Singular Limit Research Articles

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.

-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Abstract Given an uncountable cardinal $\kappa $ , we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$ -formulas that only use the cardinal $\kappa $ and sets of hereditary cardinality less than $\kappa $ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical nondefinability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of $\kappa $ of length at least $\kappa ^+$ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of $\Sigma _1$ -definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal $\omega _1$ .

On the singular limit of a boundary delayed Kuramoto-Sivashinsky-Korteweg-de Vries equation: Well-posedness and stability results

On the singular limit of a boundary delayed Kuramoto-Sivashinsky-Korteweg-de Vries equation: Well-posedness and stability results

Singular limits of invariant measures of the 3D MHD-Voigt equations

Singular limits of invariant measures of the 3D MHD-Voigt equations

Weak entire solutions of reaction–interface systems

<p style='text-indent:20px;'>In this paper, the singular limit problems arising from FitzHugh–Nagumo–type reaction–diffusion systems are studied, which are called reaction–interface systems. All weak entire solutions originating from finitely many excited intervals are completely characterized. For weak entire solutions originating from infinitely many excited intervals, periodic wave trains and time-periodic solutions are discussed. In particular, we study the dispersion relation of periodic wave trains and investigate the dependence of the propagation speed on the period.</p>

Singular limit of deterministic and stochastic reaction-diffusion systems

Singular limit of deterministic and stochastic reaction-diffusion systems

Existence and robustness results of attractors for partially-damped piezoelectric beams

This paper deals with a fully-dynamic piezoelectric beam model with a nonlinear force acting on the longitudinal displacements of the beam. The equations of motion follows a system of non-compactly coupled wave equation. As the magnetic effects are discarded by setting the magnetic permeability $ \mu $ to zero, the equations are fully decoupled to a single wave equation (electrostatic/quasi-static beam model). The existence of smooth finite-dimensional global attractors is proved by considering a damping injection to only one of the equations. Next, the existence of a finite set of determining functionals for the long-time behavior of the system is proved. Finally, to transition from the fully-dynamic beam model to the commonly-used electrostatic/quasi-static beam model, a singular limit problem $ \mu \to 0 $ is considered. This analysis allows to compare the attractors from one model to another in the sense of the upper semi-continuity of the attractor $ \mathcal{A}_{\mu} $ as $ \mu \to 0 $.

Singular limit of an Allen–Cahn equation with nonlinear diffusion

We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scaling limit of certain interacting particle systems. We investigate its singular limit and show the generation and propagation of an interface in the limit. The evolution of this limit interface is governed by mean curvature flow with a novel, homogenized speed in terms of a surface tension-mobility parameter emerging from the nonlinearity in our equation.

Topological corrections and conformal backreaction in the Einstein Gauss–Bonnet/Weyl theories of gravity at D=4

We investigate the gravitational backreaction, generated by coupling a general conformal sector to external, classical gravity, as described by a conformal anomaly effective action. We address the issues raised by the regularization of the topological Gauss–Bonnet and Weyl terms in these actions and the use of dimensional regularization (DR). We discuss both their local and nonlocal expressions, as possible IR and UV descriptions of conformal theories, below and above the conformal breaking scale. Our discussion overlaps with several recent studies of dilaton gravities – obtained via a certain singular limit of the Einstein–Gauss–Bonnet (EGB) theory – originally introduced as a way to bypass Lovelock’s theorem. We show that nonlocal, purely gravitational realizations of such EGB theories, quadratic in the dilaton field, beside their local quartic forms, are possible, by a finite renormalization of the Euler density. Such nonlocal versions, which are deprived of any scale, can be expanded, at least around flat space, in terms of the combination R Box ^{-1} times multiple variations of the anomaly functional, as pointed out in recent studies at d=4. Similar conclusions can be drawn for the proposed nonlocal EGB theory. The expansion emerges from previous investigations of the anomalous conformal Ward identities that constrain such theories around the flat spacetime limit in momentum space.

Blow-up in Coupled Solutions for a $4$-dimensional Semilinear Elliptic Kuramoto–Sivashinsky System

The existence of singular limit solutions is established for a nonlinear elliptic Kuramoto–Sivashinsky system with exponential nonlinearity and Navier boundary conditions, by means of the nonlinear domain decomposition method.

Near resonant approximation of the rotating stratified Boussinesq system on a 3-torus

Based on a novel treatment of near resonances, we introduce a new approximation for the rotating stratified Boussinesq system on three-dimensional tori with arbitrary aspect ratios. The rotation and stratification parameters are arbitrary and not equal. We obtain global existence for the proposed nonlinear system for arbitrarily large initial data. This system is sufficiently accurate, with an important feature of coupling effects between slow and fast modes. The key to global existence is a sharp counting of the relevant number of nonlinear interactions. An additional regularity advantage arises from a careful examination of some mixed type interaction coefficients. In a wider context, the significance of our near resonant approach is a delicate balance between the inclusion of more interacting modes and the improvement of regularity properties, compared to the well-studied singular limit approach based on exact resonance.

