Singular Limit Research Articles

Singular limit problem for the Keller–Segel system and drift-diffusion system in scaling critical Besov–Morrey spaces

A singular limit problem for the Cauchy problem of the Keller–Segel equation is considered in a critical function space based on Morrey spaces. A solution to the Keller–Segel system in a scaling critical function space converges to a solution to the drift-diffusion system of parabolic-elliptic type in the critical space strongly as the relaxation time tends to infinity. The key ingredient is a Banach space-valued Lorentz space in time. By the careful use of the real interpolation, a sharp estimate for the heat semigroup is obtained.

Pressure-relaxation limit for a one-velocity Baer–Nunziato model to a Kapila model

In this paper, we study a singular limit problem for a compressible one-velocity bifluid system. More precisely, we show that solutions of the Kapila system generated by initial data close to equilibrium are obtained in the pressure-relaxation limit from solutions of the Baer–Nunziato (BN) system. The convergence rate of this process is a consequence of our stability result. Besides the fact that the quasilinear part of the (BN) system cannot be written in conservative form, its natural associated entropy is only positive semi-definite such that it is not clear if the entropic variables can be used in the present case. Using an ad-hoc change of variables we obtain a reformulation of the (BN) system which couples, via low-order terms, an undamped mode and a non-symmetric partially dissipative hyperbolic system satisfying the Shizuta–Kawashima stability condition.

On minimizers of the 2D Ginzburg–Landau energy with tangential anchoring

We analyze Ginzburg–Landau minimization problems in two dimensions with either a “strong or weak” tangential boundary condition. These problems are motivated by experiments in liquid crystals with boundary defects. The imposition of classical boundary conditions on a liquid crystal surface is referred to as “strong anchoring”, while “weak anchoring” refers to the inclusion of boundary energy terms. In the singular limit when the correlation length tends to zero, we show that boundary defects will be observed for weak anchoring, while both boundary and interior vortices are possible for strong anchoring.

Lagrangian Stochastic Model for the Motions of Magnetic Footpoints on the Solar Wind Source Surface and the Path Lengths of Boundary-driven Interplanetary Magnetic Field Lines

In this work, we extend Leighton’s diffusion model describing the turbulent mixing of magnetic footpoints on the solar wind source surface. The present Lagrangian stochastic model is based on the spherical Ornstein–Uhlenbeck process with drift that is controlled by the rotation frequency Ω of the Sun, the Lagrangian integral timescale τ L, and the root-mean-square footpoint velocity V rms. The Lagrangian velocity and the positions of magnetic footpoints on the solar wind source surface are obtained from the solutions of a set of stochastic differential equations, which are solved numerically. The spherical diffusion model of Leighton is recovered in the singular Markov limit when the Lagrangian integral timescale tends to zero while keeping the footpoint diffusivity finite. In contrast to the magnetic field lines driven by standard Brownian processes on the solar wind source surface, the interplanetary magnetic field lines are smooth differentiable functions with finite path lengths in our model. The path lengths of the boundary-driven interplanetary magnetic field lines and their probability distributions at 1 au are computed numerically, and their dependency with respect to the controlling parameters is investigated. The path-length distributions are shown to develop a significant skewness as the width of the distributions increases.

Nonlocal Traffic Models with General Kernels: Singular Limit, Entropy Admissibility, and Convergence Rate

Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counter-examples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counter-example showing that the convexity assumption is necessary for our main compactness estimate.

Effective slip length for transverse shear flow over partially invaded grooves

We consider transverse shear flow over a $2L$ -periodic array of air-filled grooves, which are separated by solid slats. We address the scenario where the liquid partially invades the grooves. Following Crowdy's solution of the longitudinal problem (J. Fluid Mech. vol. 925, 2021, R2), we focus upon the simplified geometry of infinitely narrow slats. For flat menisci, the ratio $\lambda$ of the slip length to $L$ depends upon a single geometric parameter, namely the ratio $H$ of the invasion depth to $L$ . We analyse the singular limit $H\ll 1$ using matched asymptotic expansions, with an outer region on the scale of the period and an inner region on the scale of a single slat. In the outer region, we employ the complex-variable formulation of Stokes flows, obtaining the complex velocity as a divergent series. This formal representation is transformed to an admissible closed-form expression. In the inner region, the problem is equivalent to that of Stokes flow past a finite line segment. Asymptotic matching yields $\lambda = [\ln (2/{\rm \pi} H)-1]/{\rm \pi}$ .

