Showing papers in "Quarterly of Applied Mathematics in 1991"

Persistence and global asymptotic stability of single species dispersal models with stage structure

Abstract: A system of retarded functional differential equations is proposed as a model of single-species population growth with dispersal in a multi-patch environment where individual members of the population have a life history that takes them through two stages, immature and mature

Multipolar viscous fluids

Abstract: We develop a thermodynamic theory of constitutive equations of multipolar viscous fluids. The restrictions which the principle of material frame-indifference and the Clausius-Duhem inequality place on the constitutive equations are derived. Explicit forms of the viscous stresses are obtained for linear viscous fluids

Estimation of growth rate distributions in size structured population models

Abstract: We propose models for size-structured populations which allow growth rates to vary with individuals (growth rate distribution across all possible individual growth rates). A theoretical framework for the estimation of the growth rate distribution from data of sized population densities is developed. Numerical examples are presented to demonstrate feasibility of the ideas

Symmetric finite element and boundary integral coupling methods for fluid-solid interaction

Abstract: This paper presents a coupled finite element and boundary integral method for solving the timeperiodic oscillation and scattering problem of an inhomogeneous elastic body immersed in a compressible, inviscid, homogeneous fluid. By using integral representations for the solution in the infinite exterior region occupied by the fluid, the problem is reduced to one defined only over the finite region occupied by the solid, with associated non-local boundary conditions. This problem is then given a family of variational formulations, including a symmetric one, which are used to derive finite dimensional Galerkin approximations. The validity of the method is established explicitly, and results of an error analysis are discussed, showing optimal convergence to a classical solution. ••••••• '* • ' 2 < ? ~ ; ? P Q 0

The existence of travelling waves for phase field equations and convergence to sharp interface models in the singular limit

Abstract: Title The Existence of traveling waves for phase field equations and convergence to sharp interface models in the singular limit Author(s) Caginalp, G.; Nishiura, Yasumasa Citation Quarterly of Applied Mathematics, 49(1): 147-162 Issue Date 1991-03 Doc URL http://hdl.handle.net/2115/39998 Rights First published in Quarterly of Applied Mathematics in volume 49, number 1, published by Brown University. Type article File Information nishiura-83.pdf

On the quenching rate estimate

Abstract: where /? > 0, / >0, 0 < w0 < 1 is smooth. The solution u of (1.1) is said to be quenching if u reaches 1 in finite time T. Note that in this case ut blows up at the same time T. This phenomenon has been studied by many authors (see, for example, the references cited in [8] and [11]). In particular, for any /? > 0 there exists a positive constant lt = lt(P) such that u quenches for any u0 if I > lt. Hereafter we shall assume that u quenches and that u0 satisfies

Analysis of regularized Navier-Stokes equations. I, II

Abstract: A practically important regularization of the Navier-Stokes equations was analyzed. As a continuation of the previous work, the structure of the attractors characterizing the solutins was studied. Local as well as global invariant manifolds were found. Regularity properties of these manifolds are analyzed.

A comparison of certain elastic dissipation mechanisms via decoupling and projection techniques

Abstract: We study the Euler-Bernoulli elastic beam model, modified in a variety of ways to achieve an asymptotically linear relationship between damping rate and frequency. We review the so-called spatial hysteresis model and then introduce the thermoelastic/shear diffusion model

On an electromagnetic inverse problem for dispersive media

Abstract: We refine the analysis of Lerche and show that the nonlinear Riemann-Hilbert problem for his inverse problem can be easily solved in closed analytical form. Moreover, we deal with analogous problems for a semi-infinite and a finite slab correspondingly to the problems treated in the mentioned work of Beezley and Krueger and develop a method for solving these inverse problems, too

