Eigenvalue Problem For Perturbations Research Articles

Stability of internal gravity wave modes: from triad resonance to broadband instability

A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter $\mu$ , compares to the basic-state amplitude parameter, $\epsilon \ll 1$ . For $\mu ={O}(1)$ , the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that $\mu \ll 1$ but $\alpha =\mu /\epsilon \gg 1$ , this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as $\mu$ is further decreased holding $\epsilon$ fixed, higher-frequency perturbations than these two subharmonics come into play, and when $\alpha ={O}(1)$ Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when $\alpha ={O}(1)$ PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.

Membranes with thin and heavy inclusions: Asymptotics of spectra

We study the asymptotic behaviour of eigenvalues of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which the resonance frequencies of the membrane and the frequencies of thin inclusion coincide is investigated. The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces. However the so-called limit operator is non-self-adjoint and possesses the Jordan chains of length 2. Apart from the lack of self-adjointness, the operator has non-compact resolvent. As a consequence, its spectrum has a complicated structure, for instance, the spectrum contains a countable set of eigenvalues with infinite multiplicity. The complete asymptotic analysis of eigenvalues has been carried out.

On the Finite Element Approximation of Fourth-Order Singularly Perturbed Eigenvalue Problems

Abstract We consider fourth-order singularly perturbed eigenvalue problems in one-dimension and the approximation of their solution by the h version of the Finite Element Method (FEM). In particular, we use a C 1 {C^{1}} -conforming FEM with piecewise polynomials of degree p ≥ 3 {p\geq 3} defined on an exponentially graded mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at the optimal rate when the error in the eigenvalues is measured in absolute value and the error in the eigenvectors is measured in the energy norm. We also illustrate our theoretical findings through numerical computations for the case p = 3 {p=3} .

Perturbed eigenvalue problems: an overview

The study of perturbed eigenvalue problems has been a very active field of investigation throughout the years. In this survey we collect several results in the field.

Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces: The Odd Multiplicity Case

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.

Global continuation in Euclidean spaces of the perturbed unit eigenvectors corresponding to a simple eigenvalue

In the Euclidean space $\mathbb R^k$, we consider the perturbed eigenvalue problem $Lx + \varepsilon N(x) = \lambda x$, $\|x\| = 1$, where $\varepsilon,\lambda$ are real parameters, $L$ is a linear endomorphism of $\mathbb R^k$, and $N\colon S^{k-1} \to \mathbb R^k$ is a continuous map defined on the unit sphere of $\mathbb R^k$. We prove a global continuation result for the \emph{solutions} $(x,\varepsilon,\lambda)$ of this problem. Namely, under the assumption that $x_* \in S^{k-1}$ is one of the two unit eigenvectors of $L$ corresponding to a simple eigenvalue $\lambda_* \in \R$, we show that, in the set of all the solutions, the connected component containing $(x_*,0,\lambda_*)$ is either unbounded or meets a solution $(x^*,0,\lambda^*)$ having $x^* \not= x_*$. Our result is inspired by a paper of R. Chiappinelli concerning the local persistence property of eigenvalues and eigenvectors of a perturbed self-adjoint operator in a real Hilbert space.

Fractional eigenvalue problems on R N

Let N ≥ 2 be an integer. For each real number s ∈ (0, 1) we denote by (−∆) s the corresponding fractional Laplace operator. First, we investigate the eigenvalue problem (−∆) su = λV(x)u on RN, where V : RN → R is a given function. Under suitable conditions imposed on V we show the existence of an unbounded, increasing sequence of positive eigenvalues. Next, we perturb the above eigenvalue problem with a fractional (t, p)-Laplace operator, when t ∈ (0, 1) and p ∈ (1, ∞) are such that t < s and s − N/2 = t − N/p. We show that when the function V is nonnegative on RN, the set of eigenvalues of the perturbed eigenvalue problem is exactly the unbounded interval (λ1, ∞), where λ1 stands for the first eigenvalue of the initial eigenvalue problem.

Asymptotic Analysis and Numerical Method for Singularly Perturbed Eigenvalue Problems

In this paper, we study the asymptotic analysis and numerical method for singularly perturbed eigenvalue problems (SPEPs). We first make a close asymptotic analysis of the SPEP and prove that, for ...

Perturbation results involving the 1-Laplace operator

Abstract We consider perturbed eigenvalue problems of the 1-Laplace operator and verify the existence of a sequence of solutions. It is shown that the eigenvalues of the perturbed problem converge to the corresponding eigenvalue of the unperturbed problem as the perturbation becomes small. The results rely on nonsmooth critical point theory based on the weak slope.

A perturbed eigenvalue problem on general domains

The perturbed eigenvalue problem −Δu−Δpu=λV(x)u, with p∈(1,N)∖{2} and V a weight function which takes nonnegative values and may have singular points, is studied in an Orlicz–Sobolev setting on general open sets from RN with N≥3. The analysis of these problems leads to a full characterization of the set of parameters λ for which the problem possesses nontrivial solutions as being an unbounded open interval.

