Fourth-order Operator Research Articles

On Lp-Contractivity of Semigroups Generated by Linear Partial Differential Operators

This paper is devoted to the study of contraction semigroups generated by linear partial differential operators. It is shown that linear partial differential operators of order higher than two cannot generate contraction semigroups on (Lp)Nforp∈[1, ∞) unlessp=2. Ifp>1 and theLp-dissipativity criterion is restricted to the cone of nonnegative functions for differential operators with real-valued coefficients, it is proven that the criterion still fails for operators of order higher than two, except for some fourth order operators if 3/2⩽p⩽3. A class of such fourth order operators is also presented.

Fourth-order operators on manifolds with a boundary

Recent work in the literature has studied fourth-order elliptic operators on manifolds with a boundary. This paper proves that, in the case of the squared Laplace operator, the boundary conditions which require that the eigenfunctions and their normal derivative should vanish at the boundary lead to self-adjointness of the boundary-value problem. On studying, for simplicity, the squared Laplace operator in one dimension, on a closed interval of the real line, alternative conditions which also ensure self-adjointness set to zero, at the boundary, the eigenfunctions and their second derivatives, or their first and third derivatives, or their second and third derivatives, or require periodicity, i.e. a linear relation among the values of the eigenfunctions at the ends of the interval. For the first four choices of boundary conditions, the resulting 1-loop divergence is evaluated for a real scalar field on the portion of flat Euclidean 4-space bounded by a 3-sphere, or by two concentric 3-spheres.

Factorization of Helmholtz Determinant Operator for Decomposable Bi-Anisotropic Media

Conditions for the most general bi-anisotropic media in which the electromagnetic field can be decomposed in two components with respect to two six-vectors were recently derived by these authors. In the present paper it is shown that the fourth-order Helmholtz determinant operator, which appears in the basic scalar Helmholtz equation for electromagnetic fields, can be expressed in factorized form as a product of two second-order operators. These two operators are shown to correspond to the decomposition of the field. Finally, because of the factorization, the scalar Green function corresponding to the Helmholtz determinant is shown to take a numerically tractable form in terms of two finite integrations.

Plane Waves in Decomposable Bi-Anisotropic Media

The class of decomposable bi-anisotropic media was recently defined to consist of media in which any electromagnetic field can be decomposed in two individual electromagnetic fields satisfying certain conditions for the electromagnetic fields. Also it was shown that, for a medium in this class, the fourth-order Helmholtz determinant operator governing the basic electromagnetic fields can be factorized, i.e., expressed as a product of two second-order operators. In the present paper, plane-wave propagation in the general decomposable medium is studied. Analytical solutions are derived for the dispersion equation and the polarizations of the eigenwaves are also determined in analytic form. As a check the expressions are applied to a medium with previously known solutions. In an appendix, conditions for the medium dyadics of the decomposable bi-anisotropic medium are derived for the case when the medium is lossless.

Green dyadic for a class of nonreciprocal bianisotropic media

The possibility of finding an analytic Green dyadic expression for a class of bianisotropic media, defined by the relations , , and between the medium parameter dyadics, is studied. It is shown that the determinant of the dyadic Helmholtz operator, an operator of fourth order, can be expressed as a product of two second-order operators. A method for finding the solution for the Green dyadic in the form of infinite series in terms of powers of the dimensionless parameter ξζ/μoϵo is given. For small values of the parameter, a two-term approximation is seen to take a simple analytic form. As an Appendix, another approach through the Fourier transformation is briefly discussed. Dyadic formalism is applied throughout in the analysis. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 19: 216–221, 1998.

Green dyadic for a class of nonreciprocal bianisotropic media

The possibility of finding an analytic Green dyadic expression for a class of bianisotropic media, defined by the relations , , and between the medium parameter dyadics, is studied. It is shown that the determinant of the dyadic Helmholtz operator, an operator of fourth order, can be expressed as a product of two second-order operators. A method for finding the solution for the Green dyadic in the form of infinite series in terms of powers of the dimensionless parameter ξζ/μoϵo is given. For small values of the parameter, a two-term approximation is seen to take a simple analytic form. As an Appendix, another approach through the Fourier transformation is briefly discussed. Dyadic formalism is applied throughout in the analysis. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 19: 216–221, 1998.

Isospectral Sets for Fourth-Order Ordinary Differential Operators

Let L(p)u = D 4 u i (p1u 0 ) 0 + p2u be a fourth-order dierential operator acting on L 2 (0;1) with p (p1;p2) belonging to L 2 (0;1) L 2 (0;1) and boundary conditions u(0) = u 00 (0) = u(1) = u 00 (1) = 0. We study the isospectral set of L(p) when L(p) has simple spectrum. In particular we show that for such p, the isospectral manifold is a real-analytic submanifold of L 2 (0;1)L 2 (0;1) which has innite dimension and codimension. A crucial step in the proof is to show that the gradients of the eigenvalues of L(p) with respect to p are linearly independent: we study them as solutions of a non-self-ajdoint fth-order system, the Borg system, among whose eigenvectors are the gradients.

