Searcharxiv⌕ Search

arXiv subjects

Barbara Niethammer

Publications and source records attributed to Barbara Niethammer.

At least 19 recordsLinked to original sources
arXiv2026

On the shape of the positivity region for a free boundary problem describing cell polarization

In this paper we study a mass-constrained free boundary problem modeling cell polarization, in the regime where the mass is small. In the generic case of a signal with nondegenerate maxima, we prove that the solution converges locally to a global, integrable solution to an obstacle problem in the plane. We further show that the interface of the solution to the limit problem is an ellipse, the equation of which is explicit. We also study some cases where the signal has degenerate maxima, highlighting a variety of possible behaviors.

math.AP↗
arXiv2026

Long-time behaviour of a nonlocal model for electroporation

In this paper we analyze a model for electroporation, a biological process in which a cell membrane exposed to an external voltage becomes permeable due to the formation and growth of nanoscale membrane pores. We prove a local stability result for asymptotic self-similar solutions with a power-law tail. Our method relies on the analysis of an equation for the first moment as well as comparison of solutions of the full problem to solutions of a corresponding transport problem. In particular this shows that the transport term drives the long-time behaviour.

math.AP↗
arXiv2024

On an inhomogeneous coagulation model with a differential sedimentation kernel

We study an inhomogeneous coagulation equation that contains a transport term in the spatial variable modeling the sedimentation of clusters. We prove local existence of mass conserving solutions for a class of coagulation kernels for which in the space homogeneous case instantaneous gelation (i.e., instantaneous loss of mass) occurs. Our result holds true in particular for sum-type kernels of homogeneity greater than one, for which solutions do not exist at all in the spatially homogeneous case. Moreover, our result covers kernels that in addition vanish on the diagonal, which have been used to describe the onset of rain and the behavior of air bubbles in water.

math.AP↗
arXiv2024

Interface behavior for the solutions of a mass conserving free boundary problem modeling cell polarization

We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. In previous papers, we have established well-posedness of this problem and derived conditions on the initial data that imply continuity of the free boundary as $t\to 0$. In this paper we extend the qualitative study of the free boundary by considering axisymmetric data. Under additional monotonicity assumptions on the data we prove global continuity of the free boundary. On the other hand, if the initial data violate a "no-fattening" condition we show that the free boundary can oscillate as $t \to 0$.

math.AP↗
arXiv2023

A Becker-Döring type model for cell polarization

We propose a model for cell polarization based on the Becker-Döring equations with the first coagulation coefficient equal to zero. We show convergence to equilibrium for power-law coagulation and fragmentation rates and obtain a loss of mass in the limit $t \rightarrow \infty$ depending on the initial mass and the relative strengths of the coagulation and fragmentation processes. In the case of linear rates, we further show that large clusters evolve in a self-similar manner at large times by comparing limits of appropriately rescaled solutions in different spaces.

math.AP↗
arXiv2023

Qualitative properties of solutions to a non-local free boundary problem modeling cell polarization

We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. The authors have justified the well-posedness of this problem and have further proved uniqueness of solutions and global stability of steady states. In this paper we investigate qualitative properties of the free boundary. We present necessary and sufficient conditions for the initial data that imply continuity of the support at $t = 0$. If one of these assumptions fail, then jumps of the support take place. In addition we provide a complete characterization of the jumps for a large class of initial data.

math.AP↗
arXiv2023

Instability of hysteretic phase interfaces in a mean-field model with inhomogeneities

We study a system of non-identical bistable particles that is driven by a dynamical constraint and coupled through a non-local mean-field. Assuming piecewise affine constitutive laws we prove the existence of traveling wave solutions and characterize their dynamical stability. Our findings explain the two dynamical regimes for phase interface that can be observed in numerical simulations with different parameters. We further discuss the convergence to a rate-independent model with strong hysteresis in the limit of vanishing relaxation time.

math.AP↗
arXiv2022

Revisiting Shikhmurzaev's approach to the contact line problem

In this paper, we revisit a model for the contact line problem which has been proposed by Shikhmurzaev (1993). In the first part, in addition to rederiving the model, we study in detail the assumptions required to obtain the isothermal limit of the model. We also derive in this paper several lubrication approximation models, based on Shikhmurzaev's approach. The first two lubrication models describe thin film flow of incompressible fluids on solid substrates, based on different orders of magnitude of the slip length parameter. The third lubrication model describes a meniscus formation where a wedge-shaped solid immerses in a thin film of fluid.

math.AP↗
arXiv2021

Some thin film models based on Shikhmurzaev's approach

We derive several lubrication approximation models, using Shikhmurzaev's approach to the contact line problem, obtained in \cite{GNV}. The first two lubrication models describe thin film flow of incompressible fluids on solid substrates, based on different orders of magnitude of the slip length parameter. The third lubrication model describes a meniscus formation where a wedge-shaped solid immerses in a thin film of fluid.

