Homogenization of a boundary condition for the heat equation

An asymptotic analysis is given for the heat equation with mixed boundary conditions rapidly oscillating between Dirichlet and Neumann type. We try to present a general framework where deterministic homogenization methods can be applied to calculate the second term in the asymptotic expansion with respect to the small parameter characterizing the oscillations. Received August 20, 1999 / final version received March 1, 2000?Published online June 21, 2000     

1 : 3 Resonance is a possible cause of nonlinear panel flutter

A nonlinear boundary value problem modeling oscillations of a plate in a supersonic gas flow is considered. Using the normal forms method, the method of integral manifolds for dynamical systems with infinite-dimensional phase space, and asymptotic methods combined with numerical techniques, it is shown that the 1: 3 resonance of eigenfrequencies of the linearized boundary value problem can be a cause of subcritical bifurcations and hard excitation of oscillations.     

On Stability of the Inverted Pendulum Motion with a Vibrating Suspension Point

Under study is the stability of the inverted pendulum motion whose suspension point vibrates according to a sinusoidal law along a straight line having a small angle with the vertical. Formulating and using the contracting mapping principle and the criterion of asymptotic stability in terms of solvability of a special boundary value problem for the Lyapunov differential equation, we prove that the pendulum performs stable periodic movements under sufficiently small amplitude of oscillations of the suspension point and sufficiently high frequency of oscillations.     

Asymptotic Solution of the Autoresonance Problem

We study the problem of forced oscillations near a stable equilibrium of a two-dimensional nonlinear Hamiltonian system of equations. A given exciting force is represented as rapid oscillations with a small amplitude and a slowly varying frequency. We study the conditions under which such a perturbation makes the phase trajectory of the system recede from the original equilibrium point to a distance of the order of unity. To study the problem, we construct asymptotic solutions using a small amplitude parameter. We present the solution for not-too-small values of time outside the original boundary layer.     

Reflexions d'oscillations monodimensionnelles

We prove the propagation of oscillations with an asymptotic development for an oscillating initial boundary value problem of semilinear hyperbolic systems in the spirit of J.L.Joly, G.Métivier and J.Rauch. In particular we simplify the Joly-métivier-Rauch's proof for the Cauchy problem. Then, we show the new phenomenon of localised oscillations.     

Successive approximations to the solutions to nonlinear equations with a vector parameter in a nonregular case

We consider a nonlinear operator equation with a Fredholm linear operator in the principal part. The nonlinear part of the equation depends on the functionals defined on an open set in a normed vector space. We propose a method of successive asymptotic approximations to branching solutions. The method is used for studying the nonlinear boundary value problem describing the oscillations of a satellite in the plane of its elliptic orbit.     

Homogenization of low-cost control problems on perforated domains

The aim of this paper is to study the asymptotic behaviour of some low-cost control problems in periodically perforated domains with Neumann condition on the boundary of the holes. The optimal control problems considered here are governed by a second order elliptic boundary value problem with oscillating coefficients. It is assumed that the cost of the control is of the same order as that describing the oscillations of the coefficients. The asymptotic analysis of small cost problem is more delicate and need the H-convergence result for weak data. In this connection, an H-convergence result for weak data under some hypotheses is also proved.     

On parametric instability of viscous two-layer fluid in a vessel with a permeable bottom

A linear problem of parametric oscillations of a low-viscous two-layer fluid in a closed vessel partially filled with a porous medium is studied. An asymptotic solution is constructed on the basis of combined application of boundary functions and averaging methods. Approximate formulas for boundaries of instability domains in the case of subharmonic and harmonic resonances are derived.     

Locally periodic thin domains with varying period

We analyze the behavior of the solutions of the Laplace equation with Neumann boundary conditions in a thin domain with a highly oscillatory behavior. The oscillations are locally periodic in the sense that both the amplitude and the period of the oscillations may not be constant and actually they vary in space. We obtain the asymptotic homogenized limit and provide some correctors. To accomplish this goal, we extend the unfolding operator method to the locally periodic case. The main ideas of this extension may be applied to other cases like perforated domains or reticulated structures, which are locally periodic with not necessarily a constant period.     

Asymptotics of solutions of spectral problems of hydrodynamics in the neighborhood of angular points

At angular points on the boundary of a domain, we obtain an asymptotic expansion for the eigenfunctions of spectral problems that describe natural oscillations of an ideal liquid that partially fills a cavity in a solid body. We describe cases where the eigenfunctions have singularities at angular points. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 803–811, June, 1998. This work was partially supported by the Ukrainian State Committee on Science and Technology.     

Asymptotics of small spacings for subsonic nonstationary streamlined flow of a thin airfoil close to a solid boundary

In an asymptotic approximation of small spacings an analytic solution to the problem on harmonic oscillations of a thin airfoil which is moving with a subsonic velocity near a solid plane boundary is given. Results of a computation of the lifting force are given.Translated from Dinamicheskie Sistemy, No. 7, pp. 48–53, 1988.     