Nonincrease of mushy region in a nonhomogeneous Stefan problem

Abstract: For an arbitrary bounded solution of the Stefan problem the mushy region is nonincreasing in time in a sense of the theory of sets. This result takes place for the nonhomogeneous Stefan problem under some conditions on the behavior of a heat source in the mushy region. Many papers have been devoted during recent years to the study of a mushy region in the Stefan problem. This notion follows from the definition of a generalized solution of the Stefan problem. It characterizes the appearance of the set of nonzero measure, where the temperature of the material coincides with the melting temperature. The first examples of the appearance of the mushy region in a generalized solution were constructed by Meirmanov [1] and Primicerio [2]. The main cause of it was the presence of the internal heat sources. For a one-dimensional homogeneous Stefan problem, the dynamics of the mushy region were investigated by Meirmanov [3]. He found the requirements on boundary conditions, producing the disappearance of the mushy region after finite time. The measure of the mushy region in a onephase Stefan problem was estimated by Gustafsson and Mossino [4], Rogers and Berger [5] made the first attempt to investigate the behavior of the mushy region in a multidimensional two-phase Stefan problem. They proved the nonincrease of the mushy region in the homogeneous Stefan problem with constant Dirichlet data and almost uniformly continuous initial data. In this paper, we generalize the results of [5]: for an arbitrary bounded solution of the Stefan problem the mushy region is nonincreasing in time in a sense of the theory of sets. This result takes place for the nonhomogeneous Stefan problem under some conditions on the behavior of a heat source in the mushy region. We deal with a bounded generalized solution of the Stefan problem: UtA0 = f(U) in domain QT = Q x (0, T), ®la£2x(0, T) = ' (1) U\\t=0=U0(x). Received January 23, 1990. ©1991 Brown University 741 742 I. G. GOTZ and B. B. ZALTZMAN Here Q c RN is the bounded domain with piecewise smooth boundary <9Q, and © = x(U), where x is the nondecreasing Lipschitz-continuous function with X{U) = 0 when U 6 {—L, 0). By mushy zone in physical variables we shall understand a set M = {(x,t):xeQ, te(0,T), U(x,t)e(-L, 0)}, M(t0) = Mn{t = t0}. Theorem. Assume that the function / is uniformly Lipschitz-continuous: \\f{sx) -f{s2) I 0L) < 0; (ii) for every @gLoo(9Qx(0, T)), U0 £ L^Sl) the mushy region in a bounded generalized solution of the problem (1) is nonincreasing in time, M(t2) c M[t j) for every t2> tx, in a sense that \\M(t2) \\ M{tx)\\ = 0. Proof. A logical statement (ii) => (i) is obtained as a corollary of results [1] and [2] about the appearance of the mushy region in the nonhomogeneous Stefan problem. The proof of the main statement of the theorem is based on the consideration of the time-discrete Stefan problem with smooth initial data: TJn+\\ TJn —-Ax(Un+l) = f(Un+l), n = 0,...,N—r = T/N, ® lanx(o,r) ~ U° = UJx), \\ rr(n+l)_ / 0(x, t)dt, T J zn (2) with U0 e C'(£2) and 0 e L°°(dQ x (0, T)). Then we consider an auxiliary family of regularized nonlinear discrete problems (2J obtained by smoothing the function x ■ Here approximate functions x£ are twice-continuously differentiable monotonous functions such that k>x'e(s)>e, |^e(5) -x(j)| < Ce, for every e e R . The problem (2J has the classical solution Ue(x, t) U\"+l(x) for t e [r«, x{n + 1)], n = 0, 1, ... , N 1, and U\" € C2(Q/) for every y > 0, where Q' = {x e Q : p(x, dD.) > y} , if t/, < 1 . Using the standard methods we obtain estimates uniform in e , r : C\\ ' ||VO£||i2(nyX(0>t-)) < C2(y) (3) or every y > 0 . MUSHY REGION IN A STEFAN PROBLEM 743 Lemma 1. The estimate [ \\VU\"\\dx < CJy) for every «>0, e>0 (4) is valid. Proof of Lemma 1. Let us differentiate the regularized equation (2J with respect to xk : v = dUJdxk , , n+1 n w « / ' «+' \\ r1 \"+' (v -v)/t-A{x£v ) = fv . Multiplying this by

