Limit Of Small Diffusivity Research Articles

Canards in a bottleneck

In this paper, we investigate the stationary profiles of a nonlinear Fokker–Planck equation with small diffusion and nonlinear inflow and outflow boundary conditions. We consider corridors with a bottleneck whose width has a unique global nondegenerate minimum in the interior. In the small diffusion limit, the profiles are obtained constructively by using methods from geometric singular perturbation theory (GSPT). We identify three main types of profiles corresponding to: (i) high density in the domain and a boundary layer at the entrance, (ii) low density in the domain and a boundary layer at the exit, and (iii) transitions from high density to low density inside the bottleneck with boundary layers at the entrance and exit. Interestingly, solutions of the last type involve canard solutions generated at the narrowest point of the bottleneck. We obtain a detailed bifurcation diagram of these solutions in terms of the inflow and outflow rates. The analytic results based on GSPT are further corroborated by computational experiments investigating corridors with bottlenecks of variable width.

Front propagation in the shadow wave-pinning model.

In this paper we consider a non-local bistable reaction-diffusion equation which is a simplified version of the wave-pinning model of cell polarization. In the small diffusion limit, a typical solution u(x,t) of this model approaches one of the stable states of the bistable nonlinearity in different parts of the spatial domain [Formula: see text], separated by an interface moving at a normal velocity regulated by the integral [Formula: see text]. In what is often referred to as wave-pinning, feedback between mass-conservation and bistablity causes the interface to slow and approach a fixed limit. In the limit of a small diffusivity [Formula: see text], we prove that for any [Formula: see text] the interface can be estimated within [Formula: see text] of the location as predicted using formal asymptotics. We also discuss the sharpness of our result by comparing the formal asymptotic results with numerical simulations.

Brownian motion theory of the two-dimensional quantum vortex gas.

A theory of Brownian motion is presented for an assembly of vortices. The attempt is motivated by a realization of Dyson's Coulomb gas in the context of quantum condensates. By starting with the time-dependent Landau-Ginzburg (LG) theory, the dynamics of the vortex gas is constructed, which is governed by the canonical equationof motion. The dynamics of point vortices is converted to the Langevin equation, which results in the generalized Fokker-Planck (GFP) (or Smolkovski) equationusing the functional integral on the ansatz of the Gaussian white noise. The GFP, which possesses a non-Hermitian property, is characterized by two regimes called the overdamping and the underdamping regimes. In the overdamping regime, where the dissipation is much larger that the vortex strength, the GFP becomes the standard Fokker-Planck equation, which is transformed into the two-dimensional many-particle system. Several specific applications are given of the Fokker-Planck equation. An asymptotic limit of small diffusion is also discussed for the two-vortices system. The underdamping limit, for which the vortex charge is much larger than the dissipation, is briefly discussed.

Pencil-beam approximation of fractional Fokker-Planck

<p style='text-indent:20px;'>We consider the modeling of light beams propagating in highly forward-peaked turbulent media by fractional Fokker-Planck equations and their approximations by fractional Fermi pencil beam models. We obtain an error estimate in a 1-Wasserstein distance for the latter model showing that beam spreading is well captured by the Fermi pencil-beam approximation in the small diffusion limit.</p>

