Showing papers in "Quarterly of Applied Mathematics in 1997"

Constructing asymptotic series for probability distributions of Markov chains with weak and strong interactions

Abstract: Many applications arise in manufacturing systems, and queueing network problems involve Markov chains having slow and fast components. These components are coupled through weak and strong interactions. The main goal of this work is to study asymptotic properties for the probability distribution of the aforementioned Markov chains. Explicit construction of series expansions, consisting of regular part and boundary layer part or singular part, are developed by means of singular perturbation methods. The regular part is obtained by solving algebraic-differential equations, and the singular part is derived via solution of differential equations. One of the key points in the constructions is to select appropriate initial conditions. This is done by taking into consideration the regular part and the singular part together with their interactions. It is shown that the singular part decays exponentially fast. Analysis of residue is carried out, and the error bound for the remainder terms is ascertained.

Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping

Abstract: The exponential stability of the semigroup associated with the Kirchhoff plate with thermal or viscoelastic damping and various boundary conditions is proved. This improves the corresponding results by Lagnese by showing that the semigroup is still exponentially stable even without feedback control on the boundary. The proof is essentially based on PDE techniques and the method is remarkable in the sense that it also throws light on applications to other higher-dimensional problems.

An existence theorem for the multi-fluid Stokes problem

Abstract: Time-dependent flows of viscous incompressible immiscible fluids are studied in the limit of vanishing Reynolds numbers. The velocity fields associated to each fluid solve Stokes equations in a time-dependent domain. Classical immiscibility conditions on the varying fluids interfaces are taken into account by a new formulation of the problem: the viscosity solves a transport equation and the velocity field solves a Stokes problem with this nonconstant viscosity. This formulation, based on the use of a pseudoconcentration function, has already been used for numerical computations (see [9] and [4]). For this nonlinear system of equations, existence of solutions is proved, using the Schauder fixed point theorem and the concept of renormalized solutions introduced recently by R. J. DiPerna and P. L. Lions.

Dissipation of energy for magnetoelastic waves in a conductive medium

Abstract: We consider the propagation of magnetoelastic waves within a homogeneous and isotropic elastic medium exhibiting finite electric conductivity. An appropriate physical analysis leads to a decoupling of the governing system of equations which in turn effects an irreducible factorization of the ninth-degree characteristic polynomial into a product of first, third, and fifth-degree polynomials. Regular and singular perturbation methods are then used to deduce asymptotic expansions of the characteristic roots which reflect the low and the high frequency dependence of the frequency on the wave number. Dyadic analysis of the spacial spectral equations brings the general solution into its canonical dyadic form. Extensive asymptotic analysis of the quadratic forms that define the kinetic, the strain, the magnetic and the dissipation energy provides the rate of dissipation of these energies as the time variable approaches infinity. The rate of dissipation obtained coincides with the corresponding rate for thermoelastic waves. Therefore, a similarity between the dissipative effects of thermal coupling and that of finite conductivity upon the propagation of elastic waves is established.

Discontinuities in velocity gradients and temperature in the Stokes' first problem with nonclassical heat conduction

Abstract: The nonclassical heat conduction equation based on the MCF model is used to study the discontinuities in velocity gradients and temperature in fluid flows induced by impulsive or sudden heating of a plate. The influence of the thermal relaxation time in the temperature and velocity fields is investigated.

The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations

Abstract: We study the asymptotic limit of the reaction-diffusion equation E , E 1 r/ EX 1 / E. * e ut = Au 2 / ( u ) + g ( u ) \ t as £ tends to zero in a radially symmetric domain in R subject to the constraint J h(u) dx = const. The energy estimates and the signed distance function approach Q are used to show that a limiting solution can be characterized by moving interfaces . The interfaces evolve by nonlocal (volume preserving) mean curvature flow. Possible interactions between the interfaces are discussed as well.

Channel flow of a viscous fluid in the boundary layer

Abstract: where f3, called the Nahme-GrifRth number, is assumed to be large. For some positive constant a (< 2/?1/2), let fl = (0, a) x (0,T), and dfl be the parabolic boundary ([0, a] x {0}) U ({0, a} x (0, T)) of tt, where T < oo. Ockendon found that as x —> oc, u ~ t/x and tp ~ x2/2. Since (3 is very large, «-»0as2i-> 2 (31^2. Thus, the model of the channel flow of a fluid with temperature-dependent viscosity in the boundary layer is simplified to the following degenerate parabolic problem (cf. Lacey [4], Stuart and Floater [6], and Floater [3]), uxx xut — —eu in u = 0 on dfl, (1.1)

Weak solutions to a phase-field model with non-constant thermal conductivity

Abstract: We investigate the existence of weak solutions to a phase-field model when the thermal conductivity vanishes for some values of the order parameter. We obtain weak solutions for a general class of free energies, including non-differentiable ones. We also study the a;-limit set of these weak solutions, and investigate their convergence to a solution of a degenerate Cahn-Hilliard equation.

