Related papers: Singular limit of an Allen-Cahn equation with nonl…
We study the evolution of fronts in a bistable reaction-diffusion system when the nonlinear reaction term is spatially non-homogeneous. This equation has been used to model wave propagation in various biological systems. Extending previous…
In this paper, we study the asymptotic limit, as $\varepsilon\to 0$, of solutions to a vector-valued Allen-Cahn equation $$ \partial_t u = \Delta u - \frac{1}{\varepsilon^2} \partial_u F(u), $$ where $u: \Omega \subset \mathbb{R}^m \to…
Motivated by considerations from neuroscience (macroscopic behavior of large ensembles of interacting neurons), we consider a population of mean field interacting diffusions in $\mathbf {R}^m$ in the presence of a random environment and…
We derive and study a theoretical description for single file diffusion, i.e., diffusion in a one dimensional lattice of particles with hard core interaction. It is well known that for this system a tagged particle has anomalous diffusion…
When a viscous fluid partially fills a Hele--Shaw channel, and is pushed by a pressure difference, the fluid interface is unstable due to the Saffman--Taylor instability. We consider the evolution of a fluid region of finite extent, bounded…
The parabolic Allen-Cahn equation is a semilinear partial differential equation linked to the mean curvature flow by a singular perturbation. We show an improved convergence property of the parabolic Allen-Cahn equation to the mean…
We study a reaction-diffusion-convection problem with nonlinear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion…
A difficulty in the classical hydrodynamic analysis of moving contact-line problems, associated with the no-slip wall boundary condition resulting in an unbalanced divergence of the viscous stresses, is reexamined with a smoothed,…
Repulsion between individuals within a finite radius is encountered in numerous applications, including cell exclusion, i.e. avoidance of overlapping cells, bird flocks, or microscopic pedestrian models. We define such individual based…
We study the existence and uniqueness of a solution to a linear stationary convection-diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of…
We study the systematic numerical approximation of a class of Allen-Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an…
The well known scaling of the Edwards-Wilkinson equation is essentially determined by dimensional analysis. Once a drift term is added, more sophisticated reasoning is required, which initially suggests that the drift term dominates over…
How finite-sized material lines stretch in chaotic (mono-scale) and turbulent (multi-scale) flows remains a central but unresolved problem that governs mixing, transport and reaction. We show elongation is controlled by a finite-sampling…
We study a diffusion approximation for a model of stochastic motion of a particle in one spatial dimension. The velocity of the particle is constant but the direction of the motion undergoes random changes with a Poisson clock. Moreover,…
The present paper discusses the diffusion approximation of the linear Boltzmann equation in cases where the collision frequency is not uniformly large in the spatial domain. Our results apply for instance to the case of radiative transfer…
We investigate the dynamics of a single tracer exploring a course of fixed obstacles in the vicinity of the percolation transition for particles confined to the infinite cluster. The mean-square displacement displays anomalous transport,…
We consider three models of evolving interfaces intimately related to the weakly asymmetric simple exclusion process with $N$ particles on a finite lattice of $2N$ sites. Our Model 1 defines an evolving bridge on $[0,1]$, our Model 1-w an…
We consider the reduced Allen-Cahn action functional, which appears as the sharp interface limit of the Allen-Cahn action functional and can be understood as a formal action functional for a stochastically perturbed mean curvature flow. For…
Diffusive transport of a particle in spatially correlated random energy landscape having exponential density of states has been considered. We exactly calculate the diffusivity in the nondispersive quasi-equilibrium transport regime and…
The goal of this paper is to describe the metastable dynamics of the solutions to the reaction-diffusion equation with nonlinear phase-dependent diffusion $u_t=\varepsilon^2(D(u)u_x)_x-f(u)$, where $D$ is a strictly positive function and…