math.AP↗
arXiv2021

Oscillations in a Becker-Döring model with injection and depletion

We study the Becker-Döring bubblelator, a variant of the Becker-Döring coagulation-fragmentation system that models the growth of clusters by gain or loss of monomers. Motivated by models of gas evolution oscillators from physical chemistry, we incorporate injection of monomers and depletion of large clusters. For a wide range of physical rates, the Becker-Döring system itself exhibits a dynamic phase transition as mass density increases past a critical value. We connect the Becker-Döring bubblelator to a transport equation coupled with an integrodifferential equation for excess monomer density by formal asymptotics in the near-critical regime. For suitable injection/depletion rates, we argue that time-periodic solutions appear via a Hopf bifurcation. Numerics confirm that the generation and removal of large clusters can become desynchronized, leading to temporal oscillations associated with bursts of large-cluster nucleation.

math.AP↗
arXiv2020

A parabolic free boundary problem arising in a model of cell polarization

The amplification of an external signal is a key step in direction sensing of biological cells. We consider a simple model for the response to a time-depending signal, which was previously proposed by the last three authors. The model consists of a bulk-surface reaction-diffusion model. We prove that in a suitable asymptotic limit the system converges to a bulk-surface parabolic obstacle type problem. For this model and a reduction to a nonlocal surface equation we show an L1-contraction property and, in the case of time-constant signals, the stability of stationary states.

math.AP↗
arXiv2019

A local version of Einstein's formula for the effective viscosity of suspensions

We prove a local variant of Einstein's formula for the effective viscosity of dilute suspensions, that is $μ^\prime=μ(1+\frac 5 2ϕ+o(ϕ))$, where $ϕ$ is the volume fraction of the suspended particles. Up to now rigorous justifications have only been obtained for dissipation functionals of the flow field. We prove that the formula holds on the level of the Stokes equation (with variable viscosity). We consider a regime where the number $N$ of particles suspended in the fluid goes to infinity while their size $R$ and the volume fraction $ϕ=NR^3$ approach zero. We establish $L^\infty$ and $L^p$ estimates for the difference of the microscopic solution to the solution of the homogenized equation. Here we assume that the particles are contained in a bounded region and are well separated in the sense that the minimal distance is comparable to the average one. The main tools for the proof are a dipole approximation of the flow field of the suspension together with the so-called method of reflections and a coarse graining of the volume density.

math.AP↗
arXiv2019

Solutions with peaks for a coagulation-fragmentation equation. Part I: stability of the tails

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behaviour is stable. In a companion paper we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.

math.AP↗
arXiv2019

Solutions with peaks for a coagulation-fragmentation equation. Part II: aggregation in peaks

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. In a companion paper we constructed a two-parameter family of stationary solutions concentrated in Dirac masses, and we carefully studied the asymptotic decay of the tails of these solutions, showing that this behaviour is stable. In this paper we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.

math.AP↗
arXiv2019

Self-similar spreading in a merging-splitting model of animal group size

In a recent study of certain merging-splitting models of animal-group size (Degond et al., J. Nonl. Sci. 27 (2017) 379), it was shown that an initial size distribution with infinite first moment leads to convergence to zero in weak sense, corresponding to unbounded growth of group size. In the present paper we show that for any such initial distribution with a power-law tail, the solution approaches a self-similar spreading form. A one-parameter family of such self-similar solutions exists, with densities that are completely monotone, having power-law behavior in both small and large size regimes, with different exponents.

math.AP↗
arXiv2019

A bulk-surface reaction-diffusion system for cell polarization

We propose a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study stationary states of this model in certain parameter regimes in which several reaction rates on the membrane as well as the diffusion coefficient within the cell are large. It turns out that in suitable scaling limits steady states converge to solutions of some obstacle type problems. For these limiting problems we prove the onset of polarization if the total mass of protein is sufficiently small. For some variants we can even characterize precisely the critical mass for which polarization occurs.

math.AP↗
arXiv2018

Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one

We prove the existence of a one-parameter family of self-similar solutions with time dependent tails for Smoluchowski's coagulation equation, for a class of kernels $K(x,y)$ which are homogeneous of degree one and satisfy $K(x,1)\to k_0>0$ as $x\to 0$. In particular, we establish the existence of a critical $ρ_*>0$ with the property that for all $ρ\in(0,ρ_*)$ there is a positive and differentiable self-similar solution with finite mass $M$ and decay $A(t)x^{-(2+ρ)}$ as $x\to\infty$, with $A(t)=e^{M(1+ρ)t}$. Furthermore, we show that (weak) self-similar solutions in the class of positive measures cannot exist for large values of the parameter $ρ$.

math.AP↗