0 and (p G C2(Qy), we integrate it with respect to x and sum with respect to n : r m m r / (pY^(vn+i v\")sign v\"+l dx + V(^v\"+1)V(signSvn+l v (sign v sign v ) \\ dx. Jn V n=1 J Taking the limit as S —> 0, we easily obtain that lim/f> [ (|fm+1| \\v°\\)(pdx. <5-0 ~ Ja Applying integration by parts, we can rewrite the second term (/2) as follows: n+1 ifl /» TS / / n+1 • <5 «+l _ 'n ' — k /2 = T 2^ / • (V sign v ■ Vxe) ■ V + XE ■ Vu n=0 *'£i — ■ S n+1 / n+1 , . <5 n+1 _ -N , •V sign -(p-Xe'V • div(sign v -Vip)} dx = j[ + 4 + j\\ . s s Using the definition of the function sign we rewrite Jx : m ,,\"+1^ yf = T I —j-T T7T • Vv\"+1 • v/ ■ (pdx . 1 toL((vn+l)2 + df2 The function i*\"5 = vn+1d/({vn+l)2 + J)3^2 is bounded uniform in S , . \\vn+1S\\ (max(|'u'!+1|, VS))3 and lirn^p Fs(x) = 0 for every jteQ. Therefore, using that the functions Vvn+l and Vx[ are bounded in Q' we find that lim^g Jx — 0. 744 I. G. GOTZ and B. B. ZALTZMAN s s The term J1 is not less than zero, and the term /3 may be rewritten as follows: yd f | ' n+1 _ . S n+1 _ , / n+l . ■ <5 n+l\\ , J3 -t / I > V sign v -^(p + Xev Assign v \\ ax. \ =0 / Thus, we have mHI n \\imJl >~Yl,xj |V©\"+'||A^x > -C3(y) ' «_n n=0 and i2 ?— Finally, we obtain the estimate lim t > -CJy). (5 >0 2 ~ 3 [ \\vm+l\\0, £>0, L *(t/\"+1)s ^221oc(\"), « = 0,...,7V-1. which yields the estimate (4) and completes the proof of Lemma 1. With the help of (3), (4) we can take the limit as the regularized parameter e —» 0. As a result we obtain the solution UT of the time-discrete Stefan problem (2) on which the analogue to (3) holds and TM0a* II^'IIbvoj') ̂ c6(y) for every y > o. (5) This estimate follows from (4). It means that the first-order partial derivatives of the functions U' are the bounded measure in Q7. Then we shall prove the main statement [(i) =*• (ii)] of the theorem for the problem (2). Since the function x(U\"+l) satisfies the Poisson equation with a bounded righthand side, then c r;y 2, loc Thus, we have AX{UH+l) = 0 and Un+l = UH + xf{Un+X) almost everywhere in the set {x e Q : U\"+\\x) e (-L, 0)}. We denote Mn+X = {x 6 Q : Un+\\x) 6 (—L, 0), Ax{Un+l{x)) = 0}. Let x be an arbitrary point from M\"+1 . We suppose that U\"{x) {-L, 0), for example, Un(x) > 0. Then Un+\\x) > t(f(Un+\\x)) /(0)) > -xf{\\U\"+\\x)\\ and 1 < i/[ . Choosing r < 1 //, , as above, we obtain the contradiction with the latter expression. Thus, U\"(x) £ (-L, 0) and the measure of Mn+l \\Mn is equal MUSHY REGION IN A STEFAN PROBLEM 745 to zero. This completes the proof of the main statement of the theorem for the time-discrete Stefan problem (2). Let ur be a linear interpolation along time of the function Ux: / 7-rn+l / \\ — — t „ r / 1 \\ i u(x,t) = U (x) + U (x), t e [nx, {n + 1)t]. i T T The results of [6] and estimates (3), (5) yield L\\uz(x, t + At) uz{x, ?)| dx < C4(y)A?'/2 for every y>0. (6) With the help of (5), (6) we can choose the subsequence u^ such that lim u„ = lim U„ U(x, t) in the norms of the spaces CQ(0, T; L,(Q' )), a < 1/2, and L2(QT). Therefore, taking the limit as /? | 0 in the integral identity J + X{Ufi)A

