Vanishing Parameter Research Articles

Continuous dependence and convergence for a Moore-Gibson-Thompson thermoelastic problem

In this article, we investigate how the solutions of the Moore-Gibson-Thompson thermoelasticity vary after a change of the relaxation parameter or the conductivity rate parameter, although, in the second case, only for radial solutions. The results focus on the structural stability. We also obtain the convergence of the Moore-Gibson-Thompson thermoelasticity to the type III thermoelasticity and the convergence of the Moore-Gibson-Thompson thermoelasticity to the Lord-Shulman thermoelasticity in the case of radial solutions. For the structural stability results, a certain measure for the difference of solutions can be used to control by an expression depending on the square of the difference of the parameters, and, for the convergence results, a measure of the difference of the solutions is proved to be controlled by the square of the vanishing parameter. In the proof of the above results, the energy arguments are used. It is worth saying that there are no results of this kind for the Moore-Gibson-Thompson thermoelasticity. Therefore, our results are the first contributions in this sense.

Asymptotic behavior of a random oscillating nonlinear reactive transport in thin turbulent layers

We consider a nonlinear reactive transport of a scalar field in a domain composed of a bounded region lined by turbulent thin layers. We suppose that these layers display a cellular microstructure with thin varying thickness. We suppose that, within the layers, the diffusion, reaction, and velocity coefficients admit random large‐scale structures. Under mixing properties of the diffusion, reaction, and velocity processes, we study the asymptotic behavior of the problem with respect to a vanishing parameter describing the thickness of the layers. We derive the effective flux boundary condition on the surface of the fixed region which coincides with the lower surface of the layers. This flux boundary condition consists in a generalized nonlinear Robin condition involving some nonlocal random effects which emerge asymptotically due to the random fluctuations of the reactive transport parameters in the layers.

Non-Anderson critical scaling of the Thouless conductance in 1D

We propose and investigate numerically a one-dimensional model which exhibits a non-Anderson disorder-driven transition. Such transitions have recently been attracting a great deal of attention in the context of Weyl semimetals, one-dimensional systems with long-range hopping and high-dimensional semiconductors. Our model hosts quasiparticles with the dispersion ±|k|αsignk with α<1∕2 near two points (nodes) in momentum space and includes short-range-correlated random potential which allows for scattering between the nodes and near each node. In contrast with the previously studied models in dimensions d<3, the model considered here exhibits a critical scaling of the Thouless conductance which allows for an accurate determination of the critical properties of the non-Anderson transition, with a precision significantly exceeding the results obtained from the critical scaling of the density of states, usually simulated at such transitions. We find that in the limit of the vanishing parameter ε=2α−1 the correlation-length exponent ν=2∕(3|ε|) at the transition is inconsistent with the prediction νRG=1∕|ε| of the perturbative renormalisation-group analysis. Our results allow for a numerical verification of the convergence of ε-expansions for non-Anderson disorder-driven transitions and, in general, interacting field theories near critical dimensions.

Averaging of highly-oscillatory transport equations

In this paper, we develop a new strategy aimed at obtaining high-order asymptotic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature are re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models.

A mathematical model for turbulent transport through thin randomly oscillating layers surrounding a fixed domain

We consider a problem of transport of a physical entity by an incompressible velocity in a domain composed of a fixed region and thin varying layers. We suppose that these layers display a cellular microstructure with random thin varying thickness. We suppose that, within the layers, the diffusion and velocity coefficients admit random large-scale structures. Under mixing properties of the diffusion and velocity processes we study the asymptotic behavior of the problem with respect to a vanishing parameter describing the thickness of the layers. We derive the effective boundary conditions on the boundary of the fixed region. These boundary conditions reveal the effects of the random time fluctuations of the diffusion coefficient and fluid velocity.