Fixed-flux salt-finger convection in the small diffusivity ratio limit

Salt-finger convection provides a key mixing process in geophysical and astrophysical fluid flows. Because of its small characteristic spatial scale and slow diffusive time scale, this process must be parameterized in geophysical and astrophysical models, where relations linking background gradients to fluxes are required. To obtain such relations, most authors study the dependence of temperature and salinity fluxes on fixed background gradients. Using the reduced model derived by Xie et al. [“A reduced model for salt-finger convection in the small diffusivity ratio limit,” Fluids 2(1), 6 (2017)] for salt-finger convection in the limit of small diffusivity and large density ratios, this paper considers the conjugate problem where the fluxes are fixed, but the mean gradients are permitted to adjust in response. In small domains, the fixed-flux condition leads to stable single-mode solutions, which are not achievable with fixed-gradient conditions. In large domains, with statistically steady saturated states, the relations between mean fluxes and mean gradients are identical for both sets of conditions. The fixed-flux condition provides a new perspective for understanding the resulting statistically steady states by identifying two distinct regimes with the same dissipation rate. We find that the statistically steady dynamics select the state with the smaller Rayleigh ratio Ra subject to the constraint Ra > 1, ensuring that the background state is linearly unstable. The fixed-flux formulation results in a more potent restoring mechanism toward the statistically steady state, with a smaller variance, skewness, and characteristic time scale than in the fixed-gradient setup. This distinctive feature can be used as a diagnostic to determine whether in situ salt-finger convection is flux-driven or gradient-driven.

Dirac-concentrations in an integro-pde model from evolutionary game theory

<p style='text-indent:20px;'>Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Motivated by the existence of moving Dirac-concentrations in the time-dependent problem, we study the qualitative properties of steady states in the limit of small diffusion. Under different conditions on the growth rate and interaction kernel as motivated by the framework of adaptive dynamics, we will show that as the diffusion rate tends to zero the steady state concentrates (ⅰ) at a single location; (ⅱ) at two locations simultaneously; or (ⅲ) at one of two alternative locations. The third result in particular shows that solutions need not be unique. This marks an important difference of the non-local equation with its local counterpart.

Asymptotic Analysis of a Silicon Furnace Model

Silicon is produced from quartz rock in electrode-heated furnaces by using carbon as a reduction agent. We perform an asymptotic analysis of a heat and mass transfer model of an experimental pilot furnace in order to determine the dominant chemical and thermal behavior of the system. First, by prescribing a steady-state temperature profile in the furnace we explore the leading order reactions in different spatial regions, as well as early time behavior. We are able to obtain asymptotic solutions which compare well with numerical simulations. Utilizing the dominant balances found when the temperature is prescribed, we next reduce the full model to two coupled partial differential equations for the time-variable temperature profile within the furnace and the concentration of solid quartz. These equations account for diffusion, an endothermic reaction, and the external heating input to the system. A moving boundary is found and the behavior on either side of this boundary explored in the asymptotic limit of small diffusion. We note how the simplifications derived may be useful for future models and industrial furnace operation and comment on insights from the model about furnace crust formation.

Wavetrain Solutions of a Reaction-Diffusion-Advection Model of Mussel-Algae Interaction

We consider a system of coupled partial differential equations modeling the interaction of mussels and algae in advective environments. A key parameter in the equations is the ratio of the diffusion rate of the mussel species and the advection rate of the algal concentration. When advection dominates diffusion, one observes large-amplitude solutions representing bands of mussels propagating slowly in the upstream direction. Here, we prove the existence of a family of such periodic wavetrain solutions. Our proof relies on geometric singular perturbation theory to construct these solutions as periodic orbits of the associated traveling wave equations in the large-advection--small-diffusion limit. The construction encounters a number of mathematical obstacles which necessitate a compactification of phase space, geometric desingularization to remedy a loss of normal hyperbolicity, and the application of a generalized exchange lemma at a loss-of-stability turning point. In particular, our analysis uncovers log...

The Stability and Slow Dynamics of Localized Spot Patterns for the 3-D Schnakenberg Reaction-Diffusion Model