Higher-order supersymmetry in quantum mechanics and integrability of two-dimensional Hamiltonians

A new method of finding the dynamic symmetry operators in two-dimensional quantum systems is proposed. This method is based on the SUSY algebra with superchanges of higher order in derivatives. The symmetry operators arise when closing the SUSY algebra for a wide class of potentials, and, in some cases, they are of second order in derivatives. The solutions for potentials admitting symmetry operators of fourth order are also obtained. The investigation of the quasiclassical limit of the superalgebra results in new classical integrals of motion for a certain type of systems. Bibliography: 9 titles.

An application of the almost-positivity of a class of fourth-order pseudodifferential operators

Using the almost-positivity of a class of fourth-order pseudo-differential operators, we prove the inequality $$L = \sum\nolimits_{j = 1}^m {X_j^* X_j } + X_0 $$ where X0,Xj ∃ ψ phg 1 (Ω), with Xj,X complex-valued, are given.

Addendum to “A necessary and sufficient condition for a lower bound for fourth-order pseudodifferential operators”

Addendum to “A necessary and sufficient condition for a lower bound for fourth-order pseudodifferential operators”

Implicit time splitting for fourth-order parabolic equations

A coordinate-splitting economic difference scheme is proposed for generalized parabolic equations (GPE) containing fourth-order diffusion operators and the algorithm for its implementation is developed. The performance of the scheme is demonstrated for different cases, e.g. for treating bifurcation phenomena. The technique is applied to the numerical solution of Swift-Hohenberg equation describing the Rayleigh-Bénard convection and results are obtained for very large system sizes and for very long times on small computational platform.

Quantized Maxwell theory in a conformally invariant gauge

Maxwell theory can be studied in a gauge which is invariant under conformal rescalings of the metric, and first proposed by Eastwood and Singer. This paper studies the corresponding quantization in flat Euclidean 4-space. The resulting ghost operator is a fourth-order elliptic operator, while the operator P on perturbations of the potential is a sixth-order elliptic operator. The operator P may be reduced to a second-order non-minimal operator if a dimensionless gauge parameter tends to infinity. Gauge-invariant boundary conditions are obtained by setting to zero at the boundary the whole set of perturbations of the potential, jointly with ghost perturbations and their normal derivative. This is made possible by the fourth-order nature of the ghost operator. An analytic representation of the ghost basis functions is also obtained.

Conformally covariant differential operators: properties and applications

We discuss conformally covariant differential operators, which under local rescalings of the metric, , transform according to for some r if is of order s. It is shown that the flat space restrictions of their associated Green functions have forms which are strongly constrained by flat space conformal invariance. The same applies to the variation of the Green functions with respect to the metric. The general results are illustrated by finding the flat space Green function, and also its first variation, for previously found second-order conformal differential operators acting on k-forms in general dimensions. Furthermore, we construct a new second-order conformally covariant operator acting on rank-four tensors with the symmetries of the Weyl tensor whose Green function is similarly discussed. We also consider fourth-order operators, in particular a fourth-order operator acting on scalars in arbitrary dimension, which has a Green function with the expected properties. The results obtained here for conformally covariant differential operators are generalizations of standard results for the two-dimensional Laplacian on curved space and its associated Green function, which is used in the Polyakov effective gravitational action. It is hoped that they may have similar applications in higher dimensions.

Analysis of a Fourth-Order Compact Scheme for Convection–Diffusion

In, 1984 Gupta et al. introduced a compact fourth-order finite-difference convection-diffusion operator with some very favorable properties. In particular, this scheme does not seem to suffer excessively from spurious oscillatory behavior, and it converges with standard methods such as Gauss Seidel or SOR (hence, multigrid) regardless of the diffusion. This scheme has been rederived, developed (including some variations), and applied in both convection-diffusion and Navier-Stokes equations by several authors. Accurate solutions to high Reynolds-number flow problems at relatively coarse resolutions have been reported. These solutions were often compared to those obtained by lower order discretizations, such as second-order central differences and first-order upstream discretizations. The latter, it was stated, achieved far less accurate results due to the artificial viscosity, which the compact scheme did not include. We show here that, while the compact scheme indeed does not suffer from a cross-stream artificial viscosity (as does the first-order upstream scheme when the characteristic direction is not aligned with the grid), it does include a streamwise artificial viscosity that is inversely proportional to the natural viscosity. This term is not always benign. 7 refs., 1 fig., 1 tab.