A limiting viscosity approach to Riemann solutions containing delta-shock waves for nonstrictly hyperbolic conservation laws

Abstract: with initial values f (u+, u+), x > 0, M(z,0)= (1.2) I (u ,v ), x < 0. System (1.1) has the two eigenvalues Ai = u and A2 = 2u with corresponding right eigenvectors r\\ = (0,1),r2 = (1 ,v/u)T, and VAi • r 1 =0, VA2 • r2 = 2. Thus (1.1) is nonstrictly hyperbolic and Ai is linearly degenerate; A2 is genuinely nonlinear. We recall that the classical Riemann solution is composed of a shock (contact discontinuity) or rarefaction of the slower family followed by a wave of the second. But in the present situation, one finds that no classical weak solutions exist for certain states (u±,v±), and distributions of the form of Dirac delta shock waves supported on discontinuity lines are found necessary, even though for systems of conservation laws satisfying the classical assumptions (strict hyperbolicity and genuine nonlinearity), the Riemann problem breaks down for some large data [5]. For the two-dimensional system of conservation laws we refer the reader to [10]. System (1.1) with trivial difference (t —> 21) was studied by Korchinski [6] in his Ph.D. Thesis in 1977. Generalized delta-functions were used in his numerical study and in the construction of his unique solution to the Riemann problem. Afterwards, Tan, Zhang, and Zheng [11] introduced a viscosity term in the first equation of (1.1),

On extraordinary semisimple matrix N (v) for anisotropic elastic materials

Abstract: The 6x6 real matrix N(i>) for anisotropic elastic materials under a twodimensional steady-state motion with speed v is extraordinary semisimple when N(t>) has three identical complex eigenvalues p and three independent associated eigenvectors. We show that such an N(i,) exists when v ^ 0. The eigenvalues are purely imaginary. The material can sustain a steady-state motion such as a moving line dislocation. Explicit expressions of the Barnett-Lothe tensors for v ^ 0 are presented. However, N(i>) cannot be extraordinary semisimple for surface waves. When v — 0, N(0) can be extraordinary semisimple if the strain energy of the material is allowed to be positive semidefinite. Explicit expressions of the Barnett-Lothe tensors and Green's functions for the infinite space and half-space are presented. An unusual phenomenon for the material with positive semidefinite strain energy considered here is that it can support an edge dislocation with zero stresses everywhere. In the special case when p = i is a triple eigenvalue, this material is an un-pressurable material in the sense that it can change its (two-dimensional) volume with zero pressure. It is a counterpart of an incompressible material (whose strain energy is also positive semidefinite) that can support pressure with zero volume change.

Continuous dependence results for a problem in penetrative convection

Abstract: Continuous dependence inequalities are derived for a system of equations that models penetrative convection in a thermally conducting viscous fluid with a linear buoyancy law. Both the forward-in-time problem and the improperly posed backwardin-time problem are analyzed. These results indicate that solutions depend continuously on a parameter in the boundary data.

Circularly symmetric deformation of shallow elastic membrane caps

Abstract: We consider shallow elastic membrane caps that are rotationally symmetric in their undeformed state, and investigate their deformation under small uniform vertical pressure and a given boundary stress or boundary displacement. To do this we use the small-strain theory developed by Bromberg and Stoker, Reissner, and Dickey We deal with the two-parameter family of membranes whose undeformed configuration is given in cylindrical coordinates as z{x) = C(l -x7), (1) which includes the spherical cap as a special case (7 = 2 and C small). We show that if 7 > 4/3 then a circularly symmetric deformation is possible for any positive boundary stress (or any boundary displacement) and any positive pressure, but if 1 < 7 < 4/3 then no circularly symmetric deformation is possible if the stress and pressure are positive and small (or for non-positive boundary displacement and small positive pressure).

High-order essentially non-oscillatory scheme for viscoelasticity with fading memory

Abstract: In this paper we describe the application of high-order essentially nonoscillatory (ENO) finite difference schemes to the viscoelastic model with fading memory. ENO schemes can capture shocks as well as various smooth structures in the solution to a high-order accuracy without spurious numerical oscillations. We first verify the stability and resolution of the scheme. We apply the scheme to a nonlinear problem with a known smooth solution and check the order of accuracy. Then we apply the scheme to a linear problem with initial discontinuities. Discontinuity locations and strengths in the solutions of such problems can be found explicitly by making use of a pointwise estimate obtained in this paper for the Green's function of the equations, which contains two Dirac ^-functions decaying exponentially. We check the resolution of the discontinuities by the scheme. After verifying that the scheme is indeed high-order accurate, produces sharp, non-oscillatory shocks with the correct location and strength, we then proceed in applying it to the nonlinear case with discontinuous or smooth initial conditions, and study the local properties (in time) as well as the long time behavior of the solutions. We conclude that the ENO scheme is a robust, accurate numerical tool to supplement theoretical analysis to study such equations with memory terms. It should also provide an efficient and reliable practical tool when such equations must be solved numerically in applications. Received September 28, 1994. 1991 Mathematics Subject Classification. Primary 35A40, 65M06.