On a bounded three-dimensional domain $\Omega$, a hybrid asymptotic-numerical method is employed to analyze the existence, linear stability, and slow dynamics of localized quasi-equilibrium multispot patterns of the Schnakenberg activator-inhibitor model with bulk feed-rate $A$ in the singularly perturbed limit of small diffusivity $\varepsilon^2$ of the activator component. By approximating each spot as a Coulomb singularity, a nonlinear system of equations is formulated for the strength of each spot. To leading order in $\varepsilon$, two types of solutions are identified: symmetric patterns for which all strengths are identical, and asymmetric patterns for which each strength takes on one of two distinct values. The $\mathcal{O}(\varepsilon)$ correction to the strengths is found to depend on the spatial configuration of the spots through a certain Neumann Green's matrix $\mathcal{G}$. When $\mathbf{e} = (1,\dots,1)^T$ is not an eigenvector of $\mathcal{G}$, a detailed numerical and (in the case of two spots) asymptotic characterization is performed for the resulting imperfection-sensitive bifurcation structure. For symmetric multispot patterns, a leading-order global threshold in terms of $|\Omega|$ and parameters of the Schnakenberg model is obtained, below which a competition instability is triggered leading to the annihilation of one or more spots. A corresponding refined threshold is established in terms of eigenvalues of $\mathcal{G}$ in the special case when $\mathcal{G}\mathbf{e} = k\mathbf{e}$. Additionally, a local self-replication threshold for the strength of each spot is derived numerically, above which a spot splits into two. By examining $\mathcal{O}(\varepsilon)$ corrections to spot strengths, a prediction is made as to which spot will be next to split as $A$ is slowly tuned. When the pattern is stable to $\mathcal{O}(1)$ instabilities, it is shown that the locations of spots in a quasi-equilibrium configuration evolve on a long $\mathcal{O}(\varepsilon^{-3})$ time-scale according to an ODE system characterized by a gradient flow of a certain discrete energy $\mathcal{H}$, the minima of which define stable equilibrium points of the ODE. The theory also illustrates that new equilibrium points can be created when $A = A(\mathbf{x})$ is spatially variable, and that finite-time pinning away from minima of $\mathcal{H}$ can occur when $A(\mathbf{x})$ is localized. The theory for linear stability and slow dynamics when $\Omega$ is the unit ball are compared favorably to numerical solutions of the Schnakenberg PDE.

Oscillatory Survival Probability: Analytical and Numerical Study of a Non-Poissonian Exit Time

We consider the escape of a planar diffusion process from the domain of attraction $\Omega$ of a stable focus of the drift in the limit of small diffusion. The boundary $\partial\Omega$ of $\Omega$ is an unstable limit cycle of the drift, and the focus is close to the limit cycle. A new phenomenon of oscillatory decay of the peaks of the survival probability of the process in $\Omega$ emerges for a specific distance of the focus to the boundary which depends on the amplitude of the diffusion coefficient. We show that the phenomenon is due to the complex-valued second eigenvalue $\lambda_2(\Omega)$ of the Fokker--Planck operator with Dirichlet boundary conditions. The dominant oscillation frequency, $1/\mathfrak{I}m\{\lambda_2(\Omega)\}$, is independent of the relative noise strength. The density of exit points on $\partial\Omega$ is concentrated in a small arc of $\partial\Omega$ closest to the focus. When the focus approaches the boundary, the principal eigenvalue, the reciprocal of which is the mean escape time, does not necessarily decay exponentially as the amplitude of the noise strength decays. The oscillatory escape is manifested in a mathematical model of neural networks with synaptic depression. Oscillation peaks are identified in the density of the time the network spends in a specific state. This observation explains the oscillations of stochastic trajectories around a focus prior to escape and also the non-Poissonian escape times. This phenomenon has been observed and reported in experiments and simulations of neural networks.

A class of Hamilton–Jacobi equations with constraint: Uniqueness and constructive approach

We discuss a class of time-dependent Hamilton–Jacobi equations, where an unknown function of time is intended to keep the maximum of the solution to the constant value 0. Our main result is that the full problem has a unique viscosity solution, which is in fact classical. The motivation is a selection–mutation model which, in the limit of small diffusion, exhibits concentration on the zero level set of the solution of the Hamilton–Jacobi equation.Uniqueness is obtained by noticing that, as a consequence of the dynamic programming principle, the solution of the Hamilton–Jacobi equation is classical. It is then possible to write an ODE for the maximum of the solution, and treat the full problem as a nonstandard Cauchy problem.