The role of nonunique axisymmetric solutions in 3-D vortex breakdown

The three-dimensional, compressible Navier–Stokes equations in primitive variables are solved numerically to simulate vortex breakdown in a constricted tube. Time integration is performed with an implicit Beam-Warming algorithm using fourth-order compact operators to discretize spatial derivatives. Initial conditions are obtained by solving the steady, compressible, and axisymmetric form of the Navier–Stokes equations with Newton’s method. The effects of three-dimensionality on flows that are initially axisymmetric and stable to 2-D disturbances are examined. Stability of the axisymmetric base flow is assessed through 3-D time integration. Axisymmetric solutions at a Mach number of 0.3 and a Reynolds number of 1000 contain a region of nonuniqueness. Within this region, 3-D time integration reveals only unique solutions, with nonunique axisymmetric initial conditions converging to a unique solution that is steady and axisymmetric. Past the primary limit point, which approximately identifies the appearance of critical flow (a flow that can support an axisymmetric standing wave), the solutions bifurcate into 3-D time-periodic flows. Thus this numerical study shows that the vortex strength associated with the loss of stability to 3-D disturbances and that of the primary limit point are in close proximity. Additional numerical and theoretical studies of 3-D swirling flows are needed to determine the impact of various parameters on dynamic behavior. For example, it is possible that a different flow behavior, leading to a nearly axisymmetric vortex breakdown state, may develop with other inlet profiles and tube geometries.

A necessary and sufficient condition for a lower bound for fourth-order pseudodifferential operators

We give necessary and sufficient conditions for the lower bound {fx55-01} to hold for any compact setK ⊂X, an open set ofR n , andP =P* ∃ ψ 4 (X) with p(x, ξ) ~ q 2 2 + p3 + p2 + ..., q2 beingtransversally elliptic with respect to the characteristic manifold Σ =q 2 -1 (0).

An approximate factorization/least squares solution method for a mixed finite element approximation of the Cahn-Hilliard equation

We discuss in this article the numerical solution of the Cahn-Hilliard equation modelling the spinodal decomposition of binary alloys. The numerical methodology combines a second-order finite difference time discretization with a mixed finite element space approximation and a least squares formulation based on an approximate factorization of a fourth-order elliptic operator which appears in the numerical model. The least squares problem—which is linear—is solved by a preconditioned conjugate gradient algorithm. The results of numerical experiments illustrate the possibilities of the methods discussed in this article.

Viscous Cahn–Hilliard Equation II. Analysis

The viscous Cahn–Hilliard equation may be viewed as a singular limit of the phase-field equations for phase transitions. It contains both the Allen–Cahn and Cahn–Hilliard models of phase separation as particular cases; by specific choices of parameters it may be formulated as a one-parameter (sayα) homotopy connecting the Cahn–Hilliard (α=0) and Allen–Cahn (α=1) models. The limitα=0 is singular in the sense that the smoothing property of the analytic semigroup changes from being of the type associated with second order operators to the type associated with fourth order operators. The properties of the gradient dynamical system generated by the viscous Cahn–Hilliard equation are studied asαvaries in [0, 1]. Continuity of the phase portraits near equilibria is established independently ofα∈[0, 1] and, using this, a piecewise, uniform in time, perturbation result is proved for trajectories. Finally the continuity of the attractor is established and, in one dimension, the existence and continuity of inertial manifolds shown and the flow on the attractor detailed.

WILSONIAN BLACK HOLE ENTROPY IN QUANTUM GRAVITY

Using the Wilsonian procedure (renormalization group improvement) we discuss the finite quantum corrections to black hole entropy in renormalizable theories. In this way, the Wilsonian black hole entropy is found for GUT’s (of asymptotically free form, in particular) and for the effective theory for the conformal factor aiming to describe quantum gravity in the infrared region. The off-critical regime (where the coupling constants are running) for the effective theory for the conformal factor in quantum gravity (with or without torsion) is explicitly constructed. The corresponding renormalization group equations for the effective couplings are found using the Schwinger-DeWitt technique for the calculation of the divergences of the fourth order operator.

The Laguerre Type Operator in a Left Definite Hilbert Space

This article completes earlier work concerning the Laguerre type differential operator of fourth order, set in a weighted Sobolev space on [0, ∞). It is shown that the operator, which is self-adjoint in L2(0, ∞; e−x) ⊗ R, is also self-adjoint in the new space, whose inner product also involves first and second derivatives. Further, the spectrum remains unchanged, the Laguerre type polynomials are eigenfunctions, and the spectral resolution, associated with the operator, is still an eigenfunction expansion.