On rates of propagation of heat according to Fourier's theory

Abstract: Introduction. It is the aim of this paper to draw attention to certain consequences of Fourier's theory of heat conduction [1]. It might be thought that so venerable a theory could no longer be capable of generating controversy, but this is not so. The question at issue is what the theory predicts about rates of propagation of heat and, in particular, whether those rates are finite or infinite. We consider an isotropic and homogeneous conductor^ which occupies all of n-dimensional space R\"; the cases n = 1,2,3 are, of course, the ones of physical interest. A point x is identified with its position vector (x\\,..., xn) with respect to the origin, and its distance from the origin is |x| = (x\\ + ■ • • + x\\)1^2. The scalar product x ■ y, of x = (xu...,xn) and y = (ylt..., yn), is the sum xiyi H b xnyn. It will be recalled that Fourier's theory is founded upon two hypotheses, namely balance of heat and a constitutive assumption which relates the heat flux vector to the temperature gradient. Balance of heat says that

The evaluation of certain infinite integrals involving products of Bessel functions: a correlation of formula

Abstract: This analysis evaluates certain infinite integrals continuing products of Bessel functions of integer order, an exponential and a power. The integrals considered here have been previously evaluated in the literature in two different forms. In one instance they have been written in terms of complete elliptic integrals of the first, second, and third kind. Some of these integrals have also been evaluated in terms of a Legendre function of the second kind and a complete elliptic integral of the third kind. A recent result in elasticity obtained by the authors has led to a new form for the evaluations of these integrals. The integrals are still evaluated in terms of complete elliptic integrals; however, a new modulus (and parameter for the complete elliptic integral of the third kind) is used. The new form used for the complete elliptic integral of the third kind allows the integral evaluations to be written in a more convenient form than previously given. The new form for the complete elliptic integral of the third kind is also utilized in the evaluations using the Legendre function of the second kind. The new forms to the integral evaluations derived presently are correlated with existing results in the literature.

Stokes flow around a bend

Abstract: The matched eigenfunction expansion method is used for solving Stokes flow around a channel bend. The flow region is decomposed into rectangular and cylindrical subregions. This enables the stream function to be represented by means of an expansion of Papkovich-Fadle eigenfunctions in each of these two subregions. The coefficients in these expansions are determined by imposing weak C3 continuity of the stream function across subregion interfaces and then taking advantage of the biorthogonality conditions in both cylindrical and rectangular coordinates.

An integro-differential equation arising from an electrochemistry model

Abstract: In this paper, we prove the existence and uniqueness of steady-state solutions for a system of equations arising from a model in electrochemistry. The same result was established by the authors in an earlier paper under the additional assumptions that the space-dimension N = 2 and the concentrations of the charged ions satisfy an electro-neutrality condition.

On a class of limit states of frictional joints: formulation and existence theorem

Abstract: The model dealt with is a linear elastic body in frictional contact with a rigid support. Limit states of such an assemblage are characterized by deformations and forces such that a small perturbation may introduce a large change in configuration. The class of limit states considered here is specified by the possibility of superposing a time constant rigid body velocity field to a static deformation. The problem of finding such states (i.e., forces and static deformations) for a prescribed rigid body velocity is formulated, and for the case when the geometrically admissible rigid body displacements form a linear space an existence result is given. It is proved that under restrictions on the magnitude of the friction coefficient and in the case that an intuitively clear condition on the direction of the forces is satisfied, there exist a load multiplier and a corresponding static displacement.

On the force between a dielectric cylinder in a constant electric field and a conducting half space

Abstract: The force between an infinitely long dielectric cylinder in a constant electric field and a conducting half space is determined using the separation of variables technique on the Laplace equation in bipolar coordinates. The force is obtained as a series containing the relative distance between the cylinder and the half space as a parameter. This series is not uniformly convergent for the cylinder approaching the half space and the corresponding force cannot be obtained by performing the limit term per term. A special asymptotic analysis is presented leading to an analytic expression for this limiting value of the force.

The one-phase supercooled Stefan problem with a convective boundary condition

Abstract: We consider the supercooled one-phase Stefan problem with convective boundary condition at the fixed face. We analyse the relation between the heat transfer coefficient and the possibility of continuing the solution for arbitrarily large time intervals.

Convergence of the two-phase Stefan problem to the one-phase problem

Abstract: We study the limit of the one-dimensional Stefan problem as the diffusivity coefficient of the solid phase appoaches zero. We derive a weak formulation of the equilibrium condition for the resulting one-phase problem that allows jumps of the temperature accross the interface. The weak formulation consists of a regularity condition that only enforces the usual equilibrium condition to hold from the liquid phase. At the end we briefly discuss the radial problem in higher space dimensions. The main tool in order to prove the convergence are uniform bounds on the total variation of the free boundary that are derived using a regularized problem, where the equilibrium condition is substituted by a dynamical condition.