Existence of nonnegative viscosity solutions for a class of problems involving the $${\infty}$$-Laplacian

The issue of existence of nonnegative solutions for a class of problems depending on a real parameter and involving the ∞-Laplacian is considered. The problem is treated as the limiting case of a family of perturbed eigenvalue problems for the (p, q(p))-Laplacian as \({p\rightarrow\infty}\). It is shown that nontrivial nonnegative viscosity solutions for this class of problems exist if and only if the parameter is greater than or equal to the reciprocal of the maximum of the distance to the boundary of the domain.

Linear stability of magnetohydrodynamic flow in a square duct with thin conducting walls

This study is concerned with the numerical linear stability analysis of liquid-metal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow that is valid for not only high but also moderate magnetic fields. This solution shows that, for low wall conductance ratios $c\ll 1$, an extremely strong magnetic field with Hartmann number $\mathit{Ha}\sim c^{-4}$ is required to attain the asymptotic flow regime considered in previous studies. We use a vector streamfunction–vorticity formulation and a Chebyshev collocation method to solve the eigenvalue problem for three-dimensional small-amplitude perturbations in ducts with realistic wall conductance ratios $c=1$, 0.1 and 0.01 and Hartmann numbers up to $10^{4}$. As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the sidewall jets with characteristic thickness ${\it\delta}\sim \mathit{Ha}^{-1/2}$. This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as $\mathit{Re}_{c}\sim 110\mathit{Ha}^{1/2}$ and $k_{c}\sim 0.5\mathit{Ha}^{1/2}$. The respective critical Reynolds number based on the total volume flux in a square duct with $c\ll 1$ is $\overline{\mathit{Re}}_{c}\approx 520$. Although this value is somewhat larger than $\overline{\mathit{Re}}_{c}\approx 313$ found by Ting et al. (Intl J. Engng Sci., vol. 29 (8), 1991, pp. 939–948) for the asymptotic sidewall jet profile, it still appears significantly lower than the Reynolds numbers at which turbulence is observed in experiments as well as in direct numerical simulations of this type of flow.

Computational Error Estimates for Born–Oppenheimer Molecular Dynamics with Nearly Crossing Potential Surfaces

This thesis consists of two papers that concern error estimates for the Born-Oppenheimer molecular dynamics, and adaptive algorithms for the Car-Parrinello and Ehrenfest molecular dynamics. In Paper I, we study error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces. The paper first proves an error estimate showing that the difference of the values of observables for the time- independent Schrodinger equation, with matrix valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states and the electron/nuclei mass ratio. Then we present a numerical method to determine the probability to be in excited states, based on Ehrenfest molecular dynamics, and stability analysis of a perturbed eigenvalue problem. In Paper II, we present an approach, motivated by the Landau-Zener probability estimation, to systematically choose the artificial electron mass parameter appearing in the Car-Parrinello and Ehrenfest molecular dynamics methods to achieve both good accuracy in approximating the Born-Oppenheimer molecular dynamics solution, and high computational efficiency. This makes the Car- Parrinello and Ehrenfest molecular dynamics methods dependent only on the problem data.

How does a shear boundary layer affect the stability of a capillary jet?

The stability of a capillary jet with a shear boundary layer growing over its free surface is studied theoretically. The effect of an infinitely thin boundary layer is first examined from the corresponding singularly perturbed eigenvalue problem. Then, a linear stability analysis of the Navier-Stokes equations for a finite boundary layer thickness verifies the consistency of the previous asymptotic study. We show that, for sufficiently long distances from the boundary layer origin, Rayleigh's dispersion relation is recovered, which means that the layer does not affect the jet's stability in that case. However, for distances on the order of the jet's radius, the perturbation growth factor, the most unstable wave number, and the range of unstable wave numbers increase, while the convective-to-absolute instability transition delays drastically. These results establish a general framework to explain a variety of capillary jet phenomena, from the extended jetting regime of capillary jets in gas co-flows to fundamental features of the dripping faucet.

Linear stability of viscous flow induced by surface stretching

The stability of the two-dimensional flow in a semi-infinite fluid induced by the stretching of a planar surface is investigated by a normal-mode linear stability analysis. The base flow is described by Crane’s exact analytical solution of the Navier–Stokes equation. A finite-difference method and a shooting method are implemented to solve the relevant eigenvalue problem for three-dimensional perturbations that vary sinusoidally with arbitrary wave number in the spanwise direction normal to the plane of the flow. The Crane flow is found to be linearly stable, with the least-damped mode arising in the limit of two-dimensional perturbations. Eigenfunctions for the disturbance velocity are presented and discussed for selected values of the transverse wave number. The numerical results for the dispersion relation corresponding to the least-damped mode for each transverse wave number are described by a simple analytical expression. The results of the stability analysis are contextualized by superposing a planar stagnation-point flow toward the stretching surface. As the ratio of the rate of elongation of the stagnation-point flow to the rate of stretching of the surface tends to zero, the dispersion curves smoothly asymptote to those found for the Crane flow. The results for planar stagnation-point flow toward a stationary surface are recovered in the opposite limit. The analysis is extended to account for suction or injection through a porous surface undergoing in-plane stretching.