Long-wave oscillatory Marangoni instability in a horizontal liquid layer

The long-wave instability in the problem of thermocapillary convection in a horizontal layer with a free deformable boundary and a solid bottom is investigated. The transcendental equation for the main asymptotic term of the spectral parameter is written in explicit form. The main attention is paid to investigating oscillatory instability. For the frequency of neutral oscillations, simple transcendental equations are obtained that contain the Prandtl and Biot numbers. In a number of cases, exact solutions are indicated. Explicit formulae are given for the main asymptotic term of the Marangoni number. In the case of a non-heat-conducting solid wall, the relation between the critical values of the parameters for inverse Prandtl numbers is found. It is shown that, for different Prandtl numbers, the asymptotic values are in good agreement with the numerical values.     

Complex dynamic behavior during transition in a solid combustion model

Through examples in a free‐boundary model of solid combustion, this study concerns nonlinear transition behavior of small disturbances of front propagation and temperature as they evolve in time. This includes complex dynamics of period doubling, and quadrupling, and it eventually leads to chaotic oscillations. Within this complex dynamic domain we also observe a period six‐folding. Both asymptotic and numerical solutions are studied.We show that for special parameters our asymptotic method with some dominant modes captures the formation of coherent structures. Finally,we discuss possible methods to improve our prediction of the solutions in the chaotic case. © 2009 Wiley Periodicals, Inc. Complexity, 2009     

A comparison theorem

The existence of an explosive singularity in the unsteady boundary layer on a rotating disc in a counter-rotating fluid has now been shown incontrovertibly following the work of K. Stewartson, C. J. Simpson, and R. J. Bodonyi (J. Fluid. Mech.121 (1982), 507–515). Here, we develop some asymptotic results for the governing differential equations to help gain an understanding of the mechanism behind this phenomenon. No definite conclusions are possible, but the presence of inertial oscillations at the edge of the boundary layer could well play a definite role.     

Bifurcation from two equilibria of steady state solutions for non-reversible amplitude equations

In this paper, bifurcation and stability of two kinds of constant stationary solutions for non-reversible amplitude equations on a bounded domain with Neumann boundary conditions are investigated by using the perturbation theory and weak nonlinear analysis. The asymptotic behaviors and local properties of two explicit steady state solutions, and pitch-fork bifurcations are also obtained if the bounded domain is regarded as a parameter. In addition, the stability of a new increasing or decaying local steady state solution with oscillations are analyzed.     

Layer potential techniques for the narrow escape problem

The narrow escape problem consists in deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. The asymptotic formula explicitly exhibits the nonlinear interaction of many small absorbing targets. Based on the asymptotic theory for eigenvalue problems developed in Ammari et al. (2009) [3], we also construct high-order asymptotic formulas for the perturbation of eigenvalues of the Laplace and the drifted Laplace operators for mixed boundary conditions on large and small pieces of the boundary.     

Scattering by a highly oscillating surface

The boundary function method [A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, The boundary function method for singular perturbation problems, SIAM Studies in Applied Mathematics, Philadelphia, 1995] is used to build an asymptotic expansion at any order of accuracy of a scalar time‐harmonic wave scattered by a perfectly reflecting doubly periodic surface with oscillations at small and large scales. Error bounds are rigorously established, in particular in an optimal way on the relevant part of the field. It is also shown how the maximum principle can be used to design a homogenized surface whose reflected wave yields a first‐order approximation of the actual one. The theoretical derivations are illustrated by some numerical experiments, which in particular show that using the homogenized surface outperforms the usual approach consisting in setting an effective boundary condition on a flat boundary. Copyright © 2014 John Wiley & Sons, Ltd.     

Boundary voltage perturbations caused by small conductivity inhomogeneities nearly touching the boundary

We establish an asymptotic expansion of the steady-state voltage potentials in the presence of a diametrically small conductivity inhomogeneity that is nearly touching the boundary. Our asymptotic formula extends those already derived for a small inhomogeneity far away from the boundary and is expected to lead to very effective algorithms, aimed at determining location and certain properties of the shape of a small inhomogeneity that is nearly touching the boundary based on boundary measurements. Viability of the asymptotic formula is documented by numerical examples.     

关于边界层方法

本文指出传统的边界层方法(包括匹配法和Vi?ik—Lyusternik方法)的不足:不能作出边界层项的渐近展开式.提出多重尺度构造边界层项的方法,得到符合实情的结果.又与Levinson所用的方法比较,本方法能更简单地导出后一方法给出的边界层项的渐近展开式.又应用此方法研究现有的关于奇异摄动的某些成果,指出这些成果的局限性,并在一般情况下作出解的渐近展开式.     

Spectral properties of a linearized magnetohydrodynamics model

One considers the linear oscillations of a perfectly conducting nonviscous plasma in an external rectilinear magnetic field. One studies the frequency spectrum corresponding to a fixed harmonic mode. One has investigated the asymptotic behavior of the discrete spectrum at infinity and around the boundary of the essential spectrum.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 120–123, 1985.The author expresses his deep gratitude to his scientific adviser Professor M. Sh. Birman for his help and for numerous discussions and to A. E. Lifshits for consultations and for his interest in this paper.