On hysteresis–reaction–diffusion systems: Singular fast-reaction limit derivation and nonlinear hysteresis feedback

This paper concerns a general class of PDE–ODE reaction–diffusion systems, which exhibits a singular fast-reaction limit towards a reaction–diffusion equation coupled to a scalar hysteresis operator. As applicational motivation, we present a PDE model for the growth of a population according to a given food supply, coupled to an ODE for the turnover of a food stock. Under realistic conditions the stock turnover is much faster than the population growth, yielding an intrinsic scaling parameter. We present two natural models of consume rate functions such that the dynamics for the food stock converges to a generalised play operator in the associated fast-reaction limit. We emphasise that the structural assumptions on the considered PDE–ODE models are quite general and that analogue systems might describe e.g. cell-biological buffer mechanisms, where proteins are stored and used at the same time. Finally, we present a new kind of hysteresis–diffusion-driven instability behaviour caused by the nonlinear coupling between a reaction–diffusion equation and a scalar generalised play operator. We discuss in detail how this coupling with a generalised play operator can lead to spatially inhomogeneous large-time behaviour or equilibration to a homogeneous state.

Fractional diffusion limit of a kinetic equation with diffusive boundary conditions in a bounded interval

We investigate the fractional diffusion approximation of a kinetic equation set in a bounded interval with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we show that the asymptotic density function is the unique solution of a fractional diffusion equation with Neumann boundary condition. This analysis completes a previous work by the same authors in which a limiting fractional diffusion equation was identified on the half-space, but the uniqueness of the solution (which is necessary to prove the convergence of the whole sequence) could not be established.

A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels

We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in $C(L^{1}_{\text{loc}})$ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference between the solution and the nonlocal term.

The Oberbeck–Boussinesq system with non-local boundary conditions

We consider the Oberbeck–Boussinesq system with non-local boundary conditions arising as a singular limit of the full Navier–Stokes–Fourier system in the regime of low Mach and low Froude numbers. The existence of strong solutions is shown on a maximal time interval [ 0 , T m a x ) [0, T_{\mathrm {max}}) . Moreover, T m a x = ∞ T_{\mathrm {max}} = \infty in the two-dimensional setting.

Toward uniform existence and convergence theorems for three-scale systems of hyperbolic PDEs with general initial data

Uniform existence of solutions to initial-value problems and convergence of appropriately filtered solutions are proven for a special class of three-scale singular limit equations, without any restriction on the initial data. The uniform existence is proven using a novel system of energy estimates. The convergence result is based on a detailed analysis of the fastest-scale oscillations, which unlike in two-scale systems have no explicit solution formula.

Dynamical system describing cloud of particles

The analysis of the nonautonomous dynamical system modeling steady states for the interaction of the cloud of particles is carried over. Maxwell–Boltzmann and classical Michie–King statistics are taken into account. Critical mass relative to temperature phenomenon is observed. Singular limit result for admissible range of the parameters is established as well.

Phase transitions as a manifestation of spontaneous unitarity violation

Spontaneous symmetry breaking is well understood under equilibrium conditions as a consequence of the singularity of the thermodynamic limit. How a single global orientation of the order parameter dynamically emerges from an initially symmetric state during a phase transition, however, is not captured by this paradigm. Here, we present a series of symmetry arguments suggesting that singling out a global choice for the ordered state is in fact forbidden under unitary time evolution, even in the presence of an environment and infinitesimal symmetry breaking perturbations. We thus argue that the observation of phase transitions in our everyday world presents a manifestation of the unitarity of quantum dynamics itself being spontaneously broken. We argue that this agrees with the observation that Schrödinger’s time dependent equation is rendered unstable for macroscopic objects owing to the same singular thermodynamic limit that affects equilibrium configurations.

Convergence rate for the incompressible limit of nonlinear diffusion–advection equations

The incompressible limit of nonlinear diffusion equations of porous medium type has attracted a lot of attention in recent years, due to its ability to link the weak formulation of cellpopulation models to free boundary problems of Hele–Shaw type. Although a vast literature is available on this singular limit, little is known on the convergence rate of the solutions. In this work, we compute the convergence rate in a negative Sobolev norm and, upon interpolating with BV-uniform bounds, we deduce a convergence rate in appropriate Lebesgue spaces.

Bifurcation structure of traveling pulses in type-I excitable media.

We study the scenario in which traveling pulses emerge in a prototypical type-I one-dimensional excitable medium, which exhibits two different routes to excitable behavior, mediated by a homoclinic (saddle-loop) and a saddle-node on the invariant cycle bifurcations. We characterize the region in parameter space in which traveling pulses are stable together with the different bifurcations behind either their destruction or loss of stability. In particular, some of the bifurcations delimiting the stability region have been connected, using singular limits, with the two different scenarios that mediated type-I local excitability. Finally, the existence of traveling pulses has been linked to a drift pitchfork instability of localized steady structures.