A computational study of residual KPP front speeds in time-periodic cellular flows in the small diffusion limit

The minimal speeds (c∗) of the Kolmogorov–Petrovsky–Piskunov (KPP) fronts at small diffusion (ϵ≪1) in a class of time-periodic cellular flows with chaotic streamlines is investigated in this paper. The variational principle of c∗ reduces the computation to that of a principle eigenvalue problem on a periodic domain of a linear advection–diffusion operator with space–time periodic coefficients and small diffusion. To solve the advection dominated time-dependent eigenvalue problem efficiently over large time, a combination of spectral methods and finite element, as well as the associated fast solvers, are utilized to accelerate computation. In contrast to the scaling c∗=O(ϵ1/4) in steady cellular flows, a new relation c∗=O(1) as ϵ≪1 is revealed in the time-periodic cellular flows due to the presence of chaotic streamlines. Residual propagation speed emerges from the Lagrangian chaos which is quantified as a sub-diffusion process.

Escape from a potential well with a randomly switching boundary

We consider diffusion in a potential well with a boundary that randomly switches between absorbing and reflecting and show how the switching boundary affects the classical escape theory. Using the theory of stochastic hybrid systems, we derive boundary value problems for the mean first passage time and splitting probability and find explicit solutions in terms of the spectral decomposition of the associated differential operator. Further, using a more probabilistic approach, we prove asymptotic formulae for these statistics in the small diffusion limit. In particular, we show that the statistical behavior depends critically on the gradient of the potential near the switching boundary and we derive corrections to Kramers’ reaction rate theory.

Uniqueness in a class of Hamilton–Jacobi equations with constraints

In this note, we discuss a class of time-dependent Hamilton–Jacobi equations depending on a function of time, this function being chosen in order to keep the maximum of the solution of the constant value 0. The main result of the note is that the full problem has a unique classical solution. The motivation is a selection–mutation model that, in the limit of small diffusion, exhibits concentration on the zero-level set of the solution to the Hamilton–Jacobi equation. The uniqueness result that we prove implies strong convergence and error estimates for the selection–mutation model.

Time fluctuations in a population model of adaptive dynamics

We study the dynamics of phenotypically structured populations in environments with fluctuations. In particular, using novel arguments from the theories of Hamilton–Jacobi equations with constraints and homogenization, we obtain results about the evolution of populations in environments with time oscillations, the development of concentrations in the form of Dirac masses, the location of the dominant traits and their evolution in time. Such questions have already been studied in time homogeneous environments. More precisely we consider the dynamics of a phenotypically structured population in a changing environment under mutations and competition for a single resource. The mathematical model is a non-local parabolic equation with a periodic in time reaction term. We study the asymptotic behavior of the solutions in the limit of small diffusion and fast reaction. Under concavity assumptions on the reaction term, we prove that the solution converges to a Dirac mass whose evolution in time is driven by a Hamilton–Jacobi equation with constraint and an effective growth/death rate which is derived as a homogenization limit. We also prove that, after long-time, the population concentrates on a trait where the maximum of an effective growth rate is attained. Finally we provide an example showing that the time oscillations may lead to a strict increase of the asymptotic population size.

A representation of solutions to a scalar conservation law in several dimensions

We find a representation of smooth solutions to the Cauchy problem for a scalar multidimensional conservation law as a small diffusion limit of a stochastic perturbation along characteristics. It helps, in particular, to study the process of singularity formation. Further, we introduce an associated system of balance laws that can be interpreted as describing the motion of a continuum with some specific pressure term. This term arises only after the instant when the solution to the initial Cauchy problem loses its smoothness. Before this instant the system coincides partly with the one known as pressure free gas dynamics.

