Rigorous Asymptotic Analysis Research Articles

LONG TIME BEHAVIOR OF THE TWO-DIMENSIONAL VLASOV EQUATION WITH A STRONG EXTERNAL MAGNETIC FIELD

When charged particles are submitted to a large external magnetic field, their movement in first approximation occurs along the magnetic field lines and obeys a one-dimensional Vlasov equation along these field lines. However, when observing the particles on a sufficiently long time scale, a drift phenomenon perpendicular to the magnetic field lines superposes to this first movement. In this paper, we present a rigorous asymptotic analysis of the two-dimensional Vlasov equation when the magnetic field tends to infinity, the observation time scale increases accordingly. Techniques based on the two-scale convergence and the introduction of a second problem enable us to find an equation verified by the weak limit of the distribution function.

Transient Couette flows of Oldroyd's fluids under imposed torques

We study the transient Couette flow of an Oldroyd fluid that fills the gap between two circular cylinders when a constant torque is suddenly applied to the inner cylinder, the outer one being kept motionless. Contrarily to most former studies, the inertia of the moving boundary is not neglected. We give the exact solutions of this problem for a wide class of initial conditions and we present a rigorous asymptotic analysis for small gap devices when the initial state is stationary. The case of Grade 2 fluids is also considered and treated. We also show in some experimental tests, that the knowledge of the relaxation curve of the angular velocity of the rotor can be used to identify the parameters of the model.

Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution

Using the isomonodromy and Riemann-Hilbert methods, we perform a rigorous global asymptotic analysis of the Clarkson-McLeod fourth Painleve transcendent. In particular, we prove some of the Clarkson-McLeod conjectures concerning the asymptotic behaviour, as , of the solutions of Painleve IV equation which decay as . The relevant exact connection formulae are also rigorously derived.

Probability of resolution of the MUSIC algorithm

The MUSIC algorithm is well known for its high-resolution capability, and various aspects of its statistical performance have been investigated. However, rigorous asymptotic analysis of one of its most important performance measures, the probability of resolution, is not available yet. We analyze the probability of the MUSIC algorithm resolving two spatially separated signal sources in the context of array processing. By formulating the resolution problem in the framework of statistical decision theory and directly determining the probability density function (PDF) of the indefinite and singular quadratic form that defines the resolution event, we arrive at an exact asymptotic formula for the probability of resolution. This is accomplished by a multistep procedure. Computer simulations have been performed to confirm the validity of the theory. >

A numerical method for evaluating zeros of solutions of second-order linear differential equations

A numerical algorithm for computing real zeros of solutions of 2nd-order linear differential equations y + q ( x ) y = 0 y+ q(x)y = 0 in the oscillatory case on a half line is studied. The method applies to the class q ( x ) = a + b / x + O ( x − p ) q(x) = a + b/x + O({x^{ - p}}) , with a > 0 a > 0 , b ∈ R b \in {\mathbf {R}} , p > 1 p > 1 . This procedure is based on a certain nonlinear 3rd-order equation (Kummer’s equation) which plays a role in the theory of transformations of 2nd-order differential equations into each other, and was earlier introduced by F. W. J. Olver in 1950 to compute zeros of cylinder functions. A rigorous asymptotic and numerical analysis is developed by combining Borůvka’s approach to the study of Kummer’s equation and Olver’s original idea. Numerical examples are presented.

An analysis of a phase field model of a free boundary

A mathematical analysis of a new approach to solidification problems is presented. A free boundary arising from a phase transition is assumed to have finite thickness. The physics leads to a system of nonlinear parabolic differential equations. Existence and regularity of solutions are proved. Invariant regions of the solution space lead to physical interpretations of the interface. A rigorous asymptotic analysis leads to the Gibbs-Thompson condition which relates the temperature at the interface to the surface tension and curvature.

Rigorous asymptotic analysis of transmission through a curved dielectric slab

A circular cylindrical dielectric layer is an idealized but rigorously analyzable model for radome covers. With a line source located on the concave side, an exact integral formulation is derived for the field transmitted to the convex side. Alternative representations are developed therefrom in terms of discrete guided modes and continuous spectra, and of ray integrals which, asymptotically at high frequencies, yield geometric optical fields that experience multiple internal reflections between the layer boundaries and also multiple reflections on the concave side. It is then shown that the higher order multiple reflected contributions can be expressed collectively as a ray field with a weighted transmission coefficient that is equivalent to the plane wave transmission coefficient for a plane parallel layer but includes a simple curvature correction. When source and observer are close to the inner and outer layer boundaries, respectively, and are also separated by a large angular interval, guided mode effects may have to be included as well. The result is a general and novel representation of the transmitted field in terms of a certain number of ordinary multiple reflected geometric optical ray fields, a single "collective" ray field, which includes in a composite manner all of the remaining internal reflections, and, possibly, the guided modes along the layer.

Coherence of the field of a laser beam in a turbulent atmosphere

A rigorous asymptotic and numerical analysis is made of the spatial coherence function of the field of a laser beam propagating in a turbulent atmosphere. The form of the spatial coherence function is established and the maximum excess of the coherence radius of the field of a spatially bounded beam over the values of this radius for plane and spherical waves is determined under conditions of strong intensity fluctuations. The accuracy of approximations obtained earlier for the coherence radius of the field of a laser beam is estimated.