Nonlocal scalar field equations: Qualitative properties, asymptotic profiles and local uniqueness of solutions

We study the nonlocal scalar field equation with a vanishing parameter:(Pϵ){(−Δ)su+ϵu=|u|p−2u−|u|q−2uinRNu∈Hs(RN), where s∈(0,1), N>2s, q>p>2 are fixed parameters and ϵ>0 is a vanishing parameter. For ϵ small, we prove the existence and qualitative properties of positive solutions. Next, we study the asymptotic behavior of ground state solutions when p is subcritical, supercritical or critical Sobolev exponent 2⁎=2NN−2s. For p<2⁎, the ground state solution asymptotically coincides with unique positive ground state solution of (−Δ)su+u=up, whereas for p=2⁎ the asymptotic behavior of the solutions is given by the unique positive solution of the nonlocal critical Emden–Fowler type equation. For p>2⁎, the solution asymptotically coincides with a ground-state solution of (−Δ)su=up−uq. Furthermore, using these asymptotic profile of positive solutions, we establish the local uniqueness of positive solution.

The Adiabatic Theorem and Linear Response Theory for Extended Quantum Systems

The adiabatic theorem refers to a setup where an evolution equation contains a time-dependent parameter whose change is very slow, measured by a vanishing parameter $\epsilon$. Under suitable assumptions the solution of the time-inhomogenous equation stays close to an instantaneous fixpoint. In the present paper, we prove an adiabatic theorem with an error bound that is independent of the number of degrees of freedom. Our setup is that of quantum spin systems where the manifold of ground states is separated from the rest of the spectrum by a spectral gap. One important application is the proof of the validity of linear response theory for such extended, genuinely interacting systems. In general, this is a long-standing mathematical problem, which can be solved in the present particular case of a gapped system, relevant e.g.~for the integer quantum Hall effect.

Fractional Schrödinger–Poisson Systems with a General Subcritical or Critical Nonlinearity

Abstract We consider a fractional Schrödinger–Poisson system with a general nonlinearity in the subcritical and critical case. The Ambrosetti–Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

We study the leading order behaviour of positive solutions of the equation -\Delta u +\epsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\mathbb R^N, where N\ge 3 , q&gt;p&gt;2 and when \epsilon &gt;0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of p , q and N . The behavior of solutions depends sensitively on whether p is less, equal or bigger than the critical Sobolev exponent 2^\ast=\frac{2N}{N-2} . For p&lt;2^\ast the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p&gt;2^\ast the solution asymptotically coincides with the solution of the equation with \epsilon =0 . In the most delicate case p=2^\ast the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden–Fowler equation, whose choice depends on \epsilon in a nontrivial way.

A singular controllability problem with vanishing viscosity

The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? Our viscous term contains the fractional power of the Dirichlet Laplace operator and it is multiplied by a small parameter devoted to tend to zero. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to the vanishing parameter.

Asymptotic analysis of the Poisson–Boltzmann equation in biological membrane channels

The Poisson–Boltzmann equation has been increasingly used for the description of biomolecular systems in order to derive their electrostatic properties. We here consider a domain consisting of two living cells which communicate through a system of proteins which assemble at specific membrane areas building microchannels called gap junctions. We describe the asymptotic behavior of the solution of the Poisson–Boltzmann equation posed in this domain. Using Γ-convergence tools, we derive some electrostatic properties of the biological membrane with respect to a vanishing parameter which is simultaneously associated to the membrane thinness, to the diameter of the gap junction microchannels and to the Debye length parameter which characterizes the spatial scale electrostatic interactions between particles within the gap junctions.

Uniform Estimates of an Eulerian–Lagrangian Method for Time-Dependent Convection-Diffusion Equations in Multiple Space Dimensions

We prove a priori error estimates for a family of Eulerian-Lagrangian methods for time-dependent convection-diffusion equations in multiple space dimensions, which hold uniformly with respect to the vanishing parameter $\varepsilon$. We use the interpolation of spaces and stability estimates to derive an $\varepsilon$-uniform estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right-hand side data. These estimates theoretically justify the strength of Eulerian-Lagrangian methods.