Adaptation and migration of a population between patches

A Hamilton-Jacobi formulation has been established previously for phenotypically structured population models where the solution concentrates as Dirac masses in the limit of small diffusion. Is it possible to extend this approach to spatial models? Are the limiting solutions still in the form of sums of Dirac masses? Does the presence of several habitats lead to polymorphic situations? We study the stationary solutions of a structured population model, while the population is structured by continuous phenotypical traits and discrete positions in space. The growth term varies from one habitable zone to another, for instance because of a change in the temperature. The individuals can migrate from one zone to another with a constant rate. The mathematical modeling of this problem, considering mutations between phenotypical traits and competitive interaction of individuals within each zone via a single resource, leads to a system of coupled parabolic integro-differential equations. We study the asymptotic behavior of the stationary solutions to this model in the limit of small mutations. The limit, which is a sum of Dirac masses, can be described with the help of an effective Hamiltonian. The presence of migration can modify the dominant traits and lead to polymorphic situations.

Stability of cluster solutions in a cooperative consumer chain model

We study a cooperative consumer chain model which consists of one producer and two consumers. It is an extension of the Schnakenberg model suggested in Gierer and Meinhardt [Kybernetik (Berlin), 12:30-39, 1972] and Schnakenberg (J Theor Biol, 81:389-400, 1979) for which there is only one producer and one consumer. In this consumer chain model there is a middle component which plays a hybrid role: it acts both as consumer and as producer. It is assumed that the producer diffuses much faster than the first consumer and the first consumer much faster than the second consumer. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. In the small diffusion limit we construct cluster solutions in an interval which have the following properties: The spatial profile of the third component is a spike. The profile for the middle component is that of two partial spikes connected by a thin transition layer. The first component in leading order is given by a Green's function. In this profile multiple scales are involved: The spikes for the middle component are on the small scale, the spike for the third on the very small scale, the width of the transition layer for the middle component is between the small and the very small scale. The first component acts on the large scale. To the best of our knowledge, this type of spiky pattern has never before been studied rigorously. It is shown that, if the feedrates are small enough, there exist two such patterns which differ by their amplitudes.We also study the stability properties of these cluster solutions. We use a rigorous analysis to investigate the linearized operator around cluster solutions which is based on nonlocal eigenvalue problems and rigorous asymptotic analysis. The following result is established: If the time-relaxation constants are small enough, one cluster solution is stable and the other one is unstable. The instability arises through large eigenvalues of order O1. Further, there are small eigenvalues of order o1 which do not cause any instabilities. Our approach requires some new ideas: (i) The analysis of the large eigenvalues of order O1 leads to a novel system of nonlocal eigenvalue problems with inhomogeneous Robin boundary conditions whose stability properties have been investigated rigorously. (ii) The analysis of the small eigenvalues of order o1 needs a careful study of the interaction of two small length scales and is based on a suitable inner/outer expansion with rigorous error analysis. It is found that the order of these small eigenvalues is given by the smallest diffusion constant ε2(2).

Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations

Nonlocal Lotka–Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses co-exist? We will explain how these questions relate to the so-called “constrained Hamilton–Jacobi equation” and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution.

The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray–Scott Model

The dynamics and stability of multispot patterns to the Gray–Scott (GS) reaction-diffusion model in a two-dimensional domain is studied in the singularly perturbed limit of small diffusivity $\varepsilon$ of one of the two solution components. A hybrid asymptotic-numerical approach based on combining the method of matched asymptotic expansions with the detailed numerical study of certain eigenvalue problems is used to predict the dynamical behavior and instability mechanisms of multispot quasi-equilibrium patterns for the GS model in the limit $\varepsilon\to 0$. For $\varepsilon\to 0$, a quasi-equilibrium k-spot pattern is constructed by representing each localized spot as a logarithmic singularity of unknown strength $S_j$ for $j=1,\ldots,k$ at an unknown spot location ${\mathbf x}_j\in \Omega$ for $j=1,\ldots,k$. A formal asymptotic analysis is then used to derive a differential algebraic ODE system for the collective coordinates $S_j$ and ${\mathbf x}_j$ for $j=1,\ldots,k$, which characterizes the slo...