Symmetry-breaking bifurcations for compartmental reaction kinetics coupled by two bulk diffusing species with comparable diffusivities in 2-D

For a 2-D coupled PDE-ODE bulk-cell model, we investigate symmetry-breaking bifurcations that can emerge when two bulk diffusing species are coupled to two-component nonlinear intracellular reactions that are restricted to occur only within a disjoint collection of small circular compartments, or “cells,” of a common small radius that are confined in a bounded 2-D domain. Outside of the union of these cells, the two bulk species with comparable diffusivities and bulk degradation rates diffuse and globally couple the spatially segregated intracellular reactions through Robin boundary conditions across the cell boundaries, which depend on certain membrane reaction rates. In the singular limit of a small common cell radius, we construct steady-state solutions for the bulk-cell model and formulate a nonlinear matrix eigenvalue problem that determines the linear stability properties of the steady-states. For a certain spatial arrangement of cells for which the steady-state and linear stability analysis become highly tractable, we construct a symmetric steady-state solution where the steady-states of the intracellular species are the same for each cell. As regulated by the ratio of the membrane reaction rates on the cell boundaries, we show for various specific prototypical intracellular reactions, and for a specific two-cell arrangement, that our 2-D coupled PDE-ODE model admits symmetry-breaking bifurcations from this symmetric steady-state, leading to linearly stable asymmetric patterns, even when the bulk diffusing species have comparable or possibly equal diffusivities. Overall, our analysis shows that symmetry-breaking bifurcations can occur without the large diffusivity ratio requirement for the bulk diffusing species as is well-known from a Turing stability analysis applied to a spatially uniform steady-state for typical two-component activator-inhibitor systems. Instead, for our theoretical compartmental-reaction diffusion bulk-cell model, our analysis shows that the emergence of stable asymmetric steady-states can be controlled by the ratio of the membrane reaction rates for the two species. Bifurcation theoretic results for symmetric and asymmetric steady-state patterns obtained from our asymptotic theory are confirmed with full numerical PDE simulations.

Singular limits for the Navier-Stokes-Poisson equations of the viscous plasma with the strong density boundary layer

The quasi-neutral limit of the Navier-Stokes-Poisson system modeling a viscous plasma with vanishing viscosity coefficients in the half-space $$\mathbb{R}_+^3$$ is rigorously proved under a Navier-slip boundary condition for velocity and the Dirichlet boundary condition for electric potential. This is achieved by establishing the nonlinear stability of the approximation solutions involving the strong boundary layer in density and electric potential, which comes from the breakdown of the quasi-neutrality near the boundary, and dealing with the difficulty of the interaction of this strong boundary layer with the weak boundary layer of the velocity field.

Rotation-dominant three-scale limit of the Cauchy problem to the inviscid rotating stratified Boussinesq equations

In this paper, we investigate the rotation-dominant three-scale singular limit of the initial-value problem to the inviscid rotating stratified Boussinesq equations. We first prove a dispersive decay estimate for the linear propagator of that limit regime, which requires novel techniques involving an elaborate frequency cut-off since the phase function is degenerate on both horizontal and vertical directions in some areas. Using the dispersive decay estimate, we then investigate rigorously the rotation-dominant limit of local strong solutions to Boussinesq equations with ill-prepared initial data. The limiting system is shown to be the two-dimensional incompressible Euler equations coupled with a two-dimensional transport equation but in three spatial dimensions.

Asymptotic profiles and singular limits for the viscoelastic damped wave equation with memory of type I

In this paper, we are interested in the Cauchy problem for the viscoelastic damped wave equation with memory of type I. By applying the WKB analysis and the Fourier analysis, we explain the memory’s influence on dissipative structures and asymptotic profiles of solutions to the model with additionally weighted L1 initial data. Furthermore, concerning standard energy and the solution itself, we establish the singular limit between the dissipative Moore–Gibson–Thompson equation with memory and the viscoelastic damped wave equation with memory, where a crucial compatibility condition of the initial datum is found.

Singular limit problem of abstract second order evolution equations

Singular limit problem of abstract second order evolution equations

Singular limit of Lamé equations

In this paper, we study the asymptotic behavior of the solution to the Lamé equations with a parameter . We prove that the solution will converge to the solution of a Maxwell‐type system as . Meanwhile, we will show that the solution converges to the solution of a Stokes‐type system as .