Limiting behavior of micropolar flow due to a linearly stretching porous sheet

The boundary layer flow of a micropolar fluid due to a linearly stretching sheet is studied in the limit of a vanishing coupling parameter. Asymptotic expansions of the stream function and the micro-rotation are sought after. The straightforward expansions involve secular terms. This singular behavior is removed by the novel approach of replacing the coordinate, measuring distances normal to the sheet, by two strained coordinates. This makes it possible to obtain exact (series) solutions for all levels of approximation. One can obtain results, as accurate as one would wish, by retaining enough terms of the expansions. Suction and injection through the sheet are included.

Nonparametric Bootstrap Procedures for Predictive Inference Based on Recursive Estimation Schemes

Our objectives in this paper are twofold. First, we introduce block bootstrap techniques that are (first order) valid in recursive estimation frameworks. Thereafter, we present two examples where predictive accuracy tests are made operational using our new bootstrap procedures. In one application, we outline a consistent test for out-of-sample nonlinear Granger causality, and in the other we outline a test for selecting amongst multiple alternative forecasting models, all of which are possibly misspecified. More specifically, our examples extend the White (2000) reality check to the case of non vanishing parameter estimation error, and extend the integrated conditional moment tests of Bierens (1982, 1990) and Bierens and Ploberger (1997) to the case of out-of-sample prediction. In both examples, appropriate re-centering of the bootstrap score is required in order to ensure that the tests have asymptotically correct size, and the need for such re-centering is shown to arise quite naturally when testing hypotheses of predictive accuracy. In a Monte Carlo investigation, we compare the finite sample properties of our block bootstrap procedures with the parametric bootstrap due to Kilian (1999); all within the context of various encompassing and predictive accuracy tests. An empirical illustration is also discussed, in which it is found that unemployment appears to have nonlinear marginal predictive content for inflation.

Bootstrap Procedures for Recursive Estimation Schemes with Applications to Forecast Model Selection

In recent years it has become apparent that many of the classical testing procedures used to select amongst alternative economic theories and economic models are not realistic. In particular, researchers have become more aware of the fact that parameter estimation error and data dependence play a crucial role in test statistic limiting distributions, a role which had hitherto been ignored to a large extent. Given the fact that one of the primary ways for comparing different models and theories is via use of predictive accuracy tests, it is perhaps not surprising that a large literature on the topic has developed over the last 10 years, including, for example, important papers by Diebold and Mariano (1995), West (1996), and White (2000). In this literature, it is quite common to compare multiple models (which are possibly all misspecified - i.e. they are all approximations of some unknown true model) in terms of their out of sample predictive ability, for given loss function. Our objectives in this paper are twofold. First, we introduce block bootstrap techniques that are (first order) valid in recursive estimation frameworks. Thereafter, we present two applications where predictive accuracy tests are made operational using our new bootstrap procedures. One of the applications outlines a consistent test for out-of-sample nonlinear Granger causality, and the other outlines a test for selecting amongst multiple alternative forecasting models, all of which may be viewed as approximations of some unknown underlying model. More specifically, our examples extend the White (2000) reality check to the case of non vanishing parameter estimation error, and extend the integrated conditional moment (ICM) tests of Bierens (1982, 1990) and Bierens and Ploberger (1997) to the case of out-of-sample prediction. Of note is that in both of these examples, it is shown that appropriate re-centering of the bootstrap score is required in order to ensure that the tests are properly sized, and the need for such re-centering is shown to arise quite naturally when testing hypotheses of predictive accuracy. The results of a Monte Carlo investigation of the ICM test suggest that the bootstrap procedure proposed in this paper yield tests with reasonable finite sample properties for samples with as few as 300 observations.

Bootstrap Conditional Distribution Tests in the Presence of Dynamic Misspecification

In this paper, we show the first order validity of the block bootstrap in the context of Kolmogorov type conditional distribution tests when there is dynamic misspecification and parameter estimation error. Our approach differs from the literature to date because we construct a bootstrap statistic that allows for dynamic misspecification under both hypotheses. We consider two test statistics; one is the CK test of Andrews (1997), and the other is in the spirit of Diebold, Gunther and Tay (1998). The limiting distribution of both tests is a Gaussian process with a covariance kernel that reflects dynamic misspecification and parameter estimation error. In order to provide valid asymptotic critical values we suggest an extention of the empirical process version of the block bootstrap to the case of non vanishing parameter estimation error. The findings from Monte Carlo experiments show that both statistics have good finite sample properties for samples as small as 500 observations.