Numerical Study of Strong Slot Injection into a Supersonic Laminar Boundary Layer

This paper addresses the problem of strong slot injection into a supersonic laminar boundary layer within the framework of triple-deck theory. A previous numerical technique for the same problem presented a pressure sensitivity or instability due to the ineffective imposition of the downstream boundary condition and could not account for separation ahead of the slot. The present study removes these two limitations. Solutions to the triple- deck equations are presented for a wide range of wall injection velocities, and the onset of separation is indeed found to occur ahead of the slot. HE general problem of predicting the influence on the flowfield of injecting a secondary fluid into a well- developed boundary layer is of great practical significance. For example, fluid injection is widely used in the aircraft engine industry for reducing heat transfer across turbine blades and is an effective means of controlling transition or separation (blowoff) of boundary layers over airplane control surfaces. This paper addresses the particular case of strong slot injection into a high Reynolds number supersonic laminar boundary layer over a flat plate, within the framework of the triple-deck theory of Stewartson et al. M The injection velocity and the slot length are therefore assumed to be respectively of order Re-3/8Uw and Re~3/8x0. 0^ is the freestream velocity, x0 the length of the plate preceding the slot, and Re the asymptotically large Reynolds number, RC^U^XQ/V, v being the kinematic viscosity and the superscript ~ indicating physical quantities. For these conditions three regions or decks are identified around the slot: the lower, middle, and outer decks, characterized by viscous, inviscid rotational, and inviscid irrotational flow conditions and having characteristic heights of order Re-5/8x0, Re~1/2x0, and Re~3/8x0 respectively. For a detailed review of the triple-deck theory and its most im- portant applications, see Ref. 2. Smith and Stewartson3'4 consider the problem of slot as well as plate injection into a supersonic laminar boundary layer. They provide a comprehensive review of the literature, formulate a rigorous asymptotic analysis for the strong in- jection problem, and analyze the connections of their theory with the problem of weak and massive blowing. For the case of strong blowing by slot injection, of interst here, Ref. 3 provides analytical solutions for the linearized equations, valid when the injection velocity is small in the triple-deck scaling, as well as a numerical technique for the full nonlinear problem. However, such a numerical technique presents a pressure sensitivity, or instability, due to, as the authors

Bose-Einstein condensation in a three-dimensional system at constant pressure

Using techniques developed by Greenspoon and Pathria, and by Chaba and Pathria, a rigorous asymptotic analysis of the onset of Bose-Einstein condensation in a finite three-dimensional system at constant pressure, under Dirichlet boundary conditions, is carried out. Finite size corrections arising from a modification of the density of states of the system, as well as from the discreteness of the single-particle energy levels, play an important role in the critical region. At the critical temperature ${T}_{c}$, the volume of the system changes abruptly from being proportional to the total number of particles $N$ to being proportional to ${N}^{\frac{3}{4}}$, the number of particles in the condensate rises sharply from $\ensuremath{\propto}{N}^{\frac{1}{4}}$ to $\ensuremath{\propto}N$, and the specific heat ${C}_{P}$ turns out to be proportional to ${N}^{\frac{9}{8}}$. At somewhat lower temperatures, the finite size effects arising from the discreteness of the single-particle energy levels become dominant and the volume of the system, together with ${C}_{P}$, becomes $\ensuremath{\propto}{N}^{\frac{3}{5}}$.

Bose-Einstein condensation in an Einstein universe

The condensation of a non-relativistic ideal Bose gas in an Einstein universe is investigated. Explicit expressions for the condensate fraction N0/N and the specific heat are obtained by using the Poisson summation formula to express the summations as integrations plus corrections. It is shown that the finiteness of the system smoothes out the cusp-like singularity of the infinite system. A rigorous asymptotic analysis of the critical temperature and the specific heat maximum are given, and the relation with the scaling theory of finite size effects is briefly discussed.

Bose-Einstein condensation in finite noninteracting systems: A new law of corresponding states

We have carried out a rigorous asymptotic analysis of the thermodynamic behavior of an ideal Bose gas confined to an arbitrary, finite cuboidal geometry under periodic boundary conditions. Our investigation, which is based on the grand canonical ensemble, leads to the construction of an abstract, thermogeometric space in which the process of Bose-Einstein condensation appears as a "collapse" of the lattice points of the space towards its origin. In an infinite geometry, the collapse is accomplished in an infinitesimally small interval of temperature; this results in the appearance of mathematical singularities in the thermodynamic functions of the system. In a finite geometry, the "collapse" proceeds gradually and is spread over a temperature range $\ensuremath{\Delta}T$ such that $\frac{\ensuremath{\Delta}T}{{T}_{0}}=O(\frac{\overline{l}}{{L}_{<}})$, where $\overline{l}$ is the mean interatomic distance while ${L}_{<}$ is the length of the shortest side of the assembly; accordingly, the thermodynamic functions of the system remain smooth throughout. Special events, such as the specific-heat maximum, occur when the "lattice parameters" of our thermogeometric space acquire certain characteristic values which depend only on the shape of the system and not on its actual size. This leads to a new law of corresponding states for Bose-Einstein systems of finite size.

Theory of Ignition of a Reactive Solid by Constant Energy Flux

Abstract A formula for the ignition time, that Bradley obtained empirically, is derived analytically by a rigorous asymptotic analysis of the limit of large activation energy. Two terms in the expansion must be retained. The second term reveals the existence—just prior to ignition—of a reactive-diffusive zone at the surface of the solid and a transient-diffusive zone in the interior. The analysis also exhibits a universal correction factor for Bradley's formula, of order unity.