|anx(0 T) = 0, we observe the function U to be a generalized solution of the Stefan problem (1). Since the sequence UT converges to the function U uniformly over the section {t — const} we complete the proof of the main statement of the theorem for the problem (1) with smooth initial data. We may abandon a requirement as to the smoothness of initial data with the help of the following lemma. Lemma 2. The generalized solution U is stable in the space Lj(Q) with respect to the variation of the initial data: If U(x, t) is the solution of Eq. (1) with the boundary-initial data ®ld£}x(o,r) = t/|,=o U0(x) E L (£2), then [ \\U(x, t) U{x, t)\\dx < ef,t [ \\U0(x) U0(x)\\dx. Jn Jq Corollary 1. Suppose 0 s L°°(<9Q x (0, T)), U0 e L^Q.), and the requirements of the theorem and statement (i) for the function / are fulfilled. Then the mushy region of the bounded generalized solution U of the problem (1) may be described in the following way: there exists a nonnegative function G: Q. —* R U {+00} such that M {(x, t): x e M{0), 0 < t < G(x)} . For every x € M(0) on the interval t e [0, G(x)) the function U is a solution of the Cauchy problem: Ut = f(U), U\\t=0 = U0(x). The first statement of Corollary 1 follows immediately from the theorem. We prove the second statement, at first, as in the theorem, for the time-discrete Stefan 746 I. G. GOTZ and B. B. ZALTZMAN problem (2). The convergence of the sequence Ux to the solution U completes the proof of Corollary 1. Using Corollary 1 we can sometimes find the upper bound of the function G, i.e., show that the mushy region has disappeared after a finite time. Corollary 2. Let the requirements of Corollary 1 be fulfilled and let there exist s0 £ (-L, 0) and a > 0 such that f(s) > a for every 5 € (50, 0) and U0{x) > s0 for every x £ M{0). Then G(x) < -s0/a .

On two problems of electrical heating of conductors

Abstract: The thermal effects of the currents induced in a massive conductor by an external slowly varying magnetic field are studied with regard to existence and uniqueness of solutions. In the first part a theorem of existence of solution is also given for the thermistor problem with a current limiting device.

An unfolding of the Takens-Bogdanov singularity

Abstract: A complete analysis for |μ|, |υ|1 of the equation x=y, y=μx+υy+Mx 2 +Γxy describing a particular unfolding of the Takens-Bogdanov singularity is presented

A complex variable integration technique for the two-dimensional Navier-Stokes equations

Abstract: Starting from a complex variable formulation for the two-dimensional steady flow equations describing the motion of a viscous incompressible liquid, a method is developed which carries out three integrations of the fourth order system in parametric form containing three arbitrary real functions. Introduction. It is a feature of nonlinear differential equations that even when an exact solution is available it is not always possible to express the dependent variables as explicit functions of the independent variables. Clearly this can be a disadvantage when the dependent variables represent unknown physical quantities and the independent variables may be space and time. However, in some cases the solution can be parametrized in terms of derivatives, as in the elementary example x = pep , p = dy/dx . Differentiation with respect to y , followed by integration with respect to p, leads to a second equation y = {p2 p + \\)ep + c from which the net gain is a parametrization of x and y in terms of the first derivative p containing an arbitrary constant c. In this example it is possible to eliminate p to determine a relation between x, y, and c, although this is not the case with the more general equation x f(p), which can be treated by the same technique. In general parametric representation represents a powerful method for displaying solutions of nonlinear differential equations especially when the solutions bifurcate as in the case of the Navier-Stokes equations. The present paper attempts to extend this solution method to a certain class of partial differential equations and in particular the two-dimensional steady flow NavierStokes equations. Starting from a concise complex variable formulation for the flow equations first given in [1], and subsequently rediscovered by others, an integration technique is developed which exhibits solutions in implicit parametric form. The dependent variables in the complex system are the stream function y and an auxiliary function 0 associated with the Bernoulli function, or total head of pressure. One major advantage in connection with the present analysis is that the complex flow equation is quasi-linear, autonomous, and contains only z = x iy as independent variable. Received June 6, 1990. ©1991 Brown University 555

Nonexistence of periodic solutions for the FitzHugh nerve system

Abstract: x =y\x3 +x + I, y = p{a-xby), where the variable x is the potential difference through the axon membrane, y is the sodium inactivation (potassium activation), and the quantity 7 is the current through the membrane. FitzHugh investigated the system (1.1) for special values of 7 using numerical methods and phase space analysis. Several authors [4-6, 9] applied Hopf theory to (1.1), and studied the direction of bifurcation and stability of bifurcating periodic solutions of (1.1). The purpose of this paper is to give a new criterion under which the system (1.1) and a system equivalent to (1.1) have no nonconstant periodic solutions. Throughout this paper we assume that the parameters in (1.1) satisfy the conditions

Voltage-current characteristics of multidimensional semiconductor devices

Abstract: |The steady state drift-diiusion model for the ow of electrons and holes in semiconductors is sim-pliied by perturbation techniques. The simpliications amount to assuming zero space charge and low injection. The limiting problems are solved and explicit formulas for the voltage-current characteristics of bipolar devices can be obtained. As examples, the pn-diode, the bipolar transistor and the thyristor are discussed. While the classical results of a one-dimensional analysis are connrmed in the case of the diode, some important eeects of the higher dimensionality appear for the bipolar transistor.