Regular (2+1)-dimensional black holes within nonlinear electrodynamics

(2+1)-regular static black hole solutions with a nonlinear electric field are derived. The source to the Einstein equations is an energy momentum tensor of nonlinear electrodynamics, which satisfies the weak energy conditions and in the weak field limit becomes the (2+1)-Maxwell field tensor. The derived class of solutions is regular; the metric, curvature invariants and electric field are regular everywhere. The metric becomes, for a vanishing parameter, the (2+1)-static charged BTZ solution. A general procedure to derive solutions for the static BTZ (2+1)-spacetime, for any nonlinear Lagrangian depending on the electric field is formulated; for relevant electric fields one requires the fulfillment of the weak energy conditions.

A bilinear form of the variational principle with vanishing parameter

A variational principle whose Lagrangian function generates a hyperbolic heat conduction equation is exhibited. The main characteristic of this principle is that it contains two temperature fields that enter bilinearly into the Lagrangian function. These two fields are interpreted as two mutually independent approximate temperature profiles which are potentially competent to describe rationally the real physical temperature distribution. The proposed variational principle is used as a starting point for finding approximate solutions of the classical, i.e. Fourier's, heat conduction theory, by employing the vanishing parameter technique and the direct methods of variational calculus.

Reflection moveout and parameter estimation for horizontal transverse isotropy

Transverse isotropy with a horizontal axis of symmetry (HTI) is the simplest azimuthally anisotropic model used to describe fractured reservoirs that contain parallel vertical cracks. Here, I present an exact equation for normal‐moveout (NMO) velocities from horizontal reflectors valid for pure modes in HTI media with any strength of anisotropy. The azimuthally dependent P‐wave NMO velocity, which can be obtained from 3-D surveys, is controlled by the principal direction of the anisotropy (crack orientation), the P‐wave vertical velocity, and an effective anisotropic parameter equivalent to Thomsen's coefficient δ. An important parameter of fracture systems that can be constrained by seismic data is the crack density, which is usually estimated through the shear‐wave splitting coefficient γ. The formalism developed here makes it possible to obtain the shear‐wave splitting parameter using the NMO velocities of P and shear waves from horizontal reflectors. Furthermore, γ can be estimated just from the P‐wave NMO velocity in the special case of the vanishing parameter ε, corresponding to thin cracks and negligible equant porosity. Also, P‐wave moveout alone is sufficient to constrain γ if either dipping events are available or the velocity in the symmetry direction is known. Determination of the splitting parameter from P‐wave data requires, however, an estimate of the ratio of the P‐to‐S vertical velocities (either of the split shear waves can be used). Velocities and polarizations in the vertical symmetry plane of HTI media, that contains the symmetry axis, are described by the known equations for vertical transverse isotropy (VTI). Time‐related 2-D P‐wave processing (NMO, DMO, time migration) in this plane is governed by the same two parameters (the NMO velocity from a horizontal reflector and coefficient ε) as in media with a vertical symmetry axis. The analogy between vertical and horizontal transverse isotropy makes it possible to introduce Thomsen parameters of the “equivalent” VTI model, which not only control the azimuthally dependent NMO velocity, but also can be used to reconstruct phase velocity and carry out seismic processing in off‐symmetry planes.

A variational principle for linear and nonlinear wave propagation in a dissipative medium

In this paper we will give a variational principle with a vanishing parameter which provides a satisfactory variational characterization of dissipative systems. We will incorporate this variational principle into Whitham's averaged Lagrangian method to develop a formulation for the propagation of a slowly-varying wavetrain in a dissipative medium. We will then apply this formulation to describe the nonlinear evolution of a modulated gravity wave packet on the surface of a dissipative fluid and show that the modulation is stabilized by the dissipative effects.