On Some Singular Limits Arising in Fluid Dynamic Modelling

Fluid dynamic equations are used to model various phenomena arising from physics, engineering, astrophysics, geophysics. One feature is that they take place at different time and length scales and it is important to understand which phenomena occur according to the use of single scales or to the interactions of them. From a mathematical point of view, these various physical behaviours give rise to different singular limits and, consequently to a different analysis of the asymptotic state of the governing equations. In this paper we will analyse a very simplified model given by a linearised continuity equation and by the classical momentum equation which include terms that take into account of rotation and we will show, according to the values of different scales, that the asymptotic behaviour of the model will be those of an incompressible fluid or of a geostrophic flow. Finally we point out, that the set of equations analysed in the paper may also fit in the artificial compressibility approximation methods.

Hamiltonian form of Carroll gravity

We develop a Hamiltonian description of the ``Carroll'' (Levy Leblond-Sen Gupta) limit of gravity theory in the first-order formalism. Through a constraint analysis, the number of local degrees of freedom are shown to be two in this singular limit. The Hamiltonian constraint in this (``magnetic'') limit depends only on the densitized triad fields and their space derivatives. We also provide a scaling prescription within the Hilbert-Palatini theory to obtain the first-order analogue of the ``electric'' limit of canonical metric gravity. The simplicity of both these limiting Hamiltonian forms in these variables make them interesting candidates for quantization.

Singular Limit for the Compressible Navier–Stokes Equations with the Hard Sphere Pressure Law on Expanding Domains

The article is devoted to the asymptotic limit of the compressible Navier–Stokes system with a pressure obeying a hard–sphere equation of state on a domain expanding to the whole physical space $$\textbf{R}^3$$ . Under the assumptions that acoustic waves generated in the case of ill-prepared data do not reach the boundary of the expanding domain in the given time interval and a certain relation between the Reynolds and Mach numbers and the radius of the expanding domain we prove that the target system is the incompressible Euler system on $$\textbf{R}^3$$ . We also provide an estimate of the rate of convergence expressed in terms of characteristic numbers and the radius of domains.

The porous medium equation as a singular limit of the thin film Muskat problem

The singular limit of the thin film Muskat problem is performed when the density (and possibly the viscosity) of the lighter fluid vanishes and the porous medium equation is identified as the limit problem. In particular, the height of the denser fluid is shown to converge towards the solution to the porous medium equation and an explicit rate for this convergence is provided in space dimension d ⩽ 4. Moreover, the limit of the height of the lighter fluid is determined in a certain regime and is given by the corresponding initial condition.

Singular limits of a coupled elasto-plastic damage system as viscosity and hardening vanish

The paper studies the asymptotic analysis of a model coupling elastoplasticity and damage depending on three parameters—governing viscosity, plastic hardening, and convergence rate of plastic strain and displacement to equilibrium—as they vanish in different orders. The notion of limit evolution obtained is proven to coincide in any case with a notion introduced by Crismale and Rossi (SIAM J Math Anal 53(3):3420–3492, 2021), moreover, such solutions are closely related to those obtained in the vanishing-viscosity limit by Crismale and Lazzaroni (Calc Var Part Differ Equ 55(1):17, 2016), for the analogous model where only the viscosity parameter was present.

Limiting behavior of center manifolds for stochastic evolutionary equations with delay in varying phase spaces

In this article, we consider random center manifolds for a class of stochastic evolutionary equations with delay driven by multiplicative white noise. We first prove the existence and Ck smoothness of random center manifolds for the equations with delay. Then, we show the Ck smooth convergence of the center manifolds as the phase spaces approach to their singular limit.

Hyperbolic limit for a biological invasion

In a spatially heterogeneous environment the propagation speed of a biological invasion varies in space. The traveling wave theory in a homogeneous case is not extended to a heterogeneous case. Taking a singular limit in a hyperbolic scale is a good way to study such a wave propagation with constant speed. The goal of this project is to understand the effect of biological diffusion on the wave speed in a spatial heterogeneous environment. For this purpose, we consider \begin{document}$ U_t = {\varepsilon}(\gamma(s)U)_{xx}+{1\over{\varepsilon}}U(1-{U/m(x)}), $\end{document} where $ m $ is a nonconstant carrying capacity, $ s = {U\over m} $ is a starvation measure and $ \gamma(s) = s^{\tilde{k}}, \tilde{k} \ge 1 $. The diffusion is a starvation driven diffusion. We show that the diffusion speed is constant even if $ m $ is nonconstant.

Spatial segregation of multiple species: A singular limit approach

The spatial segregation of the populations occurs commonly in ecology. One typical way to understand this phenomenon is to consider strong competition in some species. In this paper, we shall consider multiple-species competition-diffusion models. Under the condition that some interspecies competition rates are large, we show that the segregation phenomenon occurs. Furthermore, we derive some two-phase Stefan-like problems appearing as the singular limit, which may provide some modeling interpretation for free boundary problems studied in the literature.