Hydrodynamic stability of Rayleigh-Bénard convection with constant heat flux boundary condition

Abstract: We study the onset of thermal instability with the heat flux prescribed on the fluid boundaries. Assuming Boussinesq fluid, the Landau equation, which describes the evolution of the amplitude of the convection cells, is derived using the small amplitude expansion technique

Creeping flow through an annular stenosis in a pipe

Abstract: The creeping flow disturbance of Poiseuille flow due to a disk can be determined by the use of a distribution of \"ringlet\" force singularities but the method does not readily adapt to the complementary problem involving an annular constriction. Here it is shown that a solvable Fredholm integral equation of the second kind with bounded kernel can be obtained for an Abel transform of the density function. The exponential decay associated with the biorthogonal eigenfunctions ensures that the flow adjusts to the presence of the constriction in at most a pipe length of half a radius on either side. Methods that depend on matching series at the plane of the constriction appear doomed to failure. The physical quantities of interest are the additional pressure drop and the maximum velocity. The lubricating effect of inlets is demonstrated by extending the analysis to a periodic array of constrictions.

Applications of generalized stress in elastodynamics

Abstract: The problem under consideration is the scattering of elastic waves by inhomogeneous obstacles. The main goal is to obtain approximation techniques which are amenable to numerical implementation. For time-periodic problems a coupling procedure involving finite elements and boundary integral equations is described. For general time-dependent problems, artificial boundary methods are studied. In both cases the concept of generalized stress, as originated by Kupradze, plays a central role. The analysis is restricted to planar two-dimensional problems since these illustrate the essential ideas.

On shakedown of elastoplastic shells

Abstract: An asymptotic theory of adaptation for elastoplastic shells under a variable loading is proposed. The hypothesis of membrane state of an elastic response is used to reduce the three-dimensional variational problems for shakedown factor to two-dimensional ones. The duality and the possibility of algebraization allow the membrane shell shakedown theory to be analytically solvable in many interesting cases. The asymptotic accuracy of the constructed membrane approximation is proved.

Foundations of Hankel transform algorithms

Abstract: A brief survey of existing Hankel (Fourier-Bessel) transform algorithms is presented along with a natural way to classify these algorithms. In several cases these algorithms were derived originally by methods that were unnecessarily complicated and not sufficiently general. By using operator notation and Radon transform methods, derivations and generalizations are straightforward. These improvements and generalizations are given at the appropriate places in the discussion

Critical lengths for semilinear singular parabolic mixed boundary value problems

Abstract: where T < oo . Also, let u be a solution of the problem: Hu = -f(u) inQ, u0 on T, u0 on-S, (1.1) where f(u) tends to infinity as u approaches c~ for some positive constant c. The length a* is said to be the critical length for the problem (1.1) if u exists globally for a < a*, and for a > a* there exists a finite time T such that max{w(x, t) : 0 < x < a} —► c~ as t —> T~ . (1.2) This finite time T is called the quenching time. In the special case that /(«) = (1 u)~x , Kawarada [9] showed that (1.2) occurred for a > 23/2. Acker and Walter [2] showed that under appropriate conditions on the forcing term /(«), there existed a unique critical length a* for the problem (1.1). This result was then extended to forcing terms of the type g(u, ux) by Acker and Walter [3], and to h(x, u, ux) by Chan and Kwong [7], Results on the behavior of the solution of the problem (1.1) with a = a* were given by Levine and Montgomery [10]. Existence of the critical length a* and its determination by computational methods were given by Chan and Chen [4] for a more general parabolic singular operator; they studied the problem:

A note on the plane vertical wavemaker in the presence of surface tension

Abstract: This paper presents another method for finding the known wave motion due to a vertical wavemaker in the presence of surface tension, either for finite constant or infinite depth, by using the Fourier integral transform technique and suitably exploiting the regularity condition of the transform.

Ignition or nonignition of a combustible solid with marginal heating

Abstract: The possibility of achieving ignition of a combustible solid with only marginal external heating is investigated. The heating barely raises the temperature to a level where reaction occurs. A parameter indicating the duration of time spent near the crucial level governs the ignition process. If the parameter is below a critical value, no thermal runaway can occur; if it is sufficiently large, ignition will be achieved