasis on periodic tame friezes of positive integers, specifying the values the growth coefficients take for any such frieze. We establish that the growth coefficients of the pair of friezes arising from a triangulation of an annulus coincide. The entries of both are shown to grow asymptotically exponentially, while triangulations of a punctured disc are seen to provide the only friezes of linear growth.

math.CO↗

arXiv2016

Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks

The quantitative convergence to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks with mass action kinetics is studied by using the so-called entropy method. In the first part of the paper, by deriving explicitly the entropy dissipation, we show that for complex balanced systems without boundary equilibria, each trajectory converges exponentially fast to the unique complex balance equilibrium. Moreover, a constructive proof is proposed to explicitly estimate the rate of convergence in the special case of a cyclic reaction. In the second part of the paper, complex balanced systems with boundary equilibria are considered. We investigate two specific cases featuring two and three chemical substances respectively. In these cases, the boundary equilibria are shown to be unstable in some sense, so that exponential convergence to the unique strictly positive equilibrium can also be proven.

math.AP↗

arXiv2015

Exponential decay towards equilibrium and global classical solutions for nonlinear reaction-diffusion systems

We consider a system of reaction-diffusion equations describing the reversible reaction of two species $\mathcal{U}, \mathcal{V}$ forming a third species $\mathcal{W}$ and vice versa according to mass action law kinetics with arbitrary stochiometric coefficients (equal or larger than one). Firstly, we prove existence of global classical solutions via improved duality estimates under the assumption that one of the diffusion coefficients of $\mathcal{U}$ or $\mathcal{V}$ is sufficiently close to the diffusion coefficient of $\mathcal{W}$. Secondly, we derive an entropy entropy-dissipation estimate, that is a functional inequality, which applied to global solutions of these reaction-diffusion system proves exponential convergence to equilibrium with explicit rates and constants.

math.AP↗

arXiv2015

Quasi-steady-state approximation and numerical simulation for a volume-surface reaction-diffusion system

The asymmetric stem-cell division of Drosophila SOP precursor cells is driven by the asymmetric localisation of the key protein Lgl (Lethal giant larvae) during mitosis, when Lgl is phosphorylated by the kinase aPKC on a subpart of the cortex and subsequently released into the cytoplasm. In this paper, we present a volume-surface reaction-diffusion system, which models the localisation of Lgl within the cell cytoplasm and on the cell cortex. We prove well-posedness of global solutions as well as regularity of the solutions. Moreover, we rigorously perform the fast reaction limit to a reduced quasi-steady-state approximation system, when phosphorylated Lgl is instantaneously expelled from the cortex. Finally, we apply a suitable first order finite element scheme to simulate and discuss interesting numerical examples, which illustrate i) the influence of the presence/absence of surface-diffusion to the behaviour of the system and the complex balance steady state and ii) the dependency on the release rate of phosphorylated cortical Lgl.

math.AP↗

arXiv2015

Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions

This paper considers quadratic and super-quadratic reaction-diffusion systems for reversible chemistry, for which all species satisfy uniform-in-time $L^1$ a-priori estimates, for instance, as a consequence of suitable mass conservation laws. A new result on the global existence of classical solutions is proved in three and higher space dimensions by combining regularity and interpolation arguments in Bochner spaces, a bootstrap scheme and a weak comparison argument. Moreover, provided that the considered system allows for entropy entropy-dissipation estimates proving exponential convergence to equilibrium, we are also able to prove that solutions are bounded uniformly-in-time.

math.AP↗

arXiv2015

Smoothness of moments of the solutions of discrete coagulation equations with diffusion

In this paper, we establish smoothness of moments of the solutions of discrete coagulation-diffusion systems. As key assumptions, we suppose that the coagulation coefficients grow at most sub-linearly and that the diffusion coefficients converge towards a strictly positive limit (those conditions also imply the existence of global weak solutions and the absence of gelation).

math.AP↗

arXiv2015

Analysis and numerical solution of coupled volume-surface reaction-diffusion systems with application to cell biology

We consider the numerical solution of coupled volume-surface reaction-diffusion systems having a detailed balance equilibrium. Based on the conservation of mass, an appropriate quadratic entropy functional is identified and an entropy-entropy dissipation inequality is proven. This allows us to show exponential convergence to equilibrium by the entropy method. We then investigate the discretization of the system by a finite element method and an implicit time stepping scheme including the domain approximation by polyhedral meshes. Mass conservation and exponential convergence to equilibrium are established on the discrete level by arguments similar to those on the continuous level and we obtain estimates of optimal order for the discretization error which hold uniformly in time. Some numerical tests are presented to illustrate these theoretical results. The analysis and the numerical approximation are discussed in detail for a simple model problem. The basic arguments however apply also in a more general context. This is demonstrated by investigation of a particular volume-surface reaction-diffusion system arising as a mathematical model for asymmetric stem cell division.

math.NA↗

arXiv2014

On the finite time blow-up for filtration problems with nonlinear reaction

We present results for finite time blow-up for filtration problems with nonlinear reaction under appropriate assumptions on the nonlinearities and the initial data. In particular, we prove first finite time blow up of solutions subject to sufficiently large initial data provided that the reaction term "overpowers" the nonlinear diffusion in a certain sense. Secondly, under related assumptions on the nonlinearities, we show that initial data above positive stationary state solutions will always lead to finite time blow up.

math.AP↗

arXiv2014

A discontinuous Poisson--Boltzmann equation with interfacial transfer: homogenisation and residual error estimate

A nonlinear Poisson--Boltzmann equation with transmission boundary conditions at the interface between two materials is investigated. The model describes the electrostatic potential generated by a vector of ion concentrations in a periodic multiphase medium with dilute solid particles. The key issue is that the interfacial transfer allows jumps and thus discontinuous solutions of the problem. Based on variational techniques, we derive the homogenisation of the discontinuous problem subject to inhomogeneous transmission interface conditions. Moreover, we establish a rigorous residual error estimate up to the first order correction.

math.AP↗

arXiv2013

Improved duality estimates and applications to reaction-diffusion equations

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.

math.AP↗

arXiv2012

A Drift-Diffusion-Reaction Model for Excitonic Photovoltaic Bilayers: Asymptotic Analysis and A 2-D HDG Finite-Element Scheme

We present and discuss a mathematical model for the operation of bilayer organic photovoltaic devices. Our model couples drift-diffusion-recombination equations for the charge carriers (specifically, electrons and holes) with a reaction-diffusion equation for the excitons/ polaron pairs and Poisson's equation for the self-consistent electrostatic potential. The material difference (i.e. the HOMO/LUMO gap) of the two organic substrates forming the bilayer device are included as a work-function potential. Firstly, we perform an asymptotic analysis of the scaled one-dimensional stationary state system i) with focus on the dynamics on the interface and ii) with the goal of simplifying the bulk dynamics away for the interface. Secondly, we present a twodimensional Hybrid Discontinuous Galerkin Finite Element numerical scheme which is very well suited to resolve i) the material changes ii) the resulting strong variation over the interface and iii) the necessary upwinding in the discretization of drift-diffusion equations. Finally, we compare the numerical results with the approximating asymptotics.

math.AP↗

arXiv2009

Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion

We present a new a-priori estimate for discrete coagulation-fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a-priori estimate provides a global $L^2$ bound on the mass density and was previously used, for instance, in the context of reaction-diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case.

math.AP↗