Vector Finite Elements and the Step by Step Eigenvalue Perturbation Methods

The electromagnetic resonant structures and their perturbations have been found their way in many applications in microwave engineering. Analysis of these structures using finite element method results in a generalized eigenvalue problem, where the eigenvalues correspond to the resonant frequencies, and the eigenvectors correspond to the resonant modes. The perturbations of resonant structures yield perturbed eigenvalue problem and can be solved by eigenvalue perturbation methods, effectively. Combining finite element method with step by step eigenvalue perturbation method yields parametric history with respect to perturbation parameter. In this study perturbation of a microwave ring resonator placed in a metalic enclosure has been examined, combining the vector finite element and the step-by-step generalized eigenvalue perturbation methods.

Stability of high-Reynolds-number flow in a collapsible channel

AbstractWe study high-Reynolds-number flow in a two-dimensional collapsible channel in the asymptotic limit of wall deformations confined to the viscous boundary layer. The system is modelled using interactive boundary-layer equations for a Newtonian incompressible fluid coupled to the freely moving elastic wall under constant tension and external pressure. The deformation of the membrane is assumed to have small amplitude and long wavelength, whereas the flow comprises the inviscid core and the viscous boundary layers on both walls coupled to each other and to the membrane deformation. Firstly, by linking the interactive boundary-layer model to the small-amplitude, long-wavelength inviscid analysis, we conclude that the model is valid only when the pressure perturbations are fixed downstream from the wall indentation, contrary to the common assumption of classical boundary-layer theory. Next we explore possible steady states of the system, showing that a unique steady solution exists when the pressure is fixed precisely at the downstream end of the membrane, but there are multiple states possible if the pressure is specified further downstream. We examine the stability of these states by solving the generalized eigenvalue problem for perturbations to the nonlinear steady solutions and also by performing time integration of the full boundary-layer equations. Surprisingly, we find that no self-excited oscillations develop in the collapsible channel systems with finite-amplitude deformations. Instead, for each point in the parameter space, with the exception of points subject to numerical instabilities associated with the boundary-layer equations, exactly one of the steady states is predicted to be stable. We discuss these findings in relation to the results reported previously in the literature.

Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I

In this paper, we study the connection between the bifurcation of diffuse transition layers and that of the underlying limit interfacial problem in a degenerate spatially inhomogeneous medium. In dimension one, we prove the existence of bifurcation of diffuse interfaces in a pitchfork spatial inhomogeneity for a partial differential equation with bistable type nonlinearity. Bifurcation point is characterized quantitatively as well. The main conclusion is that the bifurcation diagram of the diffuse transition layers inherits mostly from that of the zeros of the spatial inhomogeneity. However, explicit examples are given for which the bifurcation of these two are different in terms of (im)perfection. This is a continuation of [8] which makes use of bilinear nonlinearity allowing the use of explicit solution formula. In the current work, we extend the results to a general smooth nonlinear function. We perform detail analysis of the principal eigenvalue and eigenfunction of some singularly perturbed eigenvalue problems and their interaction with the background inhomogeneity. This is the first result that takes into account simultaneously the interaction between singular perturbation, spatial inhomogeneity and bifurcation.

Linear stability of magnetohydrodynamic flow in a perfectly conducting rectangular duct

AbstractWe analyse numerically the linear stability of a liquid-metal flow in a rectangular duct with perfectly electrically conducting walls subject to a uniform transverse magnetic field. A non-standard three-dimensional vector stream-function/vorticity formulation is used with a Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. A relatively weak magnetic field is found to render the flow linearly unstable as two weak jets appear close to the centre of the duct at the Hartmann number $\mathit{Ha}\approx 9. 6. $ In a sufficiently strong magnetic field, the instability following the jets becomes confined in the layers of characteristic thickness $\delta \ensuremath{\sim} {\mathit{Ha}}^{\ensuremath{-} 1/ 2} $ located at the walls parallel to the magnetic field. In this case the instability is determined by $\delta , $ which results in both the critical Reynolds number and wavenumber scaling as ${\ensuremath{\sim} }{\delta }^{\ensuremath{-} 1} . $ Instability modes can have one of the four different symmetry combinations along and across the magnetic field. The most unstable is a pair of modes with an even distribution of vorticity along the magnetic field. These two modes represent strongly non-uniform vortices aligned with the magnetic field, which rotate either in the same or opposite senses across the magnetic field. The former enhance while the latter weaken one another provided that the magnetic field is not too strong or the walls parallel to the field are not too far apart. In a strong magnetic field, when the vortices at the opposite walls are well separated by the core flow, the critical Reynolds number and wavenumber for both of these instability modes are the same: ${\mathit{Re}}_{c} \approx 642{\mathit{Ha}}^{1/ 2} + 8. 9\ensuremath{\times} 1{0}^{3} {\mathit{Ha}}^{\ensuremath{-} 1/ 2} $ and ${k}_{c} \approx 0. 477{\mathit{Ha}}^{1/ 2} . $ The other pair of modes, which differs from the previous one by an odd distribution of vorticity along the magnetic field, is more stable with an approximately four times higher critical Reynolds number.

Transmission Eigenvalues for Elliptic Operators

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-self-adjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.